Thermodynamic Model of Solvent Effects in Semiflexible Diblock and

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Thermodynamic Model of Solvent Effects in Semiflexible Diblock and Random Copolymer Assembly Shifan Mao,† Quinn MacPherson,‡ Chunzi Liu,† and Andrew J. Spakowitz*,†,§,∥,⊥,# †

Department of Chemical Engineering, ‡Department of Physics, §Department of Materials Science and Engineering, ∥Department of Applied Physics, and ⊥Biophysics Program, Stanford University, Stanford, California 94305, United States # Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, United States S Supporting Information *

ABSTRACT: We present a field-theoretic model to predict the equilibrium thermodynamic behavior of semiflexible diblock copolymers and random copolymers in the presence of solvent. We find that in both systems polymer−solvent contacts dramatically influence the thermodynamic behavior with decreasing the copolymer segment length (i.e., molecular weight). When a copolymer has unequal monomer composition, both polymer length and solvent concentration have a strong influence on the phase transition spinodal and magnitude of the critical wave modes. Diblock copolymers exhibit an expanded region of the lamellar phase in the phase diagram with decreasing chain length and polymer concentration. Such effects suggest a breakdown of the dilute approximation for solutions of short diblock copolymers. Random copolymer solutions also exhibit changes in the phase-transition spinodal and critical wave mode at asymmetric chemical compositions. This effect is highly relevant to most random copolymer materials, since a monomer is typically a low-molecularweight chemical unit.

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INTRODUCTION Copolymers consisting of chemically different monomers have received much attention in the past decades. Most works focus on copolymer melts that separate into different microstructures. Observation of order−disorder phase transitions in a copolymer melt at experimentally accessible temperatures is complicated by strong chemical incompatibility between different monomers, since most copolymers display strongly segregated microstructures. Since the product of Flory− Huggins parameter χ between A−B monomer pairs and number of Kuhn steps N (i.e., χABN) controls the degree of phase segregation,1−3 one can lower the degree of polymerization N to raise the phase transition χAB * and thus lower the order−disorder transition temperature. Alternatively, adding nonselective solvents to the melt also achieves the same effect of raising χ*AB by diluting the chemical incompatibility among monomers. Research has shown that solvents are critical in facilitating self-assembly of diblock copolymers.4−10 Lodge et al. show rich phase behavior of diblock copolymers in the presence of solvents with different monomer selectivity.11−13 Theoretical modeling efforts aim to determine the thermodynamic behavior of copolymers in solvents.14−25 Huang and Lodge26 use selfconsistent field theory (SCFT) to show that for flexible diblock copolymers polymer−solvent interactions are screened in the limit of concentrated polymers. In this limit, the dilute © XXXX American Chemical Society

approximation is valid, where the phase diagram of a diblock copolymer solution is identical to that of a melt by replacing χABN with χABNϕ̅ P (where ϕ̅ P is the total polymer volume fraction). Subsequent experimental work by Lodge and coworkers demonstrates that the dilute approximation fails when polymer concentration is sufficiently low.27 The blob argument to find the scaling of A−B Flory−Huggins interaction parameter χAB * on solvent concentration only partially explains the experimental observations.14,27 Apart from the present theories for copolymer solutions in the long, flexible chain limit, theoretical and computational modeling is less prevalent for copolymers in solution with finite, experimentally relevant molecular weights.28−33 In most experiments, the length of polymers usually ranges from 10 to 1000 monomers.34 Understanding the phase behavior of copolymers with experimentally relevant molecular weights (i.e., semiflexible polymers) is critical since recent applications of copolymers (e.g., in nanolithography) involve the use of short copolymers in solvents.35 Studying the phase transitions of copolymers in solution is also crucial to understanding a range of biological phenomena. Many molecular self-assembly processes in biology occur in Received: January 24, 2018 Revised: April 3, 2018

A

DOI: 10.1021/acs.macromol.8b00172 Macromolecules XXXX, XXX, XXX−XXX

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solvent-rich environments, such as protein folding and micelle organization. A random copolymer is a useful model to understand the fundamental processes underlying protein folding. Most works on random copolymers focus on either self-assembly of a single random copolymer36,37 or melts of random copolymers.38−47 Theoretical efforts address the impact of solvent on the thermodynamic behavior of random copolymers,48,49 though the role of solvents is a relatively underexplored subject for random copolymers. In this work, we study the phase behavior of solutions of diblock copolymers of experimentally relevant molecular weights and random copolymers. We employ the randomphase approximation to investigate the spontaneous concentration fluctuations and phase transitions of both copolymer solutions. We also look at the phase diagrams of diblock copolymers in the presence of solvents.

