Fast Algorithms in Monte Carlo Simulation of electrolytes near a - PowerPoint PPT Presentation

ICERM (BROWN UNIVERSITY) Fast Algorithms in Monte Carlo Simulation of electrolytes near a spherical dielectric interface Zecheng Gan Institute of Natural Sciences & Department of Mathematics, Shanghai Jiao Tong University Joint work with Zhenli Xu and Xiangjun Xing 1 2012/9/7

ICERM (BROWN UNIVERSITY) Institute of Natural sciences Suzhou 2012/9/7 2

ICERM (BROWN UNIVERSITY) Outline 1.Background 2.Model 3.Image charge approximation 4.Tree code algorithm 5.Conclution & Further works 2012/9/7 3

ICERM (BROWN UNIVERSITY) Background Soft matter(colloidal suspensions) describes a very large class of materials • whose common characteristics is that they are composed of mesoscopic particles(1nm- 1μm)dispersed into a solvent whose molecules are much smaller in size(typically of atomic dimensions). Biological materials are mainly made out of soft matter as well: membranes, • DNA, RNA, and proteins belong to this class. 2012/9/7 4

ICERM (BROWN UNIVERSITY) Background Like charge attraction observed in different systems in recent years: confocal image showing three-particle, (a) Two actin rods (green) attraction mediated four-particle and five-particle by the barium ions (red spheres). (b) A cluster.(sample:diameter:0.6μm,σ=2.7μ lamellar phase of F-actin rods formed through C/cm²,volume fraction: φ=0.0001) (with like-charge attraction.(like-charge attraction, monovalent counterions) 2004, Deyu Lu) (2008, Tata, Mohanty and Valsakumar) 2012/9/7 5

ICERM (BROWN UNIVERSITY) Background Electrostatics in equilibrium statistical mechanics: Electric double layers Gouy-Chapman model (1910,1913) ε ε 1 k T κ = 0 B ( ) ∑ 2 2 e c z i i i i Mean-field Poisson-Boltzmann(PB) Works for, e.g., dilute univalent aqueous electrolytes &weak fields far from charged surfaces 2012/9/7 6

ICERM (BROWN UNIVERSITY) Background Dielectric discontinuity: The resulting Green’s function for this geometry is, 7 Schematic drawing of a system with a planar dielectric interface 2012/9/7

ICERM (BROWN UNIVERSITY) Model Spherical cell containing one macroion and several counterions and coions. − β N exp[ U r ( )] ≡ N Maxwell-Boltzmann distribution: P r ( ) Z ∫ ≡ − β N N Partition function: Z dr exp[ U r ( )] 2012/9/7 8

ICERM (BROWN UNIVERSITY) Image charge approximation Image charge method for spherical geometry:     −∇• ε ∇Φ = πδ − ( ) r ( ) r 4 ( r r ) s the polarization potential, + ε − ε ∞ 2 n 1 n ( ) q ∑ a Φ = η o i ( ) r P (cos ) ε + ε + + ε im n n 1 ( rr ) n ( n 1) = n 0 o s i o Figure: 2D schematic illustration of a dielectric sphere with a R. Messina, J. Chem. Phys. point charge outside. The polarization effect of the charge due 117 , 11062 (2002). to the dielectric discontinuity is represented by four images (empty circles), where the closest one to the boundary is the Kelvin image 2012/9/7 9

ICERM (BROWN UNIVERSITY) Image charge approximation Image charge method for spherical geometry: By a simple derivation, the image potential can be reformulated as the sum of a Kelvin image and a line image, r q k q x ( ) ∫ Φ = + k line ( ) r dx ε − ε − im r r r x o k o 0 Parameters: γ 2 γ ε − ε ε a aq aq r = = − − σ = γ = σ = 1 r q k i o o q ( ) x ( ) ε + ε ε + ε k line k r a x r i o s i o s 2012/9/7 10

