Phase Behavior of Ionic Block Copolymers Studied by a Minimal

Jul 10, 2013 - Charlotte R. Stewart-Sloan , Rui Wang , Michelle K. Sing , Bradley D. Olsen ... Monojoy Goswami , Jose M. Borreguero , Bobby G. Sumpter...
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Phase Behavior of Ionic Block Copolymers Studied by a Minimal Lattice Model with Short-Range Interactions P. Knychała,†,# M. Dzięcielski,† M. Banaszak,†,* and N. P. Balsara‡,§,⊥ †

Faculty of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland Environmental Energy Technologies Division, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, United States § Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, 94720, United States ⊥ Department of Chemical and Biomolecular Engineering, University of California, Berkeley, California, 94720, United States ‡

ABSTRACT: We present the results of Monte Carlo lattice simulations of a model symmetric diblock copolymer wherein a fraction of segments of one block, p, corresponds to ionic species, and the other block does not contain ions. We use experimentally determined Flory−Huggins interaction parameters, χ, to quantify the interactions between ionic and nonionic monomers. Analysis of the experimental data indicate that χ between poly(styrenesulfonate) and polystyrene is about 5, a value that is orders of magnitude larger than that obtained in mixtures of nonionic polymers. Our model predicts that clustering of ionic monomers in the disordered state results in stabilization of the disordered phase and the product p2χN is well above the mean-field value of 10.5 at the order−disorder transition (N is the total number of monomers per chain). Network morphologies and hexagonally packed cylinders are observed in the ordered state at large p values while more traditional lamellar phases are found at small values of p.

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INTRODUCTION There is increasing interest in the self-assembly of ioncontaining polymers due to their relevance in the emerging clean energy landscape. Mechanically robust polymers are used to separate the electrodes in both lithium batteries and hydrogen fuel cells that are currently being considered for powering vehicles. The electrolytes in current lithium batteries that are contained within the pores of the separator comprise mixtures of flammable organic solvents that compromise cyclelife and safety. Hydrogen fuel cells rely on a hydrated singleion-conductor such as Nafion to transport protons between the electrodes but poor conductivity at elevated temperatures has proved to be a major hurdle that has thwarted their entry into the market. Block copolymers wherein mechanically robust polymer chains are covalently bonded to ion-containing chains are naturally suited for such applications. Rational design of these systems has thus far been thwarted by the lack of fundamental understanding of the interactions between the ions and polymers. There have been several efforts to understand the effect of ion-containing monomers on the self-assembly of block copolymers.1−4 These studies suggest that the theoretical framework developed for predicting the phase behavior of uncharged diblock copolymers5,6 can be used in the case of ioncontaining block copolymers provided the Flory−Huggins interaction parameter between the blocks, χ, is replaced by an effective interaction parameter that accounts for ionic interactions. Initial work that focused exclusively on the entropy of the counterions1,2 indicated that the presence of ions reduces the effective interaction parameter, i.e. their © 2013 American Chemical Society

presence stabilizes the homogeneous disordered phase. In subsequent work, Wang and co-workers3 showed that inclusion of ion solvation effects can reverse this trend, i.e. the presence of ions increases the effective interaction parameter and induces demixing of the blocks. Most of the experiments on ioncontaining block copolymers are consistent with the predictions of Wang and co-workers. In many cases, the effective interaction parameters determined experimentally are so large that one wonders if they are physically meaningful. For example, the effective χ parameter between sulfonated styrene and styrene chains, based on a reference volume of 0.1 nm3, is estimated to be about 5,7−9 which is orders of magnitude larger that χ parameters typically obtained in uncharged polymer mixtures. In this paper, we use simulations to shed light on the phase behavior of block copolymers wherein one of the monomers is extremely incompatible with the other monomers. The phase behavior of nearly symmetric poly(styrenesulfonate)-b-polymethylbutylene (PSS−PMB) as a function of sulfonation level and chain length, N, was reported in ref 9. The PSS block is a statistical copolymer of sulfonated and unsulfonated polystyrene units. We define p to be the fraction of polystyrene monomers that were sulfonated in the PSS block. Applying mean-field theory to this system requires specifying Flory−Huggins interactions parameters between the three monomers, styrene (S), styrenesulfonate (SS), and methylbutylene (MB): χS‑SS, χMB‑SS, and χS‑MB. The χ Received: January 11, 2013 Revised: June 21, 2013 Published: July 10, 2013 5724

