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14

Influence of Cohesive Energy and Chain Stiffness on Polymer Glass Formation

Wen-Sheng Xu1, ∗ and Karl F. Freed1, 2, †

1James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA2Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, USA

(Dated: May 19, 2018)

The generalized entropy theory is applied to assess the joint influence of the microscopic cohesiveenergy and chain stiffness on glass formation in polymer melts using a minimal model containinga single bending energy and a single (monomer averaged) nearest neighbor van der Waals energy.The analysis focuses on the combined impact of the microscopic cohesive energy and chain stiffnesson the magnitudes of the isobaric fragility parameter mP and the glass transition temperature Tg.The computations imply that polymers with rigid structures and weak nearest neighbor interactionsare the most fragile, while Tg becomes larger when the chains are stiffer and/or nearest neighborinteractions are stronger. Two simple fitting formulas summarize the computations describing thedependence of mP and Tg on the microscopic cohesive and bending energies. The consideration ofthe combined influence of the microscopic cohesive and bending energies leads to the identificationof some important design concepts, such as iso-fragility and iso-Tg lines, where, for instance, iso-fragility lines are contours with constant mP but variable Tg . Several thermodynamic properties arefound to remain invariant along the iso-fragility lines, while no special characteristics are detectedalong the iso-Tg lines. Our analysis supports the widely held view that fragility provides morefundamental insight for the description of glass formation than Tg.

PACS numbers: 64.70.pj, 83.80.Sg, 05.70.-a, 05.50.+q

I. INTRODUCTION

Although glasses are ubiquitous in nature and in ourdaily life, a deep microscopic understanding of the na-ture of the glass transition and the glassy state remains afundamental challenge in condensed matter physics [1–4].Typically, the dynamics of the supercooled liquid slowsdown precipitously on approaching the glass transitiontemperature Tg, while structural changes in the liquidare rather mild. Most polymers readily form glasses uponcooling or compression, and hence, provide a unique op-portunity for probing the physical mechanism of glassformation due to their distinctive molecular character-istics [5–8]. The fragility parameter m, quantifying thesteepness of the temperature dependence of the dynam-ics, and Tg constitute two of the most important quanti-ties of polymer glass formation [9, 10], governing, for in-stance, whether a polymer material can be processed byextrusion, casting, ink jet, etc. Thus, the ability for therational design of polymer materials with desired prop-erties requires an understanding of the molecular factorsinfluencing the magnitudes of m and Tg.The chemical structure of polymers has long been rec-

ognized as strongly influencing properties associated withglass formation, such as m and Tg [11]. For instance,polymers with rigid or sterically hindered backbones usu-ally exhibit larger m and greater Tg than polymers withsimple and less sterically hindered structures [11]. Thisbehavior reflects the strong impact of the backbone andside group structures on microscopic molecular proper-

∗ wsxu@uchicago.edu† freed@uchicago.edu

ties, such as the cohesive energy and chain stiffness. Re-cent experimental data indicate that the microscopic co-hesive energy and chain stiffness can even be modifiedsignificantly simply by altering the chemical species inthe side groups [12]. The shifts in molecular structurebetween different polymer materials inevitably lead tocorresponding changes in the behavior of polymer glassformation. Despite substantial experimental and simu-lational evidence [11–21], a predictive molecular theorythat describes the impact of chemical structure on poly-mer glass formation has been slow to develop.

The generalized entropy theory (GET) [22] is a mergerof the Adam-Gibbs (AG) relation between the structuralrelaxation time and the configurational entropy [23] andthe lattice cluster theory (LCT) for the thermodynam-ics of semiflexible polymers [24]. Because the LCT en-ables probing the influence of various molecular details,such as the cohesive energy (described by the nearestneighbor van der Waals interaction energy ǫ, and calledthe microscopic cohesive energy or just cohesive energyfor short), the chain stiffness (described by the bend-ing energy Eb), the molecular weights, and the monomerstructures, on the thermodynamics of multicomponentpolymer systems [24], the GET has provided initial the-oretical insights into the molecular origins of m and Tg

by describing the sensitivity of m and Tg to separate,i.e., one-dimensional, variations of the cohesive energy,the chain stiffness, and the relative rigidity of the back-bone and the side chains [22]. While the agreement withexperiment of the non-trivial predictions from the GETprovides strong validation of the theory, the goal of ra-tional design of polymeric materials requires consider-ing the additional complexities of real polymer materi-als. For instance, the previous calculations within the

2

GET [9, 10, 22, 25–30] consider the simplest model inwhich all united atom groups (i.e., the basic units of thepolymer chains) interact with a common monomer av-eraged interaction energy. However, different groups areknown to have disparate, i.e., specific, interactions whoseimplications remain to be investigated within the LCT.Moreover, the variation in structure within a monomerimplies that the monomer averaged interaction energyand the chain stiffness must all change simultaneously.While some experimental data of Sokolov and cowork-ers [12, 16] demonstrate that m and Tg for differentpolymers can be understood within the GET by con-sidering separate variations of properties with ǫ and Eb,other data exhibit quite perplexing behavior that proba-bly arises due to the competitive influences of changes inthe cohesive energy and chain stiffness between “similar”materials.

The present paper focuses on analyzing the nature ofglass formation in polymer melts as described by the LCTwith a model containing a single, monomer averaged in-teraction energy ǫ between all united atom groups in thechain. Specifically, the monomer averaged interactionmodel is used to explore the variation of m and Tg with ǫand Eb. Although a previous work briefly illustrates theseparate variation of m and Tg individually with ǫ andEb [28], the present paper emphasizes the more complexcombined influence of ǫ and Eb, thereby illuminating po-tential design concepts. Moreover, the results obtainedhere will be compared in a subsequent paper with a re-cently developed more realistic model where the LCTtreats polymer melts with specific interactions [31]. Inparticular, the chains in the more realistic model havethe structure of poly(n-α-olefins) where the terminal seg-ments on the side chains are assigned different, specificvan der Waals interaction energies with other unitedatom groups. The greater realism introduced into theLCT and the GET by this new model enables testingthe limits of validity of the present model with a singlemonomer averaged van der Waals energy [9, 10, 22, 25–30]. In addition, the subsequent work will study how thevariation of the specific interactions can be used to exertgreater control over the properties of designed materials.

