Random Block Copolymers: Structure, Dynamics, and Mechanical

Article pubs.acs.org/Macromolecules

Random Block Copolymers: Structure, Dynamics, and Mechanical Properties in the Bulk and at Selective Substrates Birger Steinmüller,† Marcus Müller,*,† Keith R. Hambrecht,‡ 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: Using computer simulations of a soft, coarse-grained model and a Lennard-Jones bead−spring model, we investigate the behavior of random block copolymer (RBCP) blends in the bulk and in the vicinity of a solid substrate. The RBCP is comprised of six random sequences of A and B blocks and adopts a microemulsion-like morphology as we increase the incompatibility between A and B segment species. The soft, coarse-grained model is used to efficiently equilibrate the morphology and molecular conformations in the melt, and its explicit molecular conformations are used to generate equilibrated starting configurations for the bead−spring model. Since the pure A and B melts are structurally asymmetric, A-rich and B-rich domains differ in their density and viscosity in the melt or mechanical properties in the glassy state, respectively. We demonstrate that the self-assembled morphology in the melt gives rise to a viscoelastic transient plateau in the stress autocorrelation function. While an analysis of the entanglement density reveals a slight increase of the number of entanglements in response to microphase separation, the viscoelasticity of the RBCP blend chiefly stems from the slow morphological relaxation and the transient trapping of blocks inside domains. Upon quenching the microphaseseparated structure below the glass transition temperature, the shear modulus increases about an order of magnitude compared to the viscoelastic plateau. The structural asymmetry of the segment species gives rise to spatially heterogeneous, local bulk and shear moduli that correlate with local composition fluctuations. The vicinity of a solid substrate that prefers one segment species gives rise to variations of the composition that propagate about three molecular extensions into the bulk. In this extended interphase, we observe an oscillatory decay of the composition and the local mechanical properties.

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INTRODUCTION Combining two different segment species into a backbone of a long flexible macromolecule, one can create a composite material that combines the attributes of both segments and yet avoid macroscopic phase separation. Typical examples are diblock or multiblock copolymers that self-assemble into spatially periodic microphases.1 If one combines multiblock copolymers with a random sequence of blocks,2−6 this random block copolymer (RBCP) blend will either form a periodic microstructure7 or remain disordered.8−10 If the number of blocks is small, the repulsion between unlike blocks is large, the individual blocks are comprised of many repeat units, and the block density is large, then fluctuation effects are less important and lamellar order has been observed in random block copolymers7 and correlated random block copolymers.11 Otherwise, this mixture of many polymer sequences exhibits a disordered microemulsion-like structure over a wide range of parameters (e.g., composition and incompatibility between the two species A and B), but the local composition exhibits strong transient heterogeneities. These fluctuating domains give rise to unique viscoelastic and mechanical properties in the molten and glassy state, respectively. In the melt, individual blocks get temporarily trapped inside segregated A-rich or B-rich domains, and this transient localization gives rise to an elastic response on © XXXX American Chemical Society

intermediate time scales. Additionally, the change of the molecular conformations in response to the formation of a microemulsion will also affect entanglements. In the glassy state, the domains exhibit different local mechanical properties, which chiefly stem from excluded volume effects and fluid-like packing of the segments. We compare the strength of the elastic properties at intermediate times in the viscoelastic melt with the glassy moduli. If one brings the system in contact with the selective substrate, these bulk heterogeneities are aligned with the external perturbation. In spite of being far away from a phase transition, however, surface effects propagate a distance of order of the end-to-end distance, Re, into the bulk. This extended region, where the solid substrate imparts changes onto the structure and dynamics of the RBCP, is denoted as the “interphase”, and we analyze its structural, dynamical, and mechanical properties. We investigate this rich behavior of these multicomponent systems by large-scale computer simulations of coarse-grained models. Our paper is arranged as follows: First we describe the models and simulation techniques and briefly summarize the Received: October 15, 2012 Revised: November 23, 2012

