Unifying kinetic approach to phoretic forces and

THE JOURNAL OF CHEMICAL PHYSICS 125, 044105 共2006兲

Unifying kinetic approach to phoretic forces and torques onto moving and rotating convex particles Martin Krögera兲 and Markus Hütterb兲 Polymer Physics, ETH Zürich, Department of Materials, Wolfgang-Pauli-Strasse 10, CH-8093 Zürich, Switzerland

共Received 1 March 2006; accepted 2 June 2006; published online 26 July 2006兲 We derive general expressions and present several examples for the phoretic forces and torques acting on a translationally moving and rotating convex tracer particle, usually a submicrosized aerosol particle, assumed to be small compared to the mean free path of the surrounding nonequilibrium gas. Point of departure is an expression of the stress tensor in terms of half-sphere integrals to be evaluated with the inhomogeneous velocity distribution function of the surrounding gas, more specifically, its approximation in terms of a finite number of moments. A worked out example covers Grad’s 13 moment approximation in and out of the so-called hydrodynamic approximation. We implement an accommodation coefficient characterizing the collision process in order to derive the tracer particle’s complete equations of motion in a form which offers the possibility for immediate numerical implementation, and the analytical exploration of the effect of particle shape and size on phoretic drift velocity. Explicit expressions for axisymmetric, spherical, cylindrical, and also cuboidal particles are presented, discussed, and successfully compared with the rarely available, previously treated special cases, thus supporting the unifying approach presented in this manuscript. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2217946兴 I. INTRODUCTION

As demonstrated recently for spherical tracer 共aerosol兲 particles, phoretic forces caused by inhomogeneities in the properties of the solvent, e.g., temperature, pressure, and velocity field, give rise to combined thermo-, baro-, and rheophoresis, respectively.1 Thermophoretic forces caused by a temperature gradient induce dust structures, are of central interest in the scavenging of aerosol particles in clouds, when droplets and/or ice crystals grow or evaporate, during the fall of hydrometeors,2,3 influence the velocity measurements in particle image velocimetry measurements,4 and have been discussed in the context of nonequilibrium thermodynamics,1,5,6 and in terms of a kinetic description,7–10 leading to a stochastic differential equation for a tracer particle in a solvent.1,11 In contrast to thermophoretic forces, barophoretic effects caused by pressure gradients,12,13 and rheophoretic forces caused by velocity field gradients have been considered less frequently and not yet elaborated in an extensive, complete, or unifying fashion for nonspherical tracers. Most aerosol particles are not spherical,3,14–16 and it is therefore of considerable interest to examine the effect of particle shape on phoretic motion.17,18 In this article we will present explicit expressions for the diverse phoretic forces on dissolved ellipsoidal, cylindrical, and cuboidal tracer particles derived from forces on surface elements.1 These results will be obtained via kinetic theory in a conceptually transparent manner for particles of arbitrary convex shape, and for arbitrary solvent velocity distribution functions by halfa兲

Electronic mail: [email protected] Electronic mail: [email protected]

b兲

0021-9606/2006/125共4兲/044105/15/$23.00

sphere integrations over solvent velocity distribution functions before and after a collision with the tracer particle. The simultaneous incorporation of thermo-, baro-, and rheophoresis is complete in the sense that it accounts for inhomogeneities in all of the independent field variables of a nonisothermal hydrodynamic description of the solvent, but our approach equally allows to study the “nonhydrodynamic” regime. As thoroughly discussed earlier,13,19 there is no simple way to account for inertia and diffusion within a hydrodynamic framework, and a kinetic approach is necessary. The calculations to be presented can be considered as “bruteforce” approach, less elegant 共but still including兲 those presented earlier,10 but involve no approximations, and may be performed, for complex shapes, with the help of a symbolic programming language.20 Particles suspended in rarefied gases, e.g., aerosols, are often nonspherical.3 Nevertheless, calculation of phoretic effects for nonspherical particles is scarce.10,21 In this article, we describe the new approach, give specific results, and compare with earlier expressions and experiments. The resulting forces and torques can be used to write down both the Langevin dynamics and the Fokker-Planck equation 共FPE兲 for the tracer particle, suitable for immediate numerical implementation.1 The same reference embedded the dynamics of the tracer particle into a wider description that encompasses the continuum equations for the nonisothermal hydrodynamics of the solvent. Beyond-equilibrium thermodynamics22 had been used to examine the phoretic effects from a nonkinetic perspective. For a large class of symmetric convex bodies such as polyhedra, the phoretic forces 共but not the torques兲 in the presence of a temperature gradient have been calculated earlier in an elegant fashion by

125, 044105-1

© 2006 American Institute of Physics

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Fernández de la Mora.10 Due to the polyhedra’s discrete “nature” the forces could be obtained without performing an integral. Surface integrals, however, enter the torques and also the forces for any continuously shaped tracer. Another special case 共thin disks and cylinders兲 had been treated in an approximate fashion21 where again a temperature gradient only is considered. Phoretic forces on ionic spherical particles were discussed,23 while Ref. 24 outlines how phoretic forces could, in principle, be calculated using modified Navier-Stokes equations. Some experimental data for nonspherical 共nonconvex兲 particle clusters subject to a temperature gradient are available.25 This manuscript is organized as follows. In Secs. II and III we provide background and extend the results obtained via kinetic theory1 to include angular motion. The approach takes into account an accommodation coefficient ␣ characterizing the collision process 共elastic versus diffusive, 0 艋 ␣ 艋 1, introduced in Ref. 26兲. Section III also summarizes the model ingredients, while Sec. IV reformulates phoretic forces and torques in a unifying fashion in terms of halfsphere integrals, tracer geometry-dependent integrals, and properties of the velocity distribution function of the gas. In Sec. V and Appendix A we present an analytic proof which solves the problem of evaluating half-sphere integrals of arbitrary complexity. Based on these findings, which also provide a brute-force algorithm towards the computation of phoretic accelerations for arbitrary shapes, we are in the position to formulate compact expressions for forces and torques in terms of basic surface integrals and replacement rules, cf. Sec. VI for the outline of the “procedure.” Section VII presents explicit results for a distribution function expanded in Sonine tensorial polynomials, which includes Grad’s 13 moment expansion in the hydrodynamic approximation as a special case. We apply the general expressions to tracer particles of various shapes in Sec. VIII and Appendix B, and offer recipes on how to generalize results to more complex situations albeit the calculations tend to become tedious beyond the 共comparably high compared to previous works兲 level of description chosen here. Next, we show how results simplify in the isotropic phase 共Sec. IX兲. We discuss and eventually visualize the findings, and illustrate the link between our results and experimental observations.

below. If one considers two extreme scenarios, 共i兲 ideally elastic collisions of the gas with the tracer particle and 共ii兲 diffusive collisions which immediately restore an equilibrium velocity 共Mawellian兲 distribution function in the bodyfixed frame, one can introduce an accommodation coefficient ␣ with ␣ 苸 关0 , 1兴 which allows to cover these extremes and all intermediate 共realistic兲 situations in an approximate fashion. The diffusively scattered particles obey a Maxwell velocity distribution function with vanishing average velocity with respect to the moving tracer particle. The temperature of the entire tracer particle coincides with the temperature of the undisturbed gas at its center of mass. The range of applicability of this assumption has been extensively discussed.27 Further taking into account the uncrossability of the tracer particle it has been shown earlier1,26 that the negative force onto a surface element, under the above mentioned assumptions, becomes55 − ⌸ · n = p关4共1 − ␣兲nn + 2␣1兴 · Z1 − ␣ p冑␲Z0n,

共2兲

with isotropic pressure p and unity matrix 1. The quantities Z0 and Z1 denote half-space integrals over dimensionless velocities ˜c of gas particles, i.e., particular “moments” of the dimensionless velocity distribution function f T* in the tracerfixed frame, Zk共n兲 ⬅ − n ·

冕

˜ ⬍0 n·c

˜ 兲d3˜c , 共 丢 k+1˜c兲f T* 共c

共3兲

where 共 丢 n˜c兲 denotes the n-fold tensor product of ˜c = 冑␤c with ␤ ⬅ ms / 共2kBT兲, mass ms of a gas particle, peculiar velocity of the gas particle c, temperature T at location r of the ˜ 兲d3˜c = 1 defines the cononuniform, tracer-free gas, and 兰f T* 共c efficient of proportionality between nondimensional f T* and dimensional f T velocity distribution functions which is usually normalized as 兰f T共c兲d3c = ns with ns the particle number density of gas particles. We further mention that the distribution function of gas particles just after colliding with the tracer particle at its surface normal n is known as soon as Z0 and Z1 were calculated.1,26

III. MODEL INGREDIENTS II. KINETIC THEORY REVISITED

The total force F and torque T acting on a convex tracer particle with surface S can be calculated as surface integrals, F=−

冕

⌸ · ndS,

共1a兲

冕

r ⫻ ⌸ · ndS,

共1b兲

S

T=−

S

where r stands for the surface position vector, n is the outward-pointing surface normal, and ⌸ = ⌸共r , n兲 the kinetic stress tensor which, for nonuniform gases, depends on the position and orientation of the surface element as shown

In order to evaluate the above formulas, we need to specify the model, i.e., 共A兲 a velocity distribution function characterized through so called ␣ tensors described below and 共B兲 a shape of the tracer particle characterized through a surface r共v1 , v2兲 parametrized by v1 and v2. (A) Velocity distribution function and its ␣ tensors. Any nonuniform distribution function of the tracer free gas f共c兲 or ˜ 兲 can be expanded into tensorial associated Laequally f *共c guerre 共Sonine兲 polynomials and thus ˜ 兲 = f *共c ˜ + ˜cT + ⌬␻ ˜ T ⫻ r兲 f T* 共c

共4兲

can be parameterized by tensorial coefficients ␣ ˜ T , ⌬␻ ˜ T兲 which are constants with respect to ˜c, but de= ␣共c pend on the peculiar velocity ˜cT and peculiar angular veloc˜ T of the tracer particle 共both quantities again reduced ity ⌬␻

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Kinetic approach to phoretic forces and torques

by 冑␤兲. The ␣ tensors are defined from the distribution func˜ 兲 of the undisturbed gas through the representation tion f *共c M

˜兲 f T* 共c

=␲

˜2 −3/2 −c

e

⬁

1

兺 兺兺c

˜ 2i

n ␣m,i 䉺m+n共 丢 nr兲共 丢 m˜c兲.

