FREE VOLUME IN POLYMER SOLUTION THERMODYNAMICSdigitool.library.mcgill.ca/thesisfile70689.pdf · THE...
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FREE VOLUME IN POLYMER SOLUTION THERMODYNAMICS
Transcript of FREE VOLUME IN POLYMER SOLUTION THERMODYNAMICSdigitool.library.mcgill.ca/thesisfile70689.pdf · THE...
FREE VOLUME IN POLYMER
SOLUTION THERMODYNAMICS
Ph.D.
David W. Dreifus
THE ROLE OF FREE VOLUME IN POLYMER
SOLUTION THERMODYNAMICS
ABSTRACT --------
Chemistry
Heats of mixing at infinite dilution, A~(coo), '1I"ere
determined calorimetrically over a temperature range of 300 to
90 0 C for sOrne fifty polymer-solvent systems containing poly-
isobutylene and polydimethylsiloxane. These datawere used to
analyse the Flory version of the general corresponding states
theory of polymer solution thermodynamics. The Flory theory
apparently fails to pred~ct the experimental results as it
incorrectly estimates the effects of free volume on.6 ~(OO) ..
The concentration dependence of the X parameter of
polyisobutylene with both heptane and 2,4-dimethylpentane was
determined with a McBain balance. The Flory model has been
analysed in the light of these data. The importance of end
effects in determining thermodynamic mixing properties is
found to be greatly over-estimated by the Flory model.
The iso-f"ree volume theor,Y of the glass transi tion
has been examined by extension of the Prigogine concept of free
volume to this region. The glass transition temperatures of .
a large number of polymer-diluent systems have been determined
by a DifferentiaI Scanning Calorimeter and have been interpreted
in terms of the Prigogine extension. The iso-free volume
concept of the glass transition has been found to hold for
diluent effects on T , although it is only an approximation for g
the corresponding effects of pressure.
THE ROLE OF FREE VOLUME IN POLYMER
SOLUTION THERMODYNAMICS
David W. Dreifus
Athesis submitted to the Faculty of Graduate Studies and Research in partial fulfilment of the requirements for the
degree of Doctor of Philosophy
Department of Chemistry, McGill University Montreal, Canada SE"~ptember, 1971
@) David VJ. Dreifus 1972
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude
to:
Professor D. Patterson for his interest,
encouragement, and guidance during the course of this
research and for his assistance in preparing this
thesis.
Dr. R.S. Chahal for his guidance and patience
in he1ping to develop the McBain balance portion of
this work.
Dr. G. Delmas for assistance in setting up
the Tian-Calvet microcalorimeter.
The Paint Techno1ogy Society (Montreal
Chapter) for the Newell T. Beckwith Fellowship
(1967 - 1969).
The Chemistry Department for a teaching
assistantship (1966 - 1970).
TABLE OF CONTENTS
PART I: A FREE VOLUME ANALYSIS OF HEATS OF MIXING OF POLYMER SOLUTIONS
CHAPTER I. THEORETICAL BACKGROUN D
INTRODUCTION . . • 0 • • · .. • • · . • •
STRICTLY REGULAR SOLUTION THEORY APPLIED TO MIXTURES OF SMALL MOLECULES • • • • • • • • o. • •
Characteristics of Strictly Regular Solutions Assumptions of Strictly Regular Solution Theory Heat of Mixing Entropy of Mixing
• • • • • •
e • · . • • • •
Free Energy and Chemical Potential
Interpretation of w as a Free Energy.
• • • •
· . • •
EXTENSION OF STRICTLY REGULAR SOLUTION THEORY TC NON-DILUTE POLYMER SOLUTIONS • • • • o •
Introductory Remarks • • • • • • • • Entropy of Mixing • • • • • • • • Heat of' Mixing
• 0 • • • • • 0 • • Free Energy and Chemical Potential ••
Separa tion of x: in to X. H and X S by Huggins
Extension of Guggenheimts Free Energy w to Po1ymer Solutions • • • • • .. • •
Failure of the Flory-Huggins Theory ••
• 0
• • • • • •
• •
· . · . • •
· .
• • • •
1
'3
'3 3 5 7 8
10
Il
Il
Il
13 14 14
16 16
Qualitative Interpretation of the U.C.S.T. and L.C.S.T. 19
THE PRIGOGINE THEORY AND CORRESPONDING STATES
Introductory Remarks
Reduced Temperature Pure Liquids • •
.. . • • .. .
• • • • • • • • • • .. .. • • • • • •
.. . • • • • • •
Liquid Mixtures •• •• •• •• •• o.
Use of the Prigogine Theory to Predict Solution Properties • • .. • • • • • • • • •
Experimental Verification of Corresponding States Evaluation of T and -V from Corresponding States Comparison of )t Determined Experimental1y with X. Calcula ted from Theory • • • • .0 ••
20
20
21 24 26
31 34 36
37
,.
USE OF LIQUID MODELS TO PREDICT SOLUTION PROPERTIES
Comparison o~ the (6-12) and (3-00) Models .. . End E~~ects •••••••••• · ..
THE FLORY THEORY AND ITS RELATION TO THE PRIGOGINE CORRESPONDING STATES THEORY •• •• •• ••
Nomenclature and Equations o~ the F10ry Model
Pure Liquids • • .. . • • • • .. . Liquid Mixtures • • • • • • • • .. . Reat o~ Mixing at In~inite Dilution .... Chemica1 Potentia1 and Activity ... e ..
PURPOSE OF WORK · .. • • .. . • • .... · ..
CHAPTER II. EXPERIMENTAL
THE CALO RI METER • • .. . .. . .. . .... · .. THE REACTION CELL • • .. . .. .. • • .... · .. PROCEDURE · .. ... · .. .. . · .. • • .. . CALCULATION OF .â ~(oo) .. . · .. .. .. .. .. .... CALIBRATION OF THE EQUIPMENT ... • • • • • •
. BLANK RUNS • • • • • • · .. · .. .. .. • •
MATE RIALS · .. • • .. . • • • • ... .. ..
Po1ymers • • • • .. .. .. . • • · .. .. . Mercury ... .. . .. . · .. · .. • • · . Solvents • • · .. • • .. .. • • .. . · ..
)8
39 41
42
43
43 45 46
49
50
51
51
54
56
58
60
60
61
61
62
62
CHAPTER III. RESULTS AND DISCUSSION
SYSTEMS INVESTIGATED • • o .. · " • •
PRECISION, ACCURACY AND SOURCE OF ERRORS .. " DISCUSSION OF RESULTS • • · .. .. . .. .
Calculation and signif'ic'ance of (s2/ sI) X12 (s2!sl)X12 as a f'unction of' Temperature
X12 as a f'unction of Chain Length ••
Quantitative Analysis of the X12 Parameter
Assessment of' End Ef'f'ects f'rom the Equations of Sta te of' n-Alkanes • • • • • •
(s2/ Xl) X12 versus a. l T ••••• ..
A ~(OO) of' Branched Alkane-Polymer Systems
A ~ f'or Squalane wi th PDMS and PIB ••
Failure of Corresponding States •• • ..
CONCLUSION S • • • • • • · .. · .. · .. CHAPTER IV
INTRODUCTION • • .. .. • • • • • • .. . EXPERIMENTAL • • · .. .. . · .. .. . · ..
Background • • • • • • · .. .. . · .. General Description of' the Apparatus • • Need for Modification • • • • • • · .. Procedure · .. .. .. · .. o .. • • .. . Determination of Solvent Vapour Pressure
Preparation of' the Ampoules ... .. .. Materials ... .. . • • • • • • • • Calibration of' the Helix " . .. .. .. .. Preparation of' the Polymer Samp1e .. .
RESULTS AND DISCUSSION • • • • • • .. . CONCLUSION S • • · .. • • " . .. . .. .
o ..
.. " · .. " .. • • ... · .. " ..
... • •
· ..
• •
· .. • •
· .. • • .. .. • • .. . .. . · .. · .. .. ..
• •
75
75
77
81
84
90
92
96
99 99
103
106
106
107
107
107
108
109
112
113
114
115
115
116
120
REFERENCES • • • • • • • • • • • • • •
NOMENCLATURE • • • • o • · . • • o • · . APFENDIX: Tables of Data o • • • • • • • · .
PART II. A FREE VOLUME ANALYSIS OF THE GLASS TRANSITION
CHAPTER V. INTRODUCTION
THE GLASS TRANSITION • 0
.0 • 0 • • • •
THEORETICAL INTERPRETATIONS OF THE GLASS TRANSITION
THE FREE VOLUME MODEL • 0
... • • • • • •
Effect of Diluents on the Tg of a Pure Material (Iso-free Volume Interpretation) ••••
Effect of Pressure on the Tg of a Pure Materia1 (Iso-free Volume Interpretation) ••••
PURPOSE OF THE PRESENT WORK • • • • .. . • •
CHAPTER VI. EXPERIMENTAL
THE DIFFEREN TIAL SCANNING CALORIMETER • • · .. Calibration • • • • • • · .. • • • • Determination of " • • g o • • • · .. • •
MATERIALS . .. • • • • • • • • • • • •
Po1ymers • • • • • • • 0 o • • • • •
Solvents ... • • • • • • • 0 • • • •
. SAMPLE PREPARATION OF NORMAL SOLVENT-POLYMER SYSTEMS
PREPARATION OF PIB-PROPANE AND PIB-BUTANE SAMPLES
122
125
130
138
138
139
141
146
148
150
152
152
156 157
157
157
158
158
159
GENERAL PROCEDURE FOR NORMAL SOLVENT-POLYMER SYSTEMS 160
GENERAL PROCEDURE FOR PIB-PROPANE AND PIB-BUTANE SAMPLES 160
EFFECT OF HEATING RATES ON T g
o .. • • • • · . 161
,-
CHAPTER VII. RESULTS AND DISCUSSION 162
APPLICATION OF THE PRIGOGINE THEORY TO THE GLASS TRANSITION OF POLYMER-DILUENT SYSTEMS .0 o • 166
GENERAL APPLICATION OF CORRESPONDING STATES THEORY TO THE GLASS TRANSITION o 0 o 0
• 0 • • 179
Correlation of Tg Values of Pure Materials 179 Pressure Dependence of Tg o • o • • • o • 185
MOLECULAR SIGNIFICANCE OF THE ISO-FREE VOLUME CONCEPT 188
CONCLUSIONS • • • • .. . • • • • o • · . 191
REFERENCES • • • • o • • • • • • • • • 194
NOMENCLATURE • • • 0 • • • • • • • • • • 197
CHAPTER VII 200
SUGGESTIONS FOR FURTHER WORK • 0 • • • • • • 200
CONTRIBUTIONS TO ORIGINAL KNOWLEDGE • • • • • 0 202
INDEX OF FIGURES
Figure Title Page
1 Monomeric Molecules Distributed on a Solution Lattice 4
2 A Po1ymer Solution consisting of Polymerie Solute
and Monomeric Solvent
3 ~he ~ Parameter as a Function of Temperature (T)
4 Contact Energy between Molecules ê (r) as a
Function of Intermolecular Distance (r)
5 General Plot of Reduced Volume versus Reduced
Temperature for Systems obeying the Principle
of Corresponding States
6 Reduced Configurational Heat Capacity versus
Reduced Temperature as predicted by F10ry and
by Prigogine
12
18
35
40
7 Measuring Element of the Microcalorimeter 52
8 Cross-Sectional View of the Reaction Cell 55
9 Apparatus used to add Mercury to the Reaction Cel1 57
10 Typical Heat of Mixing Curve 59
11 Heats of Mixing at Infinite Dilution [A~~)J as
a Function of Temperature (T) for PIS with the
n-Alkanes
12 Heats of Mîxing at Infinite Dilution [6 ~(OG)J as
a Function of Temperature (T) for PDMS with the
n-A1kanes
13 Reats of Mixing at Infinite Dilution [Â~(oo)J as
a Function of Temperature (T) for PDMS with its
Oligomers
64
65
66
Figure Title
14 Heats o~ Mixing at Infinite Dilution [6 hM~)] as
a Funetion of Temperature (T) for PIB with the
n-Alkanes [Liddell and SWinton(43 )]
15 (s2fsl)X12 as a Funetion of Temperature (T) for
PIB with the n-Alkanes
16 (s2!sl)X12 as a Funetion of Temperature (T) for
PDMS with the n-Alkanes
17 (s2!sl)X12 as a Funetion o~ Temperature (Tl for
PDMS with its Oligomers
18 (921 SI) X12 for Solutions of PIB - n-Alkanes at 2980K
as a Funetion of the Number (n) of Carbons in "the
Solvent Baekbone
19 (s2/sl)X12 at 2980 K for Solutions of PDMS-n-Alkanes
as a Funetion of the Number (n) of Carbons in the
68
78
79
80
82
Solvent Baekbone 83
* 1.. 20 (s2X12!slP2)2 as a Funetion of (l/rl +0.6) ~or
PIB vi th the n-Alkanes 88
21 [(s2/sl)(X12!P;>]-! as a Funetion o~ 1/(r1 +0.6) ~or PDMS vith the n-Alkanes at 2980 K 89
22 (s2/sl>X12 as a Funetion of aIT ~or PIB with the Alkanes
23 (s2!sl)X12 as a Funetion of aIT for PDMS vith the Alkaîles
24 (s2/sl)X12 as a Funetion o~ aIT ~or PDMS with its
Oligomers
25 (s2!sl)X12 as a Functlon o~ aIT ~or PIB with the n-Alkanes [from Liddell and SWinton(43)]
26 Sealed MeBain Balance used to Determine the
Concentration Dependence of the ?( Parame ter
93
95
97
110
Figure Title
27 Concentration Dependence (in terms of the Segment
Fraction, ~) of the )( Parameter ~t 2980 K
28 Specifie Volume (V ) versus Temperature (T) for a sp Typical Glass-Forming Substance
29 Graphical Interpretation of J"ree Volume as defined
by Williams, Landel, and Ferry; Simha and Boyer;
Turnbull and Cohen
30 The DifferentiaI Scanning Calorimeter used to
Determine the Glass Transition Temperatures of
the Polymer-Solvent Systems Investigated
31 Typical First Order Transition Curve as measured by
the DifferentiaI Scanning Calorimeter
32
33
Typical Second Order Transition Curve as measured by
the DifferentiaI Scanning Calorimeter
The Glass Transition Temperature (T ) as a Yunction g
of Weight Percent Diluent (wl ) for POlyisobutylene-
Butane and Polyisobutylene-Hexane
The Glass Transition Temperature (Tg> as a Function
of Weight Percent Diluent (Wl ) for Polyisobutylene
Octane and Polyisobutylene-Decane
35 The Glass Transition Temperature (Tg) as a Function
of Weight Percent Diluent (wl ) for Polyisobutylene
Chloroform and Polyisobutylene-Toluene
37
The Glass Transition Temperature
of Weight Percent Diluent (Wl )
Benzene and Polystyrene-Carbon
(T ) as a Function g
for Polystyrene-
Tetrachloride
The Glass Transition Temperature (T ) as a Function g
of Weight Percent Diluent (wl ) for Polyviny1
Chloride-Methylethyl Ketone and Polymethyl
Methacry1ate-Benzene
119
140
144
153
154
155
170
171
172
173
174
.. i
Figure Title
* * * * 38 A T fT as a Function of (vI IV2 ) [( T2!T1 ) - 1J g g ,sp ,sp at a Diluent Concentration of' 10% by Weight 176
* * . * * 39 AT fT as a Function of' (vI IV2 )[(T2!T1 ) - 1J g g ,sp ,sp at a Diluent Concentration of' 15% by Weight 177
* * * * *. * 40 ATg/Tg as a Function of' (P1v1,sp/P2v2,Sp)[(T2/T1) -lJ at a Diluent Concentration of 10% by Weight 178
41 -T as a Function of T for a Number of' Simple and g g
Comp1ex Substances 184
42 -Tg as a Function of' Tg for Fractionated Samples
of' Po1ystyrene 186
PART I
A FREE VOLUME ANALYSIS OF HEATS OF
MIXING OF POLYMER·SOLUTIONS
'-__ i
- 1 -
CHAPTER I. THEORETICAL BACKGROUND
INTRODUCTION
Historically, theories of polymer solution
thermodynamics have been a natural outgrowth of theories of
the thermodynamics of mixtures of small, quasi-spherical
mOlecules(l,2a,3). The thermodynamic phenomena which are to
be discussed in this thesis have had counterparts in small
molecule systems. For instance, the phase separation that
occurs in pOlymer solutions on lowering the temperature [upper
critical solution temperature (U.C.S.T.)] is also found in
small molecule systems, e.g. cyclohexane-aniline. However,
recently it has been discovered that all polymer solutions
a1so phase separate on raising the temperature(4). This
phenomenon occurs at the lower critical solution temperature
(LeC.S.T.) and is an extreme rarity in smal1 mo1ecule solutions
where it normally invo1ves hydrogen bonding. It seems clear
that there is a qualitative difference between small mo1ecule
and polymer solutions and that a new factor is required to
explain the observed phenomena. Excluding cornbinatorial entropy, (1 2a 3)
~raditiona1 polymer theories ' , have emphasized
that it is differences in the intermolecular forces and the
chemical nature of the polymer and solvent that are responsib1e
for solution thermodynamic properties; however, these concepts
,-.1
- 2 -
alone cannot explain the behaviour of macromolecular solutions.
Prigogine, Trappeniers, and Mathot(5) introduced a new factor
in the early 1950's which, it is now found, can account for the
L.e.S.T. and related phenomena. The new factor, designated as
a 'free volume effect', results from the solvent being in a
greater relative state of expansion than the polymer liquid;
this, in turn, is directly related to the difference in
molecular ,chain length between the polymer and solvent.
For instance, in a system composed of pOlyethylene
and a normal alkane such as decane, there is practically no
difference in the chemical,nature of the cornponents. The
intermolecular forces between two polyethylene molecules and
two decane Molecules are of the sarne magnitude. However, the
di:fference of chg,i.ll length between the polyethylene and the
decane leads to a :free volume difference between the poly-
ethylene melt and the decaneliquid. This fact is ignored by
traditional polymer solution theo~ies but plays an important
role in the Prigogine theory. The Prigogine theory(5,6a) is
able to predict qualitatively the solution behaviour of both
monomeric and macromolecular systems. Naturally, in monomeric
systems the free volume di:fference is small and the traditional
emphasis on chemical nature and intermolecular forces is weIl
placed. Prior to a more thorough discussion of the Prigogine
theory, a review of classical solution thermodynamics is
presented.
- :3 -
STRICTLY REGULAR SOLUTION THEORY APPLIED TO ~IIXTURES OF SMALL MOLECULES
Characteristics o~ Strict1y Regu1ar Solutions
A solution containing non-polar, monomeric Molecules
can be described by the theory of str1ct1y regu1ar
solutions(1,2a,:3). The ma1n result of th1s theory 1s t~at
ASM, the entropy of m1xing, assumes its ideal value; hence,
the excess entropy, sE, de~1ned by A SM - A SM(ideal) = sE, is
zero. On the other hand, the heat of mix1ng, A~, does not
have its ideal value of zero, but is finite. The excess
enthalpy, HE, de~ined by ~ = 6 ~
therefore a non-zero quantity.
AssumRtions of Strictly Regular Solution Theory
6. HM( ideal) = II HM is
The theory is based upon five major assumptions:
1. Themolecules of the solution occupy the sites of a rigid,
quasi-crysta1line latticee (See figure 1.)
2. Intermo1ecu1ar interactions are 1imited to nearest
neighbours. Thi s al!Jsumption 1s "Jalid 1f there are no
long-range forces. The theory is thus limited to non
e1ectrolyte solutions and probably to non-polar systems.
:3. Any net volume change on m1xing May be neg1ected.
4. Each mo1ecu1e 1s pictured as occupying one 1attice site.
This means that the mo1ecu1es are of about the same
size and shape.
5. The v1brational motion of the mo1ecules about the1r
equ11ibrium positions is not affected by mixing.
.1
- 4 -
Figure 1
MONOMERI:C MOLECULES DI:STRI:BUTED ON A SOLUTI:ONLATTICE
Monomeric Solvent
Monomeric Solute
- 5 -
Heat of' Mixing
Diff'erences between an ideal solution and one which
follows strictly regular solution theory are traced to the heat
of' mixing, since the entropy of mixing in both cases ls the
same. in a system of' small, non-polar Molecules, the heat of
mixing results l'rom energetic changes which occur when contacts
between like Molecules are broken and, simultaneously, contacts
between a number of' un11ke Molecules are f'ormed. The 'contact
energy', def'ined as the positive energy required to break a
contact between solvent (1) Molecules, is designated as ~ Il'
between solute (2) Molecules as g 22' and between solvent and
solute as € 12. The (1-2) contacts May be imagined as being
f'ormed from the (1-1) and (2-2) contacts according to the
quasi-chemical equation:
Thus the interchange energy, v, which is associated with the
f'ormation of' a (1-2) contact is given bYI
w = 'Î( € Il + g 22) - ê 12 (1-1)
An ideal solution then, where 6~ = 0, is a strict1y regular
solution in which w = O. This means that the (1-2) contact
energy is equal to the average of ~he contact energies of' the
pure components (1-1) and (2-2).
For non-polar systems the contact energy, E 12' May
'-1
- 6 -
be approximated by the geometric Mean rule € 12 = V €ll 0 €22·
This fo1lows t'rom the London theory of dispersion forces as
described by Hildebrand and Scott(7a ). The product Vé. 11· E22
is less than the arithmetic Mean of'é1l andE: 22 ; as a result w
must be positive. Equation (1-1) May be written:
and theref'ore
Ell é 22 w = --r- + --r- -,1 é ·e V Il 22
(I-2)
Thus, the interchange energy, w, :for non-polar Molecules is
predicted to be positive. Renee, the heat of mixing, given by
~HM = w x no. of' contacts between unlike MOlecules, is
theoretlcally elther zero or positive (endothermlc). If'
mixing Is completely random, the number of (1-2) contacts is
given by the product, Zxl x 2N, where Z is the lattice
coordination number, Xl and x 2 are the mole fractions of
solvent and solute respectively, and N is the number of mole-
cules on the 1attice. The heat of' mixing is then given by
or A~ kT (1-)
where k is the Boltzman constant and T the absolute temperature.
The quantity ZW/kT is termed the solvent-solute
interaction parame ter, )( , and Is a measure of the degree of'
solubility of' the solute in the solvent. The larger the value
'-
- 7 -
of' X, the more unstable is the solution. When 'X l'eaehes a
critieal value of' 2 f'or monomeric SOlutions(2b), phase
separation oceurs. Phase separation takes place in polymerie
SOlutions(2b) when Jl = t.
Entropy of Mixing
In the pure state, the solvent and solute May each
be arranged on their respective lattices in only one distinct
array. The entropy of' mixing of' these materials is assumed
to be eombinatorial only, arising from the greater number of'
geometric arrangements possible when the solution is f'ormedo
If' the solution is completely random, the number of'
distinguishable arrays, Q , 1s given by:
where N l = total number of' solvent Molecules,
N 2 = total number of' solute Molecules, and
N = Nl + N 2 •
Sinee Scomb = k ln S1 , the combinatorial entropy
both pure solvent and solute 1s zero. The eombinatorial
entropy of' mix1ng is then given byl
Ascomb = M
Therefore
6 (k ln G ) = k ln NI - k ln 1 - k ln 1 Nl' N 2'
of
(I-4)
- 8 -
Us~ng Stirling's approx~mat~on that:
ln Nt = N ln N - N
the comb~natorial entropy of m~xing becomes:
(I-5)
whereN l and N2 represent the number of Molecules of components
land 2 ~n the solution, and k is the Boltzman constant.
Free Energy and Chem~cal Potential
On combin~ng equations (I-~) and (I-5), the
expression for the free energy of m1xing of monomeric Molecules
~s found to bel
AGM kT (1-6)
Differentiat~on of (I-6) w~th respect to the number
of moles of solvent y~elds the following expression for the
change in chemical potential, ôJ1 l.:
(I-7)
where }l~ 1s the chem1cal potent1al of component l in the pure
11quid state. The f1rst term on the r1ght-hand side of
equation (I-7) represents the entropy of dilution of component 1.
The second term represents the corresponding heat of dilution.
- 9 -
If the vapour above a solution behaves as an ideal
gas,
)11 o
U1 ~ =
RT (I-8)
o where Pl is the equi1ibrium vapour pressure of component 1 in
the pure state at temperature T, and Pl is the vapour pressure
at temperature T of the same component in equilibrium with a
solution in which 1t is conta1ned. From (I-8) and (I-7):
(I-9)
where al 1s the activity of component 1. If Raoult's law of
idea1 solutions is obeyed, Xl = (P1/P~); from equation (I-9)
then it is c1ear that any deviations from ideality are due to
non-zero values of ?l(hence non-zero heats of mix1ng).
Endothermic heats cause positive deviat10ns while exothermic
heats cause negative deviations from 1deality.
The strict1y regu1ar solution theory is unab1e to
account for the positive excess entropy of mixing which is
found for Many monomeric mixtures. This apparent fai1ure of
the theory is a direct resu1t of restrictions applied in two
(Nos. 3 and 5) of the five basic assumptions stated on page 3
of this chapter. When mixing occurs, contacts between 1ike
mo1ecules are broken and the environment o~ each mo1ecu1e is
changed. This May resu1t in a loosening of the solution
,-
- 10 -
lattice about the new litee - unlike molecular contacts and a
corresponding deorease in the frequency of vibration o~ the
Molecules (relaxation of assumption 5). A volume change on
mixing May likewise acoompany this lattioe loosening
(relaxation of assumption 3). These two phenomene, the
deorease in vibrational frequency of the moleoules and the
volume ohange accompanying mixing, together May lead to the
experimentally observed positive exoess entropy of mixing.
However, strictly regular solution theory does not consider
either of these changes and ls thus unable to explain the
experimental results.
Interpretation of w as a Free Energy
GUggenheim(8) has interpreted the need to relax
res tric tions (3 and 5) in order to eXJPlain the experimen tal
results as evidence that a non-oombinatorial entropy change
takes place on mixing (in addition to the classical oom-
binatorial entropy of mixing). As a result, the parameter
w has been revised to oontain both an entropie and enthalpie
contribution .. w then assumes the character of a free energy
and May be approximated by:
(I-lO)
The introduction of Ws was ad hoc and stimulated a
molecular interpretation of li"quids by Prigogine and
i
- 11 -
co11aborators(S,6a). In the meantime, however, polymer
solution thermodynamics was developed from strictly regular
solution theory ~nd the concept of w as a free energy
introduced by Guggenheim.
EXTENSION OF STRICTLY REGULAS SOLUTION THEORY TO NON-DILUTE POLYMER SOLUTIONS
Introductory Remarks
Before 1940, deviations of polymer solutions from
idea1ity were attributed to a non-zero heat of mixing.
Experimentally, however, all po1ymer solutions exhibit negative
deviations from idea1ity req,uiring negative heats of mixing.
Endothermic heats of mixing, however, are almost always
observed experimenta11y at ordinary temperatures. K.R.
Meyer(9t lO ) was the first to make this important observation
and suggested that the non-ideal behaviour must be re1ated to
the entropy of mixing.
Entropy of Mixing
In response to Meyer's suggestion, F1ory(2a) extended
the basic assumptions of the strictly regular solution theory
to po1ymer solutions with the modification that a lattice site
was now either occup1ed by a polymer segment or a solvent
segment (or mo1ecule) as shown in figure 2. According to the
Flory theory*, the combinatorial entropy of mixing for
* The Flory theory is applicable to aIl solutions in which
(1) The polymer molecules interpenetrate to give a uniform
composition (non-di1ute SOlution).
(2) The probability of finding a polymer segment on the
1attice May be approximated by the fraction of polymer
occupied sites.