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Table 1. Sequence Distributions of Simulated Copolymers shortcut

N

p

microarchitecture

N34/S3 N34/S6 N34/S9 N34/S10 N34/S11 N34/S12 N52/S9 N52/S12 N52/S15 N68/S9 N68/S12 N68/S15 N68/S18 N68/S21

34 34 34 34 34 34 52 52 52 68 68 68 68 68

0.176 0.353 0.529 0.588 0.647 0.706 0.346 0.462 0.577 0.265 0.353 0.441 0.529 0.618

AAASAAAASAAAASAAA-17B AAASSAAASSAAASSAA-17B AASSSAASSSAASSSAA-17B AASSSAASSSSAASSSA-17B ASSSAASSSSAASSSSA-17B ASSSSAASSSSASSSSA-17B AAASSSAAAAASSSAAAAASSSAAAA-26B AAASSSSAAAASSSSAAAASSSSAAA-26B AAASSSSSAAASSSSSAAASSSSSAA-26B AAAAAASSSAAAAAASSSAAAAAASSSAAAAAAA-34B AAAAAASSSSAAAAASSSSAAAAASSSSAAAAAA-34B AAAAASSSSSAAAASSSSSAAAASSSSSAAAAAA-34B AAAASSSSSSAAAASSSSSSAAAASSSSSSAAAA-34B AAASSSSSSSAAASSSSSSSAAASSSSSSSAAAA-34B

χN exceeds 10.5, where N is the number of segments per chain and χ is an effective dimensionless interaction energy between unlike segments, A and B.15 Theories and simulations that account for fluctuations indicate that order formation occurs at a significantly larger value of χN.12,14,16,17 Recent calculations12 suggest that the critical value of χN for order formation is about 40 in the small N limit, indicating a substantial stabilization of the disordered phase. It is widely believed that the mean-field is recovered in the N → ∞ limit.13,14 We simulated melts of a symmetric diblock copolymer wherein a fraction of the segments of one of the blocks (e.g., the A block), p, have an ionic character.9,11,18−21 The other block that we will call B does not contain ionic species. Selfassembly in these copolymers is driven mainly by the strong tendency of the ionic species to segregate from the nonionic species (both A and B). Because of the low dielectric constant of the monomers of interest (styrene and methyl butylene), the ionic species are expected to exist as ion pairs. The lack of dissociation of sulfonic acid groups is important for the applicability of our model. Finally, it has to be recognized that while the Monte Carlo simulations employ a lattice model, the SCFT framework is based on the Gaussian chain model. This calls into question the correspondence between the 2 models. For example, if on the mean-field level the ODT for symmetric diblock occurs at (χN)ODT =10.5 for SCFT then what is the ODT condition and the definition of χ for the lattice model. This problem will be discussed later.

