Progress Polymer Science

7/30/2019 Progress Polymer Science

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Single-chain statistics in polymer systems

A. Aksimentieva, R. Hoystb,*

aComputational Science Department, Performance Materials R&D Center, Mitsui Chemicals, Inc., 1190 Kasama-cho,

Sakae-ku 247-8567 Yokohama, JapanbInstitute of Physical Chemistry and College of Science, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw,

Poland

Received 14 June 1999; accepted 14 July 1999

Abstract

In this review we study the behavior of a single labelled polymer chain in various polymer systems: polymer

blends, diblock copolymers, gradient copolymers, ring copolymers, polyelectrolytes, grafted homopolymers, rigid

nematogenic polymers, polymers in bad and good solvents, fractal polymers and polymers in fractal environments.

We discuss many phenomena related to the single chain behavior, such as: collapse of polymers in bad solvents,

protein folding, stretching of polymer brushes, coilrod transition in nematogenic main-chain polymers, knot

formation in homopolymer melts, and shrinking and swelling of polymers at temperatures close to the bulk

transition temperatures. Our description is mesoscopic, based on two models of polymer systems: the Edwardsmodel with Fixman delta interactions, and the LandauGinzburg model of phase transitions applied to polymers.

In particular, we show the derivation of the LandauGinzburg model from the Edwards model in the case of

homopolymer blends and diblock copolymer melts. In both models, we calculate the radius of gyration and relate

them to the correlation function for a single polymer chain. We discuss theoretical results as well as computer

simulations and experiments. 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Radius of gyration; Copolymer; LandauGinzburg model; One-loop calculations; Critical point; Orderdisorder

transition

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046

1.1. Linear polymers in good solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047

1.2. Scaling in the Edwards model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1047

1.3. Membranes, gels and fractal polymers in good solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049

1.4. Polymer melts, ideal chains and knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049

1.5. Polymers in bad solvents and the Q point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050

Prog. Polym. Sci. 24 (1999) 10451093

0079-6700/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.

PII: S0079-6700(99)00023-4

* Corresponding author. Tel.: 48-22-632-43-77; fax: 48-39-120-238.E-mail address: [email protected] (R. Hoyst)


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1.6. Protein folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050

1.7. Grafted polymers and polymer brushes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051

1.8. Polyelectrolytes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052

1.9. Stiff chains and coilrod transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052

1.10. Polymers near bulk phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10532. LandauGinzburg model for dense polymer systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054

2.1. Definition of the LandauGinzburg model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054

2.2. Single chain properties in the LandauGinzburg model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058

2.3. Upper wave-vector cutoff in dense polymer mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060

3. Homopolymer blends near the critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061

3.1. Equation for the radius of gyration of a polymer chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

3.2. Swelling and shrinking of polymer chains in homopolymer blends . . . . . . . . . . . . . . . . . . . . . . 1066

4. Copolymer melts near the orderdisorder transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072

4.1. Equations for the size of a diblock copolymer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074

4.2. Single-chain conformations in the copolymer melts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078

Appendix A. Partition function of homopolymer blends with a labeled chain . . . . . . . . . . . . . . . . . . . . . 1085

Appendix B. Ideal average of microscopic density operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086Appendix C. Partition function of the diblock copolymer melt with a labeled chain . . . . . . . . . . . . . . . . . 1088

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1090

1. Introduction

The history of polymer science dates back to the end of the 19th century when August von Kekule in1877 expressed the idea that molecules of life such as cellulose are composed of long, chain-like

molecules. The idea of molecules that are much longer than a unit cell in a crystal was not easily

accepted by the German crystallographers and until 1930 when a polymer molecule was pictured asan aggregate of smaller molecules [1,2].

Hermann Staudinger, the man who coined the term macromolecules for polymers, believed thatmacromolecules are stretched rod-like structures. His idea was based on the linear relation between

the viscosity of polymers in solutions and their molecular mass. This idea was dispelled by Kuhn and byFlory [3], who based their calculation of the shape of a macromolecule on the rotation around thecovalent bonds in the linear polymer. From their calculations emerged a new picture of the shape and

size of a macromolecule, namely, the coiled structure with the configuration resembling the trajectory ofthe Brownian particle. In particular, the size of the coil structure R in this picture scales with a molecularmass as M0.5 or with a polymerization index (number of monomers), N as N0.5 and the coil follows

Gaussian statistics. It should be noted that the Brownian trajectory is only a crude approximation of the

polymer configuration. They are similar only when viewed at a large distance scale when all themolecular details of the system are washed out. Moreover, the analogy is not quite correct since, asobserved already in 1934 by W. Kuhn and independently by E. Guth and H. Mark, a polymer chain

should avoid itself, while a Brownian trajectory can cross itself. The new and important physical effectintroduced at this point was the effect of excluded volume. At the molecular level in vacuum theexcluded volume for two monomers in a polymer chain is the volume in space inaccessible to onemonomer due to the presence of the other. For example, an excluded volume for two spheres is a sphere

of the radius twice as big as the radius of a single sphere. In a solution an excluded volume does notfollow from the quantum mechanics steric interactions described above, but is related to the

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thermodynamics of a solvent. Therefore the models should take into account not the bare interactionsbetween the monomers but rather the effective interactions between the monomers following from the

Gibbs free energy of the surrounding solvent.

1.1. Linear polymers in good solvents

The analogy between the Brownian trajectory and the polymer chain brought into the science ofpolymers two new concepts: the Feynmann path integrals [4,5] and scaling [5]. The renormalization

group calculations followed these concepts [5,6]. Another theoretical method which followed from thepath integral formulation of the polymer problem was a self-consistent field theory (SCFT) [4,5]. Thescaling relations and SCFT with excluded volume included in the Hamiltonian of the polymer chain

gave a Flory exponent for the radius of gyration in 3d, i.e. R Nn n 0:6. In general n 3=d 2,where d is the dimensionality of space. The Flory value of the exponent is close to the one obtained bythe renormalization group technique [6], n

0:588, Computer simulations [8] gave an exponent

n 0:59. In 2d space the exponent [6] is exactly equal to 3/4. In the four-dimensional space the linearchains assume the Gaussian shape (with the exponent 0.5) since the self-avoidance effects are negligible

in this case. The scaling analysis of the Edwards Hamiltonian [6] with the Fixman delta interaction [7]showed that although in principle one should have many-body effective interaction potential in a

polymer chain, only the two-body potential (and in some cases three-body potential) does not vanishin the limit ofN 3 when rescaled by the appropriate power of N. We discuss the scaling in the nextsection.

1.2. Scaling in the Edwards model

The simplest model of a polymer molecule is the Gaussian model. Each monomer in the chain has anextension given by the Gaussian probability distribution. In the continuum limit the conformationdistribution for such a chain is given by [9]:

Wr const: exp 32l2

N0

dn2rn2n

22 3; 1

where l is the Kuhn length, r(n) is the location of the nth monomer and N is the total number of

monomers. Here n is assumed to be a continuous variable measuring the length along the chain andWis the functional of all the positions of the monomers r in space. The Gaussian chain does not describecorrectly the local structure of the polymer, but does correctly describe its large length-scale properties.

In particular it gives the scaling R AN0:5

. The exponent is correctly predicted (e.g. for polymer melts)only the constant A has to be calculated for any particular local structure of the polymer. The examples ofcalculations ofA can be found in e.g. Ref. [10]. For the scaling between R and the molecular mass Mone

finds R aM0:5 (R is given in Angstrom units) where a is usually of the order of 0.10.3 [10]. Thisscaling is valid for polymer melts (to be discussed in the following subsection) but not in good solvents

where polymers experience excluded volume interactions. The part of the conformation distribution dueto the interactions is modeled in the Edwards model by the following equation:

Wintr const: exp bv

2

N0

dn1

N0

dn2drn1 rn2

2

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Here b kBT1 is the Boltzmann factor. The delta interactions introduced by Fixman [7] representsimply the fact that on the mesoscopic scale of the polymer chain the interactions are short-range and can

be approximated by the delta function. The coefficient v follows from the integration of the solvent

degrees of freedom in the full partition function for the solvent and polymer. This effective interaction isrelated to the second virial coefficient of the monomers or in general to the direct correlation function for

two monomers integrated over the volume of the system. From the discussion it follows that the model ismesoscopic and not microscopic. It should describe correctly the properties of the interacting chain on

large length scales much larger than the Kuhn length l. The model includes two-body interactions andneglects the effective three-body and higher terms. The total conformation distribution includinginteractions between the monomers of the chain is simply

