multiple glass transitions and local composition effects ... · multiple glass transitions and...

Multiple Glass Transitions and Local Composition Effects onPolymer Solvent Mixtures

JANE E. G. LIPSON,1 SCOTT T. MILNER2

1Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755

2ExxonMobil Research and Engineering, Annandale, New Jersey 08801

Received 27May 2006; accepted 8 September 2006DOI: 10.1002/polb.21023Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: Recent differential scanning calorimetry (DSC) results on polystyrene–sol-vent mixtures show two distinct glass transitions whose positions and widths vary withcomposition. Parallel work on the dynamic response in polymer blends has focused onhow segmental mobilities are controlled by local composition variations within a ‘‘cooper-ative volume’’ containing the segment. Such variations arise from both chain connectivityand composition fluctuations. We account for both using a lattice model for polymer–sol-vent mixtures that yields the composition distribution around polymer and solvent seg-ments. Insights from our lattice model lead us to simplified calculations of the mean andvariance of local composition, both in good agreement with lattice results. Applying ourmodel to compute DSC traces leads to an estimate of the cooperative volume, since alarger cooperative volume both reduces the biasing effect of connectivity, and narrows thecomposition distribution. Comparing our results to data, we are able to account for thecomposition-dependent broadening with a cooperative length scale of about 2.5 nm.VVC 2006Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3528–3545, 2006

Keywords: calorimetry; glass transition; solution properties; statistical mechanics;theory

INTRODUCTION

Discussions of the glass transition of a polymer–solvent mixture typically assume the existence of asingle glass transition temperature (Tg). Indeed, itis a traditional rule of thumb that samples exhibit-ing two Tg’s are phase-separated. Yet polymer–sol-vent mixtures can often involve two componentswith widely separated glass transitions, implyingvery different segmental mobilities at any giventemperature in this range. In addition, for suchmixtures one expects that the average local compo-sition around a segment will differ from the bulk.For the polymeric component there will be a localenrichment in self-composition because of chain

connectivity. The impact of this effect on the glasstransitions for a polymer blend was described sometime ago by Lau et al.1 and Chung et al.,2 andquantified by Lodge and McLeish.3 Their argu-ments would evidently apply as well to the poly-meric component in a polymer–solvent mixture.

For the small molecule component in a polymer–solvent mixture one can also argue for a shift ineffective local composition relative to the bulk. If theenvironment around a polymeric segment has beenenriched in polymer, it implies that the material stillavailable to fill the rest of the cooperative volumearound a segment of either type must be depleted inpolymer. Thus the averaged local environmentaround segments of both types should be shifted rel-ative to the bulk—but shifted in opposite directions.Both the local compositional difference as well asthe aforementioned difference in component mobili-ties serve to accentuate the distinction between thetwo components, and support the notion that calori-

Correspondence to: J. E. G. Lipson (E-mail: [email protected])

Journal of Polymer Science: Part B: Polymer Physics, Vol. 44, 3528–3545 (2006)VVC 2006Wiley Periodicals, Inc.

3528

metric measurements on polymeric mixtures shouldexhibit two glass transitions, reflecting two distinctlocal ‘‘points of view.’’

In principle, therefore, at any given mixturecomposition one might expect two Tg’s; however,only one is typically reported. For this reason arecent study by Savin et al.4 represents both an op-portunity and a challenge. In their work Savinet al. presented and analyzed results on mixturesacross the composition range of polystyrene (PS)with each of three substituted phthalates. Focus-ing, as did the authors, on PS/dibutylphthalate(DBP), we reproduce in Figure 1 their differentialscanning calorimetry (DSC) traces for each of thepure components and for four mixture composi-tions. Note that two transitions are identified forthe 30 wt % PS and 60 wt % PS compositions. Inaddition to the appearance of multiple transitions,we observe that the location of the transitions isclearly a function of composition for each compo-nent, and also that the DSC traces for each compo-nent in the mixture are broadened relative to thepure component trace.

We have already discussed the origin of a localcompositional average that is shifted relative tothe bulk composition. Furthermore, note that useof such an average serves as a stand-in for the fullcomposition distribution. Aside from the spacethat is occupied by segments that are covalentlybonded to a central segment, we take the rest ofthe cooperative volume around the center to befilled randomly, reflecting the relative amounts ofeach species available. A distribution of local com-positions implies a distribution of glass transi-tions, and this is the origin of increased broaden-ing in the glass transitions for mixtures, relative

to the transitions in pure components. Such amechanism was proposed some time ago by Zet-sche and Fischer.5

In this paper we propose a model that connectsthe broadening of the DSC results in the mixtureto composition fluctuations around a polymer seg-ment or a solvent molecule. Using a lattice ap-proach, we show that by accounting for polymerconnectivity and its effect on filling a specifiedlocal cooperative volume, the composition-depend-ent broadening can be predicted. We then distillthese concepts and propose a route to the contin-uum analogue of the distribution of compositionfluctuations, such that the breadth and positioningof the distribution reflect both the nature of thecomponents and the dimensions of the cooperativevolume.

In addition to the broadening of the DSC mix-ture traces, we are also interested in the composi-tion-dependent positions of the transitions. A nat-ural place to start our analysis will be the Lodge–McLeish (LM) model,3 the salient points of whichare briefly reviewed here. Before proceeding, how-ever, one might ask, Why not simply apply the LMmodel, modified to account for one componentbeing a solvent, to the problem at hand?

Chief among the reasons for developing a newmodel is that the LM approach does not make pre-dictions about the local composition distributionaround a segment. It does predict the shift in theaveraged local composition due to connectivity, butthat alone would yield predictions for the mixtureDSC traces which would be no broader than thepure component results.

Predicting the distribution of local compositionsinvolves filling a ‘‘cooperative volume’’ around agiven segment of interest, the dimensions of whichshould be on the order of a Kuhn length. In fact,the size of such a volume is an issue of current dis-cussion. The analysis in the literature has cen-tered around polymer blends and has focused onspecifics relevant to NMR and dielectric relaxationdata. Questions have been raised regarding thetemperature dependence of this volume as theglass transition is approached. Some groups havelooked at temperature dependence and by variousroutes have concluded that the cooperative lengthscale might increase by as much as a factor of 3 ingoing from well above Tg to near the glass transi-tion. References 6 and 7 provide a good overview ofthis area.

It seems reasonable to take the magnitude ofthe cooperative volume to be intermediate, that istoo large to be associated with the typical dimen-

Figure 1. DSC data from ref. 3. Traces shown are forPS-DBP mixtures with 0, 19, 39, 60, 80, and 100 wt %PS. Bars indicate the transitions as assigned by theauthors. (Vertical offset is arbitrary.)

COMPOSITION EFFECTS IN POLYMER SOLVENT MIXTURES 3529

Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb

sions of a chemical repeat unit and likely too smallto be represented as Gaussian strands. A latticemodel is well suited to describe what might be hap-pening on such a length scale. It also has theadvantage of being easy to visualize, as well asallowing for analytic counting of configurations.Although the LM model does not incorporate allthe features we appear to need, it does provide animportant starting point in the consideration ofhow the mixture glass transitions depend on theaveraged local composition. We therefore give abrief summary of the essentials here.

Lodge and McLeish describe a blend of two poly-mers A and B, which may in general have differentKuhn lengths and packing lengths. They considerthe concentration within a cooperative volume VC

of order the cube of the Kuhn length lK, surround-ing a particular segment of polymer A. Theyobserve that because of connectivity, the chain con-taining the particular segment will make a contri-bution /self to the volume fraction of Awithin VC.

They estimate /self ¼ O/l3K, where O is the dis-placed volume of a polymer strand having an end-to-end length lK. Noting that the ratio O/lK

2 scalesas the packing length lp, we see that /self scales aslp/lK. (The numerical coefficient depends upon theprecise choice of VC and the definition of lp.)