parameters between ionic and nonionic monomers (χS‑SS and χMB‑SS) were estimated in ref 9 using the domain sizes determined in the strong segregation limit (block copolymers with large N), neutron scattering profiles obtained from disordered samples (block copolymers with small N), and the random copolymer theory.10 All of these measurements suggest that χS‑SS and χMB‑SS range from 5 to 10 at 25 °C. The purpose of this paper is to explore the extent to which the phase behavior of PSS−PMB copolymers can be explained on the basis of a crude model based entirely on measured χS‑SS, χMB‑SS, and χS‑MB parameters. In an earlier paper, we that showed simulation results using this crude model were in reasonable agreement with experiments.11 That study was restricted to a single value of N and p. In this study we extend that work to include a range of N and p values. The experimental data in ref 9 show that nearly symmetric PSS−PMB block copolymers, with PSS volume fractions, ϕ between 0.45 and 0.50, exhibit phase behavior that is dramatically different from that of conventional block copolymers. Only lamellar phases have been reported in this composition in conventional block copolymers, regardless of pressure, temperature, and chain length.6,12−14 In contrast PSS−PMB systems exhibit perforated lamellae, gyroid, and hexagonally packed cylinders in addition to lamellar phases. In the present study, which is restricted to ϕ = 0.5, we seek to understand the following experimental observations: (1) What are the values of the relevant parameters, χ, p, and N, at the order−disorder transition? (2) Does the clustering of ionic groups influence phase behavior of these systems? (3) What factors promote the formation of network and hexagonal phases in these systems? (4) When a ϕ = 0.5 system forms a hexagonal phase, are the ionic groups inside the cylinders or in the matrix? (5) To what extent is the mean-field analysis that was used in ref 9 to obtain χ valid? It is obvious that more sophisticated models including dipolar interactions and charge separation will be needed for quantitative answers to the questions above. It is, however, instructive to see the extent to which experimental observations can be understood using our crude model. In the mean-field limit, probed by either conducting calculations within the random phase approximation (RPA) or by self-consistent field theory (SCFT), order formation of a symmetric A−B diblock copolymer occurs when the product

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SIMULATION METHODOLOGY Simulations were performed on copolymers consisting of 3 types of segments: A, S, and B, which correspond to effective segments of styrene (S), sulfonated styrene (SS) and methylbutylene (MB).9 We used N = 34, 52, and 68, sulfonation level in the PS block (denoted by p) from 0.176 to 0.706, as summarized in Table 1. The dielectric constant of most polymers is sufficiently low that in the absence of solvents such as water (as is the case of the system of interest here) ionic species are always present in the form of ion pairs. Since ion pairs can, to a first approximation, be considered as highly polar neutral species, the use χ to describe the interactions between S and the other segments is justified, as demonstrated in simulations of explicit ionic copolymers.22 We use the following parameters for the interaction energies: εAB = 0.042ε, εBS = 0.792ε, εAS = ε, εAA = εBB = εSS = 0, which 5725

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sites) in order to fit a fully extended 17−17 copolymer chain. The FCC lattice is generated from a simple cubic 34 × 34 × 34 lattice by removing every second site (to satisfy the condition that the sum of integer coordinates, x + y + z, is an odd integer). For the copolymer melt, the whole FCC lattice is filled with copolymer chains which means that there are na/N = 578 chains. We investigated the effect of box size on our results by performing simulations in boxes of various sizes. In particular, we carry out selected studies for the 68 × 68 × 68 box, with the number of copolymer chains 8 times greater than that for the initial box (34 × 34 × 34). At a given T*, all of the simulations in 34 × 34 × 34 and 68 × 68 × 68 boxes gave similar results. For N = 68, most of our simulations were conducted in a 34 × 34 × 34 box, while for N = 52, a 26 × 26 × 26 box size was used. Again, for selected T* values, we simulated these systems in larger boxes: 68 × 68 × 68 for N = 68 and 52 × 52 × 52 for N = 52. The morphologies obtained in the simulations were not dependent on the box sizes used. The radius of gyration of chain at high T* changes form 3.9 to 5.5 and depending on the box size the periodicity of the obtained nanostructures is from 1.5 to 3.0 times smaller than the box size. We calculate ensemble-averaged energy per lattice site, E*/ na, heat capacity at constant volume, Cv, copolymer chain endto-end distance, R2, and structure factor, S(k) for each temperature, T*. The structure factor, S(k), is calculated using the following equation:

lead to the following definitions of the reduced temperature, T*, and χ parameters T* = kT/ε, χij = zεij/kT, where i, j = A, B, and S; z is the effective number of nearest nonbonded neighbors; χAS = 1.26χ, χBS = χ (χBS is used to measure the relative magnitude of other χ′s), and χAB = 0.053χ. These values of ε are identical to those used in our earlier simulations11 but differ from the final analysis of the experimental data in ref 9. (The values of χij reported in ref 9 are χAS = 0.90χ, χBS = χ, and χAB = 0.032χ. We began our simulations before ref 9 was published and the χij values used here were gotten from a preliminary unpublished version of ref 9.) The PSS block used in ref 9 is a statistical copolymer with sulfonated and nonsulfonated segments. Indirect evidence11 suggests that the PSS block is nonrandom. The segment distributions used in the simulations (Table 1) are thus highly simplified models of the experimental system. The simulations are performed using CMA23−25 for a face centered cubic (FCC) lattice with the number of nearest nonbonded neighbors, zsim=10, and the bond length b=√2a, where a is the FCC lattice constant, used also as a length unit. The relationship between zsim, and z, used to calculate χij using the expression χij = zεij/kT is described below. Chain bonds are not allowed to be broken or stretched, and the usual periodic boundary conditions are imposed. The lattice box size is chosen to fit the chain, and the lattice sites are completely filled with chain segments; there are no vacancies. Since all lattice sites are occupied, a chain segment can move if other segments move simultaneously. An attempt to move a single segment defines a single Monte Carlo step. In contrast to our previous simulations11 where we employed standard Metropolis algorithm (SM),26 the main method applied in present simulations is parallel tempering (replica exchange) Monte Carlo (PT).27−29 Moreover, for selected runs we use also standard Metropolis algorithm (SM) to compare with PT simulations. First, we equilibrate the system in the athermal limit, that is where ε/(kT) is zero. When the system reaches its thermal equilibrium, the polymer chains assume statistical conformations, random orientations and become uniformly distributed within the simulation box. Next, we run thermal PT simulation, starting each replica with independent athermal states. In the PT method, M replicas of system are simulated in parallel, each in different temperature T*i, with i ranging from 1 to M. After a number of MCS (in our case it was 3000 MCS), we try to exchange replicas with neighboring T*i in random order with the following probability:

S(k ⃗ ) n

=

n

α α ⎯⎯⎯→ ⎯⎯⎯→ 1 ⟨( ∑ cos(k ⃗· rm))2 + ( ∑ sin(k ⃗· rm))2 ⟩thermal average na m = 1 m=1

where nα denotes the number of segments of type α and rm denotes the position of m-th segment of type α; in this research α=B. The magnitude of the wavevector, k, varies from kmin = 2π/L to kmax = 2π/b, where L is size of the cubic simulation box, and b is distance between nearest segments. Wavevectors, k, were chosen to be commensurate with the simulation box, and this constraint limits their number, orientations, and allowed lengths. In order to gain better resolution of wave vector, we doubled the simulation box used in the PT simulations at selected state points, and ran the standard (constant T*) MC simulations, performing 3 × 106 MCS, to obtain the structure factor. In all cases, the symmetry of the nanostructures obtained from the PT method were identical to those obtained from the MC simulations at constant T*. Since the systems may not be isotropic, the S(k) is calculated by averaging over all S(k) such that |k| is equal to k. The morphology of the system is determined by visual observation of the morphology in individual boxes and peaks in S(k) using well-known results for the dependence of the peak locations on morphology. Order−disorder and order−order transition temperatures were determined from peaks in the dependence of Cv on T*. Typical results obtained using our approach are shown in Figure 1, which exhibits selected simulation data for the N52/SL12 chain architecture. The heat capacity, shown in Figure 1a, exhibits three spikes at T* = 2.05, 1.73, and 1.57 corresponding to an ODT, an OOT, and lamellar reorientation, respectively. The nanostructures in the vicinity of the phase transitions are identified by S(k) and snapshots, as for example shown at representative temperatures, T* = 1.50 and 1.96, in Figure 1, parts b, c, d, e, and f. The A, B, and S segments are shown in red, blue, and green, respectively; two continuous components of B are shown in different shades of blue, and the

