Density-Functional Theory for Mixtures of AB Random Copolymer and

Article pubs.acs.org/Macromolecules

Density-Functional Theory for Mixtures of AB Random Copolymer and CO2 Xiaofei Xu Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China

Diego E. Cristancho and Stéphane Costeux The Dow Chemical Company, Midland, Michigan 48674, United States

Zhen-Gang Wang* Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States ABSTRACT: We propose a density-functional theory (DFT) to describe inhomogeneous mixtures of AB random copolymer and carbon dioxide (CO2). The statistical sequence of monomer in the polymer chain backbone is modeled by a transition matrix in a Markov-step growth process. The parameters of the theory are determined by fitting the bulk experimental data. We apply the DFT to the interfacial properties of binary mixtures of CO2 with poly(methyl methacrylate co ethyl methacrylate) (P(MMA-co-EMA)), poly(methyl methacrylate co ethyl acrylate) (P(MMA-co-EA) and poly(styrene co ethyl acrylate) (P(S-coEA)). The dependence of CO2 solubility and interfacial tension on the copolymer composition and pressure is examined. We find that higher fractions of EA or EMA result in higher solubility of CO2 at a given pressure, but also results in higher interfacial tension at a fixed CO2 content in the polymer-rich phase. Using the classical nucleation theory as a rough estimate, we examine the effect of the copolymer composition on the free energy barrier of bubble nucleation in random copolymer−CO2 mixtures. and novel materials.9,10 There is considerable interest to study their structural and thermodynamic properties. Although a number of theories have been proposed, describing the phase behavior and interfacial properties of random copolymer systems still presents an enormous theoretical challenge. The earliest theoretical descriptions of random copolymer systems are based on an extension of the Flory−Huggins theory (FHT), by assuming an interaction parameter χij for different monomer species i and j.11,12 As the theory did not consider the effect of monomer sequence in the chain, Balazs et al.13 and later Cantow and Schulz14 introduced a sequence-dependent χ parameters χijk,lmn to describe the interactions between a pair of triads of sequential monomers. Dudowicz and Freed15,16 developed a lattice cluster theory (LCT) for random copolymers, which is based on the generalized lattice model that endows the monomers with specific structures by allowing them to occupy several lattice sites with specified connectivity.17 In contrast to the FHT, the LCT treatment requires no ad hoc assumptions concerning the particular form of the sequence-dependent interaction parameters. Fredrickson et al.18−20 constructed a Landau-type density-functional theory for random copolymer and random block copolymer, in which the free energy is developed as a functional Taylor expansion in a

I. INTRODUCTION Phase separation in polymer and carbon dioxide (CO2) mixtures is a problem of great interest in the manufacturing of polymer foams.1−3 The fabrication of novel foam materials with nanoscale pores can benefit from systematic and quantitative understanding of the thermodynamics and kinetics of phase separation.4 In several recent publications, we proposed a density functional theory (DFT) for mixtures of a homopolymer and carbon dioxide,5 and systematically studied bubble nucleation in these mixtures.6,7 The theory yields results for the bulk phase behavior and interfacial tension in good agreement with experiments, and predicts the mechanistic pathways for bubble nucleation in these mixtures. In particular, we find that the presence of a metastable transition between a CO2-rich vapor and a CO2-rich liquid leads to a discontinuous drop in the nucleation barrier as a function of the CO2 supersaturation. However, experiments have shown that random copolymers made of two different types of monomers are better materials for nanofoaming than homopolymers made of either monomer.8 We are thus motivated to extend our theory to the case of random copolymers. Random copolymers consist of two or more different monomers with a statistical sequence distribution along the chain backbone. They are of considerable technological importance because of their ability to enhance miscibility or to produce a multiphase region with particular desired morphology, which has led to the development of many special © XXXX American Chemical Society

Received: May 25, 2015 Revised: July 15, 2015

A

DOI: 10.1021/acs.macromol.5b01122 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules concentration order parameter, and the sequence disorder is averaged by the replica approach. However, the PVT phase behavior and interfacial properties were not studied in these works. Density-functional theory is an attractive alternative as it can successfully capture the necessary microscopic details of random copolymer. Effects of monomer sequence in a copolymer chain and compressibility of the system are incorporated automatically in the DFT description. Compressibility effects have been shown to play an essential role in the thermodynamics of polymer and CO2 systems.5,7,21,22 Gross et al.23 developed an equation of state (EOS) for AB random copolymer by using the perturbed-chain statistical associating fluid theory (PC-SAFT). In their study, however, one of the monomer types (e.g., A) was assumed to be the minority, so that effects of the A−A bonds were neglected. In this work, we consider the mixture of AB random copolymer and CO2. We provide a simple yet reasonable approximation to model the statistical sequence of monomers, and propose a new extension of PC-SAFT EOS to the mixture of AB random copolymer and CO2. The chain sequence is described by a transition probability matrix in a Markov growth process. The parameters of the EOS are accurately determined by fitting the experimental data. We then propose a density-functional theory based on this EOS to study the phase behavior and interfacial properties of the mixture. Our resulting theory can be used to predict a host of bulk thermodynamic and interfacial properties, such the solubility of CO2 in the copolymer and the interfacial tension between CO2 and the polymer-gas mixtures, as functions of the copolymer composition, temperature and pressure. As an application of the bulk and interfacial properties, we investigate the effect of copolymer composition on the bubble nucleation barrier using classical nucleation theory.

Figure 1. Schematic of CO2 molecule. The CO2 molecule is modeled as a dimer (blue spheres). The association interaction arises from the positive and negative partial charges on the C and O atoms, respectively. We capture this interaction by assigning three interaction sites, two corresponding to the O atoms (red solid regions) and one to the C atoms (purple solid region).

