Properties of Random Block Copolymer Morphologies: Molecular

Article pubs.acs.org/Macromolecules

Properties of Random Block Copolymer Morphologies: Molecular Dynamics and Single-Chain-in-Mean-Field Simulations Birger Steinmüller,† Marcus Müller,*,† Keith R. Hambrecht,‡ Grant D. Smith,‡ and Dmitry Bedrov*,‡ †

Institut für Theoretische Physik, Georg-August Universität, 37077 Göttingen, Germany Department of Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112, United States

‡

ABSTRACT: The equilibrium structure and ordering kinetics of random AB block copolymers is investigated using a Lennard-Jones bead−spring model and a soft, coarse-grained model. Upon increasing the incompatibility a disordered microemulsion-like structure is formed, whose length scale slightly increases with segregation. The structure factor of composition fluctuations, molecular conformations, single-chain dynamics and collective ordering kinetics are investigated as a function of the segregation between A and B blocks. The harsh repulsion of the Lennard-Jones potential gives rise to pronounced fluid-like packing effects that affect the liquid structure on the length scale of the bead size and, upon cooling and increase of the local density, result in an additional slowing down of the dynamics. The soft, coarse-grained model does not exhibit pronounced packing effects and the softness of the potential allows for a faster equilibration in computer simulation. The structure and dynamics of the two different models are quantitatively compared. The parameters of the soft, coarse-grained model are adjusted as to match the long-range structure of the bead−spring model, and it is demonstrated that the soft, coarse grained model can be utilized to generate starting configurations for the Lennard-Jones bead−spring model.

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INTRODUCTION The structural properties of multicomponent polymer melts have attracted abiding interest in the field of polymer science. Experiments1−3 and theoretical models4−6 have been very successful in discovering different phases of block copolymer melts and their mixtures with homopolymers. The intricate prediction of the phase diagram of diblock copolymers and its experimental vindication gives testament to the possibilities of theoretical models and computer simulations in this field. For random block copolymers, however, theoretical, simulational, and experimental investigations are not as prevalent as for the simpler systems. Mean field calculations7−9 and integral equation theory10−12 predict a complex phase behavior encompassing macrophase-separated states and periodically ordered microstructures. The aforementioned theoretical works and experiments by Ryan et al.13 and by Eitouni et al.14 indicate that there is an order−disorder transition and that the structures formed become smaller when advancing further into the ordered region. More recent approaches have highlighted the role of fractionation between spatially homogeneous phases and microphase separated morphologies.15,16 The role of fluctuations has been considered within the Brazovskii theory17,18 and by computer simulations.19−21 The latter reveal that fluctuation effects are large and, unlike other dense polymer systems, their importance increases as the number of blocks per molecules grows large. Fluctuations prevent the formation of well-ordered, spatially periodic microphases, and the random copolymer melt adopts for a wide parameter range a disordered, microemulsionlike morphology. © 2011 American Chemical Society

At the time and length scales interesting for the dynamics of a polymer melt, which are its end-to-end radius and the time a polymer chain needs to diffuse this distance, atomic models are still too slow for such simulations. Moreover, in models with a harsh segmental repulsion (excluded volume), only very small invariant degrees of polymerization, 5̅ , are accessible and, consequently, fluctuation effects are large.22 Therefore, we resort to coarse-grained models, which grant access to the relevant time and length scales. For the structural properties of the melt, a coarse-grained model with soft interactions is sufficient. The softness of the interactions allows us to model experimentally relevant values of 5̅ and permits an efficient equilibration of the system. The soft, coarse-grained model, however, does not capture the effect of noncrossability23 and, more important for the present study, local packing effects of the segments that may lead to a glassy arrest of the dense polymer melt at low temperatures.24,25 Therefore, we additionally employ a model, which utilizes a harder interaction, i.e., a Lennard-Jones bead−spring model, and we compare the results of the two models quantitatively. This comparison of static and dynamic properties is performed at high temperatures, where we can equilibrate both model systems. Then we explore how to utilize the equilibrated configurations of the soft, coarsegrained model as starting configurations of the Lennard-Jones Received: October 19, 2011 Revised: December 6, 2011 Published: December 21, 2011 1107

dx.doi.org/10.1021/ma202311e | Macromolecules 2012, 45, 1107−1117

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incompatibility, a three-phase coexistence occurs, where the Arich and B-rich homogeneous phases coexist with a microphaseseparated, lamellar phase.15,16,20 The polymer sequences partition between the coexisting phases (fractionation); the A-rich sequences are enriched in the A-rich homogeneous phase, the B-rich sequences are preferentially located in the Brich homogeneous phase, and polymers with a small excess of A or B mainly constitute the microphase-separated morphology. Upon increasing the incompatibility, the volume fraction of the two homogeneous phases shrinks and the characteristic length scale of the microphase-separated morphology decreases in marked contrast to the behavior of diblock copolymers.8,16 Previous simulations indicate that for uncorrelated random block copolymers with symmetric composition and many blocks, thermal fluctuations are very important and result in important deviations from the mean-field predictions.20 Rather than observing the sequence of disordered phase, macrophase separation, three-phase coexistence with fractionation, and microphase separation, one only observes a completely gradual formation of local domains; i.e., the systems remain disordered and adopt a microemulsion-like structure.

bead−spring model so as to benefit from the advantages of both inherently different models. In this article, we will first introduce the two coarse-grained models used, the soft coarse-grained model, which we study by a Single-Chain-in-Mean-Field simulations,22,26 and a LennardJones bead−spring model27 that we use in conjunction with molecular dynamics (MD) simulations.28 In the following section, we use several static quantities, e.g., the structure factor and the radial pair correlation function, to characterize the morphology and single-chain conformation. We identify a correspondence between the parameters of the soft, coarse-grained model and the Lennard-Jones bead−spring model. Subsequently, we estimate the equilibration time of the system in two different ways. First, we characterize the singlechain dynamics like the diffusivity of the polymers and the endto-end vector correlation function. Second, we investigate the time the melt needs to establish a morphology in response to a quench from the disordered structure. The kinetics of structure formation is compared between the two simulation methods. Additionally, we quantify the time required for the LennardJones bead−spring model to establish the equilibrium structure when starting from equilibrated configurations of the soft, coarse-grained model which already exhibit the equilibrium structure on intermediate and large scales. The manuscript ends with a brief summary and an outlook.