p(T *i ↔ T *i + 1) = min[1, exp( −(βi − βi + 1)(Ui + 1 − Ui))]

where βi = 1/T*i and Ui is potential energy of replica at T*i. We use 24 or 36 replicas (which are simulated parallel) and perform at least 3 × 106 Monte Carlo steps (MCS). We repeat the simulation experiment at least 3 times starting with different initial states and carrying out simulations in different ranges of temperatures. All PT rounds are started with geometric series of temperatures. It is obvious that the density of temperature points is higher at lower temperatures. For a given state point, all runs yield the same type of nanostructure, and the results are averaged over all such runs. Usually, we also assume that the first half of run is needed to equilibrate the system, and the second half is used to collect the data. We perform simulations for three different lengths of polymer chain, N = 34, 52, and 68. For the shortest chain, we use a 34 × 34 × 34 FCC box (with na = 343/2 = 19652 5726

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Figure 2. p2χN for order−disorder transition in term of p for different chain lengths from simulations. The circles and squares correspond to simulation chain lengths: N = 52, and 68, respectively. The solid line is fitted to simulation points with the exception of points indicated by ∗’s. The data for p = 1 are also shown for reference. The ×’s and ▽’s show the corresponding SCFT data for N = 52, and 68, respectively.

by setting χAS = χBS = χ, and χAB = 0. In this system, the effective interaction parameter between the A/S and B blocks is p2χ. Our use of p2χN as the ordinate in Figure 2 is based on this observation. The ×’s and ▽’s show the results of SCFT calculations for N = 52 and N = 68. The value of p2χN at the ODT is about 10.5 regardless of N. (We did not consider the stability of other morphologies in our SCFT calculations; if nonlamellar morphologies were substantially more stable than the lamellar phase then p2χN at the ODT would be substantially less than 10.5. We regard this as highly unlikely.) Figure 2 also shows the value of p2χN at the ODT for N = 52 and 68 obtained from simulations. Most of the data are consistent with the solid curve shown in Figure 2 which is gotten by fitting a quadratic polynomial through all of the data except for the ones denoted by the asterisk. The equation corresponding to the curve is

Figure 1. Simulation results for the N52/SL12 microarchitectures: (a) heat capacity, Cv, as a function of T*, where the filled arrows indicate T* = 1.50 and 1.96, and the unfilled arrows indicate T* = 1.57, 1.73, and 2.05; (b) B segment structure factor, S(k), at T* = 1.50, where the arrows show peaks at 1, 3, 4 indicating the L phase; (c) B segment structure factor, S(k), at T* = 1.96, where the arrows show peaks at 1, (3/2)1/2, (2)1/2, (3)1/2, (4)1/2, (9/2)1/2, (5)1/2, and (6)1/2 indicate bicontinuous DD nanostructure; (d) snapshot of representative configuration of the L phase at T* = 1.50; (e and f) snapshots of representative configuration of the DD nanostructure at T* = 1.96 (A, B, and S segments are shown in red, blue, and green, respectively; two continuous components of B are shown in different shades of blue and A segments are not shown in part f).

p2 χN = 10.5 + 92.4p2

red A’s are not shown in part f in order to highlight the double diamond symmetry. The SCFT calculations were performed on the Gaussian chains, as presented in refs 5 and 30 (but generalized from diblocks to multiblocks), corresponding to segment sequences listed in Table 1, using χAS = 1.26χ, χBS = χ, and χAB = 0.053χ (all calculations presented here are based on these χ parameters). The order−disorder transition temperature (ODT) was obtained by comparing the free energies of homogeneous and lamellar phases. The relationship between the mean-field value of χij used in SCFT calculations and the corresponding χij used in the lattice simulations was discussed by Muller and Binder31 and more recently by Morse and Chung.32 Morse and Chung have shown that the effective χ parameter used in SCFT should be extracted from lattice simulations as follows: (1) The effective number of nearest intermolecular contacts must be calculated in the athermal state for different chain lengths, N, where this number must be extrapolated to N→∞ (or 1/N→0), yielding z. (2) This z is to be used in an equation relating χ to T* . On the basis of simulation runs in the athermal limit (see also ref 33), we estimate z ≈ 7.5. The relationship between χij and T* is thus χBS = χ = 7.5/T* (χAS = 1.26χ, χAB = 0.053χ).