Section II provides some general background concern-ing the GET. Section III begins by delineating the generalcombined influence of ǫ and Eb on the nature of polymerglass formation. Two algebraic functions m(ǫ, Eb) andTg(ǫ, Eb) are found to recapture the computed depen-dence ofm and Tg on ǫ and Eb. Section III then proceedsby introducing some important design concepts, such asiso-fragility and iso-Tg lines. The iso-fragility lines arecontours with constantm but variable Tg and other prop-erties. Our analysis demonstrates that many properties,e.g., the entropy, the polymer volume fraction and the re-laxation times at characteristic temperatures, such as Tg,remain invariant along the iso-fragility lines. By contrast,no special characteristics are found along the iso-Tg lines.Our results support the widespread view that the conceptof fragility provides more fundamental insight into glass

formation than the glass transition temperature Tg.

II. POLYMER GLASS FORMATION WITHIN

THE GENERALIZED ENTROPY THEORY

The configurational entropy plays a central role in theGET [22] as in the classic entropy theories of glass for-mation by Gibbs and DiMarzio (GD) [32] and by Adamand Gibbs (AG) [23]. These theoretical approaches buildupon the well-known fact that the rapid increase in theviscosity and structural relaxation time on cooling to-wards the glass transition temperature is accompanied bya precipitous drop in the fluid entropy [33]. The GET [22]merges the LCT for the thermodynamics of semiflexiblepolymers with the AG relation between the structural re-laxation time and the configurational entropy. Hence, theGET permits computing characteristic temperatures andfragility and addressing the influence of various molecu-lar characteristics, such as monomer structure, on poly-mer glass formation. Therefore, the GET involves a sig-nificant extension of the scope beyond that of GD the-ory [32], which effectively only focuses on the “ideal”glass transition temperature where the configurationalentropy extrapolates to zero.The LCT yields an analytical expression for the spe-

cific Helmholtz free energy f (i.e., the total Helmholtzfree energy per lattice site) of a semiflexible polymermelt [24], as a function of polymer volume fraction φ,temperature T , cohesive energy ǫ, bending energy Eb aswell as molecular weight M and a set of geometrical in-dices that reflect the size, shape and bonding patternsof monomers. The explicit expression for f can be ob-tained in ref 24 with some corrections given in ref 31. Webriefly explain the physical meaning of the key molecu-lar parameters ǫ and Eb in the LCT. The microscopiccohesive energy ǫ enters the LCT in terms of the Mayerf -function, which, in turn, is treated using a high temper-ature expansion [24, 34], and a convention that a positiveǫ describes the net attractive van der Waals interactionsbetween nearest neighbor united atom groups. The bend-ing energy Eb represents the conformational energy dif-ference between, e.g., trans and gauche conformations fora pair of consecutive bonds [24]. The trans conformationcorresponds to consecutive parallel bonds and is ascribeda vanishing bending energy, while Eb is prescribed to agauche pair of sequential bonds lying along orthogonal di-rections. Chains are fully flexible for Eb = 0, while theybecome completely rigid in the limit Eb → ∞. Also, sidechains with two or more united atom groups may have aseparate bending energy [26].Within the GET, polymer glass formation is treated

as a broad transition with four characteristic tempera-tures whose determination is accomplished by first com-puting the LCT configurational entropy density sc [35],i.e., the configurational entropy per lattice site. This scexhibits a maximum as the temperature T varies at con-stant pressure [22], an essential feature for use in the AG

3

model. Recent computations [36] also indicate that theLCT configurational entropy density sc is almost iden-tical to the ordinary entropy density s = −∂f/∂T |φ atthe same thermodynamic conditions because the latticemodel is essentially devoid of vibrational contributions.Since the ordinary entropy density very closely approxi-mates the configurational entropy density and since theformer is much easier to calculate in the LCT, the calcula-tions employ the ordinary entropy density in the presentwork. For simplicity, the ordinary entropy density is alsocalled the entropy density in the following. Also, all thecalculations are performed at constant pressure P , de-fined by

P = −∂F

∂V

∣

∣

∣

∣

Np,T

= −1

Vcell

∂F

∂Nl

∣

∣

∣

∣

Np,T

, (1)

where V is the volume of the system, Np is the number ofpolymer chains, and Vcell = a3cell is the volume associated

with a single lattice site that is set to be acell = 2.7A inall the calculations.The LCT computations for the temperature depen-

dence of the entropy density s(T ) enable the direct de-termination of three characteristic temperatures of glassformation, namely, the “ideal” glass transition temper-ature To, the onset temperature TA, and the crossovertemperature TI (Figure 1a). To corresponds to the tem-perature where s extrapolates to zero as in the GD the-ory, TA signals the onset of non-Arrhenius behavior ofthe relaxation time and is found from the maximum ins(T ). The crossover temperature TI separates two tem-perature regimes with qualitatively different dependencesof the relaxation time on temperature and is evaluatedfrom the inflection point in Ts(T ). Equivalently, TI canbe determined by finding the maximum of ∂(Ts)/∂T , asshown in the inset to Figure 1a. Although the fourthcharacteristic temperature, i.e., the glass transition tem-perature Tg, might be obtained from a Lindermann cri-terion [9], its conventional definition requires knowledgeof the temperature dependence of the relaxation time τ .To this end, the GET invokes the AG relation [23],

τ = τ∞ exp[β∆µs∗/s(T )], (2)

where τ∞ is the high temperature limit of the relaxationtime, β = 1/(kBT ) with kB being Boltzmann’s constant,∆µ is the limiting temperature independent activationenergy at high temperatures, and s∗ is the high temper-ature limit of s(T ) (i.e., the maximum of the entropydensity calculated from the LCT). τ∞ is set to be 10−13

s in the GET, which is a typical value for polymers [38].Motivated by the experimental data [38], the GET esti-mates the high temperature activation energy from theempirical relation ∆µ = 6kBTI ; more discussion of thisempirical relation appears in ref 22. Thus, the relaxationtime is computed within the GET without adjustable pa-rameters beyond those used in the LCT for the thermody-namics of semiflexible polymers. The GET then identifiesTg using the common empirical definition τ(Tg) = 100 s(Figure 1b).