共5兲

m=0 i=0 n=0

How to explicitly read off the ␣’s from any given 共approximate兲 solution of the Boltzmann equation f共c兲 in the absence of tracer particles is worked out in detail in Appendix C. All information about the distribution function to be used in 共3兲 is thus contained in ␣ and any possible distribution function 0 ˜ T兲 coefficients beis uniquely specified through its ␣m,n 共c cause of the relationship 1 0 ˜ T ⫻ r兲 · 共⵱c ␣m,n ␣m,i · r = 共⌬␻ 兲, T

共6兲

which follows from 共4兲 and 共5兲 alone if we neglect quadratic and higher order terms in ˜cT for the reason that the tracer particle can be assumed to be heavy and thus slow compared with a representative gas particle.1 In 共5兲 the symbol 䉺n ˜兲 stands for an n-fold contraction. For example, if f *共c ˜ 2兲 is the nondimensional equilibrium Maxwell = ␲−3/2 exp共−c distribution, the reader should convince her-or himself that 0 0 1 ˜ T. From 共6兲 then follows ␣1,0 ␣0,0 = 1 and ␣1,0 = −2c = ˜ T ⫻ r and all other ␣ coefficients vanish. Due to the −2⌬␻ ˜ T=0 relationship 共6兲 it is obvious, that one can consider ⌬␻ ˜ 兲. The and 共5兲 with n = 0 to derive all ␣ tensors for given f *共c remainder of this manuscript we will present worked out examples. As we will see, starting from a non-Maxwellian, nonuniform distribution function leads to phoretic motion where the peculiar translational and rotational velocities are nonzero and the tracer particle drifts in “direction” of moments 共usually temperature, pressure, flow, or other field gradients兲 of the distribution function. The exact form of the forces and torques will depend on the shape and orientation of the tracer particle. It is essential to note that the velocity distribution function of the carrier gas is not disturbed by the presence of the tracer particle, which is a reasonable assumption for large Knudsen numbers also adopted by others.26,28 (B) Geometry of tracer particle. To perform the surface integral in 共1兲 we need to define the convex geometry of the tracer particle. For a convex body, all points on the line connecting two arbitrary points on its surface must be located inside its body. Limiting the study to convex particles has important ramifications for the following calculation. Namely, solvent particles which have collided with the tracer particle will experience subsequent collisions in the surrounding gas. A more technical aspect is the fact that the directions available for incoming and scattered particles is a half sphere, which amounts to the calculations of the socalled half-sphere integrals ubiquitous below. Any surface of a convex tracer is defined by a position vector r = r共v1 , v2兲 parametrized by two variables v1 and v2, and their ranges ⳵1 and ⳵2. The surface normal n is then obtained from the surface vector s ⬅ 共⳵r / ⳵v1兲 ⫻ 共⳵r / ⳵v2兲 by normalization, n = s / s, s = 兩s兩, and one has to choose v1,2 in an order such that n points outward the convex tracer particle, cf. Table III.

IV. REFORMULATION IN TERMS OF GEOMETRYDEPENDENT INTEGRALS

Once the task has been stated through Eqs. 共1兲–共3兲, supplemented by f T* 共c兲, i.e., ␣ tensors, and a geometry r共v1 , v2兲 of the tracer particle, we rewrite it in a form which separates contributions which can be calculated analytically, and others, which require a remaining purely geometrydependent integral 共to be denoted as G below兲. Introducing the abbreviation Y for a specific parametrized integral over the tracer particle’s surface, ⬅ Yx,y,z k

冕

pZk䉺x共 丢 yn兲共 丢 zr兲dS,

共7兲

where the pressure p is under the integral and to be evaluated at the tracer particle surface, we can rewrite 共1兲, i.e, forces F and torques T, as follows: F = A共0兲 ,

共8a兲

T = ⑀:A共1兲 ,

共8b兲

with totally antisymmetric 共Levi-Cività兲 tensor ⑀ of rank 3 and A共z兲 ⬅

2共1 − ␣兲

冑␲

Y1,2,z + 1

␣

0,0,z

冑␲ Y 1

−

␣ 0,1,z Y . 2 0

共9兲

Making use of 共i兲 the half-unit-sphere integrals 共tensors ⍀m of rank m 艌 0兲 defined as ⍀m共n兲 ⬅ −

1 ␲

冕

共 丢 mcˆ 兲共n · cˆ 兲d2cˆ ,

共10兲

cˆ ·n⬍0

and evaluated analytically in Appendix A, 共ii兲 the known Gaussian integral 2

冕

⬁

2

˜ = ⌫关共n + 1兲/2兴 ⬅ 关共n − 1兲/2兴!, e−c˜ ˜cndc

0

involving the gamma function ⌫, and 共iii兲 defining purely x,y,z geometry-dependent integrals G 共Gm,n is a tensor of rank n + m + y + z − 2x兲 as x,y,z ⬅ Gm,n

冕

共 丢 nr兲⍀m䉺x共 丢 yn兲共 丢 zr兲dS,

共11兲

we can express Y and thus A and the forces and torques purely in terms of 共given兲 model parameters ␣ and 共remaining兲 geometry-dependent integrals G, since we now have, more explicitly, Yx,y,z = k

兺 n,m,i

冉

1+i+

冊

m+k x,y,z n !␣m,i 䉺n+m关p0Gm+k,n 2

˜ p兲 · Gx,y,z 兴, + 共⵱ m+k,n+1

共12兲

with

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˜p= ⵜ

J. Chem. Phys. 125, 044105 共2006兲

M. Kröger and M. Hütter

冦

1 ⵜ 共冑␤ p兲 = ⵜp − p0 ⵜ ln T if multiplied with ˜cT or ⌬␻ ˜ T, 冑␤ 2 1

ⵜp

otherwise.

We hereby also demonstrated how to incorporate pressure gradients into the formalism, albeit the dominating contribution to the force proportional in the pressure gradient will not involve a surface integral as will be seen below. In the expression for Yx,y,z k , higher than first order spatial derivatives are neglected. The triple sum over m, n, and i usually reduces to a summation over very few terms, as we will see below, and also k, z, and n can take only values 0 or 1, further x 艋 k, y 艋 2, and the ranges for m and i depend on the chosen model. Grad’s 13 moment approximation for example is covered by M = 3, m 艋 3, and i 艋 1. The notation 共13兲 arises because we had introduced—and now take care of—the velocity ˜cT which we made dimensionless by using 冑␤ ⬀ 1 / 冑T which is not a constant on the tracer’s surface in the presence of a temperature gradient. Technically, 共13兲 means that we ˜ p = ⵱p and have to recan perform all calculations using ⵱ place in the final results for forces and torques any occurrence of a 共scalar or tensorial兲 product ˜cT ⵱ p by cT ⵱ 共冑␤ p兲, or equivalently, ˜cT ⵱ p by ˜cT共⵱p − 共p0 / 2兲 ⵱ ln T兲 共similarly for ˜ T兲. As we will see below, this operation usually is rel⌬␻ evant just for a single term with m = 1, i = n = 0. To summarize, forces and torques can be calculated in a “brute-force” fashion through calculating the geometric G tensors. These tensors, in turn, are simple sums of base-surface integrals to be introduced below for which analytic expressions will be often available.

V. HALF-SPHERE INTEGRALS

In order to evaluate the G tensors we need 共i兲 the geometryies of the tracer particle, r共v1 , v2兲 and n共v1 , v2兲, in terms of surface parameters v1,2 共including their support兲 and 共ii兲 to know how to evaluate the geometry-independent ⍀’s in terms of n. Concerning 共i兲 we are free to choose a geometry but analytic results were not available for arbitrary ones. We will hence choose, for illustrative purposes, spherical, cylindrical, ellipsoidal, and cuboidal tracer particles for which G tensors can be calculated, in principle, manually. Complex geometries can be handled through a program20 which implements the strategy outline here. 共ii兲 As shown in Appendix A the ⍀k tensors are linear combinations of mixed, symmetric tensors of the kind 关共 丢 m1兲共 丢 n−2mn兲兴sym, where 关¯兴sym denotes the dyadic produced by the dots, made symmetric in all indices, and subsequently normalized. For example, 关共 丢 2e1兲共 丢 1n兲兴sym = 共e1e1n + e1ne1 + ne1e1兲 / 3. The analytical result reads int共m/2兲

⍀m =

兺

cmp关共 丢 p1兲共 丢 m−2pn兲兴sym ,

p=0

for arbitrary integers m 艌 0, with coefficients

共14兲

冧

共13兲

m!

冑␲共1 + 共m/2兲兲!p! 共− 1兲

cmp ⬅

int共m/2兲

⫻

兺

q=0

m−p

冉 冊

1 !共共m/2兲 − q兲! 2 , 共q − p兲!共2q兲!共m − 2q兲!

共− 1兲qq! q −

共15兲

where int共m / 2兲 = m / 2 and int共m / 2兲 = 共m − 1兲 / 2 for even and odd m, respectively. ⍀ tensors of arbitrary rank are thus available using basic combinatorial rules,20 some relevant contraction rules are worked out in Tables I and II. Sample values for coefficients and tensors are c00 = 1, c01 = −2 / 3, c02 = 1 / 4, c12 = 1 / 4, c03 = 0, c13 = −2 / 5, and ⍀0 = 1, ⍀1 = − 32 n, and ⍀2 = 41 共nn + 1兲, respectively. The G tensors 共11兲 are then evaluated by replacing 兰dS by the double integral 兰⳵1dv1兰⳵2dv2s with s defined in Sec. III B. Sections IV and V suffice to set up a code to compute results numerically, or symbolically, but does not provide any insight about tensorial symmetries affecting the physical mechanism. VI. COMPACT NOTATION FOR GIVEN LEVEL OF DESCRIPTION

The strategy presented in the foregoing section leads us directly to generalize the compact notation proposed in Ref. 10 and 21. Instead of directly evaluating G for a chosen application, we can use contraction rules for the half-sphere integrals 共summarized in Table I兲 still within the integrand to TABLE I. Evaluated expressions for the tensors 关共 丢 p1兲共 丢 m−2pn兲兴sym䉺x 共 丢 yn兲 of rank m − 2x + y, which appear in connection with the G tensors 共11兲. The integers m, p, x, and y are subject to constraints: m 艌 0, 0 艋 p 艋 m / 2, 0 艋 x 艋 m, and x 艋 y. A Maxwell distributions involves m 艋 2, Grad’s 13 moment expansion needs m 艋 4, i.e., m 艋 max rank共␣兲 + 1 in general. The ␮␯␬ component of the 共ini兲 and 共1n兲 tensors read ␦␮␬n␯ and ␦␮␯n␬, respectively; 共 丢 0n兲 = 1, 共 丢 1n兲 = n, 共 丢 2n兲 = nn, etc: the bold 1 symbol denotes the unity matrix. m

p

x

m 2

0 1

3

1

4

1

4

2

艌0 0 艌1 0 1 艌2 0 1 2 艌3 0 1 2 艌3

关共 丢 p1兲共 丢 m−2pn兲兴sym䉺x共 丢 yn兲 共 丢 m−2x+yn兲 1共 丢 yn兲 共 丢 2−2x+yn兲 1 y+1 n兲 + ini共 丢 yn兲 + n1共 丢 yn兲兴 3 关1共 丢 1 y−1 n兲 + 2共 丢 y+1n兲兴 3 关1共 丢 共 丢 3−2x+yn兲 关1nn兴sym共 丢 yn兲 1 y y−1 n兲 + 3共 丢 y+2n兲 + n1共 丢 y−1n兲兴 6 关1共 丢 n兲 + ini共 丢 1 y−2 y n兲 + 5共 丢 n兲兴 6 关1共 丢 共 丢 4−2x+yn兲 关11兴sym共 丢 yn兲 1 y y−1 n兲 + n1共 丢 y−1n兲兴 3 关1共 丢 n兲 + ini共 丢 1 y−2 y n兲 + 2共 丢 n兲兴 3 关1共 丢 共 丢 4−2x+yn兲

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J. Chem. Phys. 125, 044105 共2006兲

