Blockiness and Sequence Polydispersity Effects on the Phase

Article pubs.acs.org/Macromolecules

Blockiness and Sequence Polydispersity Effects on the Phase Behavior and Interfacial Properties of Gradient Copolymers Venkat Ganesan,* N. Arun Kumar, and Victor Pryamitsyn Department of Chemical Engineering, University of Texas at Austin, Austin, Texas 78712, United States ABSTRACT: We consider the influence of sequence polydispersity upon the phase behavior and interfacial characteristics of gradient copolymers. By adapting the algorithmic procedure proposed for random copolymers, we design sequences of varying blockiness and compositional polydispersities for specified composition profiles of the gradient copolymers. Using the sequences so generated, we studied the dependence of the spinodals, the phase behavior, and interfacial properties of gradient copolymers as a function of gradient strengths and blockiness of the sequences. We demonstrate that the interplay between compositional polydispersity and the overall blockiness of the sequences can play a significant role in determining the morphologies, phase behavior, and interfacial activity of gradient copolymer systems. In systems wherein the inherent blockiness of the sequences is small, such as in gradient copolymers with weak gradient strengths, the introduction of such polydispersity and blockiness effects leads to substantial changes in the self-assembly behavior and interfacial properties. In contrast, in systems for which the inherent blockiness is already large, such as in gradient copolymers with strong gradient strengths, the effects of sequence correlations upon the self-assembly characteristics and interfacial properties are seen to be much more mitigated.

I. INTRODUCTION The self-assembly characteristics of block copolymers have attracted significant attention in a variety of contexts such as semiconductor lithography, membrane materials for batteries and separations, as templates for nanoparticle synthesis, photonic crystals etc.1−7 Many such applications have been demonstrated in the context of diblock copolymers, which consist of segments of type A monomers covalently bonded to segments of type B monomers. In such systems, the selfassembly morphology is controlled only by a limited set of parameters, viz., the composition of the diblock copolymer (which is characterized by the average volume fraction of the A segments), the Flory−Huggins interaction parameter between the A and B segments χ, and the degree of polymerization of the polymer chain N.8,9 As a consequence, the morphologies of self-assembly structures which can be achieved in such systems are also limited to lamellar phases, hexagonally packed cylinders, body-centered cubic spheres, and bicontinuous gyroid phases.8,10 More recently, the desire to achieve complex morphologies and exert finer control over the parameter space of selfassembly has motivated developments in synthesis techniques by which it has now become possible to tune both the chain architecture as well as the sequence chemistry of copolymers.11−21 As a consequence, synthesis of newer classes of copolymers, such as multiblock copolymers, gradient copolymers, comb copolymers, star copolymers etc. have been achieved.17,19,22−24 In this article, we are specifically concerned with the classes of copolymers termed gradient and tapered copolymers, which are essentially random copolymers exhibit© 2012 American Chemical Society

ing a gradual change in monomer composition along the length of the polymer.25,26 Advances in living free radical polymerization techniques and ring-opening metathesis polymerization have allowed exquisite control over the sequence distributions, and has thereby facilitated the synthesis of arbitrary compositional profiles for the gradient copolymers (cf. Figure 1). Not surprisingly, the ability to tune the composition profiles of such copolymers has led to significant interest in applications of such classes of polymers as interfacial modifiers,27−29 reinforcement agents,30 damping materials,31 thermoplastic elastomers, etc.32,33 In the past decade, there have been a number of theoretical studies of the thermodynamic behavior and properties of gradient copolymer systems. Pakula and Matyjaszewski34 studied the self-assembly of diblock, gradient, and random copolymers and identified the morphological characteristics and order−disorder transition temperatures in such systems. Aksimentiev and Holyst35 studied the phase behavior of gradient copolymer melts for different composition profiles within the Landau−Ginzburg model. More recently, Shull and co-workers,36,37 Shi and co-workers,38 and Lipson, Milner, and co-workers,39 used numerical self-consistent-field theory (SCFT) to examine the microphase separation of gradient copolymers. Shull and co-workers have also used the framework of SCFT to describe the equilibrium interfacial behavior of gradient copolymers in immiscible polymer blends.40,41 Together, the preceding studies have led to a Received: June 4, 2012 Published: July 24, 2012 6281

dx.doi.org/10.1021/ma301136y | Macromolecules 2012, 45, 6281−6297

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Fredrickson and co-workers48 and other researchers49−51 have examined the phase behavior of random copolymer melts by using the quenched sequence model, and have predicted that the blockiness of the sequences can have a significant impact upon their self-assembly and phase behavior. Subsequently, Balazs and co-workers,52,53 and Dadmum and co-workers54−56 have examined the interfacial properties of random copolymers and have suggested that sequence correlations and polydispersity effects can play a significant role in influencing their interfacial properties. In our own earlier work,57−60 we have demonstrated that the use of quenched ensemble of sequences for grafted random copolymers and gradient copolymer solutions can lead to novel phenomena which cannot in general be captured by a single-sequence annealed representation of the system. In a slightly different context of diblock copolymers, the influence of both molecular weight and compositional polydispersity has attracted considerable attention in recent years.61−67 In such situations, both theoretical studies and experimental results have shown that both molecular weight and compositional polydispersity can have a significant influence upon the self-assembly characteristics of diblock copolymer melts. While the preceding studies have shed important insights into the possible richness which may emerge when compositional polydispersity and sequence correlations are included, we are not aware of any comparable study in the context of gradient and tapered copolymer systems which examine similar issues. The above considerations motivate the fundamental questions we seek to address in this article: • What is the influence of the sequence and compositional polydispersities of gradient copolymers upon their thermodynamical behavior? • What is the influence of the blockiness (i.e., the sizes of continuous sections of the chain which comprise of the same monomers) of the sequences upon the selfassembly behavior and interfacial properties of gradient copolymer melts? A resolution of the above issues will allow for a more realistic comparison between experimental results and theoretical predictions while accounting for the influence of different synthesis conditions.45−47,68,69 Moreover, an understanding developed through this study might also shed light on the use of parameters such as the blockiness of the copolymers and the sequence polydispersities resulting from synthesis conditions, as means to tune the properties and self-assembly of gradient copolymers.45−47 The rest of this article is arranged as follows. In section II, we present details of our modeling and simulation techniques. We begin by discussing the methodology used to generate the ensemble of designed sequences with specified blockiness. Subsequently, we discuss the combination of approaches we adopt to address the problem of predicting the phase behavior and interfacial properties of quenched sequence of copolymers. In section III, we present our results for the self-assembly behaviors of both annealed and quenched ensemble systems. Because of computational limitations, we restrict our consideration to only a two-dimensional physical space and present the self-assembly morphologies resulting with such symmetries. Subsequently, we discuss our results for the influence of blockiness upon the interfacial properties of gradient copolymers. We conclude with a short summary of the results and an outlook for future work.

