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Effective diffusivity of nonspherical, sedimenting particles
Effective Diffusivity of Nonspherical, Sedimenting Particles I SIMON L. GOREN Department of Chemical Engineering, University of California, Berkeley, California 94720 Received October 31, 1978; accepted April 20, 1979 The coupling of Brownian rotation with sideways drift of nonspherical particles sedimenting in response to an external force F~x results in an effective translational diffusion over and above the Brownian translational diffusion. This paper considers particles possessing an axis of symmetry and a plane of symmetry normal to this axis. The effective translational diffusivity in a plane perpendicular to the external force is shown to be @EFF= F~x(f71 --f~-l)2/64~R, where f l is the friction coefficient for translation along the axis of symmetry,f~ is the friction coefficient for translation normal to the axis of symmetry, and OR is the rotary diffusion coefficient. Because of fluctuations of velocity in the direction of F~x with fluctuations in orientation, there is also an effective translational diffusivity for this direction over and above the Brownian translational diffusivity; this is found to be twice the lateral diffusivity. Values of ~E~v/~, where ~ is the Brownian translational diffusion coefficient, have been computed for particles of several shapes. For example, a cylinder of length L very much greater than its radius yields
@Evv/~ = ( FexL/32kT) 2. INTRODUCTION
This paper calls attention to the fact that the coupling of Brownian rotation with the sideways drift of sedimenting, nonspherical particles results in effective vertical and lateral diffusion of the particles over and above that due to Brownian translational diffusion. The importance of coupling of rotational and translational effects on the transport properties of colloidal particles was recognized by Brenner and Condiff (1). They presented a general and rigorous framework for the discussion of these effects, but this was not specialized to the case considered in the present paper. Such coupling was also recognized by Gallily and Cohen (2) who concluded that even a monodisperse collection of nonspherical particles would give rise to a distribution of aerodynamic diameters as measured by sedimentation times. The calculations of 1This paper was presented at the European Mechanics Colloquium on Low Reynolds Number Flow held in Jablonna, Poland, March 29, 1978.
Gallily and Cohen are based on Monte Carlo techniques and so the results, while illustrative of the effect being considered, are not of general applicability. In the present work the long-time behavior of sedimenting nonspherical particles is examined analytically and relatively simple formulas for the effective diffusion coefficient are deduced. Knowing the effective diffusion coefficients, one could easily calculate the apparent distribution of aerodynamic particle size for a sedimenting cloud of particles or the spatial distribution of such particles. ANALYSIS
A small particle in creeping motion experiences a drag force proportional to the particle's velocity. The proportionality factor (more generally the translation tensor (3) multiplied by the viscosity /x of the suspending medium) is called the friction coefficient and is denoted by the symbolf.
209 Journal of Colloid and Interface Science, Vol. 71, No. 2, September 1979
0021-9797/79/11020%07502.00/0 Copyright© 1979by AcademicPress, Inc. All rights of reproduction in any form reserved.
