Cell Dynamics Simulation Research Articles

Morphological Investigation of Lamellae Patterns in Diblock Copolymers under Change of Thickness and Confinement in Polar Geometry

Mathematicians collaborate computationally with practical scientists to predict the novel morphologies soft materials, having potential applications in nanotechnology. This work presents the novel lamellae morphology that the confined diblock copolymer system displayed inside the polar geometry in order to achieve this, cell dynamics simulations (CDS) model is employed on a polar mesh to study the special impacts of confinement on the self-assembly behavior of diblock copolymer systems. Under the numerical formulation of the isotropic Laplacian with polar geometry employed in the CDS model, the set of 9-point stencil is considered to discretize the Laplacian operator on polar mesh system. FTCS finite difference mechanism is taken into consideration for discretization of Laplacian operator due to its computational efficiency and accuracy. Cell dynamics simulation is efficient simulation model for large-scale simulation as compared to the other models available in literature. In the simulation, evolutionary lamella morphology is presented with and without confinement for various sizes of the circular annular pores. A comparative review of lamella morphologies achieved by simulation in comparison of experimental study is presented, coupled with a thorough investigation of symmetric and asymmetric morphologies in captivity.

Implementation of Crank–Nicolson scheme in cell dynamics simulation for diblock copolymers

Many computational techniques have been designed to study block copolymer systems. Cell Dynamics Simulation (CDS) technique is one of those. It is based on solving partial differential equations (PDEs) and is computationally very fast compared to other simulation methods. In this contribution the focus is made on finite difference schemes and their use in computer simulation of PDEs involved in diblock copolymers systems. The Crank–Nicolson (CN) scheme of finite difference method is implemented in CDS method. The CN scheme is unconditionally stable but slower compared to Forward Euler’s method which is fast but not very stable.

Accelerating Cell Dynamics Simulations of Soft Materials using CUDA-GPU

Soft materials are a highly demanded class of research for predicting the characteristics of phase separation and self-assembly into nanoscale structures. One of the most well-known methods to demonstrate and simulate dynamic behavior of particles, such as particle tracking, and to consider different effects of simulation parameters is cell dynamic simulation. In fact, cell dynamic simulation as a cellular computerization can be used to investigate different aspects of morphological topographies of soft material system. To achieve accurate and efficient computational results for cell dynamic simulation method, two factors are essential and vital: first, computing/calculating time-scale; and second, simulation system size. Consequently, finding available computing algorithms and resources such as sequential algorithm for implementing a complex technique and achieving precise results is critical and expensive. Therefore, it is very important to consider a parallel algorithm and programming model to solve time-consuming and massive computational processing. This paper presents a novel GPU-based parallel acceleration approach to cell dynamics simulations (CDS) for a spherical phase diblock copolymer under a shear flow. A new achievement is based on a commodity NVIDIA Quadro K5000 graphic card and Compute Unified Device Architecture (CUDA) C language.

Rigorous embedding of cell dynamics simulations in the Cahn-Hilliard-Cook framework: Imposing stability and isotropy

We have rigorously analyzed the stability of the efficient cell dynamics simulations (CDS) method by making use of the special properties of the local averaging operator 〈〈*〉〉-* in matrix form. Besides resolving a theoretical issue that has puzzled many over the past three decades, this analysis has considerable practical value: It relates CDS directly to finite-difference approximations of the Cahn-Hilliard-Cook equations and provides a straightforward recipe for replacing the original two- or three-dimensional (2D or 3D) averaging operators in CDS by an equivalent (in terms of stability) discrete Laplacian with superior isotropy and scaling behavior. As such, we open up a route to suppress the unphysical reflection of the computational grid in CDS results (grid artifacts). We found that proper rescaling of discrete Laplacians, needed to employ them in CDS, is equivalent to introducing a well-chosen time step in CDS. In turn, our analysis provides stability conditions for phase-field simulations based on the Cahn-Hilliard-Cook equations. Subsequently, we have quantitatively compared the isotropy and scaling behavior of several discrete 2D or 3D Laplacians, thereby extending the significance of this work to general field-based methodology. We found that all considered discrete Laplacians have equivalent scaling behavior along the Cartesian directions. In addition, and somewhat surprisingly, known "isotropic" discrete Laplacians, i.e., isotropic up to fourth order in |k|, become quite anisotropic for larger wave vectors, whereas "less isotropic" discrete Laplacians (second order) are only slightly anisotropic on the whole |k| range. We identified a hard limit to the accuracy with which the discrete Laplacian can emulate the two important properties of the optimal (continuum) Laplacian, as an improvement of the isotropy, by introducing additional points to the stencil, will negatively affect the scaling behavior. Within this limitation, the discrete compact Laplacians in the DnQm class known from lattice hydrodynamics, D2Q9 in 2D and D3Q19 in 3D, are found to be optimal in terms of isotropy. However, by being only slightly anisotropic on the whole range and enabling larger time steps, the discrete Laplacians that relate to the local averaging operator of Oono and Puri (2D) and Shinozaki and Oono (3D) as well as the less familiar 3D discrete BvV Laplacian developed for dynamic density functional theory are valid alternatives.

Large scale simulation of block copolymers with cell dynamics

Cell dynamics simulation (CDS) is a fast method to simulate kinetic processes in phase separating systems of large sizes. We review this method applied to block copolymer systems and illustrate it on a set of physical phenomena occurring in these systems. Achievements of CDS demonstrate, that the method can describe such complex phenomena rather well and therefore, is a complementary method to other more elaborate but slow techniques to which it can serve as a precursor.

Influence of Electric Field on the Phase Transitions of the Hexagonal Cylinder Phase of Diblock Copolymers

We have used the cell dynamic simulations (CDS) method to study the evolution of asymmetric and symmetric diblock copolymers under electric fields. For symmetric diblock copolymers, long-range-ordered lamellar phases form readily under electric fields. For asymmetric diblock copolymers, sphere-to-cylinder phase transitions occur rapidly when strong electric fields are applied, but it takes longer for the system to form hexagonal cylinder structures. The results of these simulations suggest that the sphere phase is stable under weak electric fields, but a threshold electric intensity exists for the sphere-to-cylinder phase transition. In addition, we also studied the kinetic pathways of the transition of the lamellar phase to the hexagonal cylinder phase of the asymmetric diblock copolymers under electric fields. Hexagonal cylinder structures form when the lamellar phase is subjected to a sudden temperature jump. The scattering functions suggest that the hexagonal cylinder structures are very regular and possess very few flaws.