Wtotr const: expHtot=kBT WrWintr 3

where Htot is the mesoscopic Hamiltonian for the polymer molecule. Let us consider the properties of the

model in the long-length limit N 3 . If we do the following rescaling of our variables rn Nn

Rn Hand n Nn H we find the Hamiltonian in the following form:

Htot N2n13

2l2

10

dn H2Rn H2n H

2 32N

2dn v

2

10

dn H11

0dn H2dRn H1Rn H2 4

where dis the dimension of space. Now let us consider the order of both terms with respect to N. When

N 3 the dominant term in the Hamiltonian will have a higher exponent in N. If

2n 1 2 dn 5

then the Gaussian term dominates over the interaction term and the chain should be ideal. If we set n0:5 we find that inequality (5) is satisfied for the space dimension d greater than 4. For d 4 theinteractions scale out in the limit of N 3 . Now if

2n 1 2 dn 6

both terms in the Hamiltonian have the same order of magnitude and this equation gives the exponent for

the radius of gyration n. We find for d 3 n 0:6 and for d 2, n 0:75, which are the Floryexponents for the radius of gyration of linear polymers in good solvents. This simple scaling allows

us to consider the effect of many-body interactions on the exponent n. Let us consider an m-body term inthe Hamiltonian. Such term scales with N as Nmdnm1: For d

3 and n

0:6 we find that the two-

body term dominates over the many-body interactions and the latter scale out in the long-chain limit.Therefore they are not relevant for the large scale properties of the polymer chain. Only when theexponent n is 1/3 all the terms in the Hamiltonian are relevant for the structure of the polymer coil.

The exponent n 1=3 corresponds to the state at which the polymer chain is densely packed into a smallglobule. This happens below the Q point (discussed in one of the following sections). Finally we notethat so far we have tacitly assumed that the interaction parameter bv is of the order of unity. However athigh temperatures or at the Q point it can be much smaller than unity. For example in the case of polymerblends above the critical point we find the Gaussian behavior of the chains since the critical temperature

Tc kBb1 scales in this case linearly with N and consequently vb 1=N.



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1.3. Membranes, gels and fractal polymers in good solvents

The Edwards Hamiltonian has been also applied to polymers that are not linear, but form, for example,

a two-dimensional membrane [1115]. Let us consider a polymer as a D-dimensional object (D 1 fora linear chain, 2 for a membrane and 3 for a gel) in a d-dimensional space. The total mass of the D-dimensional object is given by LD N, where L is the linear size of the object. One can use the results ofthe previous section to obtain the Flory exponents, by replacing the one-dimensional integrals by the D-dimensional integrals. The Flory exponent n 2D=2 d; relates R and L i.e. R Ln, thereforefor a linear polymer we have 0.6, for a membrane 0.8 and for a gel 1. In a gel, additionally, we have the

usual Flory exponent 0.6 for the scaling of the distance between the crosslinking points with the numberof monomers between the points. Below dimension d1 4D=2D the self avoiding effects arerelevant. For the space dimension d d1 the self-avoiding effects can be neglected and the idealexponent follows: n 2D=2. For linear chains D 1 d1 4 as calculated in the previous section,but for membranes

D

2

for any dimension d of the space we have to include the effect of self-

avoidance. Another problem comes from the analysis of the many-body interactions. For linear chainsthe many-body interactions are irrelevant in the long-chain limit. This is not the case for membranes.

The m-body interaction term is relevant for d 2mD=2m 1D, therefore for D 2 and d 3the three-body, four-body and five-body interactions are relevant [12].

Let us now discuss the size of the polymer chain in a fractal pore of the fractal dimension df. The

distance traveled by the random walker in the fractal should scale with time as r2 tds=df where ds is thespectral dimension of the fractal describing its connectivity independently of its spatial configuration[1517]. The equilibrium configuration of the self-avoiding polymer chain in the fractal pore should

depend on both ds and df. In particular for the ideal chain in the fractal we have n ds=2df, according tothe definition of ds and analogy between the random walk trajectory and the ideal polymer chain

conformation. For a self avoiding polymer chain in a fractal pore one finds using the SCFT [18]n 3ds=df2 ds. As we see if ds df 3 we retrieve the well-known exponent 0.5 for the idealchain and 0.6 exponent for the self-avoiding walk in the ordinary three-dimensional space.

One can also imagine more complicated structures of fractal polymers. Suppose that we have a fractalpolymer of spectral dimension ds. All the above results concerning the size of the fractal polymer in the

three-dimensional space can be easily generalized if instead of D we use ds [15,16]. Let us consider anexample of the trajectory of the Brownian particle moving on another trajectory of the Brownianparticle. It can be proven rigorously [19] that in this case 6=11 ds=df 2=3 and since df 2 for arandom walk trajectory we find 12=11 ds 4=3. The best estimate [19] for ds is 6/5.

1.4. Polymer melts, ideal chains and knots

The case of a dense polymer system, without any solvent is particularly simple. In a homogeneouspolymer melt the number of contacts between the monomers does not change with swelling or shrinking

of the polymer chains. The surroundings of a given chain in the melt is the same regardless of the shapeof this chain. Although with the swelling the number of contacts of the monomers belonging to theswelling chain decreases, the number of contacts with the monomers from the other chains increases

exactly matching the former decrease. It follows that the chain should assume the Gaussian statisticswith the ideal exponent 0.5. How can we reconcile this fact with the interactions between the monomers?As already noted by Flory and calculated by Edwards [20,21,24] the effective interactions between two



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monomers belonging to the same chain are screened due to the interactions with other chains. If weintegrate out all the degrees of freedom of all the chains except for one we find that the effective

interactions between two monomers in that chain are zero when averaged over the volume of the system.

Thus the chains are ideal. Although the chains are ideal they are also strongly entangled and thenontrivial element of the chain statistics in the melts is their topology related to the knots formation.

From the standpoint of the topology a ring molecule represents a knot. Since in a melt the distancebetween the ends of a linear polymer chain is much smaller than the contour length of the chain, it can be

considered as quasi-closed and its topology can be discussed from the point of view of the knot theory(for a recent review see Ref. [22]). The problem of knots in polymer chains is known as the Delbruckproblem. One finds that the probability that the ideal chain will form a knot is 1 in the limit of N 3 .The relevant length in this case is the topological persistence length NT which is the averagedistance along the chain between two subsequent knots. For a knotted chain NT N. The knotsaffect, for example, the dynamics. When the system is rapidly quenched into the glassy state and

next the temperature is raised the long-time dynamics is associated with the untightening of tight

knots formed in the quench process [22]. The knots are also important in biological systems (see forexample Ref. [23]).

1.5. Polymers in bad solvents and the Q point

So far, we have discussed a behavior of a polymer chain in a good solvent when the effectiveinteractions between the monomers are repulsive. However as we lower the temperature the effectivetwo-body interactions will become zero at the Q temperature. At this temperature the chain will assumethe size of the Gaussian chain with an exponent 0.5 [2426]. The three-body interactions give onlylogarithmic corrections to the chain statistics. Below the Q temperature the two-body interactions

become attractive and the chains shrink very fast assuming a globular shape with R N1=3

. Thisphenomenon is called a coilglobule transition. The compact structure of the globule leads to many-body interactions between the monomers which do not scale out in the limit of the infinite chain length.

The globule consists of a dense nucleus and a relatively thin surface layer. The size of the globule settlesat such a value that the osmotic pressure in the nucleus is zero [27]. The coilglobule transition is locatedin a region close to the Q point. The size of the region on a temperature scale is of the order of N0.5.Thus in the limit ofN 3 the transition takes place exactly at the Q temperature and can be identifiedwith a true phase transition. The sharpness of the transition depends on the chain stiffness i.e. stifferchains have the sharper transition analogous to the first-order phase transitions [27]. The shape of the

globule also depends on the stiffness of the chain. For stiff chains with large persistent length the shapeof a globule is toroidal [27].

1.6. Protein folding

The coilglobule transition is closely related to the long-standing problem in biology known asprotein folding. Proteins are linear heteropolymers which are built of 20 different amino acids. Atphysiological conditions (near neutral pH at 2040C) the proteins assume a native folded structure.

