Thermodynamic Interpretation of the SEC Behavior of Polymers in a Polystyrene Gel Matrix

Macromol. Chem. Phys. 2001, 202, 1889–1901 1889

Thermodynamic Interpretation of the SEC Behavior ofPolymers in a Polystyrene Gel Matrix

Rosa García, Clara M. Gómez, Juan E. Figueruelo, Agustín Campos*

Departament de Química Física and Institut de CiØncia dels Materials, Universitat de Val�ncia, Dr. Moliner 50,E-46100 Burjassot, Val�ncia, SpainFax: + 34 96 386 4564; E-mail: [email protected]

IntroductionDuring the last decade, blending as a route to enhancedspecific material properties has been well established andexploited in a variety of commercial products.[1] The finalproperties of the polymeric blend will depend amongothers properties on the degree of miscibility between thecomponents.[2, 3] In this sense, a characteristic polymer-polymer interaction must be effective between the com-ponent polymers for them to be miscible.[1, 2] The compat-ibility of polymer blends can be determined theoreticallyas well as experimentally using different techniques suchas inverse gas chromatography,[4] thermal analysis,[5] elec-tron microscopy,[6] spectroscopy,[7] and viscosity.[8] Tomodel the miscibility behavior of polymer solutions andblends, the most widely used and best known of theincompressible-lattice models is the Flory-Huggins latticetheory[9, 10] which illustrates in a simple way the competi-tion between the entropy of mixing and the attractionforces producing liquid-liquid phase separation.

For ternary polymer systems, solvent(1)/polymer(2)/polymer(3), the Flory-Huggins theory modified byPouchly[11–13] has been used to describe the liquid-liquidequilibrium. The equilibrium composition of the threecomponents in the two phase systems has lead to thedetermination of the polymer-solvent, polymer-polymerand ternary composition-dependent interaction func-tions[14–18] instead of the concentration-independent Floryinteraction parameters, v, usually found in the litera-ture.[19]

On the other hand, size exclusion chromatography(SEC) is a widely used technique for the analysis andcharacterization of polymers and is based on the separa-tion of macromolecules by size, that is closely related totheir retention volumes. The drawback encountered isthat under real conditions, phenomena different frompure exclusion appear, i.e., secondary mechanisms,[20–22]

and the universal calibration method[23] is no longer valid.Polymer elution volumes, Ve, are shifted to higher or

Full Paper: Binary and ternary interaction functionsdependent on the composition of the components of aternary polymer system – solvent(1)/polymer(2)/poly-mer(3) – have been obtained. To attain this goal the equi-librium compositions of the three components fromliquid-liquid equilibria have been determined. The inter-action functions show a good coincidence with reportedones and have been used to better understand the SECmechanisms and the preferential solvation phenomenon.For these purposes, we first have derived an expressionthat relates the distribution coefficient of the solutebetween the mobile phase and the stationary phase in SECexperiments with the interaction functions and the compo-sition of the ternary phase. Second, an expression to cal-culate the preferential solvation coefficient for solvent(1)/polymer(2)/polymer(3) systems has been derived as afunction of the interaction functions and of the systemcomposition. A good qualitative accordance has beennoticed between the SEC distribution coefficient and thepreferential solvation parameter.

Macromol. Chem. Phys. 2001, 202, No. 9 i WILEY-VCH Verlag GmbH, D-69451 Weinheim 2001 1022-1352/2001/0906–1889$17.50+.50/0

Dependence of lnKp with V2/V1 for different ternary polymersystems: (J) THF/PBD/PS; (H) Bz/PBD/PS; (h) Diox/PBD/PS; (F) Tol/PDMS/PS; (g) Bz/PDMS/PS; (9) CHX/PDMS/PSand (6) Diox/P2VPy/PS.

1890 R. García, C. M. Gómez, J. E. Figueruelo, A. Campos

lower values with respect to those in the universal cali-bration curve when compatibility or incompatibilitybetween solute and gel appear, respectively. These phe-nomena can be quantified by means of the distributioncoefficient parameter between the mobile phase and thestationary phase, Kp, that represents the ratio of the con-centrations of macromolecules inside the pores (i.e. in thequasi-stationary phase) and in the free volume of liquid(i.e. in the interstitial mobile phase).

Moreover, the different Ve values found in SEC experi-ments can be understood by considering either adsorptionof the solute onto the gel or the opposite case of incom-patibility between solute polymer and gel. Both phenom-ena can be related with preferential or selective sorption.In this respect, the preferential adsorption coefficient, k,is an equilibrium property dealing with the solvent com-position close to and far from the polymer chain. Nega-tive values of k will indicate preferential sorption ofsolute polymer (component 2) onto the gel matrix (com-ponent 3) and can be interpreted as follows: the macro-molecules are “dissolved” within the “gel phase” (themacromolecules penetrate the grains), in these conditionsKp A 1. Contrary is the case of incompatibility betweensolute polymer and gel matrix with k A 0 (k positive andsmall values) and 0 a Kp a 1. It is clear that the compat-ibility (or incompatibility) between two polymers mustbe studied in relation to both polymers, depending on thesolvent used.

In this paper, we have obtained concentration-depend-ent binary and ternary interaction functions from theliquid-liquid equilibrium of ternary polymer systems, sol-vent(1)/polymer(2)/polymer(3). The ternary systemsstudied are: tetrahydrofuran (THF)/polybutadiene (PBD)/polystyrene (PS), benzene (Bz)/PBD/PS, 1,4-dioxane(Diox)/PBD/PS, toluene (Tol)/poly(dimethylsiloxane)(PDMS)/PS, Bz/PDMS/PS, cyclohexane (CHX)/PDMS/PS and Diox/poly(2-vinylpyridine) (P2VPy)/PS. Theobtained functions have been used for a better under-standing of the SEC separation process by relating theinteraction functions, gij, to Kp. Moreover, the interactionfunctions have been used to evaluate the preferentialsorption coefficient in these systems.