Having so estimated /self, LM then computes theeffective volume fraction /eff within the volume VC

by assuming that the fraction of VC not filled by theparticular strand of interest is filled with materialwith the bulk volume fraction / of A. That is,

/eff ¼ /self þ ð1� /self Þ/ ð1ÞEach component in the mixture will have a differ-ent /self, hence a different /eff corresponding to agiven bulk composition. The characterization ofenvironments that are specific to segments of thetwo individual components implies the appearanceof two glass transitions as follows:

1

Teffg;Að/Þ

¼ /Aeff

Tpureg;A

þ 1� /Aeff

Tpureg;B

ð2Þ

Note that if a component has /self ¼ 0, then /eff

¼ / and the cooperative volume around that segmentwill be filled using the bulk volume fractions. In thatcase eq 2 reverts to the familiar Fox equation8

1

Tgð/Þ ¼/

Tg;Aþ 1� /

Tg;Bð3Þ

The LM model is not bound to the Fox equation;other equations that provide a mixing rule for thecomposition-dependence of Tg could be used. How-

ever, the Fox equation has the advantage of requir-ing no adjustable parameters, and since the origi-nal LM paper made use of eq 3 we apply it here aswell.

Our paper is arranged as follows. UnderDescription of Model, the lattice model is describedin detail. Under Phenomenological Fits, we discussfitting of the pure component DSC traces and char-acterization of the composition dependence of theexperimentally recorded glass transitions. The sec-tion Comparison with Experimental Results com-prises our applied results, wherein we compare ourpredictions for PS/DBP mixture traces with the ex-perimental measurements, focusing on the impactof composition fluctuations on the breadth of thetransition. We also consider the composition de-pendence of the mixture transitions, returning tothe LM model and the Fox equation.8 Under thesections Self-Consistency and Lodge–McLeish andAlternative Estimate of Variance, we apply in-sights achieved from our lattice studies revisit theLodge–McLeish model and to develop an alterna-tive route for quantifying the effect of compositionfluctuations, one which bypasses the lattice calcu-lation of a composition histogram.

Under the last section, which is a summary anddiscussion of the work, we broach some of thelarger issues that have arisen in the course of ourstudies, including those related to how /self isapplied and understood, how our self-consistencyarguments affect application of the LM model, andwhat our results imply about the cooperative vol-ume at the glass transition. We conclude withsome outstanding questions that we hope thiswork will prompt future studies to address.

RESULTS

Description of Model

To predict the effect of composition fluctuations onthe breadth of the DSC transition, we model thesolvent (A)–polymer (B) system on a Bethe lattice,or tree. Some advantages of using a tree latticeinstead of an embeddable lattice, such as the sim-ple cubic, are that every site has the same numberof nearest neighbors (z) and no closed-loop configu-rations are possible. Figure 2 illustrates a z ¼ 3tree out to the third generation or ‘‘shell,’’ wherewe label the central site as the zeroth generation.Given a central site that is occupied by a segmentof polymer, two of the sites in each subsequent gen-eration must be filled with polymer connected tothe central segment; this section of polymer is rep-

3530 LIPSON AND MILNER


resented by the black segments in the figure. Forsimplicity, we choose the size of the lattice sites sothat a solvent molecule occupies one lattice site.

We begin with the filling of unconstrained sitesin the first shell. Let q be the probability of findinga solvent molecule (A) next to the polymer segment(B) at the center. The probability of finding anunconstrained B segment in such a site (that is,one that is not bonded to the B at the center) istherefore equal to 1 � q. There are z � 2 sitesavailable for filling and so the probability that thefirst shell contains i segments of B, two of whichare connected to the central B segment, is given by

pBðiÞ ¼ z� 2i� 2

� �ð1� qÞi�2qz�i ð4Þ

The factor in front represents the binomial coeffi-cient. Here and in what follows we define the bino-

mial coefficientsab

� �such that they vanish when-

ever the variables take unphysical values, such asb > a or b < 0, and so forth. (Otherwise, we wouldhave to take precautions when summing the proba-bilities over the range of values for the different indi-ces, an issue that will become important later on.)

Filling of the second shell involves considerationof how the first shell is filled, given that for eachunconstrained B segment in the first shell theremust be two constrained B segments in the secondshell. In addition, two of the second shell sites are

filled with B segments which are connectedthrough the first shell to the B in the zeroth shell.Thus the second shell contains 2i � 2 B segmentsthat are constrained to be there. If we label thetotal number of B segments in the second shell asj, then the number k of unconstrained B’s—that is,not connected to any B previously placed—whichare next to a first shell B segment may have anyvalue between 0 and j � (2i� 2).

To determine the number of second shell sitesopen for filling, consider that each of the z firstshell sites itself has z neighbors, z � 1 of which arein the second shell. However, each of the i � 2unconstrained B segments in the first shell mustbe connected to two constrained B’s in the secondshell, and therefore has only z � 3 open neighborsin the second shell available for filling. In addition,the two constrained B’s in the first shell are eachconnected to a constrained B in the second shell,and so each of these two first shell sites has onlyz � 2 open neighbors. Thus, the number of secondshell sites that are neighbors to B-containing firstshell sites is (i � 2)(z � 3) þ 2(z � 2). In eq 5 below,the first binomial coefficient accounts for fillingthis set of open sites with the k unconstrained Bsegments that neighbor a first shell B, and theprobability factor associated with this is (1 � q)k.

To determine the number of open sites remain-ing in the second shell, we note that each of theremaining z� i sites in the first shell has z� 1 sec-ond shell neighbors. Into this set we have yet toplace the remaining B segments, that is, the j � (2i� 2) � k unconstrained B’s that are neighbors to afirst-shell A. The number of arrangements is givenby the second binomial coefficient in eq 5, and theprobability of finding this set of B’s so placed isgiven by the factor of pj�(2i�2)�k. Here p is the prob-ability of finding an unconstrained B segment nextto an A segment.

Into the second shell sites that remain we placeA segments. Recall that of the (z � 3)(i � 2) þ 2(z� 2) open sites next to a first shell B,m are alreadytaken by unconstrained B segments. Therefore theprobability of finding A segments serving as neigh-bors of this set of first-shell B’s is given by the fac-tor q(z � 3)(i � 2) þ 2(z � 2)�k. Finally, there are (z � 1)(z� i) – (j � (2i� 2)� k) neighbors to first-shell A’swhich are not already filled by unconstrained B’s.The probability that these will be filled by A’s isgiven by the factor (1 � p)(z � 1)(z � i) – (j�(2i � 2)�k).Such an accounting for the assignment of con-strained and unconstrained sites is summarized inTable 1.

Putting all of this together we obtain the follow-ing expression for pB(i, j), the probability of finding

Figure 2. Schematic of a regular tree lattice (herez¼ 3 for illustrativepurposes).Dashed circles indicatefirst,second, and third ‘‘shells’’ of lattice sites. Thick line indi-cates polymer configuration passing through central site.



i B segments in the first shell and j B segments inthe second shell of a tree, the zeroth shell of whichis occupied by a B segment

pBði; jÞ ¼ pBðiÞXj�ð2i�2Þ

k¼0

ðz� 3Þiþ 2

k

� �

� ðz� 1Þðz� iÞj� ð2i� 2Þ � k

� �� ð1� qÞkpj�ð2i�2Þ�k

� qðz�3Þiþ2�kð1� pÞðz�1Þðz�iÞ�ðj�ð2i�2Þ�kÞ ð5ÞMutatis mutandis, we extend the reasoning to thethird shell and, again for the case of a B segmentat the center, we find the probability of finding i, j,and k B segments in the first, second, and thirdshells, respectively, to be

pBði; j; kÞ ¼ pBði; jÞXk�ð2j�2iþ2Þ

l¼0

ðz� 3Þjþ ð2i� 2Þl

� �

� ðz� 1Þðzðz� 1Þ � jÞk� ð2j� 2iþ 2Þ � l

� �

� ð1� qÞlqðz�3Þjþð2i�2Þ�lpk�ð2j�2iþ2Þ�l

� ð1� pÞðz�1Þðzðz�1Þ�jÞ�ðk�ð2j�2iþ2Þ�lÞ ð6ÞSo far we have given the expressions relevant tothe case of a B segment in the zeroth shell. Analo-gous results apply when the zeroth shell containsan A segment, although it is important to notethat the two sets of expressions are not symmetricwith respect to switching A and B labels because Bsegments are connected to other B’s, while A seg-ments represent individual solvent molecules. Theresulting expressions are