Figure 1. Schematic depiction of some of the composition profiles of copolymers considered in the present article. f(i) denotes the average volume fraction of A monomers at a specific location i in the polymer chain. i denotes the segment index (normalized by the total number of segments). Key: (a) diblock copolymer; (b) tapered copolymer; (c) a linear gradient copolymer; (d) a hyperbolic tangent gradient copolymer with a weak gradient (cf. Equation 5 with C = 1); (e) A hyperbolic tangent gradient copolymer with a strong gradient (cf. Equation 5 with C = 5). Note that the tapered copolymers (b) exhibit a similar profile as hyperbolic tangent copolymers with a strong gradient (e). Also, a linear gradient copolymer (c) is seen to be similar to a hyperbolic tangent gradient copolymer with a weak gradient (d).

number fundamental insights into the connection between the composition profiles of the gradient copolymers and their thermodynamic behavior. Despite the understanding which has resulted from the above-mentioned studies, much of the prior theoretical models35,36,39,40,42,43 have used technically what is termed as the “annealed” representation of the gradient copolymer (albeit, some exceptions do exist34). In such a model, all the copolymers are assumed to possess the same sequence, with, however, the identity and the interactions of each monomer in the sequence chosen to be representative of the “averaged” composition corresponding to the location of the monomer in the polymer chain. Slightly more refined versions of this annealed approach have used a “multiblock” copolymer model,38 but still within the framework of a single sequence representation of the gradient copolymer. However, a more realistic model for such gradient copolymer systems, is to use an ensemble of “quenched” sequences, whose compositions on an (ensemble) average mimic the experimentally synthesized composition profiles of the gradient copolymers. Indeed, synthesis strategies often rely on controlling the feed monomer ratios of the reacting comonomers to design gradient copolymers.44 However, such a setup does not ensure that the sequence of every copolymer so-synthesized has the same composition profile. For instance, in a recent work, Broadbelt and co-workers,45−47 used a detailed kinetic model of the polymerization process to demonstrate that even for the same specified overall composition profile of the gradient copolymer, the comonomer feed ratios of the reaction process can be used to significantly influence the sequences of the individual chains and the distribution of the segment sizes (i.e., the blockiness) in the gradient copolymers. We do note that there have been some theoretical works which have studied issues related to compositional polydispersity, quenched randomness etc. in other contexts. For instance, 6282


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the prescribed composition profile for a specified λ (the algorithm seems to be most effective for λ ≲ 0.8), if the average composition of A segments specified in the algorithm is adjusted iteratively as a parameter to match the required average composition f ̅ of A segments. To illustrate the mechanics of such a procedure, we choose as an example the target sequence of a linear gradient copolymer with a composition profile fexpt of the form (for f ̅ ≤ 0.5):

II. DESCRIPTION OF THE THEORETICAL METHODOLOGIES In this section, we provide a description of the model and the different frameworks used to address the issues motivated in the introduction. We divide our discussion into three parts: (i) the methodology which was used to generate the ensemble of sequences and the characteristics of sequences so-generated; (ii) the approach used to deduce the self-assembly behavior of the gradient copolymer systems; and (iii) the framework used to deduce the interfacial properties of the system. A. Polydisperse Sequences of Gradient Copolymers. 1. Algorithm. To pursue the issues outlined in the introduction we require a procedure for designing an ensemble of sequences of gradient copolymers with specified sequence correlations (i.e., blockiness). In the context of random copolymers of uniform composition, Fredrickson and co-workers48 presented a model which mimicked the reaction processes involved in the synthesis of such copolymers. In such a model, the different sequences were constructed through a stochastic algorithm as the outcome of a Markov time series. The input to such an algorithm was the probability that the next monomer was A (or B) as a function of the identity of the present monomer. Specifically, if pAB denotes the probability of a B monomer following A and pBA denotes the probability that an A monomer follows B (and pAA = 1 − pAB, pBB = 1 − pBA denoting respectively the probabilities that a A monomer follows A and a B monomer follows B), Fredrickson and co-workers showed that using:48 pAA = f (1 − λ) + λ ; pBB = f (λ − 1) + 1

fexpt (i) = 2f ̅

i N

(3)

where 0 ≤ i ≤ N, denotes the index of the segment i along the chain, where f ̅ denotes the overall average composition of A monomers in the chain. To generate the above composition profile for a specified blockiness parameter λ, we use the algorithm of eq 2 with f(i) however replaced by falg (i) = 2f ̅ *

i N

(4)

In the above, we treat f ̅* as a parameter which we adjust iteratively in such a manner that the overall average composition of the generated ensemble of sequences matches the required f ̅ (for small λ, our results indicate f ̅* ≃ f ̅). Using such a procedure, we found that the average composition of the sequences so-generated are in excellent agreement with eq 3 (see next section for illustrative results). The above algorithm, while not necessarily a realistic representation of the reaction scheme involved in the synthesis of gradient copolymers, is nevertheless representative of the continuous change in feed rates (as reflected through f(i) in the probabilities pAB, pBA etc.). Moreover, in the following section, we demonstrate that by using the parameters λ and the gradient strengths, we can generate ensembles of sequences with the same average composition profile but differing in their blockiness. Furthermore, in section II.B, we examine the sequence correlations and demonstrate that λ (which is independent of the monomer location i) still serves to characterize the correlations in the sequences generated through the algorithm. 2. Characteristics of Generated Sequences. In this section, we demonstrate that the algorithm proposed in the preceding section is indeed successful in generating sequences with prescribed compositional profiles. Subsequently, we characterize the influence of λ and the gradient strength on the blockiness and the compositional polydispersity of the generated sequences. To demonstrate that the above procedure is effective in generating the specified composition profiles, we consider two different classes of composition profiles (the following formulas are valid only for f ̅ ≤ 0.5): (i) The linear gradient composition profile specified by eq 3; (ii) the hyperbolic tangent gradient composition profile specified by35

(1)

along with the initial condition that the probability that the first monomer is A is equal to f, generates an ensemble of sequences with an average composition f which is uniform along the chain, and an average block size of segments (−ln λ)−1 (0 < λ < 1). The above procedure furnishes a scheme to generate random copolymers of arbitrary uniform composition and blockiness. To generalize the above methodology to gradient copolymers, we require a means to generate an ensemble of sequences with a prescribed (ensemble-averaged) nonuniform average composition f(i) at a monomer (segment) location i (cf. Figure 1). A number of options exist for accommodating such a generalization. For instance, the Markov time series analysis has itself been generalized to inhomogeneous situations to accommodate a time dependence of the probabilities (in our case, time is equivalent to the segment identity i along the chain).70 Alternatively, one may follow the procedure proposed by Broadbelt and co-workers and develop a detailed sequence-level model of the reaction processes involved in the synthesis of the gradient copolymers.45−47 In this work, we eschew such rigorous, albeit involved approaches, and instead adopt a simple generalization of the scheme proposed by Fredrickson et al.48 by rendering the probabilities a function of the monomer position (along the chain) using:

f (i ) =

pAA (i) = f (i)(1 − λ) + λ ; pBB (i) = f (i)(λ − 1) + 1 (2)

⎡ ⎛i ⎞⎤⎫ 1⎧ ⎨1 + tanh⎢Cπ ⎜ − f ̂ *⎟⎥⎬ ⎠⎦⎭ ⎣ ⎝N 2⎩

(5)

where C is a parameter which determines the gradient strength of the copolymer (see Figures 1d and e), and f*̂ is a parameter chosen such that:

We note that such a generalization breaks the Markov chain nature of the process and hence the ensemble average is not guaranteed for arbitrary λ to generate the prespecified composition profile f(i). Despite this limitation, we found that the above algorithm can indeed be used to generate an ensemble of chains whose average compositions closely mimic

1 N 6283

∫0

N

dif (i) = f ̅

(6)


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Figure 2. The points represent the average compositions of copolymer sequences generated using our algorithm. The solid lines depict the “target” composition profiles. f(i) denotes the average volume fraction of A monomers at a specific location i in the polymer chain, where i denotes the segment index. The total number of segments were fixed to N = 100. (a) Linear gradient copolymer with a target composition profile corresponding to eq 3. (b) Target composition profiles corresponding to hyperbolic tangent gradient copolymers (eq 5) with varying gradient strengths.