The structure is unique for each protein and determines its biological functions, therefore the problem ofthe protein folding received a name of the problem of the second genetic code. It should be noted that asmall protein of 100 residues has 10100 possible conformations, yet only one of them is the native



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conformation. At high temperatures or acidic or basic pH the unique folded structure unfolds (denatures)often reversibly (i.e. when cooled it returns to the same native conformation). Under such conditions the

global conformation of a globular protein is a random coil having low segment density similar to an

ordinary polymer above the Q temperature. The origin of the coilglobule transition in proteins ishydrophobic interactions with water. In the native structure the hydrophobic part of the protein issequestered into a compact core separated from the water solvent. The detailed role of different inter-actions in the final architecture of the folded protein is still a controversial issue. There are a number of

problems studied in connection with protein folding: why proteins fold into a unique state despite thefact that the number of possible configurations is enormous; how do they fold and why the foldingtransition from the open coil to the final globular (for globule proteins) state is so fast (Levinthal

paradox)? With the development of new computational techniques physicists and chemists are quiteclose to the solution of these problems [28 30]. In particular, it is now possible to predict the detailed 3D

architecture of the folded protein chain from the knowledge of the linear sequence of its chemicalconstituents using Monte Carlo simulations, but so far the reliable results have been restricted to

short chains [3133]. It should be noted that despite the fact that we know more than 400 000 proteins,the 3D folded structure is known for only 400 of them.

1.7. Grafted polymers and polymer brushes

Polymers grafted to the surface either by physical or chemical means are important from the tech-

nological point of view. For example, in the preparation of masks for the production of microchips or forcolloidal stabilization, chromatography, adhesion, spreading and wetting properties. A polymer brush

consists of long polymer chains attached to the surface with a sufficiently high coverage density (numberof polymers per unit area) so that chains stretch away from the surface. The behavior of the grafted

chains as a function of the quality of the solvent (good or bad solvents) surrounding the grafted layer andthe surface density of chains (coverage) has been studied by Alexander [34] and de Gennes [35] usingscaling arguments and blob models. They have obtained the following results. In the limit of low

coverage in a good solvent the chains have the same statistics as in the bulk solvent. In particular R N0

:6l: This scaling is correct providing that the chains at the surface do not overlap i.e. when the coverage

sR2 1 or simply sl2 N6=5

: The average concentration profile near the surface is fz sl2z=l2=3;where z is the distance from the wall and fz is the density of monomers. When sl2 N6=5 the chainsstart to stretch and the thickness of the grafted layer L Nlsl21=3: The profile fz according to deGennes is given by z2=3 up to the distance z s1=2 from the wall. From that point the profile is flat andfz sl22=3: However, as shown by Milner et al. [3640] using a self-consistent field theory, theprofile in the regime of long chains and moderate coverage is not flat but parabolic. The chain can be

pictured as a linear chain of blobs stretched normal to the wall. In the direction parallel to the wall thetypical distance covered by a single chain is of the order of N1=2: In a bad solvent the thickness of thegrafted layer is Nsl3 [34]. The small-angle neutron scattering experiments [41] confirmed these predic-

tions concerning the height of the grafted layer in good and bad solvents. In the case of a polymer oflength Nl grafted at the surface in the melt of the same polymers of length PlP N1=2 the size of thegrafted chains scales as N0:5 provided that sl2 N1 [35]. Stretching occurs when sl2 PN3=2 and in

the intermediate regime N1 sl2 PN3=2 the grafted chains overlap but this overlap does not lead tostretching. Finally for the coverage sl2 P1

=2 the mobile chains are expelled from the grafted layer[35]. The brushes that consist of two types of the chains exhibit a phase separation phenomenon [42]. A



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brush subjected to the shear flow increases its height since the flow stretches the chains [43]. The solventflowing past the grafted layer can penetrate into the layer quite deeply due to the parabolic monomer

density profile [44]. The polymer brush can also enhance the spreading [45]. Polymer brushes have been

also studied in computer simulations [4649].

1.8. Polyelectrolytes

So far we have discussed the polymer systems where the local microscopic scale does not affect thescaling properties at the mesoscopic scale. In polyelectrolytes the short-range and long-range interac-

tions are simultaneously present and details of the local chain structure can be altered by the long-rangeinteractions. This nontrivial coupling between the microscopic and mesoscopic length scale is further

complicated by the Debye screening of electrostatic interactions which introduce an intermediate lengthscale into the problem. Therefore the predictions of various mesoscopic models may differ considerably

when applied to polyelectrolytes [50]. In particular the conformations of the charged polymer depend onthe fraction of charged monomers, concentration of salt and in the case of weakly charged macromo-

lecules also on short-range interactions [24,27]. In a salt-free solvent a polymer molecule carrying a totalcharge fN (i.e. f is the fraction of charged monomers, the rest is neutral) has the size R Nf2=3lbl21=3;where lb is the Bjerrum length which characterize the strength of the electrostatic potential in the solvent

(for water at room temperature lb 7 A). For the finite fraction of the charged monomers the chargedmacromolecule assumes a rod-like configuration, while for f N3

=4l=lb1=2 it is a coil with the Gaus-sian radius of gyration (neglecting the short-range interactions). In the latter case the chains behave asneutral chains. This simple scaling follows from the comparison between the electrostatic interactions

and the Gaussian bead-spring Hamiltonian (Eq. (1)). When the salt is present two effects come into play:

the DebyeHuckel screening length l , which can vary from 10 to 1000 A, and the ion condensation(known as Manning condensation, although the phenomenon was predicted by Onsager in 1947 [27]).

The importance of l follows from the fact that on the length scale smaller than l the chains areessentially rigid and only on the larger scale the chains assume the coil configuration (of size N3=5 ina good solvent). However as shown by Odijk, Skolnick and Fixman, the electrostatic screened interac-

tions may induce a rod behavior on length scale larger than the screening length [50]. This effectintroduces a new length scale, known as the SkolnickFixmanOdijk length. The detailed discussionof various effects in polyelectrolytes are contained in the review article by Barrat and Joanny [50] and

the book by Grosberg and Khokhlov [27]. Therefore, we shall not discuss it further.

1.9. Stiff chains and coilrod transition

In the flexible molecule modeled by Eq. (1) the bond angle between neighboring segments is notrestricted and bending of the molecule does not cost any energy. As we have seen however in the case of

polyelectrolytes we may have an induced stiffness due to the electrostatic repulsion. We can also have anatural stiffness arising from the structure of the chain. In a DNA molecule two interwoven helicesprovide a mechanical stiffness. On a scale given by the persistence length the chain is stiff and start todeflect from the straight configuration on a larger length scale. For example, the DNA molecules has the

persistence length of 500 A. Such chains are quite well described by the KratkyPorod worm chain



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model [51]. The worm-chain configuration of an ideal (noninteracting) chain follows:

W

r

exp

1

2

k1

kBT N

n1un1un2 3 7

where k1 is the bending elastic constant, which is the energy penalty for the change of angle between the

nearest monomers and un rn1 rn=l is the unit vector tangent to the chain and Nl L is the lengthof the chain. The continuous version of this is defined as follows [52]:

Wr exp 12

k1

kBT

N0

dn2un2n

2

8

subject to the constraint u 2 1: The mean-square end-to-end distance is simply

R2endend

l

2 N

0

dnN

0

dn H un

u

n

H

9

where the average is taken with respect to the distribution given by Eq. (8). One finds

R2endend l2

N

3D1

1

NDexp2DN 1

10

where D 2k1=kBT1; hence P 2l=D is the persistence length. At high temperatures P 1 and thechain assume the Gaussian statistics with Rendend

N

p: At sufficiently low temperatures P Nl and

the chain assumes the rigid rod configurations [5154] with Rendend Nl: The stiff chains usuallyinteract with the typical interactions favoring the orientational ordering. Such interactions favor, atlow temperature, the orientationally ordered nematic phase. At some temperature we have the isotro-

picnematic phase transition. The transition is accompanied by the expansion of the chain length alongthe direction of the orientational ordering [53,55]. In fact just below the transition temperature the chainsstiffen exponentially with temperature. Therefore it is legitimate to say that the coilrod transitionaccompanies the isotropicnematic phase transition in the system of the semi-flexible chains [56].

Actually we may have a positive feed-back: stiffening of the chains induces the isotropicnematicphase transition and the phase transition induces stiffening [57]. The configuration of the polymer in

the nematic phase is not however a linear straight rod [53,55]. The chain makes a number of rapid U-turns which are called hairpins. Only at very low temperatures the hairpins disappear and the chains are

straight, long molecules gently fluctuating around the straight configuration. The review of the experi-mental studies of the conformation of stiff chains in various ordered liquid crystalline phases is given byCotton and Hardouin [58]. The discussion of various theoretical aspects of the conformations of the

worm-like chain with the application to DNA molecules is given by Odijk [59]. The discussion of thevarious liquid crystalline phases formed by DNA is done by Livolant and Leforestier [60]. Finally wenote that recent experiments allow to determine various elastic constants of the single DNA molecule

[61,62].

1.10. Polymers near bulk phase transitions

The phase transitions in dense polymer systems usually take place at high temperatures. For examplein the homopolymer blend the critical temperature scales linearly with the polymerization index N.