(1)

We chose 10.5 as the constant term in eq 1 for simplicity and obtained the p2 coefficient by a least-squares fit. It should be noted that eq 1 is a fit that is valid over the finite range of p, χ, and N values that we have investigated. It would be interesting to see if eq 1 holds for larger values of N. We define Δ to be Δ = (p2 χN )ODT − 10.5

(2)

where (p χN)ODT is the value of p χN at the ODT. It is clear from Figure 2 that Δ can be as high as 27.2. A noteworthy simplification is that the value of p2χN at the ODT over a wide range of parameters depends on p, not χ or N. For each value of N, the value of p2χN at the ODT deviates from eq 1 above a certain value of p and at p = 1, the results are independent of N. It is thus clear that Δ is much larger at intermediate values of p than in the p = 0 or p = 1 limits. In the p = 1 limit, we recover the conventional diblock results of Pakula and co-workers.16 An interesting conclusion of the work in refs 12 and 16 is that in the limited range covered by the simulations, χN at ODT for symmetric diblocks is more-or-less independent of N. The value of p2χN at the ODT increases in accordance with eq 1 until a critical value of p, pcrit, beyond which the simulations results deviate sharply from eq 1. The value of pcrit for N = 52, for example, is estimated to be 0.520 (Figure 2). We propose that the unexpected stabilization of the disordered phase in the 0 < p < pcrit(N) range, quantified by the large value of Δ, is due to fluctuations of S segments as the ODT is approached. This is established in Figure 3 where we plot the 2

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RESULTS AND DISCUSSION In Figure 2 we show how χ at the ODT varies with p. The simplest model for the copolymers listed in Table 1 is obtained 5727

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Figure 4. Snapshots of representative configurations (above and below T*ODT) for N = 52 in simulation box 52 × 52 × 52 and three different sulfonation levels. (a, d, and g for p = 0.346, b, e, and h for p = 0.462, and c, f, and i for p = 0.577). The A, B, and S segments are shown in red, blue, and green, respectively. Figure 3. Segment structure factor, S(k), for N = 52 and different sulfonation levels: (a) p = 0.346 at T*/T*ODT = 1.18, (b) p = 0.462 at T*/T*ODT = 1.38, and (c) p = 0.577 at T*/T*ODT = 1.11 (simulation box 52 × 52 × 52). The S−S structure factors are shifted vertically. The triangles show the shoulder due to clustering of S segments.

which exhibits the largest concentration fluctuations [SBB(k) peak value is 225], transforms into a double diamond structure. The p = 0.577 sample, which exhibits the weakest concentration fluctuations [SBB(k) peak value is 95], transforms into a perforated lamellar phase; the nonionic blue domains in Figure 4i are perforated by the ionic green/red domains. Simulation results of the kind described in the preceding paragraph were obtained over the full range of p and N values represented in Table 1, and the results thus obtained are shown in Figure 5 using the p2χN versus p format for N=34, 52, and 68. In the mean-field limit, one would expect to obtain phase behavior that is independent of N. The qualitative differences seen in the three panels in Figure 5 are an indication of the importance of nonmean field contributions that are captured by our simulations. The phase behavior seen in Figure 5 is rich, in spite of the fact that all of the nominally symmetric copolymers with the volume fraction of B segments fixed at 0.5. In the low molecular weight systems with N = 34, we obtain gyroid, hexagonal and lamellar phases (Figure 5a). Increasing N to 52 results leads to the formation of double diamond structures that are unstable in conventional diblock copolymer melts. A further increase in N to 68 results in the formation of hexagonally arranged cylinders of the nonionic phase in an ionic matrix. It is important to note that in the simulation results, both S and SS segments occupy one microphase in spite of the large χ between the monomers. This results in the formation of MBrich cylinders that appear ″bloated″ relative to cylinders found in nonionic block copolymers (see N = 68 simulations in Figure 5c). These cylinders are embedded in a S + SS matrix. In Figure 6, we show an electron micrograph of a PSS−PMB copolymer discussed in ref 9. The properties of the copolymer are: PSS and PMB molecular weights are 11.3 and 8.7 kg/mol, respectively and the PSS volume fraction is 0.49. It is evident