0.0

0.1

0.2

0.3

0.4

0.5

100 200 300 400 500 600 700

s/kB

T

(a)

PP

Nc = 8000

ǫ = 200 K

Eb = 400 K

TA

To

0.5

0.6

0.7

0.8

200 300 400 500

∂(T

s/kB)/∂T

T

TI

-10

-5

0

5

10

200 250 300 350 400

log(τ)

T

(b)

Tg

-10

-5

0

5

10

0.6 0.7 0.8 0.9 1.0 1.1

log(τ)

Tg/T

mP = ∂ log(τ)∂(Tg/T )

∣

∣

∣

∣

∣

T=Tg

FIG. 1. Illustration of the determination of various charac-teristic temperatures and the fragility parameter within thegeneralized entropy theory (GET). (a) Temperature T depen-dence of the entropy density s/kB calculated from the lat-tice cluster theory (LCT) for a melt of chains possessing thestructure of poly(propylene) (PP) with polymerization indexNc = 8000, cohesive energy ǫ = 200 K and bending energyEb = 400 K, at a constant pressure of P = 1 atm. Threecharacteristic temperatures are well defined in the curve ofs(T )/kB: the ideal glass transition temperature To where theentropy density extrapolates to zero, the onset temperatureTA where the entropy density displays a maximum, and thecrossover temperature TI which corresponds to the inflectionpoint in the curve of Ts(T )/kB. Equivalently, TI can be com-puted by finding the maximum of ∂(Ts/kB)/∂T , as shown inthe inset to Figure 1a. (b) Structural relaxation time τ forthe same melt as a function of T , calculated by combining theLCT with the Adam-Gibbs (AG) relation. The glass transi-tion temperature Tg is identified by the common empiricaldefinition τ (Tg) = 100 s. The inset to Figure 1b illustratesthe determination of the isobaric fragility parameter mP froman Angell plot [37].

Once the temperature dependence of the relaxationtime is known, other related quantities can be also cal-culated from the GET. For instance, the fragility param-eter, which quantifies the steepness of the temperaturedependence of the relaxation time, can be determined at

4

constant pressure, e.g., from the standard definition,

mP =∂ log(τ)

∂(Tg/T )

∣

∣

∣

∣

P,T=Tg

, (3)

where mP denotes the isobaric fragility parameter. Theinset to Figure 1b illustrates the determination of mP

from an Angell plot [37].

III. RESULTS AND DISCUSSION

This section begins by discussing how the propertiesassociated with glass formation vary under the combinedinfluence of the microscopic cohesive ǫ and bending Eb

energies in polymer melts with monomer averaged inter-actions. This discussion then naturally leads to the in-troduction of the concepts of iso-fragility and iso-Tg linesand the exploration of melt properties along these lines.

A. Combined influence of microscopic cohesive and

bending energies on polymer glass formation

A previous paper [28] investigates the variation offragility and Tg individually with ǫ and with Eb. How-ever, the combined variation with the cohesive and bend-ing energies yields a more detailed behavior. The calcu-lations consider chains with the structure of PP becausethis choice requires the minimal number of parameters inthe LCT. The simplest model for poly(n-α-olefins) withlonger side groups n > 1 contain separate bending ener-gies for the backbone and side groups, thus adding an-other parameter [26]. The subsequent figures all use acommon parameter set: the lattice coordination numberis z = 6; the pressure is P = 1 atm; the cell volumeparameter is acell = 2.7A; and the polymerization indexis chosen to be Nc = 8000, corresponding to a polymermelt of chains with high molecular weight.Figure 2 displays the entropy density for various co-

hesive energies with constant bending energy and forvarious bending energies with constant cohesive energy.As expected, both cohesive energy and bending energystrongly affect the entropy density. The curves for theentropy density in Figure 2 shift to higher temperaturesas either ǫ or Eb grows. Consequently, all characteris-tic temperatures elevate upon increasing either ǫ or Eb.Moreover, the magnitude of the entropy density at eachcharacteristic temperature sTα

(Tα = TA, TI or Tg) is al-tered in opposite directions as either ǫ or Eb increases.Specifically, sTα

grows with ǫ but drops with Eb. Further-more, the characteristic temperature ratio TA/To growswith increasing ǫ or decreasing Eb, implying that thebreadth of the glass-formation process can be controlledby adjusting the cohesive energy, the chain stiffness, orboth. The latter observation is important because thebreadth of glass formation is often suggested as beinggoverned by the fragility [38–40].

0.0

0.1

0.2

0.3

0.4

0.5

s/kB

(a)

Eb = 500 Kǫ, K

0.0

0.1

0.2

0.3

0.4

100 200 300 400 500 600 700

s/kB

T

(b)

ǫ = 200 KEb, K

150200250

500100015002000

FIG. 2. (a) The entropy density s/kB as a function of Tfor various cohesive energies ǫ but fixed bending energy Eb.(b) The entropy density s/kB as a function of T for variousbending energies Eb but fixed cohesive energy ǫ. Squares,circles, diamonds and pentagons designate the positions ofcharacteristic temperatures TA, TI , Tg and To, respectively.