Experimental Part

Chemicals

Polystyrene (PS) standards were purchased from TosohCorp. (Tokyo, Japan) with weight average molar mass M

—w =

37900, 355000, 2700000 and 1500000, and from PolymerStandards Service (USA) with M

—w = 6870 and 17200, all of

them with a polydispersity index between 1.01 and 1.05, asspecified by the suppliers. Polybutadienes (PBD) with apolydispersity index between 1.03 and 1.05 were purchasedfrom Pressure Chemical (PA, USA) with, M

—w = 5900,

13400, 67300 and 86500, and from Polymer Laboratories

(Shropshire, UK) with M—

w = 268000 and 1120000. Poly(di-methylsiloxanes) (PDMS) were supplied by Polymer Labora-tories (Shropshire, UK) with M

—w = 7500, 38400, 68500,

161000 and 480000 with polydispersity indexes 1.08, 1.08,1.11, 1.17 and 1.42, respectively. Poly(2-vinylpyridine)(P2VPy) samples (polydispersity index 1.02) with M

—w =

7000, 11000 and 33000 were purchased from PressureChemicals (PA, USA), and M

—w = 200000 from Polysciences

(PA, USA). Tetrahydrofuran (THF), benzene (Bz), toluene(Tol), 1–4 dioxane (Diox) and cyclohexane (CHX) fromScharlau (Barcelona, Spain) of chromatographic grade wereused as solvents. The densities of solvents and polymersused were (in g/mL): 0.862 (Tol), 0.881 (THF), 0.774(CHX), 1.027 (Diox), 0.868 (Bz), 0.960 (PBD), 1.091 (PS),0.973 (PDMS) and 1.219 (P2VPy).

Viscometric Measurements

An automatic AVS 440 Ubbelohde-type capillary viscometerfrom Schott Geräte (Hofheim, Germany) at (25.0 l 0.1)8Cwas used to perform viscometric measurements. The stocksolution was prepared by dissolving the polymer samples inthe corresponding solvent up to a concentration of approxi-mately 1.0 g/dL. At least six dilutions were obtained by add-ing the appropriate aliquots of solvent. Elution time of thesolvent was always above 100 s. For each solution, a 12 mLsample, to minimize drainage errors, was loaded into theviscometer and placed in the thermostated bath. Measure-ments started after an equilibration time of ca. 5–10 min andwere continued until several flow time readings agreedwithin 0.5%. The elution time of each solution was thendetermined as the average of several readings. The dilutionand measurements were stopped when the time difference ofthe sample solution and pure solvent dropped below 10%.

The evaluation of the intrinsic viscosity, [g], for THF/PBD, Bz/PBD, Diox/PBD, Tol/PDMS, Bz/PDMS, CHX/PDMS and Diox/P2VPy was made by extrapolation to infi-nite dilution (zero solute concentration) of Huggins plots:gsp/c = [g] + bc , i. e. gsp/c vs. c, being gsp the specific viscos-ity, c the concentration of the solution and b a viscosimetricparameter. For every system the intrinsic viscosity has beenrelated to the molar mass through the Mark-Houwink-Sakur-ada (MHS) equation, [g] = KMa, K and a being the MHSconstants. Table 1 compiles the values of K and a at 258Cfor the different binary systems studied.

Table 1. Values of Mark-Houwink-Sakurada constants, K anda, and coefficients of the linear fits logVh = n + mVe of the SECcalibration curves for the binary systems solvent(1)/polymer(2)at 25 8C.

System KmL=g

a n mmLÿ1

THF/PBDBz/PBDDiox/PBDTol/PDMSBz/PDMSCHX/PDMSDiox/P2VPy

0.01090.1120.1550.04470.05790.1590.1754

0.7600.6040.5410.6010.5720.5340.450

16.2615.0016.6216.8015.5015.8013.56

–0.411–0.333–0.360–0.426–0.373–0.376–0.277

Thermodynamic Interpretation of the SEC Behavior of Polymers in a Polystyrene Gel Matrix 1891

Chromatography

Waters liquid chromatography equipment with refractiveindex detector was used for SEC experiments at room tem-perature as previously described.[14–18] Three l-styragel 103,104 and 105 � (nominal pore size) columns from Waters(Mildford, MA, USA) were used with THF, Tol, CHX, Bz orDiox as eluent. All chromatographic experiments were per-formed at room temperature and the columns were equili-brated overnight prior to starting any experiment. chromato-grams were obtained at a flow rate of 1.0 mL/min by injectionof 100 lL. For the set of the three l-styragel columns used inthis work, the void or interstitial volume, V0, and the porevolume, Vp, were 17.7 and 18.1 mL, respectively. The totalexclusion limit (V0) was evaluated by using a polystyrenestandard of very high molar mass (M

—w = 2700000) eluted in

THF. The total (VT) permeation limit was determined withsmall molecules as THF, Tol or Bz. In this paper, we haveanalyzed, on one hand, the equilibrium composition of seventernary systems: THF/PBD/PS, Bz/PBD/PS, Diox/PBD/PS,Tol/PDMS/PS, Bz/PDMS/PS, CHX/PDMS/PS and Diox/P2VPy/PS that form two phases in equilibrium at 258C. Thechromatographic analysis of the two phases in equilibriumgives the composition of the three components in each phase,i.e. (v91, v92, v93) and (v991, v992, v993) as previously described.[14–18]

Three different runs (1–3) of six tie-lines (a, b, c, d, e and n)each one very close in composition have been determined inthis paper. On the other hand, the elution volume behavior ofTHF/PBD, Bz/PBD, Diox/PBD, Tol/PDMS, Bz/PDMS,CHX/PDMS and Diox/P2VPy has been studied. To avoidconcentration effects on the elution volumes, Ve, all solute(polymeric) samples were injected at four different concentra-tions and then extrapolated to zero concentration. This beha-vior has been summarized in terms of the universal calibrationcurves made by plotting the hydrodynamic volume, Vh, (aslog(M[g])) versus Ve. In the molar mass range studied, all theplots are linear, being the ordinates and slopes summarized inTable 1, together with their viscometric constants.

Results and Discussion

Evaluation of Interaction Functions

Under certain experimental conditions, a ternary polymersystem solvent(1)/polymer(2)/polymer(3) forms a twophase equilibrium. The analysis of each equilibriumphase by liquid chromatography[17] allows to determinethe composition of the three components in the twophases at a given temperature, that is, (v91, v92, v93) and (v991,v992, v993). As an example, Table 2 shows the equilibriumcomposition values obtained for the system Diox(1)/PBD(2)(M

—w = 86000)/PS(3)(M

—w = 17000) at 258C. Three

different runs (1–3) of six tie-lines (a, b, c, d, e and n)each one very close in composition have been deter-mined.