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SOFT COARSE-GRAINED MODEL AND SCMF SIMULATION TECHNIQUE We consider n random copolymers in a volume V. In the framework of the soft, coarse-grained model, we describe the molecular architecture of linear, flexible polymers by a simple bead−spring model with N effective segments per polymer. The bonded interactions along the chain molecule, that define the macromolecular architecture, are harmonic,

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MODELS AND TECHNIQUES System. We use Single-Chain-in-Mean-Field (SCMF) simulations of a soft, coarse-grained model and molecular dynamics (MD) simulations of a Lennard-Jones bead−spring model to study the thermodynamic equilibrium properties of random block copolymers. Random block copolymers are linear, flexible polymers that are comprised of Q = 6 blocks. Each block, in turn, consists of a sequence of M A segments or B segments, respectively. Thus, a melt of these random block copolymers contains 2Q = 64 different species of chains where we do not account for the head−tail symmetry. The total number, n = 4160, of chains in the system is much larger than the number of molecular sequences. Thus, a melt of random block copolymers can be conceived as a mixture of many (2Q) components. The probability or concentration of a specific sequence is fixed during a Markovian polymerization process.7 In this manuscript, we restrict ourselves to a symmetric sequence distribution, i.e., a sequence (e.g., AABABA) and its counterpart (e.g., BBABAB), in which all A segments are replaced by B segments and vice versa, have identical probability. Moreover, we consider the case that the blocks along a random copolymer are completely uncorrelated, i.e., an A-block is followed by another A block or a B block with equal probability. Thus, we conceive the system as a mixture of 2Q components, each fixed sequence being one component. In the bulk, all different sequences occur with equal concentration. The segment species A and B repel each other, and this interaction leads to a local unmixing. Mean-field calculations predict that, upon increasing the incompatibility, χ, between the segment species, the system will first macroscopically phase-separate into macroscopic domains. One domain will be enriched in sequences that are characterized by a large excess of A segments and the other, coexisting phase will be predominantly comprised of B-rich sequences. The segregation between the two coexisting phases, however, is small because the typical excess of A segments is only on the order of √Q along a polymer comprised of Q blocks.7 At larger

/b({ ri (⃗ s)}) = kBT

n N−1

∑∑ i=1 s=1

3(N − 1) 2R eo2

| ri (⃗ s) − ri (⃗ s + 1)|2 (1)

where kB is Boltzmann’s constant and T stands for the temperature. ri⃗ (s) denotes the coordinate of the sth segment on the polymer with index i. In the soft, coarse-grained model, we discretize the chain contour into N = 120 segments. The strength of the harmonic bond potential is chosen such that the mean squared end-to-end distance of the Gaussian chain conformations in the absence of nonbonded interactions is Reo2 = ⟨(r1⃗ (s) − rN⃗ (s))2⟩ . The nonbonded interactions cater for the near-incompressibility of the dense polymer liquid, that arises from the harsh excluded volume interactions on the atomistic scale, and the repulsion between segments of unlike species, that gives rise to structure formation. The nonbonded interaction takes the form of a density functional and only second-order virial coefficients are retained.22,29,30 ρ /nb = o kBT N −

⎡ κ oN

∫ d3r ⎢⎣

2

([1 − α]φ̂A + [1 + α]φ̂ B − 1)2

⎤ (φ̂A − φ̂ B)2 ⎥ ⎦ 4

χoN

(2)

where ρo = nN/V denotes the segment number density. The coefficient κo is inversely proportional to the isothermal compressibility of the melt. The prefactor 1 ± α is the relative normalized volume of an A segment or a B segment, respectively. For α = 0, the system is completely symmetric with respect to exchanging A ⇌ B while for α > 0 the A segments have a smaller segmental volume than B segments. 1108


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χo is the bare Flory−Huggins parameter that quantifies the repulsion between A and B segments. Equation 2 expresses the nonbonded interactions in terms of the local normalized densities, φ̂ A(r)⃗ and φ̂ B(r)⃗ . In order to employ this form of nonbonded interactions in a particle-based simulation, we express the local densities via the particle coordinates, {ri⃗ (s)}. To this end, we use a cubic collocation grid to define the local densities of a grid vertex, {ck⃗ },31 by assigning particle coordinates to a vertex according to n

φ̂A( ck⃗ |{ ri (⃗ s)}) =

N

∑∑

surroundings by the interaction of a segment with an external field, i.e.,22,26

ρ ΔL3 / SCMF nb ({ ri (⃗ s)}) = o kBT N

c⃗

wA( c ⃗) =

(4)

which state that the contributions of a particle to all cells add up to unity irrespectively of its position and that the volume assigned to each grid cell is ΔL3. Similar particle-mesh techniques have been used for calculating electrostatic interactions32,33 and in plasma physics.34 They are particularly advantageous for calculating interactions in dense systems. In the following, we utilize a linear assignment function of the form

∏

Π( c ⃗ , r ⃗) =

π(| rα⃗ − cα⃗ |)

with

α∈ {x , y , z}

⎧ | d| ⎪1 − for |d| ≤ ΔL ⎨ π(d) = ΔL ⎪ ⎩0 otherwise

(5)

i.e., a segment influences the density of the eight vertices of the grid cell in which it is contained. One advantage of this assignment function is that the densities change continuously with the segment coordinate and nonbonded forces can be defined. Using this grid-based assignment, we can rewrite eq 2 in the form ρ ΔL3 /nb({ ri (⃗ s)}) = o kBT N −

χoN 4

⎡ κ oN ([1 − α]φ̂A + [1 + α]φ̂ B − 1)2 ⎣ 2

∑⎢ c⃗

⎤ (φ̂A − φ̂ B)2 ⎥ ⎦

(7)