Knowledge of the equation of state (EOS) is also cru-cial for understanding, for instance, the pressure depen-dence of glass formation [30]. In addition, the EOS pro-vides complementary information to the entropy densityand therefore aids in understanding the joint influence ofcohesive and bending energies on polymer glass forma-tion. Hence, we additionally explore how the tempera-ture dependence of the polymer volume fraction φ is af-fected by variations of the microscopic cohesive and bend-ing energies at constant pressure. The EOS data in Fig-ure 3 clearly indicate that the temperature dependence ofφ is dominated by the cohesive energy ǫ. This anticipatedresult arises since ǫ provides the measure of the net at-tractive interactions between united atom groups in theLCT and thus a larger ǫ is expected to produce a densersystem as evidenced in Figure 3a. Hence, polymers packmore efficiently as the cohesive energy increases. Conse-quently, the volume fraction at each characteristic tem-perature φTα

(Tα = TA, TI , Tg or To) increases with ǫ.The bending energy Eb, on the other hand, has almostno effect on the temperature dependence of φ (Figure 3b).However, φTα

drops significantly as Eb elevates, a trendthat agrees with the general observations that more freevolume exists in the glassy state of more rigid polymers.

We now discuss the combined effects of ǫ and Eb onboth mP and Tg. In line with previous calculations, [28]the contour plots in Figure 4 clearly indicate that increas-

5

0.6

0.8

1.0φ

(a)

Eb = 500 K

ǫ, K

0.4

0.6

0.8

1.0

100 200 300 400 500 600 700

φ

T

(b)

ǫ = 200 K

Eb, K

150200250

500100015002000

FIG. 3. (a) The polymer volume fraction φ as a function of Tfor various cohesive energies ǫ but fixed bending energy Eb.(b) The polymer volume fraction φ as a function of T for vari-ous bending energies Eb but fixed cohesive energy ǫ. Squares,circles, diamonds and pentagons designate the positions ofcharacteristic temperatures TA, TI , Tg and To, respectively.

ing ǫ and Eb produces opposite shifts in mP , while Tg isaltered in the same direction. The trend of an increas-ing mP or Tg with Eb is of course present over a limitedrange because mP and Tg must saturate when the chainsbecome very stiff, i.e., Eb is sufficiently large (see FigureS1 in the Supporting Information). Two simple alge-braic equations fairly accurately capture the computedcombined variations of mP and Tg with ǫ and Eb,

mP =a0 + a1ǫ+ a2ǫ

2 + (b0 + b1ǫ)Eb

1 + (c0 + c1ǫ+ c2ǫ2)Eb

, (4)

Tg =u0 + u1/ǫ+ u2/ǫ

2 + (v0 + v1/ǫ+ v2/ǫ2)Eb

1 + (w0 + w1/ǫ+ w2/ǫ2)Eb

, (5)

where the fitted parameters aα(α = 0, ..., 2), bα(α = 0, 1),cα(α = 0, ..., 2), uα(α = 0, ..., 2), vα(α = 0, ..., 2) andwα(α = 0, ..., 2) can be found in the caption of Figure S1in the Supporting Information. Other forms of the equa-tion may satisfactorily describe the data as well, but eqs 4and 5 have been chosen to assure the observed saturationof mP and Tg for large Eb.Figure 5 further demonstrates that a master curve ex-

ists between the isobaric fragility parameter mP and thecharacteristic temperature ratios, implying that the com-monly used mP indeed correlates with ratios of the char-acteristic temperatures in some cases. The idea that the

150 175 200 225 250ǫ

400

800

1200

1600

2000

Eb

75

100

125

150

175

200

225

250

mP

150 175 200 225 250ǫ

400

800

1200

1600

2000

Eb

200

250

300

350

400

450

500

Tg

FIG. 4. Contour plots of the isobaric fragility parameter mP

(upper panel) and the glass transition temperature Tg (lowerpanel) in the plane of cohesive energy ǫ and bending energyEb.

50

100

150

200

250

300

0.3 0.4 0.5 0.6 0.7

mP

Tα/TA

To/TA

Tg/TA

ǫ = 150 Kǫ = 175 Kǫ = 200 Kǫ = 225 Kǫ = 250 K

FIG. 5. Correlation between isobaric fragility parameter mP

and different ratios of characteristic temperature Tα/TA. Thebending energy Eb varies from 400 K to 2000 K for each ǫ.

6

500

1000

1500

2000E

b

(a)

500

1000

1500

2000

150 175 200 225 250

Eb

ǫ

(b)

mP = 100mP = 125mP = 150mP = 175mP = 200

Tg = 250 KTg = 275 KTg = 300 KTg = 325 KTg = 350 K

FIG. 6. (a) Iso-fragility lines and (b) iso-Tg lines in the planeof cohesive energy ǫ and bending energy Eb for several repre-sentative values of mP and Tg. Solid lines in (a) and dottedlines in (b) are the results of eqs 4 and 5 with the fitting pa-rameters given in the caption of Figure S1 in the SupportingInformation.

breadth of glass formation is related to fragility is cer-tainly not new, but the plot in Figure 5 establishing itsuniversality is presented here for the first time. Our re-sults support the contention that the breadth of the glass-formation process provides a promising measure for thefragility of glass-forming liquids [22].

B. Iso-fragility and iso-Tg lines

The strong influence of the cohesive and bending ener-gies on polymer glass formation clearly demonstrates thatthe fragility and the glass transition temperature can befinely tailored by adjusting these molecular parameters.In particular, we identify two types of special lines in theǫ-Eb plane, along which either the fragility parametermP or the glass transition temperature Tg remains con-stant. These lines are naturally called iso-fragility andiso-Tg lines, respectively, and their existence is indeedquite apparent in the contour plots of Figure 4. An ex-ploration of various properties along the iso-fragility andiso-Tg lines provides better understanding of the physicalsignificance of mP and Tg in glass-forming polymers, asdiscussed below.

50

100

150

200

250

150 200 250 300 350

mP

Tg

ǫ, K Eb, K150–250

150–250200

100–300

340–565

690–390300–900

600

FIG. 7. Diverse correlation patterns between mP and Tg dueto the variations in ǫ and Eb. The lines are a guide to the eye.