A

dx.doi.org/10.1021/ma302151z | Macromolecules XXXX, XXX, XXX−XXX

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where ρ0 = nN/V denotes the segment number density. The first contribution limits fluctuations of the total density and κ0N is proportional to the inverse isothermal compressibility. ϕ̂ A(r) and ϕ̂ B(r) are the local density of A and B segments, respectively. The coefficient α accounts for different segmental volumes.10 For α > 0, A segments are smaller than B segments. The second term describes the repulsion between unlike segments, and χ0 denotes the (bare) Flory−Huggins parameter. The interaction with selective, impenetrable walls, which are located at x = 0 and x = D, are described by an additional segment-wall potential of the form13,15

mapping between the soft, coarse-grained model and the Lennard-Jones bead−spring model, which we have established in previous work.10,12 Then we discuss molecular structure and viscoelastic properties of RBCP blends in the bulk phase above the glass transition. Finally, we compare mechanical properties of RBCP blends in the bulk and at selective solid substrates in the glassy state. The paper closes with a brief summary and outlook.

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MODELS AND TECHNIQUES Random Block Copolymers with a Symmetric Sequence Distribution. In our simulations, random block copolymers (RBCP) are comprised of Q = 6 blocks. Each block, in turn, consists of a sequence of M = N/Q segments, all of which are of one type A or B, respectively.3 A melt of these RBCP is comprised of 2Q = 64 different species of chains not accounting for the head−tail symmetry of the polymers.9 Since the total number, n, of chains in the simulated system is much larger than the number of molecular sequences, a melt of RBCP can be conceived as a mixture of many components.4,6,9 The concentration of a specific sequence is fixed during a Markovian polymerization process.2 Like in previous work, we only study symmetric sequence distributions,10 i.e., a sequence (e.g., BABBBA) and its counterpart (e.g., ABAAAB), in which all A segments are replaced by B segments and vice versa, occur in equal concentrations. 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, all 64 sequences occur with equal concentration in the bulk. We consider n = 8320 ≫ 64 polymers in a simulation cell of volume V ≈ 16 × 8 × 8Reo3, where Reo denotes the mean-squared end-to-end distance in the absence of nonbonded interactions. Thus, the simulation box contains 130 molecules of each sequence. Two models are used to describe RBCP melts: (i) a soft, coarse-grained model and (ii) a Lennard-Jones bead−spring model. The former model allows for a computationally fast equilibration of the morphology at intermediate and high segregation, while the latter arrests in a glassy state and allows for the investigation of local mechanical properties. We briefly describe both models in turn. Soft, Coarse-Grained Model and Single-Chain-inMean-Field Simulations. In the soft, coarse-grained model, the contour of a linear, flexible polymer is discretized into N = 120 beads that are connected by harmonic springs. The bonded energy in terms of the segment coordinates, {ri(s)} with i = 1, ..., n and s = 1, ..., N, takes the form of a discretized Edwards Hamiltonian:10,13,14 /b({ ri (⃗ s)}) = kBT

n

N−1

∑∑ i=1 s=1

ρ /nb − wall = 0 kBT N

wall

eo

⎛ x2 ⎞ ̂ ⎟(ϕA − ϕB̂ ) exp⎜ − 2 ⎝ 2Δ wall ⎠ (3)

Like previous studies we use the value Δwall = 0.15Reo. Λ is a parameter describing the strength of the preference of the solid substrate to one of the segment species; negative values of Λ correspond to a substrate that preferentially interacts with the A species. The local segment densities and the integral, eq 2, are evaluated on a collocation grid, {c}, where each grid cell has the volume ΔL3. n

ϕÂ (c) =

N

∑∑ i=1 s=1

γi(s) ρ0 ΔL3

Π(c , ri(s))

(4)

where Π is the characteristic function of the grid cell, c, and γi(s) = 1 if the segment s on polymer i is of type A and zero otherwise. A similar expression holds for the B density. Since there is a large separation between the stiff bonded interactions, which define the molecular architecture, and the weak nonbonded interactions, which determine the microemulsion-like morphology, we use the single-chain-in-meanfield (SCMF) simulation algorithm to study the statistical mechanics of the soft, coarse-grained model.14 In SCMF simulations, the weak nonbonded interactions are temporarily replaced by external fields, wA and wB acting on the different segment species. /nb({ri(s)}) = ρ0 ΔL3 ∑ [wAϕÂ + wBϕB̂ ] kBT c

(5)

After 10 Monte Carlo steps, the external fields that mimic the interactions with the fluctuating environment of a segment are updated according to wA(c) =