Kinetic approach to phoretic forces and torques

TABLE II. Evaluated expressions for the tensors ⍀m䉺x共 丢 yn兲 cmp关共 丢 p1兲共 丢 m−2pn兲兴sym䉺x共 丢 yn兲 of rank m − 2x + y, with the coeffi= 兺int共m/2兲 p=0 cients cmp given in 共15兲. We make use of the results collected in Table I. These tensors simplify the analysis of the G tensors 共11兲 without actually evaluating an integral. The integers m, x, and y are subject to constraints: m 艌 0, 0 艋 x 艋 m, and x 艋 y. A Maxwell distributions involves m 艋 2, Grad’s 13 moment expansion needs m 艋 4, i.e., m 艋 max rank共␣兲 + 1 in general. m

x

0 1 2 2 3 3 3 4 4 4 4

0 艌0 0 艌1 0 1 艌2 0 1 2 艌3

˜ T兴 pact” form for the total force F = F关0兴 + F关0 ,˜cT兴 + F关0 , ⌬␻ ˜ T兴 exerted by the and torque T = T关0兴 + T关0 ,˜cT兴 + T关0 , ⌬␻ uniform gas,

4pA

˜ T兴 = − F关0,c

⍀m䉺x共 丢 yn兲 共 丢 yn兲 2 − 3 共 丢 y+1−2xn兲 1 y+2 n兲 + 1共 丢 yn兲兴 4 关共 丢 1 y+2−2x n兲 2共丢 2 − 5 关1n兴sym共 丢 yn兲 2 − 15 关1共 丢 y−1n兲 + 2共 丢 y+1n兲兴 2 − 5 共 丢 y+3−2xn兲 1 1 1 − 24 共 丢 y+4n兲 + 4 关1nn兴sym共 丢 yn兲 + 8 关11兴sym共 丢 yn兲 1 1 y−1 y+2 n兲 + 12 共 丢 n兲 4 关1n兴sym共 丢 1 y−2 n兲 + 3共 丢 yn兲兴 12 关1共 丢 1 y+4−2x n兲 3共丢

2

˜

冑␲ c T ·

冋

冉

冊 册

3 ␣ ␲ 具1典S + 1 − ␣ + ␣ 具nn典S , 4 4 8 共17b兲

where we use 共througout this paper兲 the definitions 具B典S ⬅

A⬅

simplify G and thus obtain compact expressions for forces and torques in terms of more basic surface integrals. Such an approach is possible if we had specified the distribution function of the surrounding gas, however, it is already sufficient having specified the range of indices m , i of nonvanishing 0 ␣m,i tensors for the problem at hand, cf. Sec. VIII and Table II. Let us therefore discuss the compact notation before turning to an explicit representation for a distribution function of the gas. In order to outline our procedure—based on Secs. IV and V—which allows to write down compact results for forces and torques in terms of a few basic surface integrals for even the most complex situations 共arbitrary ␣⬘s兲, it is completely sufficient to consider a Maxwellian distribution in the absence of angular motion only, ˜ 兲 = ␲−3/2e−共c˜ + ˜cT兲 , f *共c

共17a兲

F关0兴 = − pA具n典S ,

共16兲

0 0 = 1 and ␣1,0 which is, for cT Ⰶ c of the form 共5兲 with ␣0,0 ˜ = −2cT leading to force contributions to be denoted as F关0兴 and F关0 ,˜cT兴 共related to zeroth moment兲, respectively. Inserting these coefficients into 共8兲 and 共9兲 with 共11兲 and 共12兲, and ⍀0,1,2 from Sec. V directly produces the following “com-

冕

1 A

冕

共18a兲

BdS,

共18b兲

dS.

Here S stands for the geometry of the tracer, we will use symbols according to Table III when evaluating surface integrals. It essentially follows from 共12兲 that the remaining ˜ T兴 and all torque contributions do not require, force F关0 , ⌬␻ concerning their compact notation 共17兲, any additional calculation. This is so because we keep all averages 具¯典S 共surface integrals兲 unevaluated which is an integral part of our procedure at this stage; even those averages which seem to vanish for obvious reasons. In particular, we must write 具1典S rather than 1, and we must keep 具n典S rather than removing this term. This procedure ensures that we can use the following replacement rules, which hold not only for this particular simple example, but are valid also in the most general cases 共including the more general case which follows in Sec. VII兲, ˜ T兴 = F关•,c ˜ T兴 F关•,⌬␻

upon replacing

˜cT䉺具B典S → ⌬␻ ˜ T ⫻ 具r䉺B典S , T关•兴 = ⑀:共F关•兴兲

upon replacing 具B典S → 具Br典S ,

共19兲 共20兲

where •, 䉺, and B are placeholders for arbitrary arguments, contractions, and integrands, respectively, the cross product 共⫻兲 in 共19兲 has priority against a tensorial 共䉺兲 product, and even the bracket around F关•兴 in 共20兲 is important since it indicates that contraction inside F关•兴 are performed before

TABLE III. Surface vector r scanning the surface, surface normal n = s / s with s ⬅ 共⳵r / ⳵v1兲 ⫻ 共⳵r / ⳵v2兲, and s = 兩s兩 for various gometries: sphere of radius R, uniaxial ellipsoid with radius R and axis ratio Q, cyclindrical tube of radius R and height to diameter ratio Q储, the two flat end caps 共⫾兲 for the same cylinder, faces ␮␯␬ 苸 兵123, 132, 213, 231, 312, 321其 of a cuboid with edge lengths L1,2,3, of a cube with edge length L, and of an axialsymmetric particle aligned in e3 direction, characterized by a function ␳共v2兲 关it has no open ends if ␳共±1兲 = 0 and includes the uniaxial ellipsoid as a special case兴. The base vectors e1,2,3 define the coordinate system in which forces and torques are also expressed. We introduced the abbreviation e±iv1 ⬅ cos v1e1 ± sin v1e2. Geometry symbol Sphere 䊊 Cylindrical tube 储 Flat end caps 艚 Cuboid 䊐 Cube 䊐 Axisymmetric 両

r R关eiv1冑1 − v22 + v2e3兴 R关eiv1 + Q储v2e3兴 ±R关e±iv1 + Q储e3兴 1 2 ⑀␮␯␬共L␮e␮ + v1L␯e␯ + v2L␬e␬兲 1 2 ⑀␮␯␬L共e␮ + v1e␯ + v2e␬兲 ␳共v2兲关eiv1兴 + Qv2e3

n=s/s r/R e iv1 ±e3 ⑀ ␮␯␬e ␮ ⑀ ␮␯␬e ␮ s−1␳共Qeiv2 − ␳⬘e3兲

s = 兩s兩 R2 R R2兩v2兩, 1 2 L ␯L ␬ 1 2 2L ␳冑Q2 + ␳⬘2

v 1苸

v 2苸

关0 , 2␲兴 关0 , 2␲兴 关0 , 2␲兴 关−1 , 1兴 关−1 , 1兴 关0 , 2␲兴

关−1 , 1兴 关−1 , 1兴 关0,1兴 关−1 , 1兴 关−1 , 1兴 关−1 , 1兴

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044105-6

J. Chem. Phys. 125, 044105 共2006兲

M. Kröger and M. Hütter

the double contraction with ⑀. These rules are easily proven using the expressions of Sec. IV. They particularly reflect the 1 tensors via 共6兲. To illustrate these way we eliminate the ␣m,i rules let us give two examples. 共i兲 We have T关0兴 = ⑀ : F关0兴 still subject to replacements 共20兲, and therefore, using 共17a兲, T关0兴 = −pA⑀ : 具nr典S which vanishes for a sphere where 具nr典S is symmetric. 共ii兲 Using the replacement rules 共19兲 and 共20兲, ˜ T兴 can be related to the expression for the the torque T关0 , ⌬␻ friction force F关0 ,˜cT兴. In particular for a sphere, it follows from 共17b兲 that the relation F关0 ,˜cT兴 ⬀˜cT brings about the ˜ T兴 ⬀ ⌬ ␻ ˜ T. torque relation T关0 , ⌬␻ Replacement rules similar to 共19兲 and 共20兲 have been employed by Rohatschek and Zulehner29 for a carrier gas described by the 共equilibrium兲 Maxwellian velocity distribution. Extension to situations where the carrier gas is out of equilibrium 共e.g., Grad’s 13 moment expansion兲 requires the extended description presented in Sec. IV. Only then it becomes clear to which specific terms and shape dependent surface integrals the rules 共19兲 and 共20兲 have to be applied. Next, the applicability of the scheme is extended by generalizing 共17兲.

F关具␾11典兴 =

pA

2 冑5 冑␲

具␾11典 · 关␣具1典S + 共4 − 3␣兲具nn典S兴,

F关具␾20典兴 = 冑2pA具␾20典䉺2关共 43 ␣ − 1兲具nnn典S − 43 ␣具关1n兴sym典S兴 ,

˜ T兴 = − F关0,c

4pA

˜

冑␲ c T ·

冋

共23d兲

共23e兲 ˜ T兴 = − F关具␾11典,c

˜ T兴 = F关具␾20典,c

␣ pA ˜ T · 具nnn典S兲其, 兵具␾11典 · 共c 2 冑5

23/2 pA

冑␲

冉

再

共具␾20典 · ˜cT兲 ·

+3 1−␣+

with the nth rank tensorial, orthonormal coefficients in particular, 2

冉 冊

5 ˜2 ˜ ␾11 = 冑5 2 − c c ,

冉

冊


˜c2 ␾20 = 冑2 丢 2˜c = 冑2 ˜c˜c − 1 , 3

共22b兲

and corresponding moments 具␾nk 典 of the distribution function ˜ as explained in detail in Appendix C. By comparing f *共c +˜cT兲 from 共21兲 with 共4兲 and 共5兲 and assuming ˜cT Ⰶ˜c 共see also Appendix C兲 we see that M = 3 and the dimensionless ␣ 0 0 0 tensors read1 ␣0,0 = 1 − 冑5具␾11典 ·˜cT, ␣0,1 = 冑25 具␾11典 ·˜cT, ␣1,0 = 2 1 0 1 2 0 2 3/2 冑 冑 ˜ ˜ −2cT − 5具␾1典 + 2 具␾0典 · cT, ␣1,1 = 冑5 具␾1典, ␣2,0 = 2具␾0典 1 0 0 ˜ T, ␣2,1 ˜ T, and ␣3,0 ˜ T. Corre= − 冑45 具␾11典c = −23/2具␾20典c + 14 冑5 具 ␾ 1典c 1 sponding ␣ tensors are given by 共6兲. Using these tensors along the lines indicated in the foregoing section the complete listing of all 共twelve兲 distinct compact contributions to the total force and an equal number of contributions to the total torque on a convex tracer particle immersed in the nonuniform gas read, with accommodation coefficient ␣, F关0兴 = − pA具n典S ,

冉

共23a兲

␣ 具1典S 2

冊 册 冋冉 冊 冊

5 1 − ␣ 具nnnn典S 4

␲ ␣ 具关1n兴symn典S 8

3 3 + ␣具关1nn兴sym典S + ␣具关11兴sym典S 2 4 ˜ p,c ˜ T兴 = − 2Ac ˜T · F关⵱

共21兲

共22a兲

冋

共23f兲

3 ␲ + 2 − ␣ + ␣ 具nn典S 2 4

As an important special case of the above results, we can write down compact results for the case where the nonuniform gas is fully characterized by its first two tensorial moments, known as Grad’s 13 moment expansion 共Appendix C covers more general cases兲. Here, the velocity distribution function of the carrier gas, unperturbed by the tracer particle, reads1,22,30