Several iso-fragility and iso-Tg lines are displayed inFigure 6 as curves of ǫ vs. Eb for representative valuesof mP and Tg, respectively. The lines in Figure 6 dis-play approximations to the iso-fragility and iso-Tg linesobtained from eqs 4 and 5 with the fitting parametersgiven in the caption of Figure S1 in the Supporting In-formation. The fitted expressions clearly agree well withthe calculations along the iso-fragility and iso-Tg lines.Solutions for Eb fail to exist for Tg = 350 K on the iso-Tg

line with ǫ = 150 K because Tg saturates at a smallerlimit than 350 K for ǫ = 150 K (see Figure S1 in theSupporting Information).Figure 6a indicates that Eb grows approximately lin-

early with ǫ along an iso-fragility line and that the slopegrows with mP . Figure 6b exhibits the trend that Eb

decreases with ǫ along the iso-Tg lines, so chains mustbecome more flexible for larger cohesive energies in orderfor the system to achieve the same Tg as at a smaller ǫ.The drop in Eb with ǫ along the iso-Tg lines is quickerfor low than high ǫ.Recent experiments by Sokolov and coworkers [12]

show that modifying the chemical structures in the back-bone/side groups of polymers produces apparently differ-ent patterns for the correlations between mP and Tg. Forinstance, mP and Tg shift from 137 and 263 K for PP to189 and 304 K for poly(vinyl alcohol) (PVA) [12], i.e.,mP and Tg can both increase together for some poly-mers. Likewise, experimental data demonstrate that mP

can also decrease when Tg grows; e.g., Tg increases butmP decreases when shifting from poly(4-methylstyrene)(P4MS) to poly(4-chlorostyrene) (P4ClS). More interest-ingly, different polymers with similar mP (or Tg) mayexhibit large variations in Tg (or mP ). For instance,PVA and poly(vinyl chloride) (PVC) exhibit very sim-ilar fragilities (mP ≈ 190 for both polymers with simi-lar molecular weights), while their glass transition tem-peratures are quite different (Tg = 304 K for PVA vs.Tg = 352 K for PVC) [12]. Hence, the variation ofmP (or

7

Tg) can be also nearly independent of Tg (or mP ). Ourcalculations indeed support the contention that the pat-terns of joint variation of mP and Tg can be controlled byadjusting the cohesive energy or the chain stiffness, as il-lustrated in Figure 7, where four correlation patterns be-tween mP and Tg are depicted, including an iso-fragilityline for mP = 100 (where Eb increases from 340 K to565 K when ǫ elevates from 150 K to 250 K), an iso-Tg

line for Tg = 250 K (where Eb decreases from 690 K to390 K when ǫ elevates from 150 K to 250 K), a posi-tive correlation (where Eb increases from 300 K to 900 Kwhen the cohesive energy is fixed to be ǫ = 200 K), anda negative correlation (where ǫ increases from 100 K to300 K when the bending energy is fixed to be Eb = 600K). Figure 7 exhibits the large variations in Tg (or mP )along the iso-fragility (or iso-Tg) line. Moreover, it isclear in Figure 7 that mP can increase or decrease withTg within the GET when only a single variable Eb or ǫis altered, as first revealed in ref 28. Therefore, the GETprovides a theoretical interpretation for the experimentalobservations in ref 12.Although the concepts of fragility and glass transition

temperature have long appeared in the study of glass for-mation [41], quantitatively understanding their molecu-lar origins and their physical significance remains incom-plete for glass-forming polymers. Based on extensive ex-amination of the influence of various molecular factorson the properties associated with polymer glass forma-tion, the GET suggests that the packing efficiency deter-mines the fragility and the glass transition temperatureof polymer fluids [22]. Our identification of iso-fragilityand iso-Tg lines provides additional routes for uncoveringthe molecular significance of fragility and the glass tran-sition temperature by exploring the variation of typicalthermodynamic properties along these lines.

C. Properties along the iso-fragility and iso-Tg lines

Figure 8a displays the T -dependence of the entropydensity for different pairs of cohesive energies ǫ and bend-ing energies Eb that lie along an iso-fragility line formP = 100. As expected, the overall breadth of theentropy density curve (as measured by the characteris-tic temperature ratio TA/To) remains almost unchangedalong the iso-fragility lines because the fragility directlyprovides a measure of the breadth of glass formation, asdiscussed in Subsection 3.1 (Figure 5). In addition, themagnitudes of the entropy density at each characteris-tic temperature remain constant along the iso-fragilitylines, as highlighted by the horizontal lines in Figure 8a.The above observations suggest that the entropy den-sity along the iso-fragility lines is a unique function ofTα/T , i.e., the inverse temperature 1/T scaled by one ofthe four characteristic temperatures Tα. Figure 8b con-firms these suggestions by presenting the entropy densityalong the iso-fragility line in an Angell plot. The upperinset to Figure 8b indicates that the relaxation times col-

0.0

0.1

0.2

0.3

0.4

0.5

100 200 300 400 500 600 700

s/kB

T

(a)

ǫ,K Eb,K

mP = 100

sTA

sTI

sTg

0.0

0.1

0.2

0.3

0.4

0.5

0.4 0.6 0.8 1.0 1.2

s/kB

Tg/T

(b)

-10

-5

0

5

10

0.4 0.6 0.8 1.0

log(τ)

Tg/T

150

200

250

300

150 175 200 225 250

Tg

ǫ

150 340200 450250 565

FIG. 8. (a) The entropy density s/kB as a function of Tfor several pairs of ǫ and Eb that produce the same isobaricfragility parameter ofmP = 100. (b) Tg-scaled Arrhenius plotfor the entropy density s/kB for the same pairs of ǫ and Eb asin (a). The upper inset to (b) presents the Tg-scaled Arrheniusplot for the relaxation time τ for the same pairs of ǫ and Eb asin (a), while the lower inset to (b) depicts ǫ-dependence of Tg

along the iso-fragility line for mP = 100. Squares, circles, dia-monds and pentagons designate the positions of characteristictemperatures TA, TI , Tg and To, respectively.