With the aim of determining the interaction functionsand in the framework of the Flory-Huggins theory,[17, 18]

the phase separation of a ternary polymer system can bedescribed by the Gibbs free energy of mixing by mol ofoccupied lattice sites, DG, as a function of concentration-dependent interaction functions:

DGRT¼ ðv1 ln v1 þ

V1

V2

v2 ln v2 þV1

V3

v3 ln v3

þ v1v2g12 þ v1v3g13 þ v2v3g23 þ v1v2v3gT ð1Þ

where vi (i = 1, 2, 3) is the volume fraction of componenti in the mixture (v1 + v2 + v3 = 1); g1i (i = 2, 3) is the inter-action function between a molecule of solvent 1 and achain segment of polymer 2 or 3; g23 is the interactionfunction between a segment of polymer 2 and a segmentof polymer 3; and Vi is the molar volume of the compo-nent i. gT is the ternary interaction function, the physicalmeaning of which is still unclear, but includes interac-

Table 2. Equilibrium compositions of the two coexisting phases, expressed as volume fractions, v9i and v99i determined by SEC at25 8C, for diverse families of adjacent tie-lines in the Diox(1)/PBD-86000(2)/PS-17000(3) ternary system.

Run Tie line v91 v92 v93 v991 v992 v993

1 a 0.986 0.0123 0.0014 0.962 0.00304 0.0347b 0.975 0.0147 0.0108 0.969 0.00087 0.0298c 0.973 0.0146 0.0122 0.970 0.00073 0.0292d 0.969 0.0144 0.0169 0.974 0.00063 0.0258e 0.972 0.0130 0.0151 0.971 0.00083 0.0282n 0.986 0.0126 0.00098 0.965 0.00063 0.0340

2 a 0.957 0.0369 0.0060 0.935 0.00110 0.0643b 0.951 0.0359 0.0128 0.936 0.00063 0.0632c 0.937 0.0322 0.0305 0.941 0.00074 0.0578d 0.944 0.0308 0.0248 0.938 0.00067 0.0610e 0.953 0.0293 0.0180 0.934 0.00062 0.0653n 0.953 0.0273 0.0195 0.933 0.00076 0.0660

3 a 0.961 0.0309 0.0086 0.900 0.00074 0.0998b 0.960 0.0291 0.0110 0.898 0.00095 0.1010c 0.953 0.0280 0.0187 0.900 0.00073 0.0990d 0.956 0.0261 0.0175 0.896 0.00101 0.1031e 0.943 0.0250 0.0316 0.904 0.00073 0.0956n 0.925 0.0231 0.0516 0.918 0.00103 0.0807


tions that are ternary in nature[12, 13, 24, 25] and it has beentested that its inclusion in the expression for DG is a bet-ter description of the phase separation equilibrium.[13, 16]

Therefore, DG depends on the interaction functions thatare independent of the chain length of the polymer (molarmass) since they represent interaction energy between lat-tice sites.

Quantitative evaluation of the interaction functionsinvolved in Equation (1) can be made by using the Gibbs’two-phase equilibrium condition[26] that, at constant tem-perature and pressure, is expressed by:

Dliðv91; v92; v93Þ ¼ Dliðv991; v992; v993Þ ð2Þ

where subscripts 1, 2 and 3 refer to the different compo-nents of the ternary mixture, and the prime and doubleprime refer to the two coexisting phases. The expressionsfor the change in the chemical potentials of the solvent,Dl1 = (qDG/qn1)n2;n3 ;p;T and of the two polymers, withmean polymerization degrees x2 and x3, Dl2;x2 = (qDG/qn2)n1 ;n3 ;p;T and Dl3;x3

= (qDG/qn3)n1 ;n2;p;T, have been pre-viously reported[17, 18] with v1 = n1/(n1 + n2x2 + n3x3), vi =nixi/(n1 + n2x2 + n3x3) (i = 2, 3) and ni being the number ofmoles of component i in the ternary mixture.

Therefore, from the equilibrium condition (Equation(2)), we obtain the following unknowns: (g12)9, (g13)9,(g23)9, (gT)9, (g12)99 (g13)99 (g23)99 (gT)99, as well as their deri-vatives: (dg12/dv2)9, (dg13/dv3)9, (dg23/dv3)9, (qgT/qv3)9,(dg12/dv2)99, (dg13/dv3)99, (dg23/dv3)99 and (qgT/qv3)99. Accord-ing to this, we need to solve 16 equations to obtain these16 unknowns. So, for the three components in the twoequilibrium phases we obtain a set of three equations forone tie-line (a for instance in Table 2) that can be solvedcombined with another three times four equations (i.e. b,c, d and e tie-lines) raised by using the composition offour tie-lines in every run. This approach using the equili-brium composition of five tie-lines gives 15 equationsthat allow to determine 15 unknowns. To eliminate theone remaining unknown, the best approach[18] consists inassuming that either g12 or g13 functions change linearlywith polymer composition. Consequently, either (dg12/dv2) or (dg13/dv3) are constant at any polymer concentra-tion and do not change for the two conjugate phases of a

given tie-line neither between the different tie-lines of agiven run:

(dg12/dv2)9 = (dg12/dv2)99 = (dg12/dv2) or

(dg13/dv3)9 = (dg13/dv3)99 = (dg13/dv3)

A thorough examination of the numerical values of allthe parameters (not shown here for simplicity) yields theconclusion that there is no discrepancy between the para-meters obtained through the combination of the differenttie-lines in a run. The dependence of the different interac-tion functions and their derivatives on v1 or v3 is shown inTable 3. From the inspection of this table, it is clear thatthe dependence of g12 and g13 on polymer composition isvery small, as it was previously assumed, in contrast withthe g23 and gT functions. For the system Tol/PDMS/PS thetwo g13 values come from experiments conducted with dif-ferent molar masses and agree when v3 e 0, and also thetwo g12 values for the system Bz/PBD/PS that are equalwhen v1 e 1. For the other systems, no differences havebeen found when using different polymer molar masses inthe phase separation experiments. In the past, phase equili-brium experiments have been carried out with differentmolar masses and no dependence of the different interac-tions functions on molar mass has been detected.[14–16] Forthis reason, we conclude that the interaction functionsobtained and listed in Table 3 can be used for whicheverpolymer molar mass since the interaction functions gij

denote the energy interchange between solvent moleculesand polymer segment or between polymer segments thatoccupy neighbouring sites in the network.

Next, a comparison of the interaction functions with lit-erature data is intended. The drawback is, that these gij

values cannot be directly found in the literature. Insteadof these, the Flory-Huggins parameters, vij, are usuallyreported[19] since they can be easily evaluated from thesolvent activity in a binary solvent(1)-polymer(2) system(v3 = 0). In these systems, both parameters are related bythe following relationship:[27]

g12ðv2Þ ¼1

1ÿ v2

Z 1

v2

v12ðv2Þdv2 ð3Þ

Table 3. Values of the binary, g12, g13 and g23, and of the ternary, gT, interaction functions for the different systems studied.