Figure 3. Block size distribution ϕb(n) as a function of the number of segments in the block (normalized by the total number of segments N) for different λ: (a) Weak gradient copolymers with a hyperbolic tangent composition profile (eq 5) with C = 1 and f ̅ = 0.5; (b) Strong gradient copolymers with a hyperbolic tangent composition profile (eq 5) with C = 5 and f ̅ = 0.5.

stronger gradient copolymers possess an inherent blockiness arising from the strong gradient in the composition profile. Indeed, the extreme situation of a strong gradient copolymer is the diblock copolymer, which possess two blocks of sizes N/2 each. For the case of a strong gradient copolymer, the latter is seen to manifest as a peak in the distribution at finite block sizes, which is seen to be present even for λ = 0 in Figure 3b. In such polymers, additional blockiness arising from the stochastic nature of the generating algorithm has only a minor impact upon the distribution of blockiness. In contrast, weaker gradient copolymers lack such an inherent blockiness (as seen from the distribution of blockiness for λ = 0.0 in Figure 3a), and hence the blockiness arising from the stochastic nature of the sequences is a more significant factor. Compositional Polydispersity. To quantify the compositional polydispersity of the generated sequences, we adopt the following notation: We assign an index γ(i) to denote the identity of a monomer i on chain k, such that

In Figure 2, we display the average composition profiles for a set of 1000 chains (which corresponds to the “ensemble” we chose for the studies outlined in the later sections) for a few different f ̅ and blockiness parameters λ. It is seen that in all the cases our algorithm is able to reproduce the desired composition profiles. Blockiness of Sequences. To quantify the blockiness of the sequences generated by the above algorithm, we consider a function we term as the block size distribution, ϕb(n), representing the fraction of segments in the chain which are present in a block of (identical) segments of size n. In Figures 3a and b, we display ϕb(n) as a function of λ for two different gradient strengths (weak and strong cases) of the copolymer. We observe that for both cases, an increase in the λ parameter increases the blockiness of the sequences. The latter is seen as a reduction in the fraction of segments which are present in small blocks and the corresponding shift of the distribution to larger block sizes. However, in comparing the results for the different gradient strengths, we observe that the influence of λ is much more pronounced for the sequences with weaker gradient strengths. The latter is easily rationalized by noting that

⎧ 1 if i = A γ (k)(i) = ⎨ ⎩ 0 if i = B 6284

(7)


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Figure 4. (a) Dependence of the variance σ(i) in the average composition as a function of the location i (normalized by the total number of segments N) along the polymer chain for tangent hyperbolic composition profiles with different gradient strengths (C in eq 5) and λ parameters; (b) Probability distribution of nAcom − nBcom (denoted P(nAcom − nBcom)) as a function of the number of segments n (normalized by the total number of segments in the chain N) for different gradient strengths and λ parameters; (c) Average number of segments n̅ in the distribution of P(nAcom − nBcom) as a function of λ for different gradient strengths; (d) Variance of the distribution P(nAcom − nBcom) as a function of λ for different gradient strengths. All results correspond to ensembles with f ̅ = 0.5 and N = 100.

correlated to compositional polydispersity effects at the scales of segregation of the copolymers. 61,62,67 For weak to intermediate segregations, the latter are expected to be much larger than the segment lengths, and hence compositional polydispersity at the scale of segments are not likely to capture the complete influence of blockiness and gradient strengths upon the compositional polydispersity features responsible for self-assembly. To provide a more quantitative measure of the compositional polydispersity at larger scales we consider the distribution of the compositional center of masses of A and B segments. Explicitly, we find the compositional center of mass of A and B segments in a chain, denoted as nAcom and nBcom respectively, as follows:

We have

⟨γ (k)(i)⟩ = f (i)

(8)

where ⟨···⟩ denotes an average over the ensemble of chains. A natural measure of the compositional polydispersity would then be to use the variance in the average fraction of A segments (across the ensemble of chains) at the segment location i: σ(i) = ⟨(γ(i) − f (i))2 ⟩

(9)

In Figure 4a, we display σ(i) determined for our generated sequences, in which we observe that the gradient strength has a significant influence upon the compositional polydispersity. Weak gradient strengths are seen to possess the most compositionally polydisperse sequences, with the compositional polydispersity spanning the entire polymer, whereas, in stronger gradient strengths polydispersity effects are seen to confined to a small zone around the midlocation of the polymer chain. In Figure 4a, we observe that λ has very little influence upon σ(i), which may lead one to conclude that the compositional polydispersity is independent of λ. However, while σ(i) does provide a measure of compositional polydispersity among the chains, we argue that the self-assembly features of gradient copolymers are not strongly tied to σ(i). Indeed, σ(i) quantifies the compositional polydispersity at the scale of the segments, whereas, the self-assembly features are expected to be

N

A = ncom

∑i = 1 γ(i)i N

∑i = 1 γ(i) N

B ncom =

∑i = 1 (1 − γ(i))i N

∑i = 1 (1 − γ(i))

(10)

We note that for a system of monodisperse symmetric diblock copolymers, the distributions of nAcom and nBcom equal a delta function at the locations N/4 and 3N/4 respectively. Polydispersity in compositions is expected to broaden the distributions of nAcom and nBcom. Hence, a useful measure of the 6285


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of alternative approaches to discern the phase behavior of the quenched ensemble of gradient copolymers. As a first step, we use random phase approximation (RPA) to predict the spinodals of the self-assembly morphologies.76,77 As, we demonstrate below, it is possible to deduce an analytical expression for the correlations in the sequences generated through our algorithm, and such an expression allows RPA to be effected with quantitative accuracy. However, while RPA furnishes the spinodals for our model, it lacks in its ability to identify the morphology of self-assembly resulting beyond the spinodal. To address the latter issue, we use a method termed the self-consistent Brownian dynamics (SCBD) which is wellsuited for identifying the self-assembly morphologies.78−82 However the latter is not capable of obtaining the free energies of the morphologies, and hence cannot delineate the equilibrium characteristics with quantitative accuracy. Hence, we mainly use the SCBD approach as a tool to deduce the qualitative differences in the self-assembly morphologies between the ensemble approach and the single sequence model. 1. Random Phase Approximation for Spinodals. The random phase approximation has emerged as a popular approach for predicting the structure factor of polymer solutions and melts in the disordered phase and for deducing the spinodals of phase behavior in multicomponent polymers. Since most of the details of the RPA formalism have been described in detail in many other articles,38,61,76,83,84 we restrict our discussion below to only the final equations accompanying this formalism. We consider an ensemble of M copolymer chains each containing N segments as constituting our incompressible melt system and use the notation γ(k)(i)(k = 1···M, i = 1···N) to denote the sequence of the chains. The RPA formalism is based on the following formula for structure factor S(q) of the system:38,61,76,83,84

polydispersity and the driving force for segregation is the probability distribution of nAcom − nBcom, which we denote as P(nAcom − nBcom). In Figure 4b, we display the probability distributions P(nAcom − nBcom) for our ensemble of chains for different gradient strengths and blockiness parameters, and in Figure 4, parts c and d, we display the corresponding average and the spread in the distributions of nAcom − nBcom for different gradient strengths and blockiness parameters. Overall, we observe that the average of the distributions increase with increasing gradient strength and approaches the value expected for a monodisperse diblock copolymer. This suggests that the “effective block size” driving segregation is smaller for weaker gradient strengths. Moreover, it is seen that the spread in the distribution (Figure 4d) decreases with increasing gradient strength, suggesting that the compositional polydispersity effects are much more pronounced at weaker gradient strengths. More pertinently, it is seen that increasing the blockiness parameter λ has no perceptible effect on the average block sizes, but leads to a pronounced increase in the spread of the distribution especially for weaker gradient strength copolymers. To summarize, based on the results of Figures 3 and 4 we infer the following results. (i) Blockiness: Weak gradient copolymers do not possess an inherent blockiness, and changes in the parameter λ significantly broadens the distribution of blockiness and the sequences become more weighted in larger blocks. In contrast, strong gradient copolymers possess an inherent blockiness arising from the gradient strength. While the changes in the parameter λ broadens the distribution of blockiness, the effects are considerably less pronounced relative to the case of weaker gradient strengths. (ii) Compositional polydispersity of the chains was shown to increase with decreasing gradient strengths and/or increasing λ. However, for stronger gradient strengths, the compositional polydispersity of chains were shown to be small in magnitude even when λ was large. B. Phase Behavior of Gradient Copolymers. In the recent years, polymer SCFT has emerged as a powerful technique to examine the self-assembly behavior of multicomponent polymers.71−75 Not surprisingly, a number of prior studies have used SCFT to predict the self-assembly behavior of gradient copolymers within the framework of annealed sequence model.36,38,39 However, implementing such a framework to address the self-assembly of quenched ensemble of sequences proves computationally intensive. Specifically, the numerical iterative procedure accompanying SCFT requires the solution of a “diffusion-like” equation for every prescribed sequence of the chain. While powerful numerical schemes have emerged to facilitate the solution of the diffusion equation with high accuracy, implementation of such schemes for an ensemble of different sequences (at every step of the SCFT iteration) proves to be a significant computational undertaking in two and three physical space dimensions. Moreover, to achieve an accurate evaluation of the equilibrium morphologies we need to implement such a procedure for different cell sizes and shapes, and determine the cell size and shape corresponding to the optimal free energies, which also adds to the computational cost of the procedure. In this work, since we are primarily concerned with the impact (at a qualitative level) of features such as compositional polydispersity and blockiness upon the self-assembly morphologies, we avoid the numerical rigor and the associated computational cost of SCFT, and instead pursue a combination