lapse onto a single curve when they are plotted as a func-tion of Tg/T , a result that agrees with experiments [12]and is just a consequence of iso-fragility lines. It is alsoapparent that the relaxation times at the characteristictemperature TA, TI or Tg remain nearly constant alongthe iso-fragility lines. By contrast, Figure 9 reveals thatthe scaling behavior is absent in the temperature depen-dence of the entropy density and the relaxation timesalong the iso-Tg lines in accord with experimental datafor the structural relaxation times. For instance, poly-mers with similar Tg may have very different fragilities,resulting in different dependences of the relaxation timeon Tg/T . Such an example can be provided by comparingthe experimental data for the structural relaxation timesof PVC and poly(3-chlorostyrene) (P3ClS) [12]. Figure9a exhibits the noticeable elevation of the entropy den-sity at TA, TI or Tg with increasing ǫ along the iso-Tg

lines.

8

0.0

0.1

0.2

0.3

0.4

0.5

100 200 300 400 500 600 700

s/kB

T

(a)

ǫ,K Eb,K

Tg = 250 K

-15

-10

-5

0

0.4 0.6 0.8 1.0

log(τ)

Tg/T

(b)

60

90

120

150

180

150 175 200 225 250

mP

ǫ

150 690200 500250 390

FIG. 9. (a) The entropy density s/kB as a function of T forseveral pairs of ǫ and Eb that produce the same glass transi-tion temperature of Tg = 250 K. (b) Tg-scaled Arrhenius plotfor the relaxation time τ for the same pairs of ǫ and Eb as in(a). The inset to (b) depicts ǫ-dependence of mP along theiso-Tg line for Tg = 250 K. Squares, circles, diamonds andpentagons designate the positions of characteristic tempera-tures TA, TI , Tg and To, respectively.

Figure 10 further reveals that the polymer volume frac-tion φ along the iso-fragility lines becomes a unique func-tion of Tg/T and that the volume fraction at each charac-teristic temperature is independent of ǫ and Eb. Again,such scaling is absent from the EOS along the iso-Tg lines(data not shown). Instead, our computations indicatethat the polymer volume fraction at each characteristictemperature increases with ǫ along the iso-Tg lines (seeFigure S2 in the Supporting Information), a trend thatcan be explained by the negative correlation between ǫand Eb along the iso-Tg lines. The temperature depen-dence of the polymer volume fraction, of course, changessignificantly along both iso-fragility (see the inset to Fig-ure 10) and iso-Tg lines due to the variations in the co-hesive energy.

Figures 8 and 9 exhibit the characteristic temperaturesas depending differently on ǫ along the iso-fragility andiso-Tg lines, respectively. The general increase of Tα withǫ along iso-fragility lines arises (Figure 9a) because the

0.5

0.6

0.7

0.8

0.9

1.0

0.4 0.6 0.8 1.0 1.2

φ

Tg/T

ǫ,K Eb,K

mP = 100

0.5

0.6

0.7

0.8

0.9

1.0

100 200 300 400 500 600 700

φ

T150 340

200 450

250 565

FIG. 10. Tg-scaled Arrhenius plot of the polymer volumefraction φ for several pairs of ǫ and Eb that produce the sameisobaric fragility parameter of mP = 100. The inset depictsT -dependence of φ. Squares, circles, diamonds and pentagonsdesignate the positions of characteristic temperatures TA, TI ,Tg and To, respectively.

bending energy Eb increases with ǫ along the iso-fragilitylines. The simultaneous increase of ǫ and Eb inevitablyleads to the elevation of all characteristic temperatures(see Figure 2). Moreover, all characteristic temperaturesgrow linearly with ǫ along the iso-fragility lines withinthe parameter range investigated (see the lower inset toFigure 8b and more results in Figure S3 in the Support-ing Information). The dependence of the characteristictemperatures on ǫ appears to be complicated along theiso-Tg lines. For example, TA and TI are found to increasewith ǫ (Figures S4a and S4b in the Supporting Informa-tion), while To undergoes a slight drop with ǫ (FigureS4c in the Supporting Information). On the other hand,the fragility parameter mP monotonically diminishes asa function of ǫ along the iso-Tg lines (see the inset to Fig-ure 9b and more results in Figure S4d), a trend explainedby the GET since Eb decreases with ǫ along the iso-Tg

lines and since mP diminishes as either ǫ increases or Eb

decreases.

D. Implications of the influence of cohesive energy

and chain stiffness on polymer glass formation

Although the general variations along the iso-fragilityand iso-Tg lines of all the properties considered in Sub-section III C can be explained by analyzing the com-bined influence of the cohesive and bending energies, thescaling properties displayed by the entropy density andby the EOS along the iso-fragility lines are first derivedhere from the GET. Those remarkable scaling proper-ties along the iso-fragility lines and their absence alongthe iso-Tg lines support the well-known contention thatfragility provides more fundamental insight into glass for-mation than the somewhat arbitrarily defined glass tran-

9

sition temperature Tg. For instance, polymer fragilityhas been correlated with a variety of properties of glassformation [42–47]. By contrast, Tg is often defined em-pirically and depends on the cooling rate for some typesof experiments. Nevertheless, Tg is still of great impor-tance in characterizing glass formation because certainproperties exhibit special features around Tg. Moreover,Tg is a crucial parameter governing practical applicationsof glassy materials since Tg signals the presence of dras-tic changes in the mechanical and rheological propertiesof the materials [37]. Our calculations suggest that con-trolling the cohesive energy and chain stiffness enablesfinely tailoring the fragility and the glass transition tem-perature of glass-forming polymers and hence providesguidance towards the rational design of polymer materi-als.