ϕŜ ( r ⃗) = vS ∑ δ( r ⃗ − R⃗ k)

σαj (s)

where L is the total length of the polymers and is the chemical identify of the jth polymer at arclength position s. For instance, for diblock copolymers, σAj (s) = 1 when 0 ≤ s < L fA, σAj (s) = 0 otherwise, and σBj (s) = 1 − σAj (s), with fA as the fraction of A-segments. To capture chain semiflexibility at low molecular weights, we model the polymers as wormlike chains53,54 with chain configuration energy βEpoly =

nP

nS

∫ ∏ + rj⃗ ∫ ∏ dR⃗k

× exp[−βE int − βEpoly ]

βE int = +

2lpA

∫ d r ⃗ ϕÂ ϕB̂ +

χBS 2lpAvS

χAS 2lpAvS

nP

∫ d r ⃗ ϕÂ ϕŜ

∫

+

(2)

j=1 nP

ϕB̂ ( r ⃗) = A ∑ j=1

∫0 ∫0

ds

σjA(s)δ( r ⃗

0

(6)

(7)

np

∑j=1

∫

∫

log z P,j + nS log zS

(8)

In eq 8, the single-chain partition function zP and solvent− molecule partition function zS have all degrees of freedom of monomers and solvents z P,j =

1 LA

∫ + r ⃗ exp[−βepoly + iA ∫0

+ iA

− rj(⃗ s))

zS =

(3) L

i=1

⎛ ∂u ⃗ ⎞2 ds ⎜ i ⎟ ⎝ ∂s ⎠

∫

∫ d r ⃗ ϕB̂ ϕŜ

L

2

L

where the summation ∑α is carried over all components α = A, B, and S. The term ∏r⃗ δ(∑α ϕ̂ α − 1) enforces that the space is filled by monomers and solvents (i.e., incompressibility of the solution). The argument within the exponential of the partition function is written as χAS χ −βF = − AB d r ⃗ ϕA ϕB − d r ⃗ ϕA ϕS 2lpA 2lpAvS χBS − d r ⃗ ϕBϕS − i d r ⃗ ∑ Wα( r ⃗)ϕα( r ⃗) 2lpAvS α = A,B,S

where we choose to scale the Flory−Huggins parameters in terms of the geometric average of the volume of each mixing species. For instance, a Kuhn volume is the product of a Kuhn step 2lp and chain cross-sectional area A, and vS is the solvent molecular volume. The instantaneous volume-fraction fields of diblock copolymers are defined as ϕÂ ( r ⃗) = A ∑

i=1

nP

∑∫

⎛ ⎞ × exp[−βF ] ∏ δ ⎜⎜∑ ϕα̂ − 1⎟⎟ ⎝ α ⎠ r⃗

(1)

for a system containing nP polymer chains of length L and cross-sectional area A and nS solvent molecules of volume vS. The degrees of freedom within the partition function include polymer chain configurations rj⃗ and solvent molecule positions R⃗ k. In general, the local interactions between A-type monomers, B-type monomers, and solvent molecules are captured by pairwise Flory−Huggins parameters. We write the interaction energy as χAB

2

lp

∫ +WS+WA +WB+ϕA +ϕB+ϕS

A=

k=1

j=1

∑ βepoly =

∂ r (⃗ s)

MEAN-FIELD THEORY OF COPOLYMER−SOLVENT MIXTURES In this work, we employ field-theoretic approaches50 to determine the thermodynamic behavior of a mixture of copolymers (diblock or random) and solvents. As we are interested in segment lengths that are of intermediate to short molecular weight, it is necessary to capture the molecular elasticity in our treatment of chain conformations.46,47,51,52 The partition function of a copolymer−solvent mixture is written as 1 1 nP nP ! (LA) nS! vSnS

nP

lp

where lp is the persistence length and ui⃗ (s) = ∂i s is the chain tangent vector for the ith chain at arclength position s. In subsequent discussions, we refer to the number of Kuhn steps of each chain as chain length N = L/(2lp). This is proportional to the degree of polymerization in the experimental system, given a constant Kuhn length 2lp determined by the chemical system. From a physical perspective, the value of N defines whether the chain can be categorized as rigid (N ≪ 1), semiflexible (N ≈ 1), or flexible (N ≫ 1), since the number of Kuhn steps N gives the chain length relative to the Kuhn length 2lp. After a series of field transformations,50 we reach the following expression for the partition function in terms of field variables.