pAðiÞ ¼ zi

� �ð1� qÞiqz�i ð7Þ

pAði; jÞ ¼ pAðiÞXj�2i

k¼0

ðz� 3Þik

� � ðz� 1Þðz� iÞj� 2i� k

� �

� ð1� qÞkpj�2i�kqðz�3Þi�k

� ð1� pÞðz�1Þðz�iÞ�ðj�2i�kÞ ð8Þ

pAði; j; kÞ ¼ pAði; jÞXk�ð2j�2iÞ

l¼0

ðz� 3Þjþ 2i

l

� �

� ðz� 1Þðzðz� 1Þ � jÞk� ð2j� 2iÞ � l

� �

� ð1� qÞlqðz�3Þjþ2i�lpk�ð2j�2iÞ�l

� ð1� pÞðz�1Þðzðz�1Þ�jÞ�ðk�ð2j�2iÞ�lÞ ð9Þ

There are still several steps required to translatefrom the probability results summarized in eqs 4–9to a histogram representing a concentration distri-bution relevant for a given choice of the bulk com-position. The result we need is the probability,given a B at the center of the tree, of finding a totalof s B’s in n shells (n ¼ 1, 2, or 3), regardless of howthey are distributed; we shall also need the analo-gous result for the case of an A at the center. Theseexpressions are obtained by summing the relevantexpression for the probability associated with thenth shell over the set of filling possibilities whichare consistent with a total number of B (or A) seg-ments equal to s. (Recall that we defined the bino-mial coefficients so that unphysical values of theindices lead to coefficient values of 0, allowing us toavoid thought-provoking constraints on the sums.)

We denote the two histogram functions ashA,n(s) when the zeroth shell is occupied by an A,and as hB,n(s) when it is occupied by a B segment.The second and third shell expressions with an Aat the center are given by

hA;2ðsÞ ¼Xzðz�1Þ

j¼0

pAðs� j; jÞ

hA;3ðsÞ ¼Xzðz�1Þ

j¼0

Xzðz�1Þ2

k¼0

pAðs� j� k; j; kÞ ð10Þ

with corresponding results for hB,2(s) and hB,3(s).From such histograms, one can compute what in

the LM treatment would be the effective volume frac-

Table 1. Summary of Counting Results Used in writing Eqs 4–9

A at Center B at Center

Second Shell Third Shell Second Shell Third Shell

Constrained B sites 2i 2j � 2i 2i� 2 2j� 2iþ 2Unconstrained sitesTotal z(z � 1)� 2i z(z � 1)2 � (2j � 2i) z(z � 1)� 2iþ 2 z(z � 1)2 � (2j � 2iþ 2)Neighbors of A (z � 1)(z� i) (z� 1)(z(z � 1)� j) (z� 1)(z � i) (z � 1)(z(z � 1)�j)Neighbors of B (z � 3)i (z � 3)jþ 2i (z � 3)iþ 2 (z� 3)j þ 2i � 2



tion /eff,A which here is simply the average sitefraction of A in the cooperative volume with A atthe center (and likewise for /eff,B). We write

/ðnÞeff ;A ¼ 1�

XSn

s¼0

ðs=SnÞhA;nðsÞ

/ðnÞeff ;B ¼

XSn

s¼0

ðs=SnÞhB;nðsÞ ð11Þ

Here n is the number of shells in the cooperativevolume, and Sn is the total number of sites in thefirst through nth shells. For our regular tree latticeS1 ¼ z, S2 ¼ zþ z(z� 1), and S3 ¼ z þ z(z � 1)þ z(z� 1)2.

In fact, before we can generate the numericalresults for such a histogram we need to choose val-ues for z, p, and q. We simplify the problem byobserving that, since there is no special interactionbetween Amolecules and B segments, the probabil-ity q of finding an A next to a B segment should beequal to (1� p), the probability of finding an A nextto another A. Similarly, the probability (1 � q) offinding an unconstrained B next to a B segment isequal to p, the probability of finding a B next to anA. Thus only one choice, say that of p, is required.

Naively, one might think that p should be setequal to /, the bulk volume fraction of solvent; thatis that any unconstrained site in our lattice shouldbe filled from a reservoir of bulk material. How-ever, because of the connectivity of the polymer asopposed to the solvent, when an unconstrained sitein our lattice is filled with a B segment, it forcesfurther sites in subsequent shells to also be filledwith B segments. As a result, if we were to take p¼ /, we would end up with too many B segmentson the average in our configurations.

To make this more precise, we require a normal-ization condition, which may be used to determinethe proper choice of p(/). The condition is simplythat the average number of B segments in any shell,averaged as well over what type of segment fills thecentral site, must equal 1 � / (the bulk site fractionof B segments). The reason for this is that if we aver-age over the filling of the central site, there is noth-ing special about the sites in any shell—they arejust representative sites in the system.

It is not obvious that a single choice of p(/) willserve to satisfy the sequence of above-mentionedequations. However, explicit calculation showsthat for the first three shells, a single choice of p(/)does indeed work. For general zwe find

pð/Þ ¼ ðz� 2Þð1� /Þz� 2ð1� /Þ ð13Þ

In fact, a single choice for p(/) suffices because wehave chosen a regular tree architecture for our lat-tice. This ensures that every generation removedfrom the central lattice site has the same struc-ture; in fact this is one reason we chose such a lat-tice. The resulting function is displayed in Figure 3for z¼ 6.

With an expression for p(/) in hand, we cannow compute the histograms hn,A(s) and hn,B(s),which give the probability of finding s B segmentsin the first through nth shells, with either an A ora B segment at the center. Figure 4(a,b) displayshistograms hn,A(s) and hn,B(s) respectively, forn ¼ 1, 2, and 3, and the value / ¼ 0.61. We plot thehistograms as a function of s/Sn (recall Sn is thetotal number of sites in the first through nthshells; for z¼ 6, S1 ¼ 6, S2 ¼ 36, and S3 ¼ 186).

Note that the peaks of the distributions areshifted away from 1 � / ¼ 0.39, particularly forsmaller n values, for both A and B segments at thecenter. This shift is a consequence of connectivityconstraints on the filling of the cooperative vol-ume. When B is at the center, the shift is towardhigher fractions of B; when A is at the center, theshift is toward lower B fractions (as we argued inthe Introduction).

We also observe that the histograms narrow asthe number of shells increases. This is expected,since with more independent choices as to the con-

We write this as

S1ð1� /Þ ¼ /Xi

ipAðiÞ þ ð1� /ÞXi

ipBðiÞ

S2ð1� /Þ ¼ /Xi;j

ðiþ jÞpAði; jÞ þ ð1� /Þ

�Xi; j

ðiþ jÞpBði; jÞ ð12Þ

Figure 3. Shown is filling probability function p(/) ofeq 13 (solid curve) with 1� / (dashed line) for reference.



stituents, the fluctuations in the volume fractiondecrease. Within the lattice model, varying thenumber of shells is comparable to varying the sizeof the cooperative volume in a continuum descrip-tion such as the LM model. Similar results wereobtained by Salaniwal et al. for the case of sym-metric polymer blends, studied by lattice MonteCarlo simulations.9

Phenomenological Fits

The DSC traces of Figure 1 clearly exhibit twotransitions at intermediate concentrations, forwhich the sample contains substantial amounts ofboth polymer and solvent. By observing the loca-tions of the two transitions as a function of compo-sition, it seems clear that the transition at lowertemperature is an extension of the glass transitionin the pure solvent, while the transition at higher

temperature is an extension of the glass transitionin the pure polymer.

In principle, we may expect that two transitionsalways contribute to the experimentally observedDSC traces of such polymer–solvent blends. How-ever, depending on the difference between thepure component Tg’s the contributions may over-lay at intermediate compositions. At more extremeconcentrations where the sample consists of most-ly polymer or mostly solvent, only one transition isexpected to be visible. We would like to separatethe task of accounting for the location of these twotransitions from the problem of describing thewidth of the two transitions. To do this we proceedas follows.

We make a phenomenological fit of the DSCtraces to a fitting function, which is the sum of a lin-ear background and either one or two tanh func-tions, each tanh representing one of the two glasstransitions present in the polymer solution. Eachtanh function has an adjustable width, location intemperature, and amplitude. That is, our fittingfunction is

f ðTÞ ¼ Aþ BT þ Cs tanhT � Ts

Ws

� �

þ Cp tanhT � Tp

Wp

� �ð14Þ

where ‘p’ refers to polymer and ‘s’ to solvent; allthe coefficients are adjustable parameters. Thepurpose of this phenomenological fit is to extracttransition temperatures; we therefore fit the DSCtraces to a sum of two tanh functions when twotransitions are clearly evident, that is for the 60%PS data of Figure 1. The corresponding fits areshown in Figure 5.

