Perspective pubs.acs.org/Macromolecules 50th Anniversary Perspective: Phase Behavior of Polymer Solutions and Blends Piotr Knychała,† Ksenia Timachova,‡ Michał Banaszak,*,∥,⊥ and Nitash P. Balsara*,‡,# † Faculty of Polytechnic, The President Stanislaw Wojciechowski State University of Applied Sciences, Kalisz, Poland ‡ Department of Chemical and Biomolecular Engineering, University of California, Berkeley, Berkeley, California 94720, United States ∥ ⊥ Faculty of Physics and NanoBioMedical Centre, Adam Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland # Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, United States *S Supporting Information ABSTRACT: We summarize our knowledge of the phase behavior of polymer solutions and blends using a unified approach. We begin with a derivation of the Flory−Huggins expression for the Gibbs free energy of mixing two chemically dissimilar polymers. The Gibbs free energy of mixing of polymer solutions is obtained as a special case. These expressions are used to interpret observed phase behavior of polymer solutions and blends. Temperature- and pressure-dependent phase diagrams are used to determine the Flory−Huggins interaction parameter, χ. We also discuss an alternative approach for measuring χ due to de Gennes, who showed that neutron scattering from concentration fluctuations in one-phase systems was a sensitive function of χ. In most cases, the agreement between experimental data and the standard Flory−Huggins−de Gennes approach is qualitative. We conclude by summarizing advanced theories that have been proposed to address the limitations of the standard approach. In spite of considerable effort, there is no consensus on the reasons for departure between the standard theories and experiments. ■ INTRODUCTION mean-field theory of de Gennes and their effect on phase 4,5 Our understanding of the thermodynamic underpinnings of the behavior. We conclude by discussing the limitations of the Flory−Huggins−de Gennes approach and describe attempts to phase behavior of polymer solutions and blends began with − address these limitations. the Flory−Huggins theory.1 3 In this theory, the interactions between polymer chains of one type and another are governed by − a parameter that is related to interactions on the monomer length ■ FLORY HUGGINS THEORY scale and chain lengths. To model polymer blends, we consider Our objective is to compute the Gibbs free energy of a two types of monodisperse polymer chains labeled A and B, homogeneous mixture of two polymers, ΔG, relative to that of 1−3 with NA and NB repeat units, respectively. The monomer-scale the pure polymers, following the Flory−Huggins approach. interactions are governed by a nondimensional parameter, χ, The mixing process is conducted in two steps as shown in which is traditionally called the Flory−Huggins interaction Figure 1. In the first step, the chains are mixed at constant volume parameter. (V) and temperature (T), and in the second step, the mixture Most of the published derivations (e.g., refs 1−3) are restricted is expanded or contracted to obtain the correct volume of to the derivation of the Helmholtz free energy of mixing for the mixture, i.e., to account for the volume change of mixing. ff polymer solutions. This is usually followed by generalization to The pressure after step 1 is p1, which will di er from p due to polymer blends without a formal derivation. Research articles differences in the equations of state of the pure components and ff tend to focus on the temperature dependence of phase behavior the mixture. In our analysis, we neglect the di erence between p1 of either solutions or blends; it is rare to see both solutions and and p. blends discussed in the same paper. In this Perspective, we take a We begin by evaluating ΔS, the entropy increase due to the slightly different approach. We begin with a derivation of the mixing that occurs during step 1. The mixture comprises nA Flory−Huggins Gibbs free energy of mixing for polymer blends, indistinguishable polymer chains of type A, each with NA and solutions are treated as a special case. This is followed by segments per chain, on a lattice with nt sites. Each segment comparing predictions based on Flory−Huggins theory with occupies a volume v0. The remainder of the sites are occupied by experimentally determined phase behavior of both solutions