ICERM (BROWN UNIVERSITY) Image charge approximation Image charge method for spherical geometry: The I-point Gauss-Legendre quadrature is used to approximate the line integral, leading us to, I q q ∑ Φ = + k m ( ) r ε − im ε − r r r x = m 1 o k o m ω γ σ aq − 1/   = 1 s m q =  m  x r m 2 r m k   2 s { } ω = Where are the I-point Gauss weights and locations on the , s , m 0,1,2..., I m m interval [-1,1]. If we let : = q q 0 k { = x x 0 k I q ∑ Φ = The approximate potential reads: m ( ) r im ε − r x = m 0 o m 2012/9/7 11 11

ICERM (BROWN UNIVERSITY) Image charge approximation Image charge method for spherical geometry: Table: Truncation terms for the multipole expansion and numbers of point images(discrete images for the integral plus the kelvin image)required to obtain relative errors less than 0.1% and 0.01% of in the self energy calculation of a charge at r s . The dielectric constants inside and outside the sphere are 2 and 78.5, respectively. r s /a 0.1%error 0.01%error Multipoles Multipoles images images 1.02 177 8 235 11 1.04 90 7 120 9 1.06 61 6 81 8 1.08 47 6 62 7 1.1 38 5 50 7 1.2 20 4 27 5 1.3 14 4 19 5 1.4 11 4 15 5 1.5 10 4 12 4 2012/9/7 12 1.6 8 4 11 4

ICERM (BROWN UNIVERSITY) Image charge approximation The total potential energy of the system is then expressed as a sum of three contributions, N N N M ∑ ∑ ∑∑ = + + ms ss im U ( U U U ) i ij ik = = + = = i 1 j i 1 k i im 1 Z Z ( ) ≥ + τ M i l , for r a /2 = B i The interaction between the macroion ms { U r i i ∞ < + τ and the source ions, , for r ( a /2) i z z ( ) The interaction between source j i ≥ τ l , for r = { B ij ss r U charges, ij ij ∞ < τ , for r ( ) ij The interaction between the images δ im z z ( ) ij i j − ≥ + τ (1 ) l , for r a / 2 and source ions, B i − im 2 = r r { im i j U ik ∞ < + τ , for r ( a / 2) i

ICERM (BROWN UNIVERSITY) Tree code algorithm The electrostatic potential at x i due to all the Particle-cluster interaction particles in c is: (1) If particle x i and cluster c are well-separated, the terms in Eq. (1) can be expanded in a Taylor series with respect to y about y c : Schematic of particle-cluster interaction between particle x i and cluster c={y j }. y c : cluster center, R: particle–cluster distance and r c : cluster radius. (2) A Cartesian treecode for screened coulomb interactions, Peijun Li, H. Johnson, R. Krasny, J. Comput. Phys. 2009 2012/9/7 14

ICERM (BROWN UNIVERSITY) Tree code algorithm The tree-code has two options for computing a particle–cluster interaction. It can use direct summation or Taylor approximation as in Eq. (2). In practice, the Taylor approximation is used if the following multipole acceptance criterion (MAC) is satisfied: A simple example for Barnes-Hut algorithm in 2D and the hierarchical tree structure. Treecode algorithm for pairwise electrostatic interactions in generalized born model, haizhao Yang, phd thesis, 2010 2012/9/7 15

ICERM (BROWN UNIVERSITY) Tree code algorithm The relative error in potential is defined by: (3) Test case, particles distributed randomly on the surface of a sphere (every particle with charge +1e.) relative error in potential E, Eq. (3) computed by the tree- 2012/9/7 16 code with approximation order p.

ICERM (BROWN UNIVERSITY) Tree code algorithm Test case: Monte Carlo simulation of the spherical electric double layer. (a snap shot after equilibrium.) Fig.6: CPU time computed by the tree-code with approximation order p. 2012/9/7 17

ICERM (BROWN UNIVERSITY) Further work Further work: Two Colloidal Particle Interaction. A fast algorithm for treating dielectric discontinuities in charged spherical colloids, Z. Xu, Interdiscip. Sci. Comput. Life Sci., in press enhanced sampling technique, parallelization 2012/9/7 18

ICERM (BROWN UNIVERSITY) Thank you!!! 2012/9/7 19