3(N − 1) | ri (⃗ s) − ri (⃗ s + 1)|2 2R eo 2

∂/nb 1 ρ0 ΔL3 ∂ϕA(c)

(6)

and a similar relation holds for wB. This procedure partially restores fluctuation effects, and we have verified for selected parameters that the results of the SCMF simulations are very close to Monte Carlo simulations using eq 2. The chain conformations are updated by smart Monte Carlo (SMC) moves.16 The time scale of structural relaxation is measured in units of the time τmelt that it takes a center of mass of a homopolymer at χ0N = 0 and α = 0 in a melt of chains with identical length and density to diffuse a distance on the order of Rmelt2; τmelt = Rmelt2/Dmelt, where Dmelt denotes the self-diffusion coefficient in a homopolymer melt and Rmelt2 = 1.44Reo2 is the measured mean-squared end-to-end distance of polymers in a homopolymer melt. In the SCMF simulations we obtain τmelt =

(1)

where kBT is the thermal energy scale. The nonbonded interactions describe the near-incompressibility of the dense polymer liquid and the repulsion of the different segment species A and B.10,13,14 ⎡ κ0N ([1 − α]ϕÂ + [1 + α]ϕB̂ − 1)2 2 ⎤ χN − 0 (ϕÂ − ϕB̂ )2 ⎥ ⎦ 4 (2)

ρ / ′nb = 0 kBT N

∫ d3r Δ ΛN/R

∫ d3r ⎢⎣

B


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Figure 1. Snapshots for different values of the interaction parameter as indicated below each figure. The substrates attract only B monomers. The top row (a−c) corresponds to the Lennard-Jones bead−spring model, while the bottom row (d−f) depicts the morphology of the soft, coarse-grained model. These figures were produced using VMD.25

2.75 × 105 SMC steps, where each segment on average had the chance of one trial displacement in a SMC step. Lennard-Jones Bead−Spring Model and Molecular Dynamics Simulations. In the Lennard-Jones bead−spring model, the molecular architecture is also represented by a bead−spring model. Segments are connected via finitely extensible nonlinear elastic (FENE) springs17 /b({ri(s)}) kr =− 0 kBT 2

2

n

N−1

∑∑ i=1 s=1

The interaction between the Lennard-Jones beads and a solid substrate is modeled by a 9−3 potential18 ⎛ 2 σ9 σ3 ⎞ ULJ,wall(Δx) = εwall ⎜ − ⎟ 9 ⎝ 15 Δx Δx 3 ⎠

which is routinely used in bead−spring models and stems from integrating the Lennard-Jones interactions inside the half space of the solid substrate. Δx denotes the distance between the bead and the wall. εwall = 2 in Lennard-Jones units if not stated otherwise. Here the selectivity of the potential is not achieved through different interaction parameters, but the cutoff. For the substrate to be attractive, we choose a cutoff of Δxc = 2.5σ; when we want it to be repulsive, we choose Δxc = 0.715σ. The statistical mechanics of the Lennard-Jones bead−spring model has been studied by the program package LAMMPS.19 The equations of motion were integrated using the velocityVerlet algorithm20,21 employing an integration step of Δt = 0.005τLJ in units of the Lennard-Jones time scale τLJ ≡ σ(m/ ε) 1/2 with m being the bead mass. A Nosé−Hoover thermostat22,23 is utilized to maintain the temperature T in Lennard-Jones units, and for simulations in the isobaric NPT ensemble a Nosé−Hoover barostat23,24 fixed the pressure to P = 0. Since the vapor pressure of a polymer liquid is vanishingly small, this choice corresponds to a RBCP melt being in contact with a free surface. At constant pressure, the density of the components depends on temperature or εAA, and eventually the densification will lead to a glassy arrest. The time scale of the structural relaxation is measured in units of the time τmelt that it takes a center of mass of a homopolymer, εAA = 1.0, in a homopolymer melt of the same chain length and density to diffuse a distance on the order of Rmelt2 = 100σ. In the MD simulations we obtain τmelt = 105τLJ = 2.1 × 107 MD integration steps, Δt. In our previous study we have identified parameter such that the structure and thermodynamics of the soft, coarse-grained model resembles that of the Lennard-Jones bead−spring

