Thermodynamic Interactions in Isotope Blends: Experiment and Theory

Thermodynamic Interactions in Isotope Blends: Experiment and Theory

Buckley Crist

Department of Materials Science and Engineering, Northwestern University,Evanston, Illinois 60208-3108

Received December 23, 1997; Revised Manuscript Received June 19, 1998

ABSTRACT: Small-angle neutron scattering (SANS) studies of binary mixtures provide øNS, a measureof thermodynamic interactions between dissimilar polymer chains, one of which is usually labeled withdeuterium. For polymers differing only in isotopic substitution (isotope blends), øNS is seen to divergestrongly upward (or sometimes downward) at low concentrations of either blend component. Thisconcentration dependence seems to vanish in the limit of large degree of polymerization N. Experimentalresults can be described by øNS(æ,N) ) â + γ/Næ(1 - æ), where æ is the volume fraction of deuteratedpolymer. For SANS from a series of blends with different æ it is shown that systematic errors in Nand/or the static structure factor S(0) lead to precisely the same øNS(æ,N) when the Flory-Hugginsinteraction parameter ø is constant. While results for some isotope blend systems can be accounted forwith reasonable error estimates, others appear to have a real dependence of øNS on æ and N. It is suggestedthat these “non-Flory-Huggins” effects stem from a modified entropy of mixing that is most evident indilute blends. The concentration dependence of øNS(æ) has no practical effect on macroscopic phasebehavior.

IntroductionThermodynamic interactions between polymer chains

are often studied by small-angle neutron scattering(SANS), with one component of a binary blend labeledwith deuterium. After the suggestion by Buckinghamand Hentschel1 that isotopic substitution would lead tosmall but observable repulsive interactions in polymerblends, this effect was seen in SANS experiments byBates et al.,2 in interdiffusion studies by Green andDoyle,3 and in phase separation observed by Bates andWiltzius.4

While the “isotope effect” is known to modify interac-tions between chemically dissimilar polymers, thataspect of the problem is reasonably well understood.5,6

More perplexing are SANS results for blends of poly-mers differing only in isotopic substitution. Theseappear, at the present, to challenge our understandingof the thermodynamics of polymer-polymer blends.Consider the SANS results from Londono et al.7 fromnarrow molecular weight fractions of conventional anddeuterated polyethylene (PE/d-PE) shown in Figure 1.Most striking is that the apparent interaction parameterøNS has a strong dependence on blend composition,expressed as volume fraction of d-PE. The interactionparameter is obtained in the usual way, starting withthe general expression for coherent cross-section perunit volume I(q):

Here bi/vi is the scattering length density of the mono-mer in polymer i ()1, 2), and the static structure factorS(q) is the absolute intensity normalized by the contrastfactor in parentheses. Intensity is extrapolated to zeroscattering angle (q ) 0) to obtain S(0), which is writtenas

Ni are (weight average) degrees of polymerization, vi aremonomer volumes, æi are volume fractions, and v0 )(v1v2)1/2 is the reference volume. Equation 2 is rear-ranged to give the apparent interaction parameter fromSANS:

There is no formal restriction on the method used toextrapolate intensity or S(q) to q ) 0. This step is mostoften done with the q > 0 variant of eq 2 obtained withthe random phase approximation.8 It should be em-phasized that incompressible RPA theory, or, equiva-lently, Flory-Huggins lattice theory for a polymer-polymer blend, is invoked in the evaluation of øNS byeqs 2 and 3.

While the simplest Flory-Huggins thermodynamicmodel has the interaction parameter ø being indepen-dent of blend concentration æi and component degreeof polymerization Ni, this is not required. The relation

I(q) ) (b1

v1-

b2

v2)2

S(q) (1)

1S(0)

) 1N1æ1v1

+ 1N2æ2v2

-2øNS

v0(2)

Figure 1. Dependence of øNS on blend concentration æd for(b) isotope blends PE/d-PE, N ≈ 4400 (C2H4 monomers), T )155 °C, from Londono et al.7 and (O) model copolymer blendsHPB(97)/DPB(88), N ≈ 3200 (C2H4 monomers), T ) 83 °C, fromKrishnamoorti et al.10 The dashed line is a fit of PE/d-PE datato eq 5.

øNS ≡ v0

2 ( 1N1æ1v1

+ 1N2æ2v2

) -v0

2S(0)(3)

5853Macromolecules 1998, 31, 5853-5860

S0024-9297(97)01858-5 CCC: $15.00 © 1998 American Chemical SocietyPublished on Web 08/01/1998

between the thermodynamic interaction parameter øand the SANS result is given by

If and only if ø is independent of concentration, thenøNS ) ø. Any concentration dependence of the thermo-dynamic ø will lead to a larger change in the SANSquantity øNS. It was shown9 that the ca. 4-fold increasein øNS seen in Figure 1 corresponds to a much smaller,ca. 30%, increase in ø. Another observation is that theconcentration dependence of øNS is a function of N (ormolecular weight) and appears to vanish in the limit ofinfinite N (Figure 2). Following Krishnamoorti et al.,10

these effects can be summarized by the followingempirical expression:9

Here Ne ) N1N2/(æ1N1 + æ2N2) is the appropriateaverage degree of polymerization for blends in whichN1 * N2. The dashed line in Figure 1 is a fit of the PE/d-PE data to eq 5. It is emphasized that there is notheoretical foundation for eq 5, but the second “nonideal”term has a structure reminiscent of that deriving fromthe usual combinatorial entropy of mixing.