disordered S−S and B−B structure factors in the vicinity of the ODT of three systems with N = 52 at similar T*/T*ODT values. The data in Figure 3a were obtained at p = 0.346 [p < pcrit(N)], and in Figure 3b at p = 0.462 [also p < pcrit(N)]. The disordered state B−B structure factor is unremarkable and similar to that of conventional A−B diblock copolymers in the vicinity of the ODT. In contrast, the disordered state S−S structure factor contains two features, a low-q peak that is qualitatively similar to the B−B peak and an additional high-q peak (or shoulder) that reflects clustering of the S segments. The data in Figure 3c were obtained at p = 0.577 [p > pcrit(N)]. In this case both disordered B−B and S−S structure factors are contain a single low-q peak (no high-q feature) indicating the absence of clustering of the S segments. Selected snapshots of the systems discussed in Figure 3 at p values below and above pcrit are shown in Figure 4 for N = 52. Parts a−c of Figure 4 show S segment locations in the disordered state at p = 0.346, 0.462, and 0.577. Spatial heterogeneity is clearly seen in when p < 0.520 (Figure 4, parts a and b) but not when p > 0.520 (Figure 4c). The transformations from the disordered fluctuations into the ordered phase for the three cases can be visualized by comparing the snapshots showing all of the segments, A, B, and S in the disordered (T* > T*ODT) and ordered states (T* < T*ODT; see Figure 4d−i). The nature of S−S correlations affects microphase separation. The p = 0.346 sample, which exhibits modest concentration fluctuations [SBB(k) peak value is 180], transforms into a lamellar phase. The p = 0.462 sample, 5728

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Figure 5. Phase diagram in terms p2χN and p for three different chain lengths. The solid lines indicate the locations of the ODT obtained in simulations for N = 34 (triangles), N = 52 (circles), and N = 68 (squares). The filled triangles correspond to experimental ODT for Nexp=54. The dashed lines indicate different p’s used in simulation and the dotted lines indicate the simulation windows. Simulation snapshots emphasizing the B microphase in blue are shown in selected cases.

or not it applies to larger symmetric block copolymers with larger chain lengths is an interesting open question. It is instructive to compare the simulation results obtained at N = 52 described above, with experimental results for a PSS− PMB copolymer with experimental chain length Nexp = 54 reported in ref 9. (In this discussion we distinguish between the experimental chain length, Nexp, which is based on a monomer volume of 0.1 nm3 and theoretical chain lengths, N, as there may not be a one-to-one mapping of these quantities.) The filled symbols in Figure 5b show the experimentally determined locations of the disorder-to-order transitions for Nexp = 54. The values of p and p2χN where the transitions are experimentally obtained are shifted upward by about 25% with respect to the simulations. Moreover, the disorder-to-gyroid phase transitions were observed experimentally while disorder-to-lamellar and disorder-to-double diamond transitions are predicted by theory. It is evident that the agreement between experiment and simulations is semiquantitative at best. This is, perhaps not surprising, as differences in the experimental monomer volumes of ionic and non-ion species are not taken into account in the simulations. While the need for more refined models is clear, the present work represents an important first step in understanding the effect of concentration fluctuations on the thermodynamics of ion-containing block copolymers. When we began this study, it was not clear if the term “symmetric” block copolymers was appropriate for the copolymers listed in Table 1, as it is not clear if the “relevant” volume fraction is ϕ, or pϕ. The evidence suggests that it is, in fact, appropriate to refer to the copolymers as symmetric. First, all of the SCFT calculations indicate that p2χN at the ODT is about 10.5, a value that is obtained in the mean-field limit in symmetric block copolymers. Second, simulations and experiments (Figures 5 and 6) indicate our copolymers form two microphases, one containing A and S monomers and the other containing B monomers.