Figure 5. Phenomenological fits to data of Figure 1,using fitting function eq 14. Vertical bars denote fittedlocations of glass transitions.

Figure 4. Histograms generated from our latticemodel for case of 39% B (polymer), for first shell (trian-gles), second shell (squares), and third shell (diamonds).Solid curves are approximate Gaussian distributionsdescribed in the text. (a) A (solvent) at center. (b) B (poly-mer) at center.



This procedure allows us to extract phenomeno-logical values for the two glass transition tempera-tures Ts(/) and Tp(/), over much of the composi-tion range. The results are shown in Figure 6. Asis qualitatively evident in Figure 5, we see that thesolvent glass transition depends only weakly onvolume fraction, while the polymer glass transitionis strongly dependent on the addition of plasticiz-ing solvent.

To test the theory of the previous section, wefirst regard these transition temperature values asknown, and focus on predicting the widths of theDSC traces. Later, we return to the question ofpredicting the composition dependence of the Tg’s.

At the outset, we chose the lattice size to besuch that a single solvent molecule fills one latticesite. Given the density and molecular weight ofDBP (1.04 g/cm3 and 278.3 g/mol respectively),this implies that the volume of a lattice site is0.443 nm3, corresponding to a cube 0.76 nm on aside. Given the density and molecular weight of PS(1.05 g/cm3 and 104.2 g/mol per monomer), we findthat each lattice site can hold about 2.6 PS mono-mers. The characteristic ratio C? for PS is about9.5, which can be interpreted as the number ofmonomers per Kuhn length. Thus each Kuhn seg-ment of PS corresponds to about 3.7 lattice sites.

Comparison with Experimental Results

As a result of the efforts described in the previoussections we are now able to generate, for any bulkcomposition, distributions using one, two, or threelattice shells. In addition, we have experimental

Tg’s for the polymer for compositions ranging frompure to 60% PS and for DBP ranging from pure to40% DBP. In what follows, we shall need Tg’s forPS and for DBP for putative mixtures dilute inthe respective component of interest. To estimatethese, we extrapolate using our fits to data withinthe experimental range. Finally, we have also char-acterized each of the pure component DSC traces.

To generate DSC predictions, we sum the contri-butions from the trace associated with the PS tran-sition and that associated with the DBP transition,weighting these contributions by the volume frac-tions of PS and DBP, respectively. The trace con-tributed by each component is obtained as follows.The component of interest is taken to occupy thecenter site (zeroth shell) and a histogram of localcompositions is generated. Each composition in thehistogram contributes a pure component trace. Thetransition of that trace is shifted to the glass transi-tion appropriate for the composition and the heightof the transition is weighted by the probabilityassociated with that composition in the distribu-tion. The set of weighted, shifted traces is thensummed to give the overall trace for that compo-nent at that bulk composition. Finally, the overalltraces for each of the two components (weighted bytheir respective bulk compositions) are summed togive the predicted DSC trace to be compared withexperiment. For bulk compositions rich in one com-ponent the impact of the ‘‘dilute’’ trace will be negli-gible, and so only one transition will appear.

To determine the appropriate shifted Tg valuesas a function of the effective composition, we pro-ceed as follows. We use our fits to the extractedvalues of Tg shown in Figure 6 to compute Tg forPS and DBP as a function of the bulk average com-

Figure 6. Fitted Tg’s from Figure 5 plotted versus PSvolume fraction. Squares denote lower transition (asso-ciated with DBP); triangles denote upper transition(associated with PS). Curves are least-squares fits toquadratic (for DBP) and linear (for PS) functions respec-tively.

Figure 7. Lattice model predictions as described intext for DSC traces (solid curves), compared with datafrom ref. 3.



position. We then use the lattice model to computethe effective compositions in the vicinity of poly-mer and solvent as a function of bulk composition,from which Tg as a function of effective composi-tion for polymer and solvent may be inferred.

In Figure 7 we compare our predictions with ex-perimental data for four compositions ranging from19 to 80% PS. The arrows on the figure indicatetransition temperatures predicted using our fits tothe experimental data (Fig. 6). At the lowest compo-sition PS (19%) only the DBP transition (�85 8C) isobserved and the predicted trace has a breadthwhich agrees well with that found experimentally.(Naı̈ve extrapolation of the Tg–/ plot for PS wouldimply a PS transition near�140 8C, which is unrea-sonably low; however, within the calculation thecontribution is so weak as to be undetectable.) Forthe 39% PS sample what we observe is evidently theoverlay of the two transitions, as the temperaturesare predicted to be at about �78 8C, within a fewdegrees of each other. The predicted trace capturesrather effectively both the amplitude and thebreadth of the transition. In the results for 60% PSthe two transitions are clearly separated, that forPS occurring at �12.5 8C, and for DBP at �55.7 8C.This is the blend composition that shows mostclearly the presence of two distinct transitions. Wefind very good agreement between predicted and ex-perimental results, both for the amplitude and thebreadth of each of the transitions. The last traceshows results for 80% PS. Here the PS transition at40.3 8C is much stronger than the rather mutedpeak which, we predict, is associated with a DBPtransition temperature of �26 8C. Once again, thepredictions reproduce both the breadth and the risefound experimentally.

The set of results summarized in Figure 7 pro-vides evidence that our simple lattice model,applied using DSC results for the pure components,leads to predictions for the breadth and height ofthe mixture DSC traces that are in good to excellentagreement with that which is observed experimen-tally. It would therefore seem that the distributionof composition fluctuations predicted using the treelattice is a reasonable approximate to what is beingexperienced by a segment of a given type in the realmixtures. However, there is one aspect to the analy-sis that has not yet been discussed.

Equation 10 shows that we have the flexibility todetermine composition histograms accounting forone, two, or three shells of neighbors around a cen-tral segment. In generating the predictions shownin Figure 7 we obtain the best results by using twoshells for the DBP predictions, and three shells for

the PS predictions. Recall that Figure 4, whichshowed sample histograms associated with one,two, and three shells, illustrated the narrowing ofthe composition distribution as the cooperative vol-ume around the central segment increased. In Fig-ure 8 we show the DSC traces for each of the purecomponents, along with the respective traces formixtures with 80% of each component. Consider theimpact of adding 20% PS to pure DBP: Tg is hardlyshifted at all, but the transition is considerablybroader than that observed for pure DBP. Thisimplies that local composition fluctuations are hav-ing a relatively large impact. On the other hand,when we turn to the two analogous traces for PS wesee that, while the PS Tg is shifted significantly, thechange in breadth of the PS transition in the blendis rather modest relative to that for pure PS. Giventhat the Tg for PS is apparently very sensitive tocomposition, this implies that local concentrationfluctuations are having less impact on the PS expe-rience in the blend. To model this, we need to con-sider a larger cooperative volume, hence the incor-poration of the third shell. The choice of whether toincorporate two or three shells in generating modelpredictions has implications not just for the coopera-tive volume but also for the self-concentration of thespecies occupying the central segment. We turn tothis issue in the discussion that follows.

Self-concentration and the Fox Equation

To this point we have not tried to predict the com-position dependence of the glass transition. How-ever, the appearance of two glass transitions and

Figure 8. Shown are data of Figure 1, for pure PS(filled triangles) and 20% DBP (open triangles), and forpure DBP (filled squares) and 20% PS (open squares).Both transitions broaden upon blending, but the PStransition shifts much more in temperature than doesthe DBP transition.



the implication of self-concentration as an impor-tant element lead naturally to consideration of theLM model. As summarized in the introduction, theLM model provides a route for predicting the com-position dependence of two Tg’s in a mixture. Intheir original paper an equation is given for the cal-culation of /self; however, in applications of themodel by numerous groups this quantity has oftenbeen treated as an adjustable parameter. In arecent paper on the segmental dynamics of PS asthe dilute component in a variety of solutions andblends, Lutz et al.10 summarize the results andobserve that a considerable variation exists in theliterature for /self of PS. However, they concludethat when the difference in Tg’s between the twocomponents is larger than 20 K or so, a single value

of /self ¼ 0.35 for PS appears to do remarkablywell.