˜ 兲, ␾nk 共c

冊 册

冉

3 ␣ ␲ 具1典S + 1 − ␣ + ␣ 具nn典S , 4 4 8

˜ T兲䉺3 − 共具␾20典c

2

共23c兲

˜ p兴 = − A共⵱p兲 · 具rn典 , F关⵱ S

VII. COMPACT RESULTS FOR GRAD’S 13 MOMENT EXPANSION

˜ 兲 = ␲−3/2e−c˜ 关1 + 具␾11典 · ␾11 + 具␾20典:␾20兴, f *共c

共23b兲

+

␣

2 冑␲

冋冑 冉 册

冊

册冎

, 共23g兲

3 ␲ 1 − ␣ + ␣ 具nnr典S 4 8 ␲

2

˜ p, 具1r典S · ⵜ

共23h兲

˜ T兴 using the rule 共19兲. All the plus four contributions F关• , ⌬␻ 12 contributions to the torque are covered by the replacement rule 共20兲. In this manuscript, we choose to neglect force and torque contributions nonlinear in the field gradients of the carrier gas. The quantity pA is proportional to a transport coefficient, as will be demonstrated in the examples below. In view of 共23兲 it is thus apparent that all forces and torques are premultiplied with the transport coefficient, with the important exceptions being the terms proportional to the pres˜ p. There is clearly a distinct origin sure gradients, ⵱p and ⵱ of the effects of a pressure gradient compared to the other contributions.1 The compact result 共23兲 supplemented with 共19兲 and 共20兲 is our main result for arbitrary convex geometries, and the surface integrals 具¯典S should be evaluated, and terms should be simplified after applying the replacement rules, but not earlier! This result is a particularly useful “intermediate” result because it clarifies the tensorial symmetries involved and therefore often allows to reduce the evaluation of a force contribution by calculating an integral with an integrand which just contains a single term rather than a sum. In order to avoid any confusion in interpreting the tensorial notation

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044105-7

J. Chem. Phys. 125, 044105 共2006兲

Kinetic approach to phoretic forces and torques

when applying the replacement rules 共19兲 and 共20兲, we notice that the ␮ component of the force contributions exhibit the forms F␮关•兴 = 具C␮典S and F␮关• ,˜cT兴 = c␭具B␭␮典S with components B␭␮ and C␮ which do not depend on ˜cT. Therefore, 共19兲 ˜ T兴 = −⑀␥␭␬共⌬␻T兲␬具r␭B␥␮典S, T␮关•兴 and 共20兲 yield F␮关• , ⌬␻ ˜ T兲␥具B␥␬r␭典S, and T␮关• , ⌬␻ ˜ T兴 = ⑀␮␭␬具C␬r␭典S, T␮关• ,˜cT兴 = ⑀␮␭␬共c = −⑀␮␭␬⑀␥␨␯共⌬␻T兲␯具r␨B␥␬r␭典S, where Einstein’s summation applies. The results for the reduced levels of description, 具␾11典 = 具␾20典 = 0, can be read off from our result 共23兲 by simply removing the corresponding terms. For particles with three mutually perpendicular symmetry planes, integrals containing an odd-fold 共pure or mixed兲 tensorial product of r’s and n’s vanish; thus exactly half of ˜ T兴, to menall contributions, such as F关0兴, F关具␾20典兴, and T关c tion three of these 24 terms, vanish identically. We note that the forces F关具␾11典兴 and F关0 ,˜cT兴 in 共23兲 require calculation of exactly the same surface integrals, as was noticed already, e.g., in Ref. 10. Using rule 共20兲 we see that same holds for the torques T关具␾11典兴 and T关0 ,˜cT兴. A similar statement holds also for the forces F关具␾20典兴 and F关具␾11典 ,˜cT兴 containing 具nnn典S, and related torques which include a term of the form 具nnnr典S, respectively. These considerations further reduce the number of independent terms to be computed. Still, the number is large and we recently developed a code which allows to evaluate 共23兲 using a symbolic programming language.20 The range of applicability of Grad’s distribution 共21兲 has been discussed,31–34 and specifically also in the vicinity of boundaries.35 The equilibrium Maxwell distribution is a trivial special case of 共21兲. In the hydrodynamic approximation26,36,37 the moments 具␾nk 典 receive physical interpretation as follows: 具␾11典 =

具␾20典

2

冑5

冑␤ q p

=−

3 冑5 ␩ ⵱ ln T, 4 p 冑␤



1 p ␩⵱v 冑 = 冑2 p = − 2 p ,

共24a兲

共24b兲

with heat flux q, temperature T, macroscopic flow field v of the undisturbed solvent, hydrostatic pressure p, anisotropic
共friction兲 pressure tensor p, solvent viscosity ␩, ␤ ⬅ ms / 共2kBT兲, Boltzmann’s constant kB, mass of a gas particle
ms, and the symbol . . . denotes the symmetric traceless part of a tensor rank.38,39 For a second rank tensor p
of1 arbitrary one has p = 2 共p + pT兲 − p1 with p = 31 Tr共p兲. For an ideal gas made of rigid spheres of diameter d, the viscosity is explic5 冑 itly given as7,22,37 ␩ = 16 mskBT / ␲ / d2 and p = nskBT with the particle number density ns of the gas. Sometimes, the coefficient ␭ ⬅ 15 4 kB / ms is introduced to rewrite these relationships. The main result of this section clearly highlights which surface integrals need to be computed to obtain the force and torque on the tracer particle, depending on which physical effects are accounted for. Most surface integrals in Eq. 共23兲 do not depend on r but only on n. In contrast, as soon as the torque is calculated or effects of particle rotation and pres-

sure gradients are accounted for the surface integrals acquire additional dependencies on r, and become substantially more difficult to compute 共except for the case of a sphere兲. It is probably for that reason that theoretical calculations for these effects were only scarcely found in literature.

VIII. APPLICATIONS A. 䊊 sphere

All normalized surface integrals 具¯典S in 共23兲 for the forces and related expressions for torques 共now denoted as 具¯典䊊 when treating a sphere兲 can be evaluated using the following result, valid for a sphere of radius R: 具共 丢 ␨r兲共 丢 xn兲典䊊 = R␨具共 丢 x+␨n兲典S = R␨

冉

冊

1 1 + 共− 1兲x+␨ 关1共x+␨兲/2兴sym , 2 1+x+␨

共25兲

for all ␨ , x 艌 0, as demonstrated in Appendix B. The integral thus vanishes, as for any axisymmetric particle, for odd 共x + ␨兲. Note that 关10兴sym = 1, 关11兴sym = 1, 关12兴sym = 共11+ i1i + iiii兲 / 3, etc. Equation 共25兲 can be equivalently rewritten in a recursive fashion using 具1典䊊 = 1, 具n典䊊 = 0, 具ny+2典䊊 =

y + 1 y+2 关具n 典䊊1兴sym , y+3

共26兲

for y 艌 0. The only difficulty when inserting 共25兲 into 共23兲 arises for terms of the form 关1nn兴sym or 关r1nnr兴sym as they ˜ T兴, for example. There appear in F关具␾20典兴 共23f兲 or T关具␾20典 , ⌬␻ is some potential of further simplifying 共23兲 using the fact that 具␾20典 is symmetric and traceless, as demonstrated in detail in Appendix B. We are now in the position to generalize a result recently derived1 for purely translational degrees of freedom to the case of combined translational and rotational motions. With the area A䊊 = 4␲R2 and volume V䊊 = A䊊R / 3 of a sphere, and the coefficient ˜␨ ⬅ 4 p A , 䊊 0 䊊 3 冑␲

共27兲

with dimension of force 共because ˜cT is dimensionless兲, we obtain the following contributions to the total force F on a sphere immersed in a nonequilibrium gas characterized by its first two tensorial moments 具␾11典 and 具␾20典;40 F䊊关具␾11典兴 = ˜␨䊊

1

2 冑5

具␾11典,

共28a兲

˜ p兴 = − V ⵱ p, F䊊关⵱ 䊊

冉

共28b兲

冊

␲ ˜ T兴 = − ˜␨䊊 1 + ␣ ˜cT , F䊊关0,c 8

共28c兲

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044105-8

J. Chem. Phys. 125, 044105 共2006兲

M. Kröger and M. Hütter

˜ T兴 = − ˜␨䊊 F䊊关具␾20典,c

冑2 5

˜ p,⌬␻ ˜ T兴 = − V 䊊 F䊊关⵱

3

具␾20典 · ˜cT ,

␣R

冑␲ ⵱ 共

共28d兲

冑␤ p兲 ⫻ ⌬␻T .

共28e兲

Correspondingly, using the replacement rules and some further calculations with the help of 共23兲 and 共26兲, we obtain the total torque T on a sphere as a sum of the three terms,

˜ T兴 = ␣˜␨䊊 T䊊关具␾20典,⌬␻

˜ p,c ˜ T兴 = − V 䊊 T䊊关⵱

冋

R2

10冑2

␣R

冑␲ ⵜ 共

册

˜T , 具␾20典 · ⌬␻

冑␤ p兲 ⫻ ˜cT .

共29b兲

F䊐关具␾11典兴 =

共29c兲

˜ p兴 = − V ⵱ p, F䊐关⵱ 䊐

B. 䊐 Cube

For the case of a cubic tracer particle with square faces and edge length L, we can use a special case of the more general cuboidal shape treated in Appendix B to obtain the following normalized surface integrals: 3

"x=0,2,4,..具共 丢 xn兲典䊐 =

1 兺 共 丢 xei兲, 3 i=1

共30a兲

5L2 兺 共 丢 x+2ei兲. 36 i=1

共30c兲

Inserting 共30兲 into 共23兲 with 共19兲 and 共20兲, restricting ourself to 具␾20典 = 0, yields ˜␨ ⬅ 4 pA , 䊐 䊐 3 冑␲

All other force and torque contributions listed in 共23兲 vanish due to symmetry. The expression 共28兲 for the force simplifies to known results upon neglecting angular motion, velocity, or pressure gradients.1,10,11,16,23,26,28,41,42 When comparing with expressions available in the literature, one may replace 具␾11典 and 具␾20典 in the above equations by their hydrodynamic expres˜ p following 共13兲. sions 共24兲. In 共28e兲 and 共29b兲 we inserted ⵱ Comparison of predicted forces and experimental data by Friedlander43 for spherical particles yields ␣ = 0.497. For the case of pure shear flow, if the tracer particle is moving in the direction of the flow lines, the force F䊊关具␾20典 ,˜cT兴 tends to deflect the particle in the direction ˜cT ⫻ 共⵱ ⫻ v兲. This is opposite to the result of Saffman,44,45 which is valid for dense highly viscous solvents and, particularly different from our treatment, stick boundary conditions at the particle surface are employed. We mention in that context that opposite signs have also been found for the Magnus effect for rotating particles immersed either in a dilute gas or in a continuum stream.46,47 The term ⬀V䊊 ⵜ p is also worthy a note. It can be shown that, irrespective of the shape of the convex tracer particle of volume V, the term linear in field gradients always reads −V ⵱ p, termed “barodiffusion force”19,48 which just follows from a balance of drag and inertia.13 Even in the absence of pressure gradients the torque on a sphere does not vanish. Notice that the torque on a sphere vanishes for the case of purely elastic collisions, ␣ = 0.

