ВЫСОКОМОЛЕКУЛЯРНЫЕ СОЕДИНЕНИЯ, Серия С, 2013, том 55, № 7, с. 876-879
УДК 541.64:539.199
A MODEL FOR ROD-COIL BLOCK COPOLYMERS
© 2013 г. Stefan Dolezel, Hans Behringer, and Friederike Schmid
Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudinger Weg 7 D — 55128Mainz, Germany e-mail: dolezel@uni-mainz.de, friederike.schmid@uni-mainz.de
Abstract—A coarse-grained model for studying the phase behavior of rod-coil block copolymer systems on mesoscopic length scales is proposed. The polymers are represented on a particle level (monomers, rods) whereas the interactions between the system's constituents are formulated in terms of local densities. This conversion to density fields allows an efficient Monte Carlo sampling of the phase space. We demonstrate the applicability of the model and of the simulation approach by illustrating the formation of typical micro-phase separated configurations for exemplary model parameters.
DOI: 10.7868/S0507547513060068
INTRODUCTION
The study of nanostructure-forming materials is highly relevant from a technological point of view. For instance, the functionality of light-emitting or photovoltaic devices depends critically on the existence of nanoscale domain structures with large interface areas and optimized domain sizes that are, at least in one dimension, comparable to the exciton diffusion length. In practice, thin multilayer films are most commonly used. However, these are non-equilibrium states, whose structure depends critically on details of the process of fabrication (e.g. deposition), and whose lifetimes at higher temperatures are limited by interdiffusion. Alternative systems where the desired nano-structures form spontaneously as equilibrium states are clearly of interest. Promising candidates are block copolymers with two covalently linked components [1, 10—18]. If the stiffness disparity of the blocks is high (rod-coil block copolymer), they will microphase-separate already at relatively low molecular weight into a variety of nanostructures with morphologies that can be tuned, e.g., by varying the chemistry and the molecular weight of the coil component. This can be exploited to design functional materials with tunable properties.
The phase behavior of rod-coil block copolymer systems is much richer than that of conventional flexible block copolymers, and far less understood [1, 2]. The phase diversity results from the interplay between the conformational entropy of the coils, the "conventional" incompatibility between chemically different monomers, and local packing effects which lead to liquid crystalline or even (semi)crystalline order in the rod domains. A large number of self-assembled structures has been observed experimentally in systems of rod-coil copolymers, such as tilted and untilted smec-tic phases, wavy lamellar, zigzag, arrowhead, perforat-
ed lamellar, hockeypuck, hexagonal strip, and nano-rod phases [1]. Some, but not all of these phaseshave been predicted or reproduced in theoretical studies (see, e.g., [1, 2, 19, 20] and references therein). Computer simulations are still scarce [3—9, 21—24]. To simulate these materials, structures on very different length scales have to be taken into account. This clearly calls for a hybrid multiscale treatment.
MODEL
In the present paper, we propose a simple model for the study of rod-coil block copolymer self-assembly. To capture the relevant physics, the models must account for the possibility of local orientational and crystalline order in the block domains as well as for the conformational degrees of freedom of the coil domains. Therefore, we treat the rod and the coil blocks at different coarse-graining levels. The rods are represented as elongated impenetrable spherocylinders. The coils are modeled as flexible chains, whose monomer interactions are defined in terms of an Edwards Hamiltonian that depends on local densities [25—27]. The interactions between rods and coils is defined in terms of a density functional and will explicitly account for the extended size of the rods.
The system is built up of N rod-coil block copolymers. Each rod-coil block copolymer consists of one rod which is connected to a polymeric coil with n monomers (see Fig. 1). Each coil part consists of n - 1 Gaussian springs and is linked to its associated rod totaling in n springs for each rod-coil block copolymer. The rods are hard spherocylinders with length L and diameter D. The total energy of a three-dimensional system with N rod-coil block copolymers comprises four different contributions
P ^ = P ^ g + CC + P ^ CR +P ^ RR. (1)
A MODEL FOR ROD-COIL BLOCK COPOLYMERS
877
L
D
n - 1
Fig. 1. Cartoon of our model. Rod of diameter D, length L and unit vector u and the connected coil consisting of n monomers concatenated by springs.