In Figure 9 we show that this value of /self doesnot account for the Tg results of the PS-DBP mix-tures. The symbols represent the Tg values both forPS and DBP as a function of bulk composition, asreported by Savin et al. The upper curve in Figure9(a) is the LM prediction for the composition de-pendence of Tg for PS using /PS

self ¼ 0.35. Althoughthe composition range extends beyond the dilutelimit as considered by Lutz et al. (which went up to10 wt % PS in small molecule hosts), it is clear thatthe LM prediction using this value of /PS

self does notmatch the data over any composition range. We alsoshow the Tg data for DBP, along with the LM predic-tion using a /self

DBP¼ 0 for this small molecule compo-nent. Interestingly enough, although this curve failsto capture the solvent Tg behavior, it matches thePS data rather well. We pursue this in Figure 9(b),which shows the same two sets of experimentaldata along with LM fits to the PS results using /PS

self

¼ 0, 0.1, and 0.2. Also shown is a fit to the DBPresults using /DBP

self ¼ 0.42. The fact that the PSresults are well-described using /PS

self ¼ 0 meansthat these data follow the predictions of the Foxequation, since the Fox equation is recovered in thatlimit. This should not be overinterpreted; as long asthe cooperative volume is finite, /PS

self cannot bestrictly equal to zero. To emphasize this point, wealso show the PS prediction using LM along with/PSself ¼ 0.1, which is not too far off. However, the

quality of the fit deteriorates as /PSself is increased, as

illustrated by the curve obtained using /PSself¼ 0.2.

Before considering the implications of theseresults we wish to extend them by turning to amuch older dataset for PS-solvent mixtures.Ferry11 summarized Tg values for PS in a wide va-riety of solvents, some of the studies being doneover a considerable concentration range. In Figure10 we replot these experimental Tg results for mix-tures having up to 0.25 volume fraction of solvent.The form for the ordinate was chosen to yield astraight line with slope 1 � /self when plotting theLM equation, the limit of /PS

self ? 0 yielding theFox equation. The lines are obtained by using val-ues of /PS

self equal to (from top to bottom) 0, 0.1, 0.2,and 0.35. It is clear that although a nonzero valueof /PS

self may be used, it must be small, certainlymuch smaller than 0.35.

Self-Consistency and Lodge–McLeish

Our considerations of self-consistency between thefilling of unconstrained sites and the overall aver-

Figure 9. (a) Dotted curve is LM prediction of PS glasstransition versus DBP fraction, assuming /self ¼ 0.35;triangles are PS transitions. Dashed curve is LM predic-tion for DBP transition, assuming /self ¼ 0; squares areDBP transitions. (b) Upper solid curve is LM predictionof PS glass transition versus DBP fraction, assuming/self ¼ 0; dashed and dotted curves are for /self ¼ 0.1 and0.2 respectively; triangles are PS transitions. Lowersolid curve is LM prediction for DBP transition, assum-ing /self ¼ 0.42; squares are DBP transitions.



age volume fraction of the two components arealso applicable to the arguments leading up to theLMmodel for predicting the effect of the local envi-ronment on the Tg of a polymer segment.

Suppose for simplicity that the two polymers Aand B in a blend have the same cooperative vol-ume VC (i.e. the same Kuhn length), but differentpacking lengths and hence different values of /self.Consider the total number of pairs of A and Bmonomers such that the distance between them isless than the cooperative length scale. We maycount this total number of pairs in two ways. Wemay either take the sum of the number of A seg-ments within the cooperative volume of some Bsegment, summed over all B segments; or take an‘‘A-centric’’ view, reversing the roles of A and B.Both procedures of course must give the same an-swer. This requirement turns out to constrain theway in which unconstrained sites are filled, suchthat eq 1 must be modified.

Let /A and /B ¼ 1� /A be the bulk volume frac-tions of A and B, and /self,A and /self,B the corre-sponding self fractions. Finally, let us assume (asfor our lattice model) that unconstrained sitesregardless of what segment lies at the center of thecooperative volume are filled with B with a proba-bility p (and filled with Awith a probability 1 � p),where p is to be determined. The consistencyrequirement above translates to

/Bð1� /self ;BÞð1� pÞ ¼ /Að1� /self ;AÞp ð15Þ

In fact, this consistency requirement is equivalentto the one enforced for the lattice model, eq 12,which asserts that the average volume fraction of

B within a cooperative volume, averaged over thesegment that lies at the center, equals the bulkfraction of B. Equation 15 can be rearranged togive

/B ¼ /Bð/self ;B þ ð1� /self ;BÞpÞ þ /Að1� /self ;AÞpð16Þ

which expresses the same condition as eq 12.Solving for p, we find

p ¼ /Bð1� /self ;BÞ/Að1� /self ;AÞ þ /Bð1� /self ;BÞ

ð17Þ

Note that p ¼ /B if and only if /self,A ¼ /self,B. Oth-erwise, eq 1 should be modified to read

/eff ;A ¼ /self ;A þ ð1� /self ;AÞð1� pÞ/eff ;B ¼ /self ;B þ ð1� /self ;BÞp ð18Þ

where p is given by eq 17. Figure 11 shows theresulting dependence of /eff,A and /eff,B on /A

for /self,A ¼ 0.2 and /self,B ¼ 0.6, comparedwith the predictions of the original LM equationeq 1. Evidently, when the contrast in /self is large,the difference in the two predictions for /eff issignificant.

To see just how significant is the effect of analtered value for the effective volume fraction, wecompute the Tg’s of two hypothetical polymers Aand B for which Tg,A ¼ 200 K and Tg,B ¼ 400 K.Figure 12 shows the predicted Tg’s using the Foxequation with the effective volume fraction to pre-dict Tg. It is worth mentioning here that taking

Figure 11. Solid curves are predictions from self-con-sistent LM eq 18 of /eff,A and /eff,B as functions of bulk/A; dashed curves are corresponding predictions of origi-nal LM. Here /self,A¼ 0.6 and /self,B ¼ 0.2.

Figure 10. Symbols are glass transition data versusdiluent fraction from Ferry (ref. 11); different symbolscorrespond to different diluents, plotted as described intext. Solid line is Fox equation (LM with /self ¼ 0); dot-ted lines are for /self ¼ 0.1 and 0.2; dashed line is for /self

¼ 0.35.



the sum of the two resulting Tg’s, weighted by theeffective volume fractions, will yield a single Tg

equal to what the original Fox equation would pro-duce. This is just another way of showing that theeffective volume fractions are self-consistent; fol-lowing the same procedure using the original LMresults would not produce the Fox Tg as a functionof composition.

In fact, we can apply the result eq 18 to our lat-tice calculations. For this comparison, we need avalue appropriate to the lattice model for /self,A

and /self,B; we take /self,A ¼ 0 and /self,B ¼ 2n/Sn

(the fraction of constrained segments connected tothe B segment at the central site). Calculations of/eff,A and /eff,B using eqs 18 and 17 coincide withlattice results from eq 11. This agreement illus-trates that the lattice model and the self-consistentversion of the LM argument both properly accountfor chain connectivity even as it affects /eff of an

unconnected species (solvent) in the presence ofpolymer.

Alternative Estimate of Variance

Thus far, we have computed histograms of volumefraction within the cooperative volume (repre-sented within our lattice model as a number ofshells in a tree) by essentially explicit enumera-tion. Now we consider an alternative approach toestimating the variance in effective volume frac-tion. In a recent work, Leroy et al. have treatedthe variance as a fitting parameter in accountingfor the spectral width of dielectric relaxation datain polymer blends.12 However, Salaniwal et al.9

have addressed this problem by estimating varian-ces in local composition using the random phaseapproximation (RPA) applied to the contents of afinite spherical volume. This yields an integralexpression involving the RPA structure factor thatmust be evaluated numerically. Our simplerapproach is based on the idea that the A and B seg-ments pervading the cooperative volume have atypical size, and their presence or absence fromthe volume is determined by a set of independentdecisions.