共30b兲

3

"x=0,2,4,..具rr共 丢 xn兲典䊐 =

共29a兲

2

R ˜ T兴 = − ␣˜␨䊊 ⌬␻ ˜ T, T䊊关0,⌬␻ 2

L "x=1,3,5,..具r共 丢 n兲典䊐 = 兺 共 丢 x+1ei兲, 6 i=1 x

˜␨ 䊐

共31a兲

2 冑5

具␾11典,

共31b兲 共31c兲

冉

冊

␣␲ ˜ T兴 = − ˜␨䊐 1 + ˜cT , F䊐关0,c 8

共31d兲

5 ˜ p,⌬␻ ˜ p, ˜ T兴 = − ␣ V 䊐 ˜T⫻⵱ L⌬␻ F䊐关⵱ 6 冑␲

共31e兲

˜ T兴 = − ␣˜␨䊐 T䊐关0,⌬␻

5 2 ˜ T, L ⌬␻ 48

˜ p,c ˜ T兴 = − 2Ac ˜T · T䊐关ⵜ +

␣

冋冑 冉 册

共31f兲

冊

3 ␲ 1 − ␣ + ␣ 具nnrr典䊐 4 8 ␲

2

具rr典䊐 2 冑␲

˜ p, ·⵱

共31g兲

with A䊐 = 6L2 and V䊐 = L3, and all other force and torque contributions are equal to zero. The unevaluated surface integrals in 共31g兲 are given by 共30兲. These results compare well with the one for the sphere, mainly due to 共30兲 where the identity e1e1 + e2e2 + e3e3 = 1 can be employed. The expression for the drag force F䊐关0 ,˜cT兴 is identical to the one derived by Dahneke.16

C. ¸ cylindrical tube

We denote the orientation of the cylindrical tracer of radius R and height to diameter ratio Q by e3, cf. Fig. 1. We denote the length-to-radius ratio as Q储 = h / 共2R兲. The surface area is A储 = 2␲R2共1 + 2Q储兲 = A䊊共Q储 + 21 兲, volume V储 = ␲R2h = 2␲R3Q储. For symmetry reasons the forces 共and also torques兲 onto a cylinder with symmetry axis e3 are invariant with respect to an exchange of e1 and e2 共as for the uniaxial ellipsoid兲, and we make use of the projector P储 ⬅ e3e3 and of the expressions for the friction coefficient ˜␨储 and the cylinder volume V储, ˜␨储 ⬅ 8冑␲ pQ储R2 = 2共A储/A䊊兲 − 1 pA , 䊊 冑␲

共32兲

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044105-9

J. Chem. Phys. 125, 044105 共2006兲

Kinetic approach to phoretic forces and torques

via algebraic manipulations.20 Our result for the force contributions F储关0 ,˜cT兴 共34a兲—and therefore also F储关具␾11典兴 共34b兲—exactly reduces to the one implicitly obtained ˜ p throughout abbreviates ␤−1/2 ⵱ 共␤1/2 p兲 earlier.10 In 共34兲, ⵱ as explained in 共13兲. The hydrodynamic approximation 共24兲 can be employed to rewrite 共34兲 in terms of temperature, temperature gradients, heat flux, etc. The end caps 共symbol 艚兲 are treated separately in the subsequent section 共i兲 to facilitate the extension of the results to cylindrical particles with spherical end caps and 共ii兲 to see, how the more general results for arbitrary axisymmetric particles to be discussed below simplify to the results for the simplest axisymmetric shape. The total force onto a cylindrical tracer with flat end caps is obtained by summation of the 储 and 艚 contributions. D. Ÿ flat end caps

The two flat end caps for the cylindrical tube are parametrized in Table III. The result for the two end caps 共together兲 reads FIG. 1. Definition of quantities specifying the geometry of tracer particles. Top: cuboidal and cubic. Middle: axisymmetric with axis ratio Q / R and spherical. Bottom: cylindrical with flat end caps and length to diameter ratio Q储. Surface parametrizations are given in Table III.

V储 = 2␲Q储R ,

F艚关具␾11典兴 = ˜␨储

4 − ␣ 1 ˜ 4 − 3␣ 具 ␾ 1典 − ␨ 储 P储 · 具␾11典, F储关具␾11典兴 = ˜␨储 冑 冑 8 5 8 5

共34a兲

F储关⵱p兴 = − V储 ⵱ p + V储P储 · ⵱p,

共34b兲

冉 冉

冊 冊

R

冋 冉 冊册

共34c兲 ˜ T兴 = − T艚关0,⌬␻ +

R ˜ T兴 = − ˜␨储 共A+⌬␻ ˜ T − A −P 储 · ⌬ ␻ ˜ T兲, T储关0,⌬␻ 24

共34e兲

˜ p,c ˜ T兴 = − T艚关⵱

2

R

A 6 冑␲

−

˜ p兲 ⫻ ˜c 兲, 共共P储 · ⵱ T

共35c兲

˜␨储R2 4Q储

3 ␲ − 4 8

再冋 冉 冊册 冎 再冋 冉 冊册 1−␣ 1−

␲ − Q2储 8

共35d兲 ˜T 共1 − P储兲 · ⌬␻

␣ ˜T , ⌬␻ 2 V储R

冑␲ Q

1−␣

储

冎

˜ p ⫻ ⌬␻ ˜ T兲 , P储 · 共⵱

1 ␲ − 2 8

共35e兲

˜ p兴 ⫻ ˜c 共关共1 − P储兲 · ⵱ T

˜ p ⫻ ˜c 兴兲 + ␣Q2储 共P储 · ⵱ ˜ p兲 ⫻ ˜c − P储 · 关⵱ T T

␣R ˜ ˜ p,c ˜ T兴 = − V储 共ⵜ ˜ T储关⵱ 冑␲ p ⫻ c T 兲 − V储

共35b兲

P储 · ˜cT ,

␣共1 − 4Q2储 兲 ˜ p兲 ⫻ ⌬␻ ˜T 共P储 · ⵜ 4

+ 1−␣

共34d兲

6 冑␲

共35a兲

V储R ␣ ˜ ˜ p,⌬␻ ˜ T兴 = ˜ F艚关ⵜ 冑␲ Q 储 4 ⵱ p ⫻ ⌬ ␻ T

˜ p兲 ⫻ ⌬␻ ˜ T, A−共P储 · ⵱

− V储

· 具␾11典,

再 冋 冉 冊册 冎 再

−

␣R ˜ ˜ p,⌬␻ ˜ ˜ T兴 = V 储 ⵱ F储关⵱ 冑␲ p ⫻ ⌬ ␻ T

储

储

˜␨储 ␣ 3 ␲ 1+ 1−␣ − 4 8 Q储 4

␲−2 ˜ T兴 = − ˜␨储 1 + F储关0,c ␣ ˜cT 8 6−␲ + ˜␨储 1 − ␣ P储 · ˜cT , 8

˜ 储 4 − 3␣

˜ p兴 = − V储P储 · ⵱p, F艚关⵱ ˜ T兴 = − F艚关0,c

to arrive, for 具␾20典 = 0, at

1

储

共33兲

3

␣

具 ␾ 1典 + ␨ P 8冑5Q 8冑5Q

+ 共34f兲

where the abbreviation A± ⬅ 8Q2储 + ␣共␲ − 2兲Q2储 ± 6␣ has been used. This is the result for a cylindrical tube without end caps when disregarding the second moment, 具␾20典 = 0. Results in the presence of a velocity gradient can be obtained from 共23兲

冎

␣ ˜ p ⫻ ˜c 兲 , P储 · 共⵱ T 4

共35f兲

where Q储 ⬅ 共1 − P储兲. All other force and torque contributions not specified in 共34兲 and 共35兲 vanish due to symmetry, cf. Appendix B for details. The result for the drag force, 共34c兲 plus 共35c兲, agrees with the expression found by Dahneke.16 In the limit Q储 → ⬁, the drag force and the thermophoretic force, 共34a兲 plus 共35a兲, both agree with the expressions de-

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044105-10

J. Chem. Phys. 125, 044105 共2006兲

M. Kröger and M. Hütter

with Q储 = 1 one recovers, on average, all the expressions derived for the spherical tracer particle, cf. Sec. IX.

E.

両

axisymmetric particle

An axisymmetric, convex particle, directed along the e3 direction is parameterized by a radial function ␳共v2兲, where v2 苸 关−1 , 1兴, cf. Table III for its exact parametrization and the surface element s共v2兲. Its surface area reads

FIG. 2. Absolute value of the “friction” force F关0 ,˜cT兴 共34兲 and 共35兲, made dimensionless by ˜␨储˜cT / Q储 for flat disks 共Q储 → 0兲 共solid lines兲, and by ˜␨储˜cT for long rods 共Q储 → ⬁兲 共dashed lines兲, respectively. The enclosed angle between ˜ T ⬅ 兩c ˜ T兩兲 is denoted by the particles symmetry axis e3 and tracer velocity ˜cT 共c ␾. Results are shown vs ␾ for a range of accommodation coefficients equidistantly spaced between ␣ = 0 共ideally elastic collision兲 ␣ = 1 共ideally diffusive collision兲.

rived by García-Ybarra and Rosner21 for a cylinder with spherical caps.49 We point out that the drag force gets logarithmic corrections in the Knudsen number if at least one of the dimensions of the cylinder is larger than the mean free path in the carrier gas.50 However, this a higher order effect not of interest in the “free molecular flow” limit studied here. It is noteworthy that the torque on the tracer particle also vanishes for ␣ = 0. Results for long rods 共Q储 → ⬁兲 and thin disks 共Q储 → 0兲 obtained from the cylinder results 共34兲 plus 共35兲 converge to the ones for a rodlike and flat oblate ellipsoid, respectively, with half axis R. In the limit of flat disks, the dominant contributions are zeroth order in Q储. In contrast, for thin rods, the leading terms in the force and torque are cubic in Q储. Some illustrations of the forces are given in Figs. 2 and 3. For an artificial ensemble of randomly oriented cylinders

A 両 = 2␲

冕

s共v2兲dv2 = 2␲

冕

␳ 共 v 2 兲 冑Q 2 + ␳ ⬘ 共 v 2 兲 2 d v 2 . 共36兲

Axisymmetric particles include the cylindrical tube, sphere, and uniaxial ellipsoid as special cases. The surface is closed if ␳共−1兲 = ␳共1兲 = 0 and ␳ smooth. Convexity of the particle is ensured if ␳⬙ 艋 0. In this section we derive expressions for all surface integrals appearing in our expression for forces and torques 共23兲, but we will not insert them in order to keep the manuscript short. First, we notice from Table III that r and n can be written as r = Xreiv1 + Y re3,n = Xneiv1 + Y ne3, with v2-dependent functions Xr = ␳共v2兲, Y r = Qv2, Xn = Q / 冑Q2 + ␳⬘2, Y n = ␳⬘ / 冑Q2 + ␳⬘2, and thus n

再

n! 2 A储 具共 丢 •兲典 両 = 兺 2A k=0 k!共n − k兲! A储 n

⫻关共 丢 n−ke3兲兴sym

冕

冕

2␲

关共 丢 ke共iv1兲兲dv1兴

0

冎

1

X•kY •n−ks共v2兲dv2

共37兲

−1

for • 苸 兵r , n其. In Eqs. 共37兲, 共38兲 and also 共39a兲 the ⫻ symbol means “continue reading” and does not denote a cross product. While the last integral in 共37兲 purely depends on the shape ␳共v2兲 of the tracer, the first integral between curly brackets 关equals 具共 丢 kn兲典储 with A储 = 4␲Q兴 has been already evaluated in connection with the cylindrical tracer 共B6兲 for which n = eiv1. Thus 共37兲 vanishes for odd n 共and k is even too兲, and simplifies to n