'-
,-.1
- 12 -
Figure 2
A POLYMER SOLUTION CONSISTING OF POLYMERIC
SOLUTE AND MONOMERIC SOl.VENT
(Distribution on a Solution Lattiee)
Monomerie Solvent
Segment o~ Polymerie Solute
- 13 -
completely random mixing is
(I-ll)
J.' J.' . where ~l and ~2 are the volume fractions of solvent and polymer
defined byl
rI and r 2 are the number of segments/Molecule of solvent and
polymer respectively. In application of the theory, the
number of segments are taken proportional to the molar volumes,
v, o~ the components such that
f/J' 1
Heat of Mixing
;' 2
The heat of mixing of a polymer with a solvent of rI
segments as given by Flory's extension of regular solution
theory is
(I-12)
This expression has the same form as equation (I-3)
for the heat of mixing of monomeric Molecules. However, the
number of solvent and solute Molecules (NI and N2 respectively)
has been replaced by the number of corresponding segments; that
- 14 -
is, the moie fractions o~ solvent and solute in equation (I-)
have been replaced by volume ~ractions. ~ has been
substituted ~or ZW/kT;where w now re~ers to the interchange
energy between segments o~ polymer and solvent. The nature of'
x.. , howev~r, remains the same. The FlOry(2a) theory 1s a
theory not o~ ')(. but rathero~ the combinatorial entropy of'
mixingo~ polymer and solvent.
Free Energy andChem1cal Potential
It f'ollows that the ~ree energy of mixing is given by:
Upon di~~erentiation, the chemical potential is found:
à Ul Rf = (I-13)
where r is the ratio (r2/r1
) of the molar volumes of the polymer
and the solvent. Equation (I-l3) reduces to equation (I-7)
when r = 1 (that is, when the solvent and solute are of' the sarne
size) •
was
Separation o~ -x. into XH
and )(5 by Huggins
A somewhat more detailed analysis of' polymer solutions
presented by Huggins(l) at the same time that the Flory
~J
- 15 -
theory was proposed. A significant result emerged from the
Huggins' expression for the chemical potential. This expression
at high pOlymer dilution can be put in the same form as Flory's
equation (X-l') provided that the )( parameter takes the
following form:
whare
and -Vs = ( ST)f.) = ! /\- 8T Z
(X-14)·
On the basis of equation (X-14), the )( parameter is found to be
composed of two terms. The fir9t, )lH' i9 related to the
enthalpy of dilution by:
(X-15)
while the second, )(5' is related to the entropy of dilution by:
(X-16)
The )( 5 parameter, representing a non-combinatorial·
entropie contribution to the overall entropy of mixing, is
assumed by Huggins to take into account the error in the Flory
combinatorial entropy of mixing (deviations from completely
random mixing) and should prove to be quite small if Z assumes
- 16 -
its expected value between six and twelve. In ac tuaI fac t,
the results of Flory and Huggins are completely equivalent if
the lattice coordination number approaches infinity.
Experimental results(ll), however, indicate that )(s
is quite large and in Most cases much greater than the )(H
parameter found for polymer solutions.
Extension of Guggenheim's Free Energy w to POlymer Solutions
FlOry(2C) then extended to polymer solutions the
assumption of Guggenheim(8) that w (and hence X) has the
character of a f'ree energy [equation (I-lO)J and :foundz
Failure of' the Flory-Huggins Theory
(I-17)
(I-18)
As a result of' equation (1-18), one is f'ormally able
to predict the large values of' ?{s which are f'ound experimentally
f'or pOlymer solutions. However, the Ws term, which is a amalI
correction in the strictly regular solution theory, ls now very
large and primarily responsible f'or the final value of' the )(
parametero Ws must be taken to be negative ( )(s positive)
f'or polymer solutions, whereas for monomeric solutions it i5
,-
- 17 -
positive ( X s negative). The Flory-Huggins theory is unable
to account for this difference in sign between a polymer-solvent
contact and a monomer-monomer contact.
An even greater difficulty for the Flory-Huggins
theory arose with the discovery of the L.C.S.T. by Freeman and
Rowlinson(4) • Phase separation occurs not only on lowering the
temperature of a polymer solution to the U.C.S.T. but also on
increasing it to the L.C.S.T. This means that the )( parame ter
as a function of temperature must pass through a minimum (curve
A in figure 3) with phase separation occurring at the U.C.S.T.
and again at the L.C.S.T. when /C assumes its critical value.
According to the Flory-Huggins theory(2a), )( should be a
monotonically decreasing function of temperature [cf. equation
(I-l4)] as seen in curve B of figure 3. As a result, the Flory-
Huggins theory is not able to predict the occurrence of the
L.C.S.T.
A negative heat of dilution(l2) is found
experimentally for polymer solutions in the region of the
L.C.S.T. This ls contrary to the prediction of equation (I-l5)
that the heat of dilution of a non-polar polymer solution is
positive. The negative entropy of dilution(l3) found in the
vicinity of the L.C.S.T. is likewise not predicted. In
addition, the Flory-Huggins theory cannot account for the
volume changes which occur on mixing for most polymer
solutions.
,-
- 18 -
Figure :3
THE X PARAMETER AS A FUNCTION OF TEMPERATURE (T)
Curve A - X-
Curve B Chemical interaction portion
of the Je parame ter
Curve C Free volume portion of the
X parameter
~- , . i
- 19 -
Qualitative Interpretation of the UoCoS.T. and L.CoSoT.
It is interesting to note that at the U.C.SeT. both
the heat and the entropy of dilution are positive. Solution
instability results when ÂGM~O; since llGM = Ll~ - TAsM,
the phase separation which takes place at the U.C.S.T. is due to
the positive heat of mixing. At the L.C.S.T. the roI es of the
heat and entropy of mixing are simp1y reversed. The heat of'
mixing being negative favours solution; therefore it is the
negative entropy of mixing* which causes phase separation.
From a qualitative point of view, the L.C.S.T. can be
explained provided that the volume change on mixing ls no longer
neglected. Polymers are genera11y characterized by low
coef'ficients of' thermal expansion compared to monomeric 1iquids.
Thus, at ordinary temperatures, the 'monomer' is in a state of
much greater relative expansion (has greater f'ree volume) than
the polymer. On mixing, the expanded solvent May be pictured as
condensing back into the denser polymer. The resu1t of' this
condensation is a negative excess volume of' mixing. At the sarne
time, negative contributions are made to the heat and the
entropies of' mixing and dilution, the effect on the entropy being
* Actua1ly A SM does not have to become negative, but on1y tends
toward negativity in the region of the L.C.S.T. The ~ S, however, must be negative. For the present intuitive
discussion ASM
vas used instead of Il S.
1
- 20 -
greater than on the corresponding heat o This large negative
entropie effect causes llGM to become positive; hen.ce phase
separation occurs. Thus, by removing the restriction that the
volume of mixing is zero, it is possible to explain, at least
qualitatively, the presence of the L.C.S.T.
Before a theory can be accepted as an adequate
representation of polymer solutions, it must be able to predict
the presence of the U.C.S.T. and.the L.C.S.T.,to account for the
volume change which accompanies mixing, and to give a reasonable
prediction of aIl thermodynamic properties of the solution. The
Prigogine theOry(S,6a), based on corresponding states and applied
to polymer SOlutions(l4), is able to fulfil these requirements
at least qualitatively and shall be discussed in detail.
THE PRIGOGINE THEORY AND CORRESPONDING STATES
Introductory Remarks
In strictly regular solution thermodynamics, the rigid
lattice is not affected by temperature, pressure, or composition
changes which May occur. Prigogine et al.(S,6a) have intro-
duced a theory based on corresponding states which ls applicable
to both monomeric and polymerie solutions and which removes the
restriction of the rigid lattice. As a result, at finite
temperatures, the lattices of the solvent and polymer are found
to be in different states of expansion (the solvent being more
highly expanded). On mixing with the polymer, the highly expanded
.J
- 21 -
solvent will find itself in smaller cel1s on the solution lattice
* with a corresponding reduction in free volume , while the pOlymer
liquid will find itself in larger cells vith an increase in its
free volume. The net effect of these volume changes will not
necessarily cancel each other, so that a non-zero volume of
mixing ( AVM) will result vith corresponding effects on the â~
and 6sM• The equation of state used to predict these changes
vas developed originally by Prigogine, Trappeniers and Mathot(5).
Although the theory vas developed for monomeric mixtures, it was
extended to solutions of chain Molecules (hence polymer solutions)
vith the assumption that the segments making up the molecule are
spherical, i.e. the chain length of a segment is equal to the
cross sectional diameter of the Molecule.
Reduced Temperature
In the Prigogine theory the expansion of a material is
merely a reflection of the thermal energy of the external degrees
of freedom of the Molecules which make up the substance, i.e. the
degrees of freedom which are of lov enough frequency and high
enough amplitude to affect the volume of the system. Contributing
to these external degrees of freedom are those degrees which in
the gas phase are the translational and r-otational degrees of the
Molecule. A further contribution comes from lov frequency,
* Free volume as used by Prigogine characterizes the expansion of o a liquid above its close-packed volume at 0 K.
- 22 -
torsional oscillations o~ the chain moleculeo The total number
o~ external degrees is denoted by Jc. For a completely
~lexible chain molceule, this iSI
Jc = r + J t'or r > 1
where r is the number o~ segments in the chain Molecule.
Rence a completely flexible dimer has ~ive external degrees of'
~reedom, a ~lexible trimer has six external degrees o~ freedom,
etc. In the special case o~ a monomer, Jc is equal to three,
corresponding to the three translational degrees of freedom of
the Molecule.
The thermal energy of these external degrees, JckT,
promotes an expansion o~ the liquide This expansion, in turn,
is resisted by the liquid's intermolecular cohesive energy.
This cohesive energy i5 proportional to g* (see ~igure 4), the
contact energy at OOK between neighbouring, non-bonded segments
or molecules~ The cohesive energy is also proportional to the
number of external contacts made by the chain Molecule, qZ.
qZ = r(Z-2) + 2
where Z is the lattice coordinatlon number and q is an effective
number of segments in the Molecule.
The ratio of the thermal energy of the external
degrees of freedom to the cohesive energy per molecule defines ,."
the Prigogine reduced temperature, T. Within a constant factor,
- 2:3·-
Figure 4
CONTACT ENERGY BETWEEN MOLECULES E (1')
AS A FUNCTION OF :INTERMOLECULAR DISTANCE (r)
Mini~um in the contact energy
between adjacent Molecules.
* r Intermolecular distance when
contact energy is - E *.
,-1
- 24 -
this is:
(I-19)
:« where T is the temperature reduction parameter of the liquide
-It is the reduced temperature, T, which is used to eharacterize
the degree of expansion and hence Cree volume oC the liquide
Pure Liquids
The Prigoginetheory uses three parameters ta
* * * characterize pure liquids: €. , cfr, and r. g has been
previously deCined; * r is the intermolecular distance between _ c* Molecules or segments when the potential is at lts minimum ~
(see f'igure 4). The structural Cactor, cfr, ls a measure of'
the number of external degrees oC Creedom per segment, and is o~
great importance in polymer solutions. For a series of'
homologous liquids, this ratio assumes a value of' one for the
monomer and decreases as r approaches inf'inity. The ef'Cect oC
increasing the chain length then is to lower the value of' T [see
equation (I-l9)J~ Rence a high molecular weight polymerie
liquid will be in a reiatively smaller state oC expansion than a
shorter oligomer, and hence it will have Iess f'ree volume.
Prigogine ~ ~.(6c) assume that a single reduced
equation oC state can be used to predict the thermodynamic
propertles of' a homologous series oC dispersion force liquids.
This means that the series follows corresponding states.
At negligible pressure, the molar configurational
'-, .i
- 25 -
thermodynamic quantities are related to dimensionless reduced
quantities by the fOllowing reduction parameterso
'* -- V*(n) * V(n,T) = v (n) • V(T) = Nor(n)(v )
* (r*)3 * n is the number of carbons where v = and v is the
in the molecular backbone. hard core volume of a segmento
U( n, T) * -- lie Ile
= U (n) • U(T) U (n) = Noq(n) e:
S( n, T) >je 5(r) * Noc(n)k = S (n) • S (n) =
No 1s Avogadro's number and V * , U * and S* are the reduction
parameters for the volume, configurational energy and
configurational entropy respectively.
are functions of chain length only.
The reduction parametèrs
* S does not inelude the
combinatorial contribution to the entropy of mixing. These
reduction parameters are the values of the eorresponding thermo
dynamic quantities at OOK and the reduced quantities are the
factors by which these reduction parameters must be multiplied
-to allow for the appropriate thermal expansiono As a result V,
-like T, is a measure of the expansion or free volume of the
system. The temperature and pressure reduction parameters, T*
'* and P , are defined in terms of these three basic reduetion
parameters as follows~
lie T
U* = -s*
'* p U*
= V* (I-20)
,-.. i
- 26 -
Liguid Mixtures
A number of concentration variables are used in the
Prigogine theory. First, the segment fraction of the polymer,
for instance, is defined by:
V>2 =
, This May be compared vith the volume fraction, ~2:
"-Vl' which is the case at OaK.
A 'molecular surface fraction', X, can be defined(6f)
for the polymer by:
If each contact surface is weighted by the
* corresponding contact energy, E , the contact energy fraction
for the polymer, ~2' is given by:
tV2 = = = 1 - tPl
If solvent and polymer obey corresponding states but
belong to different homologous series, their dirferences May be
'-
- 27 -
characterized by three parameters(6g ):
1. 6 which is a measure o~ cohesive energy differences
* f,22
* E.ll 6= - 1 •
2. 9 which is ameasure o~ segmental size d1f~erences
* 9=
r 22
* r il - 1 0
3. À which represents the ~ree volume di~ferences between
polymer and solvent
I~ on mixing, there i8 no interaction between the
external degrees of freedom of the· Molecules, i.e. the external
degrees of one component are unaffected in number by the
presence of the other component, the entropy reduction parame ter
* for the solution, S , 1s simply a mole fraction average of the
pure componentso.
(I-21 )
Likewise, the volume reduction parameter of the
solution is best represented by a mole fraction average of the
volume reduction parameters of the pure components.
v*
.\
- 28 -
The con~igurational energy reduction parame ter of the
* solution, U , is only equal to
* * x1Ul + x 2U2
i~ both 9 and Ô are zero (~or a mixture o~ mo1ecules composed of
dif~erent numbers of chemical1y identica1 segments). Otherwise,
* the expression for U is given by
(1-22)
The non-1inear term in equation (I-22) represents a
reduction on mixing o~ the cohesive energy between the mo1ecules
as characterized by the V 2 parameter. This is brought about by
two ~actors:
1. Energy differences, or 6 effects, that arise from the
formation o~ (1-2) contacts which are weak in comparison
to the original (1-1) and (2-2) contacts.
2. 9 e~~ects which are a resu1t of the difference in segment
size.
If the geometric Mean ru1e(7a ) ~or interm01ecular
contact energies and the average potential model of prigOgine(6a)
are valid, then
However, in the present work and, in fact, in near1y aIl
preceding publications, ~2 is left as a parameter to be
determined empirically.
(I-2)
,-
- 29 -
In the discussion o~ results, ,,2 is related to
energy differences only, as 9 e~fects are ignored. This is a
simplification which seems justi~ied by current ideas(lS).
Combining equations (I-l9), (I-20), and (I-22), the
reduced temperature of the solution becomes:
(I-24)
It should be noted that the lV 2 parameter in the
denominator of equation (I-24) increases the reduced temperature
and the"free volume of the solution, and reduces molecular
cohesion. It therefore produces positive values of both AVM
and ~~.
In order to find the change of A on mixing (where A
has dimensions of energy and may be ei ther A~, T A SM or AGM
)
the simple equation
(I-2S)
from corresponding states is used. On substitution of equation
* (I-22) for U and on differentiation of equation (I-2S), the
partial molar quantities associated vith mixing are given by
the generai expression
(I-26)
Expressing AGI and AHl in this fashion and expanding
- 30 -
them about Tl' relationships involving )(, )(H' and )(s are
found at infini te dilution to the second order of the small
quantities ~ and 1r :
x = )CH + Xs = -
X H = (I-28)
Xs= +
'" 2 Tl 2 where ~ = [1 - ~J and U and Cp are the configurational
T2
energy and heat capacity of' the solvent. U is essentially the
negative of the energy of evaporat1on.
Each of X, X H , and X S is composed of' wo tenns.
The first represents the chemical differences between the
polymer and solvent as characterized by the interaction term
'V 2 • This term 1s comparable to the X. parame ter 1n strictly
regu1ar solut1on theory.
The second term expresses the free volume differences
'" between polymer and solvent as man1f'ested by differences in T
'" or T .. It ls the introduction of this term whlch reaily
distinguishes the Prigogine corresponding states theory from
strictly regular solution theory or the Flory-Huggins theory ..
·.1
- 31 -
Use o~ the Prigogine Theory to predict Solution Properties
At temperatures in the vicinity of the U.C.SOT8' the
free volume term (Cp T 2/2R) in equation (I-27) is usually very
smal1. Renee, phase separation which occurs at the U.C~SoTo
1s predicted prima.rily by the interaction term of' equation
(I-27) : U 2
X=-iiT'V •
As the temperature o~ the solution ls lowered, [- UJ becomes
larger. This increase of' the configurational energy coupled
with the lover temperature will force the )( parameter to
increase toward its critical value (see ~igure 3, curve A).
According to theory, the heat of' dilution should be
positive in the vicinity of' the U.C.S.T. as the interaction term
of equation (I-28) predomina tes; hence
-U + TCp RT '\)2 0
This is conf'irmed by experimental data(14).
In equation (I-29) for -X-s, the f'ree volume
con tribution virtually vanishes (T = 0) for quasi-spherical
Molecule mixtures. As a result, -x.s is negative (as assumed
by Guggenheim) 0 For polymer solutions, however, T is large
and the f'ree volume contribution is dominant even near the
U.C.S.T. Hence:X: S is positiv~ according to theory.
is also conf'irmed experimentally(ll).
This
'-i
,_.1
- 32 -
The term representing the interaction differences
between the solvent and solute in equation (I-27) CQn Qccount
for the occurrence of the U.C.S.T. As the temperature of the
solution is increased, however, the " parameter decreases
continuously if this contribution alone is considered. Thus,
the interaction term by itself cannot predict the phase
separation which oecurs at the L.CeSoT. It is the free volume
contribution of Prigogine whieh prediets the L.C.S.To for
polymer solutions. For non-pol~r, monomeric systems, in which
there are no large free volume differenees between solvent and
solute, no L.C.S.T. is predicted or observed experimentally.
If 0 and 9 are approximately zero as in a mixture of
oligomers, only the free volume term of equation (I-27) contri-
butes appreciably to the ;( parametero Bence X = (Cp /2R) l' 2
and phase separation is predicted when the temperature of the
system is raised. This is a result of the increase of Cp
toward infinity as the vapour-liquid critical point is neared.
Thus th3 free volume term alone can account for the L.C.S.T.
With no chemical difference between polymer and
sOlvent, X H = - 2~ (dCp/dT) T 2 [from equation (I-ZB>].
Sinee the heat capacity of the solvent is approaching infinity
* in the region of the L.C.S.T., dCp/dT is positive and the heat
* Aceording to equation (I-28) the dependence of Cp on temperature
ls directly responsible for determining the sign of the heat
of'dilution. At temperatures weIl below the critical pOint of'
/Contdo as a f'ootnote next page
,-
- 33 -
o~ dilution, according to theory, is negative.
con~irmed experimentally(l4).
Finally, ~or mixtures o~ oligomers,
Xs =
This is
~rom equation (I-29). ,cs is thus a positive quantity
indicating a negative contribution to the entropy of dilution.
As the temperature increases, this contribution becomes more
important, ~inally making the total ASl negative at the
L.C.S.T. The ~ree volume contributions in equations (I-27),
(I-28), and (I-29) can thus account for the presence of the
L.C.S.T. and the signa o~ the corresponding thermodynamic
quantities.
the solvent, the sign o~ this variation depends on whether the
liquid is simple or polyatomic. In simple liquids, Cp is a
monotonically increasing function o~ temperatureo As a result,
dCp/dT is a positive quantity. For Many pOlyatomic Molecules,
such as the alkanes(l6,l7) at lov temperatures far ~rom their
critical point, Cp is observed to decrease vith temperature
prior to increasing toward in~inity. The result o~ this
behaviour is that a positive heat o~ mixing is predicted, in
agreement vith experiments on mixtures o~ n-alkanes in this
region. Rence, it is this anomaly in the heat capacities of
these chain Molecules which is responsible, according to theory,
~or the positive heats of mixing of the alkanes with each other. The heats o~ mixing do become negative(l8,l9) when the
temperature is increased as dCp/dT becomes positive.
- 34 -
The ~ree volume term was introduced to polymer
solution thermodynamics by Prigogine on ~ priori grounds and
accounts for the experimentally observed results which the
Flory-Huggins theory cannot explaino In ordinary dispersion
~orce polymer solutions, both the free volume and the inter
action terms combine to give the observed values of x., X H'
and X S •
Ex~erimental Verification o~ Corresponding states
As mentioned previously, the Prigogine theory is
based upon corresponding states. Hijmans(20) veri~ied that the
n-alkanes follow corresponding states by obtaining a single
reduced volume versus reduced temperature curve for the series.
Simha and coworkers(2l,22) have shown that the principle of
corresponding states is obeyed by the cohesive energy densities
and internaI pressures of the n-alkanes. Patterson and
Bardin(23) ~ound that the thermal expansion coefficients (a)
and the compressibilities (~) o~ the n-alkanes also obey the
principle of corresponding states.
Simha and Havlik(24) have examined Many additional
systemsr polystyrene ~rom the dimer to infinite polymer, poly-
ethylene oXides, pOlydimethylsiloxanes, and fluorinated alkanes.
Data from aIl these systems fall on a sing~e curve of reduced
volume versus reduced temperature (see figure 5), thus
demonstrating the val1dity of this law.
'-
- 35 -
Figure 5
GENERAL PLOT OF REDUCED VOLUME VERSUS REDUCED
TEMPERATURE FOR SYSTEMS OBEYING THE PRINCIPLE
OF CORRESPONDING STATES
,.., T2 Reduced temperature of the polymer
-Tl Reduced temperature of the solvent
....-1
-- - -- - ------ ~E-t
N -------- ~E-t
A
1 1
- 36 -
Evaluation of T and -V from Corresponding states
Without using a model of the liquid state, it is
impossible to obtain values of the Prigogine reduction
parameters. However, the ratios of these parameters May be
determined(20) from experimental data and a corresponding states
* * treatment. For instance, the ratio of' TA/TB for two liquids,
A and B, May be obtained from a plot of l/nT ( a dimensionless
quantity that is a function of reduced temperature only) versus
log T. Data for the n-alkanes have been treated in this manner.
If the curves for the individual liquids can be shifted in the
x-direction onto the curve of one liquid arbitrarily chosen as a
* reference, it is possible to obtain the ratio of TA (corres-
* ponding to liquid A) to TR (corresponding to the reference
liquid) • The extent of the shift of the curve of liquid A in
* * the x-direction is log (TA/TR). Repeating this process with a
>le * liquid B, it is also possible to obtain TB/TRo Hence the ratio
* */ >le * * * of (TA/TR) (TB/TR> gives TA/TBo This ratio then May be used
* * to evaluate the free volume parameter, T [ T = 1 - (TA/Ta>] and
to serve as a standard test of the accuracy of various liquid
models which attempt to predict 1r.
The interaction parame ter V has been previously
defined [equation (I-23)] as:
"'\) 2 = -%= + 9 92 •
V 2 may be adequa tel y approximaOted (25) by:
,-
- 37 -
2
(~ - 3i) (I-30)
It should be noted, however, that the coef'f'icient of 9 2 t'rom
equation (1-23) has been .. altered and that a cross term, ~ 96~
has been introduced by making this approximation. The ratio
of (U;/V;> / (u~/V;)'may be determin~d from quantities hav~ng , /'
dimensions of energy/volume such as cohesive energy density,
internal pressure and 'compressibility. If the cohesive energy
density (CED) of' a reference material is known at a reduced
temperature, ~RTR' and the CED of the solvent is known at the
same reduced temperature, the ratio of CED 1 /CED ~ at aT i5 , so v re ...
If the same procedure is carried
* * >le * out on the solute, the ratio of' (u2/v2) 1 (ul/vl) is found and
hence ~2 is determined f'rom equation (1-30).
Comparison of )l Determined Experimental1y wi th -x. Calcula ted f'rom Theory
Through the use of' experimental data, then, it ls
possible to evaluate the two corresponding states parameters ~
and T • It should theoretically be possible to calculate by
equation (1-27) the value of' )( at inf'inite dilution from ,,2,
1r 2 , the configurational heat capacity of' the solvent, and the
experimentally determined energy of' vaporization. An attempt
to compare values of' )( calculated in this manner(25) and those
measured experimentally (f'or natural rubber in various
- 38 -
sOlvents(26) has been somewhat disappointinga A possible
reason for this poor agreement lies in the ~ailure of the
geometric Mean rule used in calculating ~2.
USE OF LIQUID MODELS TO PREDICT SOLUTION PROPERTIES
If empirical data as a function of temperature are not
available, the law of corresponding states May still be used to
obtain values of the reduction parameters from equatton of state
data at a Single temperature. Theoretical models of the liquid
state May be used to obtain these ~alues. If this is done, the
need for a reference liquid is obviated' and values of the
individual reduction parameters are obtained. It should be
stressed that values determined in this manner depend on the
validity of the model. However, the ratio of any wo quantities
should agree reasonably weIl vith those found empirica11y.
Prigogine ~ ~.(5) have used a smooth potentia1 mode1
of the liqu1d state based on, the Hirschfe1der-Eyring cell
partition function(27) which emp~oys a Lennard-Jones (m-n)*
dependence of the configurational energy on volume.
(1-31)
For the actua1 smooth potential model, Prigogine has
chosen m = 6 and n = 12 as in the Lennard-Jones potentia1.
The Prigogine model pred1cts that the configurational
c.p heat capacitYAof a liquid increases with temperature from
* m and n are characteristic constants.
'-1
_ •. 1
- 39 -
absolute zero in a roughly linear manner, until it curves
sharply upward in the region of' the vapour-liquid critical po~_nt.
As a result, the occurrence of' the LoC.SeT. and the proper signs
of' the entropy and enthalpy of' dilution in this region may be
predicted from equations (I-27), (I-28), and (I-29).
Flory and coworkers(28-3l ) have proposed a theory of
solutions based on Prigogine's corresponding states theory which
uses the same cell partition function as Prigogine and the Van
der Waal conf'igurational energy-volume relationship, U = - liVe
By setting m = :3 and n = ~ in equation (I-3l), the Flory model
is f'ound. Renee, f'ormally, Flory's model May be obtained as a
special case of' the m-n potential relationship*.
Comparison of' the (6-12) and (3- CIO) Modele
The configurational heat capacity calculated f'rom
Flory's (3-OQ) model is f'inite at OOKo As a result, this model
predicts a slower increase of' Cp vith temperature than the (6-12)
model of' Prigogine and is more in 1ine with experimental
** observations (see f'igure 6).
*
* ....
It is of' interest, however, that the van der WaalSdependence
of' conf'igurational energy on volume has recently received
support f'rom modern theories of' liquids(32). Theref'ore, of'
aIl the theories obtained by varying m and n, that of' Flory
should on ~ priori grounds be the MOSt satisf'actoryo
Neither model, however, is able to account f'or the anoma1y in
the temperature dependence of' the heat capacities of' the
n-alkanes mentioned previous1y on Po 3'0
- 40 -
Figure 6
REDUCED CONFIGURATIONAL BEAT CAPACITY VERSUS
REDUCED TEMPERATURE AS PREDICTED BY:
Curve A F10ry (3-00) model
Curve B Prigogine (6-12) mode1
'- , , .. -,
- 41 -
Delmas and Patterson(15) have compared predictions of'
both models, (6-12) and (3-00), vith experimentally determined
mixing ~unctions. They conclude that the (3-00 ) model 1a in
slightly better agreement vith experimental results. In
addition, they report that the reduction parameters are much
simpler to calculate vith this model.
The (6-12) and (3-00) models o~ the liquid state have
also been compared by Patterson and Bardin(23) through the use
of accurately determined thermal expansion and isothermal
compressibility coefficients of the n-alkanes measured by Flory
and orwoll(33). From a qualitative point of view both models
repro4uce this data successfully. However, both over-estimate
the temperature dependence of the expansion coefficients while
underestimating the tempe rature dependence of the com-
pressibilities. In order to compensate for these effects, the
* * * * * * * * reduction parameters '1' ('1' = U Is ) and P CP = U Iv ) in
both models are forced to va~y vith temperature. Neither '1'*
* nor P should be affected by temperature changes, and the
variation which is predicted is a reflection of weakneaaes
inherent in both modela.