IV. SUMMARY

We examine the influence of cohesive energy and chainstiffness on polymer glass formation using the general-ized entropy theory in conjunction with a minimal modelof polymer melts with monomer averaged interactions,where a single nearest neighbor van der Waals energy isemployed to describe the interactions between all pairsof nearest neighbor united atom groups. Polymers withrigid structures and weak nearest neighbor interactionsare demonstrated as being the most fragile, while theglass transition temperature Tg becomes greater when

the chains are stiffer and/or the nearest neighbor inter-actions are stronger. We find two simple algebraic ex-pressions for describing the calculated dependence of theisobaric fragility parameter mP and Tg on ǫ and Eb.The strong influence of the cohesive and bending en-

ergies naturally inspires the introduction of some impor-tant design concepts, such as iso-fragility and iso-Tg lines.Analysis of relevant properties along these special linesprovides further evidence that fragility plays a more fun-damental role in the description of glass formation thanTg. The present work clearly implies that controlling thecohesive energy and chain stiffness enables finely tailor-ing the fragility and the glass transition temperature ofglass-forming polymers and hence provides an efficientroute for guiding the rational design of polymer materi-als. Finally, we note that the results presented here willbe compared with those from more detailed models ofpolymer melts that have specific interactions for partic-ular united atom groups to study and extend the limitsof validity of the minimal model of melts with monomeraveraged interactions.

ACKNOWLEDGMENTS

We thank Jack Douglas for helpful discussions and acritical reading of the manuscript. This work is sup-ported by the U.S. Department of Energy, Office of BasicEnergy Sciences, Division of Materials Sciences and En-gineering under Award de-sc0008631.

[1] L. Berthier and G. Biroli, Rev. Mod. Phys. 83, 587(2011).

[2] M. D. Ediger and P. Harrowell, J. Chem. Phys. 137,080901 (2012).

[3] G. Biroli and J. P. Garrahan, J. Chem. Phys. 138,12A301 (2013).

[4] F. H. Stillinger and P. G. Debenedetti, Annu. Rev. Con-dens. Matter Phys. 4, 263 (2013).

[5] G. Floudas, M. Paluch, A. Grzybowski, and K. L. Ngai,Molecular Dynamics of Glass-Forming Systems: Effects

of Pressure (Springer-Verlag, Berlin, 2010).[6] C. M. Roland, S. Hensel-Bielowka, M. Paluch, and

R. Casalini, Rep. Prog. Phys. 68, 1405 (2005).[7] C. M. Roland, Macromolecules 43, 7875 (2010).[8] D. Cangialosi, J. Phys.: Condens. Matter 26, 153101

(2014).[9] J. Dudowicz, K. F. Freed, and J. F. Douglas, J. Phys.

Chem. B 109, 21285 (2005).[10] J. Dudowicz, K. F. Freed, and J. F. Douglas, J. Phys.

Chem. B 109, 21350 (2005).[11] K. L. Ngai and C. M. Roland, Macromolecules 26, 6824

(1993).[12] A. L. Agapov, Y. Wang, K. Kunal, C. G. Robertson, and

A. P. Sokolov, Macromolecules 45, 8430 (2012).[13] P. G. Santangelo and C. M. Roland, Macromolecules 31,

4581 (1998).[14] C. M. Roland and R. Casalini, J. Chem. Phys. 122,

134505 (2005).[15] J.-L. Barrat, J. Baschnagel, and A. Lyulin, Soft Matter

6, 3430 (2010).[16] K. Kunal, C. G. Robertson, S. Pawlus, S. F. Hahn, and

A. P. Sokolov, Macromolecules 41, 7232 (2008).[17] R. Kumar, M. Goswami, B. G. Sumpter, V. N. Novikov,

and A. P. Sokolov, Phys. Chem. Chem. Phys. 15, 4604(2013).

[18] R. A. Riggleman, K. Yoshimoto, J. F. Douglas, and J. J.de Pablo, Phys. Rev. Lett. 97, 045502 (2006).

[19] A. Shavit and R. A. Riggleman, Macromolecules 46, 5044(2013).

[20] W. Xia and S. Keten, Langmuir 29, 12730 (2013).[21] S.-J. Xie, H.-J. Qian, and Z.-Y. Lu, J. Chem. Phys. 140,

044901 (2014).[22] J. Dudowicz, K. F. Freed, and J. F. Douglas, Adv. Chem.

Phys. 137, 125 (2008).[23] G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965).[24] K. W. Foreman and K. F. Freed, Adv. Chem. Phys. 103,

335 (1998).[25] J. Dudowicz, K. F. Freed, and J. F. Douglas, J. Chem.

Phys. 123, 111102 (2005).[26] J. Dudowicz, K. F. Freed, and J. F. Douglas, J. Chem.

Phys. 124, 064901 (2006).[27] J. F. Douglas, J. Dudowicz, and K. F. Freed, J. Chem.

Phys. 125, 144907 (2006).[28] E. B. Stukalin, J. F. Douglas, and K. F. Freed, J. Chem.

10

Phys. 131, 114905 (2009).[29] K. F. Freed, Acc. Chem. Res. 44, 194 (2011).[30] W.-S. Xu and K. F. Freed, J. Chem. Phys. 138, 234501

(2013).[31] W.-S. Xu and K. F. Freed, J. Chem. Phys. 141, 044909

(2014).[32] J. H. Gibbs and E. A. DiMarzio, J. Chem. Phys. 28, 373

(1958).[33] L.-M. Martinez and C. A. Angell, Nature (London) 410,

663 (2001).[34] J. Dudowicz and K. F. Freed, Macromolecules 24, 5076

(1991).[35] K. F. Freed, J. Chem. Phys. 119, 5730 (2003).[36] J. Dudowicz, personal communication.[37] C. A. Angell, Science 267, 1924 (1995).[38] V. N. Novikov and A. P. Sokolov, Phys. Rev. E 67,

031507 (2003).[39] I. M. Hodge, J. Non-Cryst. Solids 202, 164 (1996).[40] E. Rossler, K.-U. Hess, and V. N. Novikov, J. Non-Cryst.