∂/nb ρoΔL ∂φA ( c ⃗) (8) and a similar equation holds for wB(c)⃗ . This computational scheme exploit the separation between the strong, rapidly fluctuating, bonded interactions, which dictate the size of a segmental movement in one Monte Carlo step, and the weak, nonbonded interactions, which only very slowly evolve in time. A SCMF simulation cycle is comprised of two parts: First, one evolves the polymer conformations in the external fields wA and wB for a small, fixed amount of Monte Carlo steps. During this Monte Carlo simulations the molecules do not interact with each other and the simulation of independent chain molecules can be straightforwardly implemented on parallel computers. In the second step, one recalculates the external fields from the instantaneous densities according to eq 8. Then the simulation cycle commences again. In the second step, fluctuations and correlations are partially restored. The quasiinstantaneous field approximation that consists in replacing the interactions via frequently updated fluctuating fields is accurate if the change of the local composition, ϕA and ϕB, between successive updates of the external fields is small.22 This property is controlled by the parameter V ε= 2 (9) nN ΔL3 which plays a similar role as the Ginzburg parameter in a meanfield calculation. In contrast to the Ginzburg parameter, ε depends on the discretization of space, ΔL, and the molecular contour, N, and these parameters are chosen such that the quasi-instantaneous field approximation is accurate. In the present simulations, the control parameter adopts the values ε = 1.8 × 10−3. For selected parameter combinations we have explicitly verified that the results of the SCMF simulations are very close to the data obtained by Monte Carlo simulations using eq 6. In order to propagate the chain conformations in the SCMF simulations, we use smart Monte Carlo moves,23,35 where the forces that act on a segment influence its trial displacement. The single-chain dynamics closely mimics the Rouse-like motion of short macromolecules in a melt for all but the first 10 SMC steps. The ‘time’ scale of the structural relaxation of the single-chain conformations and the morphology is measured in units of the time τmelt that it takes a homopolymer, χoN = 0, comprised of N segments to diffuse a distance on the order of Rmelt2, i.e., τmelt = Rmelt2/D, where D denotes the selfdiffusion coefficient in a homopolymer melt and Rmelt2 = Re2(χoN = 0,α = 0) is the measured mean squared end-to-end distance of polymers in a homopolymer melt, i.e., we apply the bonded interactions, eq 1, and the nonbonded interactions, eq 2, with χoN = 0 and α = 0 . We will use Rmelt as the relevant length scale for the rest of this article. In the SCMF simulations

γi(s)

∫ d3r Π(c ⃗ , r ⃗) = ΔL3 ∀ c ⃗

c⃗

where the external fields wA/N and wB/N that act on A and B segments, respectively, are frequently calculated from the local fluctuating densities according to

Π( ck⃗ , ri (⃗ s)) ρ ΔL3 (3) i=1 s=1 o where ΔL denotes the linear extent of a grid cell. γi(s) = 1 if the sth segment of polymer i is an A segment and γi(s) = 0 otherwise. The local normalized density of B segments is calculated by a similar relation. The assignment function Π(ck⃗ ,ri⃗ (s)) obeys the normalization conditions

∑ Π(c ⃗ , r ⃗) = 1 ∀ r ⃗ and

∑ [wA φ̂A + wBφ̂ B]

(6)

This form of the interactions is suitable for Monte Carlo simulations. The interactions of the model depend on the parameters, Reo, which sets the length scale, χoN, which parametrizes the incompatibility between polymers, α, which describes the asymmetry of segmental volumes, the polymer number density, ρo/N, and the inverse compressibility, κoN . The architecture of the random copolymers are defined by the number of blocks, Q, and the symmetric Markov process that fixes the sequence distribution. Note that neither the number M of segments per block nor the total number N of segments along the macromolecular contour is a physically significant parameter of the model; only the ratio Q = N/M has a physical meaning as the number of blocks per molecule. In Single-Chain-in-Mean-Field (SCMF) simulations, we replace the pairwise interactions, eq 6, of a segment with its 1109

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we obtain τmelt = 274500 SMC steps, where each segment on average had the chance of one trial displacement in a SMC step. The softness of the interactions allows for an efficient equilibration of the morphology and the description of dense polymer melts that are characterized by experimentally relevant, large values of the invariant degree of polymerization, 5̅ = (ρoRe3/N)2 . Here, Re denotes the measured end-to-end distance using the full Hamiltonian. Lennard-Jones Bead−Spring Model. The absence of harsh excluded-volume interactions in the soft, coarse-grained model prevents us from studying the interplay between morphology and liquid-like structure. As we reduce the temperature or increase the incompatibility, both the segregation between A- and B-rich domains becomes more pronounced and, simultaneously, the density increases and one species arrests in a glassy state. While the former effect can be described by soft interactions the latter cannot. Therefore, we turn to the Lennard-Jones bead−spring model for additional studies of the properties of random copolymers. In a model with harsh excluded-volume interactions, however, it is difficult to achieve experimentally large values of 5̅ . Therefore, we use in both, the soft, coarse-grained model and the model with harsh excluded-volume interactions, small values of the invariant degree of polymerization, ranging from 5̅ = 175 for a homopolymer melt (χoN = 0,α = 0) to 5̅ = 320 for a high incompatibility of the two types of monomers. These values are orders of magnitude smaller than in experiments; therefore, we expect that fluctuation effects are significantly larger than in experimental systems. In our model with harsh excluded-volume interactions, the nonbonded pairwise interactions act between all segments with a cutoff at a distance r = rc = 2.5σ and truncated such that the potential and the force are continuous at the cutoff distance, rc,36 n

we measure the length scale of the Lennard-Jones bead−spring model in units of σ and the energy scale in units ε. The value εAA = 1 corresponds to a melt, where A and B segments are identical. Larger values of εAA give rise to local demixing of A and B segments and, additionally, to an increase of the density of A-rich domains relative to that of B clusters. The bonded interactions are modeled as finitely extensible, nonlinear elastic (FENE) interactions /b({ ri (⃗ s)}) kr 2 =− 0 kBT 2

