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
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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).
DNA, RNA, and proteins belong to this class.
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Like charge attraction observed in different systems in recent years: (a) Two actin rods (green) attraction mediated by the barium ions (red spheres). (b) A lamellar phase of F-actin rods formed through like-charge attraction.(like-charge attraction, 2004, Deyu Lu) confocal image showing three-particle, four-particle and five-particle cluster.(sample:diameter:0.6μm,σ=2.7μ C/cm²,volume fraction: φ=0.0001) (with monovalent counterions) (2008, Tata, Mohanty and Valsakumar)
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Electrostatics in equilibrium statistical mechanics: Electric double layers Mean-field Poisson-Boltzmann(PB) Works for, e.g., dilute univalent aqueous electrolytes &weak fields far from charged surfaces
Gouy-Chapman model (1910,1913)
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2 2
B i i i i
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Dielectric discontinuity: Schematic drawing of a system with a planar dielectric interface The resulting Green’s function for this geometry is,
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Maxwell-Boltzmann distribution:
N N
N N
Partition function:
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Image charge method for spherical geometry:
( ) ( ) 4 ( )
s
r r r r ε πδ −∇• ∇Φ = −
Figure: 2D schematic illustration of a dielectric sphere with a point charge outside. The polarization effect of the charge due to the dielectric discontinuity is represented by four images (empty circles), where the closest one to the boundary is the Kelvin image
the polarization potential,
2 1 1
( ) ( ) (cos ) ( ) ( 1)
n
im n n n
i
q a r P rr n n ε ε η ε ε ε
+ ∞ + =
− Φ = + +
117, 11062 (2002).
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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,
k
r k line im
2 k s
a r r =
k s
aq q r γ = −
1
( ) ( )
k line
r aq q x a x
σ
γ
−
=
i
ε γ ε ε − = +
σ ε ε = +
Parameters:
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Image charge method for spherical geometry:
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1
( )
I k m im m
q q r r r r x ε ε
=
Φ = + − −
2
m m s
aq q r ω γ =
1/
1 2
m m k
s x r
σ
− =
I m im m
=
{ }
, , 0,1,2...,
m m
s m I ω =
Where are the I-point Gauss weights and locations on the interval [-1,1]. If we let : The approximate potential reads:
k k
q q x x = =
The I-point Gauss-Legendre quadrature is used to approximate the line integral, leading us to,
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Image charge method for spherical geometry: rs/a 0.1%error Multipoles images 0.01%error Multipoles images 1.02 1.04 1.06 1.08 1.1 1.2 1.3 1.4 1.5 1.6 177 90 61 47 38 20 14 11 10 8 8 7 6 6 5 4 4 4 4 4 235 120 81 62 50 27 19 15 12 11 11 9 8 7 7 5 5 5 4 4 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
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The total potential energy of the system is then expressed as a sum of three contributions, The interaction between the images and source ions, The interaction between source charges, The interaction between the macroion and the source ions,
1 1 1
N N N M ms ss im i ij ik i j i k i im
= = + = =
( )
, /2 , ( /2)
M i B i i i
Z Z l for r a ms r i for r a
τ τ ≥ + ∞ < +
( )
, , ( )
j i B ij ij ij
z z l for r ss r ij for r
τ τ ≥ ∞ <
( )
(1 ) , / 2 2 , ( / 2)
im ij i j B i im i j i
z z l for r a r r im ik for r a
δ τ τ − ≥ + − ∞ < +
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Schematic
particle-cluster interaction between particle xi and cluster c={y j}. yc: cluster center, R: particle–cluster distance and rc: cluster radius. The electrostatic potential at xi due to all the particles in c is: If particle xi and cluster c are well-separated, the terms in Eq. (1) can be expanded in a Taylor series with respect to y about yc : (1) (2)
Particle-cluster interaction
A Cartesian treecode for screened coulomb interactions, Peijun Li, H. Johnson, R. Krasny, J. Comput. Phys. 2009
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The tree-code has two
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
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Test case, particles distributed randomly on the surface of a sphere (every particle with charge +1e.) The relative error in potential is defined by: relative error in potential E, Eq. (3) computed by the tree- code with approximation order p. (3)
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Test case: Monte Carlo simulation
layer. (a snap shot after equilibrium.) Fig.6: CPU time computed by the tree-code with approximation order p.
Two Colloidal Particle Interaction.
A fast algorithm for treating dielectric discontinuities in charged spherical colloids,
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enhanced sampling technique, parallelization
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