Also instructive is the temperature dependence of øNS.First, note that the upward curvature of øNS(æ) isvirtually unaltered by proximity to the critical temper-ature (Figure 3). Blends of different æd each display theconventional temperature dependence øNS(T) ) A + B/T,but the coefficient A (implying an entropic contributionto the interaction) has by far the greatest sensitivity toconcentration (Figure 4).

Although the data of Londono et al.7 on PE/d-PE arethe most comprehensive in terms of æd, N, and Tdependence, similar results are seen with most isotopeblends. The first report of upward curvature of øNS(æ)was by Bates et al.11 for PEE/d-PEE and PVE/d-PVE;those data closely resemble the effect shown in Figure1. (PEE is equivalent to atactic poly(butene-1), havingan ethyl branch on every second backbone carbon; PVEhas unsaturated vinyl branches.) Krishnamoorti12 hasrecently studied isotope blends of 1,4-polyisoprene, poly-(ethylene oxide), and poly(methyl methacrylate); in each

case the interaction parameter from neutron scatteringincreases noticeably when æd e 0.2. For polystyreneisotope blends the “parabolic” curvature of øNS is down-ward, an effect confirmed by two groups.7,13 Noteworthyis the finding by Schwahn et al.13 that øNS is negativefor æd < 0.2, a point to be considered later. A summaryof these observations is presented in Table 1. Proximityto the critical point is indicated by NeøNS(0.5) < 2, andthe magnitude and sense of curvature is indicated bythe ratio øNS(0.1)/øNS(0.5). The combined error factor“c” is defined below.

Excepting PS/d-PS, the behavior in Figure 1 can beconsidered typical for weakly interacting blends, i.e.,those lacking specific interactions between componentpolymers. Expanding our scope to include model poly-olefins, where h-polymers and d-polymers differ inchemical microstructure as well as isotopic substitution,

Figure 2. øNS at T ) 155 °C versus 1/Ne for PE/d-PE blendshaving different æd; data from Londono et al.7 The concentra-tion dependence of øNS is proportional to N-1.

øNS ≡ 12

∂2(æ1æ2ø)∂æ1∂æ2

(4)

øNS ) â + γNeæ1æ2

(5)

Figure 3. øNS(æ) for PE/d-PE, N ≈ 4400, at two temperaturesabove the critical temperature Tc ) 132 °C (405 K); data fromLondono et al.7 Concentration dependencies of øNS are indis-tinguishable at T/Tc ) 1.02 (b) and at T/Tc ) 1.17 (O).

Figure 4. Temperature dependence of øNS(T) ) A + B/T forPE/d-PE of different æd, N ≈ 4400; data from Londono et al.7The increase in øNS at low æd seen in Figures 1 and 3 areassociated with larger “A”.

Table 1. Concentration Dependence of øNS in PolymerIsotope Blends

blend T (K) 10-3Ne NeøNS(0.5)

øNS(0.1)

øNS(0.5) c ref

PEE/d-PEE 299 1.50 1.3 1.78 1.4 11PVE/d-PVE 310 1.50 1 1.67 1.3 11PE/d-PE 428 4.36 1.9 1.88a 1.5 7PI/d-PI 300 1.18 0.9 1.31 1.1 12PEO/d-PEO 347 2.39 1.1 1.24a <1.1 12PMMA/d-PMMA 381 1.58 0.5 1.32 <1.1 12PS/d-PS 433 9.91 1.8 0.80b 0.9 7PS/d-PS 439 9.47 1.6 -0.65a 0.6 13

a For æd ) 0.09. b For æd ) 0.125.

5854 Crist Macromolecules, Vol. 31, No. 17, 1998

SANS behavior is generally similar. In the majority ofcases one again sees symmetric upward curvature oføNS, as illustrated in Figure 1 for the pair of saturatedpolybutadienes HPB(97)/DPB(88) from work by Krish-namoorti et al.;10 numbers in parentheses are percent1-2 additions of butadiene that lead to ethyl branchesalong the chain. In some cases this behavior is skewedby a monotonic increase of øNS with larger concentrationof the more branched component. Blends of modelalternating copolymers with the same amount of branch-ing, one methyl branched and the other ethyl branched,have øNS independent of concentration over the rangeæd ) 0.25-0.90.14 A constant ø(æ) has recently beenreported for lightly branched d-PE blended with analternating copolymer of ethylene and propylene for æd) 0.1-0.9.15

Hence SANS experiments show that the apparentinteraction parameter øNS and, by inference, the ther-modynamic ø depend on blend concentration in amanner that is not understood. In this paper weconsider certain experimental and theoretical aspectsthat appear to have been overlooked.

Experimental IssuesThe SANS result øNS is established by the difference

between a calculated quantity (the first term on the right-handside of eq 3, sometimes designated øs, the spinodal value of ø)and an experimental quantity (the reciprocal structure factormodified by v0/2). However, be reminded that øs depends onindependently measured values of Ni. Both terms increaseas the blend concentration is changed from æi ≈ 0.5, divergingas æi approaches zero. Thus øNS for dilute blends is establishedby the difference between two large numbers, leading to lessprecision in øNS when the usual random errors associated withNi and S(0) are considered.7,10

Experimental uncertainties are manifested differently whena single pair of polymers are blended to obtain different æd

and studied at one temperature, the condition for establishingøNS(æ). Here one is concerned with systematic errors, mostlikely incorrect Ni (molecular weight error) and S(0) (massdensity, deuterium fraction and/or absolute intensity calibra-tion errors). Taylor-Maranas et al.16 mentioned that these cancause a “parabolic” dependence of øNS on æ, an idea pursuedhere in more detail. The analysis can be made most simplyfor symmetric blends (N1 ) N2 ) N) in which the <1% changein molar volumes caused by deuterium labeling is ignored (v1