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A=

(5)

k=1

1 vS

∫0

L

ds σjB(s)WB( r (⃗ s))]

∫ dR⃗ exp[ivSWS(R⃗)]

L

ds σjA(s)WA( r (⃗ s)) (9)

(10)

Next, we use random-phase-approximation to expand the free energy in eq 8 in terms of field fluctuations Wα(r)⃗ and ϕα(r)⃗ from the homogeneous phase. To distinguish the different

ds σjB(s)δ( r ⃗ − rj⃗(s)) (4) B


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the polymers are sufficiently flexible, such that a Kuhn segment roughly corresponds to the reference volume of a monomer. Examples where such assumption is valid include polystyrene− polyisoprene in diethyl phthalate and dimethyl phthalate.60 For stiff chains (such as DNA and conducting polymers), this assumption is no longer valid, since the Kuhn segment is much longer than the solvent molecules. We make a few observations here about the spontaneous local variations of free energy, determined by the quadraticorder vertex matrix Γ̃ (2). For the component Γ̃(2) 11 , the enthalpic term that leads to a monomer−monomer segregation is dictated by χABNϕ̅ P at the transition, meaning the solvents effectively softens the interactions between A- and B-type monomers. For the component Γ̃(2) 22 , there is a structural correlation contribution term (scales as ∼1/ϕ̅ P) and an excluded volume term (scales as ∼1/ϕ̅ S) due to the presence of solvent. In the melt limit when ϕ̅ P → 1, the polymer−solvent concentration fluctuation is suppressed due to the infinitely stiff response of free energy with respect to finite ψ2 variations. At finite polymer concentration ϕ̅ P and in the long chain limit (large AL/vS), the excluded volume contribution again vanishes. Only when the copolymers have finite length L in the presence of solvents, both terms from structural correlation and solvent excluded volume are important. In this work, we focus on the effect of both chain length and solvent concentration on the phase behavior of copolymers. The sign of the off-diagonal terms of the quadratic vertex ̃ (2) (i.e., Γ̃(2) 12 and Γ21 ) are controlled by composition fraction fA. When A-type monomers are the minority species with fA < 0.5, ̃ (2) Γ̃(2) 12 = Γ21 > 0, indicating a thermodynamic benefit of having correlated concentration fluctuations between ψ1 and ψS = −ψA − ψB = −ψ2. This suggests that for the homogeneous phase the solvent fluctuates with the minority species of monomers. From a physical perspective, a wavelike (i.e., sinusoidal) fluctuation of the A−B volume fractions ψ1 occurs at a finite wavelength for finite N. At this wavelength, the majority species (B for fA < 1/ 2) pulls a larger fraction of the polymer into the region of excess B. The incompressibility of the solution leads to a driving force for the solvent to fluctuate toward the region of excess A. To identify the spontaneous concentration fluctuations from a homogeneous phase, we consider the local variations of free energy in response to variations of ψ. We find the eigenvectors vi (where i = 1, 2) of the quadratic vertex matrix at given Flory− Huggins parameter χAB and wave mode q with their corresponding eigenvalues λi (i = 1,2)

modes of component demixing (i.e., A−B monomer separations and polymer−solvent separations), we use a twocomponent order parameter vector ψT = [ψ1, ψ2]. The components of ψ are ψ1 =

fB ϕP̅

ψA −

fA ϕP̅

ψB

(11)

ψ2 = ψA + ψB

(12)

In Fourier space, an expansion of the free energy βF with respect to ψ̃ (q⃗) = ∫ dq⃗ ψ(r)⃗ exp(iq⃗·r)⃗ gives βF[ψ ] = βF0[ψ ] + +

1 3!

1 2

∑∫ ij

q1, q2

(2) Γ̃ ij ψψ ĩ j̃

1 (3) Γ̃ ijk ψψψ + ĩ j̃ k̃ q1, q2 , q3 4!