The polymers considered here consist of two types of monomer, A and B, which are randomly arranged along the chain backbone. We describe the chemical-sequence distributions of the chain by a Markov-step growth: The monomer identity at a given location on the backbone is determined only by the monomer identity immediately preceding it and is not affected by the prior chain sequence.18 The chain sequences can thus be statistically determined by specification of the average copolymer composition fA (mole fraction of monomer A in one chain) and a matrix p = (pij)2×2, (i, j = A, B) of transition probabilities, where pij represents the conditional probability that a monomer of type i at an arbitrary position on a chain is immediately followed by a monomer of type j. We assume pij is independent of the monomer position along a chain. Invoking conservation of species and the definition of probability, one obtains

pAA + pAB = 1 pBA + pBB = 1

II. MODEL AND THEORY A. Molecular Model. We consider a compressible mixture of polymer and CO2, where the molecular units of both components are coarse-grained as spherical particles with a hard core. Although other potentials are possible, hard spheres are the simplest and most intuitive means of capturing the excluded volume effects; the hard-sphere diameters can be readily obtained from fitting the bulk PVT behavior. In contrast to our earlier treatment, the CO2 molecule here is modeled as a dimer of two identical spheres, with a bonding potential given by exp[−βV1(ri, rj)] =

and fA = fA pAA + (1 − fA )pBA

(3)

Thus, the p matrix has only one degree of freedom not specified, which we choose to be its nontrivial eigenvalue λ = pAA + pBB − 1. Hence, the sequence of monomers is characterized by two parameters fA and λ. The parameter λ assumes values between −1 and +1: values near −1 imply a strong preference for alternation between types A and B on successive monomers (alternating copolymers), while values near +1 correspond to situations in which a homopolymer mixture of pure A and pure B is obtained. The intermediate case of λ = 0 is referred to as ideal copolymerization, where there is no correlation between the chemical identity of successive monomers. In this work, we only consider the case of ideal copolymerization whose p matrix is then given by

δ(|ri − rj| − σ1) 4πσ12

(2)

(1)

where σ1 is the diameter of the sphere, and δ is the Dirac delta function. β−1 = kBT stands for the temperature multiplied by the Boltzmann constant. To describe the associative interaction between CO2 molecules due to the partial charges on the carbon and oxygen atoms, each CO2 molecule is assigned three binding regions, two on each side of the oxygen atoms, and one region around the axis connecting the two spheres. A schematic is shown in Figure 1. Compared with our previous model,5 the treatment of CO2 molecule as a dimer with explicit associating interaction sites is a more physically realistic representation and also gives better agreement with the thermodynamic properties of actual CO2 fluid.

⎛f f ⎞ A B ⎟ p = ⎜⎜ ⎟ ⎝ fA fB ⎠

(4)

The monomer species considered here are styrene (S), ethyl methacrylate (EMA), methyl methacrylate (MMA) and ethyl acrylate (EA). The chemical structure of the homopolymer formed by these monomers is shown in Figure 2. EA and EMA have more affinity to CO2 molecules than S and MMA. We consider three copolymers: P(MMA-co-EMA), P(MMA-coB


Article

Macromolecules

As CO2 is modeled as a dimer, N1 = 2. ρ2(i) is the density of the ith unit of the polymer chain, which can be either type A or type B. The sequence is determined by the bonding potential eq 5. The ideal part of the Helmholtz free energy can be written 24,25 as

Figure 2. Chemical structure of the monomer units studied in this work. Shown here are the homopolymers made of these units. (a) Polystyrene (PS), (b) poly(methyl methacrylate) (PMMA), (c) poly(ethyl methacrylate) (PEMA), and (d) poly(ethyl acrylate) (PEA).

∫ dr N ρ1̂ (r N )[ln ρ1̂ (r N ) − 1] + ∫ dr N ρ2̂ (r N )[ln ρ2̂ (r N ) − 1] + ∫ dr N ρ1̂ (r N )βV1(r N ) + ∫ dr N ρ2̂ (r N )βV2(r N ) (9)

βF id[ρ1̂ (r N1), ρ2̂ (r N2)] = 2

EA), and P(S-co-EA). The bonding potential of a chain is given by exp[−βV2(r N2)] =

N2 − 1

∏

4πσi , i + 12

i=1

with rN2 = (r1, ..., rN2) and σi,i+1 = (σ(i) + σ(i+1))/2. N2 is the chain length and σ(i) is the diameter of the ith monomer. The interaction between two species (i.e., polymer segment and CO2 segment) is described by

N1

N2

N1

ex

N2

F = F [ρ1̂ (r ), ρ2̂ (r )] + F [ρ1̂ (r ), ρ2̂ (r )]

N1

ex βFhs =

ρ2 (r) =

i=1

N2

N2

∑ ρ2(i)(r) = ∑ ∫ dr N

2

i=1

i=1

2

(10)

∫ drϕhs [nα(r)]

(11)

n1n2 − nV 1·nV 2 1 − n3 ⎡ ln(1 − n ) ⎤⎛ n 3 ⎞ 1 3 ⎥⎜ 2 − n2 nV 2 ·nV 2⎟ +⎢ + 3 2 12πn3(1 − n3) ⎦⎝ 3 ⎠ ⎣ 12πn3

ϕhs[nα(r)] = −n0 ln(1 − n3) +

1

δ(r − ri)ρ2̂ (r N2)

2

with

N1

i=1

2

1

(7)

∑ ρ1(i)(r) = ∑ ∫ dr N δ(r − ri)ρ1̂ (r N ) 1

2

The first four terms of above equation extend the corresponding bulk terms in EOS to spatially varying systems, whereas the last term is an additional contribution due to the long-range dispersion interaction that is only nonvanishing when the system is inhomogeneous. Fex hs accounts for the excluded volume effect and is given by the modified fundamental measure theory (FMT),26,27

ρ̂1 (r N1) and ρ̂2 (r N2) are the molecular density profiles of CO2 molecule and polymer chain. They are related to the segmental densities by

ρ1(r) =

1

ex ex ex ex ex F ex = Fhs + Fch + Fassoc + Fdisp − local + Fdisp − nonlocal

(6)

The index i and j may take 1 (for CO2), A, or B. ϵij and σij are the effective energy parameter and interaction diameter, respectively . The cross interaction is given by σij = (σi + σj)/2, ϵij = (ϵiϵj)1/2 (1 − k ij) (i ≠ j). In Table 1, we list the value of these parameters for the species used in our model. The method to obtain these values is described in section II.B. B. Density-Functional Theory. The Helmholtz free energy functional of the mixture is expressed as a sum of an ideal term and an excess contribution accounting for the interand intramolecular interactions id

1

1

The last two terms in eq 9 account for the chain connectivity of molecules. In implementing the DFT, these terms serve as constraints to enforrce the monomer units to be bonded in a specified manner. In PC-SAFT-based DFT, the excess Helmholtz free energy includes the contribution from excluded-volume effect, correlation due to chain connectivity and association interaction of CO2, and dispersion interaction:

(5)

⎧ ∞ , r < σij ⎪ uij(r ) = ⎨ 6 ⎪ ⎩−ϵij(σij/r ) , r ≥ σij

2

1

δ(|ri + 1 − ri| − σi , i + 1)

1

ni = ∑j = 1, A , B ni , j (i = 0, 1, 2, 3, V1 , V2) are the weight density

functionals of Rosenfeld.26 They are defined as

(8)

Table 1. Parameters for the Species Studied in This Work species CO2b PMMA PS PEA PEMA

N/M [mol/g]

ϵ/kB [K]

σ [Å]

2/44 0.0318 0.0230 0.0352 0.0341

171.16 223.79 289.44 264.90 303.70

2.795 2.890 3.300 3.410 3.380

kij (×104)a − 1.25T 0.29T 3.20T 1.30T

− − − −

422.689 104.444 1210.080 491.595

temperature range (K) 260−450 280−380 280−470 280−390 280−390

a

kij is used to modify the cross interaction between the polymer and CO2 in the mixture. bThe energy parameter and the accessible bonding volume parameter for the association interaction of CO2 are ϵa/kB = 0.1150 K and va = 0.2141 Å3, respectively. These two parameters are used in eqs 16 and 17 to determine the free energy of CO2 association. C


Article

Macromolecules ⎛σ

⎞ − |r − r′|⎟ , ⎠

⎛σ

⎞ − |r − r′|⎟ , ⎠

n2, j(r) =

∫ dr′ ρj (r′)δ⎝ 2j

n3, j(r) =

∫ dr′ ρj (r′)Θ⎝ 2j ⎛σ ⎜

n2, j(r) πσj

⎜

∫ dr′ ρj (r′)Θ⎝ 2j

nV 2, j(r) = n0, j(r) =

⎜

, n1, j(r) =

2

where Xi(r) stands for the mole fraction of molecules at r not associated at site i. ϵa is the association energy between the oxygen site and the carbon site of the CO2 molecules, and va is a parameter characterizing the accessible volume for binding. The value of these two parameters can be found in Table 1. The dispersion term is decomposed as the sum of a local ex contribution Fex disp−local and a nonlocal contribution Fdisp−nonlocal. The local contribution from equation of state is given by

⎞ r − r′ − |r − r′|⎟ , ⎠ |r − r′|

n2, j(r) 2πσj

, nV 1, j(r) =

nV 2, j(r) 2πσj

−1 [2J1(r)β ϵij + NM (r)J2 (r)(β ϵij)2 ]σij 3 ̅

In these equations, δ is the Dirac delta function and Θ is the Heaviside step function. ρA and ρB are the density of segment A and B, respectively, defined as

∑

M(r) = 1 + N̅

(13)

i ∈ {B}

6

Jk (r) =

where {A} and {B} denotes the index set of all A segments and all B segments, respectively. The modified FMT gives an accurate description for the excluded volume effect in hard sphere fluids, including hardsphere chains.28 The use of FMT is also consistent with the treatment of chain-connectivity and dispersion-interaction contributions in the framework of PC-SAFT EOS. β Fexch due to correlation of chain connectivity of polymer and CO2 is given by ex βFch =

1 − N1 N1

n02(r)

∑

∫ dr n01(r) ln g11(r) +

1 − N2 N2

pij ln gij(r)

ai(k) = ai(0k) +

(14)

∑ Δ(r) ∫ j

∑ ∫ ∫ dr dr′ i , j = 1, A , B

(21)

where Θ is the Heaviside function. The grand potential W of the system is related to the Helmholtz free energy functional by W = F − μ1

(16)

with Xi(r) ⎡ = ⎢1 + ⎢⎣

1 4

[ρj (r) − ρj (r′)]

(15)

3

∫

(20)

Θ(|r − r′| − σij)uij(|r − r′|)[ρi (r) − ρi (r′)]

The Helmholtz free energy Fex assoc due to association of CO2 is given by ⎡ 1 1⎤ dr n01(r) ∑ ⎢ln Xi(r) − Xi(r) + ⎥ ⎣ 2 2⎦ i=1

N̅ − 1 (k) N̅ − 1 N̅ − 2 (k) ai1 + ai2 N̅ N̅ N̅

ex Fdisp − nonlocal[ρ1(r), ρA (r), ρB (r)] =

2

ex βFassoc =

(19)

where N̅ = N1x + N2(1 − x) and x is the molar fraction of CO2. The constant coefficients {a ij(k)|k = 1, 2; i = 0, 1, ..., 6; j = 0, 1, 2} are obtained by fitting the calculated binodal of the EOS with experimental data for a great number of species.29 The value of these coefficients can be found in ref.29 The weight density approximation in eq 18 alone does not sufficiently describe the long-range intermolecular attraction.30 The additional long-range dispersion contributions due to spatial inhomogeneity are included in a mean-field manner by31

n2(r) 1 1 σσ i j + 1 − n3(r) 2 σi + σj (1 − n3(r))2

(n2(r))2 1 (σσ i j) 2 18 (σi + σj) (1 − n3(r))3

k = 1, 2.

and the coefficients

where n02 = n0A + n0B. gij (i,j = 1, A, B) is the contact value of the correlation function between segments of species i and j, given by

+

∑ ai(k)(N̅ )n3i , i=0

∫ dr

i , j = A,B

gij(r) =

8n3 − 2n32

+ (1 − N̅ ) (1 − n3)2 20n3 − 27n32 + 12n33 − 2n3 4 , (1 − n3)2 (2 − n3)2

ρ2(i)(r) ρ2(i)(r).