S(q)−1 = (SAA(q) + 2SAB(q) + SBB(q)) /(SAA(q)SBB(q) − SAB 2(q)) − 2χ

(11)

In the above SAA(q) =

1 M

M

∑ k=1

1 N

N

N

∫0 ∫0

dj di γ (k)(i)γ (k)(j)

⎛ q 2 b2 ⎞ exp⎜ − | i − j| ⎟ ⎝ 6 ⎠

SBB(q) =

1 M

M

∑ k=1

1 N

N

∫0 ∫0

(12) N

dj di [1 − γ (k)(i)]

⎛ q 2 b2 ⎞ [1 − γ (k)(j)] exp⎜ − | i − j| ⎟ ⎝ 6 ⎠

(13)

and SAB(q) =

1 M

M

∑ k=1

1 N

N

∫0 ∫0

⎛ q 2 b2 ⎞ exp⎜ − | i − j| ⎟ ⎝ 6 ⎠

N

dj di γ (k)(i)[1 − γ (k)(j)]

(14)

Using the formula eq 11, the spinodals can be deduced as the critical χ for which S(q*)−1 = 0, where q* denotes the 6286


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Figure 5. (a) Representative results for P1(i,i′) = [⟨γ(i)γ(i′)⟩ − f(i)f(i′)]/[f(i)(1 − f(i))] for i > i′ as a function of i − i′ (blue points). The sequences used correspond to a hyperbolic tangent composition profile with C = 1.0, f = 0.5, λ = 0.5. Also shown by a dotted line is a best fit of the form P1(i,i′) = exp[α(i − i′)]. (b) Representative results for P2(i,i′) = [⟨γ(i)γ(i′)⟩ − f(i)f(i′)]/[f(i′)(1 − f(i′))] for i′ > i as a function of i − i′ (blue points). The results correspond to a hyperbolic tangent composition profile with C = 1.0,f = 0.5,λ = 0.5. Also shown by a dotted line is a best fit of the form P2(i,i′) = exp[α(i′ − i)]. (c) The exponents α deduced for P1(i,i′) as a function of λ. Also shown is a comparison (dotted line) with ln λ. (d) The exponents α deduced for P2(i,i′) as a function of λ. Also shown is a comparison (dotted line) with ln λ. The total number of segments in the chain was fixed to N = 100.

wavevector for which S(q) is maximum. To enable analytical calculations of the RPA formalism, we use 1 M

⟨(γ(i) − f (i))(γ(i′) − f (i′))⟩ ⎧ for i ≤ i′ ⎪ f (i)(1 − f (i)) exp[(i − i′) ln λ], ≃⎨ ⎪ ⎩ f (i′)(1 − f (i′)) exp[(i′ − i) ln λ], for i′ ≤ i

M

∑ γ (k)(i)γ (k)(j) ≃ ⟨γ(i)γ(j)⟩ (15)

(18)

where i and j on the right-hand side of eq 15 are restricted to be monomers on the same chain and ⟨···⟩ denotes an average over the ensemble of chains. To make further progress in the implementation of the RPA formalism, we require an analytical expression for ⟨γ(i)γ(i′)⟩ in eq 15. For the model of annealed sequences, we have

where f(i) and f(i′) denote the average local inhomogeneous composition at location i. Equations 11−15, supplemented with the analytical expression eq 18 provides a closed set of equations for performing RPA calculations on the ensemble of gradient copolymer chains. Although not strictly pertinent to the RPA formalism, we briefly mention a few corollaries which can be deduced from eq 18. Similar to the case of random copolymers, (−ln λ)−1 is seen to represent the correlation length of the monomer sequences generated in our algorithm. As a consequence, in the limit

k=1

⟨γ(i)γ(i′)⟩ = γ(i)γ(i′) = f (i)f (i′)

(16)

Moreover, for the model of simple random copolymers generated by the algorithm of eq 1, Fredrickson et al.48 showed that 2

⟨γ(i)γ(i′)⟩ = f + f (1 − f ) exp(|i − i′| ln λ)

( −ln λ)−1 ≪ N

(17)

(19)

the block sizes of the sequences are much smaller than the overall chain length. In contrast, when − N ln λ ≲ O(1), we expect that the blockiness of the sequences relative to the chain length to be a relevant parameter. In other words, the nondimensional “blockiness” parameter δ  |N ln λ|−1 quantifies the importance of blockiness arising from the stochastic nature of the sequences. We also note that the strong gradient copolymers possess an inherent blockiness even for λ = 0 (Figure 3b). Hence, the above considerations are to be understood to pertain to the blockiness as induced by the

Unfortunately, due to the non-Markovian nature of the algorithm used to generate our sequences, it is not possible to rigorously derive a relationship similar to eq 17 for our sequences. However, based on an empirical probing of the correlations for a variety of gradient strengths and average compositions in our chain, we were able to deduce that a simple generalization of the eq 17 holds for the sequences generated through our algorithm (see Figures 5a−d): 6287


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Figure 6. (a and b) Spinodals (χN)s as a function of the average composition of A segments f ̅ calculated from RPA for gradient copolymers with a tangent hyperbolic composition profiles (eq 5) for different values of blockiness parameter δ  |N ln λ|−1. SC denotes the case of single “annealed sequence” model (cf. Equation 16). (a) C = 1; (b) C = 20. (c) Critical wavevector of instability (for f ̅ = 0.5). (d) Order−disorder transition temperatures as a function of δ for gradient copolymers with different gradient strengths.

