A Multi-scale Simulation Method And A Large-scale Simulation Method For Electrolytes

There are large amount of soft condensed materials and biological systems whose physical properties are dominated by the electrostatic interactions between their constituents.These systems are generally called the Coulomb many-body systems.Studying the Coulomb Many-body system is a basic issue in soft matter physics and biophysics.Developing efficient and accurate computer simulation methods is the key to under-standing the Coulomb Many-Body system.However,this is a hard problem,due to the long-range nature of the electrostatic interaction.The classical methods are mainly based on the Periodic boundary condition with Fourier transform or multiple expansion.These methods achieve good accuracy and efficiency,but they also have some unphysical effect.In this thesis,we introduce two new Monte Carlo simulation methods.One is a multi-scale Monte Carlo method and the other is a GPU-based large-scale Monte Carlo method.Simulation methods with multi-scale strategies have been studied for many years,mainly for dipole systems,and a few attempts for electrolyte.In this thesis,we rigorously derive the multi-scale model for electrolyte.Starting with the partition function,we give the definition of the reaction potential energy,which is frequently used in the multi-scale simulation.Ignoring the inter-domain short-range interaction,we also show the statistical expression of it's Green's function.In addition,we find that for asymmetric electrolyte a cavity potential should be taken into account,which is not mentioned in the previous works.To compute the Green's function efficiently during the simulation,we use the linearized Poisson-Boltzmann equation to approximate the continuous medium outside the simulation cavity,and the Poisson's equation to describe the potential inside the cavity.The solution to this boundary value problem is called the Kirkwood series,which converges slowly.To deal with this problem,an efficient computing method using image charges for this Green's function is developed.Prior to our work,the image charge method for such Green's function problem is only valid in a few specific asymptotic limits.We improve the image charge method so that it can be used efficiently for spherical boundary with arbitrary size.We also find that in the simulation of the ionic fluid such as electrolyte,it is important to allow the fluctuation of total charge,since the simulation is performed in a domain with finite volume.Therefore,only with the grand canonical simulation can the simulation correctly capture the physics of infinitely large electrolyte.Combining the image charge method and the grand canonical Monte Carlo scheme,we carry out simulations for 1:-1 symmetrical electrolytes,achieving good results.Similarly,we also analyze the sources of artifacts in the simulation using periodic boundary conditions.We find that for the ionic system,the deviation originates not only from the periodic images,but also from the constraints of charge neutrality inside the simulation domain.We present a quantitative analysis of this deviation,which agree with simulation results for dilute and symmetric electrolyte.For asymmetric and dense electrolyte,the linearized Poisson-Boltzmann equation fails to capture the correct physics.In that regime we cannot ensure the correctness of the image charge method which is based on the linearized Poisson-Boltzmann.Therefore we derive the statistical expression of the Kirkwood co?efficients.The expression displays a good parallelism for computing.With a large scale canonical Monte Carlo simulation augmented by our GPU implementation,we obtained the exact value of Kirkwood coeffi-cients.This scheme makes us get rid of theory of electrolyte.We find that in the dense regime,the deviations between the higher-order Kirkwood coefficients from simulation and the ones from Poisson Boltzmann equa-tion is considerable.This deviation indicates that the linearized Poisson-Boltzmann equation is not accurate to describe the linear response properties of the electrolyte.We perform the multi-scale simulation for the asymmetric electrolytes,and analysis the source of error.Another work introduced in this thesis is a GPU-based large-scale Metropolis Monte Carlo method.GPU(Graphics Processing Unit)was originally used in the graphical rendering,and is applied to the general-purpose high performance computing.Before our work,the GPU implementations of Metropolis Monte Carlo are mainly for short-range interacting systems.For long-range interaction,previous works are based on the parallel reduction method,which is not efficient.We propose a new GPU-based parallelization scheme for Metropolis Monte Carlo,which adopts the sequential updating scheme,without mathematical approximations in the energy computation.This method reaches a remarkable four hundreds of speedup on NVIDIA Tesla K20.With this algorithm,we carry out a large-scale simulation of electrolyte with one million particles,and precisely extract the parameters that describe the far-field asymptotic behavior of the electrolyte.These parameters are the renormalized Debye length,the renormalized dielectric constant,and the renormalized charges of the constituent ions.Using these data,we verify the renormalization of these parameters as well as the internal connection between these renormalized parameters,which widely exists in the electrolytes,thereby solving the Onsager's Paradox.