210
SIMON
L. GOREN
For nonspherical particles, f depends on the shape and orientation of the particle relative to the direction of motion. In this paper we consider only particles that possess both an axis of symmetry and a plane of symmetry perpendicular to this axis. These particles are characterized by two generally different friction coefficients, fl and f~, for translation parallel to and translation perpendicular to the axis of symmetry, respectively. Examples for elongated particles are prolate spheroids and long circular cylinders (rods). With L denoting the major axis and D the minor axis, theory (4-6) shows that fi = 2~r~L/ln f~ = 47r/xL/ln
(2L/D), (2L/D)
[1] [2]
for both types of particles provided L/D is very much greater than unity. Examples of squat particles are oblate spheroids and short circular cylinders (disks). A particle of the type considered here when sedimenting at low Reynolds number in response to an external force Fex generally does not move totally in the direction of the applied force. Instead it also acquires a horizontal component of velocity that depends on the angle 0 between the direction of the applied force and the symmetry axis of the particle. In the absence of Brownian and fluid inertial effects, the particle maintains its initial orientation. It is easy to show by a force balance (3) that the sideways velocity is given by
Squaring [4] gives 1)2
y~(t)=F~x(~ x cos
f2
t I o I i sin 0(s)
O(s) sin O(r) cos O(r)dsdr. [5]
If the particle is sufficiently small it will also be subjected to Brownian diffusion and Brownian rotation; these motions are assumed to be uncorrelated. Here we are primarily interested in the interaction of the Brownian rotation with the sideways drift. The particle's orientation gradually changes due to the Brownian rotation so that the particle executes a complicated zig-zag path as it sediments. As this motion has all the characteristics of a random walk, a cloud of sedimenting particles is expected to diffuse laterally. The effective lateral diffusion coefficient may be found from the long-time behavior of y2(t) according to the well-known (7) formula
~E~v = lim E{y~(t)}/2t. [6] t--,~ E{ } represents the expected value or ensemble average of the quantity in braces. Moreover, since the vertical velocity fluctuates with fluctuations in orientation, there will be a distribution of vertical displacements about the average vertical position; this gives rise to an effective vertical diffusion coefficient. To calculate these effective diffusion coefficients the stochastic properties of the orientation angle O(t) must be known so that the long-time behavior of y2(t) can be deduced from -~- - F e x f2 s i n 0 c o s 0 . [3] Eq. [5]. The orientation of a particle can be speciAllowing for a time-dependent orientation fied by the three Eulerian angles 0, ~b, ~b. 0 but neglecting the particle's inertia, we can is the angle between a vertical axis fixed in integrate [3] to find the instantaneous dis- space and the instantaneous axis of symmetry of the moving particle and is the placement y(t) for a particle starting from angle appearing in the discussion so far. y=0. 4) is the angle in a fixed horizontal plane 1 between a fixed horizontal line and the "line y(t) = F e x ( ~ f 2 ) of nodes," this being the intersection of the horizontal plane and a plane moving with x sin O(s) cos O(s)ds. [41 the particle and normal to its axis of symmetry. The angular motion of the particle is
f2
Journal of Colloid and Interface Science, Vol. 71, No. 2, September 1979
SEDIMENTING PARTICLES governed by the well-known Euler equations of motion (Goldstein (8)) which express the law of conservation of angular momentum about three orthogonal axes moving with the particle. Using Euler's equations, it is straightforward to express the law of conservation of angular momentum about the line of nodes. For the symmetrical particles considered here subjected to no external torques other than the hydrodynamic resistance to rotation and the random bombardment of solvent molecules, the result is I
d~O dt 2
dgp d O dt dt
+ I~ - - -
-
sin 0
(I - Is) - - - - dd~b t h sin 0 cos 0 dt dt = -fR
dO dt
+ B(t).
[7]
[8]
Is is the moment of inertia of the particle about its symmetry axis. B(t) is a random, Gaussian torque of zero mean exerted on the particle about the line of nodes by the molecules of the suspending medium. Because of the symmetry of the particles, this torque does not depend on orientation. About the only important characteristic of B needed for our purposes is that the autocorrelation of B, E { B ( t ) B ( t + s)}, at two different times, t and t + s, is a very rapidly decreasing function of the time difference s. The integral I ~ E { B ( t ) B ( t + s)}ds = fl
is a constant /3, as yet unknown, and the great preponderance of the integral defining /3 is contributed near s = 0. Equation [7] is complicated by terms quadratic in the angular velocities. However, as the following estimate shows, these terms will be negligible compared to the viscous term under many applications. Since the average kinetic energy of rotation approaches that given by the equipartition of energy at long times, for order of magnitude purposes we estimate the angular velocities by (kT/I) 1/2. Then the order of magnitude of the ratio of the nonlinear inertial terms to the viscous term is kT/ fR(kT/I) l/z= (kTI)l/2/fR. Using Eq. [8] to calculate the rotary friction coefficient for a long circular cylinder whose moment of inertia about a transverse axis is 7rpD2L3/48, we write the ratio as QUADRATIC TERMS
I is the moment of inertia of the particle about a line perpendicular to its axis of symmetry and through its midsection, fR is the rotary friction coefficient which is the proportionality factor between the torque and the angular velocity about the line just described. For both prolate ellipsoids (9) and circular cylinders (5, 6) of very large aspect ratio fR = ½rclxL3/ln (2L/D).