determined using the statistics of the ensemble of the polymers simulated by the Brownian dynamics method.79 It is to be noted that the SCBD approach, due to its Brownian dynamics nature, by itself is not capable of furnishing the free energies of the systems. However, by restricting our considerations to equilibrium, the SCBD methodology becomes similar in spirit to the recently pursued “theoretically informed coarse-grained simulations” and “single-chain in mean field” approaches.86 The latter methodologies has been successfully adapted within Monte Carlo simulation schemes and is capable of furnishing free energies of the self-assembled phases. Indeed, such methodologies have been successfully adapted to study the equilibrium self-assembly of di- and multiblock copolymers, polymer blends, and random copolymers.51,87−89 In our earlier studies, we proposed and demonstrated SCBD as a useful tool to study nonequilibrium phenomena involving the coupling between flow and self-assembly in block copolymer systems.79 More recently, we applied the methodology of self-consistent Brownian dynamics simulations to study the self-assembly behavior in melts of semiflexible-flexible diblock copolymers as a function of the persistence length of the semiflexible block.82 In view of the natural “ensemble” of chains which are used in the Brownian dynamics simulations, it is evident that SCBD is especially convenient for studying a system of chains of specified polydispersity (in molecular weights and/or chemical compositions). On the basis of this idea, in this work, we propose SCBD as a methodology to deduce the equilibrium morphologies of gradient copolymer systems. However, since our main intent is to clarify the role of

stochastic nature of the sequence generating algorithm and not to the inherent blockiness (if any) of the gradient copolymers. 2. Self-Consistent Brownian Dynamics. To deduce the selfassembly morphologies of the gradient copolymer melts, we use the approach of self-consistent Brownian dynamics (SCBD) simulations,78−82 a methodology which combines Brownian dynamics simulations and polymer SCFT. To maintain brevity, we relegate the model equations and numerical details to the Appendix (Section A) and only review the most salient features of the model in this section. In brief, SCBD is a Brownian dynamics simulation approach which combines the framework of polymer SCFT within a Brownian dynamics algorithm.78 Polymer SCFT enumerates the statistical features of an interacting system of polymer chains by considering an equivalent system of noninteracting chains in the presence of pseudo chemical potential fields.85 The formalism of SCFT further specifies that these chemical potential fields are to be determined in a self-consistent manner so as to impose the inhomogeneous densities of the appropriate components. The SCBD approach is similar in spirit to Brownian dynamics algorithms in that it adopts a micromechanical model for the polymer chains and evolves them dynamically through a Langevin equation of motion. However, the SCFT approach is embedded within it by assuming that the chains are noninteracting and are instead acted upon by potential fields which account for the effects of intermolecular interactions. In SCFT, the functional form of the potential fields are determined self-consistently based on the inhomogeneous densities of the different species. In SCBD, the latter are 6288


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copolymers, increasing the blockiness parameter λ led to an enhanced compositional polydispersity (Figure 4d). In contrast, for stronger gradient copolymers, we demonstrated that the blockiness parameter has only a milder influence on the compositional polydispersity. Other works have shown (albeit, in the context of diblock copolymer melts) that increasing compositional polydispersity leads to a lowering of the critical interaction parameter and the wavevector for spinodal instability.61,62,67 The former has been argued to arise as a consequence of the segregation driven by the presence of the larger blocks of A and B segments, whereas the latter is a consequence of the reduced stretching energy accompanying polydisperse systems. It is evident that the results of Figure 6a− c when viewed in the context of compositional polydispersity are consistent with such expectations. In sum, the results presented in this section highlighted the interplay between gradient strengths and the blockiness parameter in influencing the spinodals and the critical wavevectors of instability. In all cases, the results for limit δ → 0 were seen to be consistent with the corresponding results for the single annealed sequence model. However, for δ ≠ 0, significant differences were seen between the ensemble model and the single sequence model, especially for weaker gradient copolymers. B. Self-Assembly of Gradient Copolymers. In the previous section we compared the spinodals of gradient copolymers for different gradient strengths and blockiness parameter λ. In this section, we complement the results of the previous section by presenting the self-assembly morphologies of such copolymers as a function of the gradient strengths and blockiness parameter. We also compare the results of the ensemble approach with the single chain, annealed sequence approach. We use the SCBD approach to deduce the self-assembly morphologies of the gradient copolymer systems. We note that the morphologies resulting in such simulations are likely to be influenced by the cell dimensions and metastable intermediate states. As discussed earlier, since SCBD is not capable of yielding free energies, it is not possible to deduce the morphologies which represent the true global equilibrium in free energies. As a consequence, we present results of the selfassembly behavior as deduced only by a visual inspection of the morphologies resulting in the SCBD simulations (initiated from different initial conditions). As we demonstrate below, the “phase-diagram” as deduced through such a procedure is in excellent agreement with the spinodals obtained through the RPA theory, which lends credence to the simulation methodology and the analysis we used. As mentioned in section II.B, our simulations are restricted to phase morphologies possessing 2-D symmetries. As a consequence, for the gradient copolymeric systems, the only possible morphologies (in addition to the disordered phase) are the lamellar morphologies, which corresponds to a layered arrangement of the A and B segments (Figure 7a) and the cylindrical phases (Figure 7b). In our simulations, however, we noted a number of instances in which the morphologies were a mixture of lamellar and cylindrical phases (even when the simulations were initiated from different initial conditions) (Figure 7c). Such phases could be a manifestation of the coexistence between the cylindrical and lamellar phases or alternatively be an intermediate state (such as the perforated lamellar phase) in the transition from an initially metastable phase to the final equilibrium morphology. Moreover, we note

compositional sequence inhomogeneities, we restrict our calculations only to morphologies which possess a twodimensional physical space symmetry (despite the fact that the chains can themselves move in 3-D physical space). Further details of the SCFT formalism and the SCBD methodology are furnished in the Appendix. C. Interfacial Properties of Gradient Copolymers. Gradient copolymers have been widely proposed in applications as interfacial modifiers,27−29 and hence it is of natural interest to seek the impact of compositional polydispersity and blockiness upon their interfacial properties. In the present work we do not embark on an exhaustive investigation of the interfacial properties of the gradient copolymers, such as considerations of interfacial tensions and other elastic modulii. Instead, we present a few results which illustrate the influence of compositional polydispersity and blockiness on the interfacial tension reduction of polymer blend systems achieved by gradient copolymers. The results of our analysis sheds light on the physics that emerges in comparing different gradient strengths and sequence distributions. Additionally, the insights gained through our analysis also addresses whether compositional polydispersity, an experimentally controllable parameter, be used to tune the interfacial activity of the gradient copolymers. Since calculations of interfacial tension requires consideration of free energies of the system, we cannot adapt the SCBD method to obtain such results. However, since such calculations require only a one-dimensional physical space framework, we use the SCFT approach itself to address this problem. The mathematical details of the framework we employ is very similar to that used by Matsen to delineate the interfacial properties of diblock copolymers.90−93 In the Appendix, we outline very briefly the manner by which the SCFT formalism of Matsen90 can be extended to obtain the interfacial tension of gradient copolymer systems.

III. RESULTS AND DISCUSSION A. Spinodals From RPA Analysis. In this section, we present the results for the spinodals calculated through the RPA formalism outlined in section II.B. Specifically, we choose the model of hyperbolic tangent gradient copolymers (eq 5) for which the gradient strength (represented by the C parameter in eq 5) can be independently varied. Using such a model we probed the influence of the nondimensional blockiness parameter δ|N ln λ|−1 and the gradient strength of the copolymer upon the spinodals of phase transition. Shown in Figures 6a and b are the results for the spinodals calculated from RPA for the cases of weak and strong gradients. In Figures 6c and 6d, we display the critical wavevectors and the critical interaction parameters (for f ̅ = 0.5) for different gradient strengths. Overall, we observe that for weaker gradient copolymers the parameter δ has a significant effect upon the spinodals, critical interactions and the associated wavevector of instability. Specifically, we observe that with increasing δ, the overall phase envelope and critical temperatures are shifted downward (i.e., toward lower χN), whereas the wavevector for the instability moves to smaller q*. In contrast, for the case of stronger gradients, we observe that the phase envelope and the q*s are only mildly affected by the blockiness parameter. The above influence of the blockiness parameter can be rationalized by recalling the analysis of sequence correlations and compositional polydispersities presented in Section II.A.2. In such a context, we demonstrated that for weaker gradient 6289


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Figure 7. Representative self-assembly morphologies in gradient copolymer systems. The cell sizes are 16 × 16 in Rg units. (a) f = 0.5, λ = 0, χN = 30; (b) f = 0.7, λ = 0.7, χN = 45; (c) f = 0.6, λ = 0.7, χN = 45.