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Similarly the orderdisorder transition temperature in diblock copolymer melts scales linearly with N.Hence, the values of the excluded volume interaction, v, at which the phase separation in a polymer

mixture occurs scales as (1/N). Under this condition, the mesoscopic Hamiltonian H (Eq. (4)) is domi-

nated by the part which describe the ideal chain conformations. It means that the interactions divided bykBT are small above the transition temperature and the chains should be Gaussian. However near the

phase transition the fluctuation effects become strong and they can change the shape of the Gaussianchain. Below we present the detailed analysis of the corrections to the Gaussian shape of the chain

arising from the critical fluctuations near phase transitions for homopolymer blends and copolymermelts. The model appropriate for the study of phase transitions is the LandauGinzburg mesoscopicmodel. We will present the detailed derivation of the LandauGinzburg model for polymer systems

from the Edwards Hamiltonian.

2. LandauGinzburg model for dense polymer systems

In order to describe a dense polymer mixture on a mesoscopic length scale one should introduce order

parameter fields and relate them to density operators which specify the microscopic structure of thesystem. The model is often specified by the Hamiltonian describing the interactions between themonomers, and the density distribution function of the monomers within a polymer molecule. In the

mean-field approximation the effects of the long-range concentration fluctuations on a single chain areneglected. The influence of the critical fluctuations near the phase transition can be taken into account

within the one-loop approach as will be described in the next sections.

2.1. Definition of the LandauGinzburg model

The information that N monomers are connected to form a chain is specified by the distributionfunction Wr: To describe the flexible polymer, the chain model is used, in which atoms are describedas being joined by freely rotating bonds of fixed length l. The normalized distribution function for Natoms in such a chain is given by [63]:

Wr Nn1

d un l4pl2

; 11

and is normalized asDrWr 1 12

with

Dr 1V

dr0 dr1drN: 13

Here un rn rn1 is the vector specifying the orientation of the monomer and rn is the location ofnpoint between the subsequent monomers. d is the Dirac delta function. Instead of the previous model one



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can use the Gaussian model in the discrete version:

W

r

exp

3

2l2

N

n1 rn rn1

2

2 3; 14or a continuous version of the model,

Wr exp 32l2

N0

dn2rn2n

22 3: 15

In both cases one normalizes the distributions. The same quantitative results within negligible correc-

tions of the order of1=Na; a 1 are obtained at the level of the LandauGinzburg free energy whenone uses the distribution functions (14), (15) or (11).

A flexible diblock copolymer molecule with NA and NB monomers in each block can also be described

by the distribution function (Eq. (11)) one assumes the equal segment lengths in both of the blocks (and

N NA NB).The rigid polymers are described as rigid rods, or needles, in which all bonds have not only the same

length, but also the same direction. In this case

Wr d u1 l4pl2

Nj2

duj uj1: 16

Finally, for a diblock copolymer in which a flexible chain consisting of Natoms has been joined with arod containing M atoms,

Wr Ni1

d ui l

4pl2 NM

jN 1duj uj1: 17

The case of semiflexible polymers [52] has been discussed in Section 1.9.To describe the interactions between monomers, one first notes that the typical length scale in the melt

of flexible chains is given by the radius of gyration i.e. the size of the region occupied by the chain in themelt. In the simplest Gaussian approximation one finds that it is proportional to

Nl

pand is much larger

than the monomer size l. All the interesting phenomena take place at the length scale proportional to theradius of gyration. On the other hand the range of the potential is proportional to l, and therefore is notrelevant to the phenomena occur on a mesoscopic length scale. Guided by this simple observation the

following short-range effective interaction potential has been proposed by Fixman [7] and used by

Edwards in his model [4]

vabij ri; rj wabdri rj; 18

where d is the Dirac delta function and wab is the effective interaction parameter for a and b typemonomers (e.g. A, B monomers in the binary homopolymer mixture). The effective interaction para-meter is given by the integral [4,64] of the direct correlation function [65]. In the first approximation for

the rigid molecules, such integral, does not depend on temperature and is equal to the excluded volume,the volume inaccessible to one molecule when the other is fixed in space. Apart from the repulsive forcesthere is also an attractive potential e.g. van der Waals potential. The direct correlation function is, in the



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first approximation proportional to this potential. Thus wab contains two contributions: one associatedwith the excluded volume and one with the attractive potential. Summarizing, the interactions given by

Eq. (18) are related to the microscopic interaction potential via the direct correlation function. The delta

function signifies the extreme short-range character of the potential as measured in terms of the lengthgiven by the radius of gyration.

The interaction potential for rigid, elongated molecules, depends not only on the positions but also onthe orientations of the molecules. In the case of two rods [66] with fixed orientations the excluded

volume has roughly the shape of a rectangular box. If we expand the excluded volume in terms of theLegendre polynomials of the cosine of the relative angle of two rods we obtain the constant term,proportional to wab, plus the second term related to the second Legendre polynomial, P2. If higher

order terms are neglected the anisotropic part of the potential is modeled as [67,68]:

vabi;j ri; ui; rj; uj vabdri rjP2

uiuj

ui uj

2 3: 19

The total potential is given by the sum of Eqs. (18) and (19). Please note that in general the parameters

wab and vab are not independent, since they follow from the same direct correlation function.For the sake of clarity we consider further a mixture of polymers which consists of the monomers of

two types (A and B).The density operators needed to specify the density distribution of the monomers of

type A and B in the system are as follows:

fAr 1

r0

ng1

i{A}

dr rgi 20

fBr 1r0

ng1

i{B}

dr rgi ; 21

where

{A} or

{B} denotes the summation over all A- or B-type monomers in the gth chain, r0 is thenumber density of all monomers in the system and n is a number of chains in the system. These two

operators, fAr and fBr, represent the microscopic number fraction at point r of A and B monomers,respectively. If we also consider the interactions between the molecules which depend on their orienta-

tions the next two operators should be introduced:

QAabr

1

r0

n

g1 i{A}dr rgi

3

2

ugi augi b

ug

i2

dab

22 3; 22

QBabr

1

r0

ng1

i{A}

dr rgi 3

2

ugi augi b uri

2

dab

2

2 323

They represent the microscopic nematic tensor order parameters [66] at point r for the A and B

monomers, respectively. These tensors are symmetric and of zero trace, thus each has five independentcomponents.

The total interaction Hamiltonian is obtained by summing all the interactions between monomers.



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Using the density operators it can be rewritten in the following simple form:

H HI HO; 24

where

HI r0

drwAAf2Ar wBBf2BrwABfArfBr; 25

and

HO r0

dr 13

vAAQAabrQAba r

1

3vBBQ

BabrQBba r

2

3vABQ

Aba r: 26

Here summation over repeated a and b indices is implied. Only the first part, HI, of the Hamiltonian Hisrelevant in the case of the flexible polymers since the orientation dependent part HO in this case isnegligibly small after averaging over all possible conformations of the polymer chains. The part of the

Hamiltonian, HI, leads to the macrophase separation in homopolymer blends and to the microphase (ormesophase) separation in copolymer melts providing wAA wBB 2wAB 2kBTx 0: Here x isthe FloryHuggins parameter. The orientation dependent part of the Hamiltonian, HO, is important forthe systems which contain macromolecules with rigid, elongated parts providing nematic ordering in the

systems [67,68]. There is no general recipe for the choice of the microscopic operators. The hint as to theright choice is provided by the form of the interactions between monomers and the chain architecture.

Now we can specify the mesoscopic model. Let Pi be the average values of the microscopic operators

Pi over the mesoscopic volume with the characteristic length L . The conditional partition function,

ZPi; is the partition function for the system subjected to the constraint that the microscopic operators Piare fixed at some prescribed values [69] of Pi, i.e.

ZPi

i

dPi Pi ; 27

where the average is calculated as follows:

1Z0

na1

DraWraexp H

kBT

: 28

Here Z0 is the canonical partition function, Dra denotes the measure Eq. (13). The LandauGinzburg

(LG) free energy (in the mean-field approximation), V[Pi], is given by

VP

i k

BT ln Z

P

i:

29

The conditional partition function and the LG free energy are functionals of the order parameters Pi. The

partition function of the system is given by the summation of the conditional partition function over allpossible configurations of the fields Pi:

Z

i

DPiZPi; 30

For the microscopic density operators fAr and fBr their respective mesoscopic functions fA(r)and fB(r) give the averaged values of the number fraction at point r of A and B monomers, respectively.