Consider the simplest case, that of a blend oftwo polymers A and B with the same packinglength. Given a cooperative volume VC, a strand ofeither A or B pervading VC has some volume O.There are then some number n such strandsrequired to fill VC. We may imagine filling VC byfirst generating configurations of the n strands,and then choosing among the n strands which aretype A and which are type B. The volume fractionof A within VC fluctuates depending on how manyA and B segments are chosen.

Within this simplest case, the probability ofhaving s strands of type A is

PðsÞ ¼ ns

� �xsð1� sÞn�s ð19Þ

where x is the probability that a strand be of typeA. If the total number of strands n is large, we canexpand eq 19 using the Stirling formula to approx-imate P(s) by a Gaussian,

logPðsÞ � � ðs� xnÞ22nxð1� xÞ þ const ð20Þ

From this result, we can identify that the meannumber of A segments, s, is equal to xn and thevariance, hds2i, is equal to nx(1 � x). Since the vol-ume fraction of A is s/n, the mean volume fraction

Figure 12. (a) Solid curves are predictions for polymerblends from self-consistent LM eqs 2 and 18 of Tg,A andTg,B (i.e. with A and B at center site respectively) asfunctions of bulk /A; dashed curves are correspondingpredictions of original LM. Here /self,A ¼ 0.6 and /self,B

¼ 0.2, Tg for pure A is taken to be 400 K and for pure Bis 200 K. (b) As in (a), except that Tg for pure A is takento be 200 K and for pure B is 400 K.



of A is simply x, and the variance in the volumefraction is

hd/2i ¼ /ð1� /Þn

ð21Þ

More complex is the case of polymer–solvent mix-tures, in which the typical size of strands of A andB appearing in VC have different volumes OA andOB. (Such would also be the case for polymerblends in which the two polymers had differentpacking lengths.) Then, we may imagine fillingour tree lattice from the central site out, at eachshell randomly choosing unconstrained sites to befilled with either A (solvent) or B (polymer). If wechoose B to fill a given site, it implies that con-nected monomers must be placed in subsequentshells.

Thus the process of generating configurationsmay be regarded as a sequence of choosing eitherA (with some probability x) or B (with probability1 � x), with their respective associated volumesinserted into the cooperative volume. As in thesymmetric case described earlier, we simplify mat-ters by taking a single, typical value for each of theinserted volumes OA and OB, although in the moreprecise lattice model OB would evidently varyamong B strands.

Even with the simplification of constantbut unequal values for OA and OB, an interestingthing happens when we undertake to generate asequence of choices of A or B strands to fill a givenvolume VC. We can estimate the mean total num-ber of choices required as

hNi ¼ Vc

xOA þ ð1� xÞOBð22Þ

If we were to make exactly hNi choices, the num-ber of A strands generated would fluctuate, and sothe total volume generated would also fluctuate.Since we intend to fill a fixed volume VC, thismeans the total number of choices N must fluctu-ate instead.

Let nA denote the number of A strands selected,and dN ¼ N � hNi the variation in the total num-ber of strands N. The total number of A strandsgenerated is the average number xhNi, plus thevariation dn in the number generated in hNichoices, plus the average number xdN generatedby dN ‘‘additional’’ choices (of course, dN may benegative). We may write

nA ¼ xðhNi þ dNÞ þ dn

nB ¼ ð1� xÞðhNi þ dNÞ � dn ð23Þ

The total volume so generated is nAOA þ nBOB,which must remain equal to VC. Note that the av-erage term satisfies this already, from eq 22. Thuswe may relate the number of additional choices dNto the variation dn in the number of A segmentsgenerated by the average number of choices, as

dN ¼ dnðOB � OAÞxOA þ ð1� xÞOB

ð24Þ

That is, dN times the average volume added perchoice equals the variation in volume resultingfrom dn extra strands of A.

Now the total variation in the number of A andB strands may be written as DnA ¼ xdN þ dn andDnB ¼ (1 � x)dN � dn. Using the above-mentionedresults we may write

DnA ¼ OBdnxOA þ ð1� xÞOB

DnB ¼ �OAdnxOA þ ð1� xÞOB

ð25Þ

The volume fraction of A, the variance of whichwe seek, is /A ¼ nAOA/VC, with variation d/A

¼ DnAOA/VC. The average volume fraction is

h/Ai ¼xOA

xOA þ ð1� xÞOBð26Þ

which can be inverted to give x in terms of / as

x ¼ OB/OAð1� /Þ þ OB/

ð27Þ

We can write the variance of n as hdn2i ¼ x(1 � x)hNi, as the result of hNi independent choices withprobabilities x and (1 � x). Using eqs 25–27, wecan then compute the variance of /A after somearithmetic as

hd/2i ¼ OAOB

Vc

/OA

þ ð1� /ÞOB

� �/ð1� /Þ ð28Þ

This expression evidently reduces to the previousresult eq 21 when OA ¼ OB and is symmetric underthe interchange of A and B (which impliesexchanging / and (1 � /)). Also, hd/2i vanishes inthe limits of / approaching zero or unity as itshould, since with only one component presentthere can be no variance in the volume fraction.

We may compare the predictions of eq 28 toresults from our lattice model, from which we cangenerate values of the variance in volume fractionas a function of composition. To use eq 28, we take



OA ¼ 1 (solvent occupies one lattice site) and VC

¼ Sn (the total number of sites in the first throughnth shells). Since OB varies in the lattice model, weretain it as a fitting parameter.

Figure 13 shows the variance computed fromthe lattice model for the second shell (VC ¼ S2

¼ 36), and the prediction from eq 28 with OB

¼ 2.04. Similarly good results are obtained for thefirst-shell variances with a slightly smaller valueof OB ¼ 1.45, and the third-shell variances with aslightly larger value of OB ¼ 2.44.

Thus this simple approximate approach to esti-mating the variance in volume fraction within thecooperative volume gives good agreement with themore involved lattice calculations, provided we usethe volume per strand OB as a fitting parameter.The value of about two lattice sites for OB for thesecond shell is not unreasonable, given that onlythe longest central strand pervading the secondshell is five segments long, those strands reachinginto the first shell are just three segments long,and the strands reaching only into the second shellcontribute but a single segment.

We can also compare the histograms computedfrom the lattice model to Gaussians with widthsdetermined by eq 28 and mean positions deter-mined by the self-consistent version of LM, that iseqs 18 and 17. To apply these equations to resultsof the lattice calculations, we take /self,A ¼ 0 and/self,B ¼ 2n/Sn (fraction of constrained segmentsarising from the central B site). In Figure 4, thesolid lines correspond to Gaussians so determined,which are in good agreement with the latticeresults.

In principle, this approach to estimating thevariance in volume fraction could serve as a proxy

for the lattice model histogram of volume fractionin generating DSC traces, by performing a sum ofcontributions to the overall trace over an appropri-ately discretized Gaussian distribution of effectivevolume fractions with the correct mean and var-iance. To pursue this would require a good a prioriestimate of OB.

DISCUSSION

This paper deals with the impact of local environ-ment on both the location and breadth of glasstransitions in polymer–solvent mixtures. In thecourse of our studies we have had cause to exam-ine critically notions about self-composition, thecooperative volume around a given segment, theimpact of the packing length for a polymeric com-ponent, the utility of the Fox equation,8 characteri-zation of composition fluctuations (using both alattice model and a continuum description), andpitfalls in the LMmodel.3

As a starting point we focused on recent experi-mental DSC results for PS and DBP.4 We devel-oped a simple tree lattice model as a means of com-puting the distribution of local environmentsaround a segment in the mixture, which lead tothe spread in Tg as reflected in the DSC results. Alattice description with chain connectivity natu-rally incorporates the effects of self-concentrationassociated with the presence of a polymeric compo-nent.13 It is less obvious that, having accounted forthe self-concentration, scrupulous accounting isalso required in filling the free space around sucha segment (that is, the space not taken up by anycovalently bonded segments) so as to yield the cor-rect global average composition. This consistencycondition turns out to require that the LM modelbe modified as well.