∑

(18)

with

i ∈ {A}

ρB (r) =

∫ dr n0i(r)n0j(r)

i , j = 1, A , B

(12)

ρA (r) =

∑

ex βFdisp − local = − π

∫ dr N ρ1̂ (r N ) − μ2 ∫ dr N ρ2̂ (r N ) 1

1

2

2

(22)

where μ1 and μ2 are the chemical potential of CO2 and the polymer chain, respectively. At equilibrium, the grand potential is minimized

⎤−1 δ(|r − r′| − σ1) ⎥ dr′ Xj(r′)ρ1(r′) ⎥⎦ 4πσ12

δW = 0, δρ1̂ (r N1)

Δ(r) = g11(r)[exp(β ϵa) − 1]va

δW =0 δρ2̂ (r N2)

(23)

These variational conditions yield the following Euler− Lagrange equations:

(17) D


Article

Macromolecules ⎡ δF ex ⎤⎥ ρ1̂ (r N1) = exp⎢βμ1 − βV1(r N1) − β ⎢⎣ δρ1̂ (r N1) ⎥⎦ ⎡ δF ex ⎤⎥ ρ2̂ (r N2) = exp⎢βμ2 − βV2(r N2) − β ⎢⎣ δρ2̂ (r N2) ⎥⎦

III. NUMERICAL PROCEDURE The data shown in Table 1 are obtained by the following procedure. First, the parameters for the pure CO2 are obtained by fitting the PVT and binodal data of EOS with the experimental data.32,33 Next, we obtain the parameter values for PS and PMMA by fitting the CO2 solubility in the homopolymer (PS or PMMA) using the corresponding experimental data.35,36 Finally, the parameters of PEA and PEMA are obtained by fitting the CO2 solubility in the copolymers with the experimental data. When the parameters for all the homopolymers are obtained, the phase behavior of random copolymer with any specified composition ( fA) can be predicted by the EOS. Experimental data for PS and PMMA can be found in the literature,32,33,35,36 while experimental data for copolymers involving EA and EMA are provided by the Dow Chemical Company.37 Comparison between predictions using our EOS and the experimental data are shown in Figure 3, 4 and 5. The results indicate that our EOS accurately

(24)

Using eqs 8, the functional derivative in above equations can be simplified as δF ex = δρ1̂ (r N1)

N1

∑

ex

δF = δρ2̂ (r N2)

i=1 N2

∑ i=1

δF ex δρ1(ri) δF ex δρ2 (ri)

(25)

After substituting into eqs 24 and using eqs 8 again, we have N1

ρ1(r) = exp(βμ1)

∫ dr N ∑ δ(r − ri) 1

i=1

N1 ⎡ ⎤ δF ex ⎥ exp⎢ −βV1(r N1) − β ∑ ⎢⎣ ⎥ j = 1 δρ1(rj) ⎦ N2

ρ2 (r) = exp(βμ2 )

∫ dr ∑ δ(r − ri) N2

i=1

N2 ⎡ ⎤ δF ex ⎥ N2 ⎢ exp −βV2(r ) − β ∑ ⎢⎣ ⎥ j = 1 δρ2 (rj) ⎦

(26)

The above equations can be further simplified by introducing a recursive function,34 ρ1(r) =

⎡ δβF ex ⎤ 1 ⎢ ⎥ exp − βμ ⎢⎣ 1 δρ1(r) ⎥⎦ 2πσ12

∫|r−r |=σ dr′ ′

Figure 3. Comparison between the theoretical predictions from this work (solid lines) and the experimental data (points)32,33 for pure CO2. The blue line is the liquid−vapor coexistence curve, and the other solid lines are the isotherms.

1

⎡ δβF ex ⎤ ⎥ exp⎢ − ⎢⎣ δρ1(r′) ⎥⎦

(27)

N ⎡ δβF ex ⎤ 2 ⎥ ∑ Ii(r)I N + 1 − i(r) ρ2 (r) = exp⎢βμ2 − 2 ⎢⎣ δρ2 (r) ⎥⎦ i = 1

captures the phase behavior of pure CO2 (Figure 3), and the solubility of CO2 in the homopolymers (Figure 4) and copolymers (Figure 5). We now consider the interfacial properties. We assume a planar interface with the densities varying in the z-direction. For long polymer chains, direct numerical solution of eqs 27, 28 and 29 becomes challenging. In treating a liquid−vapor interface, a simpler alternative to functionally minimizing the free energy is to use parametrized density profiles for polymer segments. Following our previous experience in homopolymer systems,5 a good choice for the density profile of polymers across a liquid−vapor interface is to assume a hyperbolic tangent function,

(28)

where the recursive function is given by ⎧1, i = 1 ⎪ ⎪ Ii(r) = ⎨ 1 ⎪ ⎪ 4π(σ (i))2 ⎩

∫|r−r |=σ ′

(i)

⎡ δβF ex ⎤ ⎥, i > 1 dr′Ii − 1(r′) exp⎢− ⎢⎣ δρ2 (r′) ⎥⎦

(29)

In our construction, the chain sequence of the random copolymer enters the theory through the recursive relation, eq 29, and through the excess free energy eq 14 (the second term) at the pair-level described by a Markov-step growth. For long chains, direct calculation of eq 29 becomes challenging. To avoid this challenging calculation, we make the simplifying assumption that the liquid−vapor density profile of long polymer chain can be described by a hyperbolic tangent function with a width to be determined variationally; see section III for details. Therefore, for the results obtained in this work, the chain sequence enters the theory only through eq 14.

1 v (ρ − ρAl )[1 + tanh(TA(z − z 0))] 2 A 1 ρB (z) = ρBl + (ρBv − ρBl )[1 + tanh(TB(z − z 0))] 2 ρA (z) = ρAl +

(30)

ρli

ρvi

where and are the bulk densities at coexistence for component i (i = A, B) in the liquid and vapor phase, E


Article

Macromolecules

Figure 4. Solubility of CO2 in PS (panel a) and PMMA (panel b). The solid lines are the predictions from this work, and the solid circles are the experimental data.35,36

Figure 5. Solubility of CO2 in P(S-co-EA) (panel a), P(MMA-co-EA) (panel b), and P(S-co-EMA) (panel c) at T = 378.15 K. The percentages shown in the legends are the weight fraction of EA or EMA component in the copolymer. The solid circles are the experimental data provided by the Dow Chemical Company.37

respectively, z0 is the location of the interface, and TA and TB are parameters that determine the interface widths. With these parametrized density profiles, minimization of the free energy functional is now simplified to one with respect to the density profile of CO2 (i.e., ρ1(z)) and the parameters TA and TB,

min W [ρ1(z), TA , TB]

at given temperature, pressure and composition fA. The coexisting densities are determined with a convergence criterion of

∑ i = 1,2

(31)

i = 1, 2

(32)

and ρAj = ρ2j fA , ρBj = ρ2j (1 − fA ),

j = l, v

(34)

Although the coexisting densities are determined to a high accuracy, the difference in the calculated grand potential in DFT between the liquid and vapor sides at coexistence may still be significant (on the order of O(10−3)). This can result in significant numerical inaccuracy in the values of the interfacial tension as a consequence of spatial integration across the interface from the vapor to the liquid side. The primary source of the numerical error occurs in computing the weight density integrals in DFT. In this work, we employ a new numerical method to compute these integrals, which greatly increases their numerical accuracy. The method is useful in DFTs involving a weight density functional. Details of the numerical method are given in the Appendix.

