Phase behavior and second osmotic virial coefficient for competitive polymer solvation in mixed solvent solutions Jacek Dudowicz, Karl F. Freed, and Jack F. Douglas

Citation: The Journal of Chemical Physics 143, 194901 (2015); doi: 10.1063/1.4935705 View online: http://dx.doi.org/10.1063/1.4935705 View Table of Contents: http://aip.scitation.org/toc/jcp/143/19 Published by the American Institute of Physics

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THE JOURNAL OF CHEMICAL PHYSICS 143, 194901 (2015)

Phase behavior and second osmotic virial coefficient for competitive polymer solvation in mixed solvent solutions Jacek Dudowicz,1,a) Karl F. Freed,1 and Jack F. Douglas2 1

The James Franck Institute and the Department of Chemistry, The University of Chicago, Chicago, Illinois 60637, USA 2 Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

(Received 28 September 2015; accepted 2 November 2015; published online 19 November 2015) We apply our recently developed generalized Flory-Huggins (FH) type theory for the competitive solvation of polymers by two mixed solvents to explain general trends in the variation of phase boundaries and solvent quality (quantified by the second osmotic virial coefficient B2) with solvent composition. The complexity of the theoretically predicted miscibility patterns for these ternary mixtures arises from the competitive association between the polymer and the solvents and from the interplay of these associative interactions with the weak van der Waals interactions between all components of the mixture. The main focus here lies in determining the influence of the free energy parameters for polymer-solvent association (solvation) and the effective FH interaction parameters { χ α β } (driving phase separation) on the phase boundaries (specifically the spinodals), the second osmotic virial coefficient B2, and the relation between the positions of the spinodal curves and the theta temperatures at which B2 vanishes. Our classification of the predicted miscibility patterns is relevant to numerous applications of ternary polymer solutions in industrial formulations and the use of mixed solvent systems for polymer characterization, such as chromatographic separation where mixed solvents are commonly employed. A favorable comparison of B2 with experimental data for poly(methyl methacrylate)/acetonitrile/methanol (or 1-propanol) solutions only partially supports the validity of our theoretical predictions due to the lack of enough experimental data and the neglect of the self and mutual association of the solvents. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4935705]

I. INTRODUCTION

Many applications involving ternary solutions of polymers find that the composition of the solvent mixture is a decisive factor governing the solubility of polymers chains and the properties of these polymer solutions. The variety of the theoretically predicted phase behaviors of these systems that may appear as the solvent composition and quality change is difficult to comprehend from simple rules of mixing. Moreover, the theoretical treatment of this important problem is inhibited by the inherent mathematical complexity arising from the presence of multiple competing interactions in ternary mixtures. For instance, equation-ofstate type theories by Wedgeworth and Glover1 apply only to simple ternary polymer solutions that are devoid of strong polymer-solvent associative interactions. In order to address this class of problems, our recent paper2 develops a simple, but reliable, theoretical approach to the thermodynamics of ternary solutions of polymers interacting with solvents through strong associations. This extension of Flory-Huggins (FH) type theory to ternary mixtures applies to solutions in which both solvents may selectively bind to the dissolved polymer, while at the same time interacting with the polymer and each other through weak van der Waals forces. The present paper a)E-mail address: [email protected]

0021-9606/2015/143(19)/194901/12/$30.00

illustrates the wide range of the phase behaviors exhibited by three different classes of these ternary mixtures: (class I) polymer solutions in which both solvents bind strongly to the polymer upon cooling; (class II) in which both bind upon heating; and (class III) in which one binds upon cooling and the other upon heating. A solution in which neither solvent binds to the polymer and only van der Waals interactions operate between all components of the solution is chosen as the reference system. The rich, often counter-intuitive, variety of phase diagrams of these ternary solutions of polymers in mixed solvents is mirrored by the behavior of the second osmotic virial coefficient. This basic measure of solvent quality vanishes at characteristic temperatures in the proximity of the multiple critical temperatures of these complex solutions. Our theory provides a framework for understanding and classifying possible patterns of miscibility for polymers in mixed solvents, a phenomenon of relevance to various methods of liquid chromatography,3–9 as discussed in more detail in our recent paper.2 Section II summarizes the most important relations of the theory, with special emphasis placed on the derivation (first detailed here) of the spinodal condition determining the phase boundaries, as well as on the derivation of expressions for the second osmotic virial coefficient B2 of ternary, strongly interacting polymer solutions. Section III describes our computations of the spinodal curves and of B2 for the

143, 194901-1

© 2015 AIP Publishing LLC

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Dudowicz, Freed, and Douglas

three classes of mixtures: those in which both solvation reactions are enhanced upon cooling, those in which both solvation processes become more favorable upon heating, and finally mixtures where one reaction is promoted upon cooling and the other one upon heating. Section III includes our analysis of the variation of B2 with the temperature and the composition of the pure solvent mixture in equilibrium with ternary polymer solutions as well as the comparison of our results for B2 with experimental data for the second osmotic virial coefficient A2 of solutions of poly(methyl methacrylate) in mixtures of acetonitrile with pentyl acetate or alcohols.10

II. FLORY-HUGGINS TYPE THEORY OF COMPETITIVE SOLVATION OF POLYMERS BY TWO SOLVENTS

J. Chem. Phys. 143, 194901 (2015) o o φoA, φC , and φoB = 1 − φoA − φC . The equilibrium volume fractions {φ A i B } = {n A i B(N + i)/Nl }, {φ BC j } = {n BC j (N + j)/Nl }, {φ A i BC j } = {n A i BC j (N + i + j)/Nl }, φ B = n B N/Nl , φ A = n A/Nl , and φC = nC /Nl are defined in terms of the numbers of molecules of each species at equilibrium (with the subscripts labeling, respectively, solvated chains { Ai B}, {BC j }, and { Ai BC j }, undecorated chains, and free solvent molecules A and C). Individual segments of all species also interact with their nearest neighbors through weaker, non-associative, attractive van der Waals energies {ϵ α β ≡ ϵ βα }, where the subscripts α, β ≡ A, B,C indicate the monomeric species of the interacting pairs. The mass conservation constraints for the individual solvents A and C are conventionally expressed2 in terms of the auxiliary quantities A ≡ φ A K A and C ≡ φC KC as

A. Lattice model for competitive solvation: Main thermodynamic relations

The solvated polymer solution comprises noB monodisperse polymer chains of species B and noA and nCo molecules of solvents A and C, respectively. The polymers and solvent molecules reside on a three-dimensional lattice with Nl lattice sites of coordination number z = 6, and no empty sites are allowed. Each polymer chain contains N identical segments connected by N − 1 consecutive bonds, while each solvent molecule of both species is modeled, for simplicity, as occupying a single lattice site (but the solvent molecules can easily be permitted to have different sizes). The composition of the mixture before solvation is specified by either pair of the three volume fractions φ Bo ≡ noB N/Nl , φoA ≡ noA/Nl , or o φC ≡ nCo /Nl . The solvation process involves the reversible formation of solvated clusters { Ai B}, {BC j }, and { Ai BC j } through the sequential association of appropriately oriented solvent molecules onto the polymer chains. The sequential association is described schematically by the reactions A + X AX,