具共 丢 n•兲典 両 =

n! A储 关具共 丢 kn兲典储 兺 2A k=0 k!共n − k兲! ⫻共 丢 共n−k兲/2P储兲兴sym

冕

1

X•kY •n−ksdv2 ,

共38兲

−1

FIG. 3. Absolute value of the “thermophoretic” force F关具␾11典兴 共34兲 and 共35兲 made dimensionless by ˜␨储兩具␾11典兩 / 共2冑5Q储兲 for flat disks 共Q储 → 0兲 共solid lines兲, and by ˜␨储兩具␾11典兩 / 共2冑5兲 for long rods 共Q储 → ⬁兲 共dashed lines兲, respectively. The enclosed angle between the particles symmetry axis e3 and 具␾11典 is denoted by ␾. The strength 兩具␾11典兩, in the hydrodynamic approximation, is proportional to the strength of the temperature gradient, cf. 共24兲.

where P储 ⬅ e3e3 and 具共 丢 2n兲典储 = 21 共1 − P储兲, for example. We do not wish to elaborate on evaluating this expression further, but rather want to show how to apply it to obtain contributions to forces and torques which involve, for example, 具nn典 両 and 具nr典 両 . Accordingly, evaluating 共38兲 for n = 2 and • = n gives

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044105-11

具nn典 両 =

J. Chem. Phys. 125, 044105 共2006兲

Kinetic approach to phoretic forces and torques

2 A储 关具共 丢 kn兲典储 兺 2A 両 k=0,2 k!共2 − k兲! ⫻共 丢 共2−k兲/2P储兲兴sym

冕

1

共39a兲

Xnk Y nn−ksdv2

51 or Eq. 共10.38兲 in Ref. 38, for vanishing order parameters. We used the notation 1 = ii to write the symmetric part in vector rather than component notation. In the isotropic phase the force contributions for a cylinder become

−1

冋冕 冋冕

=

4␲Q P储 2A 両

=

4␲Q P储 2A 両 +

1

1 Xn0 Y n2 sdv2 + 共1 − P储兲 2 −1

Xn2 Y n0 sdv2

−1

册

␳⬘ ␳

1

冕

␳

1

dv2 2 2 −1 冑Q + ␳⬘

册

共39b兲

,

with A from 共36兲. This result agrees with the calculations of Rotatschek and Zulehner29 and Fernández de la Mora.10 We kept the term Y n0 in 共39a兲 because Y n = 0 and Y n0 = 1 for a cylindrical tube. For all other cases, we resubstituted Xn and Y n according to their above definitions to obtain 共39b兲. Correspondingly, from 共39a兲 we immediately obtain, simply replacing n by r, 具rr典 両 =

冋

4␲Q Q 2P 储 2A

冕

1

1 v22sdv2 + 共1 − P储兲 2 −1

冕

1

册

␳2sdv2 ,

−1

共40兲 again, with s = ␳冑Q2 + ␳⬘2. Finally, using the same approach, we have 具rn典 両 =

冋

4␲Q P储 2A

冕

1

Y nY rsdv2

−1

冕 冕

1 + 共1 − P储兲 2 =

冋

Fiso = 储

冊 冑 冉

˜␨储

6 5

2+

冉

˜␨储 1 ␲ 具␾11典 − 1+ ␣ 3 Q储 8

+ ¯,

冊冉 冊 2+

1 ˜cT Q储 共43a兲

2

dv2 2 2 −1 冑Q + ␳⬘

Q2 共1 − P储兲 2

冕

1

4␲Q2 P储 2A

1

XnXrsdv2

−1

册

1

1 v2␳⬘␳dv2 + 共1 − P储兲 2 −1

冕

1

−1

册

␳ 2d v 2 , 共41兲

such that it should be transparent how to obtain all surface integrals appearing in our result for forces and torques, cf. Sec. VII. The above results valid for axisymmetric particles certainly reduce to the ones already stated for cylindrical tubes and spheres, when choosing ␳共v2兲 = R with constant R 共a cylindrical tube兲, for which Xr = R, Y r = Qv2, Xn = 1, Y n = 0, and ␳ = 冑1 − v22 共for a sphere兲, see also Table III and Appendix B. IX. ISOTROPIC PHASE

In order to simplify the general expressions to the case where an ensemble of axisymmetric particles is oriented isotropically 共i.e., randomly兲, we employ the useful identities 具P储典iso = 31 1, 具P储P储典iso = 51 关11兴sym =

共42a兲 1 15 共11 +

i1i + iiii兲.

共42b兲

These isotropic averages can be derived from more general expressions valid for the uniaxial phase, Eq. 共4.2–3兲 in Ref.

etc., these expressions show strong similarities with the results for a sphere. After identifying the force coefficient ˜␨储 with ˜␨䊊 the isotropic averages 共43兲 are identical to the corresponding expressions 共28兲 for a spherical tracer particle for Q储 → 1. By virtue of 共42a兲 alone, the same statement holds for the torque. For Q储 ⫽ 1 the force still differs from the one for spheres, reflecting the net effect of shape on the phoretic motion.

X. CONCLUSIONS

In this manuscript we derived explicit expressions for both the forces and torques onto moving and rotating convex tracer particles for given velocity distribution function of the unperturbed, nonuniform rarefied gas. To derive the unifying expressions, we inserted the half-sphere integrals 共A5兲 with 共15兲 and the ␣ tensors, 共C8兲 with 共C5兲 into forces and torques 共1兲 with 共11兲 and 共12兲. We then obtained compact results valid for arbitrary convex tracer particles in terms of basic surface integrals, summarized in Sec. VII which include Grad’s 13th moment approximation as a special case. Concerning geometries, we have then evaluated geometrydependent integrals for the case of both axisymmetric and nonaxisymmetric particles. The corresponding forces and torques have been stated and discussed. They enter the equations of motion for tracer particles immersed in a nonequilibrium gas. We further showed how to modify the forces and torques for the case where a higher order expansion needs to be considered. And we reduced the problem of the calculation of forces and torques for arbitrarily shaped convex tracers to the calculation of normalized surface integral tensors 共23兲 or G tensors 共11兲, in general, a task which we are going to automatize based on the results of this manuscript to treat complex situations.20 However, depending on the particle shape, a number of symmetry properties usually simplify this problem, as demonstrated here. Aerosol particles do not only experience the forces and torques as described in this manuscript. In addition, there are Brownian forces related to the calculated friction force by way of the fluctuation-dissipation theorem. Furthermore, there may be additional potential forces 共e.g., gravitational, and buoyancy兲. These can be handled via a standard Brownian dynamics simulation or Fokker-Planck equation approach.1,38,52–54 Only a few special cases of the results obtained here were spread in the literature. The presented unifying description allows us to extend the calculations to arbitrary convex shapes dissolved in an arbitrary nonuniform gas 共beyond Grad’s 13 moment approximation兲 conveniently.

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044105-12

J. Chem. Phys. 125, 044105 共2006兲

M. Kröger and M. Hütter

ACKNOWLEDGMENTS

The authors acknowledge very helpful discussions with Pedro L. García-Ybarra, Santosh Ansumali, and Hans Christian Öttinger. This project has been supported through EUNSF grant FP6-016375 共MNIBS兲 of the European Community. APPENDIX A: HALF-SPHERE INTEGRALS

In order to perform the integration over half the unit sphere 关Eq. 共10兲兴, we choose a parametrization of the unit vector y as follows: y = 兺iy iei = 冑1 − z2共cos ␾ e1 + sin ␾ e2兲 + zn, where e1 · n = e2 · n = 0 and n is the symmetry axis of the half sphere. Then 共10兲 is rewritten as 1 ␲

⍀m = −

冕 冕 2␲

0

0

共 丢 my兲zdzd␾ .

共A1兲

−1

We insert y into 共A1兲 and use binomial coefficients to arrive at m

k

1 兺兺 ␲ k=0 l=0

⍀m = −

0

0 m−k+1 y l1y k−l 2 z

−1

关共 丢 e1兲共 丢 e2兲共 丢 m−kn兲兴sym dzd␾ , l!共m − k兲!共k − l兲!/m! l

⫻

冕 冕 2␲

k−l

共A2兲

where 关A兴sym denotes a symmetrization of the tensor A in all its indices, including subsequent normalization. For example, 关共 丢 2e1兲共 丢 1n兲兴sym = 共e1e1n + e1ne1 + ne1e1兲 / 3. Making use of the identity e1e1 + e2e2 + nn = 1 allows us to rewrite 共A2兲 as 1 ␲

⍀m = − ⫻

int共m/2兲

兺 q=0

冕 冕 2␲

0

关共 丢 q共1 − nn兲兲共 丢 m−2qn兲兴sym 共2q兲!共m − 2q兲!/m!

0 m−2q+1 y 2q dzd␾ , 1 z

共A3兲

−1

where int 共m / 2兲 denotes the integer part of m / 2. The integral can be evaluated explicitly as follows: 1 ␲

冕 冕 2␲

0

0 m−2q+1 y 2q dzd␾ = 1 z

−1

共− 1兲m+1共q −

1 2

兲!关共m/2兲 − q兴!

关1 + 共m/2兲兴!冑␲

,

共A4兲 where x! is evaluated with the gamma function as x! = ⌫共x + 1兲. Finally, ⍀m is a linear combination of mixed, symmetric tensors of the kind 关共 丢 k1兲共 丢 m−2kn兲兴sym and we can introduce numerical coefficients given in 共15兲 of Sec. V of this manuscript in order to write down our explicit result for the ⍀n tensors of arbitrary rank m 艌 0 as int共m/2兲

⍀m =

兺

cmp关共 丢 p1兲共 丢 m−2pn兲兴sym ,

共A5兲

p=0

which proves 共14兲. APPENDIX B: BASIC SURFACE INTEGRALS VIA SYMMETRY CONSIDERATIONS

As we have shown in Sec. VII, alculating forces and torques for arbitrary distribution functions 共of tensorial order

M兲 effectively requires, for any given shape of the tracer, not more than the evaluation of the following basic surface integrals: 具共 丢 nr兲共 丢 xn兲共 丢 zr兲典S ⬅

1 A

冕

共 丢 nr兲共 丢 xn兲共 丢 zr兲dS,

共B1兲

for n 苸 兵0 , 1其, x 苸 兵0 , 1 , 2 , . . , M + 1其, and z 苸 兵0 , 1其 and for n = 2, x 苸 兵0 , 1 , 2 , 3其, and z 苸 兵0 , 1其 which results from 共12兲, 共11兲, and 共7兲 together with the assumption ˜cT Ⰶ c. Of course, by just changing indices, it is then sufficient to calculate the following integrals: 具共 丢 ␨r兲共 丢 xn兲典S ⬅