that even in sophisticated simulation methods and annealing strategies, similar metastable configurations have been noted.61 Because of the lack of free energies, however, we cannot resolve the nature of these phases, and hence, we have just chosen to denote such morphologies as CL in the phase diagrams discussed below. Figures 8 and 9 display our SCBD results for self-assembly of gradient copolymers for different parametric conditions and composition profiles. Specifically, parts a and b of Figure 8 present the self-assembly morphologies for the case of linear gradient copolymers (an example of weak gradients) within the ensemble approach for two different parametric values of λ. In Figure 8c, we present the results for the self-assembly morphologies of the linear gradient copolymers when a single, annealed sequence model was employed. In Figure 9, parts a and b, we compare the ensemble approach and the single sequence model for the case of gradient copolymer systems with strong gradients. In all the cases, we also display the corresponding spinodals deduced from the RPA theory presented in section IIB. Overall, from the results presented in Figures 8 and 9, we observe that the spinodals predicted by the RPA are in remarkably good agreement with the phase boundaries deduced by our simulations. In quantitative agreement with RPA theory, we observe that increasing λ for the case of weaker gradient copolymers lowers the phase boundaries. In contrast, for the stronger gradient copolymers, the phase boundaries (and the spinodals) are seen to be unaffected from changes in λ. In comparing the results for the morphologies observed in parts a and b of Figure 8 with part c of Figure 8, we observe

Figure 8. Phase diagram of self-assembly morphologies in gradient copolymer systems. (a) Linear gradient copolymer in an ensemble approach, with λ = 0. (b) Linear gradient copolymer in an ensemble approach, with λ = 0.7. (c) Linear gradient copolymer in a single sequence approach. SP denotes the spinodals determined from the RPA formalism of section II.B. D, L, CL and C denote respectively the disordered, lamellar, cylinder−lamella mixture (see text), and cylindrical phases.

significant differences between the results of the single sequence and ensemble approaches, even for the case when λ = 0.0. For instance, we see the emergence of curved phases (cylinders and the C−L phases) for f ≲ 0.25 for λ = 0.0 and even at f = 0.4 for λ = 0.7. In contrast, the results for the single sequence model show only lamellar phases for all the χN and f we probed. In contrast, by comparing parts a and b of Figures 9, 6290


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Figure 9. Phase diagram of self-assembly morphologies in gradient copolymer systems: (a) Tangent hyperbolic composition profile (eq 5 with C = 5) in an ensemble approach, with λ = 0.7; (b) Tangent hyperbolic composition profile (eq 5 with C = 5) in a single sequence approach. SP denotes the spinodals determined from the RPA formalism of section II.B. D, L and C denote respectively the disordered, lamella and cylindrical phases.

Figure 10. Distribution of volume fractions of A segments in the sequences for hyperbolic tangent composition profiles with specified λ parameters: (a) f ̅ = 0.5; (b) f ̅ = 0.3.

to the stochastic manner in which the different sequences are generated, the average volume fractions of A (and B) segments in the individual chains are expected to differ in general from f ̅. To quantify this compositional asymmetry among the sequences, in Figure 10 we display the distribution of the average volume fractions of A segments as a function of λ and gradient strengths for two illustrative values of f ̅. Consistent with other quantities discussed in Section II.A.2, we observe that the distribution of compositions of the chains become broader with increasing λ and/or decreasing gradient strengths. To explain the trends noted in Figures 8 and 9, we observe in Figure 10 that for weaker gradient strengths, and especially at larger λ, the chains include a significant population of sequences which have compositions which are substantially different from the specified f ̅. Moreover, we recall our earlier discussion in the context of the results of Figure 3a, which also demonstrated that with increase in λ there is a corresponding increase in the population of chains containing large blocks of segments. We suggest that presence of such highly asymmetric copolymers

we observe that for gradient copolymers of strong gradient strengths, the single sequence approach is able to quantitatively capture the self-assembly morphologies of the ensemble approach, even when λ = 0.7. Other studies have considered the influence of compositional polydispersity on the self-assembly of diblock copolymers and have demonstrated that increasing compositional polydispersity leads to an enhanced prevalence of curved phases.61−63,66,67,94 While the results of Figures 8 and 9 are broadly consistent with such predictions, we note that in many such studies, the compositional polydispersity occurs in conjunction with polydispersity in molecular weights. As a consequence, it is not often easy to disentangle the specific effects of compositional polydispersity alone from such results. To rationalize the results of Figures 8 and 9 in the context of our model and the characteristics of our sequences, we probed the compositional polydispersity at the level of individual chains. Explicitly, while our sequences are generated such that the average volume fraction of A segments in the ensemble of chains is set as f ̅, due 6291


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Figure 11. (a−c) Interfacial tensions Σ (normalized as the Σ = (kBT)−1σb2N1/2, where σ is the dimensional interfacial tension and b denotes the Kuhn segment length) as a function of the bulk volume fractions ϕbulk of the copolymer: (a) weak gradient copolymer (C = 1 in eq 5); (b) strong gradient copolymer (C = 5 in eq 5). SC denotes the results of the single sequence model for the same specified composition profile. DB denotes the results for diblock copolymer. (c) Comparison of the results of C = 1 and C = 5 ensemble approach for λ = 0; (d, e) Surface coverages Ω (Ω = ωRg2, where ω is the number of polymer chains at the interface per unit area) as a function of the bulk volume fractions ϕbulk of the copolymer: (d) weak gradient copolymer (C = 1 in eq 5); (e) strong gradient copolymer (C = 5 in eq 5). SC denotes the results of the single sequence model for the same specified composition profile. DB denotes the results for diblock copolymer. (f) Data of (a, b) and (c, d) replotted as Σ as a function of Ω.

are seen to be insensitive to λ and very similar to the selfassembly features of the single chain model. In sum, the above results suggest that the interplay between compositional polydispersity and asymmetry and the overall blockiness of the sequences can play a significant role in determining the morphologies and the phase behavior of gradient copolymer systems. In systems wherein the inherent blockiness of the sequences are small, such as in weak gradient copolymers, the introduction of such polydispersity and blockiness effects leads to substantial changes in the self-

(containing large blocks of segments) in the ensemble of sequences leads to the formation of curved phases such as cylindrical morphologies and mixed phases of cylinders and lamellae (CL phases in Figure 8). In contrast, for the case of stronger gradient copolymers, we observe that the width of the distribution of compositional asymmetries are much smaller, and also that the blockiness of the segments are substantial (Figure 3b) even for λ = 0.0 (termed in section II.A.2 as the “inherent” blockiness) and was only mildly modified for larger λ. Hence, the phase behavior of stronger gradient copolymers 6292


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sities are smaller, and hence, there is a correspondingly higher tendency to segregate to the interface. Since the bulk concentrations are usually underestimated in SCFT due to the inaccurate treatment of micellization phenomena,40,90 in Figure 11f, we also display the changes in the interfacial tensions as a function of the surface coverage of the copolymer. In this representation, it is seen that the weaker gradient and/or the blockier copolymers possess higher interfacial activity. Indeed, such copolymers are more likely to possess larger distribution of blocks of different sizes (cf. Figure 3a), and hence can potentially form loops to bring the A and B segments in contact with the respective bulk phase and cause a larger reduction in the interfacial tension on a per chain basis.