∑∫ ijk

∑∫ ijkl

q1, q2 , q3 , q3

(4) ψ Γ̃ ijklψψψ ĩ j̃ k̃ l̃

(13)

The exact formulas of the vertex functions for diblock copolymers are given in refs 14 and 26. For random-copolymer solutions, similar to random copolymer melts,46 we find a replica-coupling-free expression of free energy up to quadratic order expansion. For this work, we focus on the case of solvents with no enthalpic demixings from polymers with χAS = χBS = 0. The theory is easily analyzed if these enthalpic terms representing the quality of the solvent are nonzero. Therefore, in both cases of diblock and random copolymers, we write the quadratic vertex function as (2) 2 Γ̃11 = −2χAB ϕP̅ +

ϕP̅ N

∑

(2) − 1

Sα̃ 1, α2 Δα1Δα2

α1, α2 = A,B

1 (2) (2) (2) − 1 (2) − 1 ̃ ̃ ) Γ̃12 = Γ̃ 21 = [fA (SAA − SAB N (2) − 1 ̃ (2) − 1 − SAB ̃ )] + χAB ϕP̅ (1 − 2fA ) − fB (SBB (2) Γ̃ 22 =

(14)

(15)

1 (2) − 1 (2) − 1 ̃ ̃ ̃ (2) − 1] [fA 2 SAA + 2fA fB SAB + fB2 SBB NϕP̅

+ 2χAB fA fB +

2lpA ϕS̅ vS

(16)

where the α1−α2 structure factor is given by (2)

Sα̃ 1α2(q ⃗) =

1 N2

∫0

N

ds 2

∫0

N

ds1 σ α2(s2)σ α1(s1)

× ⟨exp[iq ⃗ · ( r (⃗ s2) − r (⃗ s1))]⟩

(2) v1 , v2 = eig[Γ̃ (χAB , q)],

(17)

The factor Δα takes a value of +1 if α = A and −1 if α = B. In this treatment, we use the Kuhn volume 2lpA to nondimensionalize all volume scales. The terms S̃(2)−1 α1α2 in eq 16 are obtained by inverting the ̃ structure factor matrix S(2) α1α2. We restrict α1 and α2 to only take possible letters A and B. The analytical results of S̃(2) α1α2 of wormlike diblock and random copolymers are given by our earlier work46,51,52 based on exact solutions for the statistical behavior of the wormlike chain model.55−58 Codes for their calculations are posted on our research Web site.59 Here, we make the assumption that each solvent molecule occupies the same volume of a Kuhn step, i.e., vS = 2lpA. This leads to a simplification of the excluded-volume term in the Γ̃(2) 22 term 2lpA/ϕ̅ SvS = 1/ϕ̅ S. This assumption is reasonable in cases when

where λ1 ≤ λ 2

(18)

The spontaneous concentration fluctuation at peak wave mode is then given by v1(χAB,q*), where the peak wave mode q* is determined by q*(χAB ) = argmin λ1(χAB , q) q

(19)

Because the peak wave mode varies with respect to χAB, we then determine the phase transition spinodal χAB * and the critical wave mode at phase transition q*(χ*AB) by solving * , q*(χ * )) = 0 λ1(χAB AB

(20)

To construct the phase diagrams, we use a wave mode expansion of the concentration order parameter ψ C


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Figure 1. Directions (arrows) and magnitudes (in color) of spontaneous concentration fluctuations in diblock copolymers at different equilibrium solvent composition ϕ̅ S and monomer compositions ϕ̅ A and ϕ̅ B. Top panels are for polymers with N = 1000, Flory−Huggins parameter χAB = 0 * . Bottom panels are for polymers with N = 1, Flory−Huggins parameter χAB = 0 (left), and χAB = 0.8χAB *. (left), and χAB = 0.8χAB