211
[9]
VISCOUS TERM 3kTio
- (1-6~tz2/
/ 1/2 I n
(2L/D)
(L/D)
[10]
For a circular cylinder of density 2 g/cm 3, length 10 /xm, and aspect ratio L/D of 10 immersed in water at 25°C, this ratio is computed to be 6 x 10-5, implying that to a very good approximation the quadratic terms can be omitted from Eq. [7]. Since the above ratio varies as only the inverse square root of particle size L for fixed aspect ratio L/D, the quadratic terms are negligible down to particles of molecular dimensions suspended in water. With the neglect of the quadratic terms, Eq. [7] reduces to d20 I -- dt 2
dO -JR - - + B(t). dt
[11]
The mathematical treatment of Eq. [11] from which the rotary diffusion coefficient DR can be deduced is well known (7). Key points required for the present development are summarized below. The formal solution of [11] for the angular velocity O(t) and angular position O(t) of a particle starting from rest with 0 -- 0 is Journal of Colloid and Interface Science, VoI. 71, No. 2, September 1979
212
SIMON L. GOREN
o(t)- dOd~- exp(-fRt/I)
Ii B(O I exp(fR~/I)d~
[121
and
O(t) = fot B(~) fR dE _ exp(--fRt/I)
-I Ii -B(O
exp(fR~/I)d~.
[131
If the particle is free of external torques as implied by [11], then the expected values of the angular velocity and angular position are both zero at any time. The autocorrelations of these quantities h o w e v e r are not zero, but are related to the autocorrelations of the random torque. Following (7) we find
E{O(t)b(t)} = exp(--2fRt/I) f l I o E { B ( ' ) B12( ~ ) } e x p ( f R [ , + l x ] / I ) d t z d , -
E{O(t)O(s)} =
[3 2if.
[1 -
exp(--2fRt/I],
I[ E{B(~)B(~)}dtzd( -
f l
exp(--fRt/I)
× exp(fR~/I)dtzd~ - exp(-frts/I) + exp(-frt[s +
[14]
Jl
Io' E{B(~)B(Ix)} f~
s E{B(()B(tz)} exp(fRtz/I)dtxd~
f~
t]/I) Ii Irs E{B(~)B(tz)} exp(fR[~: + tx]/I)dlxd~ .o f~
fll - - - [fRt/1 - 1 - ½ exp(-fR[s - t]/I) + exp(-fRs/I)
f~
+ exp(-fnt/I) - ½ exp(--fR(s + t)/I)], for s > t with an equation analogous to [15] for s < t. In obtaining these results use has been made of the very rapid decay of the autocorrelation of the r a n d o m torque and of exchanging the order of averaging and integrating. Since the particle eventually achieves thermal equilibrium with the suspending medium, its expected rotary kinetic energy at infinite time, ½IE{O2(~)}, must be given by the equipartition of energy ½kT which requires fl = 2kTfR. [ 16] The rotary diffusion coefficient @R' is the limit of E{O2(t)}/2t as t approaches infinity; setting s = t in [15] and taking the indicated limit results in ~r~ = lira t--,~
E{O2(t)} - kTfR.
[17]
2t
The dimensions of Or are reciprocal time. Journal of Colloid and Interface Science, Vol. 71, No. 2, September 1979
[15]
Inserting [13] into [5] gives the time dependence ofy2(t) for an individual particle. The e x p e c t e d value ofy2(t) for an ensembie of particles is
1 )2
E{y2(t)} =F~x(~ × cos
f2
t Ii Ii E{sin O(s)
O(s) sin O(r) cos O(r)}dsdr.
[18]
A theorem from statistics states that if X and Y are random Gaussian variables with zero means, then
E{exp(i[X + Y])} = e x p ( - ½ E { X 2} -
½E{
i12} _ E { X Y } ) .