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It is convenient to introduce two new fields [7072]:

CAr fAr fAr ; CBr fBr fBr ; 31

which describe the excess of the fraction of A or B monomers at the point r over their average values inthe system. It is usually assumed that the system is incompressible, i.e.

fAr fBr 1; 32

so

CAr CBr Cr: 33

For flexible polymers the field C is the only order parameter in the system. It vanishes in thehomogeneous state but is nonzero at the coexistence curve of the homopolymer blend and in the ordered

region of copolymer melts.

2.2. Single chain properties in the LandauGinzburg model

The local structure of a dense polymer mixture can be studied within the described above model byintroducing additional microscopic operators which contain information about the single chain proper-

ties [73,74]. For example, the radius of gyration of a single polymer chain

R2 12N2

N

i1 N

j1r1i r1j 2 ; 34

can be found by introducing a single chain microscopic operator

f1q 1r0

Ni1

expqr1i : 35

The radius of gyration (Eq. (34)) is obtained from the single-chain correlation function

S1q;q f1qf1q ; 36

by differentiating twice with respect to q and taking the limit of q

30. In the system composed of

different types of polymer molecules the single chain microscopic operators should be introduced foreach type of the macromolecules. If the system contains polymer molecules which have chemicallydifferent parts, the single-chain microscopic operators can be specified for each of the parts. One can alsoinvestigate orientational correlations between different macromolecules or their parts considering

single-chain microscopic operators which describe orientational properties of the macromolecules.For example, the average cosine of the angle between the segments of a polymer chain,

1

2N2

Ni1

Nj1

cos uij 1

2N2

Ni1

Nj1

uiuj

ui uj ; 37



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can be studied by considering the following microscopic operator

Q1

r

1

r0 N

i1

1

uid

r ui

:

38

The average cosine of the angle between the segments is obtained from the correlation function

S1u q;q Q1qQ1q 39

by differentiating twice with respect to q and taking the limit of q 30. In a similar way, the averagevalue of the quantity (cos uij)

2 can be obtained from the correlation function

S1J

q;q 3a1

J1a qJ1a q ; 40

where the microscopic operators are

J1a r 1

r0

Ni1

uia ui

2dr ui: 41

One should not be misled by the microscopic character of the operators which have been used to

specify the local structure of the system. The physical quantities which can be determined in thisapproach give only the information about their values which are averaged over a characteristic meso-

scopic length scale. This length scale, L , appears in the model as a cutoff for certain integrals describingthe long-range fluctuations.

The next step to find out the single-chain properties is to calculate the correlation functions such as

Eqs. (36) and (39), etc. For this purpose one can introduce an additional external field Uwhich couples tothe single chain microscopic operator f1 only and consequently add the term

drf1rUr to the

interaction Hamiltonian H. The single chain correlation function is obtained as

S1q;q d

2ZU

ZUdUqdUq U0; 42

where:

ZUZU 0 exp

dq2p3 f

1A qUq

!( ): 43

The average is given by Eq. (28). This partition function, Z[U], can be conveniently evaluated using thecumulant expansion and the Legendre transform [73,74].

At the end of the calculation one gets the following equation for the single chain correlation function:

S1q;q f1qf1q 0 corrections; 44

where 0 denotes the ideal average:

0 na1

DraWra: 45



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The corrections in the RHS of Eq. (44) are given as an infinite expansion of Eq. (42) with increasingnumbers of integrals over q. The natural small parameter in this expansion appears in connection with

the upper wave-vector cutoff 2p=L which limits the range of the integration. In the case of the homo-

polymer blends [75,76] and diblock copolymer melts [77] L Np and therefore the expansion is in the

powers of 1=

Np

: For long chains the expansion can be limited to the first term. These corrections depend

on the order parameter correlation functions, in particular, on the collective structure factor. Therefore,in order to describe the single-chain statistics near the phase transitions one should supply Eq. (44) by the

set of equations which include long-range fluctuation corrections to the order parameter correlationfunction. If one restricts the expansion of the single-chain correlation function to the first-order termsthe convenient framework to describe the order parameter correlation function is the one-loop self-

consistent (Hartree) approximation [7881].At the first step of the calculation of S1q;q we assume a certain ideal statistics for a single

polymer chain by specifying the distribution function, W[r], which gives zero-order terms in these

expansions (independent from the cutoff). The higher-order corrections perturb the ideal chain statistics

due to the long-range fluctuations (cutoff dependent). It is important to note that all the correlations at thelength scales smaller than the characteristic mesoscopic length scale L have not been ignored. They

have been taken into account within the ideal chain statistics. We have used the fact that the interactionsbetween the monomers in a dense polymer system are too weak to affect strongly the ideal chainconformations and assume that the change of the single chain statistics near the phase transition is

determined only by the long-range concentration fluctuations. In the following sections we describe inmore detail some applications of this approach to the homopolymer blends and copolymer melts.

2.3. Upper wave-vector cutoff in dense polymer mixtures

As was mentioned in the previous section the corrections to the radius of gyration due to the long-range fluctuations depend on the upper wave vector cutoff. This quantity is important, because thecorrections to the mean-field theory would dominate the properties of the system if the upper-wave

cutoff is not introduced. Although the cutoff is ubiquitous in statistical mechanics there are no readyrecipes for its choice. In low molecular mass liquids we usually have one natural length scale, whichcorresponds to the size of a molecule and the cutoff usually corresponds to this length scale. In polymermixtures the problem is more complicated since we have three different length scales: the total length of

a polymer molecule, Nl, the size of the region occupied by a polymer molecule (proportional to theradius of gyration),

Nl

p; and finally, the microscopic length scale l, corresponding to the size of a

single monomer. We assume that L is proportional to the radius of gyration, that is to

N

p. Here we

repeat the same arguments as given elsewhere [75]. At high temperatures, when the interactions between

the monomers are irrelevant and the monomermonomer correlation function decays exponentially withthe characteristic correlation length proportional to the radius of gyration. Moreover, in all our calcula-tions the specific structure of monomers has not been taken into account. Therefore the microscopic

length scale is irrelevant in the model. Finally, if we chose the microscopic length scale as the cutoff thefluctuation corrections would survive in the limit of N 3 and the mean-field FloryHuggins modeland RPA model for the scattering intensity would not be the correct description of the system in this

limit. We believe this is not the case. We note, that since the size of the polymer molecule in the blend isroughly proportional to

N

p, this choice is also in accordance with the prescription known from low

molecular mass systems, where the cutoff corresponds to the size of the molecule. Since in the polymer



17/49

system this size changes with temperature and concentration, it would be desirable to determine thecutoff consistently from the theory. The equation for the radius of gyration, R, obtained from the single-

chain correlation function (Eq. (44)) offers a simple way for a self-consistent determination of the cutoff

if we set it equal to 2pC=R where C is some constant which can only be determined from the fullmicroscopic theory. Thus, apart from this constant, the cutoff is determined from the mesoscopic theory.Therefore, the equation for the collective structure factor Sc 1=V CqCq (taken within theone-loop approximation) and the single-chain structure factor S1q;q are coupled via the self-consistently determined radius of gyration and the cutoff.

This prescription is good for the symmetric homopolymer blend. In the case of the asymmetric blendswe postulate that the cutoff should be equal to 2p=LRA;RB; where L is the symmetric function ofthe two radii of gyration for A and B chains. We assume the scaling form for L : LRA;RB RAfxwhere x RA=RB: From the symmetry properties ofL , we find the equation for f(x):1

x

f

x

f

1

x : 46We choose the following form for fx from the set of possible solutions of that equation:

fx

C11x3

1x C2x

s: 47

Here C1 and C2 are two constants which cannot be determined from the mesoscopic theory since they

depend on the microscopic details of the system. In general they are additional phenomenologicalparameters in the mesoscopic theory. We assume here that they are of the order of unity.

In the case of the diblock copolymer melts there are three independent contributions to the total size of

a diblock polymer molecule: the size of block A, RAA, the size of block B, RBB, and the distance between

the centers of mass of the blocks, RAB. We postulate that the cutoff is equal to qupper 2p=LMRAA;RBB;RAB; where LM is the symmetric function of the block sizes RAA and RBB. We assumethe scaling form for LM:

LMRAA;RBB;RAB RBBhRAA=RBB;RAB=RBB RAAhRBB=RAA;RAB=RAA: 48From the symmetry properties ofLM, we find the equation for h:

hy;x xh yx

;1

x

; 49

where x RAA=RBB and y RAB=RBB: The simplest solution of this functional equation is hx;y Cxy

1=3

; and the cutoff now has the following form:

qupper 2pC

RAARBBRAB3p ; 50

where C is the constant which depends on the microscopic details of the system.

