Showing papers by "Chun Liu published in 2014"

An energetic variational approach for ION transport

Abstract: The transport and distribution of charged particles are crucial in the study of many physical and biological problems. In this paper, we employ an Energy Variational Approach to derive the coupled Poisson-Nernst-Planck-Navier-Stokes system. All physics is included in the choices of corresponding energy law and kinematic transport of particles. The variational derivations give the coupled force balance equations in a unique and deterministic fashion. We also discuss the situations with different types of boundary conditions. Finally, we show that the Onsager's relation holds for the electrokinetics, near the initial time of a step function applied field.

A conservative finite difference scheme for Poisson---Nernst---Planck equations

Abstract: A macroscopic model to describe the dynamics of ion transport in ion channels is the Poisson---Nernst---Planck (PNP) equations. In this paper, we develop a finite-difference method for solving PNP equations, second-order accurate in both space and time. We use the physical parameters specifically suited toward the modeling of ion channels. We present a simple iterative scheme to solve the system of nonlinear equations resulting from discretizing the equations implicitly in time, which is demonstrated to converge in a few iterations. We place emphasis on ensuring numerical methods to have the same physical properties that the PNP equations themselves also possess, namely conservation of total ions, correct rates of energy dissipation, and positivity of the ion concentrations. We describe in detail an approach to derive a finite-difference method that preserves the total concentration of ions exactly in time. In addition, we find a set of sufficient conditions on the step sizes of the numerical method that assure positivity of the ion concentrations. Further, we illustrate that, using realistic values of the physical parameters, the conservation property is critical in obtaining correct numerical solutions over long time scales.

Self-Consistent Approach to Global Charge Neutrality in Electrokinetics: A Surface Potential Trap Model

Abstract: How to describe the ``electric double layer'' that is at the root of all electrokinetic phenomena such as electrophoresis and electro-osmosis? A new theoretical approach, introducing the concept of a surface potential trap and applying the constraint of global charge neutrality rigorously, answers this century-old question in the context of contemporary electrokinetics involving nanoscale systems and time-dependent electric fields.

Computation of the memory functions in the generalized Langevin models for collective dynamics of macromolecules.

Abstract: We present a numerical method to approximate the memory functions in the generalized Langevin models for the collective dynamics of macromolecules. We first derive the exact expressions of the memory functions, obtained from projection to subspaces that correspond to the selection of coarse-grain variables. In particular, the memory functions are expressed in the forms of matrix functions, which will then be approximated by Krylov-subspace methods. It will also be demonstrated that the random noise can be approximated under the same framework, and the second fluctuation-dissipation theorem is automatically satisfied. The accuracy of the method is examined through several numerical examples.

Modeling and Simulating Asymmetrical Conductance Changes in Gramicidin Pores

Abstract: Abstract Gramicidin A is a small and well characterized peptide that forms an ion channel in lipid membranes. An important feature of gramicidin A (gA) pore is that its conductance is affected by the electric charges near the its entrance. This property has led to the application of gramicidin A as a biochemical sensor for monitoring and quantifying a number of chemical and enzymatic reactions. Here, a mathematical model of conductance changes of gramicidin A pores in response to the presence of electrical charges near its entrance, either on membrane surface or attached to gramicidin A itself, is presented. In this numerical simulation, a two dimensional computational domain is set to mimic the structure of a gramicidin A channel in the bilayer surrounded by electrolyte. The transport of ions through the channel is modeled by the Poisson-Nernst-Planck (PNP) equations that are solved by Finite Element Method (FEM). Preliminary numerical simulations of this mathematical model are in qualitative agreement with the experimental results in the literature. In addition to the model and simulations, we also present the analysis of the stability of the solution to the boundary conditions and the convergence of FEM method for the two dimensional PNP equations in our model.

An energetic variational approach to ion channel dynamics

Abstract: We introduce a mathematical model to study the transport of ions through ion channels. The system is derived in the framework of the energetic variational approach, taking into account the coupling between electrostatics, diffusion, and protein (ion channel) structure. The geometric constraints of the ion channel are introduced through a potential energy controlling the local maximum volume inside the ion channel. A diffusive interface (labeling) description is also employed to describe the geometric configuration of the channels. The surrounding bath and channel are smoothly connected with the antechamber region by this label function. A corresponding modified Poisson–Nernst–Planck channel system for ion channels is derived using the variational derivatives of the total energy functional. The functional consists of the entropic free energy for diffusion of the ions, the electrostatic potential energy, the repulsive potential energy for the excluded volume effect of the ion particles, and the potential energy for the geometric constraints of the ion channel. For the biological application of such a system, we consider channel recordings of voltage clamp to measure the current flowing through the ion channel. The results of one-dimensional numerical simulations are presented to demonstrate some signature effects of the channel, such as the current output produced by single-step and double-step voltage inputs. Copyright © 2013 John Wiley & Sons, Ltd.

Transport of Charged Particles: Entropy Production and Maximum Dissipation Principle

Abstract: In order to describe the dynamics of crowded ions (charged particles), we use an energetic variation approach to derive a modified Poisson-Nernst-Planck (PNP) system which includes an extra dissipation due to the effective velocity differences between ion species. Such a system is more complicated than the original PNP system but with the same equilibrium states. Using Schauder's fixed-point theorem, we develop a local existence theorem of classical solutions for the modified PNP system. Different dynamics (but same equilibrium states) between the original and modified PNP systems can be represented by numerical simulations using finite element method techniques.