Solids 223, 207 (1998).[41] C. A. Angell, J. Non-Cryst. Solids 131-133, 13 (1991).[42] D. Huang and G. B. McKenna, J. Chem. Phys. 114, 5621

(2001).[43] R. Boehmer, K. L. Ngai, C. A. Angell, and D. J. Plazek,

J. Chem. Phys. 99, 4201 (1993).[44] D. J. Plazek and K. L. Ngai, Macromolecules 24, 1222

(1991).[45] I. M. Hodge, Science 267, 1945 (1995).[46] A. P. Sokolov, E. Rossler, A. Kisliuk, and D. Quitmann,

Phys. Rew. Lett. 71, 2062 (1993).[47] A. P. Sokolov, Science 273, 1675 (1996).

1

Supporting Information

S1 Fitting results for mP (ǫ, Eb) and Tg(ǫ, Eb)—In the main text, we propose two simple algebraic equationsthat fairly accurately capture the computed combined variations of the isobaric fragility parameter mP and theglass transition temperature Tg with the microscopic cohesive ǫ and bending Eb energies. Figure S1 displays theEb-dependence of mP and Tg for various ǫ, along with our best fits obtained from eqs 4 and 5 in the main text. Thefitting parameters are summarized in the caption of Figure S1.S2 More properties along the iso-fragility and iso-Tg lines—Figure 10 in the main text reveals that the

polymer volume fraction φ along the iso-fragility lines becomes a unique function of Tg/T and that the volume fractionat each characteristic temperature is independent of ǫ and Eb. By contrast, this behavior is not observed along theiso-Tg lines. Figure S2 indicates that the polymer volume fraction at each characteristic temperature increases withǫ along the iso-Tg lines, a trend that can be explained by the negative correlation between ǫ and Eb along the iso-Tg

50

100

150

200

250

400 800 1200 1600 2000

mP

Eb

(a)

150

200

250

300

350

400

450

500

550

400 800 1200 1600 2000

Tg

Eb

(b)

ǫ = 150 Kǫ = 175 Kǫ = 200 Kǫ = 225 Kǫ = 250 K

FIG. S1. (a) The isobaric fragility parameter mP and (b) the glass transition temperature Tg as a function of the bendingenergy Eb for various cohesive energies ǫ. Solid lines in (a) and dotted lines in (a) are fits according to eqs 4 and 5 in the maintext with the fitting parameters a0 = 1.23154×102 , a1 = −1.17539, a2 = 2.23509×10−2 , b0 = 0.606943, b1 = −1.07595×10−3 ,c0 = 1.49847 × 10−3, c1 = −5.60474 × 10−7, c2 = −4.78656 × 10−9 and u0 = 37.8879, u1 = 50325, u2 = −1.0445 × 107,v0 = 1.80486, v1 = −609.037, v2 = 8.08886 × 104, w0 = 4.7747 × 10−3, w1 = −2.09936, w2 = 282.37. As in the main text,the following parameters are used here and in the subsequent figures: the lattice coordination number is z = 6, the pressure isP = 1 atm, the cell volume parameter is acell = 2.7A, and the polymerization index is Nc = 8000.

0.40

0.45

0.50

0.55

0.60

0.65

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150 175 200 225 250

φTA

ǫ

(a)0.70

0.75

0.80

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0.95

150 175 200 225 250

φTI

ǫ

(b)

0.75

0.80

0.85

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0.95

1.00

150 175 200 225 250

φTg

ǫ

(c)0.80

0.85

0.90

0.95

1.00

150 175 200 225 250

φTo

ǫ

(d)

Tg = 250 KTg = 275 KTg = 300 KTg = 325 KTg = 350 K

FIG. S2. The volume fractions at different characteristic temperatures φTα as a function of ǫ along selected iso-Tg lines withthe indicated values of Tg. (a) φTA . (b) φTI . (c) φTg . (d) φTo .

2

300

400

500

600

700

800

150 175 200 225 250TA

ǫ

(a)

200

300

400

500

600

150 175 200 225 250

TI

ǫ

(b)

100

200

300

400

500

150 175 200 225 250

Tg

ǫ

(c)

100

200

300

400

500

150 175 200 225 250

To

ǫ

(d)

mP = 100mP = 125mP = 150mP = 175mP = 200

FIG. S3. Characteristic temperatures as a function of ǫ along selected iso-fragility lines with the indicated values of mP . (a)TA. (b) TI . (c) Tg. (d) To.

400

450

500

550

600

650

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150 175 200 225 250

TA

ǫ

(a)250

300

350

400

450

150 175 200 225 250

TI

ǫ

(b)

150

200

250

300

350

150 175 200 225 250

To

ǫ

(c)50

100

150

200

250

150 175 200 225 250

mP

ǫ

(d)

Tg = 250 KTg = 275 KTg = 300 KTg = 325 KTg = 350 K

FIG. S4. Characteristic temperatures and fragility parameter as a function of ǫ along selected iso-Tg lines with the indicatedvalues of Tg. (a) TA. (b) TI . (c) Tg. (d) mP .

lines (see Figure 6b in the main text).The lower inset to Figure 8b in the main text reveals that Tg grows linearly with ǫ along an iso-fragility line for

mP = 100. Figure S3 indicates that the linear relationship also holds for other characteristic temperatures and othermP .Figure S4 displays the ǫ-dependence of the characteristic temperatures and fragility parameter along the iso-Tg lines

with representative Tg. Both TA and TI increase with ǫ (Figures S4a and S4b), while To undergoes a slight drop withǫ (Figure S4c). On the other hand, the fragility parameter mP monotonically diminishes as a function of ǫ along theiso-Tg lines (Figure S4d), a trend explained in the main text since Eb decreases with ǫ along the iso-Tg lines (Figure6b in the main text) and since mP diminishes as either ǫ increases or Eb decreases (Figure 4 in the main text).