) v2 ) v0 ) v). We proceed by assuming that the Flory-Huggins approach to polymer blends is obeyed, i.e., for error-free experiments one obtains

Here the Flory-Huggins interaction parameter ø is indepen-dent of æ and N; under these conditions eq 3 clearly returnsøNS ) ø. But now we allow for systematic errors so that theexperimental N′ ) N/a and S′(0) ) bS(0). Evaluation of øNS

is done in terms of potentially incorrect quantities N′ and S′(0):

Substituting for N′ and S′ with eq 6 we obtain

The parallel to eq 5 is obvious, where â ) ø/b and γ ) (a -1/b)/2. Hence we have shown that systematic errors, whichmust exist for both degree of polymerization N and normalizedintensity S(0), may cause the type of concentration dependence

of øNS that is seen so often. This realistic assessment of theSANS experiment will lead to a perfectly smooth variation oføNS with æ. Random errors, e.g., those resulting from back-ground subtraction, sample thickness, etc. for blends of eachæd, will cause erratic departures from eq 8, but these appearto be small.

Equation 8 can be rewritten in terms of a combined errorfactor c ) ab as

The left-hand side of eq 9 is expressed in terms of N′ used todetermine øNS, thus corresponding to experiment. Someexamples of systematic error effects are shown in Figure 5 andFigure 6 for representative values of Nø ) 1.7 and 1.0. Theseserve to emphasize that positive or negative curvature willresult when c > 1 or c < 1, respectively. Note that negativevalues of øNS follow when c < 1 and that all curvature effectsare more pronounced for smaller Nø (lower molecular weightor substantially higher temperature).

Experiments on PE/d-PE indicate that øNS ∼ 1/N forconstant æd (Figure 2). The structure of eq 8 indicates thesame response, as illustrated in Figure 7. Not so obvious isthe effect of systematic errors on the temperature dependenceof øNS. Here we assume the thermodynamic Nø ) B/T, whichis varied (at constant N) over an appropriate range. The resultin Figure 8, again for c ) 1.2, recovers the experimental trendsshown in Figure 4. Upward shifts of parallel lines for loweræd enhance the apparent entropic component “A” of øNS(T) )A + B/T. Notice that N′øNS(0.5) ) 2 at exactly the true criticaltemperature when Nø ) 2; this point is considered below. Forthe case c < 1 the parallel lines are shifted downward, just asseen by Schwahn et al.13 with polystyrene isotope blends.

Hence it is demonstrated that the “anomalous” SANSresults from isotope blends, vis. upward or downward curva-ture of øNS that scales with 1/N and “entropic” coefficients Athat depend on æ, can be generated by systematic errors in

v2S(0)

) 12Næ1æ2

- ø (6)

øNS ≡ 12N′φ1φ2

- v2S′(0)

(7)

øNS ) øb

+ a - 1/b2Næ1æ2

(8)

Figure 5. Error model results N′øNS versus æ for Nø ) 1.7,with indicated values of c.

Figure 6. Error model results N′øNS versus æ for Nø ) 1.0,with indicated values of c.

N′øNS ) Nøc

+ c - 1c

12æ1æ2

(9)

Macromolecules, Vol. 31, No. 17, 1998 Thermodynamic Interactions in Isotope Blends 5855

either degree of polymerization (N′ * N) or structure factor(S′(0) * S(0)).

DiscussionThe concentration dependence of øNS has been most

often attributed to oversimplifications in the Flory-Huggins treatment of binary solutions. One can envi-sion two classes of refinements: (1) those involvinggeneral effects, e.g., nonrandom mixing or nonidealentropy of mixing, which may yield universal correctionsmanifested in a common øNS(æ,N), or (2) those involvingsystem specific corrections (e.g., enthalpic interactionsor volume changes varying with æ) that would allowøNS(æ,N) to have any sort of behavior, including upwardor downward curvature. A third possibility is (3) thesimplest Flory-Huggins thermodynamic model is cor-rect, and the observed øNS(æ,N) results from imprecisemeasurements, as described in the preceding section.The experimental record summarized in Table 1 pre-cludes (1), a general modification of solution thermo-dynamics. This conclusion is based not only on thequalitative distinction between PS/d-PS and the otherisotope blends but also on the various magnitudes ofupward curvature summarized in Table 1. Any generalapproach must involve a combination of (1) and (3) toaccount for varied results. While a system-specificmodification of blend thermodynamics (2) is alwayspossible, it is difficult to imagine what differentiates PSso strongly from the other polymers. We first considerthe plausibility of the systematic error model and thenreturn to theoretical implications.

A. Error Estimates. Note that the concentrationdependence of øNS in eq 9 is a function of both the error

factor c and the size of Nø. The apparent combinederror factor c associated with a particular experimentalsystem can be estimated from the observed ratio of øNSat two values of æd together with NeøNS at æd ≈ 0.5.Table 1 includes these estimates for the eight sets ofresults on seven different isotope blend systems. Wefocus for the moment on those displaying upwardcurvature of øNS(æ), here interpreted as resulting fromc > 1. The data for PI, PEO, and PMMA are allaccounted for by error factor c e 1.1, which is areasonable value. For instance, the intensity calibrationcould be high by 5% (b ) 1.05) and molecular weightslow by about the same factor (a ) 1.05). The concentra-tion dependence reported for PEE, PVE, and PE re-quires c ≈ 1.3-1.5, which is admittedly beyond therange c ) 0.9-1.1 corresponding to ca. (5% errorsusually associated with measurements of molecularweight and absolute intensities.7,10