and blends. Our use of the Gibbs free energy enables addressing both Received: December 5, 2016 temperature- and pressure-dependent phase diagrams. We dis- Revised: February 28, 2017 cuss the nature of concentration fluctuations as described by the Published: April 6, 2017 © 2017 American Chemical Society 3051 DOI: 10.1021/acs.macromol.6b02619 Macromolecules 2017, 50, 3051−3065 Macromolecules Perspective Figure 1. Schematic of mixing polymers A and B in two steps. In the first step, the polymer samples are mixed at constant V and T. The pressure after step ff 1isp1. In the second step volume of the mixture is changed to match the actual volume of the mixture at p and T of interest. The di erence between p1 and p is neglected in the analysis. fi polymer B; each B chain comprises NB segments. The overall The number of con gurations that A chains can adopt in pure A, ϕ ϕ Ω volume fractions of A and B chains, A and B, are given by A,0, is obtained by recognizing that a total number of sites in pure ϕ ϕ nANA/nt and nBNB/nt. We will assume that each lattice site has z Ais Ant, i.e., substituting Ant for nt in eq 4. nearest neighbors. If we label the nA chains from 1 to nA and NA nA ⎛ ⎞ sequentially place them in the lattice, then the total number of 1 N jNA Ω = ∏ ϕ nz A⎜1 − ⎟ configurations is given by A,0 ! A t ⎝ ϕ ⎠ nA j=1 Ant (6) n A fi Ω′=ωω... ω =∏ ω Similarly, the number of con gurations that B chains can adopt in A12nA j pure B is given by j=1 (1) NB nB ⎛ ⎞ where ω is the number of configurations available to the jth 1 N jNB j Ω = ∏ ϕ nz B⎜1 − ⎟ chain. B,0 n ! B t ⎝ ϕ n ⎠ fi fi B j=1 B t (7) The rst monomer of the rst chain has nt possibilities. However, the second monomer only has z possibilities, as do all and of the other monomers of this ideal chain. ⎛ ⎞ ΩΩAB − Δ=Skln⎜ ⎟ ω =≈nzNAA1 nzN B 1 t t (2) ⎝ ΩΩA,0 B,0 ⎠ (8) fi − The rst monomer of the (j + 1)th chain has (nt jNA) It follows that possibilities. On average, the probability of having an unoccupied N − nA ⎛ ⎞ A neighboring site for the second monomer is 1 jNA/nt. Theories ΩΩ 1/− jN n fi AB = ϕ −nA ∏ ⎜ A t ⎟ based on such averages are often referred to as mean- eld theories. ΩΩ A ⎝ 1/()− jNϕ n ⎠ − A,0 B,0 j=1 A A t The term z in eq 2 is thus replaced by z(1 jNA/nt). Thus N nB ⎛ ⎞ B N 1/−−ϕ jN n A −nB ⎜ A B t ⎟ ωjt+1AA=−()((1/))njNz − jNn t (3) × ϕ ∏ ⎜ ⎟ B ⎝ − ϕ ⎠ j=1 1/()jNB B nt (9) The number of configurations available to indistinguishable A Ω Ω′ and chains, A, is given by A/nA! nA ΩΩAB 1 N N lnlnln=−nnϕϕ − Ω = nzA(1− jN / n ) A A A B B A ∏ t A t ΩΩA,0 B,0 nA! j=1 (4) nAAn +−−−NjNnjNn[∑∑ ln(1 / ) ln(1 /(ϕ ))] The number of configurations that indistinguishable polymer B A A t A A t − j==1 j 1 chains can adopt in the remaining nt NAnA sites is nBBn nB +−−−−NjNnjNnB[∑∑ ln(1ϕϕB /t ) ln(1B /(t ))] 1 NBBN A B Ω = ∏ nz(1−−ϕ jN / n ) == B n ! t A B t j 1 j 1 B j=1 (5) (10) 3052 DOI: 10.1021/acs.macromol.6b02619 Macromolecules 2017, 50, 3051−3065 Macromolecules Perspective It can be shown (see Supporting Information)that ⎡ ϕ ϕ ⎤ ⎢ A B ⎥ ΔGnkT=++t B lnϕ ln ϕχϕϕ nAAn ⎣ A BAB⎦ NNA B (22) NA[∑∑ ln(1−−jNA / nt ) ln(1 − jNA /(ϕA nt ))] j==1 j 1 where ϕϕ =−nt BBln (11) χχ=+u ηpkT/(B ) (23) χ and Empirically it has been found that u is often a linear function nBBn of 1/T: NB[∑∑ ln(1−−ϕϕA jNB / nt ) − ln(1 − jNB /(B nt ))] B == χ =+A j 1 j 1 u T (24) = n ϕϕln χ t BB (12) In some cases u is a nonmonotonic function of 1/T. In this case, Equations 8 and 10 reduce to one could use an expression of the form B C ΔSkn=−BA[lnϕϕ + nB ln] (13) χ =+A + A B u T T2 (25) because the last two terms in eq 10 cancel each other. It is easy to While eq 24 is often used in the literature, there is no show that eq 13 is equivalent to the equation consensus on the expression that must be used in case the χ ⎡ ϕ ϕ ⎤ versus 1/T plot is not linear. In theory, this term will depend on ⎢ A B ⎥ ff ΔSkn=−B t lnϕ + ln ϕ the temperature dependence of the di erence between the ⎣ NNA B⎦ fi AB (14) speci c heats at constant pressure, Cp, of the pure components Δ 6 ∂ Δ ∂ Δ Δ and the mixture, Cp, as ( H)/ T = Cp. Equation 24 is The change in internal energy due to step 1, U, is given by Δ obtained when Cp is a constant. Equation 25 is obtained when the standard formula based on random mixing Δ the temperature dependence of Cp is dominated by a term 2 6,7 χ ΔUnz=Δϵt ϕϕAB (15) proportional to 1/T .