System g12 g13 g23 gT

THF/PBD/PS 1.15 – 0.68v1 0.56 + 0.33v3 –0.29 + 1.63v3 0.37–2.09v3

Bz/PBD/PS 0.54 – 0.05v1

1.33 – 0.84v1

0.58 – 0.15v3 0.56 + 0.007v3 –2.27 + 3.48v1 – 5.41v3

Diox/PBD/PS 0.78 – 0.18v1 0.59 + 0.04v3 –0.81 + 4.50v3 1.05 – 5.40v3

Tol/PDMS/PS 0.78 – 0.28v1 0.43 – 0.14v3

0.43 + 0.44v3

–0.33 – 1.39v3 2.48 – 2.48v1 – 0.22v3

Bz/PDMS/PS 1.31 – 0.81v1 0.58 – 0.15v3 –0.67 + 3.70v3 0.88 – 3.60v3

CHX/PDMS/PS 0.58 – 0.06v1 0.56 + 0.007v3 –1.10 – 3.20v3 1.30 + 4.50v3

Diox/P2VPy/PS 0.66 – 0.13v1 0.57 + 0.37v3 –1.20 – 2.10v3 2.10 – 4.20v3


Therefore, for the sake of comparison, Table 4 com-piles v12 values from literature,[19] the corresponding g12

functions calculated with Equation (3) and the g12 expres-sions evaluated from Table 3 by assuming that g12 is inde-pendent of v3

[12, 28] and making v1 = 1 – v2. As can beseen, the agreement between the functions evaluated inthis paper and the literature ones is rather acceptable, atleast in the tendency, and can be used to test the validityof the proposed procedure. The present calculation is notintended to fit the experimental results quantitatively, butmerely to demonstrate the qualitative comparison. It isreasonable to conclude that the calculated interactionfunctions certainly capture the trend of the bibliographicones for the present systems at least qualitatively.

Partitioning in Size Exclusion Chromatography

Chromatographic separations of macromolecules by sizerefer to the selective distribution of the macromoleculesbetween the mobile (binary phase constituted by the poly-mer solution outside the pore) and the stationary (ternaryphase formed by the mixture of polymer solution andswollen gel matrix) phases of a given chromatographicsystem. This distribution process depends on the strengthand type of the interactions, and is described by:

Ve = V0 + KDVp (4)

where Ve is the elution or retention volume, V0 the inter-stitial or void volume of the column, Vp the pore or pack-ing volume, and KD the distribution coefficient, equal tothe ratio of the solute concentration in the stationaryphase and in the mobile phase. Moreover, KD is a func-tion of the change in the Gibbs' free energy, DG, relatedto the solute partitioning between both phases, whichmay be due to different effects:

i) inside the pore, which has limited dimensions, themacromolecule cannot occupy all possible conformationsand therefore the conformation entropy, DS, decreases.

ii) when penetrating the pores, the macromolecule mayinteract with the pore walls resulting in a change inenthalpy, DH.

Consequently, depending on the chromatographic sys-tem and the chemical nature of the components, eitheronly entropic or only enthalpic interactions, or both, maybe effective. Therefore, in the general case the distribu-tion coefficient can be expressed as:

KD = KSEC Kp (5)

where KSEC is based only on entropic interactions,whereas Kp characterizes the enthalpic ones. Dependingon the magnitude of these effects, the size-exclusionmode (SEC, DS-driven) or the liquid adsorption mode(DH-driven) will be dominant.

In SEC, separation is accomplished with respect to thehydrodynamic volume (Vh = M[g]) of the macromole-cules. In the case of ideal SEC, separation is exclusivelydirected by conformational changes of the macromole-cules and then, KD = KSEC and Kp = 1. Since the confor-mational entropy decreases (DS a 0), the distributioncoefficient of ideal SEC is KSEC a 1. Accordingly, theseparation range is 0 a KSEC a 1, and the retention volumefor ideal behaviour is:

Ve = V0 + KSEC Vp (6)

However, if enthalpic effects, due to interactionsbetween the polymeric solutes and the pore walls takeplace, the distribution coefficient of real SEC is KD =KSEC Kp, and the retention volume is as follows:[21]

Ve = V0 + KSEC Kp Vp (7)

with Kp a 1, if solute-gel interactions are repulsive (i.e.electrostatic type) or Kp A 1 if they are attractive (reversi-ble adsorption of polymer onto the matrix packing).

In this regard, recently, we have derived an expressionto evaluate Kp from a thermodynamic starting point, thatat v20 e 0 (the usual condition in SEC experiments) canbe written as:[18]

ln Kp ¼ ÿV2

V1v3f1þ g23 ÿ g13 ÿ g12 ÿ ðv2 ÿ v1ÞgTg

¼ ÿ V2

V1v3g�chr ð8Þ

or by assuming that (v2 – v1) l –1 when v2 e 0 and v3 e

0 (since the gel fraction dissolved in the ternary phase isvery low), as:

ln Kp ¼ ÿV2

V1v3f1þ g23 ÿ g13 ÿ g12 þ gTg

¼ ÿ V2

V1

v3gchr ð9Þ

Table 4. Comparison of calculated and experimental sol-vent(1)-polymer(2) interaction parameters at 25 8C.

System v12a) g12

b) g12c)

Tol/PS 0.44 – 0.35v2 0.27 – 0.18v2 0.43 – 0.14v2

Tol/PDMS 0.44 + 0.37v2 0.61 + 0.18v2 0.50 + 0.28v2

Bz/PS 0.48 – 0.27v2 0.34 – 0.13v2 0.58 – 0.15v2

CHX/PDMS 0.41 + 0.18v2 0.45 + 0.042v2 0.52 + 0.06v2

Bz/PDMS 0.48 + 0.37v2 0.67 + 0.19v2 0.50 + 0.81v2

a) From literature data (ref.[19]).b) From ref.[19] and Equation (3).c) Experimental values from Table 3 by assuming that they are

binary parameters.


The values of g*chr and gchr functions compiled in Table5 are obtained from the interaction functions previouslycalculated (Table 3), and come out as a quadratic or lineardependence on v3, respectively.