Data for PS/d-PS are revealing in this regard. Thissystem is unique in many ways; it was studied bydifferent groups, shows downward curvature of øNS, andwas investigated in the glassy state. Results fromSchwahn et al.13 and Londono et al.7 are plotted inFigure 9, along with data for a second, higher Ne PS/d-PS system.7 Data from Londono et al. are describednicely by c ) 0.9, while those from Schwahn are quitedifferent and require c ≈ 0.6 as a rough assessment ofthe combined errors if it is assumed that ø is constant.Note that this latter factor implies systematic errors ofthe same absolute size as those for PEE, PVE, and PE(c ≈ 1.4), considerably beyond the conventionally ac-cepted range of 0.9 e c e 1.1. In this instance, however,later studies showed that the apparent molecularweights of both conventional and perdeuterated poly-styrene were nearly 30% too large.17 With revised N′the first term on the right side of eq 7 is larger andøNS(æ) closely resembles the Londono results (c ≈ 0.9);øNS is no longer negative for low æd. It is not known ifthe downward curvature of øNS(æ) in PS/d-PS glassesis real or derives from some residual errors. The pointto be made is that, in at least one case, large systematicerrors do occur.

Systematic errors are by nature more difficult todetect and assess than random errors. It is fair to askif another experimental result can be brought to bearon the problem. One immediately thinks of macroscopic

Figure 7. Error model results øNS versus 1/N for differentvalues of æ. Here c ) 1.2 and the constant ø ) 6 × 10-4.Compare to Figure 3.

Figure 8. Error model results N′øNS versus 1/T for differentvalues of æ. Here c ) 1.2 and Nø ) 600/T. Compare to Figure4.

Figure 9. Concentration dependence of øNS in PS/d-PS glassesannealed at T ≈ 160 °C. NeøNS ) 1.75 system as analyzed byLondono et al.7 (9). Higher molecular weight NeøNS ) 2.5system as analyzed by Londono et al.7 (b). Results fromSchwahn et al.13 (2) with original (incorrect) molecular weights.Dashed lines are approximate fits to the error model with c )0.9 and c ) 0.6.


phase separation or “cloud point” measurements, mostunambiguously done with blends having the criticalconcentration æc ≈ 0.5. Unfortunately, this test willreturn a result completely consistent with “erroneous”SANS measurements of øNS in the one-phase state.Consider the extreme example c ) 1.5 when the appar-ent molecular weight is 33% too low (a ) 1.5) or theless likely condition when intensities are 50% too large(b ) 1.5). Equation 9 becomes N′øNS ) (Nø + 0.25/æ1æ2)/1.5, which equals 2.0 at precisely the true critical point(Nø ) 2 and æ1 ) æc ) 0.5). Figure 8 presents anotherexample for c ) 1.2. Hence phase separation of criticalblends will occur at exactly the temperature predictedfrom measurements of øNS(T). Experiments with off-critical blends can yield volume fractions and/or phaseconcentrations18,19 that can in principle test the con-centration dependence of the thermodynamic ø andhence the significance of øNS(æ). However, the shapeof the binodal is scarcely altered, even when ø has a(small) æ dependence consistent with the (large) con-centration dependence of øNS in Figure 1,9,10 meaningthat phase-separated structures cannot practically dis-tinguish between real and error-generated concentrationdependence of øNS. While this situation is frustratingfrom the perspective of assessing the reality of øNS(æ),it also means that phase behavior can be interpretedconfidently by the Flory-Huggins model with interac-tion parameters (constant ø ) øNS(0.5)) measured forsymmetric blends.

We believe that it is more than coincidental that theempirical form of øNS(æ,N) summarized in eq 5 is exactlythe same as eq 8 expected for systematic errors. Atissue are two questions: Why are all curvatures up-ward, except for PS/d-PS? Can the factor c be as largeas 1.5? Possible answers involve details of molecularweight measurement and intensity calibration that arebeyond the scope of this work. It is suspected thatmolecular weight measurements may vary by more thanusually quoted. We refer to the 26% change in Ne forPS/d-PS mentioned previously.17 PE/d-PE, for whichøNS has the largest upward curvature, has molecularweights determined by gel permeation chromatography(GPC). The author is aware of very careful measure-ments on a standard reference material PE for whichMw from GPC was variously reported as 170k20 and140k,21 a 20% difference that is two-thirds that neededto account for c ) 1.5 for that system (Table 1).

The exact role of experimental errors in evaluationof øNS cannot be established at present. Reasonablevalues of c ) 0.9-1.1 account for the downward curva-ture in PS and upward curvature in PI, PEO, andPMMA; such “moderate” changes of øNS with æd are notreliable evidence for deviations from the Flory-Hugginsmodel. Systems with stronger upward curvature (PE,PVE, PEE, and model polyolefins) can also be accom-modated by systematic errors, but only if c g 1.3, whichis beyond the normally accepted range. It is likely (butnot certain) that the SANS result øNS(æ,N) is real inthese cases. However, the size of changes of øNS withæd and N are subject to expected effects from conven-tional c ) 0.9-1.1. Even for these systems it isdisquieting that all features of “non-Flory-Huggins”behavior are self-consistently reproduced by the errormodel.