X ≡ B, { Ai B}, {BC j }, { Ai BC j } (1)

C + X C X,

X ≡ B, { Ai B}, {BC j }, { Ai BC j }, (2)

and

where X designates either a pure B polymer, an arbitrary cluster of B with A, with C, or with both A and C. The equilibrium constants K A = exp[−(∆h A − T∆s A)/k BT] and KC = exp[−(∆hC − T∆sC )/k BT] apply for Eqs. (1) and (2), respectively. The enthalpies ∆h A and ∆hC and the entropies ∆s A and ∆sC of solvation by the solvents A and C are adjustable free energy parameters. The (N − 2) interior polymer chain segments are permitted, for simplicity, to bind only a single solvent molecule A or C, provided these solvent molecules adopt specific orientations with respect to the polymer and the rest of the solvent. The distributions of clusters { Ai B}, {BC j }, and { Ai BC j } and the equilibrium concentrations of unsolvated chains B and of unassociated solvent molecules A and C are governed (for given free energy parameters ∆h A, ∆s A, ∆hC , ∆sC , and N) by the temperature T and any pair of the initial volume fractions

φoA =

N −2 A A + φoB KA N 1+A+C

(3)

o φC =

C C N −2 + φoB . KC N 1+A+C

(4)

and

Equations (3) and (4), in conjunction with the final expression for the Helmholtz free energy F of the system2 F ≡ f = φ Ao ln φ A + φCo ln φC + φ Ao − φ A Nl k BT φ Bo φo + φCo − φC + B ln N (1 + A + C) N −2   1 2(z − 1)2 N − 1 + ln + − ln(z − 1) φ Bo + f ϵ , N zN N (5) where f ϵ is the dimensionless FH interaction energy density of the system before solvation, z  2 2 2 fϵ = − ϵ A A(φ Ao ) + ϵ CC (φCo ) + ϵ B B(φ Bo ) 2k BT  + 2 ϵ AB φ Ao φ Bo + 2ϵ AC φ Ao φCo + 2ϵ BC φ Bo φCo , (6) provide the computational framework for the FH type theory of polymer solvation in mixed solvents.2

B. Spinodal curves

The condition for the existence of a homogeneous stable phase for a constant volume, ternary mixture has the form2  2 ∂2 f ∂2 f ∂2 f D≡ | | − | > 0, o o ∂φ Bo ∂φ Ao Nl ,T ∂φ Bo 2 Nl ,T ,φ A ∂φ Ao 2 Nl ,T ,φ B (7) where the volume fractions φ Ao and φ Bo are chosen as the independent composition variables. The derivatives of the Helmholtz free energy f with respect to compositions from Eq. (7) simply follow from Eqs. (5) and (6) and are substituted into the spinodal constraint D = 0 to give

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Dudowicz, Freed, and Douglas



J. Chem. Phys. 143, 194901 (2015)

  N −2 1 1 ∂φC C 1 + − | − o N φ Bo φC ∂φ Bo Nl ,T ,φ A N 1+A+C  1 ∂φ A 1 × | o − φ A ∂φ Ao Nl ,T ,φ B φC  −

1 ∂φ A 1 ∂φC | | o − o + χ AB − χ BC − χ AC φ A ∂φ Bo Nl ,T ,φ A φC ∂φ Bo Nl ,T ,φ A

The phase boundary for a ternary system is the surface T = T(φ Ao , φ Bo ) that is deduced from Eq. (8) when the interaction parameters χ AB, χ BC , and χ AC are expressed in terms of the absolute temperature T and the nearest neighbor van der Waals interaction energies {ϵ α β } (α, β = A, B,C) as z [ϵ A A + ϵ B B − 2ϵ AB] , 2k BT z [ϵ B B + ϵ CC − 2ϵ BC ] , = 2k BT

χ AB = χ BC

N −2 1 ∂φ A A − 2 χ BC o | N l ,T ,φ o φ A ∂φ B N 1+A+C A  ∂φC | − 2 χ AC o ∂φ Ao Nl ,T ,φ B

(9) (10)



2 = 0.

associative interactions are absent,11   1 1 + − 2 χ BC N φoB 1 − φoA − φoB   1 1 + − 2 χ AC × φoA 1 − φoA − φoB  2 1 − + χ − χ − χ = 0, AB BC AC 1 − φoA − φoB

(8)

(19)

o by setting φ A ≡ φoA, φC ≡ φC , A = C = 0,

and χ AC =

z [ϵ A A + ϵ CC − 2ϵ AC ] , 2k BT

(11)

o and when the derivatives {Dγδ ≡ ∂φγ/∂φδo | Nl,T,φ σ,δ } of Eq. (8),

D AB ≡

∂φ A ∂φC | DC B ≡ | o, o, ∂φ Bo Nl ,T ,φ A ∂φ Bo Nl ,T ,φ A

∂φ A ∂φC DAA ≡ | o, o | N l ,T ,φ o , and DC A ≡ ∂φ A ∂φ Ao Nl ,T ,φ B B

(12)

∂φoA | N ,T,φ Bo = 1, ∂φoA l

D AB →

∂φoA | N ,T,φ Ao = 0, ∂φoB l

DC A →

o ∂φC | N ,T,φ Bo = −1, ∂φoA l

DC B →

o ∂φC | N ,T,φ Ao = −1. ∂φoB l

and

are determined from the mass conservation conditions of Eqs. (3) and (4). Differentiating Eqs. (3) and (4) with respect to the initial volume fractions φoA and φoB produces a set of four linear equations with four unknowns, 0 = D AB(1 + b) − cDC B + a,

(13)

1 = D A A(1 + b) − cDC A,

(14)

− 1 = DC B(1 + e) − dDC A − gD AB,

(15)

− 1 = DC A(1 + e) − gD A A,

(16)

with the coefficients a, b, c, d, e, and g that are defined as a ≡ N2A/W, b ≡ N2φ Bo K A(1 + C)/W 2, c ≡ N2φ Bo KC A/W 2, d ≡ N2C/W,

DAA →

(17)

e ≡ N2φ Bo KC (1 + A)/W 2, g ≡ N2φ Bo K A C/W 2, (18) N2 ≡ (N − 2)/N, W ≡ 1 + A + C, and that are determined from the numerical solution to the set of Eqs. (3) and (4) for A and C. The subsequent solution of Eqs. (13)-(16) provides D A A, D AB, DC A, and DC B and, along with Eq. (8), the boundary of phase stability (spinodal). Equation (8), as it must, simplifies into the two wellknown spinodal condition for A/B/C polymer solutions when