N

dULJ(r ) dr

r = rc

(10)

where the prime indicates that the sum runs only over pairs of particles. r = |ri⃗ (s) − rj⃗ (t)| is the distance between the interacting particles, and ULJ(r) denotes the Lennard-Jones potential

⎛ σ12 σ6 ⎞ ULJ(r ) = 4εIJ ⎜ 12 − 6 ⎟ ⎝r r ⎠

(12)

with the maximal bond lengths, r0 = 1.5σ, and the spring constant, k = 16. The parameters are chosen such that the combination of bonded and nonbonded interactions prevents the chain molecules from crossing through each other in the course of their motion in molecular dynamics simulations. The repulsive part of the Lennard-Jones potential also imparts a local stiffness on the chain conformations. Since the nonbonded interactions differ for A and B species, the equilibrium bond length slightly differs for A and B blocks and changes with segregation. Each molecule is comprised of N = 60 LennardJones interaction centers. We studied the properties of the Lennard-Jones bead−spring model by molecular dynamics simulations using the program package LAMMPS.37 The equations of motions were integrated using the velocity-Verlet38,39 algorithm with an integration step of Δt = 0.005τ in units of the Lennard-Jones time scale τ. The temperature was fixed at T = 1 in Lennard-Jones units using a Nosé−Hoover thermostat40,41 with a damping parameter of 0.7 in Lennard-Jones units. Simulations in the isobaric ensemble (NPT) at P = 0 employed a Nosé−Hoover barostat41,42 with a pressure damping of 5 in Lennard-Jones units. In analogy to the SCMF simulations, the “time” scale of the molecular dynamics simulations is calculated from the number of integration steps that it takes a homopolymer in a pure homopolymer melt to diffuse a distance on the order of Rmelt2, i.e., τmelt = Rmelt2/D, where D denotes the self-diffusion coefficient in a homopolymer melt and Rmelt2 is the measured mean squared end-to-end distance of polymers in a homopolymer melt, which we will use as the relevant length scale in the following text. In the MD simulations we obtain τmelt = 21000000 integration steps, i.e., τmelt = 105000τ .

/nb({ ri (⃗ s)}) 1 ∑ ′ ∑ ′ULJ(r) − ULJ(rc) = kBT 2 i,j=1 s,t=1 − (r − rc)

⎛ | r (⃗ s) − ri (⃗ s + 1)|2 ⎞ ln⎜⎜1 − i ⎟⎟ r02 ⎝ ⎠ i=1 s=1 n N−1

∑∑

■

RESULTS

Static Properties. The snapshots shown in Figure 1 illustrate how the morphology of the melt changes with growing incompatibility of the two different monomeric repeat units. Already for small incompatibility, see Figure 1a and Figure 1d, A and B segments are not randomly mixed but they show a preference to form clusters, albeit with small differences in composition. The internal “interfaces” of these clusters of high concentration of one type of segments are rather broad. For the next set of parameters, εAA = 1.3 and χoN = 100, a microemulsion-like structure has formed, cf. Figure 1, parts b and e, where the domains are segregated more strongly and the interfaces can now be clearly determined. In the final set of parameters, depicted in parts c and f of Figure 1, the microemulsion-like structure has fully formed, and therefore the internal interfaces between the domains are rather sharp. The morphology is characterized by a well-defined, finite, characteristic length scale but no long-range order like in

(11)

σ characterizes the spatial range of the potential, and the depth of the potential, εIJ with I,J being A or B, depends on the segment species. The repulsive part of the Lennard-Jones interaction limits the compressibility of the dense polymer melt and dictates the liquid-like packing of the Lennard-Jones segments, while the attractive part of the Lennard-Jones interaction results in a realistic equation of state and a densification of the liquid upon cooling. The difference in the attractive interactions gives rise to the incompatibility between A and B segments and drives the structure formation in the random copolymer melt. In the following, we set εAB = εBB ≡ ε, which defines the energy scale; εAA is varied to control the incompatibility between A and B segments. In the following, 1110


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Figure 1. Snapshots for different values of the interaction parameter as indicated below each figure. On the top, the snapshots are from the Lennard-Jones bead−spring model while the bottom figures show snapshots obtained with the soft, coarse-grained model. These figures were produced using VMD.43

In order to match the equilibrium structure of the soft, coarse-grained model to the Lennard-Jones bead−spring model, we have adjusted the parameters incompatibility, χoN, and the segmental asymmetry, α, keeping the inverse compressibility, κoN = 600 fixed. Varying α does not strongly influence the AB pair correlation function, but chiefly affects the AA and BB pair correlation functions. We have adjusted the value of α to match the behavior of the total correlation functions, gA and gB at large distances to the results of the Lennard-Jones bead−spring model. Increasing α, we increase the difference of local densities in A- and B-rich domains, i.e., the density of A increases and the density of B decreases. The density difference becomes notable in the Lennard-Jones bead−spring model for large values of the incompatibility. In the inset of Figure 2c, we highlight the behavior of gA and gB at large distances in the Lennard-Jones bead−spring model and compare the data with the results of the soft, coarse-grained model for two values of α. When one changes the value of χoN, it results in a shift of position and change of magnitude of the peak in the pair correlation function. A greater value of χoN shifts this peak to larger length scales and increases its height. In Figure 3 the pair correlation functions are plotted for three sets of parameters, which show similar structure behavior on a mesoscopic length scale. The analogue data for the structure factor are displayed in the insets. Because of the harsh repulsion (excluded volume) of the Lennard-Jones bead−spring model, as compared to the soft interactions of the soft, coarse-grained model model, which allow segments to overlap, the two structural properties of both models cannot be reconciled on short length scales for the pair correlation function and large q-values for the structure factor, respectively. This difference manifests itself as a peak of the structure factor for MD simulations around (qRmelt)2 = 5000, which corresponds to (qσ)2 = 50, and the packing structure of the pair correlation function. At intermediate to large length

a lamellar phase is established on the time scale of our simulations.45 Although these snapshots are generated by simulating different models, where there is no analytical expression to identify matching parameters, they show a striking similarity in their mesoscopic structure in a statistical sense. The first goal of our study is to identify parameters of the two different models to produce this structural similarity. We chose Rmelt as the length scale, which allows us to compare the results of these two models. The values are Rmelt = 1.2Reo in the soft, coarse-grained model used in the SCMF simulations and Rmelt = 10.0σ in the Lennard-Jones bead− spring model used in MD simulations, respectively. The quantities, which we used to characterize the static structure, are the radial pair correlation function of A and B segments, gAB(r), as well as the structure factor, S(q) . The structure factor is defined as n