Figure 6. Transmission electron microtomography of a PSS−PMB discussed in ref 9. The properties of the copolymer are: PSS and PMB molecular weights are 11.3 and 8.7 kg/mol, respectively, ϕ = 0.49, N = 99. The PSS phase is darkened by staining. The black dots are gold nanoparticles added to the sample to enable the tomographic reconstruction (see ref 9 for details).

that bloated cylinders are observed experimentally in relatively high molecular weight PSS−PMB copolymers. In addition, the experiments show that the S+SS monomers (dark regions in the Figure 6) are located in the matrix, in agreement with simulation results. Returning to Figure 5, we find that the boundary between disorder and order is peaked due to the clustering effects described above, and network phases tend to form near in the vicinity of the peak, regardless of chain length. More traditional lamellae, perforated lamellae and cylinders form on either side of the peak. Regions where two morphologies are noted (L/ PL) indicate that the simulations starting from different initial conditions resulted in two different results. This is probably due to the similarities of the free energies of these morphologies. The location of the ODT of the N = 34 samples lie below the curve in Figure 2. It is thus evident that the simple scaling Δ = 92.4p2 implied by eq 1 only applies to N = 52 and 68. Whether

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CONCLUDING REMARKS Monte Carlo simulations were used to obtain insight into factors that govern the phase behavior of symmetric block 5729

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copolymers containing ionic monomers such as PSS−PMB. One of the blocks in our model polymers is a statistical copolymer of ionic and nonionic monomers while the other block is entirely nonionic. Our model assumes that the interactions between ionic and nonionic species can be described by large Flory−Huggins interactions parameters that are greater than 5. The model provides definitive answers to the questions posed in the introduction: (1) When p is less than pcrit, the values of χ, p, and N at the ODT are such that the product p2χN is given by eq 1. In this regime p2χN increases with increasing p. When p is greater than pcrit, p2χN decreases with increasing p and approaches a value of 19.4 at p = 1. (2) The increased stability of the disordered phase seen at intermediate values of p appears to be correlated with clustering of the ionic groups. (3) More traditional lamellar phases dominate the phase behavior when p < pcrit, while network and hexagonal phases form when p > pcrit. (4) The hexagonal phase at ϕ = 0.5 is found for all N and the nonionic groups are located inside the cylinders. (5) The stabilization of the disordered phase observed in the simulations indicate that p2χN at the ODT is greater than 10.5. It is thus obvious that the Leibler structure factor will not apply to scattering profiles from disordered PSS−PMB copolymers. We expect the true magnitude of the χ parameter between ionic and nonionic monomers to be smaller than that obtained in ref 9. While the proposed model does provide qualitative explanations for many of the unexpected thermodynamic properties of PSS−PMB copolymers, it is obvious that models that include dipole−dipole interactions are essential for quantitative understanding of the phase behavior of ion-paircontaining block copolymers.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (M.B.) [email protected]. Present Address

́ Higher Vocational State School in Kalisz, ul. Nowy Swiat 4, Kalisz, Poland.

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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS P.K., M.D., and M.B. gratefully acknowledge the research grant from the Polish NCN No. DEC-2012/07/N/ST4/00293, and the computational grant from the Supercomputing and Networking Center (PCSS) in Poznan, Poland. N.P.B.’s work was supported by the Electron Microscopy of Soft Matter Program supported by the Office of Basic Energy Sciences, Materials Sciences and Engineering Division, U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

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REFERENCES

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dx.doi.org/10.1021/ma400078y | Macromolecules 2013, 46, 5724−5730