assembly behavior. In contrast, in systems for which the inherent blockiness is already large, such as in strong gradient copolymers, the effects are seen to be much more mitigated. C. Interfacial Activity of Gradient Copolymers. In this section, we present results which quantify the changes in the interfacial tension of a symmetric A−B polymer blend arising from the addition of gradient copolymers. Again, our interests are centered on examining the influences of compositional polydispersity and blockiness upon the interfacial activity of gradient copolymers. By using the SCFT method (cf. Appendix), we are able to quantify such properties as a function of both the bulk concentration and the interfacial surface coverages of the copolymers. To maintain brevity of the article, we restrict our results to only the case of gradient copolymers possessing hyperbolic tangent composition profiles, with f ̅ = 0.5 (a symmetric gradient copolymer), and with the number of segments in the A, B components of the polymer blend set identically equal to the number of segments in the gradient copolymer. The results for the interfacial tensions of the polymer blend systems are displayed in parts a and b of Figure 11 as a function of the bulk concentration of the gradient copolymer for both weak and strong gradient strengths for different blockiness parameters λ. In this representation, it is seen that increasing the blockiness of the sequences leads to a reduction in the interfacial activity (i.e., the amount of lowering of the interfacial tension between the blend components brought about by the gradient copolymers at a given bulk concentration) of the gradient copolymers. Moreover, it is also evident that the diblock copolymers serve as more efficient compatiblizers when compared to gradient copolymers. As a result, it is also not surprising that the stronger gradient copolymers serve as better compatiblizers when compared to weaker gradient copolymers (cf. Figure 11c). Furthermore, it is seen in all cases that adopting the single sequence approach proves quantitatively inaccurate in predicting the interfacial activity of gradient copolymers. To rationalize the above trends, in Figure 11, parts d and e, we consider the behaviors of the surface coverages of the copolymers as a function of the bulk concentrations. For a specified bulk concentration of the copolymer, it is seen that the diblock copolymers have the highest surface coverage. Among the gradient copolymers, the blockier sequences and/or weaker gradient copolymers are seen to have lower surface coverage in comparison to the less blockier sequences and stronger gradient copolymers. From these results, it is evident that the reduced interfacial activity of the weaker and blockier gradient copolymers arise as a consequence of the enhanced solubility (and correspondingly reduced interfacial segregation) of the gradient copolymers in the bulk phases. Such a behavior can be explained based on the polydispersity in the volume fractions of A (and B) segments in the individual sequences. Indeed, in Figure 10 we demonstrated that weaker gradient copolymers and blockier sequences exhibit significant polydispersity in the volume fractions of A and B segments among the different sequences. Hence, in any given ensemble, it is likely to find a substantial fraction of copolymers which are enriched in A (or B) components. Such copolymers are more likely to be dissolve in the bulk phases of the respective blend component, thereby enhancing the bulk solubilities and reducing the tendency of the copolymer to segregate to the interface. In contrast, for stronger gradient copolymers (and at the extreme, diblock copolymers), compositional polydisper-

IV. SUMMARY In this work, we considered the influence of sequence polydispersity upon the phase behavior and interfacial characteristics of gradient copolymers. Specifically, by adapting the algorithmic procedure proposed for random copolymers, we were able to design sequences of varying blockiness and compositional polydispersities for specified composition profiles of the gradient copolymers. Using the sequences sogenerated, we studied the dependence of the spinodals, the phase behavior and interfacial properties as a function of gradient strengths and blockiness of the sequences. It was seen that the interplay between compositional polydispersity and the overall blockiness of the sequences plays a significant role in determining the morphologies, phase behavior and interfacial activity of gradient copolymer systems. In systems wherein the inherent blockiness of the sequences was small, such as in weak gradient copolymers, the introduction of such polydispersity and blockiness effects led to substantial changes in the selfassembly behavior and interfacial properties. In contrast, in systems for which the inherent blockiness was already large, such as in strong gradient copolymers, the influence of compositional polydispersity and sequence effects were seen to be much more mitigated. The results presented in the article highlight that comparing the theoretical predictions of the properties of gradient copolymers to experimental results should be done within the combined context of the gradient strengths of the copolymers and the blockiness of the sequences. From an alternative perspective, our results suggest that controlling the reaction conditions to influence the blockiness of the sequences (for the same prescribed average composition profile), can serve as an additional means to tune the phase behavior and interfacial activity of weaker gradient copolymers.

■

APPENDIX A: SCFT AND SCBD DETAILS In this section, we provide details of the simulation schemes and models employed in our work. Most of the simulation methodologies have been used extensively and elaborated in other contexts.36,38,40,78−82,90 Hence we restrict our discussion to only the most pertinent details and the modifications needed to adapt such methodologies to the system of gradient copolymers. 1. SCFT Framework for Gradient Copolymers

We consider a mixture of M gradient copolymer molecules inside a volume V. Each molecule is assumed to consist of a total of N segments. The gradient copolymer is modeled as a flexible Gaussian chain with a statistical segment length b and segmental volume ρ−1 0 . Conformations of the polymers are 6293


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represented by the function Rk(s), where k = 1···M indexes the different polymers and s is a variable running from 0 to N. On the basis of these notations, the local microscopic densities of A and B monomers, denoted respectively as ρ̂A(r) and ρ̂B(r) can be identified as:73,75 M

ρÂ (r) =

∑∫

N

Qk =

(k)

ds δ(r − R k(s))γ (s)

0

k=1

in which the individual chain partition functions Qk are given by

ρB̂ (r) =

∑∫

N

ds δ(r − R k(s))[1 − γ (k)(s)]

0

k=1

where γ (s) denotes the identity of the monomer s on chain k (eq 7). For the annealed sequence model, we use: (A3)

where f(s) denotes the average composition of A monomers at the location s in the chain. The bonded interactions in the Gaussian thread model are represented by an elastic interaction potential given by75,95 U0[R k(s)] =

3kBT 2b2

M

∑∫ k=1

0

N

⎛ d R (s ) ⎞ 2 ds ⎜ k ⎟ ⎝ ds ⎠

(A4)

Non-bonded intermolecular interactions are described by a standard quadratic form of microscopic monomer densities:75 k Tχ U1 = B ρo

∫

dr ρÂ ρB̂

ϕA (r) =

(A6)

By following the steps outlined in Fredrickson et al.73,75 we convert the above model involving conformational path integrals of the chains into a field theory where the fundamental degrees of freedom are fluctuating chemical potential fields. In such a framework the partition function can be expressed as, Z=

∫ D[w] ∫ D[π ] exp(−H[w , π ])

(A12)

1 M

M

∑ k=1

V Qk

∫0

1

ds qk (r, s)qk†(r, 1 − s)γ (k)(s)

and ϕB(r) =

1 M

M

∑ k=1

V Qk

∫0

1

ds qk (r, s)qk†(r, 1 − s)

[1 − γ (k)(s)]

(A7)

(A14)

In the above, the function qk(r, s) satisfies the diffusion equation and initial condition

∫

∂ q (r, s) = ∇2 qk (r , s) + [γ (k)(s)[π (r) + w(r)] ∂s k

(A8)

+ [π (r) − w(r)][1 − γ (k)(s)]]qk (r, s)

where M

Q=

(A10)

(A13)

where, the Hamiltonian H[w, π] can be expressed as, −ρo ρ H[w , π ] = d r w 2(r ) − o dr π (r ) + ln Q 2 N χN

∫

ds [π (R(s)) + w(R(s))]γ (k)(s)

where, ϕj(r)(j = A,B) denotes the average individual (microscopic) compositions and is defined as ⟨ρ̂j⟩/ρ0, where ⟨···⟩ is an average over the statistics of non-interacting chains. The above development furnishes all the equations required for use in the SCBD method detailed in the following section. Interfacial Activity of Gradient Copolymers. For the implementation of SCFT to quantify the interfacial activity of gradient copolymers, we complete the above discussion on the formalism of SCFT for gradient copolymers. In SCFT,73,75 the above eqs A11 and A12 are rendered self-consistent by expressing the volume fractions ϕj(r) as a function of the statistics of the polymer chains in the external potential fields w(r) and π(r). Explicitly, the volume fractions of A and B monomers ϕA(r) and ϕB(r) can be calculated as

(A5)

∫ D[R]δ(ρÂ + ρB̂ − ρo ) exp[−β(U0 + U1)]