S(q) =

N

1 ⟨| ∑ ∑ (2γi(s) − 1) exp(iq ⃗ ri (⃗ s))|2 ⟩ nN i=1 s=1

(13)

with γi(s) used as in eq 3. It measures fluctuations of the Adensity on different length scales. Since the fluctuations of the A-density chiefly arise from composition fluctuations, we can compare the data of the soft, coarse-grained model and the strongly repulsive Lennard-Jones bead−spring model although their isothermal compressibilities differ. The choice of values for q is constrained by our finite simulation box. Since it is cubic, we can only calculate S(q) for values of q = (2π)/ (L)((kx2 + ky2 + kz2)1/2) where all ki are integers. The A-B pair correlation function is defined as gAB(r ) =

∫ d3r1 ∫ d3r2⟨ρA( r1⃗ )ρB( r2⃗ )⟩δ(r − | r1⃗ − r2⃗ |) ∫ d3r1 ∫ d3r2⟨ρA( r1⃗ )⟩⟨ρB( r2⃗ )⟩ 4πVr

2

(14)

where ρA,B(r)⃗ denote the local densities. 1111


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Figure 2. Comparison of the radial pair correlation function of A (gA = (gAA + gAB)/2) and B (gB = (gBB + gAB)/2) monomers for three different sets of parameters between the two different models. The density of A-rich domains becomes larger than that of B-rich domains at higher incompatibility. In the soft, coarse-grained model, κoN was always set to 600, α and χoN are as follows: (a) α = 0.01, χoN = 30; (b) α = 0.02, χoN = 100; (c) α = 0.05, χoN = 240.

Figure 3. Comparison of the radial A−B pair correlation function and the structure factor for three different sets of parameters. In the soft, coarsegrained model, κoN was always set to 600, α and χoN are as follows: (a) α = 0.01, χoN = 30; (b) α = 0.02, χoN = 100; (c) α = 0.05, χoN = 240.

between unlike segments. The reason are the strong fluctuation effects due to the small value of 5̅ . We summarize the results of this section in Figure 4, plotting the pairs of χoN and εAA, for which we found good agreement of the static structure at mesoscopic length scales. Conformations. In the previous section, we have demonstrated that one can identify parameters such that the morphology of the soft, coarse-grained model and the Lennard-Jones bead−spring model at intermediate and long length scales coincide. In this section we investigate to what extent this structural agreement also holds for conformational properties of individual molecules. Chain conformations have been analyzed by examining the mean square radius of gyration, Rg2, and the mean squared endto-end distance, Re2. In this analysis we have sorted all chains based on the overall chain composition. Since all chains consist of 6 blocks there are 7 different types of chains with 0, 1, 2, 3, 4, 5, or 6 blocks of type B. Conformational properties were averaged for each chain composition. Figure 5 shows the change of conformations as a function of the interaction parameter, εAA and χoN, respectively, for different chain composition. As the interaction strength increases, the conformations of chains with different composition show dissimilarities. With the formation of a microemulsion-like structure, homopolymer chains become more compact as is evident from the reduced values of Rg2 in Figure 5(a), compared to the homopolymer melt with εAA = 1.0 and χoN = 0, respectively. The shrinking of the molecular extension is likely a consequence of the confinement into corresponding A or B domains.

scales, however, the equilibrium structure agrees very well for each set of parameters. Let us discuss the three different sets in turn. For the smallest incompatibilities, χoN = 30 and εAA = 1.1, we observe a correlation hole in the AB pair correlation function, Figure 3a, but no peak appears, neither in the pair correlation function, nor in the structure factor, which is displayed in the inset. As the snapshots already suggested, this value of interaction strength fails to impose a strong segregation onto the system. When we increase the interaction strength to χN = 100 and εAA = 1.3, see Figure 3b, we observe a formation of a structure on a scale of 1.2Rmelt in the pair correlation function and a peak in the structure factor at (qRmelt)2 ≈ 14, which corresponds to a wavelength of 1.7Rmelt. The position of this peak of the structure factor corresponds to a point of inflection in the pair correlation function. Increasing the interaction strength further to χN = 240 and εAA = 1.6, cf. Figure 3c, the peak of the pair correlation function shifts to 1.3Rmelt and the peak in the structure factor to (qRmelt)2 ≈ 11 . Again, the position of this peak in the structure factor, with a wavelength of 1.9Rmelt, and the point of inflection of the pair correlation function coincide. Mean-field theory predicts that at intermediate segregation, random block copolymers form a microphase separated morphology with a characteristic wavelength that decreases with increasing incompatibility.7 This prediction differs from our simulation results. First, the morphology, which we observe in the simulations, lacks long-range order. Second, the characteristic length scale increases as we increase the incompatibility 1112