Here p denotes the inverse temperature, i.e. p = 1/kBT with kB being Boltzmann's constant. The first term
N n
ß * g = 2 H(R
a=1 i=1
a, i-1
■ R a, i )
(2)
models the Gaussian chain statistics of the polymer coils and provides the connectivity [25, 26]. Each polymer chain consists of n "point-like" monomers at positions RaJ, i = 1,2,...,n with the polymer index a = 1,2,..., N. The monomer positions R ai can take any value in the three-dimensional volume V (simulation box, off-lattice model). The monomer with index i = 0 is associated with a "virtual" monomer, fixed to one end of the spherocylindrical rod of the given copolymer (see Fig. 1). This monomer only contributes to p M g (via bond/spring 0 ^ 1) in Eq. (1), but not to the monomer density which determines p M CC and p M CR (see below). The strength of the Gaussian bond energy is given by the spring constant k which is related to the averaged bond length b by k = 3/b2 in three space dimensions.
The second term in Eq. (1) models the excluded volume interactions of the monomers and is given by the first term in the corresponding virial expansion
ß * CC =X2C \dr (Pc(r))2
(3)
with pC(r) denoting the local monomer density related to a given configuration [25, 26]. The parameter x CC needs to be positive to have a stable polymer configuration. The rod-coil interaction is modeled by hard-body constraints (a monomer may not enter a sphero-cylinder) and an additional density-dependent interaction which describes soft interactions of longer range.
ß * CR =
œ
: monomers enter rods XCR Jdr pC(r)pr(r) : otherwise (4)
The field pR (r) denotes the rod density and the parameter xCR the strength of the monomer-rod interaction.
Finally, the rod-rod interactions are pure hard-body interactions between spherocylinders.
ß * RR =
œ : rods overlap 0 : otherwise
(5)
We note that the expression p M does not contain an explicit temperature dependence. The connectivity is provided by chemical bonds which are basically unbreakable for the temperature range of interest. The remaining energy parameters xCC and x CR are effective parameters that stem from integrating out microscopic degrees of freedom. They consequently exhibit a temperature dependence which is however (assumed to be) weak in the range of interest. We will therefore investigate the phase behavior in dependence of these x -parameters instead of looking explicitly at the temperature.
The phase behavior is investigated by means of Monte Carlo simulations. A new polymer conformation is generated by a set of trial moves. These comprise simple translational moves for the monomers of the coil parts whereas new rod configurations are proposed in a two-fold manner, namely by spatial translations and by rotations. We will now briefly discuss how a discretized version of the energy terms (3) and (4) can be generated for the use in such a particle-based Monte Carlo simulation [28—30]. To this end the configuration of monomers and rods has to be converted into density fields pC and pR. The conversion is based on the introduction of a regular rectangular grid of cells so that for each cell a corresponding discretized density can be computed. For a cell I with center node rf the local monomer density for all points inside is approximated by pC(rf) = pCI = Cf / Vf with Cf being the number of monomers in the cell f. Here, Vf denotes the volume of cell f accessible to the monomers (see below, all cells have the same form and volume Vu = 8x5y5z with 5 j being the discretization size of the mesh in j-direction). The excluded volume energy of the polymers is then given by
\2
(6)
^ ^r (Pc(r))2 = ^ XvJ Cl
2 V 2 I V y
Let us just remark that the conversion of the monomer positions to a local density field introduces a further model parameter via the discretization length ô j or the
1
n
u
0
2
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(a) (b)
Fig. 2. Snapshots of rod-coil block copolymer systems in different states. The (dark grey) coil parts are displayed in a tube-representation of the bonds. The reduced density in both cases is p* = 0.59. (a) Rods with aspect ratio L / D = 3 attached to chains of n = 40 monomers. The cell size is 8j = 1.5. (b) Rods with aspect ratio L/D = 2 attached to chains of n = 60 monomers. The cell size is 8 j = 2.
discretization volume Va. These parameters can be interpreted as effective monomer size or volume, respectively. The contribution of a given monomer to the local field is here obtained by a simple "binning" procedure. We note, however, that higher order schemes can be applied which lead to a smoother assignment to several neighboring cell nodes, for details see e.g. [30].
The local rod density associated with a given cell I is just the volume fraction occupied by rods. This volume fraction OI is determined by means of a Monte Carlo integration scheme: For a cell a number of random positions is generated, the fraction of random positions inside rods is then taken to be the volume fraction, i.e. the rod density pRI of cell I is just OI. It is not necessary to generate the distribution of random points in a
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