The above equation can be solved by iteration with guessed initial values of TA and TB and the CO2 profile. The interfacial tension of a planar interface is calculated from the excess grand potential per unit area, i.e., γ = (W−Wbulk) /A, where Wbulk is the grand potential in bulk phase, and A is the surface area. Calculation of the interfacial tension requires accurate determination of the coexisting densities {ρli,ρvi |i = 1, A, B}. These are determined by searching the chemical potentials such that μi (ρ1l , ρ2l ) = μi (ρ1v , ρ2v ),

|μi (ρ1l , ρ2l ) − μi (ρ1v , ρ2v )| < 10−10

(33) F


Article

Macromolecules

Figure 6. Interfacial tension between the polymer-rich phase and the CO2-rich phase as a function of pressure. The solid lines are the DFT predictions, and the solid circles are the experimental data.38,39

Figure 7. Interfacial tension between the polymer-rich phase and the CO2-rich phase as a function of the CO2 content at T = 373.15 K. The percentages shown in the legends are the weight fraction of EA or EMA component in the copolymer. The pressure range for each curve can be read off from the data given in Figure 5.

IV. RESULTS AND DISCUSSION We first test the performance of our newly constructed DFT by calculating the interfacial tension at coexistence for homopolymer and CO2 mixtures. In Figure 6, we compare the predicted values with experimental data for both PS−CO2 and PMMA− CO2 mixtures. Good agreement is clearly seen for both cases. Experimental data for mixtures of copolymers (the species studied here) and CO2 are not available in literature. Therefore, we can only present the theoretical results. Of particular interest is the effect of copolymer composition (weight fraction wEA or wEMA) on the interfacial tension. Since copolymer composition affects the CO2 solubility, a useful comparison is the tension as a function of the CO2 content for different copolymer compositions; this is shown in Figure 7. The general trend is that the interfacial tension increases with increasing wEA or wEMA. An interesting exception is the P(S-co-EA) system at

low CO2 content: the tension decreases with increasing wEA. Another useful comparison is the tension as a function of the copolymer composition at a given CO2 content. Figure 8 shows the interfacial tension as a function of wEA or wEMA in the copolymer chain at a CO2 weight percent of 15%. For the P(MMA-co-EA) and P(MMA-co-EMA) copolymers, the tension is a monotonically increasing function of wEA or wEMA. For the P(S-co-EA) copolymer, the tension is the highest for pure PEA, but a minimum occurs at wEA = 9.52%. These predictions await experimental validation. The solubility of CO2 in the copolymer and the interfacial tension between the copolymer-CO2 mixture and CO2 vapor have direct consequence on the bubble nucleation behavior. We discuss the implications of our results on bubble nucleation using P(MMA-co-EA) as an example. From Figure 5b, we see that at a fixed pressure, CO2 solubility increases with increasing G


Article

Macromolecules

Figure 8. Interfacial tension between the polymer-rich phase and the CO2-rich phase as a function of the weight fraction of EA or EMA at T = 373.15 K. The CO2 content in the polymer-rich phase for all cases is fixed at 15% by weight. The pressure drops as wEA increases from 0 to 1; the ranges of the pressure variation are (a) 27.29 to 8.80 MPa, (b) 24.02 to 8.80 MPa, and (c) 24.02 to 15.68 MPa. .

Figure 9. Density profiles across the interface between the P(MMA-co-EA)-rich phase and the CO2-rich phase at T = 373.15 K. The bulk pressure in panels a−c are 24.02, 12.87, and 8.80 MPa, respectively, corresponding to a fixed CO2 content in polymer-rich phase at 15% by weight.

wEA; PMMA has the lowest solubility. Thus, with increasing wEA, lower pressure is needed to achieve the same solubility of CO2 than in pure PMMA. For instance, to achieve a CO2 solubility of 15%, the bulk pressure for PMMA is 24.02 MPa, while it is 15.18 MPa when the copolymer contains 40% EA. Figure 9 shows the density profiles of the different segments across the interface between the copolymer-rich and CO2-rich phase at CO2 content of 15%. It can be seen that the CO2 density in the vapor side decreases with increasing wEA, consistent with the decrease of the pressure with increasing wEA to maintain the same CO2 solubility. The low CO2 in the vapor side will result in a higher interfacial tension. Therefore, for fixed CO2 content, the interfacial tension increases with increasing the weight fraction of EA in the copolymer chain. That a copolymer with high wEA has a higher CO2 solubility at a given pressure means that it is easier to initiate bubble nucleation in the such systems. Moreover, high CO2 content usually results in higher porosity for the foamed materials. However, at the same CO2 content in the polymer-rich phase, copolymers with high wEA has a higher interfacial tension at coexistence, which is unfavorable for nucleation. In previous work, we studied the bubble nucleation in homopolymer−CO2 mixtures by combining DFT with the string method.6,7 In

principle, similar calculations can be performed on the copolymer-CO2 studied here. However, the focus of the current work is the construction of the DFT and its validation on equilibrium bulk and interfacial properties. Here we will briefly discuss the qualitative implications of these equilibrium properties on bubble nucleation using the classical nucleation theory (CNT). In CNT, the nucleation barrier is given by W = 4πR2γ − (4π /3)R3ΔP

(35)

where ΔP = Po−Pbulk is the pressure difference between the center of a well-developed bubble and the metastable bulk phase, γ is the interfacial tension between the nucleus and the parent bulk phase, often approximated by the planar interfacial tension at coexistence. The competition between the interfacial and volume terms in eq 35 gives rise to a critical size R* = 2γ/ |ΔP|and a free energy barrier W* = (16πγ3) /(3|ΔP|2) for the nucleation. Figure 10 shows the free energy barrier of CO2 bubble nucleation in P(MMA-co-EA) at a fixed CO2 content at T = 373.15 K. Nucleation is initiated by a pressure drop from a high initial pressure at the coexistence phase (corresponding the given CO2 content in the mixture) to the ambient pressure 0.1 MPa. As is shown in Figure 10, PMMA (wEA = 0) has a nucleation barrier of 5.08kBT, while the barrier for PEA (wEA = H


Article

Macromolecules

Figure 10. Free energy barrier of bubble nucleation in P(MMA-co-EA) and CO2 mixture at T = 373.15 K calculated using the classical nucleation theory. The content of CO2 in the metastable bulk phase is 15%. The corresponding bulk pressure changes from 8.80 MPa (wEA = 1) to 24.02 MPa (wEA = 0) . Nucleation is initiated by a pressure drop from a high initial pressure that results in the 15% solubility at the coexistence to the ambient pressure 0.1 MPa. The inset shows the pressure difference between the center of a well-developed bubble and the bulk phase.