At this point, we can investigate the dependence of Kp

on the fraction of crosslinked polymer or gel matrix, v3,in the ternary phase constituted by the mixture of polymersolution and gel according to Equation (8) and (9) at threehydrodynamic volumes, Vh, 106, 107 and 108 selectedarbitrarily for this work. For this reason, we previouslyneed to determine the molar volume of the polymer, V2,corresponding to every molar mass (or Vh) selected. So,having in mind the MHS relationship, Vh = M2[g]2 =K(M2)1+a, then: V2 = M2/d2 = (Vh/K)1/1+a/d2 where M2 and[g]2 are the molar mass and intrinsic viscosity of the poly-mer; K and a the MHS constants (in Table 1) and d2 is thedensity of the polymer (given in the experimental part).As an example, Figure 1 shows plots of lnKp vs v3 (Equa-tion (9)) for several systems at two Vh values, 106 (part a)and 107 (part b) calculated with the gchr function. Anincrease of Kp with v3 has been noticed in most of the sys-tems. In some others, however, there is an initial decreaseof Kp with a subsequent increase. An increase of Kp withv3 would be expected[18] since an increase of the gelvolume fraction in the ternary phase indicates more inter-actions between segments of the polymeric solute 2 andthe gel 3, and consequently a greater value of Kp. Thistrend could indicate that the experimental (Kp, v3) valuesshould be in the upward curve. That is to say, the slope ofthe plot lnKp vs v3 should be positive. In order words, theslope of the curve is dlnKp/dv3 = –(V2/V1)(a0 + 2a1v3 +3a2v3

2) when gchr is given in a polynomial form: a0 + a1v3

+ a2v32. For the slope to be positive, it follows that (a0 +

2a1v3 + 3a2v32) a 0 which is fulfilled for small v3 values

which is our case, as we will see later.To evaluate the experimental v3 values involved in

SEC experiments we first need to know the values of thecorresponding Kp. As has been said above, the coefficientKSEC is evaluated at every Vh by combining Equation (6)with the corresponding elution volume obtained from thecalibration plot in Table 1 (logVh = n + mVe) as:

KSEC ¼nÿmV0 ÿ log Vh

mVp

ð10Þ

A value of KSEC is obtained for every Vh with respect tothe system considered as the reference system, that is, theone in which the separation process is exclusively direc-ted by conformational changes, in the present case THF/PBD/PS. Following this, the experimental values of Kp

for each system and Vh are obtained from Equation (7) asfollows:

Kp ¼Ve ÿ V0

KSECVp

ð11Þ

where Ve is obtained from the corresponding calibrationcurve (Table 1), and are compiled in Table 6. Also inTable 6 experimental Kp values (Equation (11)) are givenat the three Vh selected for the different ternary polymersystems. Altogether, Kp A 1 and increases as Ve or Vh

increases, which indicates that the polymer greatly inter-acts with the gel matrix and secondary mechanismsappear in the chromatographic separation process retard-ing the polymer elution.

Table 5. Values of the chromatographic functions g*chr (Equa-tion (8)) and gchr (Equation (9)) for the ternary polymer systemsstudied.

System g*chr gchr

THF/PBD/PS 0.05 – 1.84v3 + 2.09v23 0.05 – 1.47v3

Bz/PBD/PS 0.33 – 6.29v3 + 8.89v23 0.33–5.08v3

0.33 – 5.50v3 + 8.89v23 0.33 – 4.29v3

Diox/PBD/PS 0.05 – 2.17v3 + 5.4v23 0.05 – 1.12v3

Tol/PDMS/PS –0.26 + 0.15v3 – 2.26v23 –0.26 + 0.15v3

–0.26 + 0.74v3 – 2.26v23 –0.26 + 0.74v3

Bz/PDMS/PS 0.13 –1.44v3 + 3.60v23 0.13 – 0.56v3

CHX/PDMS/PS 0.12 – 0.067v3 – 4.5v23 0.12 + 1.233v3

Diox/P2VPy/PS 0.80 – 8.90v3 + 4.2v23 0.80 – 6.80v3

Figure 1. Dependence of lnKp on v3 (Equation (9)) at twohydrodynamic volumes, Vh: a) Vh = 106; and b) Vh = 107; for dif-ferent ternary polymer systems: (J) THF/PBD/PS; (H) Bz/PBD/PS; (h) Diox/PBD/PS; (F) Tol/PDMS/PS; (g) Bz/PDMS/PS; (9)CHX/PDMS/PS and (6) Diox/P2VPy/PS.


On the other hand, to investigate the influence of themolar mass, M, (or solute size) on Kp we plot lnKp againstV2/V1 in Figure 2. A good linear dependence is observedfor every system, but according to Equation (8) and (9)no intercept should be expected, the slope should be equalto either –v3g*chr or –v3gchr and independent of M. Thisalso indicates that g*chr or gchr should be independent of Mfor a given solvent/polymer/PS(gel matrix) system. Tointerpret this conclusion, we need to point out that thesolute/gel interactions occur at the molecular segmentlevel and, therefore, they do not vary with increasing M,and, consequently, neither does the v3 (gel fractioninvolved in the ternary phase). However, in contrast withEquation (8) and (9) an intercept is attained, which couldbe a consequence of the approximations[18] made to obtainthese equations such as neglecting the terms multipliedby (v2 – v20) (v20 is the volume fraction of polymer(2) inthe binary phase solvent(1)/polymer(2)). By includingthis term, it yields:

ln Kp ¼V2

V1

ðv2 ÿ v20Þð2g12 ÿ 1Þ ÿ V2

V1

v3g�chr ð12aÞ

or

ln Kp ¼V2

V1ðv2 ÿ v20Þð2g12 ÿ 1Þ ÿ V2

V1v3gchr ð12bÞ

According to these expressions, the first member of theright-hand side remains constant for each system (as seenin Figure 2) since as the molar mass increases (or V2/V1),the value of the polymer involved in the ternary phase(v2) should decrease. Therefore, from Equation (12a) and(12b) and recalling[18, 29] that Kp = v2/v20, we can solve thecomposition of the ternary system (v2 and v3) from theordinates and slopes of Figure 2 for every system. Thevalues of v2 and v3 calculated with Equation (12a) and(12b) are compiled in Table 7 at the three hydrodynamicvolumes selected where only real and positive values inthe range 0 f v3 f 0.25 have been included since they arethe ones with physical meaning. An increase of Kp as v3

increases is observed, which confirms the previous state-

ment made when analysing the dependence of Kp on v3 inFigure 1.

It is worthwhile to mention that with the simplifiedequations proposed in ref.[18] (Equation (8) and (9)) it isonly possible to obtain v3 values, whereas with Equation(12a) and (12b) both v2 and v3 can be evaluated being thevalues of the later magnitude similar to those obtainedthrough the simplest expressions. Some systems cannotbe completely solved, as for example CHX/PDMS/PS inwhich the mathematical solution is only attained withEquation (12b). Other systems, as Diox/PBD/PS andDiox/P2VPy/PS are not feasible the highest solute size(Vh = 108).