Equation (8) is contrasted with its counterpart12,13 for solutions of solvated polymers B in a single solvent A, ∂ 2 f ( AB) ∂φ A

o2

| Nl ,T =

[1 + N2 A/(1 + A)] 2 1 o + N φ B A/K A + N2φ Bo A/(1 + A)2 − 2 χ AB = 0,

(20)

where A is determined from the modification of Eq. (3) in which C = 0 and where the superscript (AB) labels A/B polymer solutions. Equation (8) also becomes numerically equivalent to Eq. (20) when the single bead molecules of solvents A and C are thermodynamically indistinguishable, i.e., when ∆h A = ∆hC , ∆s A = ∆sC , χ BC = χ AB, and χ AC = 0. If the solvents A and C are also taken as energetically equivalent ( χ BC = χ AB, and χ AC = 0), the spinodal condition for A/B/C polymer solutions interacting exclusively through the weak van der Waals forces [see Eq. (19)] likewise transforms into the standard spinodal condition for A/B polymer solutions in a single bead solvent, 1 1 + − 2 χ AB = 0. N φoB 1 − φoB

(21)

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J. Chem. Phys. 143, 194901 (2015)

C. Second osmotic virial coefficient B2

The osmotic pressure Π for an equilibrium mixture of two solvents A and C and a solvated polymer B is defined14 in terms of the differences in the chemical potentials µoo A and µ A for solvent A, µoo µA Πvcell = A − , k BT k BT k BT

(22)

or, alternatively, in terms of the chemical potentials µC and µoo C for solvent C, µoo Πvcell µC = C − , k BT k BT k BT

(23)

where the subscripts A and C label the solvent species and oo the superscripts oo on µoo A and µC refer to mixtures of pure solvents A and C that are in full equilibrium contact with the polymer solution through an osmotic membrane (permeable

only to the solvents) and where vcell is the unit volume associated with a single lattice site. Both definitions of the osmotic pressure for a ternary polymer solution A/B/C provide necessary relations2 for our calculation scheme below. After evaluating the chemical potentials µ A and µC from the free energy densities of Eqs. (5) oo and (6) and the chemical potentials µoo A and µC from our FH type theory specialized to binary mixtures of two monomeric species A and C that interact exclusively through weak van der Waals forces, and then substituting the expressions for µ A, oo µC , µoo A , and µC into Eqs. (22) and (23), the latter equations may be presented as polynomials in the volume fraction oo φoo A = 1 − φC of solvent A in the pure solvent mixture, the volume fractions φ A, φC of free (unassociated) solvents A and C, and the initial composition of the system (φoA and φoB). Virial coefficients may then be extracted from these lengthy expansions. Adding these two new equations yields

φo Πvcell 1 1 oo oo oo = ln[φoo ln(φ A φC ) + φ A + φC + B − 1 A (1 − φ A )] − χ AC [1 − φ A ]φ A − k BT 2 2 N −

1 1 1 χ AB[φoB(1 − 2φoA)] − χ BC [φoB(−1 + 2φoA + 2φoB)] + χ AC [φoB + 2φoA(1 − φoA − φoB)], 2 2 2

while subtracting one equation from the other produces the o o relation between φoo A , φ A, φC , φ A, and φ B , φoo A + χ AC [1 − 2φoo ln A ] 1 − φoo A = ln

Eq. (26) with Eq. (28) and Eq. (27) with Eq. (29) gives o o 2 φ A = φoo A + (A1 + λ 1)φ B + (A2 + λ 2)(φ B ) + · · ·

o o 2 φC = 1 − φoo A + (C1 − λ 1 − 1)φ B + (C2 − λ 2 − 1)(φ B ) + · · ·.

(31) (25)

Equations (24) and (25) provide the basis2 for calculating the osmotic virial coefficients B1 and B2 for ternary A/B/C systems in the dilute polymer limit φoB → 0. The resulting expression for the osmotic pressure of Eq. (24) is fully symmetric with respect to solvents A and C. In the dilute limit of vanishing polymer volume fraction φoB, the equilibrium volume fractions φ A and φC can be represented as a power series in φoB, φ A = φoA + A1 φoB + A2(φoB)2 + · · ·

(26)

The coefficients A1, A2, C1, C2, λ 1, and λ 2 are determined by first substituting the power series from Eqs. (30) and (31) into the mass conservation conditions in Eqs. (3) and (4) and into the additional constraint of Eq. (25), expanding all terms in the denominators and the arguments of the logarithms that are in the form of power series expansions in φoB (around φoB = 0), and finally noting terms of a given power of φoB on the left- and right-hand sides of Eqs. (3), (4), and (25) must be identical. Collecting the individual coefficients of powers of (φoB)n (n = 0, 1, 2, . . .) term by term leads to the relations2 A1 = −N2K A φoo A /W,

and o φC = φC + C1 φoB + C2(φoB)2 + · · ·,

=

+

λ 1φoB

+

λ 2(φoB)2

+···

2 + φoo A KC (A1 + C1 − 1)]/W ,

(28)

(33)

oo + φC K A(A1 + C1 − 1)]/W 2,

− KC )/W − χ] − oo 1 − 2 χ AC φoo A φC χ ≡ χ AB − χ BC − χ AC ,

λ1 =

(29)

with the coefficients A1, A2, C1, C2, λ 1, and λ 2 being functions of φoo A , K A, KC , and { χ α β } of Eqs. (9)-(11). Combining

(32)

C2 = N2 KC [−(1 + K A)(C1 − λ 1 − 1)

and φCo = 1 − φ Ao − φoB,

oo C1 = −N2 KC φC /W,

A2 = N2 K A[−(1 + KC )(A1 + λ 1)

(27)

o where the initial volume fractions φoA and φC are likewise expanded for internal consistency as power series in φoB,

φoo A

(30)

and

φA + χ AB φoB − χ BC φoB − χ AC [2φoA + φoB − 1]. φC

φoA

(24)

and

oo φoo A φC [N2(K A

(34) φoo A

,

(35)

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λ2 =

J. Chem. Phys. 143, 194901 (2015)

oo oo oo oo 2 oo 2 −A2φC + C2φoo A + (1/2)φ A φC {[(A1 + λ 1)/φ A ] − [(C1 − λ 1 − 1)/φC ] } , oo 1 − 2 χ AC φoo A φC

oo where φC = 1 − φoo A , and N2 and W are defined by Eqs. (18). An identical procedure is applied to Eq. (24) for the osmotic pressure to enable determining the first and the second osmotic virial coefficients as the coefficients multiplying, respectively, the linear and quadratic terms in φoB. Thus, B1 and B2 are found analytically as2 ( )  1 A1 + λ 1 C1 − λ 1 − 1 + + A1 + C1 B1 = − oo 2 φoo φC A  1 oo + − 1 + ( χ/2 − χ AC λ 1)(φoo − φ ) (37) A C N

and ( ( ) )2 1 A1 + λ 1 1 A2 + λ 2 C2 − λ 2 + + oo 2 φoo φC 4 φoo A A ( )2 1 C1 − λ 1 − 1 + A2 + C2 + oo 4 φC

B2 = −

oo 2 + λ 1 χ − χ BC − χ AC [λ 2(φoo A − φC ) + λ 1]. (38)

When the polymer solution contains only a single solvent A, oo o i.e., when φC = φC = φC = 0, KC = 0, and χ AC = χ BC = 0, Eqs. (37) and (38) simplify, respectively, to the well-known relations12 B1( AB) =

1 N

(39)

and B2( AB) = − χ AB +

( )2 1 KA 1 + N2 , 2 1 + KA

(40)

which is now valid for infinitely dilute polymer solutions in a single solvent. The superscript (AB) on B1(AB) and B2(AB) are used to distinguish the single solvent quantities from B1 and B2 of Eqs. (37) and (38). Equations (37) and (38) demonstrate that the osmotic virial coefficients B1 and B2 of ternary polymer solutions are functions of the equilibrium constants K A and KC , the composition (φoo A ), of the solvent mixture in osmotic equilibrium with a polymer solution, and, of course, the interaction parameters χ AB, χ BC , and χ AC , defined by Eqs. (9)-(11).