End Ef'f'ects
It should be pointed out that although both models
are qualitatively adequate, neither by itself is able to account
for the positive heats of mixing which are f'ound for mixtures of
(28-':tl) n-alkanes at low temperatures. The Flory model ~ can
'-, 1
- 42 -
predict positive heats if end effects are introduced. According
to Flory, end effects arise as a result of the weaker force
fields which surround the methyl end groups of the n-alkanes
compared to those which surround the interior methy1ene segments~
The net result of these end effects 1s a positive contribution
to the heat of mixing. Flory took as justification for these
effects the variation of p* with chain length found from an
analysis of bis equation of state applied to the n.-alkanes 0
* Theoretically, of course, P should remain constant.
Assumlng no particular liquid model, Patterson and
Bardin(23) showed not only that the normal alkanes obey the
princ1ple of corresponding states but also that p* is
approximately constant for the series. They attribute the
variation of p*, found by Flory, to the approximate character
of the (3-~) model. The experimental evidence for a
significant difference of force fields between end and interior
groups i5 lacking.
THE FLORY THEORY AND ITS RELATION TO THE PRIGOGINE CORRESPONDING STATES THEORY
The Flory theory, or (3-~) model, has been
extensively tested(28-31 ,33-39) with data on systems of the
following types: mixtures of approximately spherical mole
cules(31); solutions of hydrocarbons with cyclic alkanes(31);
monomer-dimer mixtures(31) (e.g. benzene vith diphenyl) and
solutions of hydrocarbons with fluorocarbons(31 ). In addi ti on,
,-
- 4:3 -
a number of polymer-solvent systems have been examined:
PIB_n_alkanes(:35,36), PIB-benzene(38), PIB-cYClOhexane(39), and
natural rubber-benzene(37). The predictions of the Flory
the ory are similar to those made by the general Prigogine the ory
with other choices of the parameters m , n in the conf'igurational
energy-volume relationship. Nevertheless, it seems most
important to test this particular theory because of the success
it has demonstrated in previous work and because of its relative
simplicity and convenience.
Nomenclature and Equations of the Flory Model
Pure Liquids:
In this section, sorne of the special nomenclature
and equations (to the extent that they will be needed in this
work) of the Flory model for pure liquids are introduced.
Essentially the same parameters characterize a chain
Molecule in both the Flory and Prigogine theories. However,
an auxiliary parameter, s, is introduced by Flory; s is defined
to be the number of external, intermolecular contacts made by a
segment. The product rs, the total number of' contacts, is a
measure of molecular surface. It is related te sm' the number
of contacts per interior segment of the chain and s , the e
additional number of' contacts available at the chain ends
through:
'-___ i
- 44 -
In a lattlce model, sm = (z-2) and se = 2.
The molar conflguratlonal energy, as lndlcated
*1-prevlously, ls glven bya U = -u v. The reduction parameter,
* U , as deflned by Flory iSI
u* =
where * 1) 12v characteri zes. the s trengthof' an in tersegmen tal
contact. Using equation (1-"), the p* parameter ls given bya
p* u* = v* =
(1-34)
Some of the equation of state propertles of' the liquid are
required in this work. The Flory equation of' state ln reduced
fonn is:
--~ T
vl/3 1 = --~----- - --Vl/'J _ lVT
(1-35)
At zero or negligible pressure, the reduced temperature and
volume are related byz
T= (1-36)
The thermal expansion coef'f'icient, ~ = l/v<av/aT)p' at constant
negligible pressure, May be f'ound through differentiation of
e·quation (1-36). Rewrlting thls result in terms of Vl/3 gives
.1
- 45 -
the fo11owing equationl
nT (1-37) 3(1 + nT)
From a know1edge, then, of an experimenta11y measured -value of n at a temperature T, V May be ca1cu1ated according to
equation (1-)7). , - * From V thus eva1uated, T May be determined
from equation (1-,6) and V* found from the experimenta1 mo1ar
volume of the 1iquid.
* The P parameter is determined from data on the
isotherma1 compressibi1ity of the 1iquid, ~ = - l/V(av/sP)T'
which in the F10ry mode1 becomesl
(1-'8)
Liquid Mixturesa
On extending the F10ry theory to 1iquid mixtures, the
parameter ~2 occurring in equation (1-2,) is rep1aced by
(1-'9)
and the Prigogi,ne surface fraction, X2 ' is rep1aced by 9 2 = X2 •
Assuming that the quantity rs is proportiona1 to the surface
area of the mo1ecu1e, 8 2 , the site fraction, is defined bya
, .. ,
Heat of Mixing at Infinite Dilution:
- 46 -
In the present work, heats of mixing of polymers with
solvent are obtained to very high dilution of the polymer.
Eichinger and Flory give an expression for the heat of mixing at
infinite dilution of polymer, expressed per mole of polymer
repeat units (equation 42 of reference 37). Writing
for this quantity, expressed per gram of polymer, the
corresponding expression is:
A~(-) = (I-40)
* Rere v is the volume reduction parameter per gram of p01ymer 2,sp
* and s2/sl is the segment surface ratio •
* The quantity s2/s1' which appears in equation (I-40) represents
the ratio of the number of surface sites per unit hard core
volume of solute to that of solvent. When the solute 1s a
polymer, s2 is chosen to represent the number of surface sites
per unit hard core volume of a monomeric repeat unit. There
are essentially two basic methods of evaluating the ratio s2/sl.
1. It is possible to obtain s2/sl by casting shadows of accurately
constructed molecular models of the solvent and solute along
thethree molecular axes. These shadows are traced on paper
and the projected areas of the solvent (s;> and solute (s~> , *, * are measured. SI is then divided by VI and s2 by V2 in order
to ob tain the ratio s2/sl [see equation (I-4l)] for segments
of equal characteristic hard core volume.
(I-4l)
/Contd. as a footnote next page
- 4"1 -
It can be shown that equat~on (1-40) May be obtained
~rom the genera1 Pr~gogine corresponding states theory.
Equation (1-25) can be written, putting U = A and us~ng equation
* (1-22) ~or U of the solut~on, aSI
(1-42)
-- ,.., U(T) May now be expanded around Tl with on1y the ~irst term in - - ...., ,.., (Tl - T) reta~ned ~s T approaches Tl at in~in~te dilution.
Simi1ar1y, higher powers o~ X2 than the ~irst are omitted. At
in~inite dilution 4~(-) is found to bel
4~( .. )
* * P v 2 2,sp (1-43)
The F10ry termlno1ogy has been lntroduced by writing
* 2 * s2 E.11 ...., / e. 22 = S ~2· 1
2. It is a1so possible, where suitable data exists, to determine
s2/s1 ~rom crystal10graphic determined dimensions o~ the
materia1s. In the case o~ n-a1kane mixtures, the molecu1es
are treated as cylinders and their sur~ace areas (si and s~)
ca1cu1ated ~rom these dimensions. A~ter division o~ S~by * , * / V2 and sl by VI' the ratio o~ s2 sl is ~ound by using (1-41).
- 48 -
Equation (I-4o) 1s immediate1y obtained on substitution
- ""-1 - N"" 1 [ of U = V and Cp = nV- both of which are va1id for the (3-00)
model] in equation (I-/.f'3).
Equation (I-43) i5 composed of two contributions: a
free volume term and an interaction terme The interaction term,
makes an overall positive contribution to A ~(oo) since
s2 *
(- sI X12) is multiplied by the negative expression in brackets •
To determine the sign of the free volume term, it is
** -- -necessary to expand U(T2 ) about Tl and substitute this result
into equation (I-43). The free volume term then reduces to:
•
This indicates that the sign of the free volume contribution to
~ ~~) is opposite in sign to dCp(Tl ) /dT1 • Calculations ,.., --' ~
based on Prigogine's (6-12) model show dCp (T1 ) /dTl to be
positive (cf. figure 6). Hence, the free volume contribution
to A ~(OO) should be negative.
contraction on mixing.
This corresponds to a
* ~-The expression in brackets is negative because U(Tl ) is a
~,.J
negative quantity and the positive product TlCp is
sub.tracted from i t.
** Taylor series expansion.
'-
Chemica1 Potential and Activity:
- 49 -
The chemica1 potentia1 and activity of the solvent
are quantities of importance to this and Most previous work in
po1ymer solution thermodynamics. They are considered to arise
from combinatorial and non-combinatoria1 effects. Thus
o Aî\U
T1 __ U1 U1 __
- RT'" = (ln al) comb + (ln al) non-comb (I-44)
The combinatorial contribution May be obtained from
~I 1/~' the F1ory-Huggins theory, i. e. (ln al) comb = ln y 1 + (1 - r) 't' 2.
" and "2 origina11y were taken to be volume fractions. . 1
resu1t, however, is now considered more exact if segment
fractions are used instead.
This
The second contribution, (ln a 1 )non-comb' is obtained
from corresponding states theory. F10ry finds that
(I-45)
This expression May be immediate1y obtained from the genera1
Prigogine corresponding states theory by use of equation (I-26),
- "'-1 2 1 * - - -1/"3 setting A = G, U = -V , V = ~2 Pl and S = "3T1 ln (V -1).
i .i
./
- 50 -
PURPOSE OF WORK
The purpose of this work is to test: (1) the l'lory
theory for the thermodynamics of polymer solutions, and (2) the
general corresponding states theory of which Flory's model
appears to be a special case. This is done by obtaining heats
of mixing to infini te dilution, â~(DO), of polymers in various
homologous series of solvents, i.e. changing the free volume of
the solvent by varying its molecular chain length. The sarne
experiments are then repeated as a function of temperature, i.e.
the free volume of the solvent ls now varied by changing the
temperature. In the course of these tests, it was necessary
to examine the molecular nature of the X12 paramet~r and to
obtain activity data.
· i
- 51 -
CHAPTER II. EXPERIMENTAL
THE CALORlMETER
The heats of mixing of non-polar, polymer-solvent
systems are generally small and are evolved or absorbed over
long periods of time. As a result, a very sensitive instrument
is required to measure these heats. A Tian-Calvet differential
microcalorimeter purchased from DAM, Lyon, France, capable of
detecting heating rates as small as 0.001 cal/hr, was used for
this purpose.
Within the calorlmeter, 496 chromel-constantan
thermocouples provide a path for the flow of heat and electricity
between a large constant temperature aluminum block and a
cylindrical chamber which holds the reaction cell (see figure 7).
Tvo of the cells, diagonally situated and wired in opposition,
are used as a single measuring unit; one cell contains the
pol ymer and solvent to be mixed while the other contains a
non-volatile material and acts as a dummy.
The internaI temperature of the calorimeter is
detected by a copper-constantan thermocouple planted directly in
the isothermal block. The temperature of the outer shell of
the calorimeter is controlled to an accuracy of better than
O.loC by a DAM proportional temperature controller, RT64.
Successive shells minimize the temperature variations 'feIt' by
the central blück. Because of the high sensitivity of the
- 52 -
Figure 7
MEASURING ELEMENT OF THE MICROCALORIMETER
A Isothermal aluminum block
B Chromel-constantan thermocouples
C Reaction cell chamber
,-1
- 53 -
calorimeter, however, it is not possible to control the
temperature of the block well enough to prevent thermal
fluctuations from having an effect on the experimental deter
minations. As a result, the two cells, used as a single
measuring unit, are wired differentially so that any thermal
variations which occur are effectively eliminated.
After the reaction cell is inserted into the
calorimeter and thermal equilibrium attained, the polymer and
solvent are mixed. As a result of the mixing, heat is evolved
or absorbed and a small temperature difference develops between
the reaction cell and the isothermal temperature block. In a
typical experiment the temperature dlfference would be __ 10-3
degrees. The heat is conducted by the thermocouples connecting
the block to the cell chamber (the direction of thermal flow
depending on whether the heat of mixing is exothermic or
endothermic)~
Accompanying this heat flow ls an e.m.f. proportional
to the temperature difference between the cell and aluminum
block. This e.m.f. is detected by a galvanometer recorder
(Graphispot, also obtained from DAM) and a curve representing
the temperature difference as a function of time is traced.
The heat of mixing in calories/gram is determined by integration
of this curve and a knowledge of the calibration constant of the
reaction celle
- 54 -
THE REACTION CELL
A cross-sectional view o~ the reaction cell is shown
in ~igure 8. The viscous pOlymer i5 placed on the te~lon
~loat (A). A portion o~ a stainless steel syringe needle (B)
passes through this ~loat and is held in the threaded te~lon
plug (C). The base o~ the needle (B) has a t~lattened' cross
section (D) which ~its into a special lock (E) at the bottom o~
~loat (A). This prevents the ~loat ~rom being buoyed upward
when mercury is added to the system. A thin piece o~ piano
wire (F) is used to fire the cell (release the float so that it
can rise into the solvent). This wire, which ~its tightly into
the nylon plug (G), passes compl~tely through the needle (B) and
emerges at its base. At this point it is bent upward and
manoeuvered into a special sleeve (H) on the teflon float (A).
When the nylon plug (G) is turned a ~ew degrees, the wire (F)
causes the float to rotate independently of the needle (B).
Hence the float is released, and the cell is fired.
The plug (G) is locked into position by the wire (I)
to prevent premature firing. Mercury and solvent are introduced
into the cell through the hole (J) which is then closed with a
teflon stopper. A~ter filling, the cell is attached to a
specially threaded holder (K) through which passes a thin wire
(L) that fits tightly into the top of the nylon plug (G). The
wire (I), used to prevent this plug trom rotating9 is removed
and the holder and cell are placed into position in the
- 55 -
Figure 8
CROSS-SECTIONAL VIEW OF THE REACTION CELL
A TeC10n C10at
B Stain1ess steel syringe need1e
C T.hreaded teC10n p1ug
D F1attened base oC the stain1ess steel
syringe need1e
E Lock at the base oC the teC10n C10at
F Piano wire
G Nylon p1ug
H 'Special Sleeve' on the teC10n C10at
l Wire used to prevent premature mixing oC the
solvent and solute
J Ho1e (c1osed by a teC10n p1ug) through which
Mercury and solvent are introduced into the ce11
K Threaded cel1 ho1der
L Vire used to Cire the ce11
,-.. 1
'-
G 1-
c
l
B
> ~. > < K > <
A
L
H
- 56 -
calorimeter. A~ter thermal equilibrium is attained, the wire
(L) is turned, causing the nylon plug to rotate and ~ire the
cell.
PROCEDURE
In a typical heat o~ mixing determination, a sample
of polymer is ~irst degassed. A~ter this is done, 0.150 grams
of the polymer are placed on the teflon ~loat. The cell is
then assembled and taken to the vacuum apparatus shown in
figure 9, where it is attachedto a special holder (A). The
* stopper in hole (J) of the cell is removed and a special piece
o~ nylon tubing (B) connected to a burette (C) is placed into
(J). The system is then closed and e~acuated by an ordinary
vacuum pump. A~ter vacuum has been applied to the cell ~or
about ~ifteen minutes, three ml o~ Mercury are added from
burette (C). The entire system is kept under vacuum ~or an
additional twenty minutes in order to remove any air which May
** have been entrapped by the Mercury •
*
**
A~ter this has been done, the cell is removed ~rom the
See ~igure 8.
I~ this is not done, the entrapped air could ~orm a vapour
space above the solvent when mixing occurs and a small
endothermic heat would result from the evaporation of
solvent into this space.
,-i
,-.1
- 57 -
Figure 9
APPARATUS USED TO ADD MERCURY TO THE REACTION CELL
A Threaded brass cell holder
B Burette containing Mercury
C Nylon tube connecting the
reaction cell to the burette
D Tube leading to vacuum pump
B
D
- 58 -
vacuum apparatus and filled by addition of approximately 8.6 ml
* of' solvent • The teflon stopper is placed into the hole (J)
and the cell is ready for insertion into the calorimeter.
When hea ts of mixing are measured a t tempera tu-res in
excess of' 300 C, the filled cell is preheated with the hole (J)
open prior to insertion into the calorimeter. This is done for
essentially two reasons:
1. It allows for removal of excess solvent. This excess
solvent is expelled from the cell by the expansive
forces of the Mercury and solvent. If this is not
done, a pressure build-up within the cell could cause
serious leakage problems and an accurate determination
of the heat of mixing would be impossible.
2. It preheats the cell; this essentially reduces the time
required for thermal equilibrium.
After the filled cell has been placed in the
calorimeter, a time of three to five hours is normally required
to reach equilibrium. Thermal stability of the system is
easily verified by observation of the baseline of the recorder.
If the baseline is completely linear, showing no deviation,
equilibrium is attained and the cell is fired.
CALCUlATION OF A ~(QO)
The area of the heat of mixing curve generated when
the cell 1s fired (see figure 10) is measured vith a planimeter
* Hence, after mixing, the final solution contains no more than
one or two percent polymer by weighto
· :
- 59 -
Figure 10
TYPICAL HEAT OF MIXING CURVE
Temperature difference ( AT) between
the ce11 and the isotherma1 b10ck as
a function of time (t)
'-
- 60 -
to within ! 0.1 cm2 • This area (A), the calibration constant
( ~) f'or thecell in calories/ cm2 , and the number of' grams of'
polymer (w) dissolved are used to calculate the heat of' mixing
in cal/gm according to
A~(-) = )..A/W
CALIBRATION OF THE EQUIPMENT
One of' the cells which is supplied with the calorimeter
contains a special resistance used to calibrate the individual
cell chambers. Af'ter this cell has been placed in one of' the
* f'our chambers of' the calorimeter, a precisely determined current
is passed through for a known amount of' time. The curve
corresponding to this input is recorded.
multiplied by the time the current f'lows through the system is
used to obtain the heat emitted at the resistance. This
quantity is divided by the area of' the recorded curve to give
the calibration constant in calories/cm2 f'or the particular cell
chamber used.
BLANK RUNS
It should also be mentioned that a small amount of'
heat is produced when the cell is f'ired as a result of' f'riction
* The value of' this current is determined through the use of' a
standard ten ohm resistance, an accurate potentiometer, and
the relation l = E/R.
- 61 -
developing between the Mercury and the rotating teflon ~lüat.
Several blanks (cells loaded without polymer) vere run to
determine the magnitude of this effect. It was found, however,
that the heat produced in this manner is negligible.
MATERIALS
POlymers
The polyisobutylene (PIS) studied in these experiments
is the same polymer used in previous work by Delmas et al.(14).
The polymer was obtained originally as Vistanex LM-MH-22S, a
gift from the Enjay Company. The fractionated sample used had
a viscosity average molecular weight of 30,000 and had been
previously fractionated by G. Delmas(14) fOllowing the method
outlined by F1Ory*.(40)
*
The polydimethylsiloxane (PDMS) studied was a gift
In brief, the PIS was fractionated as follows:
About 2S grams of the polymer vere dissolved in three liters
of benzene. Acetone was added to this solution in slight
excess of the amount required to cause incipient clouding.
The solution was then warmed and stirred simultaneously to
insure homogeneity. The flask containing the solution was
next placed in a thermostatted bath and allowed to cool
slowly. After several .hours the precipi tated polymer was
recovered by decantation and dissolved in benzene once more.
The entire precipitation was repeated, and the final
fractionated precipitate was heated and dried at 70 0 C under
vacuum until a constant weight of polymer had been achieved.
- 62 -
from the Dow Corning Company of Midland, Michigan. The
unfractionated sample used had a number average molecular
weight of the order of 30,000(41).
Mercury
Engelhard triple distilled Mercury has been used in
aIl runs without further purification.
Solvents
Pentane, hexane, heptane and octane were obtained
from Fisher Scientific Company and were all of spectrograde
quality. Decane and hexadecane were purchased from the
Aldrich Chemical Company and were 99% pure. Except for drying
with sodium sulfate, no attempt vas made to purify further the
linear alkanes used. in this work.
The oligomers of PDMS (dimer to pentamer) vere given
to us by Dow Corning. Prior to use they were distilled and
dried .. The level of impurities in the oligomers was found by
gas-liquid chromatographie analysis to be less than 0.2%.
The branched and cyclic alkanes were purchased from
the Aldrich Chemical Company, Most being of a purissimum
quality. Several of these solvents were tested for p~~~ty by
passing them through a gas-liquid chromatograph and were found
to contain less than 1% impurity. The branched and cyclic
alkanes vere used without further purification. (They were,
of course, dried with sodium sulfate.>
1 .-<
- 63 -
CHAPTER III. RESULTS AND DISCUSSION
SYSTEMS INVESTIGATED
Heats of' mixing at inf'inite dilution [A~(DO)J were
determined f'or the t'olloving sets of' systems: PIB - the
n-alkane series f'rom pentane to hexadecane, PDMS - the n-alkane
series t'rom pentane to hexadecane, and PDMS vith its oligomers
o 4 0 ° f'rom the dimer to the pentamer at temperatures of' 30, 0, 55
The limiting temperature f'or any solvent was
determined by its boiling point (non-volatile solvents vere used
up to 90°C).
The results of' this york are shown in f'igures Il, 12
and 13. These f'igures also include data at 250 C measured
previously(14,42).
In aIl these f'igures, the A ~(oo) becomes more
negative as the temperature of' the system is increased. The
negative 6~(OO) f'ound at 25°C f'or Pla vi th the n-alkanes and
PDMS vith its oligomers can be qualitatively explained in terms
of' equation (I-40). In both cases the chemical dif'f'erence
between polymer and solvent is very small, and it is the
negative f'ree volume term which de termines the sign of' A~(OO).
On the other hand, values of' A~(oo) at 250 C f'or PDMS with the
n-alkanes are positive (except f'or PDMS - n-pentane). In terms
of' equation (I-40) this is a result of' a large, positive
chemical dif'f'erence between polymer and solvent outweighing a
- 64 -
Figure 11
BEATS OF MIXING AT INFINITE DILUTION [â~(oO) ]
AS A FONCTION OF TEMPERATURE CT) FOR PIS
WITH THE n-ALKANES
0 PIS - Pentane
0 PIS - Hexane
'\l - PIS - Heptane
• - PIS - Octane
• - PIS - Decane
• - PIS - Hexadecane
Data shown in Table l (P. 70)
"-
.. i
o .-1
•
o . o
•
1 ~
o .-1
1
• ~
t>
J J
o . N
1
t>
0
o C"")
1
•
0
0
/ 0
o ..;:t
1
,-
0 C"") C"")
,.-... ~
0 '-"
0 N E-l C"")
0 .-1 C"")
0 0 C"")
0 0"\ N
- 65 -
Figure 12
BEATS OF MIXING AT INFINITE DILUTION [AhM(~) ]
AS A FUNCTION OF TEMPERATURE (T) FOR PDMS
VITH THE n-ALKANES
0 PDMS - Pentane
0 - PDMS - Hexane
\l - PDMS - Heptane
• - PDMS - Octane
À PDMS - Decane
!II PDMS - Hexadecane
Data shown in Table I (P. 71)
1 --1
•
o o o . . . N
•
o . o
o . N
1
- 66 -
Figure 13
BEATS OF M'IXING AT INFINITE DILUTION [A~(OO) ]
AS A FUNCTION OF TEMPERATURE (T) FOR PDMS
WITH ITS OLIGOMERS
o PDMS - Dimer
a PDMS - Trimer
~ - PDMS - Tetramer
• PDMS - Pentamer
Data shown in Table l (P. 72)
, .. i
.-<'
o . . o o
• <J
• <J a
e<l 0
lIî . o
1
o .-1
1
o
o
lIî
.-1 1
o C"J
1
o C"J E-4 ("t')
o o ("t')
o 0"1 C"J
- 67 -
- -small, negative ~ree volume term (i.e. the v and T ~or both
polymer and solvent are very similar, hence the ~ree volume term
is almost negligible). As the tempe rature ls increased, the
~ree volume term should become more important and cause the
A~(oO) ~or PDMS vith the n-alkanes to become negativeo This
is observed ~or PDMS-hexane at 550 C (see ~igure 12) where the
heat has become exothermic.
Figure 14 shows a plot o~ the A~(oo) ~or PIB-pentane,
hexane, and octane as measured by Liddell and SWinton(43)
covering a vider temperature range than the present investigation
(a high pressure cell was used for this purpose). Although the
same general behaviour of Il hx(oD) vi th tempera ture is observed
in their work, the values o~ the A hx(oo) are not in good
agreement with the data presented in this thesis. At this time
it is not possible to account for this discrepancy which is of
the order of l joule/g (the data of Liddell and Swinton being
more endothermic). The qualitative conclusions which are
reached, however, are independent of which set of data is used.
It should be mentioned that the data presented in this
study are consistent with previously published results for the
sys tems: PIB _ n-alkanes ( 14), PDMS _ n-alkanes (42), PIS _
heptane(44), and PDMS _ heptane(45).
Heats of mixing at infinite dilution at 300 C vere also
measured for PIB and PDMS vith a number of branched alkanes to
de termine the ef~ect of additional solvent end-groups on
Il. ~(OG).
- 68 -
Figure 14
HEATS OF MIXING AT INFINITE DILUTION [Il ~(c.o) ]
AS A FONCTION OF TEMPERATURE (T) FOR PIS WITH
THE n-ALKANES [LIDDELL AND SWINTON (43) ]
o PIS - Pentane
o - PIS - Hexane
• - PIB - Octane
[Data shown in Table l-a, columns 1, 2 and 3 of the Appendix]
'-i
• 1 • 1 •
o . o
•
0
/ Cl
1 0
o . N
1
0
o ..::t
1
/ 0
0
01
o \0
1
o 00
1
Cl
o . o ,.......1
1
,-i
0 ("')
..::t
0 ,.......1 .
..::t
0 ~ ("')
0 r--("')
,,-.. :::.:::
0 0 "-" Lt"'t ("') E-c
0 ("') ("')
,'1:
0 ,.......1 ("')
0 ~ N
- 69 -
In addition, heats of mixing at infinite dilution
were obtained at ~OoC for both polymers vith the following
cycllc alkanes: cyclopentane, cyclohexane, cycloheptane, cyclo-
octane, and methyl cyclopentane.
In Table l a complete list of all of the heats of
mixing measured is presented. The values of A~(oo) reported
in the fourth column of this table are a numerical average of
all the heats measured for a given system.
For Pla with the n-alkanes, PDMS vith the n-alkanes
and PDMS vith its oligomers, aIl measurements vere at least
duplicated and, in Most cases, repeated three or four times.
For a few of the polymer-branched and cyclic alkane systems,
hovever, only one measurement vas made. The number of times a
given system was examined i5 listed in column three of Table 1.
PRECISION, ACCURA.CY AND SOURCE OF ERRORS
In Table l the average A ~~) for each polymer-
solvent system is presented (column 4). The amount of scatter
in these heats, represented by the average absolute deviation
(column 5), reflects the precision of the experimental
measurements. The precision of these results 1s of the same
order as reported prevlously by Delmas
vary appreciably vith temperature.
t 1 (14)
.L !Le and does not
In Table II a compilation of the average absolute
deviation of the heats for each polymer-solvent series
investigated is presented. The average absolute deviatlons for
'-
- 70 -
TABLE I
Solvent Temp. No. of' times Ave. A~(OO) Ave. Absolute system measured Deviation. 6 *
Oc joules/g jou1es/g
PIB - n-a1kanes
Pentane** 25 -).59 Hexane** 25 -2.55 Heptane** 25 -1.79 Octane** 25 -1.21 Decane** 25 -0.55 Hexadecane*'" 25 +0.05
Resu1ts of' this work
Pentane 30 2 -3.87 0.10 Hexane 30 3 -2.57 0.08 Heptane 30 3 -1.84 0.06 Octane 30 3 -1.22 0.01 Decane 30 3 -0.67 0.01
Hexane 40 3 -3.)4 0.02 Heptane 40 3 -2.10 0.06 Octane 40 3 -1.43 0.05 Decane 40 3 -1.09 0.02
Heptane 55 ) -2.27 0.07 Octane 55 3 -1.86 0.01 Decane 55 4 -1.18 0.12
Octane 90 2 -).29 0.08 Decane 90 3 -2.15 0.06 Hexadecane 90 3 -0.72 0.09
/Contd.
"'Average absolute deviation, 6,: 6 = L 1 X - X 1/ N
**
where X is the numerical average of' the individua1 à ~(..o)
measurements f'or a given system, X ls the actua1 A~(CoC) f'or
each indlvidua1 measurement, and N ls the number of' times
each system was lnvestigated.