Now we could calculate the numerical values of g*chr

and gchr that come out as a function of v3. Most of theseparameters are negative indicating Kp A 1 values accord-ing to Equation (8) and (9) and in good accordance withthe experimental values (Table 6). Kp increases as gchr

decreases, and in the limit, gchr e 0, it holds that Kp L 1.

Preferential Solvation

In the chromatographic system we are using, two oppo-site phenomena can be taken into account: the adsorption

Table 6. Values of polymer elution volumes, Ve, polymer(2) molar volume, V2, from the MHS relationship and experimental Kp

values from Equation (11). The values of KSEC at the three Vh are: 0.40, 0.27 and 0.14, respectively.

System Vh = 106 (mL/mol) Vh = 107 (mL/mol) Vh = 108 (mL/mol)Ve

mLV2

mL=molKp Ve

mLV2

mL=molKp Ve

mLV2

mL=molKp

THF/PBD/PSBz/PBD/PSDiox/PBD/PSTol/PDMS/PSBz/PDMS/PSCHX/PDMS/PSDiox/P2VPy/PS

24.9627.0329.5025.3525.4726.0627.24

34 82722 44627 3359 507

41 29327 79237 420

1.0061.2931.6361.0611.0771.1591.318

22.5324.0226.7223.0022.7823.4023.64

128 85094 314

121 80040 056

178 650124 680183 182

0.9921.2981.8531.0891.0431.1711.215

20.1021.0223.9420.6620.1120.7520.04

476 740396 290542 730168 760772 920559 370896 228

0.9581.3252.4901.1810.9621.2170.923

Figure 2. Dependence of lnKp with V2/V1 for different ternarypolymer systems: (J) THF/PBD/PS; (H) Bz/PBD/PS; (h) Diox/PBD/PS; (F) Tol/PDMS/PS; (g) Bz/PDMS/PS; (9) CHX/PDMS/PS and (6) Diox/P2VPy/PS.


of the polymeric solute(2) onto the gel(3) or the incom-patibility between solute and gel. Both behaviors can berelated to preferential or selective sorption. It is wellknown that the preferential solvation phenomenon quanti-fies which component, (1) or (2), is preferentiallyadsorbed by component (3).

In this respect, and on the basis of the thermodynamicconcept of the preferential solvation phenomenon, wethink that the representative coefficient, k, may have asimilar meaning as the chromatographic Kp coefficient, inthe sense that an equilibrium between a binary and tern-ary phase is also established. Therefore, both coefficientsare useful to interpret the experimental chromatographicbehavior. In other words, the k values could serve to cor-roborate the obtained Kp values since when Kp A 1 (com-patibility between polymer(2) and gel(3)), the k valueshave to be negative (k a 0) denoting a preferential solva-tion of component(2) towards component(3). On the con-trary, k A 0 means preferential solvation of component(1)by component(3) and the polymer (2) would be elutedearlier being Kp a 1.

In practice, the preferential solvation coefficient, k,that represents the volume of solvent(1) (in mL) adsorbedby mass (in grams) of polymer(3) is expressed at finitepolymer composition v3 by: [30]

kðu1; v3Þ ¼ B1ðu1Þv3 þ B2ðu1Þv3v3 ð13Þ

where v3 is the partial specific volume of component(3)which constitutes the gel matrix, and B1 and B2 as fol-lows:

where B1 ¼ ÿM13

M11;

B2 ¼ ÿM13M11

� �2

M111 ÿ 2 M13M11

M113 þM133

2M11

and the corresponding Mij and Mijk are defined as:

q2DGu

quiquj

� �u3e0

¼ Mij;q3DGu

quiqujquk

� �u3e0

¼ Mijk

(see Appendix for the complete development of theseexpressions).

As it can be seen, the calculation of the k coefficientthrough Equation (13) requires the knowledge of thefunctions M11, M13, M111, M113 and M133, which are defin-ed by the expressions given in the Appendix. All of themdepend on the system composition (values in Table 7)and on the interaction functions, gij, compiled in Table 3,as well as on the corresponding derivatives easilyobtained from there.

Figure 3–5 show, for all the systems here studied, thevariation of k with v2 according to Equation (13);together with the single value of k (outlined by a singlepoint) at a given composition (v3 in Table 7) derivedfrom the corresponding experimental Kp value (in Table 6and applying Equation (12a) and (12b)). To draw thesefigures, we have chosen in the calculation the composi-tions, v3 values at each Vh, with physical meaning fromall the mathematical solutions compiled in Table 7.

The main purpose of the preferential solvation data isto corroborate from a thermodynamic viewpoint theexperimental chromatographic behavior. For this reason,we next compare the Kp values in Table 6 with these fig-ures in more detail. Figure 3 plots the variation of k withv2 and Vh for PBD/PS in different solvents: THF (part a),Bz (part b) and Diox (part c). As seen in Figure 3a, k

shows positive values but very low, near zero, in accor-dance with values of Kp e 1. This behavior in THF isgreatly illustrative since it corroborates the fact that thiseluent is recommended for SEC experiments in these col-umns. Moreover, if THF is a good solvent for the poly-mer and does not preferentially solvate the gel (k L 0) the

Table 7. Values of v2 and v3 from Equation (12a), and of v*2 and v*3 from Equation (12b) for the different ternary polymer systemsstudied.

System Vh = 106 (mL/mol) Vh = 107 (mL/mol) Vh=108 (mL/mol)v2 v3 v*2 v*3 v2 v3 v*2 v*3 v2 v3 v*2 v*3