III. CALCULATIONS

Equations (8) and (38), respectively, are used to calculate the phase boundaries and the second osmotic virial coefficient B2 for solutions of solvated polymers B in binary mixtures of the solvents A and C. As already mentioned, the spinodal condition for a ternary A/B/C mixture is actually defined by a surface T = T(φoA, φoB) which transforms into a curve T = T(φoB,r o = const.) where r o is the ratio of the initial o volume fractions φoA and φC of the pure solvents A and C. The

(36)

solvation of polymer (B) is driven by the temperature and a set of phenomenological free energy parameters {∆h A, ∆s A } and {∆hC , ∆sC } which define, respectively, the equilibrium constants K A and KC for the reversible solvation reactions in Eqs. (1) and (2). The second osmotic virial coefficient B2 is, for a given set of ∆h A, ∆s A, ∆hC , ∆sC and interaction parameters χ AB, χ BC , and χ AC , a function of two variables, the temperature T and the composition of the solvent mixture (φoo A ) in osmotic equilibrium with the polymer solution. The oo oo oo ratio r oo = φoo A /φC (where φC = 1 − φ A ) provides a more convenient independent composition variable than the volume o fraction φoo A . The parameter r o approaches r oo when φ B → 0, i.e., when the solutions of polymer B are infinitely dilute. All illustrative calculations are performed using the identical absolute values of enthalpies of solvation |∆h A | = |∆hC | = 35 kJ/mol and identical absolute values of entropies of solvation |∆s A | = |∆sC | = 105 J/mol K that have been used in the majority of our previous studies of selfassembly.15–18 Choosing a single absolute value for the enthalpy of solvation and a single absolute value for the entropy of solvation still leaves the possibility of three different combinations of the two (AB and CB) solvation processes: Both processes are enhanced upon cooling (∆h A = ∆hC = −35 kJ/mol, ∆s A = ∆sC = −105 J/mol K), both processes are promoted upon heating (∆h A = ∆hC = 35 kJ/mol, ∆s A = ∆sC = 105 J/mol K), and one process is favorable upon cooling (∆h A = −35 kJ/mol, ∆s A = −105 J/mol K), while the other one is promoted upon heating (∆hC = 35 kJ/mol, ∆sC = 105 J/mol K). The polymerization index N of the polymer B is chosen as N = 100, and the interaction parameters χ AB and χ BC (characterizing the effective weak van der Waals interactions between the polymer and the solvents) are assumed, for simplicity, to reduce the number of adjustable parameters to be identical ( χ AB = χ BC ) and equal to 300/T, unless otherwise specified. Notice that the polymerization index N strongly affects the phase boundaries but does not influence B2. Figures 1 and 2 present the spinodals and the temperature variation of the second virial coefficient B2(T), respectively, for these three classes of solvation processes for the simplest case where the A–C interaction parameter χ AC vanishes, i.e., when these solvents are completely miscible over the whole range of temperatures. The choice of χ AB = χ BC and the equality of the equilibrium constants K A and KC [for the reaction in Eqs. (1) and (2)] along with the assumption of vanishing interaction parameter χ AC makes this limiting model useful as a starting point for investigating general trends in competitive polymer solvation and for testing the theory derived in Sec. II. The first conclusion emerging from Fig. 1 confirms our previous finding13 that strong solvation of the polymers (i.e., strong associative interactions A–B and C–B) promotes miscibility of polymer solutions. The upper critical solution temperature (UCST) phase boundary (with Tc(o) ≈ 500 K) for

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FIG. 1. Spinodal curves T = T (φ oB ) evaluated from Eq. (8) for ternary A/B/C polymer solutions in which both solvation reactions are promoted upon cooling (upper red curve), upon heating (lower blue curve), and when associative solvation of polymers B is absent (dotted black curve). Phase boundaries are absent for A/B/C solutions in which one solvation reactions proceeds more favorably at low temperatures and the other at high temperatures because these mixtures are totally miscible over very wide ranges of T and r o for N = 100 (see Figs. 9 and 11). The free energy parameters ∆h A, ∆h C , ∆s A, and ∆s C are provided in the text, and the interaction parameters are χ AB = χ BC = 300/T and χ AC = 0. The choice of { χ α β } renders the o . The initial volume fraction spinodals insensitive to the ratio r o = φ oA/φ C φ oB refers to polymers B whose polymerization index N is fixed as N = 100. Unless specified otherwise, the spinodals in all other figures apply to N = 100.

A/B/C polymer solutions without associative solvation (the dotted black curve in Fig. 1) shrinks into a closed loop (the upper red curve in Fig. 1) when the two solvation reactions are enhanced upon cooling and to a considerably smaller UCST phase diagram (the lower blue curve in Fig. 1), with the critical temperature Tc ≈ 300 K significantly lower than Tc(o), when both solvation processes are promoted upon heating. Second, these two spinodals are completely insensitive to the relative

FIG. 2. Temperature variation of the second osmotic virial coefficient B2 of ternary A/B/C polymer solutions as evaluated from Eq. (38) when both solvation reactions are enhanced upon cooling (red curve), both upon heating (blue curve), one reaction is promoted upon cooling and the other upon heating (orange curve), and when solvation of polymers B through associative interactions is absent (dotted black curve). The interaction parameters { χ α β } oo are chosen as in Fig. 1. The orange curve refers to r oo = φ oo A /φ C = 1, while the other three curves are insensitive to r oo. The theta points are indicated by crosses.