From resu1ts of' ref'erence 14.
,-1
,-
- 71 -
Solvent Temp. No.of" times Ave. A~ (000) Ave.Abso1ute system devia tion, 6
Oc measured jou1es/g joul.es/g
PIB - branched a1kanes
2,4-Dimethy1pentane 30 2 -1.92 0.02
2,3-Dimethy1pentane 30 3 -2.08 0.03
3-Methy1hexane 30 4 -1.91 0.06
3-Methy1heptane 30 3 -1.67 0.05
2,2,4-Trimethylpentane 30 3 -0.58 0 0 05
2-Methy1heptane 30 2 -1.39 0.00
2,5-Dimethy1hexane 30 2 -1.02 0.09
3,4-Dimethy1hexane 30 1 -1.49
2,2-Dimethy1hexane 30 1 -1.17
2,4-Dimethy1hexane 30 1 -1.57
Squa1ane 90 1 -0.54
PIB - cI:c1ic a1kanes
Cyc10pentane 30 3 -5.43 0.24
Cyc10hexane 30 2 -0.82 0.03
Cyc10heptane 30 1 -0.43
Cyc100ctane 30 1 +0.12
Methy1cyc1opentane 30 3 -3.46 0.05
PDMS - n-a1kanes
Pentane* 25 -0.95
Hexane* 25 +0.68
Heptane* 25 +1.96
Octane* 25 +2.57
Decane* 25 +3.85
Hexadecane* 25 +5.55
Resulte of" this york
Pentane 30 2 -1.32 0.04
Hexane 30 4 +0.50 0.05
Heptane 30 5 +1061 0.03
Octane '21\ 3 +2.40 0.07 JV
Hexane 40 2 +0.09 0.01
Heptane 40 3 +1.58 0.05
Octane 40 3 +2.17 0.08
Hexane 55 2 -0.10 0.02
Heptane 55 3 +1.50 0.09
Octane 55 :3 +1.97 0.08
Decane 55 4 +3.47 0.16
1", __ td 1 v", •• - •
'* From results of" ret'erence 42.
- 72 -
Solvent Temp. No.of' times Ave. Ll~(-) Ave.Absolute system deviation,6
Oc measured jou1es/g joules/g
Results of' this york (contd. )
Octane 90 3 +1.12 0.13 Decane 90 3 +3.04 0~16 Hexadecane 90 3 +5.16 0.17
PDMS - branched a1kanes
2,4-Dimethylpentane 30 3 +0.96 0.05 2,3-Dimethy1pentane 30 6 +1.09 0.03 3-Methy1hexane 30 3 +1.23 0.02 2-Methy1hexane 30 1 +1.17 3-Ethylpentane 30 1 +0.97 2,2,4-Trimethy1pentane 30 4 +1.53 0.08 3-Methylheptane 30 3 +2.01 0.07 2-Methy1heptane 30 2 +1.60 0.07 Squalane 90 3 +2.28 0.01
PDMS - cxclic a1kanes
Cyclopentane 30 2 +1.04 0.00 Cyc10hexane 30 2 +5.00 0.06 Cyc10heptane 30 1 ~6.06 Cyc100ctane 30 2 +6.96 0014 Methylcyc10pentane 30 3 +1.66 0.06
PDMS - olis:omers
Dimer* 25 -1.18 Trimer* 25 -0.59 Tetramer* 25 -0.43 Pentamer* 25 -0.28
Resu1 ts of' this work
Dimer 55 3 -1.62 0.05 Trimer 55 3 -0.82 0.04 Tetramer 55 3 -0.49 0.04 Pentamer 55 2 -0.33 0.02
Trimer 90 3 -1.16 0.12 Tet:"!::.mer 90 3 -0.80 0.02 Pentamer on :3 -0.62 0.03 ",-
* From resu1ts of' reference 42.
System
PIB - n-alkanes
PIB - branched alkanes
PIB - cyclic alkanes
PDMS - n-alkanes
- 73 -
Table II
No. of' systems investigated
15
10
5 14
PDMS - branched alkanes 10
PDMS - cyclic alkanes 5 PDMS - oligomers 7
Ave. Absolute Deviation, 6
joules/g
0.08
0.09
0.12
0.13
0.08
0.06
0.07
PDMS with the a1kanes and PIB vith the cyc1ic alkanes are
somewhat higher th an the other results but seem very reasonab1e.
Although the experimental measurements are
reproducible, a small reduction in precision(46) may have
resu1ted f'rom:
1. A slight shif't of' the recorder baseline taking place while
a measurement i3 in progresse In systems in which the
A~~) is small, a signif'icant lack of' precision would
natura1ly result f'rom this shif't.
2. An uncertainty of' the order of' 2.5% in the value of' the
calibration constant f'or each of' the cell chambers used
in this work. This, in turn, May result f'rom
uncertainties inl
a. the rate at which heat is supplied to the cell chamber
during calibration.
'-
- '74 -
b. the resistance of the calibration celle
c. the area of the calibration curve.
d. the value of the current passed through the
calibration cell.
e. the time in which this current flows through the cell.
3. The fact that the planimeter used to measure the
curves can only reproduce the areas of these curves to
2 + 0.1 cm • -While it should be stressed that reproducibility of the results
was quite good, uncertainties in the above three factors
certainly contributed to the overall scatter of the results.
It is felt that the accuracy of the experiments is of
about the same order as .the precision. This is a result of the
calorimeter being calibrated directly using known heats. Of
course, any heat which results trom starting the reaction
(turning the float in Mercury) or stirring the solution would
affect the results. However, blank runs showed these heats to
be negligible. Systematic errors whichmay influence the
accuracy(46) of the measurements are listed below:
1. Any irregular distribution of the thermocouples around
the cells.
2. Possible heat losses from the upper portion of the reaction
cell that is not surrounded by thermocouples.
3. Thermoelectric effects which result from faulty electrical
contacts.
4. Sensitivity of the Graphispot to external magnetic fields.
, .i
-c'
- 75 -
DISCUSSION OF RESULTS
Calculat10n and signi~icance
o~ (s2!sl)Xl2
Values o~ (s2!sl)XI2 have been determined by fitting
the Flory equation (I-40) ~or À~(oo) to the experimentally
determined heats.
determined ~rom experimentally measured values of a., Y (the
thermal pressure coe~~icient), 9 (the density) o~ the
pOlymer(47,48) and solvent()),50 ,SI) byequations (I-)6),
(I-)7), and (I-)8).
Theoretically, the temperature reduction parameter,
* T , is temperature independent. However, because the Flory
* theory predicts a dependence o~ a. on T which 1s too great, T
varies in this model.
* * * In ail o~ these calculations it is assumed that P2 and v do sp
not vary with temperature. In equation (I-4o) it may be
* * noted that v sp and P both act as multiplication factors for
the entire expression in brackets on the right hand side vith
* P2 also appearing in the denominator of the interaction terme
* * The actual variation of P and v over the tempe rature range sp *
exam:lned :ls small in
* * product (v )(p) is
comparison to the changes in T , and the
practically constant within + 2% over -* Changes in T with sp
the temperature range :lnvestigated.
temperature are instrumental in determining the magnitude of
the free volume terme Renee, A~(OG) is much more sensitive
* to changes in T vith temperature. It ls for these reasons
* * that T is treated as a variable vith temperature, while v sp
* and Pare considered to be constants.
.1
..,,'
- 76 -
equation (I-40) for PIS and PDMS with the n-alkanes, experimental
values of a at the temperature at which A~(oo) was measured
* * were used to determine Tl and T2 as a function of temperaturee
For PDMS with its oligomers, experimental values of a as a
function of temperature for the oligomers are unavailable. As
a result, values of T~ at 25 0 C were taken to be constant.
Values of al were then calculated according to equations (1-)6)
and (I-)7).
In the case of PDMS with its oligomers, the values of
(s2/sl)X12 are quite small (less than 0.25 joules/cc at 25 0 C).
The interaction term, X12 ' thus makes a very small contribution
to A ~(oo). For PDMS with its dimer, the total à~(~), the
free volume term, and the contact interaction term are
respectively -1.18, -1.41, and +0.23 joules/gram at 250 C. This
result is consistent with the previous conclusions of Patterson
et ~.(49) in which it was found that the Flory theory, with
X12 = 0, could be used to explain the heats of mixing results of
dimethylsiloxane chain Molecules amongst themselves. Thus the
value of A~(oO) is determined by the f"ree volume contribution
in equation (I-40), as there is essentially no difference of a
chemical nature between the PDMS and its oligomers.
When (s2/sl)X12 is calculated from equation (1-40) for
systems of PIS and PDMS with the n-alkanes, the magnitude and
importance of this quantity is seen to change markedly. For PIS
with pentane at 25 0 C, the A~(co), free volume term, and
interaction term are -3.6, -10.5, and +6.9 joules/gram. Here
'-
- 77 -
the free volume contribution to A~(oo) is very exothermic and
the interaction term is vital in establishing the total heat.
o The corresponding quantities for PIB- hexadecane at 25 C are
+0.04, -1.22, and +1.26 joules/gram. It will be argued later
that the Flory theory overestimates the free volume term, thus
forcing the fitted (s2/sl)X12 to be too large.
(s2/sl)X12 as a function of Temperature
Figures 15, 16, and 17 show that (s2/sl)X12 changes
considerably as the temperature of measurement is increased.
For PDMS vith the n-alkanes (figure 16), a decrease of (s2/s1)X12
with temperature is noted while, for the other two systems, PIB
with the n-alkanes and PDMS with its oligomers (s2/sl)X12 is
found ta increase. We emphasize that X12 is a molecular
parameter, in principle independent of temperature. Its
variation indicate$ a failing of the Flory theory. The greatest
changes of (s2/sl)X12 occurs for the solvents of shortest chain
length. This is presumably a reflection of the relatively high
coefficient of thermal expansion of these materials which results
in free volume terms that increase too rapidly. When this
occurs, (s2/sl)X12 must increase to compensate for the
overestimation of the free volume expression.
The data of Liddell and SWinton(43 ) was determined
over a much greater temperature range than the present work.
Using their data, the (s2/sl)X12 for PIB with hexane is found to
- 78 -
Figure 15
(S2/S1)X12 AS A FONCTION OF TEMPERATURE (T)
FOR PIB WITH THE n-ALKANES
0 PIB - Pentane
0 PIB - Hexane
" - PIB - Heptane
ce PIB - Octane
 - PIB - Decane
• PIB - Hexadecane
[Data shown in Table 2-a, co1umns 1, 3 and 4 of' the Appendix]
•
•
t> • \ \
0"" 1>\ ~ 0 0
o o o o o . . . . . "" li') ('t"')
•
44 \ ~
o .
o li') ('t"')
0 ..::t ('t"')
0 ('t"') ('t"')
0 c--J ('t"')
0 .-1 ('t"')
0 0 ('t"')
0 0"1 c--J
;-...
:::.::: 0 '-"
E-t
- 79 -
Figure 16
(S2/Sl)X12 AS A FONCTION OF TEMPERATURE (T)
FOR PDMS WITH THE n-ALKANES
0 PDMS - Pentane
0 PDMS - Hexane
\l - PDMS - Heptane
• PDMS - Octane
.6 - PDMS - Decane
• PDMS - Hexadecane
[Data shown in Table 3-a, columns 1, 3 and 4 o~ the Appendix]
Il
o . \0
•
o LI"l
• t> Jr;;J
o o . . ~
,-
•
0 ('1"')
0 ('1"') -.. ~
0 '-'
0 E-! N ('1"')
0 0 ....-1 ('1"')
00 0 , , 0
00 ('1"')
0 ~ N
o . N
J
- 80 -
Figure 17
(s2!sl)X12 AS A FUNCTION OF TEMPERATURE (T)
FOR PDMS WITH ITS OLIGOMERS
o PDMS - Dimer
o PDMS - Trimer
6. . - PDMS - Te tramer
• - PDMS - Pentamer
[Data shown in Table 4-a, columns l, 3 and 4 of' the Appendix]
'- 1 .. 1
o 0
o li"') o . . . . N o
oo<lt
o . o
o o ('\")
'-.-1
- 81 -
increase by a ~actor o~ ~ive ~rom 300 to 150o c. This is
consistent with resu1ts determined in this work and is
comp1ete1y incompatible with the idea that X12 is a temperature
independent parameter.
X12 as a ~unction o~ Chain Length
The ~act that X12 varies with temperature raises the
question o~ how this parameter varies within an homo10gous ~
series at a single temperature; this amounts to holding T2 ~
constant but changing Tl. Figures 18 and 19 show X12 p10tted
as a ~unction o~ the chain 1ength, n, ~or PIB and PDMS with the
alkanes. The values of {s2/s1)X12 ~or these systems were
ca1cu1ated ~rom values of the heats of mixing previous1y
determined at 250 C.
1ength ~or PDMS with the a1kanes (figure 19), the end e~~ects
(X12) increase as the chain 1ength o~ solvent increases. This
is in direct contrast to the situation for PIB with the n-alkanes.
Both o~ these resu1ts can be exp1ained in terms o~ the mo1ecular
geometry o~ PIB and PDMS i~ it is assumed that a terminal solvent
methy1 group interacts 1ess than either an interior solvent
methy1ene segment or a methy1 group attached to the backbone of
a PIB-mo1ecu1e.
PIB May be pictured as a compact rod simi1ar in nature
to p01ymethy1ene. Flory and EiChinger( 36) have found that a
plot o~ X12 versus l/n for the n-a1kanes in PIB extrapo1ates to
a value of X12 = 0.84 joules/cc. This 10w value of X12 means
- 82 -
Figure 18
(S2/S1)X12 FOR SOLUTIONS OF Pia - n-ALKANES AT 2980 K
AS A FUN CTION OF THE N UMBER (n) OF
CARBONS IN THE SOLVENT BACKaONE
[Data shown in Table 2-a, co1umns 1, 4 and 5 of the Appendix]
,-
•
o \0
o
•
o . ..::t
o . C"")
•
o . C"J
•
o ~
o ~
00
- 83 -
Figure 19
(S2!S1)X12 AT 2980 K FOR SOLUTIONS OF PDMS - n-ALKANES
AS A FUN CTION OF THE N UMBER (n) OF
CARBON S IN THE SOLVENT BACKBONE
[Data shown in Table 3-a, co1umns 1, 4 and 5 of the Appendix]
o .
• "-.~
o C"")
•
o . N
'" .-1
Ln .-1
..::t
.-1
C"")
.-1
N .-1
.-1
.-1
0 .-1
0"\
00
"-
'" Ln
- 84 -
that contacts between a CH2 group in polymethylene and the
backbone methyl group of a PIB Molecule are almost indis-
tinguishable from contacts of the like species amongst
themselves. Hence, when PIB is mixed with an n-alkane, it 15
only when the number of solvent end-groups of the n-alkanes
become relatively important that Xl2 increases.
On the other hand, PDMS is considered to be a very
flexible Molecule that is surrounded by relatively weak force
* fields • A low value of Xl2 is found when PDMS is mixed with
an n-alkane having a relatively high percentage of end-groups.
~2 should increase with the chain length of the solvent until
it approaches sorne limiting value as the fraction of chain ends
of the solvent approaches zero.
Quantitative Analysis of the Xl2 Parame ter
Qualitatively, the type of argument presented in the
last section is attractive, but a quantitative analysis of the
molecular significance of the Xl2 parameter and its variation
with the chain length of the solvent Molecule is required.
* This can be made by writing an expression for U for the
solution in analogy to equation (I-33) for a pure component.
* Assuming random mixing, U May be related to energy parameters
* This is a result of the large number of methyl groups attached
to the polymer backbone.
- 85 -
"11' 1}22 and 1)12 characterizing 1-1, 2-2, and 1-2 contacts.
One can then show that
u*
* where rs is an average number of contacts per mole of solution,
and
2 The ~ parameter of the Prigogine theory is obtained from
equation (I-22) giving
V 2 : ..An-1)11
The F10ry parame ter, X12 ' is given by
(III-l)
In general, it May be assumod that a mo1ecule has
different types of sites. If the fraction of sites on
component i of type ip is a ip and the interaction of these
sites with those of component j of type jp is characterized
by " ipjp' then âl) may be wri tten
(III-2)
* Mole fraction average.
,-.1
- 86 -
If cross terms are e1iminated by use of the Berthelot
equation (geometric Mean ru1e) such that (1] eml 7Jm)2 = 1) e/~ m'
then equation (111-2) May be wri tten for the case of' a binary
mixture in which there are on1y two types of sites, ends (e) and
midd1es (m). If 11 1e1e is wri tten as "1] le' etc., then
( 111-)
In po1ymer-so1vent systems n 2e = o. When equations
(111-3) and (1-34) are substituted into equation (111-1), the
fo11owing resu1t is obtained.
= ('1 lm - 1} 2m)
[
t t t 12
- 7J Im)J (111-4)
nIe' the fraction of solvent end sites May be written:
(111-5)
For the a1kanes, F10ry and orwOl1(33) have taken:
(1) r 1 = n + 1, where n is the number of carbon atoms in
the a1kane.
s Is e m = 0.6 as determined from crysta110graphic data.
r = e 0.0, where re i5 the number of segments having
an end or methyl character.
This means that the on1y methy1 sites are the extra
- 87 -
Se sites at the ends of the chain. A value of re = 2 wou1d
certain1y seem, however, to be a more reasonab1e choice.
According to equation (III-5) when the solvent chain
length is large, ale becomes sma11 and X12 ref1ects the
difference in interaction energy between midd1e sites of po1ymer
and solvent [see equation (III-4)J. The variation of X12 with
solvent chain 1ength depends main1y on the energetic difference
between solvent ends .and po1ymer midd1es, since the term
(~tm - ~~> is essentia11y constant. From equation (111-4),
a graph of (s2X12!slP;r!- versus 1/(r1 +se/sm> shou1d resu1t in a
straight 1ine vith a slope of [(''lIe - 1J lm)!" tu] (re + sel Sm>
and an intercept of' (1') 1m/ 1}2m>1- - 1. From this intercept, one
can obtain 111m/ 1)2m' the relative strength of midd1e contacts
in solvent and po1ymer mo1ecu1es. This, coup1ed vith the value
of the slope enab1es one to ca1cu1ate the parame ter r , given by
f= [slope /(l + intercept)] 2
(III-6)
In view of' the above discussion, an ana1ysis of the X12
parame ter in te~s of' equation (IXI-4) is presented. Figures
* À 1 20 and 21 show plots of (s2X12!slP2)2 versus (r1 + 0.6)-
calculated from the data of' De1mas, Patterson, and SOmCynsky(14),
De1mas, Patterson, and Boehme (42) , and Liddell and Swinton(43).
'-
.1
.1
- 88 -
Figure 20
FOR PIB WITH THE n-ALKANES
8- PIB -0 n-alkanes at 298 K. (De1mas ~ &.(14»
[Data refers to abscissa and 1eft-hand ordinate]
~- PIS - 0 n-alkanes at 303 K. (Liddell and Swin ton ( 43) )
[Data refers to abscissa and right-hand ordinate]
a- PIB - n-alkanes at 348oK. (Liddell and Swinton (43) )
[Data refers to abscissa and right-hand ordinate]
0- PIB - n-alkanes at 4230 K. (Liddell and Swinton (43) )
[Data refers to abscissa and right-hand ordinate]
[Data shown in Table 5-a of the Appendix]
:".-
~
.<1 ...-l a 0
.{/ 0
("1")
...-l a 0- 0
-II 0
...-l
...-l 0 """ 0 \.0
0 • <1 0 0 0
/ + 0'\ ...-l 0 lo-I • 0 '-' 0 .........
/ ...-l
r--. !! 0
1 0
0 Il 1
Il ~ 0
0
0
("1") 0
0
0
N \.0 0 ~ 00 N 0 0 ...-l ...-l ...-l N 0 0 0 0 0 0
0 0 0 0 0 0 1 1 1 1 1 1
~(~dls/Z1XZs)
, i
- 89 -
Figure 21
.Jo.
(S2 ~2)2 AS A FUNCTION OF 1 6 FOR PDMS sl P rl+o.
2
(Delmas, Patterson, and Boehme(42»
[Data shown in Table 5-a of the Appendix]
Ln
Il ..-1 · 0
("t")
..-1 · 0
,,-... ..-1 ~ ..-1 . · 0
0
+ ..-1
'" $-1 0 '-" · --• 0 ..-1
1"-0 · 0
• tr)
0 · 0
("t")
0 · 0
I.f") Ln tr) I.f") I.f") I.f")
r-. 00 '" 0 ..-1 N
0 0 0 ..-1 ....-! .-l
. . . . . . 0 0 0 0 0 0
~(~dlS/GIXlS)
- 90 -
If the Flory model is correct, these plots should be linear.
The linear dependence is approximately observed for both PDMS
and PIB with the n-alkanes at 2So C. For PIB - n-alkane
systems, hovever, deviations frorn linearity do occur in the
region of hexane and pentane. The values of the 1) lm/1) 2m
parameter are respectively 0.96 and 1.37 for the PIB - n-alkane
and PDMS - n-alkane systems. The slopes of both lines are
similar and give r = 0.3. This value of r May be compared
with those obtained from an analysis of the n-alkane equation
of state(23) <r= 0.09) and from the thermodynarnics of n-alkane
mixtures(34) <f= 1.4).
Assessment of End Effects from the Equations of State of n-Alkanes
Orwoll and FIOry(34 ) have interpreted equation of
state data for the n-alkane series by using their theory. They
* obtain values of P from the experimental data and find that
these values decrease as the chain length of the solvent is
shortened. * Since P has the signifieance of a cohesive
energy/molecular volume, Flory attributed the decrease of p*
vith solvent chain length to weaker force fields surrounding
the methyl end groups relative to those around the interior
* Methylene groups. The same deerease in P would.theoretically
be observed if a linear n-alkane vere replaced by a branehed
alkane of the sarne molecular weight.
Unfortunately, the quantitative applieation of the
Flory theory gives a negative value to the "le/1') lm parameter
- 91 -
[c~. equation (III-6)] which is, o~ course, unacceptable.
However, it should still be possible to assess the end ef~ects
through their e~~ectson the heat of mlxing of n-alkane systems.
I~ the n-alkane Molecules vere merely chains o~ identical
segments, X12 would be zero, and the Flory theory would predict
negative values o~ 6~ because o~ a negative free volume terme
o The experimental heats o~ mix1ng, however, are positive at 25 C
so that XI2 must be given a positive value which ls related to
the ~act that end segments are of a different chemical nature
than the interior methylene groups. orwoli and Flory find
reasonable agreement vith the experimental AHM of n-alkane
mixtures when~ is introduced and assigned a value o~ 1.4.
This ls a very large value which re~lects an enormous dl~~erence
in ~orce ~ields between end and middle segments.
Pa tterson and Bard'in (23) have made an extensive
analysis of the n-alkane equatlon of state data. They find
that the corresponding states principle is followed by the
n-alkanes but that the Flory theory and man y other simple
theoretical models do not satis~actorily reproduce the data.
* These theories predict, as an artefact, the decrease o~ P with
decreasing chain length. According to Patterson and Bardin, al<
P is almost constant .~rom polyethylene to butane. The concept
o~ a large di~~erence in force fields between the end and middle
'-
segments thus obtains no support from the equatlon of state data.
The Patterson-Bardin assessment of the end effects gives
r = 0.09. Such a value of r is insufficient to give positive
- 92 -
values o~ ~HM ~or the n-alkane mixtures when used in the Flory
theory. It may be shown, however, that the positive tl. ~1 of
n-alkane mixtures may still be predicted(17) by a corresponding
states treatment which avoids the simple liquid model used by
Flory and others.
Because of this, we view the introduction of large
end effects as a theoretical device required to correct an
approximate theory. This view receives support from the fact
that con~licting values of ~ are required (1) in this analysis
of polymer - n-alkane systems (~ = 0.3) and (2) in application
of the Flory theory to n-alkane mixtures (1' = 1.4).
Further confirmation is af~orded by the data of
Liddell and SWinton(43) plotted over a temperature range of
30 0 - l50 0 C in ~igure 20. This graph shows that the slope of
the (s2X12/slP;)i vs. (rI +0.6)-1 curve (which should
theoretically remain constant) actually increases quite rapidly
with temperature and that the linear dependence predicted by
theory is not fulfilled for the lower alkanes.
Additional information on the effects of solvent chain
length on X12 may he determined ~rom a plot of (s2/sl)X12 versus
The quantity aIT is a measure of the reduced temperature,
-T19 or ~ree volume of the solvent. In ~igures 22, 23, and 24
(s2!sl)X12 is plotted for the different polymer-solvent syptems
as a function of reduced temperature, aIT. The Flory the ory
- 93 -
Figure 22
(s2/s1)~2 AS A FUNCTION OF a 1 T FOR
PIB WITH THE ALKANES
0 PIB - Pentane
0 PIB - Hexane
\J - PIB - Heptane
• - PIS - Octane
 - PIB - Decane
il - PIB - Hexadecane
f) - PIB - Branched Heptanes
CI - PIB - Branched Octanes
[Data shown in Table 2-a, co1umns 1, 2, 3 and 4 and
Table 6-a, co1umns 1, 2, 3 and 4 of the Appendix]
'-
N 1./")
· 0
0
• 00 ..j"
· 0
..j"
..j"
· 0
~ 0
~ ..j" ·
~ 0
E-! ~-
, ~QQ \0 C")
e .~ ~ · • 0 • 'e \~ N
C")
~ · 0
00 N · 0
..j" N
· 0
o o o o o o . . . . 1./") ..j" N
(~JjSalnoÇ) ZlX(lSjZS)
- 94 -
Figure 23
(S2/Sl>X12 AS A FONCTION OF aIT FOR
PDMS VITH THE ALKANES
0 PDMS - Pentane
0 PDMS - Hexane
\l - PDMS - Heptane
• PDMS - Octane
 - PDMS - Decane
III PDMS - Hexadecane
C> PDMS - Branched Heptanes
(J PDMS - Branched Octanes
[Data shawn in Table 3-a, columns l, 2, 3 and 4 and
Table 7-a, columns l, 2, 3 and 4 of the Appendix]
1
N Lf"\
· 0 0
.01 00
0 ..:t · 0
..:t
..:t · 1>
0
0 ..:t · 0
ei) E-I
e.e "" '6-
M
e · 0
e 8 N M · 0
00 N
· 0
• ..:t N
· 0
o o o o o o . . . . . . r- "" ..:t M N
- 95 -
Figure 24
(S2/SI)XI2 AS A FONCTION OF aIT FOR
PDMS VITH ITS OLIGOMERS
o PDMS - Dimer
o PDMS - Trimer
I::!:. - PDMS - Te tramer
• PDMS - Pentamer
[Data shown in Table 4-a, co1umns 1, 2, 3
and 4 o~ the Appendix]
,-
C".J \.0 · 0
o 00 IJ"'I · 0
"" IJ"'I · 0
0 IJ"'I · 0
\.0
"" · E-c 0
\S
C".J ...::t · 0
00 C"')
· 0
0
\~ ...::t C"')
· @
0
0 C"')
· 0
IJ"'I IJ"'I IJ"'I 0 0 r- ""
C"') . . . ,.-l 0 0 0
- 96 -
predicts that each polymer-solvent system should be represented
by a horizontal straight line in these graphs; Xl2 then should
not change with ~lT for any particular solvent but should vary
from solvent to solvent. However, (s2/sl)Xl2 is found to be
an increasing function of ~lT for PIB with the n-alkanes and
PDMS with its oligomers, while it is found to decrease for
PDMS with the n-alkanes. These trends are shown regardless
of whether ~lT varies through a change of temperature T or a
- -change of solvent (Tl) at constant T2 - This suggests that
values of (s2/ sI) ~2 are detennined when the Il ~(oo) is fi tted
to an expression containing an incorrect free volume terme
Figure 25 shows (s2!sl)~2 calculated from the data of Liddell
and Swinton(43) plotted in the same manner as figures 22, 2),
and 24. AltpoUgh the values of (s2!sl)XI2 are somewhat
higher than those calculated in the present york, the same
trend is observed, indicating that something is wrong with the
Flory model. On this basis, it 1s felt that the large end
effects proposed by Flory for the n-alkanes cannot be justified.
