A Brownian dynamics algorithm for calculating the hydrodynamic friction and the electrostatic capacitance of an arbitrarily shaped object Huan‐Xiang Zhou, Attila Szabo, Jack F. Douglas, and Joseph B. Hubbard Citation: The Journal of Chemical Physics 100, 3821 (1994); doi: 10.1063/1.466371 View online: http://dx.doi.org/10.1063/1.466371 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/100/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in SU-E-T-22: A Novel Sub-Pixel MTF Calculation Algorithm for Arbitrarily Shaped QA Phantoms Med. Phys. 41, 226 (2014); 10.1118/1.4888352 Comparison of Brownian dynamics algorithms with hydrodynamic interaction J. Chem. Phys. 135, 084116 (2011); 10.1063/1.3626868 Sharp scalar and tensor bounds on the hydrodynamic friction and mobility of arbitrarily shaped bodies in Stokes flow Phys. Fluids 17, 033602 (2005); 10.1063/1.1852315 Brownian dynamics algorithm for bead-rod semiflexible chain with anisotropic friction J. Chem. Phys. 122, 084903 (2005); 10.1063/1.1848511 Brownian dynamics with hydrodynamic interactions J. Chem. Phys. 69, 1352 (1978); 10.1063/1.436761
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A Brownian dynamics algorithm for calculating the hydrodynamic friction and the electrostatic capacitance of an arbitrarily shaped object Huan-Xiang Zhou and Attifa Szabo National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892
Jack F. Douglas and Joseph B. Hubbard Polymers Division and Biotechnology Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
(Received 28 October 1993; accepted 16 November 1993) An algorithm originally devised for calculating the diffusion-controlled reaction rate toward an arbitrarily shaped object is adapted to calculate the scalar translational hydrodynamic friction and the electrostatic capacitance of the object. In this algorithm Brownian particles are launched from a spherical surface enclosing the object. Each particle is propagated until it either hits the enclosed object or crosses the starting surface. In the latter case the particle is allowed to escape to infinity with an analytically known probability. If the particle does not escape to infinity, it is put back on the starting surface with the correct distribution density and the process is repeated. The scalar friction or capacitance of the "probed" object is proportional to the fraction of particles that hit the object. This algorithm is illustrated on a dumbbell made of two equal-size spheres, a cube, and a phantom spherical shell having random distributed beads embedded in its surface.
I. INTRODUCTION
Different branches of physical science often have intimate connections due to the similarity among the equations that arise in different contexts. A simple example is provided by properties of a sphere (with radius a) in reaction kinetics, in electrostatics, and in hydrodynamics. The diffusion-controlled reaction rate of particles (with diffusion constant D) wandering toward an absorbing sphere is l k=41TDa. In electrostatics the capacitance of a conducting sphere, i.e., the total charge on the sphere that is maintained at unit potential with respect to infinity, is 2 C=a. The translational friction coefficient of a sphere in a viscous fluid with viscosity 'Y/, under the stick boundary condition, is 3 5= 61T7Ja. All three quantities scale as the radius of the sphere. These relations can be generalized for arbitrarily shaped objects. The connection between the diffusioncontrolled rate k toward an object and the capacitance C of the object is simple as their calculations involve solving the Laplace equation subject to equivalent boundary conditions. One has k=41TDC.
(1)
The connection of k or C to the translational friction coefficient is more subtle because of the complexity of the hydrodynamic equations. For an arbitrarily shaped object, the friction coefficient is a tensor rather than a scalar. In the case of a nonskew object (for which the translationrotation coupling vanishes), the drift velocity v of the object is related to a given drag force F through the relation 4 V=5-I'F,
(2)
where 5 is the translational friction tensor. If the object assumes random orientations in the fluid, then the angularly averaged (AA) velocity is (V)AA=(5- 1)AA·F.
(3a)
(5 -1)AA=tTr( 5 -1)!=t(5i 1+52 1+53 1)!=5- 11, (3b) where 51' 52' and 53 are the three principal components of 5, one can relate the AA velocity to the drag force through a "scalar friction coefficient" 5:
(4) The scalar friction coefficient measures the average resistance-to-translation experienced by an object undergoing Brownian motion. 4 For an arbitrary nonskew object with a stick boundary condition, by invoking angular averaging of the Oseen tensor and incorporating the conservation of linear and angular momenta, Hubbard and DouglasS have recently shown that the scalar friction 5 is related to the capacitance C to an excellent approximation by (5) 5
This result is exact for triaxial ellipsoids (of which a sphere is a special case) and is expected to be accurate to within a few percent, for particles with arbitrary shape. Traditionally the friction coefficient of an arbitrarily shaped object has been calculated by covering the surface of the object with a shell of small beads6•7 or dividing the surface into small elements. 8,9 These approaches require delicate tailoring of the methods to best fit individual
J. Chern. Phys. 100 as (5),indicated 1 March in1994 This article is copyrighted the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 3821 129.6.154.112 On: Mon, 30 Nov 2015 20:00:22
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Zhou et al.: Brownian dynamics for arbitrary objects
surface n is absorbing. For this choice of boundary condition on n, the quantity f3 00 can be interpreted as the fraction of Brownian particles (started at r=b) that hit the surface of the object. With the aforementioned choice the electrostatic potential ¢J(r) satisfies the Laplace equation and has value 0 at infinity and 1 on the surface n. Since
1 - /3~
C=-(41T)-1
fn
du-V¢J(r) = (41T)-1
fn
du-Vp(r)
and k=Dfn du-Vp(r),
Eq. (1) holds. When U(r)=O, we have k(b) = 41TDb and thus
r=b
(7) FIG. 1. A schematic diagram that illustrates the quantity {3", .