1 A

冕

共 丢 ␨r兲共 丢 xn兲dS,

共B2兲

with ␨ 苸 兵0 , 1 , 2 , 3其 and x 苸 兵0 , 1 , 2 , . . , M + 1其 covering all cases 共B1兲. In the following we will evaluate the integrals 共B1兲 or equivalently 共B2兲 for some geometries 共the axisymmetric particle is treated in Sec. VIII E兲. With these expressions at hand, the forces and torques are available through 共23兲 which contain the still unevaluated integrals. To us it seems that this approach provides the most compact and transparent presentation of explicit results, while at the same time facilitating the generalization to other geometries. In order to get a flair on how to make use of this appendix, let us show why the following contributions to the torque vanish: 共i兲 T关具␾20典兴 for a sphere, 共ii兲 T关0兴 for a cylinder, and 共iii兲 T关具␾11典 ,˜cT兴 for a cube. Case 共i兲: according to 共23兲 with 共20兲 we have T关具␾20典兴 = c1⑀ : 共具␾20典 : 具nnnr典S兲 + c2⑀ : 共具␾20典 : 具关1n兴symr典S with constant coefficients c1,2, to be evaluated for a sphere. Here, 具nnnr典䊊 ⬀ 具nnnn典䊊 关due to Eq. 共B3兲兴 which is symmetric in the last two indices, ⑀ : A vanishes if A is symmetric, and thus the term with c1 vanishes. Similarly, 具␾20典 : 具关1n兴symr典䊊 ⬀ 具␾20典 : 具共1nn + inin + niin兲典䊊 simplifies to 2具␾20典 共which is again symmetric兲 because 具␾20典 is also traceless by definition, 共具␾20典 : 1 = 0兲, and 具nn典 ⬀ 1 关due to Eq. 共B4兲兴. Case 共ii兲: we derived T关0兴 ⬀ ⑀ : 具nr典S for arbitrary geometries. For a cylinder, r = c1n + c2v2e3 holds 共cf. Table III兲, thus T关0兴 ⬀ ⑀ : 具nv2典储e3 since nn is symmetric and does not contribute. Finally, because n, for a cylinder, is independent of v2 and v2 苸 关−1 , 1兴 共torque with respect of center of mass for a homogeneous cylinder兲, the surface integral vanishes. Case 共iii兲: we have T关具␾11典 ,˜cT兴 ˜ T · 具nnnr典S兲兴, in general. For a cube, 具nnnr典䊐 ⬀ ⑀ : 关具␾11典 · 共c 3 ⬀ 兺i=1 Li共 丢 4ei兲 关using Eq. 共B20兲兴 is symmetric in the last two indices 共to be double contracted with the antisymmetric ⑀兲 and thus vanishes. 1. 䊊 sphere of radius R

The surface of a sphere of radius R is parametrized by r = Rn, normal vector n = 共cos v1e1 + sin v1e2兲冑1 − v22 + v2e3 with v1 苸 关0 , 2␲兴, v2 苸 关−1 , 1兴, and surface element s = R2, cf. Table III. Accordingly, surface area A = 兰dS = 4␲R2. Since 具共 丢 ␨r兲共 丢 xn兲典䊊 = R␨具共 丢 x+␨n兲典䊊

共B3兲

holds, the task is reduced towards calculating the symmetric tensor 具共 丢 xn兲典䊊. For a sphere this tensor is certainly isotropic, i.e., 具共 丢 yn兲典䊊 ⬀ 关1y/2兴sym, and 共thus兲 vanishes for odd y

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044105-13

J. Chem. Phys. 125, 044105 共2006兲

Kinetic approach to phoretic forces and torques

= x + ␨. We hence need to calculate the coefficient of proportionality, or equivalently, a single component of the tensor. We choose the “simplest” component, which is 具n3y 典䊊, with n3 ⬅ n · e3 = v2, and obtain 具n3y 典䊊 = 具vx2典䊊 =

1 A

=

1 2

冕 冕

v2y sdv1dv2 v2y dv2 =

冉

冊

1 1 + 共− 1兲y , 2 1+y

which provides the solution to the task 共B2兲 for the sphere as follows:

冉

冊

1 1 + 共− 1兲x+␨ 关1共x+␨兲/2兴sym . 具共 丢 ␨r兲共 丢 xn兲典䊊 = R␨ 2 1+x+␨

共B4兲

Note that 关1 兴sym = 1, 关1 兴sym = 1, 关1 兴sym = 共11 + i1i + iiii兲 / 3, etc. Equation 共B4兲 can be equivalently rewritten in a recursive fashion using 具1典䊊 = 1, 具n典䊊 = 0, 具ny+2典䊊 = 共y + 1兲共y + 3兲−1关具ny+2典䊊1兴sym for y 艌 0. 0

1

2

具rr共 丢 xn兲r典储 = R3共 丢 x+3n兲 +

+ e3具共 丢 x+1n兲典储e3 + 具ne3共 丢 xn兲典储e3兴. 共B10兲 We have thus traced back all integrals 共B2兲 to the the integral 具共 丢 xn兲典储 for which a closed form solution can be written down as a series in terms of integrals of the form 具关共 丢 n1兲共 丢 mP储兲兴sym典储. Because it will be often sufficient for practical purposes, we presented the alternate recursive approach 共B6兲 with 共B7兲. 3. Ÿ flat end caps for cylinder of radius R and height h = 2RQ¸

The two end caps are parametrized by r / R = ⫿ v2 cos v1e1 + v2 sin v1e2 ± Q储e3, v1 苸 关0 , 2␲兴 and v2 苸 关 −1 , 1兴, yielding n = ± e3 and s = R2兩v2兩. To make use of the results we prefer using the area A = 4␲R2Q储 of the cylinder when normalizing the integrals. Thus 具共 丢 xn兲典艚 =

2. ¸ cylindrical tube of radius R and height h = 2RQ¸

Here, r / R = cos v1e1 + sin v1e2 + Q储v2e3 with v1 苸 关0 , 2␲兴 and v2 苸 关−1 , 1兴, thus n = cos v1e1 + sin v1e2, s = R, A = 4␲R2Q储, V = ␲R2h = 2␲R3Q储, and height to diameter ratio Q储. Again, we take into account a basic symmetry consideration to evaluate 共B2兲 and trace integrals with ␨ ⬎ 0 or z ⬎ 0 back to integrals of lower rank, in the light of the representation r = R共n + v2Q储e3兲. All integrals can be rewritten in terms of 1 and p储 ⬅ e3e3 since p储 = 1 − e1e2 − e2e2. These considerations lead, for x = 2 and x = 4, for example, to the ansatz 具共 丢 2n兲典储 = c11 + c2P储 ,

共B5兲

具共 丢 4n兲典储 = d1关11兴sym + d2关1P储兴sym + d3关P储P储兴sym ,

共B6兲

where the coefficients are manually calculated, cf. previous section, via c1 = 具n21典储 = 具cos2 v1典储 = 21 ,

d1 = 具n41典储 = 具cos4 v1典储 = 83 , d2 = 具n23n31典储 − d1 = 共c1 + c2兲c1 − d1 = − d1 , d3 = 具n43典储 − d1 − d2 = − d1 − d2 = 0,

共B7兲

and so on for x 艌 6. Further, we have, for the cylindrical tube, using r = R共n + v2Q储e3兲, 共B8兲

具r共 丢 xn兲典储 = R具共 丢 x+1n兲典储 , because 具v2共 丢 n兲典储 = 具v2典具共 丢 n兲典储 = 0, x

x

具rr共 丢 xn兲典储 = R2具共n + v2Q储e3兲共n + v2Q储e3兲共 丢 xn兲典储

冋

=

= R2 具共 丢 x+2n兲典储 +

2

册

Q储 P储具共 丢 xn兲典储 , 3

where the denominator stems from

具v22典储 = 31

1 关1 + 共− 1兲x兴共 丢 xe3兲 A

R2兩v2兩dv2dv1

关1 + 共− 1兲x兴 x 1 共 丢 e3兲 = 共 丢 x/2P储兲, 2Q储 Q储

具rr共 丢 xn兲典艚 =

共B11兲 共B12兲

R4 关␲共1 − P储兲 + 4␲Q2储 P储兴共 丢 x/2P储兲, A

共B13兲 共B14兲

具rrr共 丢 xn兲典艚 = 0,

where it is understood that 共 丢 x/2p储兲 vanishes if x is odd. 4. 䊐 cuboid of size L1,2,3

A cuboid oriented along the e1,2,3 axes has six faces which we shall denote as F = 123, F = 132, F = 213, F = 231, F = 312, and F = 321 to simplify the analysis. Doing so, face F = ␮␯␬ of the cuboid is parametrized by

冉

冊

L␮ L␯ L␬ e␮ + v1 e␯ + v2 e␬ , 2 2 2

共B15兲

where a summation convention is not used, ⑀␮␯␬ are the components of the Levi-Civita tensor, and v1 苸 关−1 , 1兴 and v2 苸 关−1 , 1兴. From 共B15兲 we immediately derive the normal vector n共F兲 = ⑀␮␯␬e␮ and surface element s共F兲 = L␯L␬ / 4 for face F. The area and volume of a cuboid are A = 兰dS = 2共L1L2 + L1L3 + L2L3兲 and V = L1L2L3, respectively. Unlike for the sphere, for which r ⬀ n, we cannot simplify the task 共B2兲 in a fashion corresponding to the case of a sphere. We have to go through a more tedious calculation. For the single face F = ␮␯␬ we have 兰dS共F兲 = 兰s共F兲dv1dv2 = 4s共F兲 = L␯L␬, and we have to sum over 共F兲 具共 丢 xn兲典䊐 =

共B9兲

and, analogously,

冕

具r共 丢 xn兲典艚 = 0,

r共F兲 = ⑀␮␯␬

c2 = 具n23典储 − c1 = − c1 ,

R3Q2储 关P储具共 丢 x+1n兲典储 3

1 x ⑀ 共 丢 xe ␮兲 A ␮␯␬

冕

dS共F兲 =

1 x ⑀ 共 丢 xe␮兲L␯L␬ , A ␮␯␬ 共B16兲

where

⑀␮x ␯␬

denotes the xth power of the number ⑀␮␯␬, and

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044105-14

J. Chem. Phys. 125, 044105 共2006兲

M. Kröger and M. Hütter

共F兲 具r共 丢 xn兲典䊐 =

1 1+x ⑀ A ␮␯␬

冕冉

L␮ L␯ L␬ e␮ + v1 e␯ + v2 e␬ 2 2 2

⫻共 丢 xe␮兲dS共F兲 =

V 1+x 1+x ⑀ 共 丢 e␮兲, 2A ␮␯␬

冊

k

Lm k 共x兲

i=0

共B17兲

etc. for all faces 共notice: ⫻ in 共B17兲 is not a cross product兲 to arrive at "x=1,3,5,..具共 丢 xn兲典䊐 = 0, because 共B16兲 is symmetric in ␯␬ and 3

具共 丢 xn兲典䊐 = =

兺 具共 丢 xn兲典䊐共F兲 ␮,␯,␬=1 2 关共 丢 xe1兲L2L3 + 共 丢 xe2兲L3L1 + 共 丢 xe3兲L1L2兴, A 共B18兲

for even x 艌 0. More generally, we find 具共 丢 ␨r兲共 丢 xn兲典䊐 = 0

if ␨ + x is odd.

共B19兲

Further, while skipping intermediate steps, we arrive at 3

具r共 丢 xn兲典䊐 =

V 兺 共 丢 1+xei兲, A i=1

共B20兲


˜ 2兲 丢 n˜c, ␾nk = lnk Ln+1/2 共c k

冉冑

␲ k!共1 + 2n兲!! 2 共k + n + 共1/2兲兲!n!

冊

1/2

.