In the light of 1:his, da ta on .ô~(oo) of PIB and PDMS wi th
branched alkane solvents would be of interest.
à~(I'JIO) of Branched Alkane
Polymer Systems
The  ~(ooo) of certain branched alkanes vi th PIB and
PDMS have been measured at 30°C. These heats are very similar
'-.1
- 97 -
Figure 25
(S2/s1)X12 AS A FUNCTION OF a 1 T FOR PIB VITH
THE n-ALKANES [FROM LIDDELL AND SWINTON(43)]
o PIB - Pentane
a PIB - Hexane
• PIB - Octane
[Data shown in Table 1-a, co1umns 1, 3, 4 and 5 o~ the Appendix]
o
o . C"-I C"-I
o 00 ~I
8
o . o r·~
'-
· o
· o
\,()
· 0
lf"'\ · 0
0 0
.\ ...::t · 0
G
Ct")
· 0
- 98 -
* to those found for the sarne polymers with n-alkanes having
an identical carbon number. If aIl methy1 carbons are
considered to be equivalent, the fraction of sites having an
end character in these Molecules is drastically changed. In
equation (III-5) r would represent the total number of alkane e
methyl groups and Se/sm would be replaced by 0.3 x total number
of methyl groups. This would Mean that a branched alkane
having two tertiary carbons should have a value of X12 which is
about four times the value found when the normal alkane is used
as a solvent. In the case of PDMS with the same solvent, X12
should be approxi~ately zero. In actual fact, however, the
values of·Xl2 for these branched alkanes with PIB and PDMS are
very similar to what is calculated for the corresponding n-
** alkane-polymer systems • These facts supply additional
support for the idea that ~2 is merely acting to compensate
for an incorrect free volume term and is not the strong
reflection of end effects which Flory has proposed.
*
**
Actually heats of mixing at infinite dilution of PDMS -
branched systems are, in general, slightly less than those
observed for PDMS with the corresponding n-alkanes. The
difference, however, is not significant.
The(s2/sl)~2 for the branched alkanes with PIB and PDMS are
shown plotted in figures 22 and 23.
· i
- 99 -
t:. ~ f'or Squa1ane wi th PDMS and PIB
o has a1so been measured at 90 C f'or squa1ane
(C30H62 ) with Pla and PDMS to determine if' a high1y branched,
long-chain alkane wou1d show the ef'f'ects of' chain bran ching on
A ~(-) as predicted by F1ory. For Pla with squa1ane, the
measured A~(~) is exothermic and slight1y 1ess negative than
the corresponding A~(-) with hexadecane. Since ~~(oo)
becomes more endothermic f'or Pla - n-a1kane systems as the
solvent chain 1ength is increased, the magnitude of' the 6~~)
of' squa1ane with Pla can be rationa1ized as resu1ting f'~om the
increased chain 1ength of' the solvent. According to F1ory,
however, the large amount of' branching present in squa1ane
shou1d 1ead to a much greater endothermic ~ ~(I>O) than is
observed.
In the case of' PDMS with squa1ane, the large chain
1ength of' the sol ven t shou1d tend to make the ~ ~(oO)
endothermic, while the large number of' branches should
signif'icantly reduce this endothermicity according to Flory.
However, a very large endothermic A~(oG) is observed. As a
result, it is not possible to rationa1ize this f'act with the
strong end ef'f'ects which F10ry assumes.
~~lure of' Corresponding states
Due to an error in the f'ree volume term, the Flory
model is unab1e to exp1ain quantitative1y the values of' ~ ~(~)
- 100 -
presented in this thesis without assuming large end effects.
However, as shown, these end effects are not only too large,
but also they are forced to vary with temperature for the model
to be consistent with experimental data. From a fundamental
point of view, this temperature variation cannot be explained.
As a result,. the question arises as to whether another liquid
model consistent with corresponding states May be used to
describe the observed A~(-).
Equation of state data on the n-alkanes do follow
corresponding states. The data predict that F = 0.09 for the
n~alkanes. This is consistent with the assumption of small
end effects. However, these small end effects do not permit
the Flory model to predict the positive excess heats of mixing
of the n-alkanes amongst themselves.
If a small value of )? is accepted, the HE of
n-alkane mixtures may be explained by use of a general corres-
ponding states theory without any specifie model, provided that
dCp/dT (or dCp/dTl ) for the n-alkanes is negative.
This negative sign has been confirmed(17) by the
experimental configurational heat capacities of the alkanes.
""/-The sign of dCp dTl in the present study can be determined from
* differentiation of the general corresponding states expression
*
i
- 101 -
,., for b~(,.o) with respect to Tl to give~
(III-7)
Experimentally the left-hand side of this equation
found to be negative. For PIB with the s2 X12 n-alkanes -- --- ls s p*
1 2 * * very small compared to (1 - Tl !T2 ); hence, the first term in
15
brackets on the right-hand side of equation (III-7) is s X
negative. The quantity (J s2 p;2) 1 (d Tl) as calculated from 1 2
this end effect analysis i8 small but positive, making the
second term of equation (III-7) negative as weIl. Thus,
I!}ëpl S 1\ must be positive to be consistent vith the observed
temperature dependence of A~(oo) for PIS with the n-alkanes e
This, however, is contrary to what has been found for mixtures
of the n-alkanes amongst themselves(17) and is interpreted to
mean that the principle of corresponding states is not
completely obeyed for PIB with the n-alkanes and probably as
weIl for PDMS with the n-alkanes.
The difficulty seems therefore to be much more
fundamental than something which can be corrected by a change
of the model of the liquide Partial confirmation cf this
comes from the Il ~(oo) of PIB vi th the following cyclic alkanes ~
,, ..l
- 102 -
cyclooctane, cycloheptane, cyclohexane, and cyclopentane. The
values of à~(co) are respectively +0.12, -0.43, -0.82 and
-5.43 joules/go The slow progression of heats from cyclo-
octane to cyclohexane is expected, since it parallels the
behaviour of PIB - n-alkane systems and reflects the
increasingly important free volume terme The drastic change
in the value of A ~(oo) between cyclohexane and cyclopen tane is
unexpected however, and can only reflect the change of molecular
shape from globular to plate-like. The same sudden change of
A~(~}, although much less marked, is seen for PDMS with the
following cyclic alkanes: cyclooctane, cycloheptane, cyclo-
hexane, and cyclopentane. The values of A~(oo) are
respectively +6.96, +6.06, +5.05 and +1.04 joules/go
Any theory based on the corresponding states principle
predicts the values of A~(~) from the equation of state
proverties of the pure components as well as the Xl2 parameter.
In the series of cyclic alkanes investigated, there are no
sudden changes in boiling point, critical temperature, heat of
* * vaporization, or in the T and P parameters. The free volume
term should thus progress slowly from cyclooctane to cyclo-
pentane for any theory. It is also difficult to maintain that
the intermolecular forces suddenly change in intensity between
cyclohexane and cyclopentane; the X12 parame ter thus changes
in a regular fashion. Some factor seems to be missing from
the corresponding states theories in general; perhaps it is
the possibility of preferred orientations of the cyclopentane
' , .i
- 103 -
in the presence of PIB. In the case of PIB - n-a1kanes, there
are no sudden changes of solvent molecular structure which
would reveal an inadequacy of the theory in a qualitative
fashion. This, however, does not preclude a small failure of
the corresponding states princip1e as apparent1y indicated in
the present work.
CON CLUSION S
Heats of mixing at infinite dilution were determined
for PIB with the n-alkanes, PDMS with the n-alkanes, and PDMS
with its oligomers over a temperature range from 300 to 90 0 Co
In aIl cases the A~,(~) became more negative as the temperature
of measurement was increased. In addition to these systems,
A~(bO) were also determined at 300 0 for PIB and PDMS with a
number of branched and cyclic alkanes.
The experimental data were used to test the Flory
model of the liquid state by analysingthe fitted values of the
~2 parameter. This parameter depends on the intermo1ecular
forces and, in principle, should not vary with temperature.
Experimental results, however, indicate that Xl2 varies with
temperature for aIl systems investigatedo For PDMS with its
oligomers and PIB with the n-alkanes, Xl2 increases with
temperature, while for PDMS with the n-alkanes it is found to
decrease.
The value of Xl2 for a polymer with a homologous
series of solvents can, in principle, vary with the length of
,-
- 104 -
the solvent molecu1e because of solvent end effects, i.e. the
force fields around the end segment are different fTom those
* J.. around the interior segments. A plot of (s2Xl2/slP2)2 versus
1/(r1 + 0.6) was used to determine the value of ~ for the n-
alkanes. (r is actua11y a measure of end effec ts 0 ) The
value of ~ = 0.3 found in this work is much less than the value
of r = 1.4 used by Flory to exp1ain the Il ~ of n-alkane
mixtures. Furthermore, either value corresponds to a large
difference between force fields around the methy1 ends and
methy1ene interior segments. This is contrary to a finding(23)
of re1ative1y sma11 end effects made without the aS5umption of
a mode1 for the liquide
Vith a large value of -e , X12 and à~(QO) should
increase or decrease significant1y with the degree of chain
branching of the solvent. The values of il ~(Odt) and X12 for
the branched alkanes vith the above polymers are of the same
order of magnitude as found for corresponding n-alkanes of the
same carbon number. This also suggests that the Flory
interpretation of X12 as a measure of end effects i5 not
correct.
The Flory theory isqualitative1y successfu1 in
attempting to describe solution properties. The theory,
however, incorrectly estimates the free volume contribution to
A ~(QO) and, as a result, introduces an X12 parameter which 15
forced to vary if theory and experiment are to agree. A11 of
the results of this work indicate then that X12 cannot be
'- , .. 1
- 105 -
considered as a ~undamental molecular parame ter as Flory has
ascribed. It seems clear tbat a better model of the liquid
state is required.
It bas also been found that the corresponding states
theory, on whicb the Flory model is based, is not exact. For
mixtures of the n-alkanes, dCp/dTI , according to the theory~
is negative while for mixtures of PIB with the n-alkanes - / ..., dCp dTI is positive. -/The discrepancy in the sign of dCp dT1
in the same homologous series points out that the principle of
corresponding states fails to sorne extent.
'-1
- 106 -
CHAPTER IV
IN TRODUCTION
In the Flory model, it is assumed that the X12
parame ter for mixtures of alkanes vith PIB increases
significantly when the solvent is branched. For example,
according to equation (111-4), the X12 parameter for a solution
of PIB and 2,4-dimethylpentane is approximately four times
greater than the corresponding parame ter for a solution of PIB
and n-heptane. In the previous chapter it vas pointed out,
* however, that values of (s2!sl)XI2 , as determined from
experimental A~(oO), are approximately the same for both the
n-alkanes and their isomers vith PIB.
A very thorough test of the (3- 00) model and, in
particular, FloryOs concept of end effects would be provided
by a comparison of experimentally determined solution
properties of PIB in both linear and branched alkanes with
those calculated from theory. The concentration dependence
of the " parameter, as determined experimentally by vapour
sorption, has been chosen as the quantity on which this
comparison is based. According to the Flory model, the
concentration dependence of 1L is expressed as:
* The ratio s2/s1 does not vary appreciably when an isomer ls
substituted for a linear alkane.
,-i
•..• -1
- 107 -
(IV-l}.
where (ln al) b is given by the Flory theory [see non-corn
equation (I-45)J.
EXPERIMENTAL
Background
If the segment fraction (~) of solvent in a polymer
solution is known as a function of the solvent vapour pressure
above the solution, the concentration dependence of the ~
parameter for the system May be determined experimental1y from
equation (IV-2) (be1ow). A McBain balance was used to obtain
the necessary data for the systems, PIB - n-heptane and PIB -
2,4-d1methy1pentane.
o
1n(1 - 'lJ 2) + (1- ;);2 + X~~ = ( "liT U1 ) \1: ln (IV-2)
B is the second v1r1a1 coefficient of the solvent vapour.
o Pl is the equil1brium vapour pressure of pure solvent at
temperature T, and Pl is the equi1ibrium vapour pressure of
solvent above a solution of pOlymer and solvent at the sarne
temperature T.
General Description of the Apparatus
In genera1, a McBain balance used in vapour sorption
studies 1s made of a cylindrical glass chamber (the temperature
i .,
- 108 -
of which is accurately controlled) attached to a high vacuum
line. A calibrated quartz helix is suspended from a hook
within this chamber. The sample to be studied ls placed in a
small pan which is hung from the bottom of the helixe After
the system is evacuated, the extension of the helix caused by
the combined weight of the sample and pan ls measured with the
aid of a cathetometer. At an appropriate moment a known
amount of solvent is introduced into the system. The
additional extension of the quartz helix which results is a
direct measure of the amount of solvent that has been absorbed
by the sample. From this data and the calibration constant of
the helix, the segment fraction of solvent in the solution May
be determined ..
At the same time that the concentration of solvent
in solution is determined, the corresponding equilibrium
vapour pressure of solvent in the McBain chamber is measured
by means of a manorneter. Using this data in equation (IV-2)
in conjunction vith the measured vapour pressure of the pure
solvent at the experimental temperature, the ~ parameter for
the system is determined as a function of concentration.
Need for Modification
Because of the relatively low sensitivity of the
available helix, it was necessary to work with large (100 mg)
samples of PIB. With large samples such as these, several
days were required for equilibrium. In this long period of
time, however, leaks developed within the vacuum system
,-
- 109 -
because the solvent vapour attacked the grease surrounding the
stopcocks. Hence it was necessary to work in a completely
sealed system.
A diagram of the sealed McBain is shown in figure 26~
The body (A) of the McBain is made from heavy-walled glass
tubing 25 cm in length. At its base are six 'legs' (B), each
approximately 10 cm long, which are used to hold sealed
ampoules of solvent. The neck (C) of the McBain is
approximately 15 cm long; the helix (D) and the sample (E)
are moved into position through this neck before each rune
Procedure
Prior to use, the empty McBain is washed with hot
benzene several times and dried in a vacuum oven. Ampoules
(F), containing precisely determined quantities of solvent,
are then introduced into the system; each ampoule is fitted
into a single leg of the McBain. After this is completed, a
bar magnet (G) is carefully placed in a horizontal position at
the base of the chamber.
An aluminum pan containing degassed polymer is hung
from the bottom of the calibrated quartz helix. The helix
and pan, in turn, are suspended from a glass hook (H). This
system of hook, helix and pan is then lowered into the
slightly tapered neck of the McBain. The helix and pan are
able to pass through into chamber (A). The hook, however, is
too large and is held in position at the base of the neck&
· i
- 110 -
Figure 26
SEALED Mc BAIN BALANCE USED TO DETERMINE
THE CONCENTRATION DEPENDENCE OF THE )lPARAMETER
A McBain chamber
B Legs of McBain used to hold solvent ampoules
C Neck of McBain
D Helix
E Sample pan
F Solvent ampoules
G Bar magnet
H Glass hook used to support the helix
l Point at which the McBain is evacuated
-.i
c
A
G
B
- III -
The neck of the Mc Bain is then sealed at the top and the
system evacuated through a glass joint located at point (I)o
After two or three hours under high vacuum, the McBain is
tested for leaks. If none can be located, the glass joint at
point (I) is sealed and the system isolated.
The sealed McBain is then placed in a temperature-
controlled water bath, the temperature of the bath being
regulated by a Haake pump. Two reference points are chosen
so that the helix extension can be measured with a cathetometer
when solvent is introduced into the system. One reference
point remains fixed throughout the entire run while the other
point is free to move up or down, depending on the amount of
solvent absorbed. The equilibrium distance between these
points is measured when the McBain chamber contains no solvent
vapour, and this distance is used as a reference for aIl
future readings.
To introduce solvent into the system, the bar magnet
(G) is moved into one of the McBain legs by a large magnet
which is lowered into the water bath. When this large magnet
is removed, the magnet (G) is released and the solvent ampoule
within the leg is broken. The solvent contained by this
ampoule escapes into the evacuated system as vapour and the
absorption process begins.
Generally after a period of three or four days,
* equilibrium is achieved and the final distance between the two
* Equilibrium is reached when the extension of the helix becomes constant with time.
- 112 -
reference points is recorded. The difference between this
reading and the one taken prior to introduction of solvent
into the system is related through the calibration constant of
the helix to the amount of solvent absorbed by the polymer.
Thus the concentration of the polymer solution in the pan i5
determined. After this is done, another ampoule of solvent
is broken; the process is continued until aIl remaining
ampoules are used.
Dêtermination of Solvent Vapour Pressure
An ordinary manometer is used to de termine the
vapour pressure of the pure solvent (po). However, it is not
possible to measure directly the vapour pressure of the
solvent above the solution in the sealed McBain. Instead,
the 'manometer pressure' is calculated from the virial
equation of state with aIl terms of p2 or higher neglected.
The volume of the empty McBain, V, is accurately
determined by filling the system with water from volumetric
flasks. n, the number of moles of solvent which remain in
the vapour phase, is merely the difference in the amount of
(IV-3)
solvent originally introduced into the McBain and the amount
absorbed by the polymer. B, the second virial coefficient,
- 113 -
* is given by equation (IV-4) :
T Tc 2 T 4.5
B=VcCO.43-0.886 (;> -0.694 (T) -0.0:315 (n-1)( ;> ] (IV-4)
where Vc and Tc are the corresponding critical volume and
temperature of the solvent and n is the number of carbon atome
in the solvent.
Pressures ca1cu1ated in this manner, which correspond
to the measured concentrations of solvent in the solution, are
used in equation (IV-2) to determine the concentration
dependence of the )( parame ter.
Preparation of the Ampoules
A piece of glass tubing approximately 10 cm in
length vith a diameter of S mm is sea1ed at one end and, at
sorne point along its 1ength, a short capi11ary is drawn. The
hourg1ass shaped tube is then dried in a vacuum oven and
carefu11y weighed.
After solvent has been added to the upper portion
(A) of the tube, the lover portion (B) is gent1y warmed in a
bunsen burner f1ame which causes air within (B) to expand.
*This equation, from the work of McG1ashan and Potter(SI),
applies to aIl n-alkanes from propane to octane. For
the purposes of thi~ York, equa~ion (IV-4) has been used
to calculate B for n-heptane. In addition, it has also
been used in an approximation to calculate B for
2,4-dimethylpentane.
.. ;
- 114 -
As a result, a portion of the air bubb1es through the solvent
and escapes. When no more air can be removed from (B) in this
manner, the tube is cooled to room temperature. As this
occurs, the solvent is forced into (B) to replace the air
which has been expelled. After section (B) is filled with
solvent, any excess solvent remaining in (A) is removed. The
tube is then lowered into a dewar of liquid nitrogen with the
opening in the tube stoppered to prevent any water from
entering and condensing in the solvent. After the solvent is
frozen, the bottom portion (B) is sealed and separated from
the remainder of the tube. A very fine tip is left on the
sealed ampoule so that it can be easily broken in the McBain.
The sealed ampoule and portion (A) are dried and weighed.
The weight of solvent contained within each ampoule is then
accurately determined as the difference between this quantity
and the weight of the empty tube.
The ampoule is then placed under high vacuum for
several hours. If no leaks are detected, it is loaded into
the McBain.
Materials
The PIS examined in this study ls a fractionated
sample having a viscosity average molecular weight of 30,000.
It is the same material previously used in experimental
measurements on the heats of mixing (cf. Page 61).
The solvents, heptane and 2,4-dimethylpentane, were
'-
- 115 -
distilled prior to use. The vapour pressures of the purified
solvents were measured with a Mercury manometer.
Calibration of the Helix
In order to determine the helix calibration constant,
a small aluminum foi1 pan is placed onto the bottom hook of the
helix .. Standard known weights of 100 and 200 mg are then
added to the pan vith the corresponding def1ection of the he1ix
being measured by means of a cathetometer. In this manner the
number of centimeters of extension, corresponding to loads of
100, 200 and )00 mg, is determined. Extension of the he1ix
is found to be linear with respect to weight up to loads of
)00 mg.
Preparation of the Polymer Samp1e
The PIB used in this study is first disso1ved in
pentane. A small amount of this solution is then added drop-
wise to an a1uminum pan that has previous1y been weighed and
dried. The pan is very gently heated unti1 Most of the
pentane is removed. This procedure is continued until
approximate1y 100 mg of pOlymer are deposited. At this point,
the pan is put into a f1ask and p1aced under vacuum for several
hours. Periodical1y, the flask is heated to speed up the rate
of solvent evaporation. The degassing procedure is continued
unti1 the weight of the po1ymer and pan becomes constant. The
samp1e is then ready for insertion into the McBain.
', i
- 116 -
RESULTS AND DISCUSSION
The concentration dependence of the )( parame ter was
determined from e~uation (IV-2) for PIB with 2,4-dimethyl
pentane at 30.20 and 34.90 0 and for PIB with heptane at 15010
o and 28.1 C. " is essentially independent of temperature
over the small temperature interval in which these measurements
were made. As a result, aIl calculations of the concentration
dependence of )( , using the Flory model, were based on the
thermal expansion coefficients of the PIB(48) and the
solvents(50 ) at 20 0 C.
The cathetometer used to de termine the deflection of
the McBain's ~uartz helix could be read to ± 0.05 mm. This
corresponds to an error of + 0.02 mg in solvent uptake and a
maximum error of + 0.02 in the " parameter. Similarly an
error of 0.05 mm in the pressure reading of the pure solvent
would correspond to a maximum error in iK of ± 0.02.
The change in chemical potential of the solvent is
given by the expression:
A.I:l RT
where the term B(p - p~) acts as a correction for non-ideal
behaviour of the solvent vapour. The virial coefficient B is
obtained from e~uation (Iv-4) given by McGlashan and Potter(5l ).
This correction in the chemical potential is quite small for
'-.... i
- 117 -
the systems investigated. In fact, if B is neglected, the
maximum error in the "'}( parameter is no greater than + 0.04.
Table III lists the segment fraction, ~2' of the
pOlymer and the corresponding ~ parameter for PIS with heptane
and 2,4-dimethylpentane as calculated from (IV-2).
In figure 27 a plot of )( versus ~2 for the polymer-
solvent systems studied 1s presented. The solid curves
represent experimental results, while the two bottom dashed
curves are the Flory model predictions basad on Xl2 for PIB
with heptane and 2,4-dimethylpentane as calculated from
A~(~} data. The values of X12 used for the two solvents
with PIS are 5.50 joules/cc (heptane) and 5.98 JOUles/cc
(2,4-dimethylpentane). It should be noted that the general
shape of the experimental curves is reproduced quite weIl by
the Flory model. The model seems to underestimate slightly
the experimental results, but agreement is quite satisfactory
in both cases.
If equations (III-4) and (III-5) of the Flory model
are used to determine the X12 parameter of 2,4-dimethylpentane
with PIB and this value (23.92 joules/cc) is used in equation
(IV-2) to calculate the concentration dependence of the )l
parame ter, the top dashed curve in figure 27 is found. Here
it is obvious that the Flory theory drastically over-estimates
the magnitude of the ?l parameter and its concentration
dependence. This signifies once more that the X12 parameter
is not a measure of chain end effects.
,-
- 118 -
Table III
EXPERIMENTAL DATA
PIB - heptane PIB - 2, 4-dimethy1pen tane
f!J 2 X ~2 ""X-
0.887 0.84 0.803 0.75
0.720 0.66 0.717 0.74
0.718 0.68 0.577 0.66
0.471 0.60 0.415 0.64
0.463 0.58 0.348 0.63
0.334 0.56
0.323 0.57
- 119 -
Figure 27
CONCENTRATION DEPENDENCE (IN TERMS OF THE
SEGMENT FRACTION, ~) OF THE ')( PARAMETER AT 298 0 K*
For:
*
• - PIB - heptane (full curve): experimen tal
 _ PIB - 2,4-dimethylpentane (full curve): experimental
~ - PIB - heptane (dashed curve): calcula ted from the
Flory model with (s2/sl)X12 = 50:35 joules/cc
• - PIB - 2,4-dimethylpentane (dashed curve): calculated
from the Flory model with (s2/sl)X12 = 5.98 joules/cc
o - PIB- 2,4-dimethylpentane (dashed curve): calculated
from the Flory model with (s2/sl)XI2= 2:3.92 joules/cc
AlI data used in these calculations to determine Il are based
on values of thermodynamic properties measured at 29:3°K.
[Data shown in Table a-a of the Appendix]
o " , ,
" , , '0
\ \
\ \
\ \
."", ~ ., "" \ \
\ o
.. ~ \ \ \ \ • • ••
~ \ \ \
\ \
\\ ~ ~ \\ .. \ 411\ \ \ \ \ \ ~\ , \
<I .. \ \ \ \ , 1 1
<b
----~----~~~- ~ L-L __ -1-__ 0 00. 0 0
~ 0 o ~ . ~
o . ~
- 120 -
Experimental data clearly indicate that values of
)( for PIe vith 2,4-dimethylpentane are somewhat higher than
the corresponding value of )l for PIe with heptane. This May
be accounted for in the Flory theory if small end effects are
accepted.
Evidence for these small end effects is likewise
provided by A~(oO) data. If the large end effects predicted
by Flory are accepted, neither the A~(~) nor the concen
tration dependence of ?l can be calculated correctly for
branched alkane - PIe sys tems. Hence Flory's concept of strong
end effects must be incorrect.
CONCLUSION 5
The concentration dependence of the X parame ter for
PIB-heptane and PIB-2,4-dimethylpentane vere experimentally
determined with the aid of a McBain balance. Results vere
compared vith predictions of the Flory model. In general, )(
parameters for Pla with 2,4-dimethylpentane are somewhat higher
than those for Pla vith heptane. Qualitatively, these results
are predicted by the Flory model, and very good agreement
between experiment and theory is found if values of Xl2 from
A ~(OCI) data for these systems are used in equation (IV-2).
If the Flory concept of strong end effects (hence
large XI~) is used for 2,4-dimethylpentane, both the magnitude
and the concentration dependence of the ~ parameter are
drastically over-estimated. This indicates that Flory's
'-, .. Î
,-.. 1
- 121 -
pred~ction o~ large end e~~ects cannot be justified on the
basis o~ the experimental data. The Flory model is a
reasonably good qualitat~ve device, but it fails to stand up
to rigorous quant~tative testing.
- 122 -
REFERENCES
1. H. Tompa, 'Polymer Solutions', Butterworth's, London; Academic Press, New York, 1956, Chapter 4.
2. (a) P.J. Flory, 'Principles of Polymer Chemistry', Corne11 University Press, Ithaca, New York, 1953, Chapter 12e
(b) Ibid., P. 544; (c) Ibid., P. 510.
3. H.A. Morawetz, 'Macromolecules in Solution', Interscience Publishers, New York, 1965, Chapter 2.
4. P.I. Freeman and J.S. Rowlinson, Polymer, l, 20 (1959).
5. I. Prigogine, N. Trappeniers, and V. Mathot, Discussions Faraday Soc., 12, 93 (1953)0
6. (a) I. Prigogine (with A. Bellemans and V. Mathot), ~The Molecular Theory of Solutions', North Rolland Publishing Co., Amsterdam, and Interscience Publishers, New York, 1957.
(b) Ibid., P. 53; (c) Ibid., Chapters 16 and 17;
(d) Ibid., P. 325; (e) Ibid., P. 352;
(f) Ibid., P. 350; (g) Ibid., P. 43.
7. (a) J.H. Hildebrand and R.L. Scott, 'Regular Solutions', Prentice Hall, Englewood Clif'f's, New Jersey, 1962, P. 67.
(b) Ibid., P. 92.
8. E.A. Guggenheim, Discussions Faraday Soc., 12, 24 (1953).
9. K.H. Meyer, Relv. Chem. Acta., ~, 1063 (1940).
10. K.R. Meyer and A.J.A. van der Wyk, Helv. Chem. Acta., . ~,488 (1940).
r
Il. G. ~hultz and H. Doll, Z. Elecktrochim., 22, 248 (1952). c
12. G. Shultz and A. Horbach, Z. Physik. Chim. (N.F.), 22, 377 (1959).
13. C. Baker, W.B. Brown, G. Gee, J.S. Row1inson, D. Stub1ey, and R.E. Yeadon, Polyme~~, 215 (1962).
14. G. De1mas, D. Patterson, and T. Somcynsky, J. Polymer Sci., 21.., 79 (1962).
- 123 -
15. G. De1mas and Do Patterson, Discussions Faraday Soc., to be pub1ished.
16. J.S. Row1inson, 'Liquids and Liquid Mixtures', Butterworth's, London, 1959, P. 290.
17. s. Bhattacharyya, D. Patterson, and T. Somcynsky, Physica, 2Q, 1276 (1964).