3. Homopolymer blends near the critical point

A homopolymer blend consists of two types of linear macromolecules (say A and B) which are mixed



18/49

together without a solvent. The state of this system is fixed by two parameters: the concentration of theA-type polymer molecules and temperature. At high temperatures the system is a homogeneous viscous

polymer liquid. If one decreases the temperature below the critical temperature the difference in the

interactions between the monomers A and B drives the system to the phase separation (macrophaseseparation). The system separates into the A-rich and B-rich phases.

The critical behavior of the homopolymer blend belongs to the Ising universality class [82,83].Nonetheless in the limit of very long chains the Ginzburg region around the critical point decreases

and eventually in the limit of infinitely long chains the mean-field behavior is recovered even infinitelyclose to the critical point. Despite the well-known facts about the global properties of the system (such ascritical behavior) there are still some unresolved issues concerning the local structure of the blend. In

particular it is interesting to know how do the single chain conformations change when the systemapproaches the phase separation?.

The blend where the polymerization index of the first component is much smaller than the polymer-ization index of the other component resembles the system of polymer chains in a solvent. Indeed, a

single polymer chain ofNA-type monomers in an incompatible melt of B-type chains with P monomersin each chain Pp N exhibit a coilglobule transition [84] similar to a collapse of a single polymerchain in a poor solvent. At high temperatures, when the difference between the interactions of distinct-type monomers are negligible in comparison to kBT, the polymer blend should have the same propertiesas a melt of the same-type polymers. In the blends of long and short chemically identical polymer chains

the long chains are swollen if their index of polymerization N P2, where P is the polymerization indexof the short chains [85]. A single polymer ring in a melt of linear polymer chains has also swollen

conformations due to the topological constraint of being unknotted with itself [86,87]. Moreover, ringsof A monomers can be compatible with linear chains of B monomers, even when linear chains of A and

B are not compatible. It is also known that the melt of ring polymers have statistics intermediate between

those of collapsed and Gaussian chains [86]. A FloryHuggins treatment suggests that the radius ofgyration of such rings scales with the index of polymerization as R N2=5.

The conformations of the macromolecules in blends and solutions are studied experimentally bylight-, X-ray-, or neutron-scattering. These experimental techniques allow to measure the radius ofgyration, R, molecular weight, and the FloryHuggins interaction parameter, x, (or the second virial

coefficient, A2) [85,8890]. Small angle neutron scattering (SANS) experiments [58] typically involvethe measurement of scattering from dilute labeled (deuterated) chains in a matrix of unlabeled (proto-nated) chains. For example, the radius of gyration, R, is determined from the asymptotic form of the

scattering intensity in the limit of small q:

S

q

r0N

1 q2R2=3

;

51

where r0 is the monomer number per unit volume. The whole form factor

Pq 1N2

N;Ni;j

exp iqri rj 52

allows to determine the chain conformation by fitting the experimental data to various models of P(q)(for a recent review see Ref. [58]).

The calculation of the single-chain conformations in a dense homopolymer blend on a microscopic



19/49

level is a difficult task. Therefore a number of mesoscopic models [20,72,9193] and computersimulation techniques [9499] are used to study this problem. In both cases a very complicated inter-

action potential between the monomers is approximated by some simple, tractable form.

The FloryHuggins lattice model is often used in the Monte-Carlo computer simulations. In thismodel the polymer chains are arranged on a lattice. Each site of the lattice can be occupied by the A- or

B-type monomer. The chains are represented by a self-avoiding random walks with the number of stepsequal to the polymerization index. The interactions between the monomers are modeled within some

simple approximation. For example, the system gains the energy 1 if two neighboring sites of the latticeare occupied by the monomers of the same type [94]. There are three different ways of modeling theMonte-Carlo motions of the lattice chains [95]. The first one is applicable for the system where there is a

significant number of non-occupied sites. This method includes a drawing of a chain segment and anearest neighbor position, and, if the nearest site position is vacant, the motion is performed. If themotion is not possible, the old conformation is retained. At higher densities the randomly chosen

nearest neighbor position is rarely vacant, and it is better to draw a vacancy and a nearest neighbor

position. If this nearest neighbor is a suitably arranged chain segment, the motion is performed. Finally,when all sites on the lattice are occupied, only a collective rearrangement motions can take place

[97,100].Despite the great benefits of the computer simulations [96], they can be practically performed only for

the blends with small polymerization indices. On the other hand, the mesoscopic descriptions for

polymer blends which start from the Gaussian model are practical in the limit of long chains. Anexample of such mesoscopic model is the Random Phase Approximation (RPA) developed by

de Gennes [24]. The theories based on the generalization of the reference interaction-site model(RISM) integral-equation theory to homopolymer systems [64,101,102] also use the Gaussian

model for single chain conformations as in RPA. The large concentration fluctuations in the

system with finite chain lengths near the phase separation may affect the Gaussian statistics. In orderto study this problem we derive the equation for the radius of gyration of a single polymer chain with

the explicit inclusion of the fluctuations. As we show in the following sections the critical fluctuationsdo not perturb the Gaussian statistics in the homopolymers blends near the critical point. Largedeviations from the Gaussian statistics occur at low temperatures and for the disparate sizes of the A

and B chains.

3.1. Equation for the radius of gyration of a polymer chain

Let us consider a mixture ofnA A-type polymers, with NA monomers in each molecule, and nB B-type

polymers, with NB monomers in each molecule, inside a volume V. We will use the chain model in whichmonomers are joined by freely rotating bonds of fixed length (flexible chains). The distribution functionfor N monomers in such a chain is given by Eq. (11). The part of the interaction Hamiltonian corre-sponding to the mixture is given by the following expression [4,71] (see also Eq. (25)):

HI r0 dq

2p31

2wAA fAq 2

1

2wBB fBq 2 wABfAqfBq

!; 53

where r0 nANA nBNB=V is the density of monomers in the system, fAq and fBq are Fourier



20/49

transforms of the microscopic concentration operators:

fAq

1

r0 nA

a1 NA

i1exp

qr

ai

;

54

fBq 1

r0

nBb1

NBi1

expqrbi : 55

In the case of the homopolymer blend the conditional partition function, Eq. (27), is the partition

function for the system subject to the constraint that the microscopic operators fgq g A; B arefixed at some prescribed values [69] fgq; i.e.

ZfA;fB

N0

nA

a1 nB

b1 DraDrbWAraWBrb gA;B dfgq fgq exp HI

kBT : 56Here N0 is a constant, Dr

g denotes the measure (13), the interaction Hamiltonian HI is given by Eq. (53),

WA and WB are given by Eq. (11), and fgq g A; B are Fourier transforms of the concentrationsfAr and fBr; respectively.

The radius of gyration of an A-type polymer chain [9],

R2A 1

2N2ANAi1

NAj1

r1i r1j 2 ; 57

can be found from the single chain correlation function SAA

q;q

by differentiating twice with respectto q and taking the limit of q 3 0 (see Section 2.2). the single chain correlation function,

SAAq1;q2 f1A q1f1A q2 ; 58

with the single chain microscopic operator defined as

f1A q 1

r0

NAi1

expqr1i 59

is obtained from the partition function Z[UA] (Eq. (43)) which can be conveniently written in the

following form [73,74,76]:

ZUA

DfA

DfB exp

HI

kBT

DJA

DJB

exp i dq

2p3 fAJA i dq

2p3 fBJB FnAA JA; UA FnBB JB

& ': 60

Fngg is the free-energy for the system ofng non-interacting Gaussian chains in the external fields Jg, UA,

and is given in the form of a cumulant explanation [71,75] (see also Appendix A). One can integrate outthe fields JA, JB in a saddle point [75], take the Legendre transform to the fields (31) and write the



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partition function Z[UA] as

ZUA DCA DCB exp HI

kBT GACA; UA GBCB

gA;B

1detd2FngJg=dJgq1dJgq2

q ; 61where Jg defines the saddle point:

dFngJgdJgq

iCgq: 62

Now we take the functional derivative ofZ[UA] twice with respect to UA(q), put UAq 0, and theneliminate all the terms that vanish in the thermodynamic limit. Additionally we impose the condition of

incompressibility CA CB C (see Eq. (32)). In this way the approximate equation for the singlechain correlation function is obtained, i.e.:

f1A qf1A q f1A qf1A q 0 1

2

dk2p3 G

A2 k;k GA2 k;k

1

V CkCk 1

!

f1A qf1A qf1A kf1A k 0 f1A qf1A q 0 f1A kf1A k 0:63

means the average with partition function (56), and 0 means the average (45). The RHS of Eq.(44) contains the zero-order term f1A qf1A q 0 and the first-order corrections. These corrections

are of the order of1=Np since the upper wave-vector cutoff has been chosen as const:=R; were R is theradius of gyration, proportional to Np .In order to obtain the correct description of the collective structure factor CqCq

V=G2q;q near the critical point the set of the self-consistent one-loop equations [75] have beenused, i.e.:

G2q;q G02 q;qDG02 q;q1

2

dk2p3

G04 q;q; k;kG2q;q

1

2

dk2p3

G3q; k;q k2G2q;qG2q k;q k

; 64

G3q; p;q;p G03 q; p;q p DG03 q; p;q p

1

2

dk2p3

G3q; k;q kG3p;k;p kG3q p; p k; q kG2k;kG2q k;q kG2p k;p k

1

2

dk2p3

G05 k;k; p; q;p qG2k;k

: 65

They have been derived using the loop expansion [78] of the partition function of the system, Eq. (30).