The normalization coefficients arise from the orthogonality m features of the Laguerre functions, 兰⬁0 e−xxmLm k 共x兲Lk⬘共x兲dx = ␦k,k⬘共k + m兲! / k!, and the one for irreducible tensors,

兰S1 丢 n˜c 丢 m˜cd2cˆ = 4␲␦n,mn! / 共2n + 1兲!!⌬共n兲, where ⌬共n兲 is a
projector with the feature38 ⌬共n兲共 丢 ncˆ 兲 = 丢 n˜c. The symbol cˆ denotes the unit vector c / c. The lowest order “moments” therefore are given by 共22兲. We should mention that the requirements 具c典 = 0 and ms / 2具c2典 = 23 kBT immediately yield 具␾01典 = 0 and 具␾10典 = 0. The remaining two lowest order moments are then 具␾11典 and 具␾20典. Assuming a Ⰶ c, thus neglecting terms of order 共a / c兲⬎1, we have ⬁

f共c + a兲 = f 0共c + a兲

具␾nk 典䉺n␾nk 共c + a兲 兺 n,k=0

aⰆc

˜ · ˜a兲 兺 lnk Ln+共1/2兲 ˜ 2 + 2c ˜ · ˜a兲 共c ⬇ f 0共c兲共1 − 2c k

3

n,k



˜ 共 丢 n−1˜c兲 + 丢 n˜c兲䉺n具␾nk 典 ⫻共na

3

V + 兺 Li共 丢 x+2ei兲, 3A i=1

共B21兲

= f 0共c兲

for even x 艌 0, where e4 ⬅ e1 and e5 ⬅ e2 have been introduced to simplify notation and finally, Eqs. 共B18兲–共B21兲 are the solutions to all tasks posed by 共B2兲 except the one for 具r2共 丢 xn兲r典䊐 which we refrain from writing down, albeit conceptually as “simple” to obtain as the stated results.

Any velocity distribution function f共c兲 can be expanded in terms of orthonormal 共scalar product defined by f 0兲, dimensionless moments ␾nk 共c兲 as follows: ⬁

冊

f共c兲 = f 0共c兲 1 + 兺 具␾nk 典䉺n␾nk , n,k

f共c兲␾nk d3c,

with particle number density ns for convenience, because c will have dimension of velocity and the spatially averaged f should be dimensionless. The isotropy of f 0 requires that ␾’s are linear in the irreducible tensors made of c’s. For the case 2 2 that f 0 ⬀ e−c˜ , more precisely, f 0共c兲 = ns共␤ / ␲兲3/2e−c˜ with ˜c ˜ 兩, the orthonormal polynomials are the asso= 冑␤c and ˜c = 兩c ciated Laguerre 共or Sonine兲 functions Lnk defined as

冊 冊

冉

k

˜ · ˜a 兺 lnk 兺 + 2c i=0

n,k

共k + n + 共1/2兲兲! 共i − 1兲!共k − i兲!共n + i + 共1/2兲兲!

册


⫻共− ˜c2兲i 丢 n˜c䉺n具␾nk 典 ,

共C5兲

˜ / c兲a and the following rewhere we used the notation ˜a = 共c sult: aⰆc k

˜2 Lm k 共c

共C2兲

冉


˜ 2兲 na ˜ 共 丢 n−1˜c兲 lnk Ln+共1/2兲 共c k


2 + 3 1 − ˜c · ˜a 丢 n˜c 䉺n具␾nk 典 3

共C1兲

where f 0共c兲 denotes the isotropic part of the distribution function, and the average is defined as

冋兺 n,k

APPENDIX C: ORTHONORMAL EXPANSION AND ␣ TENSORS

冕

lnk ⬅

共C3兲

共C4兲

1 具rr共 丢 xn兲典䊐 = 兺 L2i 共 丢 x+2ei兲 12 i=1

具␾nk 典 = ns−1

共k + m兲! 共− x兲i , i!共k − i兲!共m + i兲!

5 3/2 i.e., Lm 0 = 1, L1 共x兲 = 2 − x, and we have

for odd x 艌 1, and

冉

=兺

˜ · ˜c兲 ⬇ 兺 + 2a i=0

共k + m兲! ˜ · ˜c兲 共− ˜c2兲i共1 + 2ia i!共k − i兲!共m + i兲! k

˜ 2兲 + 2 兺 = Lm k 共c i=0

共k + m兲! 共i − 1兲!共k − i兲!共m + i兲!

⫻共− c 兲 a · ˜c . ˜ 2 i˜

共C6兲

We can further make use of38

˜c · ˜a 丢 n˜c = ˜a · 丢 n+1˜c +


n ˜a · ⌬共n兲䉺n−1 丢 n−1˜c , 2n + 1

共C7兲

and similar relationships to directly read off the coefficients ␣ defined through

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044105-15

J. Chem. Phys. 125, 044105 共2006兲

Kinetic approach to phoretic forces and torques

f共c + a兲 = f 0共c兲 兺 ˜c2i␣n,i共a兲䉺n共 丢 n˜c兲

共C8兲

n,i

by comparing 共C5兲 with 共C8兲. Notice that for the case n 艋 1 which covers Grad’s 13th
moment approximation, one can remove the anisotropic . . . symbols in 共C5兲. M. Hütter and M. Kröger, J. Chem. Phys. 124, 044511 共2006兲. O. Dupont, F. Dubois, and A. Vedernikov, Meas. Sci. Technol. 11, 331 共2000兲. 3 F. Zheng, Adv. Colloid Interface Sci. 97, 253 共2002兲. 4 A. Melling, Meas. Sci. Technol. 8, 1406 共1997兲. 5 A. Pérez-Madrid, J. M. Rubí, and P. Mazur, Physica A 212, 212 共1994兲. 6 J. M. Rubí and P. Mazur, Physica A 250, 253 共1998兲. 7 L. Waldmann, in Thermodynamik der Gase, Handbuch der Physik, Vol. 12, edited by S. Flügge 共Springer, Berlin, 1958兲, p. 373. 8 P. S. Epstein, Proc. Phys. Soc. London 47, 259 共1945兲. 9 E. A. Mason and S. Chapman, J. Chem. Phys. 36, 627 共1962兲. 10 J. Fernández de la Mora, J. Aerosol Sci. 33, 477 共2002兲. 11 J. Fernández de la Mora and J. M. Mercer, Phys. Rev. A 26, 2178 共1982兲. 12 I. Goldhirsch and D. Ronis, Phys. Rev. A 27, 1616 共1983兲. 13 J. Fernández de la Mora and D. E. Rosner, J. Fluid Mech. 125, 379 共1982兲. 14 B. E. Dahneke, J. Aerosol Sci. 4, 139 共1973兲. 15 M. L. Laucks, G. Roll, G. Schweiger, and E. J. Davis, J. Aerosol Sci. 31, 307 共2000兲. 16 B. E. Dahneke, J. Aerosol Sci. 4, 147 共1973兲. 17 H. J. Keh and C. L. Ou, Aerosol Sci. Technol. 38, 675 共2004兲. 18 H. J. Keh and H. J. Tu, Colloids Surf., A 176, 213 共2001兲. 19 J. Fernández de la Mora, Phys. Rev. A 25, 1108 共1982兲. 20 M. Kröger and M. Hütter, Comput. Phys. Commun. 共in press兲. 21 P. García-Ybarra and D. E. Rosner, AIChE J. 35, 139 共1989兲. 22 H. C. Öttinger, Beyond Equilibrium Thermodynamics 共Wiley, Hoboken, NJ, 2005兲. 23 X. Chen, J. Phys. D 33, 803 共2000兲. 24 H. Brenner and J. R. Bielenberg, Physica A 355, 251 共2005兲. 25 F. Zheng and E. J. Davis, J. Aerosol Sci. 32, 1421 共2001兲. 26 L. Waldmann, Z. Naturforsch. A 14, 589 共1959兲. 27 Y. Sone, Kinetic Theory and Fluid Dynamics, Modeling and Simulation in Science, Engineering and Technology 共Birkhäuser, Boston, 2002兲. 28 L. Waldmann, in Rarefied Gas Dynamics, Proceedings of the Second International Symposium on Rarefied Gas Dynamics, edited by L. Talbot 共Academic, New York, 1961兲, pp. 323–344. 1 2

29

H. Rohatschek and W. Zulehner, J. Colloid Interface Sci. 119, 378 共1987兲. 30 S. Hess, in Vielteilchensysteme, Bergmann-Schaefer, Lehrbuch der Experimentalphysik, Vol. 5 共W. de Gruyter, Berlin, 1992兲, p. 41. 31 A. N. Gorban and I. V. Karlin, Physica A 360, 325 共2006兲. 32 H. Struchtrup and M. Torrilhon, Phys. Fluids 15, 2668 共2003兲. 33 S. Ansumali, I. V. Karlin, and H. C. Öttinger, Europhys. Lett. 63, 798 共2003兲. 34 A. N. Gorban and I. V. Karlin, Physica A 206, 401 共1994兲. 35 S. S. Chikatamarla, S. Ansumali, and I. V. Karlin, Europhys. Lett. 74, 215 共2006兲. 36 S. Chapman and T. G. Cowling, The Mathematical Theory of NonUniform Gases, 3rd ed. 共Cambridge University Press, Cambridge, 1970兲. 37 L. E. Reichl, A Modern Course in Statistical Physics 共Wiley, New York, 1998兲. 38 M. Kröger, Models for Polymeric and Anisotropic Liquids, Lecture Notes in Physics Series, Vol. 675 共Springer, Berlin, 2005兲. 39 M. Kröger, Phys. Rep. 390, 453 共2004兲. 40 The expression for F关具␾20典 ,˜cT兴 corrects our previous result for a force contribution presented in Ref. 1. 41 A. Einstein, Z. Phys. 27, 1 共1924兲. 42 F. Zheng, Adv. Colloid Interface Sci. 97, 255 共2002兲. 43 S. K. Friedlander, Smoke, Dust and Haze 共Wiley, New York, 1997兲. 44 P. G. Saffman, J. Fluid Mech. 22, 385 共1965兲. 45 P. G. Saffman, J. Fluid Mech. 31, 624 共1968兲. 46 C.-T. Wang, AIAA J. 10, 713 共1972兲. 47 S. G. Ivanov and A. M. Yanshin, Fluid Dyn. 15, 449 共1980兲. 48 S. N. Semenov and M. E. Schipf, Phys. Rev. E 72, 041202 共2005兲. 49 Our expressions for F关1 ,˜cT兴 and F关具␾11典兴 are both smaller by a factor of 冑␲ than the results of Garcia-Ybarra and Rosner Ref. 21. If their expressions are evaluated in the sphere limit, their results also seems to differ by 冑␲ from Waldmann 共Refs. 26 and 28兲, while we recover Waldmann’s 共correct兲 expressions exactly. 50 J. R. Dorfman and J. V. Sengers, Physica A 134, 283 共1986兲. 51 M. Kröger and H. S. Sellers, J. Chem. Phys. 103, 807 共1993兲. 52 J. Honerkamp, Stochastic Dynamical Systems 共Wiley, New York, 1993兲. 53 H. C. Öttinger, Stochastic Processes in Polymeric Fluids 共Springer, Berlin, 1996兲. 54 M. Hütter and H. C. Öttinger, J. Chem. Soc., Faraday Trans. 94, 1403 共1998兲. 55 A few minus signs in front of terms containing Zk and ⌸ · n were misprinted in Ref. 1, but actually had no effect on the results. The current manuscript should correct these misprints.

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