18. T. Ho11eman, Physica, 11, 49 (1965).
19. J.A. Friend, J.A. Larkin, A. Maroudas, and M.L. McG1ashan, Nature, ~, 683 (1963).
20. J. Hijmans, Physica, gz, 433 (1962).
21. V.S. Nanda and R. Simha, J. Phys. Chem., 68, 3158 (1964).
22. V.S. Nanda, R. Simha, and T. Somcynsky, J. P01ymer Sci., flg, 277 (1966).
23. D. Patterson and J.M. Bardin, Trans. Faraday Soc., 66, 321 (1970).
24. R. Simha and A.J. Hav1ik, J. Am. Chem. Soc., 86, ':3507 (1964).
25. D. Patterson, J. P01ymer Sei., Q12, 2379 (1968).
26. G.M. Bristow and V.F. Watson, Trans. Faraday Soc., ~, 1731 (1958).
27. H. Eyring and J. Hirsch~e1der, J. Phys. Chem., 41, 249 (1937).
28. P.J. F10ry, R.A. Orwo11 , and A. Vrij, J. Am. Chem. Soc., 86, 3507 (1964).
290 P.J. F10ry, R.A. Orwo11, and A. Vrij, J. Am. Chem. Soc., 86, 3515 (1964).
30. P.J. F10ry, J. Am. Chem. Soc., ~, 1833 (1965).
31. A. Abe and P.J. F10ry, J. Am. Chem. Soc., ~, 1838 (1965).
32. J.S. Row1inson, Discussions Faraday Soc., ~, 1 (1970).
33. R.A. Orwo11 and P.J. F10ry, J.Am. Chem. Soc.,.§.2., 6814 (1967).
34. R.A. Orwo11 and P. J. F10ry, J. Am. Chem. Soc.,.§.2., 6822 (1967).
35. B.E. Eichinger and P.J. F10ry, Trans. Faraday Soc., 64, 2066 (1968).
- 124 -
36. B.E. Eichinger and P.J. F10ry, Macromolecules, 1:., 279 (1968) •
37. B.E. Eichinger and P.J. F10ry, Trans. Faraday Soc. , 64, 2035 (1968).
38. B.E. Eichinger and P.J. F10ry, Trans. Faraday Soc. , 64, 2053 (1968).
39. B.E. Eichinger and P.J. F10ry, Trans. Faraday Soc., 64, 2061 (1968).
40. P.J. F10ry, J. Am. Chem. Soc., ~, 372 (1943).
41. A.J. Barry, J. App1. Phys., 17, 1020 (1946).
42. G. De1mas, D. Patterson, and D. Boehme, Trans. Faraday Soc., ~, 2116 (1962).
43. A. Liddell and F. swinton, Discussions Faraday Soc., to be pub1ished.
44. H. Daoust, C. Watters and M. Rinfret, Cano J. Chem., ~, 1087 (1960).
45. s. Morimoto, J. Po1ymer Sei., !!, 1547 (1968).
460 E. Ca1vet and H. Pratt, 'Recent Progress in Microca10rimetry', Macmillan Co., New York, 1963, P.54.
47. T. Kataska and s. Veda, Po1ymer Letters, 4, 317 (1966).
48. B.E. Eichinger and P.J. F10ry, Macromolecules, 1, 285 (1968).
49. D. Patterson, s.N. Bhattacharyya, and P. Picker, Trans. Faraday Soc., 64,. 648 (1968).
50. G. Allan, G. Gee, and G. Wilson, Po1ymer, 1, 456 (1960).
51. M.L. McG1ashan and D.J.B. Potter, Proc. Royal Soc. (London), A267, 478 (1962)0
1 ... 1
- 125 -
NOMENCLATURE
A: Area of heat of mixing curve 60
AA: Type of energy such as TAS, AHM, AGM, etco 29
al Activity
B: Virial coefficient
3c: Number of externa1 degrees of freedom of an r-mer (Prigogine)
Cp: Configurational heat capacity
El Voltage
âGMI Free energy of mixing per mole of solute
AGIl Partial molar free energy of mixing of component l
9
107
22
30
60
8
29
AHM: Heat of mixing per mole of solute :3
à~(~)i Heat of mixing at infinite dilution per gram of pol ymer 46
AHI : Enthalpy of dilution 15
AHl : Partial molar heat of mixing of component 1 29
I: Curren t 60
k: BOltzman's constant 6
m: Characteristic constant 38
No: Avogadro' s number 25
n: Number of atoms in molecular backbone 25
n: Number of moles 112
nI Characteristic constant 38
N: Total number of mo1ecu1es 6
po: Vapour pressure of a pure 1iquid 9
~--
- 126 -
p: Vapour pressure o~ a component in a solution 9
PJ Pressure 25
q: E~~ective number o~ segments 22
qZ: Number o~ first neighbours ~or an r-mer 22
RI Gas constant 8
R: Resistance 60
ri: Number of segments making up an r-mer 13
r: Ratio of the molar volumes of solute to solvent 14 ...
r J Characteristic distance between segments or
Molecules at OOK 23
r: Number o~ segments which have an end character 86 e
rsJ A mole fraction average of the total number
of contacts per mole of solution
6~1 Entropy of mixing
SI Con~j.gurational entropy
A5l : Entropy of dilution
SI Average number of external contacts made by
a segment of component i
S 1 Average number o~ external contacts made by m
85
3
25
15
an interior segment 43
se: Average number of external contacts made by
an end segment in addition to s 43 m , s: Area of curves projected from molecular models 46
T: Absolute temperature 6
T: Critical temperature 113 c
Ua Configura tional energy 25
V: Cri tical volume 113 c
'-.1
- 127 -
AVMI Volume change on mixing
VI Volume, on a molecular basis
V: Volume o~ empty McBain balance
v : Speci~ic volume sp
'le'
VI Volume, on a segmental basis
w: Interchange energy
W: Weight o~ polymer
XI Mole ~raction
XI Surface ~raction
X12 = Enthalpy interaction parame ter
Z: Lattice coordination number
GREEK LETTERS
~: Thermal expansion coe~~icient
~ 1 Fraction o~ end sites e
~: Isothermal compressibility
~: Thermal pressure coe~ficient
~: Heat of mixing parameter for an homologous
series
&: Parameter characterizing energetic dif~erences
between components in solution
E: Contact energy
~ 0 €: Contact energy at 0 K
~I Parameter characterizing the Mean inter
molecular interaction between two segments
1
"l em' "lm
Parameters characterizing the Mean inter
molecular interaction between two end groups,
an end and Middle group, and two Middle
groups respectively
21
13
112
46
25
5
60
6
26
45
6
34
86
34
75
87
27
5
22
44
86
- 128 -
e: Site fraction
À: Calorimetrie calibration constant
À: Parame ter representing free volume difference
between po1ymer and solvent
b~i: Change in chemica1 potentia1
~: Parameter characterizing chemica1 difference
between solvent and solute
g: Density
9: Parame ter characterizing size differences
between components
T: Parameter characterizing differences in the
temperature reduction parameters of
solvent and solute
~: Segment fraCtion
~': Volume fraction
~: Po1ymer-so1vent interaction parameter
~H: Enthalpic contribution to X
~S: Entropic contribution to }(
tp: Contact energy fraction
n: Number of distinguishab1e arrays
SUBSCR'IPTS
H: Entha1py
R: Reference 1iquid
SI Entropy
1: Solvent
2: Solute
45
60
27
8
28
75
27
30
26
13
6
15
15
26
7
10
37
10
5
5
'-i __ •• J
- 129 -
SUPERS CRI PTS
Comb: Combinatoria1
E: Excess thermodynamic quantities
*: Red.uction parameter
-: Reduced quantity
7
3
21.:-
22
'-
, i
APPENDIX
TABLES OF DATA
'-
- 130 -
Table 1-a
(1) ( 2) (3) (4) (5 )
Solvent A~(eo) Temperature a.1T (sl/ s 2)X12
jou1es/g oK joules/cc
Pentane -2.84 303 0.509 8.26
-3.43 333 0.673 14.68
-4.52 352 0.829 20.96
-5.48 365 0.978 27.20
Hexane -1.80 303 0.430 5.38
-2.09 324
-2.93 348 0.574 9.00
-3.68 373
-5.31 393 0.825 16.69
-6.65 ,,.08
-8.95 423 1.12 26.19
-9.92 433
Octane -0.29 303 0.358 3,,75
-0.71 324
-0.88 348 0,,452 5.33
-1.05 373
-1.34 393 0.590 8.80
-3.09 423 0.730 Il.50
- 131 -
Table 2-a
. (1) ( 2) (3 ) ( 4) (5)
Solvent a.1T Temperature (sl/ s 2) X12 Carbon No.
with PIB of solvent
oK joules/cc
Pentane 0.487 298 6.66 5
0.509 303 7.30
Hexane 0.414 298 4.10 6
0.430 303 4.58
0.459 313 4.74
Heptane 0.372 298 3 0 03 7
0.382 303 3.14
0.404 313 3.47
0.434 328 4.09
Octane 0.347 298 2.58 8
0.357 303 2.80
0.377 313 2.98
0.405 328 3.28
0.494 363 4.25
Decane 0.313 298 2.08 10
0.319 303 2.06
0.335 313 1.85
0.356 328 2.11
0.425 363 2.48
Hexadecane 0.263 298 1.29 16
0.353 363 1.54
- 132 -
Table 3-a
(l) ( 2) (3) (4) (5)
Solvent CllT Temperature (sl/s2)X12 Carbon No.
with PDMS of' solvent
OK joules/cc
Pentane 0.487 298 2.02 5
Oe509 303 2.00
Hexane 0.414 298 2.29 6
0.430 303 2.28
0.459 313 2.08
0.500 328 2.16
Heptane 0.372 298 3.05 7
0.382 303 2 .. 80
0.404 313 2.78
0.434 328 2.78
Octane 0.347 298 3.38 8
0.357 303 3.27
0.377 313 3.04
0.403 328 2.92
0.494 363 2.36
Decane 0.313 298 4.57 10
0.356 328 4.14
0.425 363 3.74
Hexadecane 0.263 298 6.35 16
0.353 363 5.65
(1 )
Solvent
Dimer
Trimer
Te tramer
Pentamer
- 133 -
Table 4-a
( 2)
aIT
0.417
0.519
0.372
0.457
0.599
0.349
0.421
0.544
0.331
0.398
0.508
( 3)
Temperature
OK
298
328
298
328
363
298
328
363
298
328
363
( 4)
(51 /52) X12 with PDMS
joules/cc
0.25
0.88
0.18
0.63
1.70
0.06
0.04
0.29
1.68
- 1')4 -
Table 5-a
* .!. Solvent 1 (S2X12/ sl P2) 2 Temperature
r l + o. (; ~or solvent with oK
( a) (b) PIS PDMS
Pentane 0.151 -0.122 0.077 298
Hexane 0.1')1 -0.096 0.082 298
Heptane 0.116 -0.082 0.094 298
Octane 0.104 -0.076 0.100 298
Decane 00086 -0.068 0.116 298
Dodecane 0.074 -0.065 298
Tetradecane 0.064 -0.058 298
Hexadecane 0.057 -0.054 0.1')6 298
Pentane * 0.151 -0.1')4 ')0')
* Hexane 0.1')1 -0.110 ')0')
* 0.104 -0.094 Octane ')0 ')
Pentane * 0.151 -0.214 ')48
* -0.141 ')48 Hexane 0.1')1
* 0.104 ')48 Octane -0.110
* Hexane 0.1')1 -0.250 42')
* 0.104 -0.160 42') Octane
*Data ~rom Liddell and Swinton. (4')
,-
- 135 -
Table 6-a
(1 ) ( 2) (3) ( 4)
Solvent a.lT Temperature (sl/s2)X12
with PIB
oK joules/cc
2,4-Dimethylpentane 0.385 303 3.23
2,3-Dimethylpentane 0.370 303 2.45
3-Methylhexane 0.373 303 2.73
2,2,4-Trimethylpentane 0.360 303 3.60
3-Methylheptane 0.349 303 2.66
2-Methylheptane 0.345 303 2.23
2,5-Methylhexane 0.367 303 3.38
3,4-Dimethylhexane 0.342 303 2.00
2,2-Dimethylhexane 0.370 303 3.36
2,4-Dimethylhexane 0.348 303 2.16
- 136 -
Table 7-a
(1) ( 2) (3) ( 4)
Solvent OolT Temperature (sl/s2)'S.2
with PDMS
oK joules/cc
3-Ethylpentane 0.370 303 1.83
2,3-Dimethylpentane 0.370 303 1.96
2,4-Dimethylpentane 0.385 303 2.02
3-Methylhexane 0.373 303 2.16
2-Methylhexane 0.382 303 2.22
2,2,4-Trimethylpentane 0.360 303 2.34
3-Methylheptane 0.349 303 2.75
2-Methylheptane 0.345 303 2.25
Solvent
Heptane
- 1:37 -
Table 8-a
Segment ~raction, fJ 2 , o~ polymer
0.90
0.70
0.50
o. :30
0.20
0.89
0.72
0.72
0.47
0.46
0.:3:3
0.:32
")C. ~rom X f'rom the Flory theory experiment with X12 = 5.51
0.84
0.68
0.66
0.60
0.58
0.56
0.57
joules/cc
0.68
0.61
0.57
0.5:3
0.52
2,4-Dimethylpentane X~rom the Flory theory with X12 =
0.90
0.70
0.50
0.:30
0.10
0.80
0.72
0.58
0.42
0.:35
0.90
0.70
0.50
0.:30
0.75
0.75 0.66
0.64
0.6:3
(a) 5.98 (b) 2:3.92 joules/cc joules/cc
0.75 0.68
0.64
0 .. 61
0.60
1.14
1.12
0.96
0.84
1
.. 1
PART II
A FREE VOLUME ANALYSIS OF THE
GLASS TRANSITION
- 138 -
CHAPTER V. INTRODUCTION
THE GLASS TRANSITION
In chapters l through IV, it was shown that the
Prigogine theory, employing the concept of free volume, could
be applied with semi-quantitative success to properties of
monomeric and polymerie solutions. In this chapter, the
Prigogine theory is extended to the phenomenon of the glass
transition to determine if the free volume concepts of the
theory can be used to improve on other free volume models in
giving an explanation of this transition.
A liquid will become a glass on cooling when a
barrier to crystallization exists, such as asymmetry of
molecular structure and/or an absence of sufficient thermal
energy to permit molecular reorganization into a crystalline
structure. The transformation from the liquid to the glassy
state is known as the glass transition. In addition to being
discussed in several te:x:tbooks(l,2,3), this phenomenon has been
the subject of recent review articles(4,5,6).
A large variety of substances, ranging from simple
liquids to highly comple:x: polymers, undergo this transition.
In this work, only polymerie materials are experimentally
investigated.
The Most widely accepted method of measuring the
- 139 -
glass transition is dilatometry. Rere the specifie volume of
a material is determined as a function of temperature. When
the glass transition temperature (Tg> is reached, there is a
dramatic change in the slope of the speci,f'"',c volume (V ) sp
versus temperature (T) curve as shown in figure 28. T is g
marked as the point of intersection of the V - T line of the sp ,
liquid wi th the V sp - T 1ine of the glass. In recent years a
number of rather sophisticated pieces of apparatus have been
used to study the glass transition. A discussion of some of
these devices May be found in reference 4.
THEORETICAL INTERPRETATIONS OF THE GLASS TRANSITION
Severa1 theories have been proposed as to the
fundamenta1 nature of the glass transition. These theories
have attempted to describe this phenomenon in terms of:
1. An iso-free volume mode1(3,7-12).
2. A kinetic theOry(13,14).
3. A second order thermodynamic transition(15-l8 ).
The free volume approach to the glass transition is
MOst intuitively simple. Furthermore, it is able qua1itatively
to exp1ain, among other things, the effects of di1uents and of
pressure on Tg. A11 subsequent discussion of the glass
transition then will be in terms of an iso-free volume model.
This is not meant to imp1y that the kinetic and/or the
thermodynamic theories of the glass t~ansition are incorrect,
'-1
- 140 -
Figure 28
SPECIFIC VOLUME (V ) VERSUS TEMPERATURE (T) sp
FOR A TYPICAL GLASS-FORMING SUBSTANCE
,-
,-.. i
- 141 -
as both o~ these theories are qualitatively acceptable as weIl.
THE FREE VOLUME MODEL
A ~ree volume model o~ the liquid state was ~irst
developed by Eyring(19) and later extended by Fox and Flory(7)
to the glass transition. Results o~ a study o~ ~ractionated
polystyrene samples by Fox and Flory allowed these authors to
hypothesize that the glass transition represents an iso-~ree
volume state, i.e. the ~ractional ~ree volume f = V~/(VO+Vf)' was said to be the same at Tg ~or all materials. (Vo is the
volume occupied by the molecules themselves and V~ is the total
~ree volume associated with the molecules.) Below T , ~ does g
not change. At temperatures above Tg' the temperature
dependence o~ the fractional free volume was taken to be:
where = fractional free volume at T • g
~L = thermal expansion coefficient of the liquid.
~g = thermal expansion coefficient of the glass.
(V-l)
In a later investigation, Doolittle(20) found the
following empirical equation relating viscosity,~ , to free
volume:
(V-2)
Both A and B are constants.
- 142 -
This resu1t has been derived on a theoretica1 basis
by Bueche(3) and by Turnbu11 and COhen(9,10,12) for mo1ecular
transport occurring in a 1iquid mllde up of hard spheri cal
mo1ecu1es. The importance of the Doo1itt1e equation 1s that
it provides a theoret1ca1 bas1s for the Yi11iams, Lande1, and
Ferry(ll) (W.L.F.) equat1on.
Though orig1na11y deve10ped on pure1y emp1r1cal
grounds, the W.L.F. equation 1s very s1m11ar to the Doo1ittle
equation. If the Doo11tt1e equation 1s rearranged, the
fo11owing genera1 resu1t 1s foundc
B T-T g
f' 1 r:J.f' + (T - T ) g g (V-:3)
where a T 1s the rat10,
T and T •
llT/1\T ' of viscos1 ties at temperatures g
g
Th1s equation is similar to the empir1ca1 resu1t f'or
a T eva1uated at Tg by W.L.F.:
T - T = -11.44 g 51.6 + (T - T ) g
Doo1itt1e found that B 1n equat10n (V-3) vas
(V-4)
approximate1y equa1 to one. A comparison of equations (V-3)
and (v-4) by W.L.F. provided a means of evaluating f'g. r:J. f vas
4 -4 0 -1 taken to be .8 x 10 C for a11 mater1a1s which undergo a
glass transition, whi1e f'g vas ca1cu1ated to be 0.025. Hence
,-.i
- 143 -
the fractiona1 amount of free volume possessed by a11 materials
at T was eva1uated to be 2.5% of the total volume of the g
materia1. Be10w T , the W.L.F. equation is no longer g
applicable since there is no further co11apse of free volume.
The W.L.F. equation has been extende.d to Many aspects of the
glass transition with marked success, e.g. the work of Ferry
and stratton(21). In fact, the work of W.L.F. is the most
successfu1 effort made to describe the glass transition in
terms of an iso-free volume mode1. The W.L.F. free volume is
i11ustrated graphica11y in figure 29 taken from a review by
Shen and Eisenberg(4).
volume at
Another method of eva1uating the fractiona1 free
T vas presented by Simha and Boyer(8). These g
authors found, in support of the princip1e of corresponding
states described in chapters 1- IV, that the product a:fT is g
* approximate1y constant (0.113) at T for the po1ymer systems . g
they investigated [the product ClLTg was a1so found to be near1y
constant (0.164)]. This resu1t means that the percentage of
free volume at T is constant and has a value of 11.3% compared g
The difference in these two values is
due to the differences in the definitlon of free volume taken
by the two sets of authors. The Simha-Boyer free volume ls
a1so :lllustrated in figure 29 ..
* It should be recalled that in the Prigogine theory the product
aT is a measure of reduced temperature which ls related
direct1y to t'ree volume.
- 144 -
Figure 29
GRAPHICAL INTERPRETATION OF FREE VOLUME AS
DEFINED BY:
(1) Williams, Landel, and Ferry(ll) (W-L-F)
(2) Simha and BOyer(8) (S-B)
(3) Turnbul1 and Cohen(9,lO,l2) (T-C)
Shown sChematica1ly in a plot of specifie
volume (Vsp ) versus temperature (T)
[Taken from a review by Shen and Eisenberg(4)]
>
\ .\
'\ \ " \ " \
ds A
'\. \.
" \ '\ \ " .\
,,~
, "-
1
- 145 -
An additional model applicable to the glass
transition was proposed by Turnbull and Cohen(9,lO,12). A
segment o~ a Molecule, or the Molecule itself depending on its
size, was assumed to be located in a cage o~ radius R. The
potential UeR), i.e. the negative o~ the work required to
remove the molecule ~rom the centre of its cage at OaK to
vacuum, is described by a Lennard-Jones ~unction. The
potential ~unction, UeR), is at a minimum at some particular
value o~ cage radius, Ro. As the temperature increases, the
average R o~ the cage radii likewise increases. When R is
not much greater than Ro' the total expansion o~ the Molecule
i5 proportional to R - Ro' thus 1
v sp
where Vo = volume of' a Molecule.
v = speci~ic volume o~ the liguid expressed per sp Molecule.
Av = v -sp v o ' termed the excess volume.
A = proportionality constant.
The potential at Ro i8 a minimum when this excess
volume is distributed uni~ormly amongst aIl the cages, while a
non-uni~orm redistribution increases the potential energy.
Near Ro this increase is greatest but decreases when R becomes
larger than Ro and ~inally starts to approach zero when R is
in the linear region o~ UeR). Turnbull and Cohen then have
de~ined that portion o~ the excess volume that can be
- 146 -
redistributed without any accompanying energy change as the
free volume, v f ' such thatl
AVe is the expansion of the material expressed per molecule
due to the anharmonieity of molecular vibration.
At low temperatures v f = 0 and the glass expands like
a erystalline solide When the exeess volume reaehes some
eritieal value Avg , whieh corresponds to R in the linear
region of U(R), the free volume is added to the material and
the material undergoes a glass transition.
Although there is no precise definition of free
volume as yet agreed upon, there is no confliet between
existing definitions. The basic concept of free volume is
the same for aIl theories.
The simple free volume approach can be extended to
predict, inter alia, the effects of diluent concentration and
pressure on T • g
Effeet of Diluents on the
Pure Material (Iso-free Interpretation)
Volume
The effect of a diluent on the T of a polymer can g
be expressed very simply in terms of an iso-iree volume model.
The method of Kelly and Bueche(22) is followed in which
equation (V-l) is written for the polymer and diluent
respectively as:
'- 1 .1
- 147 -
where ~g = 0.025 and T 2 and T re~er to the glass g, g,l
transition temperature o~ the pure po1ymer and diluent.
Kelly and Bueche assumed that the ~ree volumes o~
the constituents were additive and ~ound the fo1lowing
expression ~or the ~ractiona1 ~ree volume, ~, o~ the mixture:
(V-5)
where .J. 2' T and t~ are taken to be the volume ~raction of
po1ymer and· diluent respectively.
At T o~ the p1asticized system, T = T and r = 0.025. g g
Substituting these two conditions into the above expression
yie1ds:
(v-6)
-3 0 -1 To obtain Tg,l' a f ,l was set equa1 to 10 C and
equation rv-6) was then solved ~or Tg ,l by ~itting it to actua1
experimenta1 data. A1though the value o~ T obtained in g,l
this way seems reasonab1e, it remains an essential1y ~itted
parame ter. Equation (v-6) was tested by Kelly and Bueche and
found to be reasonab1y accurate. A review o~ the treatlnent
o~ po1ymer-di1uent systems is given by reference 23.
- 148 -
E~~ect o~ Pressure on the Tg
a Pure Materia1 (Xso-~ree Volume Inte~retation)
o~
Intuitively one expects that the application of
pressure to a 1iquid should resu1t in a reduction in the amount
o~ 'free volume' vithin the ~luid. This reduction in ~ree
volume then should raise the Tg of the 1iquid. Experimentally
this has been confirmed by several investigators(24-26). The
e~fects o~ pressure on Tg are easily exp1ained in terms of an
iso-free volume model. The ~ree volume fraction at any
temperature above Tg has been given previously by Fox and
FlOry(7) :
If the compressibility o~ the free volume i8 ~~,
where ~f = ~L - ~g' this result at pressure P and temperature
T beeomesl
Tg(O) refers to Tg at zero pressure. At Tg under any pressure,
~T,P = ~ g' and this result becomes
a.fCTg - Tg(O)] = ~fP
On differentiation thenl
(V-7)
'-1
.. 1
- 149 -
Equation (V-7) has been reported by several
authors(2l,24). A summary of dT /dP data for a large number g
of glass forming materials has been presented by O'Reil1y(24).
In addition, Bianchi(27) has tabulated dT /dP data for po1y-g
styrene, po1yviny1 acetate, and polymethy1 methacry1ate
measured by himse1f and other workers. From the tabulation
of Bianchi, it becomes evident that ~f/~f over-estimates
(dTg/dP)f by a factor of about two. Therefore, in genera1,
a free volume approach app1ied to the pressure dependence of
the glass transition is able to predict the increase of T g
which resu1ts when pressure is app1ied. From a quantitative
point of view, however, the theory is not successfu1.
It shou1d be mentioned that in a11 these discussions
no attempt has been made to review the large body of experi-
mental data on the glass transition which exists for Many
systems, nor have empirica1 or theoretical treatments, other
than those of the iso-free volume approach, been discussed.
A1though the iso-free volume concept has been
eminent1y successfu1 in interpreting the glass transition
phenomenon, recent evidence(S,28-32) seems to indicate that the
concept of a critica1 free volume May not be universa1.
Mi11er(28-30 ), in app1ying a modified Arrhenius treatment to
the viscosities of po1ystyrene and p01yisobuty1ene, showed that
the fractiona1 free volume at Tg increases slight1y with
mo1ecu1ar weight. A simi1ar resu1t was a1so found by
Kanig(32 ). In a recent review Boyer(S) noted, in addition to
-1
- 150 -
its variation with molecular weight, that the fractional free
volume at T also varies vith degree of cross linking, g
copolymer composition and other structural details such as
bulkiness of side-groups and chain stiffness, etc.
PURPOSE OF THE PRESENT YORK
In the first four chapters of this thesis, it vas
shown that the Prigogine theory, based on the principle of
corresponding states, can be applied vith success to monomeric
and polymerie solutions. In a previous article(33), the
Prigogine theory vas extended to the glass transition and it
vas suggested that the ratio of free volume to total volume
for polymer liquids is a constant at T • g
* Since V - V is
* identified in the Prigogine theory vith the free volume of
the liquid, the ratio of free volume to total volume is thus
* 1 ,." ,-(V - V ) V =(V - 1) V. If this ratio 1s to be the same for aIl -liquids at T , V must have a universal value. g g For T g
occurring at ordinary pressure (negligibly different from 0), -this means that T should be a constant. g
This suggestion is
in agreement vith the idea that the glass transition represents
a state of iso-free volume.
In this investigation a number of polymer-solvent
systems are to be examined in order to obtain data on the
* As in chapters I-IV, V refers to the volume of the liquid,
* -V to the liquid's reduction parameter, and V to the reduced
volume of the liquide
- 151 -
effects of diluent concentration on the T of a pure polymero g
These results are to be compared vith the theoretical
predictions of the Prigogine theory.
Experimental data on a large number of substances
are to be examined to see if they follow corresponding states
at the glass transition temperature.
i
- 152 -
CHAPTER VI. EXPERIMENTAL
THE DIFFERENTIAL SCANNING CALO RIME TER
(See f'igure 30)
The glass transition temperatures of' nineteen
po1ymer-di1uent systems are determined as a f'unction of' solvent
concentration. The T of' each of' these systems is measured by g . .
a Perkin-E1mer Dif'f'erentia1 Scanning Ca10rimeter - 1 (DeS.C .. ) ..