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Due to the cutoff the loop expansion is also the expansion in the small parameter 1=

Np : Here we denote

the ideal term corrections coming from the denominator of Eq. (61) as DG02 and DG03 : The long-range

concentration fluctuations are taken into account in the set of equations (64) and (65) as one-loop

corrections [75]. The critical temperature and concentration determined from these equations are shiftedin comparison to the mean-field values, but in the limit of N 3 the FloryHuggins theory and RPAresults are recovered.

The relevant change of the radius of gyration of an A-type polymer chain RA in comparison to the

Gaussian value R0

NAp

lA=

6p

is given by the equation:

R2A

R20 1 0:062

NAp 1 1

ff

lA=lB34 5

R0=LdxAfxAgDxA

GA2 xAG2xA

1

4 5; 66

where f nANA=nANA nBNB is the average fraction of A monomer in the system, xA

NAp

qlA=6p

; and L is a symmetric function of the two radii of gyration for A and B chains which

determine the upper wave vector cutoff (see Section 2.3). Our assumption that the mixture is incom-pressible is expressed in the following form:

V

nANAl3A nBNBl

3B

1: 67

The functions gD(x), f(x) and the other ones needed to calculate the vertex functions in Eqs. (64) and (65)

are given in Appendix B.

3.2. Swelling and shrinking of polymer chains in homopolymer blends

Although as we shall see the effect of swelling and shrinking is very small in homopolymer blends oflong chains, it is still instructive to find out where and why it occurs. We will also see that even a verysmall change of the single chain statistics can lead to a large increase of the scattering intensity.

Fig. 1 shows the behavior of the radius of gyration as a function of the FloryHuggins parameter x(proportional to the inverse of the temperature) and the fraction of the A-type monomers f nANA=nANA nBNB for the mixture with equal indices of polymerization NA NB 1000:Thesolid lines indicate the cross-over between the system with swollen and the system with shrunk chains.

On these lines, the chains of A-type (line A) and B-type (line B) have the Gaussian radii of gyration.These lines have been obtained from the self-consistent solutions of Eqs. (64)(66) for A and B-type

chains. The mean-field spinodal curve

1NA

f 1

NB1 f 2x 0; 68

is plotted as a dotted line. The minimum value ofxon the spinodal curve corresponds to the mean-field

critical point (shown as an empty triangle). If one takes into account the long-range fluctuations, its true,shifted position can be found [75] (shown as a filled triangle). This location is defined by the self-consistent system of Eqs. (64) and (65) i.e. by G20; 0 0 and G30; 0; 0 0:

The general features of the behavior of the radius of gyration behavior are as follows. In the limit of aninfinite temperature, when the interaction parameter x approaches zero, the chains of both types areswollen. Then, when we lower the temperature, the size of the chains decreases. The chains less



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abundant in the blend pass through the Gaussian size first. So we can have the chains of one type swollenand of the other shrunk. The chains of both types are shrunk when we approach the spinodal. There is one

point on the phase diagram where the chains of both types are Gaussian (similar to the Q point in thetheory of polymer solutions). This point is located at the intersection of the solid lines.

The relative shrinking of the chains changes as 1=

Np

which follows immediately from the Eq. (66)since the corrections to the Gaussian chain are proportional to 1=

N

p: This result has been confirmed by

the SCFT calculations and Monte-Carlo simulations [93]. The change of the end-to-end distance in the

symmetric mixture

f

0:5

with the temperature and the index of polymerization resulting from these

calculations are shown in Fig. 2.The calculations of the radii of gyration based on the self-consistent determination of G2q;q are

valid in some region around the critical point where the fluctuation corrections are important. Far awayfrom the critical point at high temperatures this is not the case and the RPA prediction for G2q;q

Gmf2 q;q r0

NAf

1NA

f

NB1 f

2 3 2xNA f 1

l2B

f

l2A1 f

2 3NAl

2Aq

2

6

4 5; 69

is used in Eq. (66). From the substitution of this expression in the equation for the radius of gyration (66)


Fig. 1. Swelling and shrinking of polymer chains in the symmetric mixture NA 1000; NB 1000: The lines A and linesB show where the polymer chains of A- and B-type have the Gaussian size R20A NAl2A=6 or R20B NBl2B=6: Above theselines chains are shrunk, and below are swollen. These lines have been computed using the self-consistent one-loop system of

equations. The mean-field spinodal is plotted as the dotted line. The empty triangle presents the mean-field critical point, filled

triangle presents the real critical point determined by the self-consistent one-loop system of equations.


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follows that the chains at x 0 are always swollen and have the same radii of gyration independently ofthe concentration for NA NB: This result is quite reasonable, because the swelling at x 0 is causedonly by the constraint of incompressibility. If one calculates the radius of gyration directly as an average

of the squared inter-monomer distances with the constraint of incompressibility given by Eq. (12), i.e.

R2

nA

a1 nB

b1 DraDrbWAraWBrbdfAq fBq

1

2N2A

NA

i1 NA

j1

r1i

r1j 21=

nAa1

nBb1

DraDrbWAraWBrbdfAq fBq; 70

one exactly obtains Eq. (66) without the term GA2 xA=G2xA in square brackets under integration. In thelimit of infinite chain length, N 3 , the Gaussian size for the polymer chains is obtained.

The changes of the radii of gyration in the symmetric mixture with different concentration of A- andB-type monomers depend on the FloryHuggins parameter x as shown in Fig. 3. All calculations have

been performed using the self-consistent set of equations (64)(66). The change of the radius of gyrationdirectly depends on the value of the monomer concentration. The chains less abundant in the blend have

bigger relative shrinking. When the concentration of one component is very small, the shrinking ofchains of this type can be very large at low temperatures. For the systems with concentrations differentfrom f 0:5 one can observe some pretransitional effects. In the regions where the chains exhibit asharp change of their size, the scattering intensity Iq 1=G2 also increases very sharply. In Fig. 4, wepresent such a sharp increase of the scattering intensity. For the same system with fraction of A-monomers different from f 0:5 (here we choose f 0:2) we change FloryHuggins parameter x(from bottom to top on Fig. 4:x 0.88, 0.89, 0.90, 0.91, 0.92). The jump occurs near some special point,which we call a local demixing point (in our case for f 0:2 and x 0:91). At this point the correlationlength jumps to a higher value. It is preferable for the chains to be surrounded by the chains of the same


Fig. 2. The end-to end distance at f 0:5 as a function of the temperature and chain length. The lines present the results of theSCFT calculations for chain length N 16; 32, 64, 128, 256 and 512 (from bottom to top), while the symbols display the resultsof the Monte-Carlo simulations for chain length N 16: Reprinted with permission from Macromolecules 1998;31:904457. 1998 American Chemical Society [93].


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type, so they can demix locally. This effect is weak when the concentrations of A- and B-type monomersare comparable, and becomes stronger when we increase the difference between the A- and B-type

monomer concentrations.The relative shrinking of chains with N 1000 is very small even close to the spinodal line. The more

pronounced shrinking of the chains with minor number fraction has been observed in the computedsimulations based on the soft ellipsoid model for polymer melts and mixtures [99] and performed for

short chains N 50:

The averaged squared radius of gyration of both the minority A chains andmajority B chains f 0:1 as a function of the excess AB interaction parameter d (proportional to theFloryHuggins parameter) is shown on Fig. 5. The chains A exhibit significantly larger shrinking.


Fig. 3. The radius of gyration in the symmetric mixture NA 1000; NB 1000 depends on the monomer concentrationfa f; and the FloryHuggins parameter. Please note the regions where the chains exhibit sharp points (although small)shrinking (see Fig. 4).

Fig. 4. A jump of the scattering intensity I(q) for the symmetric mixture NA NB 1000 at the fixed concentration, f0:2: The solid lines correspond to the different FloryHuggins parameters: from bottom to top: x 0:88; 0.89, 0.90, 0.91, 0.92.Please note the sudden increase in I(q) for x 0:90 0:91: This is the indication of the local demixing point characterized bythe sharp shrinking of chains (see also Fig. 3).