J. Chem. Phys. 143, 194901 (2015)

initial composition r o of solvents A and C. The identity between the spinodals in Fig. 1 for ternary A/B/C mixtures and those for binary A/B mixtures is a consequence of the assumption that ∆h A = ∆hC , ∆s A = ∆sC , χ AB = χ BC , and χ AC = 0, i.e., that the solvents A and C are thermodynamically indistinguishable. In fact, the three spinodals of Fig. 1 coincide with the corresponding curves found for solvation of polymers B by a single solvent A and with that typical for polymer solutions A/B without solvation. The latter spinodal curves have been generated in our previous studies13,19 of polymer solvation by taking |∆h| = 35 kJ/mol (or |∆h| = 25 kJ/mol),19 |∆s| = 105 J/mol K, and χ AB = 300/T. Phase boundaries are absent in Fig. 1 for A/B/C solutions in which one solvation reaction proceeds more favorably in low temperatures and the other at high temperatures because these mixtures are totally miscible over very wide ranges of T and r o for N = 100 (see Figs. 9 and 11 below). The trends emerging from Fig. 1 are examined in Fig. 2 which presents B2 as a function of temperature T. The ternary mixtures in which both solvation reactions proceed more favorably upon cooling or upon heating exhibit, respectively, two theta temperatures TΘ(1) and TΘ(2) (see the red curve in Fig. 2) and a single theta temperature TΘ (see the blue curve in Fig. 2). In analogy to the spinodal curves in Fig. 1 that are insensitive to r o , the curves for B2(T), as well as the theta temperatures TΘ(1), TΘ(2), and TΘ, do not depend on r oo. On the other hand, polymer solutions in which one solvation process is promoted upon cooling and the other one upon heating are characterized by a positive B2, the lack of the theta temperature(s) (see the orange curve in Fig. 2), and by a dependence of B2 on r oo. The latter feature stems from different associative interactions of the polymer with the two solvents for all temperatures. (The orange curve in Fig. 2 applies for r oo = 1.) A negative B2 over a very wide range of temperatures for A/B/C solutions without solvation (see the dotted black curve in Fig. 2) is consistent with the spinodal (exhibiting a large immiscibility window) for this mixture (see the dotted black curve in Fig. 1). The theta temperature TΘ(o) = 2a = 600 K (where a = χ ABT) is the limit of the critical temperature Tc(o) for solutions of high molar mass polymers. Since the polymerization index N is taken as N = 100, the critical temperature Tc(o) ≈ 500 K is much smaller than TΘ(o). While the interaction parameters χ AB and χ BC are chosen as positive quantities because otherwise the binary A/B and B/C solutions would be completely miscible over the whole ranges of composition, temperature, and the polymerization index of the solute, the interaction parameter χ AC can generally be of either signs. Our analysis in the present paper is restricted, however, to the systems with a non-negative χ AC . For clarity of presentation, the description of the calculations is partitioned into three separate subsections, depending on the signs of the free energy parameters ∆h A, ∆hC , ∆s A, and ∆sC . The final subsection D presents a comparison with experimental data10 along with an analysis of the dependence of B2 on the composition r oo of the pure solvent mixture. The nature of this dependence commonly is taken as a signature for the solvation of polymers in mixed solvents, a signature that follows from our calculations and that has been identified in many experiments.10,20,21

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A. Two solvation processes upon cooling (class I solutions of solvated polymers)

As already mentioned, when both solvation processes are favored upon cooling, the phase boundaries for A/B/C solutions exhibit a closed loop shape (see Fig. 1) with upper and lower critical temperatures. Since the spinodals in Fig. 1 apply only for identical interaction parameters χ AB and χ BC and a vanishing interaction parameter χ AC , it is relevant to establish how removing these two constraints affects the mixture’s miscibility and its variation with r o when the other adjustable parameters of the theory are held constant. Figure 3 addresses one of these questions by displaying the spinodal curves obtained for variable χ AC but fixed and identical χ AB and χ BC . Not surprisingly, a large positive χ AC implies a strong enhancement of miscibility. This behavior is not, however, caused by solvation since simple A/B/C polymer solutions, devoid of strong interactions, also exhibit this type of cosolvency phenomena.11 The spinodals in Fig. 3 for χ AC , 0 are generally sensitive to the relative ratio r o of the initial solvent compositions, but those corresponding to r o = a are identical to those for r o = 1/a. The temperature variation of the second osmotic virial coefficient B2 in Fig. 4 unambiguously confirms the enhancement of miscibility due to increasingly repulsive interactions between the two solvents, a phenomenon emerging from Fig. 3 and existing for arbitrary r o . In particular, the theta temperatures TΘ(1) and TΘ(2) approach each other as the interaction parameter χ AC increases. Finally, when χ AC becomes sufficiently large (i.e., χ AC ≥ 250/T; see the orange curve in Fig. 4), B2 is positive over the whole temperature range, and both theta temperatures cease to exist. Large departures of χ AC = 250/T from the value ≈67/T for which the closed-loop phase boundary in Fig. 3 vanishes arise from huge difference between the polymerization index N = 100 of the polymer and a “critical” value of N corresponding to the constancy of phase boundaries and the critical temperatures.

FIG. 3. Spinodal curves T = T (φ oB ) for ternary A/B/C polymer solutions in which both solvation reactions become favorable upon cooling and the interaction parameters χ AB and χ BC are fixed as χ AB = χ BC = 300/T . Different curves correspond to the interaction parameters χ AC indicated in the figure. The solvent composition ratio r o is fixed as r o = 1. (The spinodal for χ AC = 0 is insensitive to r o .)

FIG. 4. Temperature variation of the second osmotic virial coefficient B2 of ternary A/B/C polymer solutions as evaluated from Eq. (38) for the three cases considered in Fig. 3 (green, red, and blue curves). The orange curve refers to χ AC = 250/T for which the high molar mass ternary A/B/C polymer solutions are miscible over the whole temperature range. The solvent oo composition ratio r oo = φ oo A /φ C is chosen as r oo = 1. (The B2(T ) curve for χ AC = 0 is insensitive to r oo.) The theta points are indicated by crosses.

The enhancement of miscibility in Fig. 3 is accompanied by growth of B2 with increasing χ AC at all temperatures. Consistently, all curves for B2(T) in Fig. 4 depend on r oo, except for the special case χ AC = 0, and curves generated for symmetrical ratios r oo = a and r oo = 1/a are indistinguishable. The figure captions provide more details concerning the values of r o and r oo employed in Figs. 3 and 4. Comparing the spinodals for r o = 1 and for variable χ AC from Fig. 3 with those referring to r o , 0 and variable χ AC leads to the conclusion that the maximal sensitivity of miscibility to the interaction parameter χ AC occurs when the ternary polymer solution contains identical amounts of solvents A and C. A similar conclusion concerning the dependence on r o of the miscibility of solutions of solvated polymers can be drawn from the spinodal curves generated for variable r o but fixed χ AC , 0. The existence of maximal miscibility for r o = 1 coincides with our finding for regular A/B/C solutions where solvation of polymer chains is absent. This behavior appears for solvated polymer mixtures only when the interaction parameters χ AB and χ BC and the enthalpies and entropies for the two solvation processes are identical (i.e., χ AB = χ BC , ∆h A = ∆hC , and ∆s A = ∆sC ). Figure 5 illustrates the transformation of phase boundaries of solvated polymer solutions A/B/C with changes in the polymerization index N of the polymer for fixed { χ AB, χ BC , χ AC , and r o }. As N increases, the lower and upper critical temperatures Tc(1) and Tc(2) depart from each other and approach, respectively, the theta temperatures TΘ(1) and TΘ(2) (indicated by the dotted lines in Fig. 5) in the limit of high molar mass polymer. A complete analysis requires examining how asymmetry in the interaction parameters χ AB and χ BC alters the general trends emerging from Fig. 3 and from the calculated spinodal curves for r o , 1 (not shown), which all refer to symmetrical systems with χ AB = χ BC to demonstrate that miscibility improves with increasing χ AC for fixed r o and achieves a maximum for r o = 1 when χ AC is held constant. Consider,

