Variational multiscale models for charge transport

Abstract

This work presents a few variational multiscale models for charge transport in complex physical, chemical and biological systems and engineering devices, such as fuel cells, solar cells, battery cells, nanofluidics, transistors and ion channels. An essential ingredient of the present models, introduced in an earlier paper (Bulletin of Mathematical Biology, 72, 1562-1622, 2010), is the use of differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain from the microscopic domain, meanwhile, dynamically couple discrete and continuum descriptions. Our main strategy is to construct the total energy functional of a charge transport system to encompass the polar and nonpolar free energies of solvation, and chemical potential related energy. By using the Euler-Lagrange variation, coupled Laplace-Beltrami and Poisson-Nernst-Planck (LB-PNP) equations are derived. The solution of the LB-PNP equations leads to the minimization of the total free energy, and explicit profiles of electrostatic potential and densities of charge species. To further reduce the computational complexity, the Boltzmann distribution obtained from the Poisson-Boltzmann (PB) equation is utilized to represent the densities of certain charge species so as to avoid the computationally expensive solution of some Nernst-Planck (NP) equations. Consequently, the coupled Laplace-Beltrami and Poisson-Boltzmann-Nernst-Planck (LB-PBNP) equations are proposed for charge transport in heterogeneous systems. A major emphasis of the present formulation is the consistency between equilibrium LB-PB theory and non-equilibrium LB-PNP theory at equilibrium. Another major emphasis is the capability of the reduced LB-PBNP model to fully recover the prediction of the LB-PNP model at non-equilibrium settings. To account for the fluid impact on the charge transport, we derive coupled Laplace-Beltrami, Poisson-Nernst-Planck and Navier-Stokes equations from the variational principle for chemo-electro-fluid systems. A number of computational algorithms is developed to implement the proposed new variational multiscale models in an efficient manner. A set of ten protein molecules and a realistic ion channel, Gramicidin A, are employed to confirm the consistency and verify the capability. Extensive numerical experiment is designed to validate the proposed variational multiscale models. A good quantitative agreement between our model prediction and the experimental measurement of current-voltage curves is observed for the Gramicidin A channel transport. This paper also provides a brief review of the field.

Figure 1

Illustration of ion channel and its multiscale simplification. (a) Atomic view of the Gramicidin A channel in the membrane and aqueous environment; (b) A cross section of the multiscale representation of the system.

Figure 2

Illustration of surface function S and solvent characteristic function 1 – S in a 1D setting.

Figure 3

Consistency of electrostatic free energies of 10 proteins among the PB, LB-PB and LB-PNP predictions (protein IDs are listed in Table 1).

Figure 4

Surface representations for protein 1ajj. (a) Molecular surface generated by the MSMS package with probe radius 1.4 and density 10; (b) Variational surface extracted from the isovalue of S = 0.5 based on the LB-PNP model.

Figure 5

Comparison of surface electrostatic potentials computed at ρ₀ = 0.1molar for protein 1ajj. (a) Surface electrostatic potential profile obtained from the LB-PNP model; (b) Surface electrostatic potential profile obtained from the LB-PB model.

Figure 6

Convergence history of total free energy (kcal/mol), electrostatic energy (kcal/mol) and scaled volume (ų) at ρ₀ = 0.1 molar for protein 1ajj.

Figure 7

Surface representations of the Gramicidin A channel protein structure. (a) MSMS Surface with probe radius 1.4 and density 10; (b) Surface extracted from the LB equation with S = 0.7.

Figure 8

Concentration profiles along the cross section of the Gramicidin A channel at different numbers of iterations with ρ₀ = 2.0 molar, Φ₀ = 0 mV. Two vertical dashed lines indicate the channel region. (a) Cation; (b) Anion.

Figure 9

Convergence history of the peak concentration value along the cross section of the Gramicidin A channel with ρ₀ = 2.0 molar, Φ₀ = 0 mV. (a) Maximal concentration values for cations; (b) Minimal concentration values for anions.

Figure 10

Comparison of cross sections of electrostatic potential and concentration profiles with Φ₀ = 0mV and ρ₀ = 0.1molar for Gramicidin A channel. The concentrations of cations and anions are labeled with green and yellow dots, respectively. Two vertical dashed lines indicate the channel region.. (a) Electrostatic potential profiles; (b) Concentration profiles.

Figure 11

Comparison of surface electrostatic potential profiles with Φ₀ = 0mV and ρ₀ = 0.1molar for Gramicidin A channel. (a) Surface electrostatic potential profile of the LB-PB model; (b) Surface electrostatic potential profile of the LB-PNP model.

Figure 12

Comparison of surface electrostatic potential cross sections with Φ₀ = 0 mV and ρ₀ = 0.1molar for Gramicidin A channel. (a) Surface electrostatic potential profile of the LB-PB model; (b) Surface electrostatic potential profile of the LB-PNP model.

Figure 13

Comparison of cross sections of electrostatic potential and concentration profiles with Φ₀ = 100mV, ρ₀ = 0.5molar. The concentrations of cations and anions are labeled with green and yellow dots, respectively. (a) Electrostatic potential profiles; (b) Concentration profiles.

Figure 14

Comparison of the top views of surface electrostatic potentials with Φ₀ = 100mV and ρ₀ = 0.1molar. (a) Surface electrostatic potential profile of the LB-PBNP model; (b) Surface electrostatic potential profile of the LB-PNP model.

Figure 15

Comparison of the cross sections of surface electrostatic potentials with Φ₀ = 100mV and ρ₀ = 0.1molar for Gramicidin A channel. (a) Surface electrostatic potential profile of the LB-PBNP model; (b) Surface electrostatic potential profile of the LB-PNP model.

Figure 16

Electrostatic potential and concentration profiles with Φ₀ = 150mV for Gramicidin A channel. (a) Electrostatic potential profiles; (b) Concentration profiles.

Figure 17

Electrostatic potential and concentration profiles with ρ₀ = 0.1molar for Gramicidin A channel. (a) Electrostatic potential profiles; (b) Concentration profiles.

Figure 18

A comparison of simulated I-V curves and experimental data from Ref. for Gramicidin A channel.