Figure 11. Free energy barrier for bubble nucleation in P(MMA-co-EA) and CO2 mixture at T = 373.15 K calculated using the classical nucleation theory. Nucleation is initiated by a pressure drop from an initial pressure of 16 MPa at the coexistence to the ambient pressure 0.1 MPa. Inset a shows the profile of pressure difference between bubble inside and the metastable bulk, and inset b shows the weight fraction of CO2 in the metastable bulk phase.

foaming condition at T = 373.15 K. This temperature is typical of those used for microcellular foam production. The result is quite different if instead of the CO2 content, the initial pressure is fixed. In that case, since the CO2 solubility increases with wEA at a given pressure, and the interfacial tension drops with increasing wEA, the nucleation barrier decreases with wEA. For sufficiently high initial pressure, the nucleation barrier can become reasonable for all copolymer compositions. Figure 11 shows the free energy barrier for an initial pressure of 16 MPa at T = 373 K. In this case, the free energy barrier changes moderately from 24kBT to 16kBT, as wEA

1) is 218.65kBT. The rather sharp increase in the nucleation barrier with wEA can be understood as arising from two factors. First, as mentioned earlier, the interfacial tension increases with increasing wEA. Second, as shown in the inset in Figure 10, the driving force for nucleation−the pressure difference between bubble nucleus and the metastable bulk phase−decreases with increasing wEA. Both factors work to increase the free energy barrier since W* = (16πγ3)/(3|ΔP|2)). For a specified CO2 content in the polymer-rich phase, by simultaneous considering the effect of solubility and interfacial tension, P(MMA-co-EA) with a small fraction of EA (wEA) should give the optimal I


Article

Macromolecules

Figure 12. (a) Free energy barrier for bubble nucleation in P(MMA-co-EA) and CO2 mixture at T = 308 K calculated using the classical nucleation theory. Nucleation is initiated by a pressure drop from an initial pressure of 30 MPa at the coexistence to the ambient pressure 0.1 MPa. (b) Weight fraction of CO2 in the metastable bulk phase. (c) Pressure difference between bubble inside and the metastable bulk. (d) Interfacial tension between the polymer-rich phase and the CO2-rich phase at the coexisting point.

goes from 0 to 1. In contrast to the case of fixed CO2 content discussed in the previous paragraph where both the driving force and the interfacial tension raises the nucleation barrier as wEA is increased, here the decrease of the free energy barrier with wEA is due primarily to decrease in the interfacial tension; the pressure difference between the nucleating bubble inside and bulk phase remains essentially unchanged (it decreases only slightly with increasing wEA); see inset a. The above discussion is valid for temperature conditions close to those used for microcellular foams. Nanocellular foams are usually produced at lower temperatures and higher pressures, for instance close to the critical point of CO2 in PMMA-CO2 or PS-CO2 mixtures.7 Under such conditions, we do not expect the assumptions in the CNT to hold.43 Here we simply use CNT to obtain the general trend. Figure 12 shows the free energy barrier for an initial pressure of 30 MPa at T = 308 K. In comparison with Figure 11, the nucleation barrier is much reduced at all compositions. Indeed with nucleation barriers comparable to or less than kT, nucleation is no longer kinetically limited. However, our earlier work has shown that CNT greatly underestimates the barrier for the bubble nucleation,6 so the actual barrier can be significantly higher (by an order of magnitude or more) than shown in Figure 11. Nevertheless, we expect the general trend that (1) the barrier decreases with increasing wEA and (2) the barrier is greatly reduced upon increasing the initial pressure and decreasing the temperature, will hold. In particular, we note that the nucleation barrier drops sharply with the EA content in the range of wEA < 0.2. It is quite possible that beyond wEA = 0.2 nucleation barrier is sufficiently low that it is no longer an issue. In that case, other factors must be considered. Specifically, nanofoaming relies on the onset of glass transition to stabilize the cell structure.2−4 As wEA increases, the glass transition temperature of copolymer decreases sharply with EA level; see Figure 13. The optimal wEA between 5 to 15% may reflect a compromise between the requirements for nucleation and for stabilization.44 Similar conclusions can be drawn for other copolymers such as P(MMA-co-EMA).

Figure 13. Glass transition temperature of P(MMA-co-EA) as a function of wEA. The red solid circles are the experimental data, and the blue solid line is the result of a model prediction of the glass transition temperature for copolymers using a modified version of the Barton and Johnston theory.40−42

V. CONCLUSIONS In this work, we have developed a density-functional theory (DFT) to describe the structure and thermodynamics of inhomogeneous random AB copolymer-carbon dioxide (CO2) mixtures. The statistics of the monomer sequence in the copolymer chain is modeled as a Markov-step growth process using a matrix of transition probabilities. The parameter values for three copolymer systems P(MMA-co-EMA), P(MMA-coEA), and P(S-co-EA) and CO2 mixtures, have been determined by fitting a limited set of bulk experimental data. As an improvement over our previous work on polymer−CO2 mixtures, we have modified our model for the CO2 by explicitly accounting for the associating sites on the molecule. Furthermore, we have proposed a new numerical method for accurately calculating the weighted density integrals, which is J


Article

Macromolecules

are calculated similarly by the way of eq 37, the difference in the grand potential between the coexisting liquid and vapor can be reduced to O(10−12), giving an accurate value for the interfacial tension. It should be mentioned that eq 37 is useful in all versions of DFT that uses a weight density functional. It is particularly useful when treating a wetting or drying problem at a substrate, where the grand potential requires to be determined with a high accuracy in the numerical calculation.

essential for the calculation of interfacial tension in any DFTs using weighted density functionals. Our newly developed DFT allows accurate and systematic predictions of the phase behavior and interfacial properties in random copolymer−CO2 mixtures. Of particular interest are the CO2 solubility in the polymers and the interfacial tension between the CO2 gas and the CO2 saturated polymers, as functions of the copolymer composition. For example, copolymer with high weight fraction of EA or EMA (wEA or wEMA) has higher solubility for CO2; however, at the same CO2 content in polymer-rich phase increasing wEA or wEMA also increases the interfacial tension between polymer-rich and CO2-rich phases. This type of information is useful for the various applications of polymer-CO2 mixtures, e.g., in the manufacture of polymer foams using CO2, in particular for the production of novel nanocellular materials expected to possess unique mechanical and thermal characteristics.45,46

■

Corresponding Author

*(Z.-G.W.) E-mail: [email protected]. Notes

The authors declare no competing financial interests.