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of polymers B with solvent C [in the FH type expression for the free energy of mixing ∆ f mix obtained from Eqs. (5) and o (6) by subtracting terms linear in φ Ao , φoB, and φC ]. A more positive ∆ f mix reflects the diminished ability of components to mix. Similar arguments imply that the opposite trend, i.e., a reduction of miscibility with increasing r o is predicted when χ AB > χ BC . B. Two solvation processes upon heating (class II solutions of solvated polymers)

FIG. 5. Spinodal curves T = T (φ oB, r o = 1/4) for ternary A/B/C polymer solutions with both solvation reactions enhanced upon cooling. The interaction parameters are fixed as χ AB = χ BC = 300/T and χ AC = 50/T , and the different polymerization indices N are indicated in the figure. The theta temperatures (denoted by dashed lines) provide the high N limits for the upper and lower critical solution temperatures.

for instance, systems with χ AC = 0 and χ BC > χ AB, i.e., with the effective interactions between monomers of the polymer B and single-bead solvent C more repulsive than the effective interactions between segments of the polymer B and singlebead solvent A. Consequently, miscibility worsens relatively to the miscibility implied by the closed loop diagram generated for the same χ BC = χ AB = 300/T in Fig. 1. Increasing χ AC still enhances miscibility when χ BC > χ AB. Elevating χ BC above χ AB induces, on the other hand, a qualitative change in the variation of miscibility with the ratio r o . More specifically, miscibility improves monotonically with increasing r o , as shown in Fig. 6 which displays several spinodal curves for variable r o and fixed χ AB = 300/T, χ BC = 400/T, and o χ AC = 0. This behavior arises because a smaller r o = φoA/φC o translates to a higher volume fraction φC , which in turn implies o that the term φoB φC χ BC becomes more positive. This term provides a quantitative measure of the nonathermal mixing

FIG. 6. Spinodal curves T = T (φ oB ) for ternary A/B/C polymer solutions in which both solvation reactions are favored upon cooling. The interaction parameters are fixed as χ AB = 300/T , χ BC = 400/T , and χ AC = 0. The solvent composition ratios r o are indicated in the figure for each of the curves.

The main difference between the solvation of polymers upon cooling and upon heating emerges because the latter does not induce qualitative changes in the upper critical temperature phase diagram of the corresponding regular polymer solutions devoid of solvation (see the blue and dotted curves in Fig. 1). More specifically, the solvation of the polymer upon heating leads to a much lower upper critical temperature Tc and to phase boundaries with a characteristic flatness13 (see Fig. 1). The decrease of Tc and an accompanying increase in the miscibility of the polymers in mixed solvents are direct consequences of the decoration of the polymer chains by molecules of solvents A and C at high temperatures, a reversible process that almost entirely “dies” at low temperatures, causing the immiscibility of bare polymer chain solution with the solvent species. Figures 7 and 8 display the spinodal curves and the temperature variation of the second osmotic virial coefficient B2, respectively, for variable χ AC but fixed interaction parameters χ AB = χ BC = 300/T and fixed ratio r o = 1 [r oo = 1 for B2(T)]. The improvement of miscibility upon increasing χ AC in Fig. 7 is, however, much smaller than that emerging from Fig. 3 which refers to polymer mixtures in which both solvation processes are promoted upon cooling. In fact, the counterparts of spinodals from Fig. 3 for the interaction parameter χ AC = 50/T, χ AC = 60/T, and χ AC = 65/T would be barely distinguishable by the naked

FIG. 7. Spinodal curves T = T (φ oB ) for ternary A/B/C polymer solutions in which both solvation reactions are enhanced upon heating. The interaction parameters are χ AB = χ BC = 300/T . The different interaction parameters χ AC are indicated in the figure. The solvent composition ratio r o is fixed as r o = 1. (The spinodal for χ AC = 0 is insensitive to r o .) The figure becomes the counterpart of Fig. 3 when both solvation processes are promoted upon heating.

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FIG. 8. Temperature variation of the second osmotic virial coefficient B2 of ternary A/B/C polymer solutions for the three cases considered in Fig. 7 (green, orange, and red curves). The magenta curve refers to χ AC = 800/T and A/B/C polymer solutions which are miscible the whole temperature oo range. The ratio r oo of solvent compositions φ oo A and φ C is taken as 1 (only the red curve for χ AC = 0 is insensitive to r oo). The theta points are designated by crosses.

eye when plotted in Fig. 7. Increasing χ AC induces an enhancement of B2 that changes sign at most once. If the interaction parameter χ AC is large enough to ensure complete miscibility over the whole range of N, then B2 is positive at all temperatures (see the magenta curve in Fig. 8). The variations of miscibility with r o are not illustrated since they generally follow the pattern described in Subsection III A for both symmetrical and asymmetrical interaction parameters, χ AB and χ BC . C. One solvation process favored upon cooling and the other upon heating (class III solutions of solvated polymers)

More careful inspection of Fig. 1 suggests that solutions of polymer B in mixed solvents A and C would likely exhibit nontrivial miscibility patterns if one solvation reaction is enhanced upon cooling and the other one upon heating. The hypothetical miscibility patterns may range from a closedloop phase boundary to upper critical temperature phase diagrams, depending on the composition r o which is the o ratio of the initial volume fractions φoA and φC of solvents A and C, respectively. For example, when the polymerization index N = 100, χ AB = χ BC = 300/T, χ AC ≥ 0, and the o ratio r o = 1 (i.e., φ Ao = φC ), the ternary A/B/C solution is totally miscible, which is consistent with the positive sign of the second osmotic virial coefficient B2 over the whole temperature range (see the orange curve in Fig. 2) when oo φ Aoo = φC (i.e., r oo = 1). Also, physical intuition prompts the expectation that equal amounts of solvents A and C enable both solvation processes to participate equally, and, thus, to cooperate in full synergy in the determination of the phase diagram. This suggestion and preliminary results from Figs. 1 and 2 encourage the examination of the implications of these competitive solvation processes more quantitatively. Again consider an AB solvation reaction in Eq. (1) that is promoted upon cooling, while the BC solvation process in Eq. (2) is facilitated upon heating. If the polymerization index