B. Theoretical Approaches. Accepting that theobserved øNS(æ,N) are correct (not the result of system-atic errors) in at least some systems, behavior appears

dominated by entropic effects in the free energy ofmixing. Symmetric changes around æd ) 0.5 andvertical shifts of øNS(1/T) both testify to this, as doesthe fact that “anomalous” effects vanish as 1/N (eq 5).This entropic nature of “non-Flory-Huggins” resultswas noted by Londono et al.7 but has not been empha-sized in the many theoretical studies that address theconcentration dependence of øNS. Those have focusedon nonrandom mixing associated with concentrationfluctuations near the critical point and on densityfluctuations (compressibility), both of which are ignoredin the Flory-Huggins approach used to define øNS (eq3). We make no attempt to review comprehensively thelarge volume of literature that has addressed theseissues for polymer blends but consider those treatmentsmost applicable to isotope blends (structurally identicalchains), with emphasis on the divergence of øNS at lowconcentrations.

With the classical model, concentration fluctuationsin the one-phase liquid are predicted to diverge in bothamplitude and length scale as the critical point (or anypoint on the spinodal) is approached, typically bydecreasing temperature. These are characterized bydeviations from random mixing and the formation oflarge concentration gradients. The former modify bothenthalpy and entropy of the mixed state, and the latterleads to an additional enthalpic gradient energy penalty.The consequence is to damp the magnitude of concen-tration fluctuations, meaning that S(0) is reduced fromits classical value, hence øNS(0.5) is less than the Flory-Huggins ø, which represents the enthalpy of binaryinteractions. At a fixed temperature these fluctuationsdiminish as global concentration is changed from æc ≈0.5, and one expects øNS to rise toward ø as æ is reduced.Related expectations are that the æ dependence of øNSbecomes smaller as the temperature is increased fromTc and that the importance of excess energy associatedwith fluctuations becomes smaller for larger chains. Therole of fluctuations in incompressible polymer blendshas been addressed by two groups. In one case øNS ispredicted to have a very small downward curvature,22

while the other analysis has øNS diverging upward,11

similar to experiments in Figure 1. The fact that twotreatments of the same problem lead to qualitativelydifferent results, neither of which is expected from ageneral consideration of fluctuations, is disturbing.23 Wenote also that experiments on PE/d-PE show that øNS(æ)is unchanged by proximity to the critical temperature(Figure 3), indicating that the observed effect does notderive from fluctuations of this sort.

The role of compressibility in blend thermodynamicshas led to some confusion. At issue is the effect of voidsor free volume, a third component with volume fractionφv ≈ 0.1, on the blend free energy and øNS. For isotopeblends one expects φv to be effectively independent ofæd because compressibilities of the pure polymers andthe blend are very similar. A number of treatments ofcompressible isotope blends have indicated that øNS,defined by the “incompressible” approach embodied ineq 3, diverges either upward24-26 or downward27-29 atlow concentrations of either component. It is likely thatthese calculated results, which indeed mimic experi-mental øNS(æ,N) effects, result from subtle problemsencountered when describing compressible and incom-pressible blends with lattice models. In Appendix 1 itis shown that a constant volume fraction of voids doesnot lead to divergent behavior of øNS when S(0) and øs


are defined consistently, as is done with analysis ofexperimental data. This finding is in accord with recentanalyses of systems in which compressibility is allowedto vary a small amount, as in isotope blends. Taylor-Maranas et al.16 show with a lattice-fluid model thatcompressibility does not modify the SANS intensity atsmall æ; the only way to account for observed I(0) orS(0) is to increase the thermodynamic ø for diluteblends. Kumar et al.30 use general thermodynamicarguments to show that øNS is similarly unaffected byfree volume of the magnitude seen in “weakly interact-ing” polymer blends. Hence compressibility itself (com-ponents and blend of same or similar compressibilityor void content) cannot account for the observed con-centration dependence of øNS in isotope blends.

Fluctuations and compressibility are both included inthermodynamic approaches more sophisticated than theFlory-Huggins model. Singh et al.31 have employed thepolymer reference interaction site model (PRISM) toaddress the concentration dependence of øNS in struc-turally symmetric (isotope) blends. Density or voidfraction is independent of blend concentration, as ex-pected for weakly interacting systems, in this off-latticecalculation. They find that øNS invariably increases asthe blend concentration is changed from æc ) 0.5. Themagnitude of the change depends on model details butis smaller for larger N and less compressible (moredense) systems. For N g 1000 and the highest liquiddensity, the calculated ratio øNS(0.1)/øNS(0.5) ≈ 1.1,which is much smaller than the reliable experimentaleffects in Table 1. More importantly, there is noindication that øNS(æ) diverges at low æ, contrary toobserved behavior. Quite interesting in this regard isthe recent work of Melenkevitz32,33 using off-latticeoptimized cluster theory (OCT) to model isotope blends,specifically the PE/d-PE system of Londono et al.7 Hereagain fluctuations and constant compressibility areincluded. The model mimics experiment in many ways,with øNS(æ,N) described by the empirical eq 5; both thedivergence of øNS(æ) and the 1/N dependence of øNS(N)are captured. Calculations were done at two tempera-tures, permitting one to determine that the apparentincrease in øNS has entropic character (A increases whenæ is altered from 0.5). Changes of øNS in the model are,however, about 10% of the observed ones, i.e., øNS(0.1)/øNS(0.5) ≈ 1.1 rather than the ratio of 1.8 in Figure 1.Such calculated effects would be virtually impossible tomeasure with confidence, as they are of the same sizeas those generated by a combined error factor c < 1.03.