We focus on the spontaneous concentration fluctuation at the peak wave mode v1(q*) that results in the smallest local variational increase of free energy βF. In Figure 1, the arrows in each triangle show the direction of such spontaneous concentration fluctuations at a given equilibrium composition condition. Note that within the triangle any fluctuation in the horizontal direction indicates a monomer−monomer only fluctuation with ψ2 = 0, and any fluctuation in the direction along the constant fA lines corresponds to polymer−solvent only fluctuation with ψ1 = 0. The underlying color at each point within the triangle gives the magnitude of concentrationconcentration correlation in response to spontaneous concentration fluctuation ⟨v1·v1⟩ = 1/λ1(χAB,q*). The top two panels of Figure 1 show the spontaneous concentration fluctuations at peak wave mode v1(q*) of long diblock copolymer solutions with N = 103. The top left panel is obtained when Flory−Huggins parameter χAB = 0. The top right panel is when Flory−Huggins parameter is close to the mean-field transition χAB = 0.8χ*AB. Along the two sides of the triangle when polymers are homopolymers with only B- or Atype monomers ( f B = 1 or fA = 1), the spontaneous concentration fluctuations are in the direction of polymer− solvent segregations. At all other conditions, the horizontal arrows indicate only monomer−monomer segregations with ψ2 = 0. The same directions of spontaneous concentration *. fluctuations are seen when χAB is increased to 0.8χAB Noticeably, the magnitudes of the spontaneous concentration fluctuations are stronger near the polymer melt limit approaching the bottom of the triangle with ϕ̅ S near zero. The directions and magnitudes of spontaneous fluctuations suggest that the presence of the solvent only softens the

n

ψ1( r ⃗) = a1 ∑ exp[ir ⃗·(q ⃗ − qk⃗ *)] + c.c. k=1

(21)

n

ψ2( r ⃗) = a 2 ∑ exp[ir ⃗·(q ⃗ − qk⃗ *)] + c.c. k=1

(22)

where |q*k⃗ | = q* determined from eqs 19 and 20. The sets of critical wave modes for the lamellar (LAM), cylindrical (CYL), and body-centered-cubic (BCC) phases are found in ref 3. For instance, a lamellar phase will correspond to a single plane-wave decomposition with n = 1. We then insert the Fourier representations of the order parameters composed of planar waves into eq 13. In the evaluation of the free energies, we leverage our analytical solutions of the vertex matrices Γ̃ (n) (n = 2, 3, 4) of wormlike chains. The codes for calculating the vertex matrices are included on our research Web site.59 Above the * , the stable microstructure at given χAB is phase transition χAB the phase that gives the minimum free energy with respect to their amplitudes of the order parameters a1 and a2. The order− order transition χAB is determined by matching the free energies of different ordered microstructures βFn1 = βFn2.

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DIBLOCK COPOLYMER SOLUTIONS We first examine the spontaneous concentration fluctuations of ψ of diblock copolymer solution at a range of equilibrium solvent concentration ϕ̅ S = 1 − ϕ̅ P, A-monomer concentration ϕ̅ A = fAϕ̅ P, and B-monomer concentration ϕ̅ B = 1 − ϕ̅ S − ϕ̅ A. Each point in the triangles of Figure 1 represents such an equilibrium condition defined by the composition triad (ϕ̅ S, ϕ̅ A, ϕ̅ B). D


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Figure 2. Spinodals of diblock copolymer solutions for chains with N = 1000 and N = 1 and polymer compositions ϕ̅ P varying from 0.10 (blue) to 1.00 (red). The polymer composition dependence is too weak to give any noticeable differences in spinodal when N = 1000.

monomer−monomer interactions by rescaling χ to χϕ̅ P, without explicitly affecting the way monomers segregate. In comparison, the bottom two panels of Figure 1 show the spontaneous concentration fluctuations of solutions of short diblock copolymers with N = 1. In the bottom left panel when χAB = 0, the directions of concentration correlations are highly conditioned on equilibrium composition. Near the limit that ϕ̅ S is close to 1.0, the spontaneous concentration fluctuations point along the constant-fA lines with ψ1 = 0, leading to preferred polymer−solvent segregations. Approaching the melt limit with smaller ϕ̅ S, monomer−monomer segregation becomes more favorable. The arrows point in the directions toward the bottom vertices B and A of the triangle, with respect to the constant fA condition lines, suggesting preferred allocation of the solvent molecules to the minority species. In the limit of polymer melts with ϕ̅ S = 0.0, there only exists monomer− monomer segregations. The bottom right panel shows the concentration correlations at χAB = 0.8χAB * , the arrows now indicate that concentration correlations are now in favor of monomer−monomer segregations. We show the spectrum of concentration correlations ⟨v1·v1⟩ and ⟨v2·v2⟩ over range of q at different chain length and χAB in the Supporting Information. Next, we consider the phase transitions of diblock copolymer solutions at χAB = χ*AB. The top row of Figure 2 shows the phase transition spinodal χAB * Nϕ̅ P and critical wave mode at the phase transition Rq*(χAB * ), with increasing solvent concentration ϕ̅ S, where R is the average end-to-end distance of the polymer