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Dynamic Properties. In this section, we compare the single-chain dynamics of the soft, coarse-grained model and the Lennard-Jones bead−spring model. In Figure 6, we present the mean square displacement (MSD), and the end-to-end vector autocorrelation, C(t), as a function of time. The analysis of the static properties in Figure 2 revealed structural differences on the length scales ≤0.25Rmelt . The differences in the local, fluidlike structure on the segment scale give rise to different local friction coefficients in the soft, coarse-grained model and the Lennard-Jones bead−spring model. The latter exhibits a local, glass-like dynamics at high segregation. To account for these differences, the time scale is adjusted by the diffusion coefficient, D, see Table 1 and Table 2. The diffusion coefficient is calculated from the MSD of the polymers’ centers of mass, g3(t),44 by D = limt→∞((g3(t))/(6t)) . The vertical dashed line in the figure indicates the time scale, on which the segments have diffused several segment sizes. Beyond this time and length scale the static structure agrees and the single-chain dynamics can be compared. The diffusion coefficient decreases by a factor of 2 when we compare the disordered melt, χoN = 0, 30, with the microemulsion like, χoN = 100, 240, systems. We can deduce that the formed structure severely restricts the mobility of the chains. When we compare the segmental MSD, the A segments show a slightly smaller value for large length scales, see parts a and b of Figure 6. Using this matching of time scales, the dynamics of the slower A-segments nicely agrees between the soft, coarse-grained model and the LennardJones bead−spring model, while there are some differences in the dynamics of B-segments and the center-of-mass at small and intermediate times. The end-to-end vector autocorrelation function in Figure 6d shows that the time needed for the end-to-end vector to decorrelate undergoes a larger change than the diffusion coefficient. This decoupling of translational and rotational dynamics due to domain formation in the Lennard-Jones bead−spring model is quantitatively captured by the soft, coarse-grained model. This time grows by a factor of 5 when we compare χoN = 30 to χoN = 100, and a factor of 10 when we compare χoN = 30 to χoN = 240 . The influence of the formation of domains and their boundaries becomes more pronounced in this quantity, while also being noticeable in the MSD.

Figure 4. Model parameters for similar equilibrium morphologies of the two different models. In the soft, coarse-grained model, κN was always set to 600, α = 0.01 for 30 ≤ χoN ≤ 75, α = 0.02 for 100 ≤ χoN ≤ 180, and α = 0.05 for χoN = 240 .

Interestingly, after a well-segregated morphology forms, εAA ≥ 1.3 and χoN ≥ 100, the conformations of homopolymer chains are almost independent of the strength of the interaction and the corresponding changes in structure observed in Figure 3. Chains that contain both A and B blocks become more extended, compared to a homopolymer melt, as the interaction strength increases. This molecular stretching is the strongest for chains with 50−50 composition. We also calculated the ratio of ⟨Re2⟩/⟨Rg2⟩ to examine if the stretching/shrinking of chains will affect their Gaussianity. For Gaussian chains, this ratio is 6. Figure 5b shows that chains with 50−50 A/B compositions noticeably deviate from the Gaussian behavior with their ratio being larger than 6.0 as a sign characteristic for extended confirmations. Chains with 33 or 67% of A segments have ratios very close to 6 indicating that, on average, their conformations are similar to a Gaussian chain. Finally, homopolymer chains (A or B) and chains with large composition asymmetry have ⟨Re2⟩/⟨Rg2⟩ ratio smaller than 6.0 indicating a more coiled conformations compared to Gaussian chains. We emphasize that the same parameters that result in an agreement of the morphology between the Lennard-Jones bead−spring model and the soft, coarse-grained model additionally give rise to excellent agreement of the conformation properties.

Figure 5. Conformational properties of the two different models as a function of interaction strength and the number of B blocks of the chains. The mean squared radius of gyration, Rg2, normalized by the mean squared end-to-end distance of the homopolymer melt, is shown in the left figure (a) to investigate the extension of the polymers. The figure to the right (b) is used to examine the Gaussianity of the chains by plotting ⟨Re2⟩/⟨Rg2⟩ . Both figures show these results for the soft, coarse-grained and the Lennard-Jones bead−spring models. 1113


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Figure 6. Dynamic properties of both models. The MSD of A-segments, B-segments and the center of mass is depicted in panels a, b, and c. Panel d displays the end-to-end vector autocorrelation function. The time is multiplied with the diffusion coefficient of the random block copolymer melt and divided by the diffusion coefficient of the homopolymer melt. The different dynamics for short time scales of the two models are clearly visible. The end-to-end vector autocorrelation function decorrelates on the same time scale for MD and SCMF simulations for each set of parameters.

short time scales. The difference in the MSD of single segments decreases with longer time scales. In Figure 6d, we investigate the autocorrelation function of the polymers’ end-to-end vector. As in the case of the MSD, this dynamic quantity implies the same slower dynamics of the LJ melt when compared to the soft, coarse-grained model. Equilibration Times after Quench. We investigate the structure formation in response to a quench from the disordered phase in SCMF simulations. For each set of parameters, we take an equilibrated homopolymer melt as the starting configuration, then distribute the polymer types randomly, and let the system equilibrate, now with a different α and χoN. In Figure 7, the time evolution of the pair correlation function is presented for three different values of χoN and, in the inset, the time evolution is presented for the structure factor. For the smallest interaction strength, χoN = 30, see Figure 7a, the pair correlation function and the structure factor show that the system equilibrates very fast; within 0.18τmelt, its structure has reached an equilibrated state. When we look at the structure formation at the intermediate interaction strength, χoN = 100, we estimate from the pair correlation function, Figure 7a, that it takes 0.36τmelt to equilibrate the system. This equilibration time is compatible with the time evolution of the structure factor in Figure 7b, but it is more difficult to extract a time scale from S(q). In case of the highest interaction strength, χoN = 240, it takes even longer to achieve an equilibrated

Table 1. Diffusion Coefficient of the Polymer Chains in SCMF Simulations, Calculated for Different Parameters of the Soft, Coarse-Grained Model χoN D/Dmelt

0 1.0

30 0.88

100 0.49

240 0.46

Table 2. Diffusion Coefficient of the Polymer Chains in MD Simulations, Calculated for Different Parameters of the Lennard-Jones Bead−Spring Model εAA D/Dmelt