It appears unlikely that the large concentrationdependence of øNS(æ) seen for example in Figures 1 and3 results from either fluctuations or compressibility.Stipulating that such experiments are not stronglycompromised by systematic errors, one is led to examineøs or the term in parentheses in eq 3. Experimentalresults suggest that this term, which derives from theideal entropy of mixing, is inadequate, with discrepan-cies being larger at low concentrations. Flory34 clearlyidentified limitations of the lattice model for calculating∆Sm in dilute solutions, where local concentrations arequite different from the global volume fraction æ. Inblends one expects a similar situation at concentrationsbelow æ* ) 3v/(4πN1/2l3), where one (labeled) chain withradius of gyration Rg ) N1/2l is completely solvated bychains of the other type (unlabeled), but effects may beseen at higher æ. The point is that in dilute blends thereare additional concentration fluctuations, not accounted

for in Flory-Huggins or related theories, that arisepurely from chain connectivity.

The SANS result from eq 3 is based on the idealentropy of mixing calculated per lattice site or liquidvolume element v0:

This expression equals zero for either pure component(æi ) 0) and furthermore is the maximum entropy for arandomly mixed blend with concentrations æ1 and æ2) 1 - æ1. Any modification for dilute blend effects willreduce the magnitude of ∆Sm below that for ∆Sm

id whenæi ∼ æ*, and ∆Sm must approach 0 when æi ) 0. Giventhese constraints, the concentration derivative of en-tropy should be less than that for the ideal case: ∂∆Sm/∂æi < ∂∆Sm

id/∂æi. The second derivative is needed forcalculating S(0). We assert without proof that thesecond concentration derivative of ∆Sm in dilute blendsis smaller than the classical øs; an example based onthe empirical form of øNS(æ,N) is given in Appendix 2.There it is also shown that downward curvature of øNS-(æ) is inconsistent with this thermodynamic argument.

Systems in Table 1 have æ* in the range 0.015 (PE/d-PE) to 0.085 (PEE/d-PEE). Since effects are seen atconcentrations up to 10æ* in some cases (e.g., Figure1), a gradual crossover is implied. There is no apparentrelation between æ* and the magnitude of øNS(æ)changes. It is possible that the strength of the “isotopeeffect” is operative here. Flory34 demonstrated yearsago that repulsive interactions in a polymer/solventsystem can lead to an apparently ideal entropy of mixingthat defines the “Θ condition”. Similar argumentsshould apply to polymer-polymer blends, and thedifferent upward curvatures, assuming they are real,may reflect different amounts of “compensation” towardthe equivalent of ideal mixing. We are not aware thatthis approach to the concentration dependence of øNShas been taken, although it may be present in the OCTstudy mentioned above. Melenkevitz and Curro32 findthat nonrandom mixing in their model of the PE/d-PEsystem extends over ∼3Rg for a blend near the criticalpoint (æ ) 0.5), which is expected from fluctuations.More interesting is that nonrandom mixing has decayedby only ∼50% when æ ) 0.1. As this dilute blend is farfrom the spinodal, nonrandom mixing could be a mani-festation of the postulated dilute blend effect.

Conclusions

There is no doubt that the popularity of the Flory-Huggins approach to polymer thermodynamics derivesin large measure from its simplicity. While this modelis not expected to apply to all systems, isotope and otherweakly interacting blends would appear to conform mostclosely to the assumptions of equal monomer volumesand a concentration independent interaction parameterø. On this basis it is surprising that SANS experimentson such “model systems” indicate strong departuresfrom regular solution behavior.

Taken at face value, the concentration, chain length,and temperature dependencies of øNS in isotope blendshave every characteristic of originating from the entropyof mixing. It is unfortunate that unavoidable, and inmany cases reasonable, systematic errors in the SANSexperiment generate exactly the same effects. The most

-∆Smid

k)

æ1

N1ln æ1 +

æ2

N2ln æ2 (10)


conservative position is that the magnitude and eventhe sign of changes in øNS(æ) are uncertain. Neverthe-less, the dominance of experimental reports of upwardcurvature of øNS(æ) implies that such responses aregenuine reflections of the thermodynamics of the mixedstate. This sort of “non-Flory-Huggins” behavior doesnot affect the relation between øNS measured for criticalblends and macroscopic phase separation.9,10

We are aware of no thermodynamic model thataccounts for the type and magnitude of øNS(æ,N) inisotope and other weakly interacting polymer blends.A qualitative case is made for the modified entropy ofmixing associated with chain connectivity, which isexpected to be most important in dilute blends. Anyapproach must account for system specific effects, as theconspicuous curvature of øNS(æ) appears to vary frommoderate (PI, PEO, PMMA) to strong (PE, PEE, PVE).We conclude by reiterating that the symmetric concen-tration dependence of øNS in isotope blends, where theonly “asymmetry” is the substitution of deuterium forhydrogen, presents an ongoing challenge to our under-standing of the thermodynamics of polymer mixtures.There are apparent departures from the ideal entropyof mixing which, although they vanish in the limit ofinfinite N, are system specific. Our ability to handlemore complex mixtures10,14,15 is compromised in theabsence of an explanation for this “isotope effect”.

Acknowledgment. The author is indebted to R.Krishnamoorti, J. D. Londono, and D. Schwahn forsharing data and insights into the SANS experiment.Discussions with K. Freed, W. Graessley, S. Kumar, J.Melenkevitz, M. Olvera, and W. Stockmayer wereinvaluable.