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The case of asymmetric mixtures NA 400; NB 2000 is shown in Fig. 6. The general featuresare the same as in the case of symmetric mixtures, but the intersection of the Gaussian lines is moved tothe right on the figure. The incompleteness of the solid lines is caused by some technical difficulties in

the numerical computations. In the limit ofx3 0; the longer chains swell more than the shorter ones.This resembles the well-known situation when the chains consist of the same type monomers, and the

short chains can be considered as a solvent for the long ones with the partially screened excluded volumeinteractions [24,103].


Fig. 5. The averaged squared radius of gyration of both the minority A chains fA 0:1 and the majority B chains fB 0:9as a function of the excess AB interaction parameter d. The simulation volume contained 4000 particles representing chains of

N 50: According to Ref. [99].

Fig. 6. Swelling and shrinking of polymer chains in the asymmetric mixture NA 400; NB 2000: The lines A and linesB show where the polymer chains of A- and B-type have the Gaussian size R20A NAl2A=6 or R20B NBl2B=6 (see the legend ofFig. 1).


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Fig. 7. Swelling and shrinking of polymer chains in the mixture with equal indices of polymerization NA 1000; NB 1000 but different segment lengths lA and lB. The lines A and lines B show where the polymer chains of A- and B-type haveGaussian size R20A NAl2A=6 or R20B NBl2B=6: Above these lines the chains are shrunk, and below are swollen. The solid linescorrespond to lA lB; dashed to lA 2lB; dotted to lA 5lB: The empty squares present the intersection of the Gaussian linescomputed from the self-consistent determination ofG2, The mean-field spinodal is plotted as the dotted line.

Fig. 8. The magnitude of the chains shrinking as a function of the asymmetry of the indices of polymerization. The lines Ra or

Rb correspond to the relative differences of the radii of gyration of A- or B-type chains in the critical points, Rcr, in

comparison to its Gaussian radii of gyration, R0. We change the number of segments in B-type chains, NB, while the number

of segments in A-type chains, NA, is constant NA 200:


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The case of the mixture with the same indices of polymerization NA NB 1000 but differentmonomer volumes is shown in Fig. 7. This means, that the bond lengths lA and lB for each type of chains

are different in Eq. (11). The solid lines correspond to lA lB; the long-dashed lines correspond to lA 2lB; and the dotted lines correspond to lA 5lB: The mean-field spinodal is shown as a dot-dashed line.Q-points are plotted as empty squares. The increase of the ratio lA/lB moves the Gaussian point toward

higher fractions of monomers with the longer segments.In Fig. 8 it is shown how the shrinking of the chains at the critical point depends on the asymmetry of

the indices of polymerization. The points on the lines Ra or Rb correspond to the relative changes ofthe radii of gyration of A- or B-type chains at the critical point, Rcr, in comparison with the Gaussian radiiof gyration R0. In the symmetric case these changes are equal. When we increase the index of poly-

merization in one-type (B-type) chains, shrinking of the shorter (A-type) chains becomes smaller, butshrinking of the longer (B-type) chains increases. When asymmetry is very big our system behaves as asystem of polymer chains mixed with a solvent. If, additionally, the concentration of the longer chains is

very small, they shrink very rapidly as x increases. This phenomenon is similar to the collapse of a

polymer chain in a strongly incompatible melt [84].Summarizing, the critical fluctuations have a very weak influence on the radius of gyration in homo-

polymer blends. The Gaussian statistics is a very good approximation for the polymer statistics near thecritical point. We can expect significant deviations from the Gaussian statistics when the sizes of themolecules are very disparate and/or the temperature is very low.

4. Copolymer melts near the orderdisorder transition

The simplest copolymer system is a diblock AB copolymer melt. It composes of the diblock

copolymer molecules which contain a sequence of A-type monomers chemically joined to the sequenceof B-type monomers. The state of the system of mixed diblock copolymer molecules with a givenarchitecture is determined by temperature, only. In contrast to the homopolymer blend, in the copolymer

melt A-homopolymer and B-homopolymer are chemically joined in a block, so the system of diblockscannot undergo a macroscopic phase segregation. Instead, a number of orderdisorder phase transitionstake place in the system between the isotropic phase and spatially ordered phases in which A- and B-richdomains, of the size of a diblock copolymer, are periodically arranged in space. One can expect

the following types of ordering in such a system [70,79,104112]: the lamellar mesophase(LAM); hexagonally packed cylinders (HEX); spheres arranged at the sites of a bcc lattice

(BCC); bicontinuous double-gyroid structure with Ia 3d space group symmetry (G); hexagonallymodulated lamellae (HML) and hexagonally perforated layers (HPL). The size of the domains in

the ordered structures can vary from tens up to hundreds of angstrom and is controlled by thesize of the diblock macromolecules.

The phase diagram of the AB diblock copolymers is often divided into two regions: the weak

segregation limit (where the boundaries between the A- and B-rich domains are diffuse, i.e.comparable to the size of the domains) and the strong segregation limit (ordered mesophaseswith sharp boundaries between A- and B-rich domains). Deep in the disordered region the

conformations of a single chain are the same as the conformations of an isolated noninteractingchain (Gaussian coil) and the size of a polymer chain scales with the polymerization index as

R lN1=2: The scattering experiments manifest the characteristic correlation length [70]



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Fig. 9. A schematic illustration of the proposed real-space morphology of a symmetric diblock copolymer melts. GST and ODT

correspond to the Gaussian- to stretched-coil transition and orderdisorder transition, respectively. Taken from Ref. [116].

Fig. 10. The dependence of the SANS peak position q on the polymerization index N. According to Ref. [116].


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(correlation hole) in the disordered melt due to the connectivity of the two blocks and lowcompressibility of such melts. In the strong segregation limit the chains are significantly stretched

and lN2=3 scaling for the radius of gyration is valid instead of the Gaussian scaling [113115].

Therefore, the crossover between the Gaussian- and stretched-coil conformations is expected tobe somewhere around the orderdisorder transition (ODT) (see illustration in Fig. 9).

The first direct observation of the chains stretching in the disordered region was performed by Almdalet al. [116] in SANS experiments. They observed two different scaling behaviors of the peak position

q 2p=R in the scattering intensity with the index of polymerization of the diblock macromolecules ata fixed temperature (see Fig. 10). They found a slope d 0:49^ 0:02q Nd in the disorderedregion characteristic for the Gaussian conformation of the chains, but near the ODT the scaling behavior

changed to d 0:8^ 0:04:It is important to note that concentration fluctuations play a decisive role in the nature of the ODT in

the diblock copolymer system. For example, the mean-field theory [70] predicts a second-order phase

transition from the disordered state to the lamellar structure in the symmetric melt (the lengths of block

A and block B are equal). However, as it was shown [79,80], increasing the volume in the reciprocalspace of important field fluctuations modify the nature of ODT and instead of the second-order transition

the fluctuations induce the weak first-order transition. Also the single-chain conformation in the disor-dered state are strongly affected by large concentration fluctuations as observed in the experiments[116,117].

In Section 4.1 we derive the set of equations which describe the single chain conformations in thediblock copolymer melts. In Section 4.2 we discuss how the local structure of the diblock melt depends

on the architecture of the chains and temperature in the disordered region before the phase transition tothe ordered state. In Section 4.2 we also discuss the single-chain properties of different copolymer

systems such as random copolymers [118120], gradient copolymers [97,121] and the melts of the

copolymer rings [122,123].

4.1. Equations for the size of a diblock copolymer

We consider a mixture ofn polymer molecules inside a volume V. Each of the molecules consists of

one block ofNA monomers of type A and the second block of NB monomers of type B. The architectureof the polymer molecules is specified by the composition, f NA=NA NB and distribution function,Wr; r {r1; ; rN} (Eq. (11)) where N NA NB is the index of polymerization of the entirediblock macromolecule. Here we assume the equal segments lengths in both blocks of different species.

The interaction Hamiltonian with the specified short-range interactions between the monomers (18) isgiven by Eq. (53), where r

0 nN

AN

B=V is the number density of monomers in the system, f

Aqand fBq are the Fourier transforms of the microscopic concentration operators

fAq 1

r0

na1

NAi1

expqrai ; fBq 1

r0

nb1

NA NBiNA 1

expqrbi : 71

Due to the connectivity of the blocks this part of the Hamiltonian leads to the microphase (mesophase)

separation in the diblock copolymer melt providing wAA wBB 2wAB 2kBTx 0; where x is theusual FloryHuggins parameter.

The conditional partition function, ZfA;fB; is specified in the similar way as for a homopolymer



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blend (Eq. (56)) with the micro

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