With our lattice histograms providing a meansof accounting for local composition distributions,along with analysis of pure component DSC traces,we were able to predict the composition-dependentspread of the mixture DSC results. In doing so werequired a means of calculating the shift in Tg

expected for a given composition, which lead usboth to apply and then ponder the form of the Foxequation. Finally, recognition that the latticemodel is not a model of polymer solvent interactionor structure, but merely a mechanism by which adesignated amount of space is to be filled by twodifferent components, leads us to develop a simplecontinuum analogue. Our continuum descriptionyields results that overlap those of the lattice

Figure 13. Solid curve is variance in composition com-puted from lattice model for second shell, versus volumefraction. Dashed curve is prediction of eq 28 as describedin text. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]



approach, and under certain conditions, we believeit represents a clean route to accounting for localcomposition fluctuations.

The above-mentioned generalities are intendedto give the casual reader a sense of the scope anddepth of what we have covered in this paper. Moredetailed remarks follow.

General Principles

In this work, we have assumed that the contribu-tion of a molecular segment within a polymer–sol-vent mixture to its glass transition is a function ofthe local concentration within some small coopera-tive volume containing the segment. To model this,we require a connection between the local concen-tration and the contribution of the segment to theglass transition, as well as a way of computing thedistribution of local concentrations.

Within such a mixture, the average local poly-mer concentration in a small volume surroundinga polymer segment will be enriched relative to thebulk average, because of the connected polymersegments obliged to cohabit the volume.2,4,5 Like-wise, the polymer concentration in a small volumesurrounding a solvent segment will be depleted,because some of the polymer segments are boundto their connected brethren. These shifts in theeffective concentration will lead to shifted glasstransitions for two mixed components, relative totheir locations if the local concentration wereeverywhere equal to the bulk value.

Beyond such effects of the average local concen-tration, there will be fluctuations in the local con-centration around polymer and solvent segments.These variations in concentration, taken togetherwith the concentration dependence of the Tg foreach component, imply that the glass transition ofeach component will be broadened relative to itswidth in a pure sample.

Controlling both the shifts in the average localconcentration and the breadth of the concentrationfluctuations are two physical parameters: the sizeof the cooperative volume VC and the packinglength of the polymer lp. As VC grows, the averagelocal concentration approaches the bulk concentra-tion and the fluctuations diminish. The packinglength lp is a measure of the length scale overwhich a given chain makes a significant contribu-tion to the local concentration. Bulkier and moreflexible polymers have a larger packing length. To-gether, lp and VC determine /self, the self-concen-tration a polymer contributes in the vicinity of oneof its own segments.

If we regard lp as a known material parameterand VC as a parameter to be determined by fittingto experiment, we see that VC can be determinedin two ways: by choosing it so that the compositionfluctuations give predicted widths of DSC tracesthat match experiment, or by choosing it so that/self in combination with eq 2 predicts the composi-tion dependence of Tg. We shall return to this pointshortly.

Our Lattice Model

We have developed a lattice description of thepacking of polymer and solvent in a small volume,enabling us to compute the probability distributionof local concentration in a cooperative volume sur-rounding either a polymer or a solvent segment. Alattice model is simple to define, naturally containsshort-distance cutoffs, and is amenable to enumer-ation. Use of a tree lattice eliminates polymer over-laps through avoidance of loops. Instead of thephysical parameters VC and /self, we can choosethe (integer) number of generations or ‘‘shells’’ inthe tree lattice, and the (integer) coordinationnumber of the lattice, as proxies for adjusting VC

and /self.Combining the above with a recipe for how the

Tg of polymer or solvent depends on effective com-position, as well as DSC data for the glass transi-tion in pure polymer and pure solvent, we predictthe shapes of DSC traces for polymer–solvent mix-tures. These traces are the composition-weightedsum of contributions from the polymer and the sol-vent, each such contribution reflecting a differentlocal composition distribution. We assume thateach element of the distribution comprises the cor-responding pure DSC trace for that component,shifted to its own local Tg, and having an ampli-tude proportional to mass fraction. We applied thisrecipe to the data on PS/DBP mixtures of Savinet al.4 with good results.

In generating the DSC traces we initially side-stepped the problem of predicting the dependenceof Tg on effective composition by fitting the data ofSavin et al. for Tg of each component as a functionof bulk composition. We then used the latticemodel to compute the effective compositions in thevicinity of polymer and solvent as a function ofbulk composition, so as to translate from the bulkto the effective composition dependence of the ex-perimental Tg values. As noted earlier, our analy-sis leads to two routes for the determination of VC.First, we adjusted the number of shells used in



generating the DSC traces to give the best fit; thisamounts to varying VC to control the compositionfluctuations so as to match the widths of the DSCtraces. Later, we determined values of /self that,when used with the LM prescription eq 2, gavereasonable fits to the Tg–composition data for thepolymer and for the solvent. This amounts toadjusting VC (and hence /self) to match Tg(/).

For PS solutions in DBP we found both thewidths and the locations of the glass transitionsassociated with the PS component to be well-described by a small value of /self, of less than 0.1or so. For the widths, we base this statement onthe need to include up to n ¼ 3 shells in our treelattice to achieve a sufficiently narrow PS glasstransition in the mixture. For the locations, thisconclusion reflects the reasonable fit of eq 2 to thePS glass transition versus bulk composition with/self < 0.1. The significance of such a small valueof /self relative to other published results is dis-cussed toward the end of the paper.

Finally, in assuming random filling we areasserting that enthalpic interactions between seg-ments are not strong enough to affect filling on thelocal scale. This is certainly true of polymer–poly-mer blends without strong attractions. It is eventrue of polymer blends near phase separation,because there v is of order N�1 and so v is a smalleffect on the scale of the Kuhn segment. However,for polymer–solvent blends the condition for phaseseparation is v ¼ 0.5, which is not small for effectson the scale of the Kuhn length. Thus for marginalsolvents filling need not be random; such repulsiveinteractions would lead to concentration fluctua-tions that are enhanced beyond those for athermalsolvents.

Comparison to Lodge–McLeish

Lodge and McLeish3 hypothesized as we have thatthe glass transition is a local phenomenon con-trolled by the concentration in a small cooperativevolume VC surrounding a given molecular seg-ment. LM assumed VC to be of order the Kuhnlength cubed, though many subsequent workershave used /self and thus in effect VC as a fitting pa-rameter. We have taken similar advantage, adjust-ing the number of shells in our lattice calculationsto give a reasonable fit to the widths of DSC tracesfor the PS/DBP data.

LM adopted the Fox equation to compute the Tg

for a segment from the local ‘‘effective’’ concentra-tion. Initially, we extracted the dependence of Tg

on composition from the DSC data itself; later, we

followed LM and used the Fox equation, motivatedby the observed concentration dependence of thepolymeric glass transition in the PS/DBP mix-tures.

The LM model makes a particular assumptionabout how the effective composition is enriched (eq1). We have presented a self-consistent LM model,such that site-averaged effective compositionsequal the bulk composition. This self-consistentLM model is in excellent agreement with results ofthe lattice calculations. Finally, no account istaken of concentration fluctuations in the LMmodel, and so it makes no prediction as to thebroadening of glass transitions in polymeric mix-tures.

Continuum Approach

We initially used the tree model to compute thecomposition distribution in a small volume of poly-mer solution. The filling of such a volume is theresult of a number of independent decisions aboutthe identity of segments occupying the space. Thisallows for an estimate of the variance of composi-tion by elementary counting arguments.

Pursuing this approach, we developed a simpleexpression for the variance of composition thatdepends on knowing only the ‘‘typical’’ length of apolymeric segment within the cooperative volume.We showed that the composition distributions com-puted from the lattice model are well described byGaussians constructed using the variance given bythis argument eq 28, along with mean compositionvalues given by our self-consistent form eq 18 ofthe LMmodel.

From this perspective the lattice model servesas a test bed to verify these routes to the effectivecomposition, variance, and mean, and also as away of computing the distribution of polymericstrand lengths within the cooperative volume. Forthe regular tree lattice with z ¼ 6, it can be shownthat the weight-averaged strand length hsi is inde-pendent of bulk composition, and approaches 7/3� 2.33 in the limit of a large number of shells (forn ¼ 2, hsi ¼ 2.19; for n ¼ 3, hsi ¼ 2.29). The vigor-ous branching of the regular tree leads to a pre-ponderance of short strands.