THF/PBD/PS

Bz/PBD/PS

Diox/PBD/PS

Tol/PDMS/PSBz/PDMS/PS

CHX/PDMS/PSDiox/P2Vpy/PS

0.05360.07500.03170.28270.01790.01930.13940.0376

––

0.02730.0382

0.03390.0000960.0650.07690.04450.0000940.0001570.00016

––

0.1180.000014

0.05690.07500.03390.28880.01850.01930.13940.00960.03760.06270.0290.0382

0.027980.0000960.0570.06740.02440.0000950.0001570.1380.000160.1560.0000140.094

0.02610.01780.010.18150.0035

–0.05670.0237

––

0.0080.0118

0.03380.0002050.0650.07690.0463

–0.0003050.000128

––

0.1180.000051

0.02610.01780.01120.18880.0036

–0.05670.00400.02370.01440.01180.0086

0.027860.0002060.0570.06730.0263

–0.0003050.13740.0001280.1560.0000510.094

0.00330.04340.00250.13990.0006

–0.0201

–––––

0.03380.0001760.0650.07690.0459

–0.000270.000102

––

0.1180.000043

0.000530.04340.00290.14860.00062

–0.0201

––

0.0027––

0.027890.0001760.0570.06730.0259

–0.000270.13740.0001030.1560.0000430.094


additional secondary mechanisms will be avoided. For Bz(Figure 3b), k values are always negative in agreementwith the Kp A 1 values obtained at any Vh. Absolute k

values decrease as v2 does and with the increase of Vh. Inthis system, the observed trend is even more noticeabledenoting that the gel matrix (PS) is preferably solvated bythe polymer PBD and, consequently, it will be latereluted. In Figure 3c, for Diox as eluent, the same beha-vior is noticed although enhanced, k values are alwaysnegative but their absolute values are higher than in Bz,in good accordance with the highest values obtained forKp. Again, if Diox does not exhibit preferential solvation

towards the PS gel, the PBD will be greatly retained onthe column as it has been experimentally evidenced.

Figure 4 shows the variation of k for the systemPDMS/PS in Tol (part a), in Bz (part b) and in CHX (partc). The general trend is that k decreases when v2

decreases and Vh increases, contrary to the variation ofKp for this system (see Table 6). Specifically, Figure 4agives k e 0 for Vh = 106 and becomes negative for Vh =107 and 108, in accordance with the respective Kp values:1.06; 1.09 and 1.18. This behavior is similar in Bz (partb) and in CHX (part c), but quantitatively enhanced, thatis, as Kp increases (higher Ve) when changing the eluentfrom Tol to CHX (at constant Vh), k becomes more nega-

Figure 3. Plot of the preferential solvation coefficient, k, cal-culated with the volume fraction of PS, v3, as a function of thevolume fraction of polymer(2), v2, at three hydrodynamicvolumes, Vh, for different ternary polymer systems: a) THF/PBD/PS; b) Bz/PBD/PS and c) Diox/PBD/PS.

Figure 4. Plot of the preferential solvation coefficient, k, cal-culated with the volume fraction of PS, v3, as a function of thevolume fraction of polymer(2), v2, at three hydrodynamicvolumes, Vh, for different ternary polymer systems: a) Tol/PDMS/PS; b) Bz/PDMS/PS, and c) CHX/PDMS/PS.


tive denoting an increasing preferential solvation of thepolymer PDMS towards the PS gel.

Finally, Figure 5 outlines the k behavior for the systemDiox/P2VPy/PS at the two compositions given in Table7. At the relatively high composition v3 = 0.118 (part a)the values of k are negative and their absolute valuesincrease with increasing Vh, as in the preceding systems,but they do not correspond to a diminution in the Kp

values. Moreover, the observed decrease in k with v2 isvery sharp which gives unreliable values. However, atlower volume fractions of gel (part b), the positive valuesof k are in agreement with the decrease in Kp whenincreasing Vh, as expected for this system. Therefore, itcan be concluded that among the possible mathematicalsolutions, those with physical meaning are those that cor-rectly predict the experimental trend.

ConclusionsInteraction functions for seven ternary solvent(1)/poly-mer(2)/polymer(3) systems have been derived from theFlory-Huggins theory. These functions show a good qua-litative accordance with the reported ones and have beenused to evaluate the gel fraction involved in SEC experi-

ments, as well as the preferential solvation phenomenon.Experimental values of the distribution coefficient of thesolute between the mobile phase and the stationary phasein SEC experiments, Kp, have been determined from thecalibration curves. They show values, in general, higherthan unity, denoting the existence of reversible adsorptionas a secondary effect. The experimental Kp values com-bined with some expressions derived in our laboratory asa function of the interaction functions have been used todetermine the fraction of gel matrix, v3, and the fractionof polymer(2) in the ternary phase, v2. The interactionfunctions, v3 and v2 have served to obtain the preferentialsolvation coefficient, k, through an expression derivedfor solvent(1)/polymer(2)/polymer(3) systems. The trendof k values shows a good qualitative coincidence with theKp values, which may indicate that the observed adsorp-tion secondary effects could be attributed to the preferen-tial solvation of solute polymer(2) towards the gel matrixor polymer(3).

Appendix

Derivation of the Preferential Solvation Coefficient forSolvent(1)/Polymer(2)/Polymer(3) Systems

Considering a ternary system, solvent(1)/polymer(2)/polymer(3), composed by the mixture of N1, N2 and N3

molecules of each component, we can distinguish twophases: a ternary phase with n1 molecules of solvent, n2

molecules of polymer(2) and N3 molecules of polymer(3);and a binary phase with the remainder solvent and poly-mer(2), that is, (N1–n1) and (N2–n2) molecules of solvent1 and polymer(2), respectively. The Gibbs free energy ofsuch a system is the result of adding the relative to themixture,

Pi ni; li, plus the elastic free energy, Ge, origi-

nated by a change in pressure. Therefore, we can write:

DG ¼ ðN1 ÿ n1Þl�1 þ ðN2 ÿ n2Þl�2 þ n1l1

þn2l2 þ N3l3 þ Ge ðA1Þ

where l*1 and l*2 are the chemical potentials of compo-nents 1 and 2 in the binary phase, outside the polymerdomain; and li (i = 1, 2, 3) is the chemical potential ofthe three components in the ternary phase.

The equilibrium condition implies that at any infinitesi-mal composition variation dDG = 0, then:

ÿl�1 þ l1 þqGe

qn1

� �n2

!þ dn1

þ ÿl�2 þ l2 þqGe

qn2

� �n1

!dn2 ¼ 0 ðA2Þ

where

Figure 5. Plot of the preferential solvation coefficient, k, cal-culated with the volume fraction of PS, v3, as a function of thevolume fraction of polymer(2), v2, at three hydrodynamicvolumes, Vh, for Diox/P2VPy/PS, and different gel composition:a) high v3 values; b) low v3 values.


qG2

qn1

� �n2

¼ v1qGe

qV¼ v1Pe

and

qGe

qn2

� �n1

¼ v2qGe

qV¼ v2Pe

Pe being the variation of elastic free energy in the tern-ary phase. The fulfillment of Equation (A2) requires that:

–l*1 + l1 + V1Pe = 0(A3)

–l*2 + l2 + V2Pe = 0

where the partial molar volumes, vi have been replacedby the molar volumes, Vi.[11, 31]

However, the phase composition is commonlydescribed by the volume fractions. In this sense, vi0 (i = 1,2) is the volume fraction of the binary phase; vi (i = 1, 2,3) is the volume fraction of the ternary phase (the domainof the coil), and ui (i = 1, 2) is the volume fraction of theliquid mixture of components 1 and 2 inside the coil, ui =vi/(1–v3) (i = 1, 2). Then, Equation (A3) can be written interms of volume fractions instead of number of moleculesand taking into account[11] that gi = li/Vi:

g1(u1) – g*1 (v10) = g2(u1) – g*2 (v10) (A4)

The solvent composition in the ternary phase is usuallydifferent from the one in the binary phase giving rise tothe preferential solvation phenomenon, which basicallyconsists on the fact that the solvent composition changesin the ternary phase. This difference can be quantified bye, the excess volume fraction of one of the components inthe ternary phase with respect to its composition in thebinary phase: e = u1–v10 = v20–u2.

