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Multiscale Simulation for Entangled Polymer Systems

Polymers are macromolecules which are composed of monomers and they have string like structures. Dynamics of polymers are strongly affected by such structures. Even for cases of relatively short polymers, dynamic properties are different from monomeric systems. If the polymerization index is larger than a characteristic value, motion of polymers are constrainted by surrounding other polymers. Such topological constraints are called "entanglements".

It is difficult to perfome molecular dynamics (MD) simulations for entangled polymer systems due to their large polymerization indexes. Thus several coarse-grained models are proposed. For example, the Doi-Edwards theory gives a simple interpretation based on the picture of a polymer in a tube. The primitive chain network (PCN) model enables fast simulations dynamic properties by using a description of networks composed of entanglement points.

However, these coarse-grained models are introduced phenomenologically and their theoretical basis is not fully understood even they have achieved many successful results. Besides, we need several phenomenological parameters to perform simulations based on these models. Usually the parameters are determined experimentally. If we want to combine or compare a coarse-grained simulation and a microscopic simulation, it will be natural to use parameters derived from microscopic simulations.

We are developping a method to combine microscopic MD simulations and the PCN model. Recently methods to extract primitive path information which describes entanled polymer systems are proposed. Using the methods we can calculate several statistical values. We are calculating parameters used in the PCN model from MD simulation data. We are also comparing PCN simulations and MD simulations.

We are also developping a coarse-grained simulation model which is suitable for embedding to a macroscopic simulation model. To simulate macroscopic flow behaviors of entangled polymers, fast rheological simulators based on coarse-grained models are required. We are implementing the fast rheology simulator based on the single chain slip-spring model, and accelerating it by a graphic processor unit (GPU).

Density Functional Theory for Block Copolymer Systems

Block copolymers are polymers which consists of several blocks. Each blocks contains different monomer species, and they are connected chemically (Figure 1). (Polymers which consists of only one monomer species are called homopolymers.) Polymers which consist of different monomer species are known to be immiscible and cause phase separation, just like mixtures of water and oil. However, in block copolymer systems, the macroscopic phase separation cannot occur because of the chemical junction between blocks. As a result, we can observe the microscopic phase separation structures (microphase separation structures) of which scale is about 10 nm - 1 μm (Figure 2).

diblock.png triblock.png
A diblock copolymer
A triblock copolymer
Figure 1: Block Copolymers

diblock_lam.png diblock_hex.png
(a) A lamellar structure
(b) A cylinder structure
Figure 2: Microphase separation structures
formed by diblock copolymers

Microphase separation structures formed by block copolymers are affected by several parameters; strengthes of interaction between monomers (called &H967; parameters), polymerization index or architecture of block copolymers, volume fractions (for blend systems), and so on. If we can predict the bahaviour of block copolymer systems by theories or simulations from tehse parameters, it is quite useful to understand the physics of block copolymers or to design polymerica materials

The self consistent field (SCF) theory is widely used to study block copolymer systems. The SCF is very accurate and can be applied for arbitrary sytems and fast algorithms for the SCF simulations are developped. The SCF simulations achieved successful results to create phase diagrams, to predict phase behaviours for complicated systems (such as triblock copolymers), or to study dynamics of block copolymers. However, the SCF requires large computational costs and we need powerful computer environments such as parallel PC clusters for simulations for large systems such as 3 dimensional systems.

We have developped the density functional theory (DF), which requires much less computational costs. Thus large simulations based on the DF can be perfomred on a normal PC. While the accuracy of the DF is less than the SCF, the DF is sufficient to study the behaviour of the system qualitatively. However, the traditional DF is limited to several systems such as homopolymer blends or diblock copolymer melts (it cannot be applied to arbitrary systems which can be handled by the SCF). We developped the DF which can be applied for arbitrary systems just like the SCF (all the parameters required for our DF is same as the SCF).