In the long-chain limit, diblock copolymer solutions assume approximately the same dimensionless spinodal χAB * Nϕ̅ P. This is consistent with wearlier work, which use dilute approximation to replace χ*ABN with χ*ABNϕ̅ P.11,14,26 The critical wave mode of instability RqAB * (χAB * ) at the spinodal is also independent of polymer concentration. For solutions of short diblock copolymers (i.e. N = 1), however, when the diblock copolymers are asymmetric with fA ≠ 0.5, the spinodal χAB * Nϕ̅ P decreases with increasing solvent concentration ϕ̅ S. Correspondingly, the critical wave mode of instability is also altered when copolymers have asymmetric composition. Notably, the onset of concentration fluctuation occurs on bigger domain sizes when solvents concentration is raised. This is because solvent is preferably fractionated into the minority monomer regions, expanding the characteristic domain sizes of monomer segregations. This nonuniversal spinodal and critical wave mode dependence on polymer concentrations, as suggested in the Theory section, shows the failure of dilute approximation for short copolymer solutions. A universal spinodal is recovered at symmetric composition. When fA = 0.5, solvent molecules equally partition into the domains of each monomer species, thus only softening the monomer−monomer interactions. We recently used quartic-order free energy expansion to find phase diagrams of semiflexible diblock copolymer melts51,52 (see Supporting Information for a comparison of our theory with results for diblock copolymer melts from SCFT30). We now examine the microstructural orderings of diblock copolymer solutions above the phase transition. In constructing the phase diagrams of diblock copolymer solutions, we compare the free energies of the classical microstructures, BCC, CYL,

⎡ ⎤1/2 1 1 2 1/2 R = ⟨[ r (⃗ L) − r (0)] ⟩ = 2lp⎢N − + exp(− 2N )⎥ ⃗ ⎣ ⎦ 2 2 (23) E


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Figure 3. Phase diagrams of diblock copolymer solutions. Disordered (DIS), body-centered-cubic (BCC), cylindrical (CYL), and lamellar (LAM) phases form depending on A-segment fraction fA and interaction parameter χABNϕ̅ P.

and LAM phases above mean-field spinodal χAB * . The freeenergy landscapes of each microstructure are included in the Supporting Information, which we use to find the order−order transition conditions for χAB. Figure 3 shows the phase diagrams of diblock copolymer solutions at a range of chain lengths in Kuhn steps N = 100, 50, and 10 and polymer concentrations ϕ̅ P = 0.75, 0.50, and 0.25. In each subplot, the solid black lines are the spinodals for phase transitions. Above the spinodals BCCs are always the first stable phase except at fA = 0.5 where the lamellar phase is stable. Above the solid blue lines, CYL phases are stable. At asymmetrical compositions fA ≠ 0.5, LAM phase becomes stable above the red solid lines. The dashed lines in each subplot are the mean-field phase diagrams for diblock copolymer melts (i.e., ϕ̅ P = 1.00 at each chain length N, identical to mean-field phase diagrams shown in ref 51).

At each length of diblock copolymers, the spinodals are lowered with increasing solvents concentration ϕ̅ S. Additionally, the lamellar phase regions expand above the phase transition with addition of solvents, most noticeably for short diblock copolymers. This is due to the expansion of the minority monomer regions in the presence of the solvent, inducing comparable domains sizes even at unequal chemical compositions and stabilizing lamellar phases over other microstructures. We note that the dominance of LAM phase above the phase transitions is qualitatively similar to the fluctuation-corrected phase diagrams we presented in ref 51 considering fluctuation corrections to mean-field phase diagrams. These results suggest one can preferentially engineer lamellar microstructures of diblock copolymers by considering polymers with different aspect ratios (see ref 51), polymer lengths, and solvent F


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Figure 4. Directions (arrows) and magnitudes (in color) of spontaneous concentration fluctuations in ideal random copolymers (λ = 0) at different equilibrium solvent composition ϕ̅ S and monomer compositions ϕ̅ A and ϕ̅ B. From top to bottom rows are results for NM = 100, NM = 1.00, and NM = 0.01. The left column is at χAB = 0.0, and the right column is at χAB = 0.8χAB *.