An expression to evaluate e as a function of the ther-modynamic properties and of the different volume frac-tions can be deduced from the equilibrium condition(Equation (A4)), but it only can be solved in the limit v3

e 0. In this condition, the magnitudes g1(u1) and g2(u1)can be expanded in a Taylor's series as a function of u3

and e, yielding:

g1ðu1; u3Þ ¼ g�1ðv10Þ þ eqg1

qu1

� �u3¼0

þ e2

2q2g1

qu21

� �u3¼0

þu3qg1

qu3

� �u1 ;u3e0

þ ðA5Þ

u23

2q2g1

qu23

� �u1 ;u3e0

þeu3q2g1

qequ3

� �u3e0

þ:::

A similar expression can be obtained for g2 (u1, u3). Byincluding these series expansions in Equation (A4) andmaking some rearrangements, it yields:

eqðg1 ÿ g2Þ

qu1

� �u3e0

þ e2

2q2ðg1 ÿ g2Þ

qu21

� �u3e0

þu3qðg1 ÿ g2Þ

qu3

� �u1 ;u3e0

þ u23

2q2ðg1 ÿ g2Þ

qu23

� �u1 ;u3e0

þeu3q2ðg1 ÿ g2Þ

qequ3

� �u3e0

þ::: ¼ 0 ðA6Þ

On the other hand, for calculation purposes we defineDGu as:

DGu ¼DG

1ÿ v3

¼X

i

vi

1ÿ v3

gi ¼X

i

uigi ðA7Þ

and which derivatives are expressed as:

qDGu

qu1

� �u3e0

¼ qðu1g1 þ u2g2Þqu1

� �u3e0

¼ g1 ÿ g2 ðA8Þ

For calculation purposes, we denote:

q2DGu

quiquj

� �u3e0

¼ Mij;q3DGu

quiqujquk

� �u3e0

¼ Mijk ðA9Þ

With this nomenclature, Equation (A6) can be rewrit-ten as a second order equation in e:

eM11 þe2

2M111 þ u3M13 þ

u23

2M133

þ eu3M113 þ ::: ¼ 0 ðA10Þ

from where e can be deduced. However, e can be moreeasily obtained making a series expansion as a functionof u3, as follows:

e ¼ ÿ M13

M11u3 ÿ

M13M11

� �2

M111 ÿ 2 M13M11

M113 þM133

2M11

u23 þ ::: ¼ B1u3 þ B2u2

3 ðA11Þ

B1 and B2 being the coefficients of u3 and u23, respec-

tively, which depend on solvent composition (u1). Bymultiplying Equation (A11) by (1–v3), it holds:

eð1ÿ v3Þ ¼ B1ðu1Þu3ð1ÿ v3Þ þ B2ðu1Þu23ð1ÿ v3Þ

¼ B1ðu1Þv3 þ B2ðu1Þv23 ðA12Þ


In this regard, the preferential solvation coefficient, k,is related with e through:

k ¼ limv3e0

eð1ÿ v3Þv3

v3

¼ limv3e0

B1ðu1Þv3v3 þ B2ðu1Þv23v3

v3

¼ B1ðu1Þv3 ðA13Þ

at infinite dilution. In the case of systems with finite poly-mer(3) concentration, the values of k will be:

kðv3Þ ¼ B1ðu1Þv3 þ B2ðu1Þv3v3 ðA14Þ

Finally, the calculation of k values through Equation(A14) requires the knowledge of the corresponding Mij

and Mijk given in the coefficients B1 and B2. These magni-tudes are derived from the Flory-Huggins theory in termsof the binary, gij, and of the ternary, gT, interaction func-tions, and are given by the expressions:

M11 ¼RTV1

1v1

þ sv2

ÿ 2g12 þ 2ðv2 ÿ v1Þ�

dg12

dv1

� �þ v1v2

d2g12

dv21

� ��ðA15Þ

M13 ¼RTV1

ÿ1þ sÿ ðv2 ÿ v1Þg12 ÿ v1v2

dg12

dv1

� ��

þ g013 ÿ g0

23 þ ðv2 ÿ v1Þg0T þ v1v2

qgT

qv1

� �0�ðA16Þ

M111 ¼RTV1

ÿ 1v2

1

þ sv2

2

ÿ 6dg12

dv1

� ��

þ 3ðv2 ÿ v1Þd2g12

dv21

� �þ v1v2

d3g12

dv31

� ��ðA17Þ

M113 ¼RTV1

2ðg12 ÿ g0TÞ ÿ 2ðv2 ÿ v1Þ

dg12

dv1

� ��

ÿ qgT

qv1

� �0�ÿ v1v2

d2g12

dv21

� ��

ÿ q2gT

qv21

� �0��ðA18Þ

M133 ¼RTV1

½1ÿ sþ 2ðv2 ÿ v1Þg12 þ 2v1v2

dg12

dv1

� �

ÿ2g013 þ 2

dg13

dv3

� �0

þ2g023 ÿ 2

dg23

dv3

� �0

ÿ4ðv2 ÿ v1Þg0T þ 2ðv2 ÿ v1Þ

qgT

qv3

� �0

ÿ4v1v2

qgT

qv1

� �0

þ2v1v2

q2gT

qv3qv1

� �0�ðA19Þ

where the superscript “0” indicates that these parametersare evaluated at infinite dilution (v3 e 0); and being s =V1/V2 the ratio of solvent and polymer(2) molar volumes.

Acknowledgement: Financial support from Dirección Generalde Enseæanza Superior (Ministerio de Educación y Cultura,Spain) under Grant No. PB98-1435 is gratefully acknowledged.

Received: May 17, 2000Revised: August 2, 2000

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Thermodynamic Interpretation of the SEC Behavior of Polymers in a Polystyrene Gel Matrix

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