⎛ |r (s) − ri(s + 1)|2 ⎞ ⎟ ln⎜1 − i r0 2 ⎝ ⎠ (7)

with the maximal bond lengths, r0 = 1.5σ, and the spring constant, k = 16. Each molecule is comprised of N = 60 Lennard-Jones beads. The nonbonded interactions take the form /nb({ri(s)}) 1 = 2 kBT

n

⎛ ′ ⎜⎜ULJ(r ) − ULJ(rc) s,t=1 ⎝ N

∑′ ∑ i,j=1

− (r − rc)

dULJ(r ) dr

⎞ ⎟ ⎟ r = rc ⎠

(8)

where the prime indicates that the sum runs only over pairs of particles. r = |ri(s) − rj(t)| is the distance between the interacting particles, and ULJ(r) denotes the Lennard-Jones potential ⎛ σ 12 σ6 ⎞ ULJ(r ) = 4εij⎜ 12 − 6 ⎟ ⎝r r ⎠

(10)

(9)

σ = 1 sets 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. In the following we fix εBB = εAB = ε = 1 in LennardJones units, and we vary εAA. C


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model. We have found very good agreement for εAA ≤ 1.6, but the agreement between the morphologies of both models deteriorated for εAA = 2.0 and 2.3.10 In Figure 1, we illustrate this agreement for a RBCP blend in contact with a solid substrate that attracts one segment species. We use the same parameters as for the bulk, and we choose the strength of the surface interaction sufficiently attractive to one of the components for the surface layer composition be almost pure. Figure 1 demonstrates good visual agreement between the two different models, which is corroborated by the profiles of the A and B densities as a function of the distance from the interface. In addition to matching of the bulk parameters, we tuned the preference of the substrate for the B species such that the substrate is almost completely covered with the preferred component. A further increase of the surface preference has only little effect because the surface potential directly affects the segregation only in the vicinity of the substrate. The equilibration time of the soft, coarse-grained model, however, is at least one order of magnitude less than that of the LennardJones bead−spring model. The computational advantage of the soft, coarse-grained model increases as we increase the value of εAA because larger values of εAA not only give rise of an increased incompatibility between A and B species but also result in a densification of the A-rich domains. The latter effect eventually leads to a glassy arrest of the morphology in the Lennard-Jones model. This glassy behavior is illustrated by the mean-square displacements of a homopolymer melt in Figure 2, where all

Jones model with N = 60 by representing the center of mass of two neighboring soft beads of an equilibrated configuration of the soft, coarse-grained model by a single Lennard-Jones particle. As in the previous study of bulk properties, we initially restrict the bead motion during the short equilibration of local bead overlaps. Since the soft, coarse-grained model correctly captures the correlation hole and structural correlations on the length scale Re and beyond, the relaxation requires only of the order 0.3τmelt. (ii) The so generated, equilibrated configuration at kBT = 1 is quenched over 106 MD steps to kBT = 0.1 or 0.3. The system is further equilibrated for another 4 × 106 MD steps in the isothermal−isobaric ensemble at P = 0. Measuring Local Mechanical Properties in the Glassy State. In order to measure the local compression and shear moduli in the glassy state, we apply computational techniques that have been devised by Lutsko28,29 and Yoshimoto et al.30 and apply them to the glassy morphologies of our LennardJones RBCP melts. When a system is linearly deformed, each position r is transformed according to r′ = Jr, and the strain tensor is given in terms of deformation matrix, J, by 1 eij = (JTil Jlj − δij) (11) 2 where i, j, l denote Cartesian coordinates and a summation over double indices is implied. The local mechanical constants, Cijkl, are defined by the first derivative of the local stress, ⟨τ̂(r)⟩ij, with respect to the strain, εkl. ∂⟨τ(̂ r′)⟩ij

Cijkl(r) = lim

∂ekl

J → 3

r = Jr

′ The local stress at position r can be computed via28−30 τiĵ (r) =

∑

pα , i pα , j m

α

δ(r − rα) +

(12)

⎛ ∂U ⎞ rαβ , irαβ , j ⎟⎟ ⎝ ∂|rαβ| ⎠ |rαβ|

∑ ⎜⎜ α

Random Block Copolymers: Structure, Dynamics, and Mechanical

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