1

ϕA(r) + ϕB(r) − 1 = 0

where, χ denotes the Flory−Huggins interaction parameter between the A and B monomers. The system is assumed to be incompressible and is imposed through a delta functional constraint δ (ρ̂A + ρ̂B − ρ0), which restricts the sum of the microscopic individual densities, ρ̂A(r) + ρ̂B(r), to be equal to the average total monomer density ρo. Henceforth, we use a notation in which the arc length variable “s” is nondimensionalized by N and all physical space dimensions are nondimensionalized by Rg = N1/2b/√6 which denotes the unperturbed radius of gyration of the polymer chain. The partition function for the system is written by assembling the above three interactions as, Z=

∫0

⎛ d R (s ) ⎞ 2 ds ⎜ k ⎟ ⎝ ds ⎠

and can be interpreted as the partition function of a single polymer in a potential field. In effect, the equilibrium properties of the polymer chains have been mapped onto the properties of non-interacting polymers in the presence of a potential − π(r) − w(r) acting on the A monomers and − π(r) + w(r) acting on the B monomers. In the above formulation w(r) is an exchange potential conjugate to ρ̂A − ρ̂B and π(r) is the chemical potential field conjugate to the total monomer density ρ̂A + ρ̂B. While the above formulation is exact, polymer self-consistent field theory corresponds to a saddle point approximation of the above field theory. In this framework, the saddle point values of the fields w(r) and π(r) are given by the solution of the following coupled equations:73,75 χρ w*(r) = o [ϕA(r) − ϕB(r)] (A11) 2

(A2)

(k)

γ (k)(s) = f (s)

+

∫0

1

⎤ + [π (R(s)) − w(R(s))](1 − γ (k)(s)]⎥ ⎥⎦

(A1)

and M

∫

⎡ −1 DR k exp⎢ ⎢⎣ 4R g 2

∏ Qk k=1

(A15)

qk (r, s = 0) = 1

(A9) 6294

(A16)


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The auxiliary function q†k (r, s) satisfies

additional component representing the external potential forces (if any) acting on the polymers. In the SCBD framework, the potential field W(r) constitutes a unique feature and is derived from the self-consistent field theory formulation. Specifically, the field theory developed in the preceding section reduces to a partition function of a single chain in a mean field potential. We borrow this idea to the Brownian dynamics framework and specify that at any given instant each bead (statistical segment) is acted on by a potential which is given by the SCFT as,

∂ † q (r, s) = ∇2 qk†(r , s) + [γ (k)(1 − s)[π (r) + w(r)] ∂s k + [π (r) − w(r)](1 − γ (k)(1 − s)]]qk†(r, s) (A17)

and qk†(r, s = 0) = 1

(A18)

Equations A11−A18 furnish the framework of SCFT as implemented for a gradient copolymer melt containing M chains, each of which can be of potentially different sequence. In the saddle point approximation, eq A8 furnishes the free energy of the system at equilibrium. To determine the interfacial activity of gradient copolymers, we consider a ternary blend of gradient copolymers with a binary, symmetric mixture of A and B homopolymers. In a comprehensive article,90 Matsen developed and applied the SCFT formalism for determining the interfacial tensions and elastic constants of diblock copolymer monolayers at the interface between A and B homopolymers. In this work, we adapt his formalism, while restricting our consideration to only the interfacial tension effects arising from the gradient copolymers. Since Matsen’s paper presents a detailed description of the accompanying formalism,90 to maintain brevity, we refrain from repeating the details here. We note that the only differences from Matsen’s formalism are that partition function and the density fields accompanying the diblock copolymers in his work are replaced by the appropriate expressions for the partition function (eq A9) and the density fields (eqs A13 and A14) of the gradient copolymers.

W (R ik) = [−π (r = R ik) − w(r = R ik)]γ (k)(i) + [−π (r = R ik) + w(r = R ik)][1 − γ (k)(i)]

To complete the above formulation, the Brownian dynamics simulations need to be supplemented by a prescription for the potentials fields w(r) and π(r) in eq A20. Instead of an adiabatic evolution slaving the potential to the instantaneous volume fractions as in eqs A11 and A12, we propose a phenomenological prescription for the evolution of the potentials such that they follow the evolutions of the composition fields and take their saddle point values at equilibrium. Explicitly, the potential fields are evolved on the discretized lattice by the following equations,

In the SCBD approach,78−82 we embed the above selfconsistent potential field idea underlying eqs A10−A12 into a Brownian dynamics framework.95 Explicitly, we model the polymer chains each as N beads connected linearly by N − 1 springs, with sequences specified by our sequence-generating algorithm. Within the context of the single sequence, annealed representation, we consider an ensemble of identical chains, wherein the identity and interactions of each segment is chosen to be partially A and B based on the average composition at that location along the chain. Since our objective is to probe the equilibrium properties of the system, the exact nature of the dynamical evolution is less relevant as long as the evolution is guaranteed to go to equilibrium. The dynamical equation of motion for the beads are then specified by the Langevin equation95 dR ik = Fs + FP + FR dt

⎡ χρ ⎤ dw(r) = Γ1⎢ o (ϕA(r, t ) − ϕB(r, t )) − w(r)⎥ ⎣ ⎦ dt 2

(A21)

dπ (r) = Γ2[ϕA(r, t ) + ϕB(r, t ) − 1] dt

(A22)

where, Γ1, Γ2 are fictitious mobilities that drive the potential evolution appropriately such that, the collective fields (w(r), π(r)) are evolved on slower time scales than the motion of individual beads. To determine the volume fractions ϕA(r, t) and ϕB(r, t) in our system which consists of many copies of discrete bead−spring chains, we adopt a space fixed lattice discretization of the volume to determine the volume fractions ϕk(r, t), by explicitly counting the number of beads of type A and B, at any given instant t.

2. Self-Consistent Brownian Dynamics

ζ

(A20)

3. Numerical Parameters

SCBD Simulations. All our simulations were effected with an ensemble of 1000 chains consisting of 100 segments each. We used a cell of size 8Rg × 8Rg and discretized them as 64 × 64 lattice units. The parameters Γ1, Γ2 in eqs A21 and A22 where chosen as 0.05 and 0.1, respectively. We did check that the properties computed did not exhibit any significant variations with these parameters. Of course fluctuations arise due to the fact that the number of chains per unit volume is finite (which contrasts with the idea behind self-consistent field theory), but by choosing a reasonably dense system of chains per unit volume, we minimize such effects. The time step for the Brownian dynamics was chosen as 0.005. We use convergence of the composition profile to monitor whether equilibrium has been achieved. SCFT Calculations. All our calculations were effected with an ensemble of 1000 chains consisting of 100 segments each. We used an one-dimensional lattice of 512 units and a physical space discretization of 0.04. We used the pseudospectral scheme of Rasmussen and co-workers to solve the diffusion equations eqs A15−A18.96

(A19)

Rik

where, denotes the coordinate of the ith bead of kth chain and ζ represents a phenomenological monomeric mobility coefficient (assumed to be a constant in this work). In the above equation, the term on the left hand side represents the frictional force experienced by a bead. The right hand side represents the other internal (and external) forces acting on the bead, which includes the following contributions: FR is a random force mimicking the random collisions of the solvent molecules on the polymer beads, Fs is the spring force 2 3kBT[Ri+1 + Rik + Ri−1 k k ]/b acting on each bead (where, b denotes the Kuhn segment length) and FP = −∇W(r) an 6295


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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Prof. Kenneth Shull for discussions which motivated the present work. We also acknowledge discussions with Prof. Michael Baldea on inhomogeneous Markov processes. We thanks Keith Gallow and Prof. Lynn Loo for their comments on the preprint version of this article. This work was supported in part by a grant from Robert A. Welch Foundation (Grant F1599), the US Army Research Office under Grant W911NF10-10346, and the National Science Foundation (DMR 1005739).

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