cases and are time consuming. The relation of Hubbard and Douglas, Eq. (5), opens up a new approach. The purpose of this paper is to make this approach practical for obtaining the scalar friction of an arbitrarily shaped object. This is done through connecting the scalar friction 5 with the diffusion-controlled rate k (via the capacitance C) and taking advantage of the existence of a wealth of Brownian dynamics (BD) simulation techniques for studying diffusion-controlled reactions. The BD algorithm recently devised by Luty, McCammon, and Zhou 10 is particularly suitable for calculating the capacitance and the scalar friction of an arbitrarily shaped object. In the remainder of this paper we first describe the adaptation of this algorithm and then illustrate its application on a dumbbell made of two equal-size spheres, a cube, and a phantom spherical shell having randomly distributed beads embedded in its surface.
r=ro+ (2Dtlt) 1I2R
(8)
until it either hits the object n or crosses the starting surface. Here R is a vector of Gaussian random numbers with the following properties for its three components: (Ra) =0,
(9a)
(RaR{J) =8 a {J'
(9b)
and can be easily generated by the Box-Muller method. 12 If the particle crosses the b surface, it will either escape to infinity or move back to the starting surface r=b. Suppose that the particle is at a distance roe > b) from the origin, the probability that the particle will escape to infinity is given by!3 b
II. BROWNIAN DYNAMICS ALGORITHM
Pesc=I-- =1-a.
In the original design of their algorithm, Luty, McCammon, and Zhou 10 treated a general boundary condition for the probability density per) of diffusing particles on the surface n of the object and included an interaction potential U(r) between the object and the diffusing particles. The probability density of the diffusing particles at infinity was fixed at 1, and their reaction rate toward the surface n, i.e., the total flux across n, was given by k=k(b)f3oo .
To obtain the hit fraction f3 00' we launch Brownian particles uniformly from the spherical surface r= b. These initial positions can be achieved by picking the cosines of the polar angles randomly between - 1 and 1 and the azimuthal angles randomly between 0 and 21T. Each particle is then propagated according toll
(6)
In this expression (see Fig. 1) k (b) is the reaction rate toward a spherical surface r= b that encloses the object n, and f3 00 is the probability that particles that start on r= b will react with n (reactivity specified by the boundary condition on n) rather than escape to infinity. It is the quantity f3 00 that is obtained through BD simulations. The electrostatic potential ¢J(r) suitable for the capacitance calculation is related by ¢J(r) = I-p(r) to the probability density per) in a diffusion-controlled reaction problem in which the interaction potential U(r) =0 and the
( 10)
ro
The fate of the particle is determined by generating a random number f!ll with a uniform distribution between 0 and 1. If f!ll < Pesc' then the particle is stopped and labeled as having escaped to infinity. Otherwise the particle is put back on the b surface with its location chosen from the distribution density w(O,cp)
41T[1-2a cos O+a 2]3/2
(11)
and the process is repeated. Here (b,e,cp) denote the spherical coordinates of the new position when the old position (with distance ro to the origin) is put on the polar axis. The distribution in Eq. (11) is the hit density on the b surface for a Brownian particle starting from the outside (see derivations of Ref. 10). It can be realized by generating cos through
e
cos
e
1 +a 2
(l_a 2 )2
2a
2a(1-a+2af!ll) ,
(12)
J. Chern. Phys., Vol. 100, No.5, 1 March 1994 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 129.6.154.112 On: Mon, 30 Nov 2015 20:00:22
Zhou et al.: Brownian dynamics for arbitrary objects
where f!lI is a random number uniformly distributed between 0 and 1, and picking
(a,{3, I da 0 cosh {3-cos a a{3 f3=f3 ' o
5
(16)
Similarly, the problem of finding the capacitance of a dumbbell made of two overlapping spheres can be solved analytically using toroidal coordinates. 15,16 For two equalsize spheres, the exterior electrostatic potential is given by
2 ¢>(a,{3,