To show the spontaneous concentration fluctuations, we first focus on the ideal random copolymers with λ = 0. The top rows of Figure 5 show the spontaneous concentration fluctuations at peak wave mode v1(χAB,q*) of random copolymers with length of each monomer NM = 100 at χAB = 0 (top left) and χAB = 0.8χ*AB (top right). The use of arrows and colors is the same as in Figure 1. In this case, the directions and magnitudes of spontaneous fluctuations are very similar to those of diblock copolymer solutions with N = 1000 in Figure 1, with all concentration correlations dominated by monomer−monomer correlations. Note that the peak wave mode of instability occurs at q*(χAB) = 0 for ideal random copolymers at all points in the triangle. In experiments, random copolymers are made from monomers of only a few Kuhn steps long. Thus, NM is typically of the order of magnitude of unity. The middle row of Figure 4 shows the directions and magnitudes of spontaneous

concentrations. In the future, we will extend the current theory to include fluctuation effects. Qualitatively, we expect that the dominance of LAM phase in diblock copolymer solution will be further elevated by concentration fluctuations.

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RANDOM COPOLYMERS SOLUTIONS In our previous papers,46,47 we discussed the impact of chain semiflexibility on phase transitions of random copolymer melts. In this work, we model random copolymers as chains made of monomers with Markovian sequence statistics. We find that chain semiflexibility has dramatic influence on the phase transition χAB and critical wave mode. Here, we examine the phase behavior of random copolymer solutions. Following the same notation as our previous work,46 NM as the number of Kuhn steps of each monomer, RM as the average end-to-end distance of a monomer, and fA as the fraction of A-type monomers and λ the degree of chemical correlation, G


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Figure 5. Spinodals 4fA f BχAB * NMϕ̅ P and critical wave modes RMq(χAB * ) of random copolymers versus different degrees of chemical correlations λ and A-type monomer fraction fA. The top two rows are results for monomer lengths NM = 100 and NM = 0.01. The bottom row shows the spinodals and critical wave modes at fA = 0.5 at different NM shown by the solid and dashed lines in the figures above. All results are shown at total polymer concentration ϕ̅ P = 0.50.

concentration fluctuations at the peak wave mode v1(χAB,q*) of random copolymers with NM = 1.00. At this NM, the spontaneous concentration fluctuations are comparable to that of diblock copolymers with N = 1.00, with comparable monomer−monomer and polymer−solvent segregations at χAB = 0. Approaching the phase transition at χAB = 0.8χAB *, monomer−monomer segregation starts to dominate, with solvents preferentially aggregate in the region of minority monomers. In the limit of random copolymers with very short monomers NM = 0.01, when the Flory−Huggins parameter χAB = 0 (bottom left), all arrows (except in the case of polymer melt when ϕ̅ S = 0) are parallel to the constant fA lines, showing * , the dominant polymer−solvent segregations. At χAB = 0.8χAB bottom right panel of the phase diagram shows shift in directions of concentration correlations to monomer−mono-

mer correlations. At this Flory−Huggins parameter, the intensity of monomer−monomer concentration correlations are not as strong as the polymer−solvent concentration correlations approaching the homopolymer limits. We now look at the phase transitions of random copolymers solutions. To show the effect of solvent, we focus on equal composition of polymers and solvents with ϕ̅ P = 0.5. Figure 5 shows the spinodal 4fA f BχAB * NM ϕ̅ P and critical wave mode * ) at spinodal for polymers with NM = 100 and 0.01 RMq*(χAB where RM is the average end-to-end distance of a monomer (given by eq 23 with NM replacing N). The unphysical regime of (fA, λ) are left white constrained by the Markovian chemical sequence.46 Namely, for a given chemical correlation λ, the fraction of A-type monomers f A needs to satisfy λ 1 − 1 − λ ≤ fA ≤ 1 − λ . The bottom row of Figure 5 shows the spinodal and critical wave mode at symmetric composition fA = H


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copolymer solutions are dramatically different from what one would expect in polymer melts. Our theoretical work can also be extended to study the role of fluctuations on diblock copolymer phase diagrams. Our previous work51 predicts that semiflexibility causes a breakdown in the universal phase behavior that was previously found for flexible polymers.61,62 Such a breakdown is also likely to arise in the presence of solvent, and our theory would serve as a basis for exploring the interplay between semiflexibility, solvent, and concentration fluctuations. A replica-field theory can be used to consider the phase diagrams of random copolymer solutions. Our previous work46 predicts replica coupling at the quartic order in concentration fluctuations. Thus, fluctuation effects need to be incorporated into the treatment to predict the impact of quenched disorder on the phase behavior. Future work will aim to address these effects.