The basic measuring unit of' the D.S.C., the head (A),
consists of' two sma11, cy1indrica1 sample ho1ders (B) into
which a ref'erence materia1 and the samp1e to be studied are
placed. The tempe rature of' the head is raised or lowered to
the desired starting 1eve1, af'ter which f'if'teen minutes are
a110tted f'or thermal equi1ibrium. Once equi1ibrium is
attained, the temperature of' both of' the samp1e ho1ders is
increased at the same rate. When a transition occurs in the
samp1e, the amount of' power required to keep the samp1e holders
at the same temperature is changed. This change in the rate
of' power supp1ied to the ho1ders (in mi11icalories/sec) is
recorded as a f'unction of' time. A f'irst order transition
produces a sharp curve (see f'igure 31) whi1e a second order
transition is detectab1e on1y as a change in the slope of' the
recorder base1ine as i11ustrated in figure 32.
A two-pen Leeds and Northrup Speedomax W recorder is
,-
- 153 -
Figure )0
THE DIFFERENTIAL SCANNING CALORIMETER USED
TO DETERMINE THE GLASS TRANSITION TEMPERATURES
OF THE POLYMER-SOLVENT SYSTEMS INVESTIGATED
A Head o~ the instrument
B Reference and sample holders
C Temperature indicator
D Sensit1vity control
E Slope control
F Average temperature control
G DifferentiaI temperature control
H Manual control used to change temperature
l Toggle switch used to increase or decrease temperature
J Power 11ghts
.1
1-)
o~
o 0 f><
0"
='-1
i .·1
- 154 -
Figure 31
TYPICAL FIRST ORDER TRANSITION CURVE AS MEASURED
BY THE DIFFERENTIAL SCANNING CALORIMETER
[Differentia! rate of power supp1ied to the samp1e
and reference ho1ders as a function of time (t)]
~I
· 1
c ________ __ -------
- 155 -
Figure 32
TYPICAL SECOND ORDER TRANSITION CURVE AS MEASURED
BY THE DIFFERENTIAL SCANNING CALORIMETER
[Differentia1 rate of power supplied to the sample
and reference holders as a function of time (t)]
A Recorder baseline
B Change in rate of power supp1ied to the head
A' New recorder baseline
', . ..1
,1
-~
- 156 -
supplied with the D.S.C. One pen 1s used to record the
difference in the rates of power supplied (ordinate) to the
two sample pan holders as a function of time (abscissa). The
second pen is used to indicate, by a series of pips, the
o temperature of the head in K. A recorder sensitivity of four
millicalories/see (corresponding to a half-scale deflection) is
emp10yed throughout this work.
Dried helium gas is kept flowing through the head at
a rate of )0 ml/min in order to prevent any condensation of
water vapour from occurring.
Experimental measurements are made over a temperature
A special dewar adapter is needed
to ob tain the low temperatures required to investigate the PIB-
diluent systems. By addition of 1iquid nitrogen to the dewar
(which fits directly over the head of the D.S.C.), it is
possible to lover the temperature of the head to -1500 C.
Calibration
The temperature scale of the D.S.C. 1s ca1ibrated
from the melting pOints of several reference materialsl highly
purified tin and octane obtained from Perkin-Elmer, spectro-
grade benzene and carbon tetrach10ride purchased from Fisher
Scientific, and ~aphthalene from Fisher Scientific containing
less than 0.5~ impurities. Each of these materials is heated
in the D.S.C. at a rate of 10oC/min through its melting point;
, i
- 157 -
this is the same heating rate* used in a11 experimenta1
determinations. For calibration purposes, the temperature
reading on the D.S.C. control panel (figure 30, part c)
corresponding to the first point of deviation from the recorder
base1ine is set equa1 to the melting point of the materia1. A
plot of these D.S.C. temperatures versus the actua1 me1ting
points of the five reference materia1s proved to be 1inear.
Determination of Tg
In the case of the glass transition temperature, the
point at which the first significant change of slope of the
recorder's base1ine occurs is not a1ways distinct. As a
resu1t, the Tg is taken to be the point of intersection of the
recorder's base1ine (line A in figure 32) vith the 1ine
representing the change in the rate of power supp11ed (line B
of figure 32).
MATE RIALS
P01ymers
Four pOlymers are studied extens1ve1y in this
investigation.
1. A fractionated samp1e of p01ystyrene from Pressure Chemica1
Company having a mo1ecu1ar weight of 97,200 and a M lM w n of 1.06.
* The e~feet of the heating rate on the value of the transition
1s discussed 1ater in this section.
.1
- 158 -
2. A ~ractionated sample of polyvinyl ch10ride given by Dr. Lo
utrac'i o~ Gulf Canada Research Laboratory having a Mv of
60,000 and a Mw/Mn of 1.42.
3. An unfractionated po1yisobuty1ene samp1e from Naphtachimie,
Lavera, France, having a M of 8,000. v 4. A fractionated samp1e of po1ymethy1 Methacrylate from Ur. P.
(1)
Rempp's Laboratory, Centre de Recherches sur les
Macromolecules, Strasbourg. This samp1e had a Mv of 5 2.06 le 10 •
Solvents
The ~ollowing spectrograde solvents were purchased
from Fisher Scientific: pentane, hexane, heptane, octane,
benzene, to1uene, ch1oroform, and carbon tetrach1oride.
The fo11owing reagent grade solvents were purchased
from Fisher Scientificl 1,2-dich1oroethane, ethy1 ether, and
methy1ethy1 ketone.
Decane, 99% pure, and methy1cyclohexane of purissimum
qua1ity vere obtained from Aldrich Chemicals.
A11 of the above solvents are dried vith sodium
sulfate prior to use. Hovever, no furtner attempt is made to
purify these materia1s.
(4) Propane and butane vere obtained as instrument grade
gases in lecture bott1es from Matheson and used as such.
SAMPLE PREPARATION OF NORMAL SOLVENT-POLYMER SYSTEMS
A11 samp1es, except PIB vith propane and butane, are
prepared in the fol1owing manner. A known weight of polymer 1s
- 159 -
dissolved in a large excess of solvent. The solution is then
placed in a vacuum oven and gently heated until the desired
concentration of solvent is achieved. The vial contain1ng the
solution is then stoppered, weighed, and put aside until needed~
PRE~ARATION OF PIB-PROPANE AND PIB-BUTANE SAMPLES
Preparation of solutions of PIB vith propane and
butane present special problems because of the physical state
(gaseous) of the diluents. About 0.3 grams of PIB are added to
a clean, dry glass tube which has been weighed. The tube 1s
weighed again to determine the precise amount of PIS present
and then connected to a vacuum line along with a lecture bottle
of the propane or butane. After the system is thoroughly
evacuated, a small amount of the gas is introduced and collected
in a small calibrated flask attached to the line which is placed
in a dewar of liquid nitrogen. Once a predetermined amount of
the gas has condensed, the vacuum pump isdisconnected, although
the vacuum within the system is maintained. The dewar of
liquid nitrogen is placed around the tube containing the polymer.
After the liquified solvent has boiled away from the flask and
condensed in the PIB tube, vacuum is applied once more and the
tube is sealed. Both sections of the tube (the sealed portion
and the portion still attached to the vacuum line) are
thoroughly cleaned, dried and veighed. In this vay the amount
of propane or butane contained in the sealed tube is determined.
- 160 -
Each tube is stored for at 1east one week before use to al10w
for equi1ibration of the solution.
GENERAL PROCEDURE FOR NORMAL SOLVENT-POLYMER SYSTEMS
In a typica1 experiment, a glass vial containing one
of the solutions to be studied is first weighed to determine if
any 1eakage of the diluent h~s occurred. If none can be
detected, 30 mg of the solution are removed from the vial and
p1aced in a sma11 a1uminum pan. The pan is then crimped about
an a1uminum 1id p1aced over the samp1e. An empty a1umlnum pan
with a 1id is p1aced in one of the samp1e ho1ders of the D.S.C.
head to act as a reference whi1e the filled samp1e pan ls placed
in the other ho1der. The head is then coo1ed at a rate of
10oC/minto the desired starting temperature. After this is
done, 15 or 20 minutes are a110tted for the samp1e to reach
thermal equilibrium. At the end of this time, the samp1e and
reference are heated at a rate of 10°C/min through the
solution's T • g
Each indivldua1 sample pan is used on1y one
time. However, eacb solution is run at 1east three times.
GENERAL PROCEDURE FOR PIB-PROPANE AND PIB-BUTANE SAMPLES
Prior to use, the sealed tubes or ampoules of PIB
wlth propane and vith butane are weighed to determine if any
diluent leakage has occurred. If none can be detected, the
ampoules, one at a time, are p1aced in a dewar of 1iquid nitrogen
'-. i
~I
- 161 -
for fifteen or ~enty minutes. After remova1, the tip of each
tube is broken and a stopper tight1y inserted to prevent
evaporation of the diluent. The tube ls then placed back into
the 1iquid nitrogen unti1 it is needed. At the appropriate
time, it is removed and a110wed to warm up gradua11yo The
stopper is withdrawn for a moment and a spatu1a is inserted to
remove a piece of the frozen PIB solution. Meanwhi1e, the tube
is stoppered and p1aced back in the 1iquid nitrogen. The
frozen solution is p1aced in a samp1e pan and a cover quick1y
app1ied. The samp1e pan is then placed in the precooled D.S.C~
and heated to determine its T • g This procedure is repeated at
least three times for each sample.
EFFECT or HEATING RATES ON TG
Tg is a function of the heating rate used to measure
it (T g tends to increase with the heating rate of the D.S.Co)o
Thus, in order to obtain T g at a standard heating rate of
OOC/min, it is necessary to ob tain values of T g at several rates
and extrapo1ate these back to a 'zero G rate. The Tg at a zero
heating rate is thus determined and found to be approximately
two degrees 1ess than the value measured for the same system
at the experimental rate of 10o C/min. As a resu1t, this
correction has been applied to a1l experimental values of T g
listed in this work.
1 .1
- 162 -
CHAPTER VII. RESULTS AND DISCUSSION
The following polymer-diluent systems were
investigated: PIB vith propane, butane, pentane, hexane,
heptane, octane, decane, toluene, methylcyclohexane, chloroform,
carbon tetrachloride, and ethyl ether; polystyrene with
toluene, benzene and carbon tetrachloride; polyvinyl chloride
with 1,2-dichloroethane and methylethyl ketone; and polymethyl
methacrylate vith chloroform and benzene.
AIl measurements vere made on a Perkin-Elmer
DifferentiaI Scanning Calorimeter -·1 to within a precision of
At diluent concentrations 01' greater than 25% by
weight, it vas generally impossible to detect T on the D.S.C. g
for the systems investigated. Our results for polystyrene
with benzene, toluene and carbon tetrachloride do not agree vith
the prev10usly published york of Jenckel and Heusch(34). These
authors found a far greater concentration dependence of T than g
reported here. At present no reason for this apparent
discrepancy can be proposed.
A compilation of data on aIl systems investigated is
presented in Table I. Here it is noted that, as the diluent
concentration increases, the glass transition is reduced. From
this data, a relative comparison of the effects of the d1fferent
diluents on T can be made. g
,-
- 163 -
Table I
POLYMER-DILUENT SYSTEMS
POLYSTYRENE VITH a
Benzene
'\ft. ~ diluent Tg ATg
0.0 100.0 0.0 4.1 97.5 -2.5 6.6 88.0 -12.0 8.5 85.0 -15.0 8.5 83.5 -16.5
12.1 72.0 -28.0 16.4 62.0 -38.0
Carbon Tetrach10ride
0.0 5.8 7.4 9.0
14.9 17.0 18.6
100.0 93.5 90.0 88.0 73.5 66.0 74.5
POLYVINYL CHLORIDE ViTHa
Methylethy1 Ketone
0.0 8. 1.,.
11.9 15.2 17.9 22.0
83.0 37.0 34.0 20.0 16.0 17.0
0.0 -6.5
-10.0 -12.0 -26.5 -34.0 -25.5
0.0 -46.0 -49.0 -63.0 -67.0 -68.0
'\ft.
To1uene
% diluent T âT g g
0.0 100.9 0.0 1.5 9900 -1.0 4.0 94.0 -6.0 4.5 92.5 -7.5 7.6 92.0 -8.0
11.0 88.5 -11.5 ·12.2 82.5 -17.5 16.0 83.5 -1605 17.8 78.5 -2105 23.3 75.0 -25.0
1,2-Dich1oroethane
0.0 7.0
10.0 15.2 19.3 22.0
83.0 47.0 42.0 31.0 26.0 25.0
0.0 -36.0 -41.0 -52.0 -57.0 -58.0
IContd.
· i
- 164 -
POLYI:SOBUTYLENE Wl:TH:
Ethyl Ether To1uene
1ft. ~ diluent T AT lit. % diluent T AT g g g g
0.0 -67.0 0.0 0.0 -67.0 0 .. 0 4.0 -76.5 -9.5 4.:3 -70.0 -3.0 7.5 -8:3.0 -16.0 7.4 -7:3.0 -6 .. 0 8.5 -85.5 -18.5 9.7 -8200 -15.0 9.8 -88.0 -21.0 12.6 -8'7.0 -20.0
1:3.7 -95.0 -28.0 14.9 -88.0 -21 .. 0 15.5 -100.0 -:3:3.0 19.0 -90.0 -23.0
23.0 -92 0 5 -25.5
Ch1oro:form Carbon Tetrach10ride
0.0 -67.0 0.0 0.0 -67.0 0.0 5.0 -75.0 -8.0 5.2 -74.0 -7.0
10.6 -80.5 -1:3.5 10.5 -79.0 -12.0 16.5 -84.0 -17.0 14.5 -8:3.0 -16.0 26.:3 -98.0 -:31.0 20.0 -91.0 -24.0
Propane Butane -0.0 -67.0 0.0 0.0 -67.0 0.0 :3.0 -78 • .0 -11.0 2.2 -70.5 -3.5 8.5 -90.0 -2:3.0 4.4 -75.0 -8.0 9.4 -92.5 -25.5 5.2 -77.0 -10.0 9.6 -9:3.0 -26.0 7.1 -82.0 -15.0
10.0 -96.0 -29.0 9.0 -86.0 -19.0 13.0 -94.0 -27.0 12.7 -91.0 -24.0 1:3.5 -98.0 -:31.0 14.7 -9:3.0 -26.0 14.8 -100.0 -:3:3.0 15.5 -99.0 -:32.0
17.4 -96.0 -29.0
Pentane Hexane
0.0 -67.0 0.0 0.0 -67.0 0.0 2.7 -70.0 -:3.0 2.6 -72.0 -5.0 4.2 -7:3.5 -6.5 :3.4 -76.0 -9.0 5.8 -79.0 -12.0 4.5 -74.0 -7.0 7.7 -8:3.0 -16.0 5.2 -76.0 -9.0
11.2 -88.0 -21.0 7.:3 -80.0 -13 .. 0 1:3.0 -91 .. 5 -2405 9 .. 2 -86.0 -19.0 14.:3 -9:3.5 -26.5 10.4 -88.0 -21.0 20.0 -93.0 -26.0 13.0 -93.0 -26.0 24.5 -9:3.5 -26.5 14.4 -99.0 -32.0
1 Contd.
'-_ -1
- 165 -
POLYISOBUTYLENE VITH:
Heptane " Octane
Wt. ~ diluent T ATg lit" % diluent T 6T g g g
0.0 -67.0 0.0 0.0 -67.0 0.0
2.8 -72.5 -5.5 2.5 -68.0 -1,,0
4.0 -74.0 -7.0 5.5 -73.0 -6.0
6.8 -83.5 -16.5 6.8 -74.0 -7.0
8.2 -81.5 -14.5 8.0 -78.0 -11.0
8.2 -83.5 -16.5 10 .. 7 -82.0 -15.0
10.4 -86.0 -19.0 13.6 -8600 -19.0
14.1 -90.5 -23.5 14.8 -86.0 -19 .. 0
15.1 -90.5 -23.5 18.0 -89.0 -22.0
18.0 -94.0 -27.0 20.6 -97.0 -30.0
Decane Methylcyc10hexane
0,,0 -67.0 0.0 0.0 -67.0 0.0
2.5 -69.5 -2.5 2.0 -68.0 -1,,0
'6.5 -71.5 -4.5 5.5 -77.0 -10.0
10.7 -81.5 -14.5 6.2 -73.0 -6.0
14.7 -85.0 -18.0 7.0 -72.0 -5.0 8.5 -82.0 -15.0 9.5 -84.5 -17.5
10.1 -87.5 -20.5 Il.0 -86.0 -19.0 13.1 -88.0 -21.0 18-.7 -91.5 -2L!-.5
21.9 -89.0 -22.0
POLYMETHYL METHACRYLATE VITH:
Benzene Ch1oro:form
0.0 95.0 0.0 0.0 95.0 0.0
9.5 82.0 -13.0 7.9 80.0 -15.0
12.0 66.0 -29.0 12.0 73.0 -22.0
12.2 78.0 -17.0 14.8 68.0 -27.0
12.2 62.0 -33.0 21.7 50.0 -45.0
17.0 60.0 -35.0 22.0 58.0 -37.0
20.6 52.0 -43.0 25.0 46.0 -49.0
- 166 -
It becomes evident that as the chain 1ength o~ the
diluent is increased (e.g. in PIB-n-a1kane systems and poly
styrene vith benzene and to1uene) the reduction o~ the glass
transition produced by the diluent is decreased.
In the case o~ po1yviny1 ch10ride with ~,2-dich10ro-
ethane ~nd methy1ethy1 ketone, remarkably large changes in the
glass transition temperature occur with increasing diluent
concentration.
APPLICATION OF THE PRIGOGINE THEORY Ta THE GLASS TRANSITION OF POLYMER-DILUENT SYSTEMS
According to the Prigogine theory, V and T, through
the equation o~ state, are measures o~ ~ree volume. The iso--~ree volume hypothesis there~ore suggests that T shou1d have the
same value at T for the mixture as T2 wou1d have at the T o~ g g
the pure po1ymer. I~ Tg,M ls the value o~ Tg ~or the mixture
and T that o~ the pure po1ymer, g,2
To ~ind the lowering o~ T M . g,
or ~ T g,M
be10w T 2' we require the g,
* * lowering o~ T ~or the solution be10w T2 o~ the pOlymer. ,..,
The T o~ a po1ymer (2) - diluent (1) system is given by
equation (I-24):
'-, .i
- 167 -
where the concentration parameters X2 ' tt' 1 and tV 2 have been
defined on page 26.
For small'valu~s of diluent concentration, i.eo
~l c::< 1, we have
Keeping only first powers of tt'1' equation (I-24) is writtent
In terms of the temperature reduction parameters, this result
becomes
1 +
Ve next define ATg
~ T g,M
AT =1+--E-_
T -g,M
and reca11 from page 26 that:
tVl =
T g,2 = T g,M
AT 1 + -=--.g
T g,2
[It shou1d be recalled that xi is the mole fraction of a
particular component.]
(VII-1)
( VII-2)
1 .1
- 168 -
Introducing the weight fractions w1 and w2 and the
* * specific volume reduction parameters v and v 2 ' l,sp ,sp
~l = * * wlPlvl ,sp ( VII-y)
Equations (VII-Il, (VII-2) and (VII-) are combined
to yield the following result&
AT .::......s. = T g,2
( VII-4)
The lowering of Tg is thus proportional to the weight
fraction of diluent. The effic1enty of a diluent in lowering
* Tg depends on the difference between its T an4 that of the
polymer, i~e. This difference plays the same role
as the difference between the T of polymer and diluent in the g
Kelly-Bueche theory but May be determined from equation of
state data, so that an ~ priori prediction of AT is possible g .
with no fitted parameters. The difference between the
chemical nature of the polymer and diluent is characterized by
2 1 * the \) parameter (identical to X12 Pl in the Flory terminology,
cf. page 45). The parameter plays an important role in
determining the mixing functions, A~ and AHM' but here the
effect is almost negligible, since \)2 = X12/P~« [(T;/T~) - lJ.
* * In Most polymer-diluent systems, (T2 /Tl ) - 1 is of the order of
'-1
1
- 169 -
In the PIB-a1kane systems of Part I, a typical value of
* X12!P1 wou1d be ~ 0.02, while a maximum value of thisparameter
might be ::::! 0.1. Neg1ecting V/2 in equatlon (VII-4) should
introduce an error of considerably 1ess than 10% in usual
* systems. The P2 parameter ls obtained from compressibillty
measurements and ls on1y aval1ab1e for a few polymers. However,
* * Pl !P2 is within the limits of 0.8 and 1.2 for almost aIl
systems. Based on these two arguments, ve tentatively take:"
* v l.sp
* (VII-S)
v 2,sp
The T* of the polymer and the T* of the diluent are
calculated from the thermal expansion coefficients (a) and
densities <g) of the materia1s according to equation (I-36).
* * ,-v ls ca1cu1ated according to v = v v. a and 9 of the ç sp ~
di1uents are obtained from the data of Allen et a1.(3S) a and -- , 9 of po1yisobuty1ene from the york of Eich1nger and F10ry(36),
a and 9 of po1yviny1 ch10ride from He11we,;e II .!!.. (26) and a
and 9 of po1ystyrene and polymethyl Methacrylate from Fox and
Loshaek(37).
Plots of Tg versus weight fraction of diluent are
presented for some of the systems 1nvestigated in figures
The solid curves are an approximation of the
experimenta1 resu1ts while the dashed 1ines represent the
theoretica1 prediction of equatlon (VII-S) for each system.
.1
- 170 -
Figure 33
THE GLASS TRANSITION TEMPERATURE (Tg> AS A
FUNCTION OF WEIGHT PERCENT DILUENT (v1) FOR:
POLYISOBUTYLENE-BUTANE (Upper Plot)
POLYISOBUTYLENE-HEXANE (Lover Plot)
Dashed curve in each plot represents the theoretica1
prediction of equation (VII-S)
Solid curve represents the actua1 experimenta1 resu1ts
[Data compi1ed in Table 1]
1 • 1
.i
IJ") IJ")
(-> M M
1 C""') @ C"")
~ M .-l
/ /
/ .....:t r-l M M
M ::: /
0 / 0'\
~/ 0 0/ .......
1
1 / /
0
(l/ ï
/ / @
@ ï le 1/ M
f.) j)
IJ") Ir) IJ") IJ") IJ") IJ") ...... co 0'\ \0 ...... co 1 1 1 1 1 1
(Jo) .L
- 171 -
Figure 34
THE GLASS TRANSXTXON TEMPERATURE (Tg> AS A
FUNCTXON OF WEXGHT PERCENT DXLUENT (vl) FOR:
POLYXSOBUTYLENE-OCTANE (Upper Plot)
POLYXSOBUTYLENE-DECANE (Lower Plot)
Dashed curve in each plot represents the theoretical
prediction o~ equation (VXX-S)
Solid curv~ represents the actual experimental results
[Data compiled in Table X]
.1
•
1 •
Lt"I 00
1
•
'" ..-1
• 1 l
/ /
,-1
Lt"I 00
1
'" .-!
.....
.-!
Lt"I .-!
~ .-!
1 1
- 172 -
Figure 35
THE GlASS 'l'BAN Si'1'iON TEMPERATURE ( Tg) AS A
FUNCTiON OF WEiGHT PERCENT DiLUENT (w1) FORt
POLYiSOBUTYLENE-CHLOROFORM (Upper Plot)
POLYiSOBUTYLENE - TOLUENE (Lower Plot)
Dashed curve in each plot represents the theoretical
prediction of equation (VII-5)
Solid curve represents the actual experimental resu1ts
[Data compiled in Table IJ
'-
If)
"-1
•
If) co
1
If)
0"\ 1
co .-1
..::t
.-1
If)
\0 1
II')
'" 1
If)
co 1
\0 N
N N
co .-1
..::t
.-1
'-
~
- 17:3 -
Figure :36
THE GLASS TRANSITION TEMPERATURE (Tg) AS A
FUNCTION OF WEIGHT PERCENT DILUENT (W1 ) FOR:
POLYSTYRENE - BElqZENE (Upper Plot)
POLYSTYRENE ... CARBON TETRACHLORIDE (Lower Plot)
Dashed curve in each plot represents the'theoretica1
prediction o~ equation (VII-S)
Solid curve represents the actua1 experimenta1 resu1ts
[Data compi1ed in Table 1]
'-.. i
, .i
• N N
N N
• • 00 . 00
/ .-1 .-1
/ • / /
...,. ...,. .-1
.-1 .-1 ~
Y ~
• 0 0 .-1 .-1
Ô U • U \0
\0
;; ~
Ô ~
N N
o 0 0 0 0 0 00 ...... \0 lI"\ ...,. cr)
lI"\ lI"\ lI"\ 11"\ 00 ...... \0 lI"\
- 174 -
Figure 37
THE GLASS TRANSITION TEMPERATURE (Tg> AS A
FONCTION OF WEIGHT PERCENT DILUENT (v1 > FOR:
POLYVINYL CHLORIDE - METHYLETHYL KETONE (Upper Plot)
POLYMETHYL METHACRYlATE - BENZENE (Lower Plot)
Dashed curve in each plot represents the theoretica1
prediction of equation (VII-S>
Solid curve represents the actua1 experimenta1 resu1ts
[Data compi1ed in Table I]
o .-1 .-1
o o .-1
••
o 0 0'1 00
o 0 " \.0
o .-1 .-1
o 0 o 0'1 .-1
o 0 0 00 " \.0
- 175 -
The fact that good agreement is found between theory and
experiment ls quite signi~icant, since there are no empirica1
constants fitted to the data. Hence, from a knowledge of a
and 9 of both the p01ymer and diluent, it ls possible to
predict quite satisfactori1y the reductlon of the glass
transition which occurs when diluent is added to a po1ymer.
The quantlty âT ls plotted versus g .
* * v T !,Sp [ ; _ IJ
v T 2,sp 1
in
figures 38 and 39 for polymer-diluent systems at welght
concentrations of 10 and 15%. It appears as if the princip1e
of corresponding states is obeyed ~or a11 systems except PIS
vith propane. The experimenta1 polnts in both graphs 11e
close to the theoretical 1ine (dotted) predicted by the
Prigoginetheory. This seems to support the assumptlon made
ear1ier of the iso-free volume nature of the glass transition.
It shou1d be noted that a large value of ~Tg is
predicted by the theory for the PVC-diluent systems. However,
the experimental values are still considerab1y 1arger. This
May be due to the polar nature of both components and a
specifie interaction between them.
. p*/ * If the ratio, 1 P2 , ls taken into account in
equation (VII-4), agreement between theory and experiment is
not signlflcant1y altered (see ~igure 40). Renee the
* * * assumption that (P1!P2) ~ 1 is valld. Values of P have been
calculated from compresslbi1i~y data on p01yisobutylene,
polystyrene, po1ymethyl Methacrylate and the varlous diluents
,-
.1
- 176 -
Figure 38
b-T ...:::::..Jl. AS A FONCTION OF
Tg
... v 1,sp
... v 2,sp
DILUENT CONCENTRATION OF 10% BY WEIGHT
For Po1yisobuty1ene with. For Po1ystyrene vith.
1)
2)
3) 4)
S) 6)
7) 8)
9) 10)
Il)
12)
Propane 13) Benzene Butane 14) To1uene
Pentane lS) Carbon Tetrach10ride
Hexane
Heptane For Po1ymethy1 Methacrylate vith:
Octane 16) Ch1oroform
Decane 17) Benzene
Ethy1 Ether
Methy1cyc1ohexane For Polyvinyl Chloride"with:
To1uene
Ch1orofonn 18) Methylethyl Ketone
Carbon Tetrachloride 19) 1,2-Dichloroethane
curve represents the theoretical
prediction of equation (VII-S)
[Data compiled in Table IIJ
0.14.1- 18 / V / 1
/ 0
0.12 t- 19 / V /
8 / o 3 /
0.10 1- 0/ 2
~ / 0 -. 9 5 / bO
10 0 0 / E-t ....... bO 0.08 0 / E-t q
6 / -11 7 o /13 17
0.06 ~ 0 0 / DA 12 / 0 / 14
15/Â 0
0.04 l- .~ 16
/ /
/
/1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9
* * [* * (v1 ,sp/v2,sp) (T2/T1) - ~ w1 x 10
r
- 177 -
Figure 39
AT ....:::....B. AS A FUN CTION OF
Tg
v
v
* 1,sp
* 2,sp
* [ T
2* ] - 1 AT A Tl
DILUENT CONCENTRATION OF 15~ BY WEIGHT
For Po1yisobuty1ene withc For P01ystyrene with:
1)
2)
3) 4)
5) 6)
7) 8)
9)
10)
11)
12)
Propane 13) Benzene
Butane 14) To1uene
Pentane 15) Carbon Tetrach10ride
Hexane
Heptane For Po1ymethy1 Methacrylate with:
Octane 16) Ch1orof'orm
Decane 17) Benzene
Ethy1 Ether
Methylcyc10hexane For Po1yviny1 Ch10ride with:
To1uene
Ch1oroform 18) Methylethyl Ketone
Carbon Tetrach10ride 19) l,2-Dichloroethane
curve represents the theoretical
prediction of equation (VII-5)
[Data compi1ed in Table II]
\,
,.... f:-ItJlJ .......