Appendix 1

The effect of compressibility (presence of voids) on theanalysis of thermodynamics and scattering is consideredfor a system of unit volume composed of polymer 1(volume fraction φ1), polymer 2 (φ2), and voids (φv) on alattice of Nc cells of volume vc ) 1/Nc. Following Freed,24

one can write the free energy of mixing per lattice cellas

Here â ) 1/kT, Ni is degree of polymerization of polymeri, and φ1 + φ2 + φv ) 1. The øij are the usual normalizedexchange energies for interactions between dissimilarmonomers, here including voids. Assuming for simplic-ity that void fraction φv is independent of blend concen-tration (expected to be approximated closely for isotopeblends), the static structure factor is given by35

The same compressible system (polymers plus voids)can be treated on an “incompressible” lattice of the sameunit volume but having fewer cells Ninc ) (1 - φv)Nc oflarger volume vinc ) vc/(1 - φv). The free energy ofmixing ∆F is unchanged, but it is written in terms ofthe incompressible lattice as

Here the blend concentration is expressed in terms ofnominal volume fractions æi ) φi/(1 - φv) where æ1 +æ2 ) 1 and ø12

inc represents the free energy in excess ofthe combinatorial entropy of mixing. While the thirdterm in eq 13 is associated with pairwise contactsbetween polymer 1 and polymer 2, ø12

inc is not the sameas ø12 in eq 11. The static structure factor is written asin Flory-Huggins theory, assuming ø12

inc is independentof blend concentration, as

Either eq 12 or eq 14 may be used to evaluate ø12 orø12

inc, respectively. The relation between them is ob-tained by rewriting eq 14 as

Substitution from eqs 12 and 14 leads to

or, using the right-hand sides of eqs 12 and 14:

With the identity vc ) (1 - φv)vinc, eq 16 returnsunequal structure factors for the two models, i.e. Sc(0)) (1 - φv)2Sinc(0). This inequality of structure factors,which has not been generally appreciated, derives fromthe “contrast” factor in parentheses in eq 1 of the text.For a compressible model the scattering length densitiesare based on “hard core” volumes (vc) rather than thelarger nominal molar volumes (vinc) of monomers.Either side of eq 16 is consistent with a sample havinga coherent cross section per unit volume I(0), which canbe analyzed according to compressible or incompressiblelattice models.

The only difference between the “incompressible”result ø12

inc and the “compressible” ø12 is the factor (1 -φv) in eq 17. This is readily understood in terms of theheat of mixing from 1-2 contacts in the two models,â∆Hm ) Ncφ1φ2ø12 ) Nincæ1æ2ø12

inc, which leads directlyto eq 17. Because voids are not considered explicitly,the factor Nincæ1æ2 in the incompressible model overes-timates the number of 1-2 contacts, so the interactionparameter ø12

inc must be reduced to correct for thisovercounting.

In summary, the mere presence of a void population(constant φv > 0) alters neither the thermodynamics norscattering of a binary blend in a significant manner. Anycomparison of analyses by compressible and incompress-ible models should account properly for differences instructure factors S(0) and cell or reference volumes v.When this is done, the concentration independentinteraction energy embodied in ø (or øNS) is affected onlyby a constant factor (1 - φv). There is no divergence oføNS(æ) that can be attributed to compressibility of an

â∆FNc

) φv ln φv +φ1

N1ln φ1 +

φ2

N2ln φ2 + φ1φ2ø12 +

φ1φvø1v + φ2φvø2v (11)

vc

Sc(0))

∂2(â∆F/Nc)

∂φ12

) 1N1φ1

+ 1N2φ2

- 2ø12 (12)

â∆FNinc

)æ1

N1ln æ1 +

æ1

N1ln æ1 + æ1æ1ø12

inc (13)

vinc

Sinc(0))

∂2(â∆F/Ninc)

∂æ12

) 1N1æ1

+ 1N2æ2

- 2ø12inc

(14)

∂2(â∆F/Ninc)

∂æ12

)Nc

Ninc

∂2(â∆F/Nc)

∂φ12 (∂φ1

∂æ1)2

(15)

vinc

Sinc(0)) (1 - φv)

vc

Sc(0)(16)

ø12inc ) (1 - φv)ø12 (17)


experimental blend. It is obvious, however, that mixingelements of the two approaches, e.g., using a compress-ible vc/Sc(0) on the right-hand side of eq 3, when thecombinatorial entropy terms are based on in incom-pressible model, will cause a meaningless divergence of“øNS” at low concentrations. In fact, this is just anotherexample of a systematic error (from theory, not experi-ment) affecting S(0) and hence øNS(æ). The need forconsistent application of a particular model has beenemphasized by Taylor et al.26

Appendix 2The empirical expression for øNS(æ,N) in eq 5 can be

integrated twice over æ1 to obtain the following expres-sion for the thermodynamic ø consistent with the Flory-Huggins model:9,10

The second term can be assigned to the departure fromideal entropy of mixing, leading to a phenomenologicalequation for symmetric blends (N1 ) N2 ) N):

This modified version of the entropy of mixing is plottedin Figure 10 for γ ) 0.17 (equivalent to c ) 1.5 for PE/d-PE) and γ ) -0.055 (equivalent to c ) 0.9 for PS/d-PS). Be reminded that there is no theoretical basis forthe particular form of eq 19, which derives fromexperimental measurements of øNS(æ,N). For γ > 0(upward curvature) this model reflects the reducedentropy of mixing expected from general considerationof chain connectivity. Downward curvature is hereassociated with γ < 0, implying that ∆Sm > ∆Sm

id,which seems impossible if the ideal entropy is indeedthe maximum. This is another reason for believing thatnegative curvature of øNS(æ) has causes other thanthermodynamic ones (e.g., systematic errors).

References and Notes

(1) Buckingham, A. D.; Hentschel, H. G. E. J. Polym. Sci., Polym.Phys. 1980, 18, 853.

(2) Bates, F. S.; Wignall, G. D.; Koehler, W. C. Phys. Rev. Lett.1985, 55, 2425.