0.5, indicated by the solid and dashed black lines in the above figures. At NM = 100, Figure 5 shows that the phase transition spinodal decreases with degree of chemical correlation. This is because the average size of a block becomes bigger with larger λ. Thus, the phase transition occurs at smaller Flory−Huggins parameter χAB. Correspondingly, Figure 5 shows that the critical wave mode of instability also decreases with increasing λ because of bigger block size. For a given λ, we find a universal dependence of the phase transition 4fA f BχAB * NMϕ̅ P and critical wave mode RMq(χAB * ) on λ in the physical regions of λ. A single crossing of the critical wave mode RMq(χ*AB) to zero at the Lifshitz point is found at λL = −0.27. Such a universal spinodal is identical to that of a melt.46 These results suggest the validity of dilute approximation for flexible random copolymers, similar to the case in diblock copolymers. We now look at the phase behavior of random copolymers made of short or stiff monomers NM = 0.01. Similar to the case of random copolymers with long monomers NM = 100, we observe a decrease of the spinodal χ*ABNMϕ̅ P with increasing chemical correlation λ. At symmetric composition fA = 0.5, the phase transition χAB * NMϕ̅ P and critical wave mode RMq(χAB * ) are unaltered from the melt case,46 shown in the bottom rows of Figure 5. For asymmetric compositions fA ≠ 0.5, the spinodals of rigid random copolymers do not collapse to a universal curve. Note that the critical wave mode is only weakly dependent on λ until they eventually become zero beyond the Lifshitz points. When fA ≠ 0.5, the critical wave mode sharply transitions from a finite value to zero. This suggests that for short random copolymers, the Lifshitz point depends on fA. At sufficiently low and high fA, the Lifshitz points vanish with zero wave mode instability at all physical values of chemical correlations λ.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b00172. Figures S1−S6 (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (A.J.S.). ORCID

Andrew J. Spakowitz: 0000-0002-0585-1942 Notes

The authors declare no competing financial interest.

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CONCLUSIONS In this paper, we analyze the mean-field phase behavior of semiflexible diblock and random copolymer solutions. As in our previous work on copolymer melts,46,47,51,52 we find the polymer semiflexibility and finite molecular weight play a significant role in the thermodynamic behavior. We find that for semiflexible copolymers solutions excluded volume effects of the solvent molecules become important and significantly alter the phase transition conditions and the self-assembled morphology. Specifically, the concentration fluctuations induce prominent polymer−solvent segregations. The solvent concentration preferentially fluctuates with the minority species of the copolymers. As a result, asymmetric diblock copolymers have different spinodal χAB * and critical wave modes depending on Atype monomer composition and total polymer concentration ϕ̅ P. Above the phase transition at sufficiently large χAB, lamellar phase becomes the dominant phase in solutions of shorter polymers and lower polymer concentrations. The phase transition spinodal χAB * and critical wave modes of random copolymers are also altered by excluded volume effects of solvents. Specifically, Lifshitz points λL depend on A-type monomer fraction fA for random copolymers with short monomers. These results emphasize the importance of solvents when working with copolymer solutions at experimentally relevant molecular weights. In particular, the dilute approximation breaks down even at high polymer concentration ϕ̅ P for polymers with lower chain length N. The phase behavior of

ACKNOWLEDGMENTS We are grateful to Jian Qin, Zhen-Gang Wang, David Morse, and Jay Schieber for valuable discussions. Shifan Mao was supported by the NSF Interfacial Processes and Thermodynamics Program, Award 1511373. Quinn MacPherson and Andrew Spakowitz acknowledge funding support from the TomKat Center for Sustainable Energy at Stanford University.

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REFERENCES

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Thermodynamic Model of Solvent Effects in Semiflexible Diblock and

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