[19]
I am grateful to Professor N o r m a n Kaplan for calling my attention to this theorem. Since the trigonometric functions can be represented in terms of exponentials with
SEDIMENTING PARTICLES imaginary argument and since O(t) is a random Gaussian variable with zero mean, use of [19] reduces [18] to
E{yZ(t)} = x
VsF~x( 1_ 1) ~ \fl A
f f2
These two effective diffusivities are over and above those due to Brownian translational diffusion. For comparison, the Brownian translational diffusion coefficient @ for a particle free to assume all orientations randomly is
exp(-2[E{O2(s)}
= kT(f-f 1 + 2f~-1)/3.
[20] ~Erv,x
The three functions E{O2(s)}, E{O2(r)}, and E{O(s)O(r)} appearing in the integrand are all known from Eq. [15]. We expect the coupling of Brownian rotation with sideways drift to result in lateral diffusion, so E{y2(t)} is expected to increase linearly with t for sufficiently long times. In accordance with [6], the effective diffusivity is found from the limit of E{y2(t)}/2t as t becomes infinite. Application of L'Hospital's rule twice to the indefinite quotient E{y2(t)}/2t as t ~ oo with E{y2(t)} given by [20] does indeed yield the following "constant" as the limit: ~ E F F ~- F 2 x ( f l
1 --
f~1)2/32~R.
[24]
On dividing [22] by [24] and replacing DR by [17] we find
+ E{O2(r)})[exp(4E{O(s)O(r)}) - exp(-4E{O(s)O(r)})]dsdr.
213
[21]
1 {Fex ]2 3(f11 _ f~-x)~fR - -~\'-~] ( f l x T 2f~-1)
[25]
For the special case of prolate spheroids or circular cylinders of very large aspect ratio L/D, use of [1], [2], and [8] reduces [25] to
~EFr,x _ 1._}__(F¢xL / 2 " @ 1024\ kT ]
[26]
For the special case of oblate spheroids or circular cylinders of very small aspect ratio, inserting known values (2) offl,f2, andfa for circular disks into [25] yields
@EVr,x __
l___~(FexD] z 2048\ kT ] "
[27]
For intermediate values of the aspect ratio, @~vv,xl~ depends on L/D and the specific geometry of the particle. This dependence is illustrated in Fig. 1 for several geometries. These results were calculated from [25] for oblate and prolate spheroids using the friction coefficients derived by Overbeck as reported in Fuchs (4) and Gans (9), for circular cylinders using the friction coefficients derived by Batchelor (6), and for toroids using the friction coefficients ~EFF,x = ~EFF,y derived by O'Neill and Goren (10). It is also = F~x(fi-' --f~l)2/64~R. [22] possible to calculate the effective diffusivity for dumbbells (i.e., two spheres of each of By an analysis following the same lines as above, it is possible to show that the effec- diameter D connected by an infinitely thin tive diffusivity in the vertical direction aris- rigid rod of length (L - 2D) using the exact ing from fluctuations of the settling velocity translational frictional coefficients of Goldwith fluctuations in orientation is just twice man et al. (11) and an approximate rotary friction coefficient of Szu and Hermans (12). the lateral diffusivity, i.e., Here we find the surprising result that for ~EFF,z = F 2 x ( f ~ -1 - f~-l)2/32~R. [23] large separations the effective diffusion Because the angle 0 always lies in a plane containing the vertical axis but this plane is free to rotate about the vertical axis, the lateral displacement computed above is a radial displacement &. Since &2= x = + y2 and we have no reason to distinguish between diffusion in the x and y directions, the effective diffusivity in the lateral direction is one-half that given by [21] or
Journal of Colloid and Interface Science, Vol. 71, No. 2, September 1979
214
SIMON L. GOREN 0.05 ~
_0.05
I 0.04~o
I I
,.,
~-c~|
CirculorCylinders ,, L . _ " J ~
/ for discs - -
_
-
I, ~
,I
J J /JProlate
/"
~-
\Ob,o~e Spheroids
Toroids
/
\
f
,.