Diffuse Interface Methods for Multiple Phase Materials: An Energetic Variational Approach

Abstract: In this paper, we introduce a diffuse interface model for describing the dynamics of mixtures involving multiple (two or more) phases. The coupled hydrodynamical system is derived through an energetic variational approach. The total energy of the system includes the kinetic energy and the mixing (interfacial) energies. The least action principle (or the principle of virtual work) is applied to derive the conservative part of the dynamics, with a focus on the reversible part of the stress tensor arising from the mixing energies. The dissipative part of the dynamics is then introduced through a dissipation function in the energy law, in line with the Onsager principle of least energy dissipation. The final system, formed by a set of coupled time-dependent partial differential equations, reflects a balance among various conservative and dissipative forces and governs the evolution of velocity and phase fields. To demonstrate the applicability of the proposed model, a few two-dimensional simulations have been carried out, including (1) the force balance at the three-phase contact line in equilibrium, (2) a rising bubble penetrating a fluid-fluid interface, and (3) a solid particle falling in a binary fluid. The effects of slip at solid surface have been examined in connection with contact line motion and a pinch-off phenomenon.

An Energetic Variational Approach for ion transport

Abstract: The transport and distribution of charged particles are crucial in the study of many physical and biological problems. In this paper, we employ an Energy Variational Approach to derive the coupled Poisson-Nernst-Planck-Navier-Stokes system. All physics is included in the choices of corresponding energy law and kinematic transport of particles. The variational derivations give the coupled force balance equations in a unique and deterministic fashion. We also discuss the situations with different types of boundary conditions. Finally, we show that the Onsager's relation holds for the electrokinetics, near the initial time of a step function applied field.

Dynamics of Multi-Component Flows: Diffusive Interface Methods With Energetic Variational Approaches

Abstract: The diffusive interface method is an approach for modeling interactions among complex substances. The main idea behind this method is to introduce phase field labeling functions in order to model the contact line by smooth change from one type of material to another. The approach has been widely used and successfully incorporated to numerous practical applications, including models of phase transitions, contact line dynamics in complex fluids, cell motility. and many other problems in science and engineering. In this review article, we describe how the diffusive interface method can be used for the modeling of complex fluids and point to some main results in the field.

Blow-up criterion for compressible visco-elasticity equations in three dimensional space

Abstract: In this paper, we prove a blow-up criterion for 3D compressible visco-elasticity in terms of the upper bound of the density and the deformation tensor. Due to the special structure of the equation, we get the cancellation to the derivatives of the density and transform tensor, which brings us the desired results.

Behavior of different numerical schemes for population genetic drift problems

Abstract: In this paper, we focus on numerical methods for the genetic drift problems, which is governed by a degenerated convection-dominated parabolic equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary points as time evolves. In order to find a \emph{complete solution} which should keep the conservation of total probability and expectation, three different schemes based on finite volume methods are used to solve the equation numerically: one is a upwind scheme, the other two are different central schemes. We observed that all the methods are stable and can keep the total probability, but have totally different long-time behaviors concerning with the conservation of expectation. We prove that any extra infinitesimal diffusion leads to a same artificial steady state. So upwind scheme does not work due to its intrinsic numerical viscosity. We find one of the central schemes introduces a numerical viscosity term too, which is beyond the common understanding in the convection-diffusion community. Careful analysis is presented to prove that the other central scheme does work. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity must be avoided.

A variational formulation for dissipative fluids in inhomogeneous temperature

Abstract: We propose a formalization for dissipative fluids in inhomogeneous temperature from the viewpoint of a variational principle. The dynamics of the fluids satisfy the conservation laws of mass, momentum, angular momentum, and energy, and obey the second law of thermodynamics. By Noether’s theorem, these conservation laws are associated with the corresponding symmetries. On the other hand, the equation for entropy gives a nonholonomic constraint. Then Lagrangians and this nonholonomic constraint of the fluids satisfy these symmetries. The dynamics of fluid are given as a weak solution, minimizing an action subject to this constraint. For the existence of the weak solution, all the surface terms appearing in the variational calculus have to vanish with the aid of the nonholonomic constraint. In this study, we show that the equation for entropy is determined to be consist to “the second law of thermodynamics”, “the symmetries”, and “the necessary condition of the existence of the weak solution”. Our method gives a general scheme to derive the equation of motion for fluids in inhomogeneous temperature, and gives an explanation of some nontrivial thermal effects on the rotation of liquid crystal, vaporization of a one component fluid, and dissolution of a two component fluid.

A variational formulation for dissipative fluids with interfaces in an inhomogeneous temperature field

Abstract: We propose a formalization for dissipative fluids with interfaces in an inhomogeneous temperature field from the viewpoint of a variational principle. Generally, the Lagrangian of a fluid is given by the kinetic energy density minus the internal energy density. The necessary condition for minimizing an action with subject to the constraint of entropy yields the equation of motion. However, it is sometimes to know the proper equation of entropy. Our main purpose is to obtain it by using the three requirements, which are a generalization of Noether's Theorem, the second law of thermodynamics, and well-posedness. To illustrate this approach, we investigate several phenomena in an inhomogeneous temperature field. In the case of vaporization, diffusion and the rotation of a chiral liquid crystals, we clarify the cross effects between the entropy flux and these phenomena via the internal energy.