(3) Green, P. F.; Doyle, B. L. Phys. Rev. Lett. 1986, 57, 2408.(4) Bates, F. S.; Wiltzius, P. J. Chem. Phys. 1989, 91, 3258.

(5) Rhee, J.; Crist, B. J. Chem. Phys. 1993, 98, 4147.(6) Graessley, W. W.; Krishnamoorti, R.; Balsara, N. P.; Lohse,

J. D. Macromolecules 1993, 26, 1137.(7) Londono, J. D.; Narten, A. H.; Wignall, G. D.; Honnell, K.

G.; Hsieh, E. T.; Johnson, T. W.; Bates, F. S. Macromolecules1994, 27, 2864.

(8) de Gennes, P. G. Scaling Concepts in Polymer Physics; CornellUniversity Press: Ithaca, NY, 1979.

(9) Crist, B. J. Polym. Sci., Part B: Polym. Phys. 1997, 35, 2889.(10) Krishnamoorti, R.; Graessley, W. W.; Balsara, N. P.; Lohse,

D. J. J. Chem. Phys. 1994, 100, 3894.(11) Bates, F. S.; Muthukumar, M.; Wignall, G. D.; Fetters, L. J.

J. Chem. Phys. 1988, 89, 535.(12) Krishnamoorti, R. Unpublished results.(13) Schwahn, D.; Hahn, K.; Streib, J.; Springer, T. J. Chem. Phys.

1990, 93, 8383.(14) Lin, C. C.; Jonnalagadda, S. V.; Balsara, N. P.; Han, C. C.;

Krishnamoorti, R. Macromolecules 1996, 29, 661.(15) Maurer, W. W.; Bates, F. S.; Lodge, T. P.; Almdal K.;

Mortensen K.; Fredrickson, G. H. J. Chem. Phys.1998, 108,2989.

(16) Taylor-Maranas, J. K.; Debenedetti, P. G.; Graessley, W. W.;Kumar, S. K. Macromolecules 1997, 30, 6943.

(17) Schwahn, D. Personal communication. Muller, G.; Schwahn,D.; Eckerlebe, H.; Rieger, J.; Springer, T. J. Chem. Phys.1996, 104, 5326.

(18) Crist, B.: Nessarikar, A. Macromolecules 1995, 28, 890.(19) Scheffold, F.; Eiser, E.; Budkowski, A.; Steiner, U.; Klein, J.

J. Chem. Phys. 1996, 104, 8786.(20) “Reference Standard Polyethylene Resins and Piping Materi-

als”, GRI-86/0070, Gas Research Institute, Chicago, IL, 1987.(21) “Reference Standard Polyethylene Resins and Piping Materi-

als”, GRI-87/0326, Gas Research Institute, Chicago, IL, 1988.(22) Olvera de la Cruz, M.; Edwards, S. F.; Sanchez, I. C. J. Chem.

Phys. 1988, 89, 1704.(23) Equation 11 of ref 11 adds to the standard Flory-Huggins

free energy of mixing a term that reduces to 2x2/πN3/2 for æ) 0 or æ ) 1. Hence the free energy of mixing does not vanishin the limits of zero concentration, a thermodynamicallyimpossible result that leads to the upturn of øNS at low orhigh æ.

(24) Freed, K. F. J. Chem. Phys.1988, 88, 5871.(25) Bidkar, U. R.; Sanchez, I. C. Macromolecules 1995, 28, 3963.(26) Taylor, J. K.; Debenedetti, P. G.; Graessley, W. W.; Kumar,

S. K. Macromolecules 1996, 29, 764.(27) Dudowicz, J.; Freed, K. F. Macromolecules 1990, 23, 1519.(28) Dudowicz, J.; Freed, M. S.; Freed, K. F. Macromolecules 1991,

24, 5096.(29) Dudowicz, J.; Freed, K. F.; Lifschitz, M. Macromolecules 1994,

27, 5387.(30) Kumar, S. K.; Veytsman, B. A.; Maranas, J. K.; Crist, B. Phys.

Rev. Lett. 1997, 79, 2265.(31) Singh, C.; Schweizer, K. S.; Yethiraj, A. J. Chem. Phys. 1995,

102, 2187.(32) Melenkevitz, J.; Curro, J. G. J. Chem. Phys. 1997, 106, 1216.(33) Melenkevitz, J. Macromolecules, in press.(34) Flory, P. J. Principles of Polymer Chemistry; Cornell Univer-

sity Press: Ithaca, NY, 1953; Chapter 12.(35) Equation 12 is not exact, even for the case of constant volume

fraction of voids considered here. The complete expressionfor the inverse structure factor in a compressible blend (seeeq 2.11 of ref 27) contains additional terms that may bewritten as

2øcorr )[1 - N2φ2(ø12 - ø1v + ø2v)]

2

N2φ2[(2ø2v - fs) - 1]

With values appropriate for SANS experiments on isotopeblends (Nø12 ≈ 1, ø1v ) ø2v ≈ 2, and fs )1/φv ≈ 10) thecorrection term øcorr is of order ø12/N and may safely beneglected.

MA971858K

Figure 10. Entropic contribution to the free energy of mixingas a function of blend concentration for ideal mixing (s) andthe empirical eq 12 for γ ) 0.167 (- - -) and for γ ) -0.055(‚-‚).

ø ) â - 2γNeæ1æ2

(æ1 ln æ1 + æ2 ln æ2) (18)

-N∆Sm

k) (1 - γ)(æ1 ln æ1 + æ2 ln æ2) (19)


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Thermodynamic Interactions in Isotope Blends: Experiment and Theory