0.21 1-
0.18 1-
0.15 1-
~tJlJ <1 0.12 1-
'-'
0.09 1-
0.06 1-
l 1
0.2
1 19 /0
/ /
1 1
1
1 18
/1 V
V 1 8 3 / . 0
I~ ~ 2 o
l0~l 5
12 0 / ~ o /07 06
13
1 17
11° L .616
IJ.~ 1
1
14 o
1
1.0
1
1.8
1
2.6
(v Iv ) (T IT) - 1 w x 10. * * [* * 9 1,sp 2,sp 2 1 1
1 o
- 178 -
Figure 40
* * AT Pl -=.....K AS A FUNCTION OF
T * g P2
v 1,sp
* v 2,sp
AT A
DILUENT CONCENTRATION OF 10% BY WEIGHT
For P01yisobuty1ene ri 'th., For Po1ystyrene with:
1)
2)
)
4)
5) 6) 7) 8)
9) 10)
Pentane
Hexane
Heptane
Octane
Decane
Methy1cyc1ohexane
To1uene
Ethy1 Ether
Ch1orof'onn
Carbon Tetrach10ride
11) Carbon Tetrach10ride
12) To1uene
1) Benzene
For Po1ymethy1 Methacrylate with:
14} Ch1orof'orm
15) Benzene
curve represents the the"oretica1
prediction of' equation (VII-S)
[Data compi1ed in Table II]
,-
0.12 1-
0.10 1-
-. ~bD 0.08 t-
......... bD
E-t <J '-'
0.06 t-
0.04 ....
1
0.1
/
t 1 o
6 3 b / o 7 0 /
o /
4 / 9 5 0 /13 o 0 / 0
10 o / 11 /.6 0 0/ 1412
/ /
1
0.5
1
15 .6
0.9
/ /
/
/ /
/
1
1.3
(P*v* /P*v* ) [(T* /T*) - il w x 10 1 1,sp 2 2,sp 2 1 :J 1
- 179 -
investigated. The data appearing in figures )8 - 40 are
compiled in table II.
GENERAL APPLICATION OF CORRESPONDING STATES THEORY TC THE GLASS TRANSITION
The Tg values of' the polymer-diluent systems studied
support the idea that the reduced value of' T is a constant at g
least over the concentration range investigated f'or each
system. However, as we will now show, the simple iso-f'ree
volume approach is not quantitatively successf'ul in relating
the T values of dif'f'erent polymers or in predicting the g
pressure dependence of' Tg.
Correlation of' Tg Values
of' Pure Materials
!je -Values of' Tg' aL' and Tg f'or thirteen pure materials
are llsted in table III. It is apparent on inspection that
-Tg does not have a universal value. This implies, as weIl,
that the quantity aLTg , introduced by Simha and Boyer(8), can
only be approximately constant. ,."
In f'igure 41 a plot of' Tg
versus T ls presented. g
,.., It is apparent that T tends to
g ,.,
higher values as Tg increases, i.e. that Vg , or the relative
f'ree volume at Tg' increases with Tg. This result is -inconsistent with the constancy of' Tg f'ound in the polymer-
* -Values of' T g
* = Tg/T were calculated f'rom equations (I-)6) and
(I-)7).
,-
- 180 -
Table II
At a Concentration o~ 10%
POLYISOBUTYLENE VITHe
Propane Butane Pentane Hexane Heptane Octane Decane Ethy1 Ether Methy1cyc1ohexane To1uene Çh1oro~orm
Carbon Tetrach10ride
POLYSTYRENE WITH:
To1uene Benzene Carbon Tetrach10ride
1.91 1.28 1.00 0.84 0.74 0.65 0.54 0.93 0.58 0.50 0.35 0.32
0.68 0.78 0.43
POLYMETHYL METHACRYLATE WI'rH:
Ch1oro:form Benzene
POLYVINYL CHLORIDE VITH:
Methylethy1 Ketone 1,2-Dich1oroethane
0.53 0.86
1.20 0.67
V
v
* 1.sp
* 2,sp
13.09 9.70
10.19 9.21 8.73 6.79 6031
10.67 8.73 8.25 6.31 5.33
4.85 6.31 4.37
4.85 6031
13.58 11.64
IContd.
- 181 -
At a Concentration o~ 15%
* ole
5 x 10 2 v [T; -1 l!!S;E
T * g v Tl 2,sp
POLYISOBUTYLENE WITH:
Propane 2.70 15.60 Butane 1.92 12.90 Pentane 1.50 1).50 Hexane 1.26 12.75 Heptane 1.10 12.00 Octane 0.98 10.05 Decane 0.82 8.70 Ethy1 Ether 1 .. 40 14.70 Methylcyc10hexane 0.86 10.)5 To1uene 0.74 10 0 05 Ch1oro~orm 0.52 7.65 Carbon Tetrach10ride 0.48 8.70
POLYSTYRENE WITH:
To1uene 1002 5.55 Benzene 1.15 9.)0 Carbon Tetrach10ride 0.64 6.75
POLYMETHYL METHACRYLATE WITH:
Ch1oro~orm 0.78 7.40 Benzene 1.26 9.15
POLYVINYL CHLORIDE WITH:
Methy1ethy1 Ketone 1.82 17.10 1,2-Dich1oroethane 1.02 14.10
IContd.
~I
- 182 -
At a concentration o~ 10%
POLYISOBUTYLENE WITHI
Pentane Hexane Heptane Octane Decarie Methylcyclohexane To1uene Ethy1 Ether Chloro~orm Carbon Tetrach10ride
POLYSTYRENE WITHI
Carbon Tetrach10ride To1uene Benzene
POL YMETHYL METHACRYLATE 1
Ch1oro~orm Benzene
~ x 10 2 T g
10.19 9.21 8.73 6.79 6.:31 8.73 8.24
10.67 6.:31 5.:33
4.:37 4.85 6.:31
4.85 6.:31
* ~ * P2
i .' 1
* * v [T; 1) 1 2 9 12 '* v Tl 2,sp
0.91 0.80 0.71 0.64 0.54 0.55 0.62 0.86 0.48 0.39
0.39 0.6:3 0.79
Table III
'" Substance T Ref. aL Ref. T
..!. g
(oK) (oK-1 )
1) Sucrose 336 38 -4 38 0.040 5.02 x 10
J 2) Po1yviny1 Acetate 302 4 -4 25 0.042 5.93 x 10
3) Bis(m-(m-phenoxy lphenoxy) pheny1 ether) 267 39 4 -4 7.7 x 10 39 0.045
4) m-Bis(m-phenoxy pnenoxy benzene) 250 39 7.19 x 10 -4 39 0.040
5) tri-o-to1y1phosphate 231 39 -4
39 0.036 7.15 x 10
-4 .... 6) Diisobuty1 phtha1ate 188 41 7.73 x 10 40 0.031 (Xl
\,,)
7) Squa1ane 182 39 6 -4 .93 x 10 39 0.028
8) Po1yisobuty1ene 207 4 4 -4 5 • .5 x 10 36 00027
9) Po1ydimethy1si1oxane 150 41 -4· 4 8.85 x 10 0.027
10) Isopropy1benzene 125 39 -4 35 0.024 9.95 x 10
Il) Methy1cyc1ohexane 98 39 1.10 x 10-3 35 0.019
12) 1-Propano1 100 38 0.96 x 10-3 35 0.018
13) 2,3-Dimethy1pentane 85 38 1.22 x 10-3 35 0.017
- 184 -
Figure 41
-Tg AS A FUNCTION OF Tg FOR A NUMBER OF
SIMPLE AND COMPLEX SUBSTANCES
1) Sucrose
2) Polyvinyl Acetate
3) Bis(m-(m-phenoxy phenoxy) phenyl ether)
4) m-Bis(m-phenoxy phenoxy benzene)
5) tri-o-tolylphosphate
6) Diisobutyl phthalate
7) Squalane
8) POlyisobutylene
9) Po1ydimethylsiloxane
10) Isopropylbenzene
Il) Methylcyclohexane
12) 1-Propanol
13) 2,3-Dimethy1pentane
[Data compiled in Table III]
,-, i
.-l0
0
"'" 0 . 0
, " 00 0
" \0 0 '0 r--.
"
0 t"") 0 . 0
~J. -
0 N 0 . 0
o 00 N
o 00
~I
- 185 -
-diluent systems. IC Tg oC the polymer-diluent system is
-varied with Tg in the same manner as Tg for the pure materials,
the pred! cted lowering of Tg is increased by a factor of 1.5 - 2.
This is undoubtedly too large to be consistent ~th experiment.
A similar variation of Tis found in the molecular g
weight dependence of the glass transition temperature of poly-
styrene. -Figure 42 shows Tg for polystyrene as a function of
Tg values for diCferent molecular weights of the polymer,
values at T* having been obtained from a(M) values of Fox and
Loshaek ( :)'7) •
function of T • g
methacrylate.
-Aga!n it is seen that Tg is an increasing
Similar results are obtained with polymethyl
J Pressure Dependence of Tg
,., The concept cf a constant V gives the Collowing
g
expression for the pressure dependence of T : g
( VII-6)
where ~L and aL refer to the liquid polymer. Values of dT /dP /- g
calculated in this manner a're typically too· large by a factor
of approximately 4. In actual fact, the value of V at which g
the glass transition takes place is observed to decrease as T g
increases through the application of pressure. Equation
(VII-6) then becomes the result given by Bianchi(27):
- 186 -
Figure 42
~
Tg AS A FUNCTION OF Tg FOR FRACTIONATED
SAMPLES OF POLYSTYRENE
[From the work of Fox and FIOry(7)
and of Fox and LOShaek(37)]
[Data compiled in Table IV]
-'
.i
., 0 • - r--.
" C"') .,
• 0
" - ..0
C"')
• " "- - 0
Lr\ C"')
" ..-.. ~
"- 0 0 '-"
• - ...::t bD C"') E-c
" • 0
" - C"')
C"')
" " 0
- N C"')
". 0 - r-I C"')
1 1 1 ..0 Lr\ ...::t ...::t ...::t ...::t 0 0 0 . . . 0 0 0
~.L ,..,
,-
- 187 -
Table IV
,.., x 10 2 Mo1ecu1ar Weight of T T
Po1ystyrene Sample g g
oK
:3,000 :315 4.33
4,000 3:32 4.42
5,000 339 4.46
10,000 :356 4.55
15,000 )62 4.58
25,000 :366 4.60
100,000 :371 4.62
:373 4.63
~ dP =
- 188 -
.l:..~ V dT
g g
(VII-7)
This result greatly improves the predicted pressure
dependence of Tg" The sign dV 1 dT in equa tion (VII-7) is g g
opposite to that found from either the molecular weight
dependence of Tg or the correlation of Tg versus Tg found for
the thirteen different materials. The effect of pressure on
Tg' then, May be compared to that of a diluent on Tg; pressure
reduces the free volume of the polymer whereas the diluent
increases it and May be considered to have an effect similar to
a negative pressure. It is therefore perturbing to find that ,., V is apparently a constant in the case of polymer-diluent
g
systems while it varies in the case of the pressure dependence
of T • g
MOLECULAR SIGNIFICANCE OF THE ISO-FREE VOLUME CONCEPT
In partial solution of these difficulties, we offer
the fOllowing discussion of the molecular meaning of the 'iso-
free volume' concept. We acceptthe view that a liquid
becomes a glass when certain molecular motions become too weak
to overcome potential barriers that are largely interm~lecular
in origino The simplest point of view is to associate the
'molecular motions' vith c effective segments for intermolecular
motion. The molecular volume corresponding to such a segment is;
- 189 -
molecular vOlume/segment '* kT
* P
The thermal movement of this length of chain has an energy ~kT
which at Tg is equal to the barrier height. The total inter-
mo'~cular potential/mole ls characterized by u* or the hard core
* * volume V times P. The barrier height must be characterized by
* the molecular volume associated with the aegment times P , i 8 e.
* *, * -P (V Noc) = kT and by a factor f(V) which depends on
volume. Thus the condition for T would be g
* -kT ~ kT f(V) g g
and we have
-T g
,., f(V )
g
- ,., -
the free
At P = 0, V is a function of T ., g g
,., hence T g and V must be
g
constant for all materials, i.e. the iso-free volume condition
is obtained. The essential point seems to be the assumption
that the barrier height ls directly proportional to the
* segmental volume, V, or to T •
The effect of pressure on Tg comes about through a
- -decrease of V or free volume and an increase of the factor f(V) • ..,
Since V also changes with temperature, we have
or
dT g
= T* S! ,., dV
i . ,
- 190 -
-dT g
=.9!. -dV (O)~, dT J c9TJ- g
P
-, -Solving for dT dP and returning to unreduced variables,
it can be shown that
dT --& = dP
a. -
s 1 dV _ ---E.
V dT g g
where dV IdT ls the change of the specifie volume of the g g
polymer at the glass transition temperature with change of
pressure. This ls Blanchl's equation which has been shown to
give ~easonable results. The purely iso-free volume approach
leading to
dT --& = dP
gives only approxlmate results. When the corresponding
argument is applled to polymer-diluent systems, it ylelds
AT ---1! = T g
where a sol 15 the thermal expansion coefficient of the solution.
This result would lncrease the predicted AT IT ratio by about g g
a factor of 2. The conclusion is th en reached that the free
- 191 -
volume mode1 May be made to give the correct pressure dependence
of T , but at the expense of destroying the agreement for the g
diluent dependence. Essentia11y, this is because the theory
treats the diluent effect as equiva1ent to that of a negative
pressure in increasing the free volume of the system. It is
apparent that the simple free volume picture cannot give a
quantitative treatment of all of the possible effects on T • g
At present a more detailed molecular picture, which might be
able to do so, seems to be lacking.
CONCLUSION S
The Prigogine theory, employing the concept of free
volume, is extended to the phenomenon of the glass transition.
By assuming that the glass transition represents an iso-free
volume state, an equation predicting the effects of varied
diluent concentrations on the ~lass transition temperature (T ) g
of a pure material is derived from the theory. To test this
equation, values of T of some nineteen different polymerg
diluent systems are determined at various concentrations with a
Perkin-Elmer Differential Scanning Calorimeter-l. Agreement
between theory and experiment is found to be quite good, thus
supporting the iso-free volume concept of the glass transition.
According to the Prigogine theory which is based upon and assuming the iso-free volume concept
corresponding state~, all materials should have the sarne reduced
-glass transition temperature (Tg). In order to test the
,., -constancy of T , a plot of T versus T i9 made for thirteen g g g
,-.. 1
- 192 -
materials ranging from very simple substances to highly complex
polymers. According to theory, a horizontal line should
result. In actual fact, a line having a significant slope 1s
-found. In a s1milar manner, T varies with T for samples of g g
polystyrene having different molecular weights.
-Rence T 1s not a constant for aIl glass-fo~ing g
substances as the Prigogine theory assumes. The assumption of
-a constant T , however, is made in developing the equation g
relating the depression of Tg to the concentration of diluent in
the glass-forming material. When this assumption ia relaxed,
agreement betveen theory and experiment is worsened by a factor
of 1.5 to 2.0.
When the free volume concept of Prigogine is applied
to the pressure dependenee of Tg' very poor agreement is found
between actual experimental results and theoretical predictions.
If the iso-Cree volume assumption is relaxed, agreement is
considerably improved.
The iso-free volume concept of Tg adequately
describes the effect of diluents on the T oC a pure material g
but Cails to predict the proper pressure dependence of Tg. IC
the iso-Cree volume assumption is relaxed, the pressure
dependence oC T can be explained adequately but at the expense g
oC explaining the dependence oC Tg on diluent concentration.
Renee it is obvious that the iso-Cree volume concept cannot be
treated in a quantitative manner but rather must be considered
'- , .i
/
- 193 -
as a qualitative representation only. A theory capable of
quantitatively describing the effects of diluents and pressure
on T ls currently lacking. g
, .1
.1
- 194 -
REFERENCES
1. J.D. Ferry, 'Viscoe1astic Properties o~ Po1ymers', John . Wi1ey and Sons, New York, 1961, Chapter Il.
2. F. Bi11meyer, 'Textbook o~ Po1ymer Science', Interscience pub1ishers, New York, 1962, Chapter 6.
~. F. Bueche, 'Physica1 Properties o~ Po1ymers', Interscience Pub1ishers, New York, 1962, Chapters 4 and 5.
4. M. C. Shen and. A. Eisenberg, 'Progress in Solid State Chemistry', H. Reiss, Editor, Volume III, Pergamon Press, New York, 1966, Chapter 9.
5. R.F. Boyer, Rubber Chem. Tech. (Rubber Reviews), 12, 1)0~ (196~).
6. A. Eisenberg and M.C. Shen, Rubber Chem. Tech. , !ll, 156 (1970) •
7. T.G. Fox and P.J. F1ory, J. App1. Phys., !!.' 581 (1950).
8. R. Simha and R.F. Boyer, J. Chem. Phys., H, 100) (1962).
9. D. Turnbu11 and M.H. Cohen, J. Chem. Phys. , n, 1164 (1959).
10. D. Turnbu11 and M.H. Cohen, J. Chem. Phys., .l.!, 12'0 (1961).
11. M.L. Williams, R.F. Lande1, and J.D. Ferry, J. Am. Chem. Soc., ZZ, )701 (1955).
12. D. Turnbu11 and M.H. Cohen, J. Chem. Phys., ~, 1049 (1958).
1~. T. A1~rey, G. Go1d~inger, and H. Mark, J. App1. Phys., 14, 700 (194~).
14. A.S. Kovacs, J. Po1ymer Sei., ~, 1)1 (1958).
15. J.H. Gibbs, J. Chem. phys., ~, 185 (1956).
16. J.H. Gibbs and E.A. Di Marzio, J. Chem. Phys., !!!, )9) (1958).
17. E.A. Di Marzio and J.H. Gibbs, J. POlymer Sei., Al, 1417 (1963).
1 - ~ • 1
- 195 -
18. E.A. Di Marzio, J. Res., Nat. Bur. Stds., 68A, 611 (1964). 19. Ho Eyring, J. Chem. Phys., 4, 283 (1936).
20. A.K. Doo1ittle, J. Appl. Phys., ~, 1471 (1951). 21. J.D. Ferry and R.A. Stratton, Kol1oid-Z., 1Z!, 107 (1960). 22. F.N. Kelly and F. Bueche, J. Po1ymer Sei., ~, 549 (1961). 23. M.C. Shen and A.V. Tobo1sky, Adv. in Chem. Series,
48, 27 (1965). 24. J.M. O'Rei11y, J. Polymer Sei., j!, 429 (1962). 25. G. Allen, G. Gee, D. Mangaraj, D. Sims, and G.J. Wilson,
Polymer, ~, 467 (1960). 26. K.H. Hellwege, W. Knappe, and P. Lehmann, Ko11oid-Z.,
~, 110 (1962). 27. U. Bianchi, J. Phys. Chem., ~, 1497 (1965). 28. A.A. Miller, J. Po1ymer Sei., ~, 1857 (1963).
29. A.A. Miller, Ibid. ~, 1865 (1965). 30. A.A. Miller, Ibid. ~, 1095 (1964). 31. M.H. Litt and A.V. Tobo1sky, J. Macromo1. Sei., Phys.,
l, 433 (1967). 32. G. Kanig, Ko11oid-Z., ~, 1 (1963).
33. T. Somcynskyand D. Patterson, J. Po1ymer Sei., ~, 5151 (1962).
34. E. Jencke1 and R. Heusch, Ko11oid-Z., ~, 89 (1953). 35. G. Allen, G. Gee, and G.J. Wilson, Po1ymer, ~, 456 (1960). 36. B.Eo Eichinger and P.J. F1ory, Macromolecules,
1, 285 (1968). 37. T.G. Fox and S. Loshaek, J. Po1ymer Sei., 1i, 371 (1955). 38. W. Kauzmann, Chem. Revs., ~, 219 (1948).
39. A.J. Barlow, J. Lamb, A.J. Matheson, P.R.K.L. Padmini and J. Richter, Proc. Roy. Soc. (London), Sere A298, 467 (1967).
- 196 -
40. A.J. Barlow, J. Lamb, and A.J. Matheson, Proc. Roy. Soc. (London), Sere A292, 322 (1966).
41. T. Kataska and S. Veda, POlymer Letters, 4, 317 (1966).
'-
A:
f:
f ~ g
M 1 n
M : v
M ~ w
P:
RI
R : o
T:
T : g
U{R):
V : sp
V : o
v : sp
- 197 -
NOMENCLATURE
Character~stic constant
Ratio of viscosit~es at two different temperatures
Character~stic constant
Fractional free volume
Fractional free volume at Tg
Number average mo1ecu1ar weight
Viscosity average mo1ecu1ar weight
Weight average mo1ecu1ar weight
Pressure
Cage radius
Cage radius when potentia1 is a minimum
Temperature
Glass transition temperature
Potential function
Volume
Specifie volume
Total volume occupied by the mo1ecu1es themse1ves
Volume occupied by a single mo1ecule
Total free volume associated with the Molecules
Specifie volume of the liquid expressed per molecule
Free volume associated vith each mo1ecule
Excess volume
Expansion of a materia1 expressed per molecu1e caused by anharmonicity of mo1ecu1ar vibrations
141
142
141
141
141
157
158
157
148
145
145
139
139
145
150
139
141
145
141
145
146
145
146
'- 1
- 198 -
WI Weight fraction
XI Mole fraction
X: Surface fraction
X12' Entha1py interaction parame ter
GREEK LETTERS
~L: Thermal expansion coefficient of a liquid
~g' Thermal expansion coefficient of a glass
~f' aL - ~g
~L' Isotherma1 compressibi1ity of a 1iquid
~g' Isotherma1 compressibi1ity of a glass
~f' ~L - ~g
1} 1 Viscosi ty
\) 1 Parameter characterizing chemica1 di1'ference between solvent and solute
:1..' • "t". Vol ume frac ti on
tif 1 Contact energy fraction
SUBSCRIPTS
16~
167
167
168
141
141
141
148
148
148
141
166
147
166
1': Re1'ers to the fractiona1 free volume of a materia1 141
gl Glass 139
LI Liquid 141
MI Mixture 166
P: Pressure 148
Sol: Solution 190
TI Temperature 142
Tg' Glass temperature 142
- 199 -
SUPERS CRI PTS
*: Reduction parameter
-: Reduced parame ter
150
150
, !
- 200 -
CHAPTER VIII
SUGGESTION S FOR FURTHER YORK AND
CONTRIBUTIONS 10 ORIGINAL KNOWLEDGE
SUGGESTIONS FOR FURTHER YORK
1. Heats or dilution shou1d be determined for
po1yisobuty1ene (PIS) and po1ydimethy1si1oxane (PDMS) with the
various solvents investigated within this thesis. This data,
in conjunction with the heats of mixing at infini te dilution
for these systems, cou1d be used to eva1uate any mode1
applicable to po1ymer solutions.
2. Heats or mixing at infini te dilution, â~(oo), should
be measured for p01yo1efins other than PIB (for examp1e
po1y-1-pentene) vith the solvents studied in this work. In
this way the effects on A~(oo) of the longer pendant groups
of the po1ymer cou1d be determined.
The thermodynamic properties or po1ymer-so1vent
systems be10w the glass transition of the pure po1ymer should
be investigated. The data cou1d then be used to examine how
we1l 1iquid theories can be app1ied to this region.
4. The concentration dependence of the J( parame ter
should be determined over large temperature ranges for certain
- 201 -
solvent-so1ute systems by use of the modified McBain balance
discussed in this thesis. This wou1d provide a relative1y
simple method of determining the concentration dependence oC
the )( parameter in the vicinity of the L.e.S.T. This data
cou1d then be used to eva1uate 1iquid mode1s which have been
extended to this region.
The equation deve10ped in this thesis that predicts
the effects of di1uents on the glass transition temperatures
of pure p01ymers shou1d be extended to solutions in which the
diluent has a higher glass transition than the po1ymer (for
examp1e, a system such as po1ydimethy1si10xane with di-n-octy1
phtha1ate). This wou1d serve not only to test the equation
but also to shed 1ight on whether the change in the glass
transition is a result of a free volume change or whether it
is a mechanical effect re1ated to the size of the diluent
mo1ecu1es.
6. The amount of pressure that must be applied to a
polymer-di1uent system in order-to keep the glass transition
temperature constant, hence compensating for the reduction in
the glass transition temperature produced by the presence of
the diluent, should be measured. It wou1d be of interest to
determine whether, for a constant glass transition temperature, -the reduction in V found in the presence of the diluent was g -equa1 to the increase of Vg found vith the application of
pressure to the system. This data eould also be used as a
,-
- 202 -
means oC evaluating any model proposed to describe the
phenomenon oC the glass transition.
CONTRl:BUTION S TO ORIGJ:NAL KNOWLEDGE
The important results oC this investigation have
been given in detail at the conclusion oC Chapters III, IV and
VII. These Cindings are summarized below.
1. Heats oC mixing at inCinite dilution, 6~(~), vere
o 0 determined over a temperature range oC )0 to 90 C Cor the
Collowing sets of polymer-solvent systems: Pla - n-alkanes,
PDMS - n-alkanes, and PDMS vi th i ts oligomers. In addi ti on,
values oC A~(~) vere also dete~ined at )OoC Cor Pla and
PDMS vith several branched and cyclic alkanes. This data has
been used to test the Flory model oC the liquid state. This
analysis has revealed thata
Ca) The Flory model, while qualitatively acceptable,
incorrectly estimates the Cree volume contribution
to A~(~) and at least a part oC the Xl2 parame ter
is required to compensate Cor this error.
(b) The large diCCerence in the strength oC the Corce fields
surrounding methylene interior and methyl end groups,
required by Flory to explain the heats of mixing of
the n-alkanes amongst themselves, must be rejected on
the basis of experimentally determined values of
A~(-) :for both branched and normal alkanes vi th the
same polymer.
2.
- 20) -
(c) The principle of corresponding states is not perfectly
obeyed. -, "'" The sign of dC p dTl for the ~-alkanes must
be positive in order to be consistent with the A~(oo)
of PIS vith the n-alkanes and negative to be consistent
with the heats of mixing of the n-alkanes amongst
themselves.
The concentration dependence of the j( parame ter
was determined experimentally for PIS - n-heptane and PIB-
2,4-dimethylpentane. According to the Flory model, the role
played by the 'extra' methyl groups of the 2,4-dimethylpen!:,ane
Molecule should significantly change its thermodynamic mixing
properties compared with those of n-heptane. No experimental
evidence for this was found, however.
). The Prigogine theory, uSing the assumption that the
glass transition can be described in terms of an iso-free
volume state, has been extended to this phenomenon. The glass
transition temperatures of nineteen polymer-diluent systems at
varying concentrations have been measured by Differential
Scanning Calorimetry. The predictions of the Prigogine theory
on the effects of diluents on T compare quite favourably with g
the experimental results, providing the iso-free volume model
is assumed.
- -A single T (V) versus T curve has been obtained g g g
for a large number of glass-forming materials. According ta
~I
- 204 - .1
the Prigogine theory, wh1ch 1s based on corresponding states,
- -th1s curve should have a slope of zero (T and V should be g g
constant); in aetual fact a large slope is observed. As a
result of th1s fact and support1ng studies on the pressure
dependence of Tg' 1t vas concluded that any theory based on
the assumpt10n that the glass transition represents an iso-free
volume state must be taken as an approximate representation.
FREE VOLUME IN POLYMER
SOLUTION THERMODYNAMICS
,-