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is N = 100 and the interaction parameter is χ BC = 300/T, the spinodal for the binary B/C system (chosen as a reference system) coincides with that presented in Fig. 1 for a ternary mixture in which both solvation reactions are enhanced upon heating, N = 100, χ AB = χ AC = 300/T, χ AC = 0, and r o is arbitrary (see the blue curve in Fig. 1). Thus, the upper critical temperature phase diagram from Fig. 1 represents simultaneously the spinodal for the reference system specified by r o = 0. Allowing r o to increase successively from zero to unity (i.e., in the range where the initial concentration of solvent C exceeds that for solvent A) by the addition of solvent A to the binary C/B mixture, and assuming that solvation of the polymer B by the molecules of solvent A is promoted upon cooling, we determine phase boundaries for a series of these ternary A/B/C solutions. Figure 9 collects resulting phase diagrams for these mixtures, along with the phase diagram for the reference system, and reveals that increasing r o from r o = 0 (the dotted curve) to r o = 0.25 (the magenta curve) leads to a significant improvement of miscibility. Figure 10 demonstrates that the miscibility is fairly insensitive to the magnitude of the interaction parameter χ AC . More surprising is, however, the reduction of miscibility with growing χ AC , a trend completely opposite to those found when both solvation processes are enhanced upon cooling or upon heating, as well as to the behavior observed for simple ternary polymer solutions devoid of strong interactions. When one solvation reaction is favorable as temperature is decreased and the other when temperature is increased, the examination of the miscibility patterns becomes more complicated when one solvent, say A, is in the majority, i.e., when the ratio r o ≫ 1. This complexity arises because ternary solutions of solvated polymers are found to become totally miscible for the rather small polymerization index N = 100, χ AB = χ BC = 300/T, χ AC = 0, and r o ≫ 1. Figure 11 illustrates this behavior and reveals that closed-loop phase boundary is predicted to disappear for N < 122 when r o = 100, χ AB = χ BC = 300/T, and χ AC = 0. The spinodal reappears, however, when N is increased, e.g., to N = 1000,

FIG. 9. Spinodal curves T = T (φ oB ) for ternary A/B/C polymer solutions in which one solvation reaction is promoted upon cooling and the other one upon heating. The interaction parameters are fixed as χ AB = χ BC = 300/T and χ AC = 0, and variable solvent composition ratios r o are indicated in the figure.

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FIG. 10. Spinodal curves T = T (φ oB, r o = 1/20) for ternary A/B/C polymer solutions in which one solvation reaction is enhanced upon cooling and the other one upon heating. The interaction parameters are fixed as χ AB = χ BC = 300/T . Each curve corresponds to a different interaction parameter χ AC as indicated in the figure.

as presented in Fig. 12 which displays the phase boundaries of these systems for variable r o . In particular, Fig. 12 manifests that phase boundaries vanish when r o ≈ 44, thereby indicating a very nontrivial feature of the solvated polymer mixtures. Adding a very small amount of solvent C that dresses the polymer B upon heating to a solution of polymer B in a solvent A that decorates polymer chains upon cooling leads to a conspicuous improvement of the miscibility of the mixture. This improvement is sensitive to the magnitude of the interaction parameter χ AC and surprisingly becomes maximal when χ AC = 0. D. Dependence of the second osmotic virial coefficient B2 on the solvent composition: Direct comparison to experiment

While the relevance of temperature variation of the second osmotic virial coefficient B2 to the miscibility and competitive

FIG. 11. Spinodal curves T = T (φ oB, r o = 1/20) for ternary A/B/C polymer solutions in which one solvation reaction is favorable upon cooling and the other one upon heating. The interaction parameters are fixed as χ AB = χ BC = 300/T and χ AC = 0, and various polymerization indices N are indicated in the figure.

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FIG. 12. Spinodal curves T = T (φ oB ) for ternary A/B/C polymer solutions in which one solvation reaction is promoted upon cooling and the other one upon heating. The interaction parameters are χ AB = χ BC = 300/T and χ AC = 0. Different curves refer to different solvent composition ratio r o as indicated in the figure. The polymerization index N of the polymers B is N = 1000 for all curves.

solvation of polymers has already been discussed, the present subsection describes the influence of the solvent composition (φoo A or r oo) on B2 at fixed temperatures. Figure 13 depicts the osmotic virial coefficient B2 as a function of the volume fraction φoo A of solvent A in a solvent mixture in osmotic equilibrium with the polymer solution for which both solvation reactions are promoted upon cooling and whose spinodals for r o = 0.25 (or r o = 4) are considered in Fig. 5. (Notice that the spinodals of Fig. 5 are the same for the “symmetrical” values of r o = a and r o = 1/a, and quantitative comparisons between spinodals and B2 are only adequate for solutions of high molar mass polymer.) Each curve in Fig. 13 corresponds to a different temperature T as indicated in the figure. When the temperature T lies below a lower theta temperature TΘ(1) (T < TΘ(1)) or ranges between lower and upper theta temperatures, TΘ(1) < T < TΘ(2), the virial

FIG. 13. The second osmotic virial coefficient B2 of ternary A/B/C polymer solution as a function of the composition (φ oo A ) of the pure solvent mixture in osmotic equilibrium with A/B/C solutions in which both solvation reactions are enhanced upon cooling. The interaction parameters are χ AB = χ BC = 300/T and χ AC = 50/T . The different temperatures are indicated in the figure. The theta points are denoted by crosses.

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coefficient B2 remains positive or negative, respectively, over the whole range of φoo A , thus reflecting regions of miscibility and immiscibility for high molar mass polymers (see the dotted magenta curve in Fig. 5). The second virial coefficient B2 at T = TΘ(1) changes its sign (see Fig. 13) when φoo A = 0.2 and φoo = 0.8, i.e., when r = 0.25 and r = 4, respectively. oo oo A The values of r oo then coincide with the values of r o specified for ternary A/B/C polymer solutions in Fig. 5. Replotting B2 of Fig. 13 as a function of r oo instead of φoo A leads to the appearance of a maximum of B2(φoo A ,T = const) for r oo = 1. The parabolic shape of B2(φoo A ) also appears robustly when both solvation reactions proceed upon heating as well as when one solvation reaction occurs upon cooling and the other one upon heating. In the latter case, the parabola can be very steep and asymmetric. Comparisons of B2 evaluated from the lattice type theory require establishing a relation between B2 determined from Eq. (38) and the experimental10 second osmotic coefficient A2 in units of [cm3 g−2 mol]. The former is defined as the coefficient of the second power of φoB in the power series for the dimensionless osmotic pressure, Πvcell = B1φoB + B2(φoB)2 , φoB → 0, (41) k BT while data for A2 estimated from measurements of the osmotic pressure Π are generally extracted from the relation c Π = + A2 c2 , c → 0. (42) RT Mw The polymer concentration c in the experimental sample is usually expressed in g/cm3, Mw denotes the molecular mass of the polymer in the sample, and R is the gas constant, i.e., the product of Boltzmann’s constant k B and Avogadro’s number N Av . First, Eq. (41) is converted to the form B1 o Π B2 o 2 = φB + (φ ) , φoB → 0, (43) RT Vmol Vmol B where Vmol = vcell N Av designates the volume associated with a mol of lattice sites. Comparing Eq. (42) with Eq. (43) leads to the desired relation, A2 = B2