■

■

ACKNOWLEDGMENTS The Dow Chemical Company is acknowledged for funding and for permission to publish the results. X.X. expresses thanks for the support of the National Natural Science Foundation of China (No. 21404078).

APPENDIX: NUMERICAL METHOD TO COMPUTE THE WEIGHTED DENSITY INTEGRALS IN DFT The accumulation of numerical errors in computing the integrals of DFT can result in large inaccuracies in the value for the interfacial tension. Let us illustrate this by an example of calculating the Rosenfeld functionals. In one dimension, the first two lines of eqs 12 can be written as n2, j(z) = πσj

z + σj /2

∫z−σ /2

■

z + σj /2

∫z−σ /2

ρj (z′) dz′

[σj2/4 − (z − z′)2 ]ρj (z′) dz′

j

(36)

In most of this work, we used a grid of the size Δz = 0.025 nm for the computation. However, the radius of CO2 monomer is σ1/2 = 0.13975 nm, which is not an integer number of grid size (0.13975/0.025 = 5.59). If we choose the nearest integer number 6 to approximate the ratio of the monomer radius to the grid size (5.59), σ1/2 takes a value of 6 × 0.025 = 0.15 nm on the grid. After thousands of iterations, the accumulated error of using the approximate value 0.15 nm for σ1/2 instead of its real value 0.13975 nm, can become substantial in the result for the interfacial tension. To summarize, the numerical inaccuracy arises in the difference σj/(2Δz) − R(σj/(2Δz)), where the function R(x) denotes the nearest integer of x. Because of this difference, n2,j and n3,j do not reduce exactly to their bulk value in a numerical integral method (such as Simpson method). To solve this problem, we write the integral as πσj 2

n2, j(z) ≈

n3, j(z) ≈

πσj3 3

(σj)

σj

z + σj /2

∫z−σ /2

z + σj /2

∫z−σ /2

REFERENCES

(1) Binder, K.; Müller, M.; Virnau, P.; MacDowell, L. G. Polymer plus solvent systems: Phase diagrams, interface free energies, and nucleation. Adv. Polym. Sci. 2005, 173, 1−104. (2) Lee, S. T.; Ramesh, N. S. Polymeric foams: Mechanisms and materials; CRC Press, Taylor & Francis Group: Boca Raton, FL, 2004; pp 1−318. (3) Lee, S. T.; Park, C. B.; Ramesh, N. S. Polymeric foams: Science and technology; CRC Press, Taylor & Francis Group: Boca Raton, FL, 2007; pp 1−218. (4) Costeux, S. CO2-blown nanocellular foams. J. Appl. Polym. Sci. 2014, 131, 41293. (5) Xu, X.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. Densityfunctional theory for polymer-carbon dioxide mixtures: A perturbedchain SAFT approach. J. Chem. Phys. 2012, 137, 054902. (6) Xu, X.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. Bubble nucleation in polymer-CO2 mixtures. Soft Matter 2013, 9, 9675−9683. (7) Xu, X.; Cristancho, D. E.; Costeux, S.; Wang, Z.-G. Discontinuous bubble nucleation due to metastable phase transition in polymer-CO2 mixtures. J. Phys. Chem. Lett. 2013, 4, 1639−1643. (8) Costeux, S.; Zhu, L. Low density thermoplastic nanofoams nucleated by nanoparticles. Polymer 2013, 54, 2785−2795. (9) Yoshida, E. Nanospheres prepared by self-assembly of random copolymer in supercritical carbon dioxide. Int. J. Polym. Sci. 2012, 54, 592759. (10) Mansky, P.; Liu, Y.; Huang, E.; Russell, T. P.; Hawker, C. Controlling polymer-surface interaction with random copolymer brushes. Science 1997, 275, 1458−1460. (11) ten Brinke, G.; Karasz, F. E.; MacKnight, W. J. Phase behavior in copolymer blends: Poly(2,6-dimethyl-1, 4-phenylene oxide) and halogen-substituted styrene copolymers. Macromolecules 1983, 16, 1827−1832. (12) Kambour, R. P.; Bendler, J. T.; Bopp, R. C. Phase behavior of polystyrene, poly(2,6-dimethyl-1,4-phenylene oxide), and their brominated derivatives. Macromolecules 1983, 16, 753−757. (13) Balazs, A. C.; Sanchez, I. C.; Epstein, I. R.; Karasz, F. E.; MacKnight, W. J. Effects of sequence distribution on the miscibility of polymer-copolymer blends. Macromolecules 1985, 18, 2188−2191. (14) Cantow, H. J.; Schulz, O. Effect of chemical and configurational sequence distribution on the miscibility of polymer blends. Polym. Bull. 1986, 15, 449−453. (15) Dudowicz, J.; Freed, K. F. Molecular influences on miscibility patterns in random copolymer/homopolymer binary blends. Macromolecules 1998, 31, 5094−5104.

j

n3, j(z) = π

AUTHOR INFORMATION

ρj (z′) dz′

j

[(σj)2 /4 − (z − z′)2 ]ρj (z′) dz′

j

(37)

with ⎛ σj ⎞ ⎟ σj ≡ 2ΔzR ⎜ ⎝ 2Δz ⎠

The form of the above equations ensures that n2,j and n3,j reduce exactly to their corresponding value in the bulk phase for a homogeneous system. When all the integrals in the DFT K