One might expect that for a nonregular, space-filling tree the average strand length would be lon-ger, which would mean that for a given size of VC

the number of independently chosen segments fill-ing VC would be smaller, and hence the composi-tion fluctuations larger. We expect that for a morerealistic (embeddable) tree lattice the required size



of VC to match the width of the DSC traces wouldbe somewhat larger than we found earlier.

Values of /self

As summarized earlier our results show that forPS solutions in DBP, both the widths and the loca-tions of the glass transitions associated with thePS component are well-described by a small valueof /self, of less than 0.1 or so. Correspondingly, wefound it optimal to use three shells in modeling thepolymer DSC transitions. We translate our resultfor the best-fitting number of shells in our latticemodel into a size for the cooperative volume as fol-lows. Since the tree lattice does not corresponddirectly to a compact, space-filling structure, wecount the number of sites in the shells and convertthis to a compact volume. Thus for the PS coopera-tive volume, which was best represented by n ¼ 3shells, we have 186 lattice sites, each site having avolume of one DBP molecule as described previ-ously. The cooperative volume is thus estimated tobe equivalent to a sphere of radius 2.6 nm.

This reflects a somewhat larger cooperative vol-ume than has been found by other authors in ana-lyzing results for PS solutions. In particular,Ediger and coworkers10 concluded that a value of/self of about 0.35 describes NMR segmental relax-ation data for dilute PS solutions in various sol-vents. However, it must be noted that these dataare limited by the frequency range of NMR to tem-peratures about 60 8C above the PS-related glasstransition. Several authors have argued for anincrease in the cooperativity length (i.e. the cuberoot of VC) as the glass transition is approached.This has recently been framed in the context ofAdam-Gibbs theory, which describes the diver-gence of the cooperativity volume as the tempera-ture decreases to somewhat below Tg.

7

Here, we turn to estimate how large an increasein the cooperativity length is implied if we are torationalize the present results with Ediger et al.At issue is how /self should scale with VC. If weassume that the polymer strand making the self-contribution is essentially straight, then /self

should scale as VC�2/3. If we instead assume that

the polymer strand may be regarded as a randomwalk, /self should scale as VC

�1/3. (The latter scalingis consistent with the LM conclusion that /self

scales as lp/lK.) If the polymer strand is ‘‘of theorder of’’ a Kuhn length, the scaling is presumablyin the crossover between these two limitingresults.

The ratio of values for VC implied by two differ-ent values of /self could then be as large as (VC

0 /VC)

¼ (/self0 //self)

3 (if random-walk scaling holds). Thusthe linear dimension of the cooperative volumewould be larger by a factor of at least (0.35/0.1)¼ 3.5 under the conditions of the DSC measure-ments (i.e. near to Tg) relative to the experimentsof Ediger et al., 60 8C or so above Tg. This magni-tude of increase in the cooperative length scale astemperature approaches Tg over such a range isnot inconsistent with other published estimates.7

In contrast, the DBP glass transition variesonly weakly with PS concentration; thus, when wefit to eq 2, a large value of /self,A (about 0.42) isrequired. Unlike our results for the PS transition,this is not consistent with the number of shells (n¼ 2) we found gave the best account of the DSCline shapes for the DBP transition. An apparentcontradiction emerges: a large value of /self forDBP implies a very small cooperative volume (sothat the single A site at the center dominates theconcentration). But the relative narrowness of theDBP transition suggests a cooperative volumelarge enough so that composition fluctuations aresomewhat suppressed.

In a very recent study Ediger and coworkersfound that dilute tetracene in a variety of glass-forming hosts, including some that were poly-meric, exhibited an effective glass transition thatwas equal to the matrix transition.14 However,these results were once again associated withNMR measurements, and the Tg’s were extrapo-lated results from data that were collected at tem-peratures well above the transition. It may be thatthe Fox equation is not a good phenomenologicaldescription of the effect of local composition on theglass transition of small molecules. Or, perhaps asimplified lattice description of local compositionfluctuations may be inadequate to describe mole-cules with a small cooperative volume. Or, ‘‘clus-tering’’ of the solvent molecules because of attrac-tive interactions may lead to a larger /self. Clearlymore work is needed to understand in detail theglass transition of the solvent in polymer–solventmixtures. These issues may be confounded withthe fact that the solvent is the low-Tg componentin the Savin systems. It would be interesting tostudy a polymer–solvent system in which the poly-mer was the low-Tg component.

Remaining Questions

Several key questions remain unanswered. We havefound that a larger cooperative volume for PS, hencea smaller value of /self, is required to fit the DSCdata, because the experiment explores temperatures



approaching Tg, rather than being limited to tem-peratures well above Tg as is the case for NMR stud-ies, for example. This should be tested with otherDSC studies, of mixtures of PS with other solvents.We would also expect that DSC studies of suitablepolymer–polymer blends would show the effects of alarger cooperative volume, both in the location andbreadth of observed transitions.

Use of the Fox equation to relate local composi-tion to Tg remains ill-defended. It is curious thatsuch a widely used expression should have such amurky provenance. We know of no derivation ofthe Fox equation in the literature. Indeed, theoriginal reference8 is an abstract in the Bulletinof fthe American Physical Society, which does notgive any theoretical argument for that particularform.

There are ambiguities as well regarding theclassical literature data on PS–solvent mixtureglass transitions, as summarized by Ferry.11 It isunknown whether these polymer–solvent mix-tures exhibit two glass transitions in DSC traces;if so, it is unclear whether the single Tg’s reportedare to be associated with the polymer, or insteadrepresent some weighted average of the transi-tions associated with the polymer and the solvent.At least on the polymer-rich end of the compositionrange, whatever transition was measured must bepredominately related to the polymer.

So it is noteworthy that the data presented byFerry is, while evidently scattered, reasonably con-sistent with the unmodified Fox equation, that iswith small self-concentration for PS. It would beuseful for a new and comprehensive set of data tobe collected, given the PS/DBP results and emer-gence of a ‘‘local’’ view of the glass transition.Indeed, given the unresolved questions regarding/self and the cooperative volume, we believe itwould be extremely useful to have new data thatencompasses a set of glass-forming polymers span-ning a range of packing lengths, dissolved in a va-riety of solvents.

Expanding the scope of miscible blend dynamicsstudies to polymer–solvent mixtures broadens theset of available systems for study. This work,which focuses on the glass transitions in PS DBPmixtures, presents new computational approachesthrough which the relationship between local seg-mental environments and dynamic heterogeneitymay be clarified.

We thank Dan Savin and Tim Lodge for useful discus-sions, and we further benefited from Professor Savin’sgenerosity in sharing unpublished data. J. E. G. Lipsonacknowledges the support of the National Science Foun-dation [DMR-0502196].

REFERENCES AND NOTES

1. Lau, S.-F.; Pathak, J.; Wunderlich, B. Macromole-cules 1982, 15, 1278.

2. Chung, G.-C.; Kornfield, J. A.; Smith, S. D. Macro-molecules 1994, 27, 5729.

3. Lodge, T. P.; McLeish, T. C. B. Macromolecules 2000,33, 5278.

4. Savin, D. A.; Larson, A. M.; Lodge, T. P. J Polym SciPart B: Polym Phys 2004, 42, 1155.

5. Zetsche, A.; Fischer, E. W. Acta Polym 1994, 45, 168.6. Kant, R.; Kumar, S. K.; Colby, R. H. Macromolecules

2004, 26, 10087.7. Cangialosi, D.; Schwartz, G. A.; Alegria, A.; Colme-

nero, J. J Chem Phys 2005, 123, 144908.8. Fox, T. G. Bull Am Phys Soc 1956, 1, 123.9. Salaniwal, S.; Kant, R.; Colby, R. H.; Kumar, S. K.

Macromolecules 2002, 35, 9211.10. Lutz, T. R.; He, Y.; Ediger, M. D. Macromolecules

2005, 38, 9826.11. Ferry, J. D. Viscoelastic Properties of Polymers, 3rd

ed.; Wiley: New York, 1980; Ch. 17, p 488.12. Leroy, E.; Alegria, A.; Colmenero, J. Macromolecules

2003, 36, 7280.13. Colby, R. H.; Lipson, J. E. G. Macromolecules 2005,

38, 4919.14. Ediger, M. D.; Lutz, T. R.; He, Yiyong. To be pub-

lished.



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