1.0 1.0

1.1 0.83

1.3 0.25

1.6 0.11

When looking at the same quantities, MSD and the end-toend vector autocorrelation function in the MD simulations, see Figure 6, the dynamic quantities show a significantly different behavior as a function of the interaction strength. The diffusion coefficients in Table 2, calculated from the mean-squared displacement of the polymers’ centers of mass, Figure 6(c), decrease faster in the MD simulations with increasing interaction strength. We examine the MSD of single segments in parts a and b of Figure 6 for A and B segments, respectively. For short times, an increase of the interaction strength does not affect the movement of B segments; their freedom becomes constricted only at larger time scales, when the collective motion of the entire chains dominates. For A segments, a larger interaction strength also gives rise to a reduction of mobility at 1114


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Figure 7. Time evolution of the radial A−B pair correlation function and the structure factor for three different sets of parameters in the soft, coarsegrained model. κN was always set to 600, α and χoN are as follows: (a) α = 0.01, χoN = 30; (b) α = 0.02, χoN = 100; (c) α = 0.05, χoN = 240.

Figure 8. Time evolution of the radial A−B pair correlation function and the structure factor for three different sets of parameters in the MD simulations. The parameter εAA was set as follows: (a) εAA = 1.1; (b) εAA = 1.3; (c) εAA = 1.6.

inside the well-segregated A-domains will increase and eventually lead to a vitrification. Note that the glass transition occurs in similar Lennard-Jones bead−spring models around 1/εAA ≈ 0.42 . Since the slowing-down and the glass transition are related to the local, fluid-like packing, which is not captured by the soft, coarse-grained model, packing effects do not contribute to an increase of the equilibration time in the SCMF simulations. Equilibration after Mapping the Soft, Coarse-Grained Model onto the Lennard-Jones Bead−Spring Model. Since the Lennard-Jones bead−spring model and the soft, coarse-grained model agree in their equilibrium structure at intermediate and large length scales, and the SCMF simulations are computationally more efficient because the interactions are softer and the time scale is not affected by the liquidlike packing of segments, it is tempting to use equilibrated configurations of the SCMF simulations for generating starting configurations for the Lennard-Jones bead−spring model. It is important to note that the configurations obtained by Single-Chain-in-Mean-Field simulations capture intermolecular correlations on the scale of the molecule’s extension, e.g., the correlation hole in the intermolecular pair correlation function. Otherwise, a full single-chain relaxation time, τmelt, would be required to establish those correlations.46,47 Given an equilibrated soft, coarse-grained configuration with chain length N = 120, we construct a staring configuration of the Lennard-Jones model with N = 60 by representing the center of mass of two neighboring soft beads by a LennardJones particle. Since the soft beads may overlap, also the Lennard-Jones particles will overlap resulting in excessively large forces, and we cannot simply switch on the harsh

structure. From the evolution of the pair correlation function, see Figure 7c, we estimate that the system takes about 0.73τmelt to equilibrate. The failure to reach equilibrium at 0.36τmelt can, for this large incompatibility, also be inferred from the structure factor, cf. Figure 7c. In the MD simulations, we adhered to the following simulation setup. We equilibrated a melt at εAA = 1.0 and then quenched the system to the desired value of εAA. After the quench, we continued the simulation of the system in the NPT ensemble. We studied the time evolution for the three different values of εAA = 1.1, 1.3, and 1.6, which we investigated in the last section. In Figure 8, the time evolution of the pair correlation function is shown for these three different values of εAA, and the insets present the corresponding data for the structure factor. In qualitative agreement with the SCMF simulations, the time to form the equilibrium morphology increases with incompatibility, χoN. At εAA = 1.1, both measures of structure formation show an equilibration within 0.23τmelt see Figure 8a. For the intermediate incompatibility, εAA = 1.3, the equilibration takes 0.7τmelt, as depicted in Figure 8b, and for the highest incompatibility, εAA = 1.6, there is a further increase in the equilibration time to 1.7τmelt. Quantitatively, the soft, coarse-grained model attains the equilibrium morphology faster than the Lennard-Jones bead− spring model when measured in units of the relaxation time of a single molecule in the disordered state. Moreover, the ratio of time scales [τeq,MD/τmelt,MD]/[τeq,SCMF/τmelt,SCMF] increases from 1.28 at low incompatibilities to 2.33 at high incompatibility. The relative slowing-down of the Lennard-Jones model is partially explained by the increase of the density in the segregated A-rich clusters. Upon further increase of εAA, the density 1115


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Figure 9. Time evolution of the A−B pair correlation function and structure factor in MD simulations. This simulation used equilibrated configurations of the soft, coarse-grained model as starting configurations. In the soft, coarse-grained model, κoN was always set to 600, α and χoN are as follows: (a) α = 0.01, χoN = 30; (b) α = 0.02, χoN = 100; (c) α = 0.05, χoN = 240.

with a system of this size, but only 36 h when we use the SCMF method.

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SUMMARY AND OUTLOOK We have investigated the structure and dynamics of random block copolymer melts as a function of the incompatibility using a soft, coarse-grained model and a Lennard-Jones bead− spring model. Upon increasing the incompatibility, both models gradually form a microemulsion-like structure. When we compare our results for a higher incompatibility to mean-field predictions7,8 for random block copolymer melts, we find a significant difference in the structure formed. (i) we do not observe well-ordered, periodic microphases and (ii) the characteristic length scale of the morphology does not decrease with growing incompatibility. These discrepancies are rooted in fluctuation effects, which are particularly strong in our system because the Lennard-Jones model can only describe systems with a modest value of 5̅ . We have characterized the molecular conformations and morphology and the single-chain dynamics and identified a mapping between the parameters of the two different models such that they exhibit similar structure on intermediate and large length scales. On short length scales, the Lennard-Jones model is characterized by strong liquid-like packing effects that are absent in the soft coarse-grained model and give rise to a pronounced slowing down of the dynamics in the LennardJones bead−spring model at low temperatures. Although the short-ranged structural properties of the two systems are inherently different, we can use the quickly equilibrated SCMF simulations to create starting configuration for the Lennard-Jones bead−spring model, which reduces the equilibration time needed as compared to a random starting configuration by at least 1 order of magnitude. When comparing the dynamic properties of the models, we see that the dynamics, as exemplified by the mean-squared displacement and the end-to-end vector correlation function, are significantly different for the two models. In the simulations of the soft, coarse-grained model the slowing-down of the dynamics is less pronounced for higher incompatibilities than for the harder Lennard-Jones bead−spring model because the local fluid structure is independent from the incompatibility. This method for acquiring equilibrated configurations for a Lennard-Jones bead−spring model grants better access to equilibrium properties of random block copolymer melts. For example, one can use these results to probe mechanical properties of random block copolymer melts. Thereby, creating