vcell N Av [Mw(mon)]2

,

(44)

where vcell is taken as the “standard” value (2.7 Å)3 used in many of our lattice model calculations,22 and Mw(mon) is the molar mass of a single monomer of the polymer. Figure 14 compares B2 evaluated from Eq. (38) with experimental data10 for A2 of poly(methyl methacrylate) (polymer B) dissolved in a few binary solvent mixtures at t = 25 ◦C. Blue, red, and green circles denote the data for A2 as a function of the composition of solvent mixtures of acetonitrile (MeCN) (solvent A) with pentyl acetate, 1-propanol, and methanol (the series of solvents C), respectively, The curves are fits to the data. All fits are generated for the same values, ∆h A = ∆hC = −33 kJ/mol and ∆s A = ∆sC = −105 J/mol K, to reduce the number of adjustable parameters. The interaction parameters χ AB, χ BC , and χ AC determined from the fits are specified in the figure caption for each system. The use of a common value of the

FIG. 14. Comparison between B2 calculated from Eq. (38) (curves) with experimental data10 (circles) for the second osmotic virial coefficient A2 of poly(methyl methacrylate) (PMMA) in a few binary solvent mixtures at t = 25 ◦C. The data for B2 are converted into units of cm3 g−2 mol. Blue, red, and green circles represent mixtures of acetonitrile (MeCN) with pentyl acetate, 1-propanol, and methanol, respectively. The volume fraction φ MeCN refers to the concentration of acetonitrile in the reservoir containing the pure solvent mixture and corresponds to the volume fraction φ oo A in Eq. (38). The blue, red, and green curves are fits to the experimental data by taking { χ AB = χ BC = χ AC = 270/T }, { χ AB = 270/T, χ BC = 320/T, χ AC = 350/T }, and { χ AB = 270/T, χ BC = 400/T, χ AC = 270/T }, respectively. The values of free energy parameters are chosen as common adjustable parameters and equal ∆h A = ∆h C = −33 kJ/mol, and ∆s A = ∆s C = −105 J/mol K. Both solvation processes are assumed to be promoted upon cooling, and PMMA is the polymer component B, acetonitrile is solvent A, and pentyl acetate, 1-propanol, or methanol represent solvent C.

interaction parameter χ AB is consistent with the presence of a common solvent (MeCN) in the three experimental samples.10 The quality of the fits in Fig. 14 is very good, as the only noticable discrepancies appear for binary systems (i.e., when the volume fraction φMeCN is zero or unity). The fits in Fig. 14 are not unique, however, and fits of similar quality have been also obtained when both solvation reactions are assumed to proceed upon heating. A serious concern lies, however, in the use of Eq. (38) [that has been derived for solutions of solvated polymer B in mixed solvents exhibiting neither selfassociation nor mutual association] to describe experimental data for poly(methyl methacrylate)/acetonitrile/methanol (or 1, propanol) mixtures.

IV. DISCUSSION

The most striking feature of ternary A/B/C polymer solutions is an enhancement of their miscibility upon increasing the net repulsive van der Waals interactions between the molecules of solvents A and C. The origin of this behavior lies in the fact that the effective interaction parameter χ AC enters into the expression for the system’s free energy of mixing with a negative sign.11 The competitive solvation of polymers B by small-molecule liquids A and C strongly affects the miscibility patterns of these mixtures, but does not necessarily influence the intrinsic relation between the magnitude of χ AC and the miscibility. When both polymer solvation processes in ternary A/B/C polymer solutions are promoted either upon cooling or upon heating,

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the improvement of miscibility with growing interaction parameter χ AC is qualitatively similar to that occurring for regular polymer A/B/C solutions devoid of strong associative solvation (see Figs. 3 and 7). o The ratio r o = φoA/φC of the initial concentrations φoA o and φC of the small-molecule liquids A and C is the other important parameter besides χ AC controlling the miscibility of ternary solutions. Depending on the relative magnitudes of the interaction parameter between the polymer B and a solvent (i.e., of χ AB and χ BC ), miscibility can be driven in both directions upon changing r o (see, e.g., Fig. 6 where miscibility improves upon elevating r o because χ BC significantly exceeds χ AB). The occurrence of a maximal miscibility for r o = 1 and of identical spinodal curves for the symmetric values r o = a and r o = 1/a is associated with the proximity of the interaction parameters χ AB and χ BC . The general trends illustrated in Figs. 3, 6, and 7 for a wide class of polymer solutions in which both solvation processes are facilitated either at low or at high temperatures remain valid when no polymer solvation takes place. As has been already hinted, the least intuitive and nontrivial patterns of miscibility emerge from our calculations when polymer solvation by one liquid is enhanced upon cooling, while “dressing” the polymer by the other solvent is promoted upon heating. In strong contrast to regular polymer solutions and to the ternary polymer mixtures described above (and in Sections III A and III B), increasing the interaction parameter χ AC leads to the worsening of the miscibility (see Fig. 10). No less intriguing is, however, the enormous sensitivity of the phase diagram to the addition of a small amount of a second solvent (A or C) to a binary polymer solution (C/B or AB, respectively). For instance, Fig. 9 demonstrates that blending of the C/B solution with a small amount of solvent A (about 20% of solvent C) shifts the upper critical solution temperature from 300 K to below 200 K. A more striking example of the sensitivity of miscibility to the small-molecule additives is even found when a tiny amount of solvent C (whose strong association with the polymer is favorable at high temperatures) is added to binary A/B polymer solutions that exhibit a closed-loop reentrant phase diagram (see Fig. 12). More specifically, the addition of about 2% of solvent C eliminates the phase boundaries and, thus, ensures complete miscibility over the whole temperature range. Examples illustrated in Figs. 9 and 12 suggest a wide spectrum of potential applications to the field of chromatography.3–6,8,9 One of the most important outcomes of our theory is that it predicts the second osmotic virial coefficient B2 as a function of both temperature T, and the composition (r oo) of the solvent mixture. The theory provides the framework for analyzing the relation between the spinodal curves and the plots of B2 versus T and r oo and for classifying polymer solutions in terms of the matrix of possibilities depending on whether polymer

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solvation by the two solvents is preferred upon cooling or heating. An example of this analysis is presented in Section III D. The favorable comparison of B2 with experimental data for poly(methyl methacrylate)/acetonitrile/methanol (or 1, pentanol) solutions in Fig. 14 has been obtained without considering the mutual and self-association of the solvents and this type of association must be incorporated in future modeling of polymers in mixed solvents. Moreover, the mean field nature of the theory prevents its applicability infinitely dilute polymer solutions where conformational transitions of the polymer chains induced by solvent association within the chain may occur.23,24 ACKNOWLEDGMENTS

This research is supported, in part, by National Science Foundation (NSF) Grant No. CHE-1363012. 1J.

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