Block Copolymer Micelles / Vesicles

If the block copolymers of which blocks have different affinity to the solvents into selective solvents, micellar structures such as micelles or vesicles are formed (Figure 3). This is similar to the case of the aquous solutions of surfactants.

diblock_micelle.png diblock_vesicle.png
(a) Micelle
(b) Veislce
Figure 3: Micellar structures in block copolymer solutions

Block copolymer micelles and vesicles can be used as micro capsules, thus they are expected to be useful for the drug delivery system (DDS). Currently block copolymer micelles and vesicles are studied well experimentally, but theoretical works or simulation works are not studied so well.

Recently the SCF simulations are applied to block copolymer micelles and vesicles. By using our DF, fast simulations for block copolymer micellar systems will be acheved.

Mesoscale Dynamics of Block Copolymers

Phase separation structures formed by block copolymers are known to depend several parameters such as temperatures. For expamle, if we quench homogeneous block copolymer melts at high temperature into low temperature, block copolymers undergo phase separation and finally form equilibrium microphase separation structures.

The SCF or the DF is originally designed to predict the equilibrium structures, and they cannot predict dynamical processes. Dynamics can be handled by using the extension of the SCF or the DF into dynamics (so called the dynamic density functional theory (DDF)). The DDF theory is combined with the SCF or the DF to study the dynamics of block copolymer systems.

However, popularly used DDF is based on the phenomenological model, and thus it is not clear how the microscopic equation of motion of monomers and the mesoscopic dynamic equation are related. Besides, at the characteristic scale of the microphase separation structures, the effect of the thermal fluctuation is quite large and it is unrealistic to neglect the fluctuation effect. Recently the theories to derive the mesoscopic dynamic equation from the microscopic equation of motion have been developped. By combining these theories with our DF, we expect that realistic and fast simulations for mesoscopic dynamics of block copolymers will be possible.

In fact, we reproduced the block copolymer vesicle formation process by using the dynamic equation with sufficiently large noise (such a process cannot be reproduced by previous phenomenological models). In the vesicle formation process, the processes which are driven by thermal fluctuations (such as the collision and coalescence process of micelles) is quite important (Figure 4). We expect that our model will be useful for several situations where the effect of the thermal noise is large.

Figure 4: The formation mechanism of a block copolymer vesicle

Development of Simulators

We need many parameters to perform DF simulations, and we need to perform complicated calculations for each systems. Thus it is inefficient to develop or reconstruct simulators for new systems, and simulators for generic systems will be useful.

I have developped the DF simulator which can be read parameters for arbitrary block copolymer systems. Input parameters are, for example, the system size, the number of lattice points, architectures of block copolymers, interaction between monomers, volume fractions of polymers. By developping such a simulator, we can perform simulations for many systems with ease.

I consider that it is not good to make such simulators to be closed source, especially for academic use. If the simulators are closed, we have to reconstruct new simulators if we want to perform the simulations (of course, the commercial simulators can be closed source). For the development of refined thories or simulation methods, the simulators should be used freely. Thus I distribute the simulator as the free software under the GNU general public license (GPL).

Fast and Accurate Algorithms

It will be clear that the numerical costs required for the simulations and the accuracy of the simulations depend on the model used. They also depend algorithms used. Therefore, algorithms are very important for simulations

However, in soft matter simulations, algorithms are despised because there are so many models and we have to choose appropriate algoritms case-by-case. Besides, there are not sufficient works about algorithsm for strongly non-linear field models or stochastic differential equations.

We also study algorithms while we are making simulators. We choose or construct algorithms proper to the simulations, or studying the validity or accuracy of existing algorithms. These works are rather primary compared with exparts' work, but such works will improve the numerical efficiency and the accracy of our simulations.

Recently, there are multi-core CPUs or special purpose co-processors are developped and sold. Such new processor architectures are interested in the point of view of algorithms. I am helping the study of algorithms on such special architectures.

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Last modified: Mon Jul 4 12:24:44 JST 2011