Article

Macromolecules (16) Dudowicz, J.; Freed, K. F. 1996. Molecular origin of the free energy dependence on the monomer sequence in random copolymer systems. Macromolecules 1996, 29, 7826−7837. (17) Freed, K. F.; Dudowicz, J. A lattice-model molecular theory for the properties of polymer blends. Trends Polym. Sci. 1995, 3, 248−255. (18) Fredrickson, G. H.; Milner, S. T. Thermodynamics of random copolymer melts. Phys. Rev. Lett. 1991, 67, 835−838. (19) Fredrickson, G. H.; Leibler, L. Nematic to isotropic transition in chemically disordered or multicomponent liquid-crystalline polymers. Macromolecules 1990, 23, 531−539. (20) Fredrickson, G. H.; Milner, S. T.; Leibler, L. Multicritical phenomena and microphase ordering in random block copolymer melts. Macromolecules 1992, 25, 6341−6354. (21) DeFelice, J.; Lipson, J. E. G. Polymer miscibility in supercritical carbon dioxide: Free volume as a driving force. Macromolecules 2014, 47, 5643−5654. (22) Müller, M.; MacDowell, L. G.; Virnau, P.; Binder, K. Interfacial properties and bubble nucleation in compressible mixtures containing polymers. J. Chem. Phys. 2002, 117, 5480−5496. (23) Gross, J.; Spuhl, O.; Tumakaka, F.; Sadowski, G. Modeling copolymer systems using the perturbed-chain SAFT equation of state. Ind. Eng. Chem. Res. 2003, 42, 1266−1274. (24) Chandler, D.; Pratt, L. R. Statistical mechanics of chemical equilibria and intramolecular structures of nonrigid molecules in condensed phases. J. Chem. Phys. 1976, 65, 2925−2940. (25) Pratt, L. R.; Chandler, D. Interaction site cluster series for the Helmholtz free energy and variational principle for chemical equilibria and intramolecular structure. J. Chem. Phys. 1977, 66, 147−151. (26) Rosenfeld, Y. Free-energy model for the inhomogeneous hardspheres fluid mixture and density-functional theory of freezing. Phys. Rev. Lett. 1989, 63, 980−983. (27) Yu, Y. X.; Wu, J. Z. Structures of hard-sphere fluids from a modified fundamental-measure theory. J. Chem. Phys. 2002, 117, 10156−10164. (28) Wu, J. Z. Density functional theory for liquid structure and thermodynamics. Struct. Bonding (Berlin, Ger.) 2009, 131, 1−73. (29) Gross, J.; Sadowski, G. Perturbed-chain SAFT: An equation state based on a perturbation theory for chain molecules. Ind. Eng. Chem. Res. 2001, 40, 1244−1260. (30) Tarazona, P.; Cuesta, J.; Martinez-Raton, Y. Density functional theory of hard particle systems. Lect. Notes Phys. 2008, 753, 247−341. (31) Ebner, C.; Saam, W. F.; Stroud, D. Density-functional theory of simple classical fluids. 1. surfaces. Phys. Rev. A: At., Mol., Opt. Phys. 1976, 14, 2264−2273. (32) Mantilla, I. D.; Cristancho, D. E.; Ejaz, S.; Hall, K. R. P−ρ−T data for carbon dioxide from (310 to 450) K up to 160 MPa. J. Chem. Eng. Data 2010, 55, 4611−4613. (33) Shin, H. Y.; Wu, J. Equation of state for the phase behavior of carbon-dioxide-polymer systems. Ind. Eng. Chem. Res. 2010, 49, 7678− 7684. (34) Xu, X.; Cao, D.; Zhang, X.; Wang, W. Universal version of density-functional theory for polymers with complex architecture. Phys. Rev. E 2009, 79, 021805. (35) Rajendran, A.; Bonavoglia, B.; Forrer, N.; Storti, G.; Mazzotti, M.; Morbidelli, M. Simultaneous measurement of swelling and sorption in a supercritical CO2-poly(methyl methacrylate) system. Ind. Eng. Chem. Res. 2005, 44, 2549−2560. (36) Sato, Y.; Yurugi, M.; Fujiwara, K.; Takishima, S.; Masuoka, H. Solubilities of carbon dioxide and nitrogen in polystyrene under high temperature and pressure. Fluid Phase Equilib. 1996, 125, 129−138. (37) Experimental data were obtained by Magnetic Suspension Balance measurements [The Dow Chemical Company]. (38) Otake, K.; Kobayashi, M.; Ozaki, Y.; Yoda, S.; Takebayashi, Y.; Sugeta, T.; Nakazawa, N.; Sakai, H.; Abe, M. Surface activity of myristic acid in the poly(methyl methacrylate)/supercritical carbon dioxide system. Langmuir 2004, 20, 6182−6186. (39) Jaeger, P.; Eggers, R.; Baumgartl, H. Interfacial properties of high viscous liquids in a supercritical carbon dioxide atmosphere. J. Supercrit. Fluids 2002, 24, 203−217.

(40) Nguyen, G.; Nicole, D.; Matlengiewicz, M.; Roizard, D.; Henzel, N. Relation between microstructure and glass transition temperature of poly[(methyl methacrylate)-co-(ethyl acrylate)]. Polym. Int. 2001, 50, 784−791. (41) Barton, J. Relation of glass transition temperature to molecular structure of addition copolymers. J. Polym. Sci., Part C: Polym. Symp. 1970, 30, 573−597. (42) Johnston, N. W. Sequence distribution: Glass transition effects. J. Macromol. Sci., Polym. Rev. 1976, C14, 215−250. (43) Oxtoby, D. W.; Evans, R. Nonclassical nucleation theory for the gas-liquid transition. J. Chem. Phys. 1988, 89, 7521−7530. (44) Costeux, S.; Khan, I.; Bunker, S.; Jeon, H. Experimental study and modeling of nanofoams formation from single phase acryllic copolymers. J. Cell. Plast. 2015, 51, 197−221. (45) Notario, B.; Pinto, J.; Rodríguez-Pérez, M. A. Towards a new generation of polymeric foams: PMMA nanocellular foams with enhanced physical properties. Polymer 2015, 63, 116−126. (46) Notario, B.; Pinto, J.; Solorzano, E.; de Saja, J. A.; Dumon, M.; Rodríguez-Pérez, M. A. Experimental validation of the Knudsen effect in nanocellular polymeric foams. Polymer 2015, 56, 57−67.

L


Density-Functional Theory for Mixtures of AB Random Copolymer and

Recommend Documents