Figure 10. Position of the first maximum of the A−B pair correlation function. This quantity is used to gauge the time needed to equilibrate the system in SCMF simulations, the MD simulations with a random starting configuration and the MD simulations with the equilibrated SCMF configuration used as starting configurations.

repulsion of the Lennard-Jones potential. Therefore, we first relax the system in the microcanonical ensemble and restrict the maximum movement per bead and integration step to 0.05σ. At the same time, we rescale the velocities to put the temperature to kBT/εBB = 1 at the end of this simulation part. After 100 integration steps with Δt = 0.005τ, we switch on the thermostat and equilibrate the system for t = 0.05τmelt. Finally, we simulate the system in the NPT-ensemble at vanishing pressure, P = 0. The time evolution of the pair correlation function and the structure factor, cf. Figure 9, show that the structural equilibration of the system is very fast. Only the local fluid-like packing has to be established by the equilibration procedure. Both the morphology of A and B domains and the conformations on intermediate and large length scales as well as the correlation hole in the intermolecular pair correlation function are already captured by the SCMF simulations. For all three incompatibilities the desired structure is safely attained within three million integration steps which is equivalent to 0.14τmelt. The results of the last two sections can be summed up by Figure 10. It depicts the position of the first peak of the AB pair correlation function as a function of the simulation time. We can deduce from this figure that the equilibration of the Lennard-Jones bead−spring model with the help of an equilibrated starting configuration takes only a small fraction of the time compared to using a random starting configuration. To put this graph and the time scales into perspective, on a single CPU of the type Intel XEON X5570 at 2.93 GHz, 1 τmelt takes roughly 3600 h of computing time when using LAMMPS 1116


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(30) Daoulas, K. C.; Müller, M. Adv. Polym. Sci. 2010, 224, 197−233. (31) Müller, M.; Daoulas, K. C. J. Chem. Phys. 2008, 128, 024903. (32) Eastwood, J. W.; Hockney, R. W.; Lawrence, D. N. Comput. Phys. Commun. 1980, 19, 215−261. (33) Deserno, M.; Holm, C. J. Chem. Phys. 1998, 109, 7678−7693. (34) Dawson, K. A.; March, N. H. Phys. Lett. A 1983, 96, 460−462. (35) Rossky, P. J.; Doll, J. D.; Friedman, H. L. J. Chem. Phys. 1978, 69, 4628−4633. (36) Sewell, T. D.; Rasmussen, K. O.; Bedrov, D.; Smith, G. D.; Thompson, R. B. J. Chem. Phys. 2007, 127, 144901. (37) Plimpton, S. J. J. Comput. Phys. 1995, 117, 1−19. (38) Verlet, L. Phys. Rev. 1967, 159, 98. (39) Verlet, L. Phys. Rev. 1968, 165, 201−214. (40) Hoover, W. G. Phys. Rev. A 1985, 31, 1695−1697. (41) Melchionna, S.; Ciccotti, G.; Holian, B. L. Mol. Phys. 1993, 78, 533−544. (42) Hoover, W. G. Phys. Rev. A 1986, 34, 2499−2500. (43) Humphrey, W.; Dalke, A.; Schulten, K. J. Mol. Graphics 1996, 14, 33−38. (44) Paul, W.; Binder, K.; Heermann, D. W.; Kremer, K. J. Chem. Phys. 1991, 95, 7726−7740. (45) In very long simulation runs of the soft, coarse-grained model, t > 50τmelt, we observe a gradual change of the morphology to a defected lamellar structure without a change of the characteristic spacing. (46) Hömberg, M.; Müller, M. J. Chem. Phys. 2008, 128, 224911. (47) Local packing and density fluctuations have to be reequilibrated. Since the length scale of local packing effects as well as the correlation length of density fluctuations in a dense melt is on the order of the segment size, these local effects can be equilibrated on a time scale that is much shorter than the single-molecule relaxation time. The compressibility of the soft, coarse-grained model is sufficiently small for a correlation hole to develop in the intramolecular chain correlations function,22 thus intermolecular chain conformations and the large-scale molecular structure are correctly captured by the soft, coarse-grained model. Long wavelength density fluctuations, however, differ in the Lennard-Jones model and the soft, coarsegrained model, which are characterized by different isothermal compressibilities, and large scale diffusion is required to equilibrate these long wavelength fluctuations of the total density. Since fluctuations of the total density and the composition approximately decouple in dense multicomponent polymer melts and fluctuations of the total density are much smaller than composition fluctuations even in the soft, coarse-grained model, density fluctuations will not affect the morphology.

independent equilibrated configurations in order to perform statistically relevant measurements of said properties has become more efficient computationally.

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AUTHOR INFORMATION Corresponding Author *E-mail: (M.M.) [email protected]; (D.B.) [email protected].

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ACKNOWLEDGMENTS Financial support by the DFG priority program “polymer-solid contacts: interfaces and interphases” under Grant MU 1674/ 9-1 and computing time at the GWDG Göttingen, the HLRN Hannover/Berlin, and Jülich Supercomputing Centre are gratefully acknowledged. The support of the National Science Foundation through NSF MRSEC grant DMR-1536145 and allocation of computing time at the University of Utah Center for High Performance Computing are gratefully acknowledged.

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Properties of Random Block Copolymer Morphologies: Molecular

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