e, Spheroids L
o.,
-
o.o3
%-~'.%
=1"
//
/~/
O.OI ~
o.o,
-0.04
,.o
-
,o
,oo
0.01
,ooo
L/D Fro. 1. Plot of a dimensionless ratio proportional to the effective diffusivity against aspect ratio for circular cylinders, prolate spheroids, oblate spheroids, and toroids. coefficient becomes independent separation and is given by @~vv,x _
9
(FexD] 2
of the
[28]
with @ = kT/6~rpJ) as L / D --+ ~. F o r L / D = 4.7622, the numerical value of the coefficient in [28] is reduced to 1.87 × 10 -~ as c o m p a r e d to 2.20 × 10-8 for L/D--+ ~; thereafter, the coefficient reduces more rapidly as the spheres approach and for touching spheres it is estimated to be 2.6 × 10 -4. DISCUSSION In order to characterize the dispersion of nonspherical particles due to the interaction of Brownian rotation and sideways drift by diffusivities independent of time, it is necessary for sufficient time to elapse so that the particles execute many changes of orientation. Thus the elapsed time should significantly e x c e e d the time 1/~R characteristic of rotary diffusion. H o w e v e r , if one is dealing with a large n u m b e r of particles whose initial orientations are random, then the slurry probably can be described by a time-independent diffusivity much more quickly than if all orientations Journal of Colloid and Interface Science,
Vol.71, No. 2, September1979
were initially the same. F o r particles of a specified geometry, 1/@a varies as the volume of the particle or the third p o w e r of particle size so that the time before these results b e c o m e applicable increases rapidly as particle size increases. The Brownian diffusivity decreases with the inverse first p o w e r of particle size. H o w e v e r , Eq. [25] for the ratio of effective diffusivity to Brownian diffusivity is extremely sensitive to particle size. If the external force Fex is a body force, then @ErF/@ varies as the eighth p o w e r of particle size! F o r sufficiently small particles the effect discussed here will be negligible c o m p a r e d to Brownian diffusion, but for sufficiently large particles this effect will dominate Brownian diffusion. This is illustrated in the following table for circular cylinders of aspect ratio L / D = 10. The cylinders are taken to have a density of 2 g/cm ~ and to be sedimenting through water (density = 1 g/cm 3, viscosity = 10-z P, absolute temperature = 300°K) in response to gravity. F o r L = 1 /~m, @ = 1.17 × 10-s cm2/sec and ~ E F F / ~ = 1.06 × 10 -7, showing that @EFt is completely negligible. H o w e v e r at L = 10 ~m, @ is reduced to 1.17 x 10-9 cm2/sec but @Ev~/@ is increased to 10.6, giving @Err = 1.24 x 10-s cm2/sec; now the Brownian
215
SEDIMENTING PARTICLES
diffusion is negligible. For the example chosen, a fiber 10 /zm long and 1 /zm in diameter has an effective diffusivity comparable to the Brownian diffusivity of a fiber 1/xm in length and 0.1/zm in diameter, provided the time for sedimentation exceeds about 136 sec or provided one is dealing with large numbers of particles whose initial orientations are largely random. It is worth pointing out the complete parallel of this diffusive process with the well-known phenomenon of Taylor dispersion. For laminar flow in a pipe as an example, the coupling of axial convection with radial diffusion gives rise to an effective axial diffusivity (after sufficient time) in addition to the molecular diffusivity 9. This is given by V2D2/192~, where D is the pipe diameter and V is the fluid velocity. A molecular species of very high @ samples all radial positions sufficiently rapidly so that it is convected down the pipe with the average fluid velocity; the only axial dispersion would be due to molecular diffusion. Compare this with Eq. [22] or [23] which shows an effective diffusivity proportional to the square of a characteristic particle velocity, Fex(fi -1 - f~-l), and inversely proportional to the Brownian rotary diffusivity @R. A particle with very high @a samples all orientations sufficiently rapidly that it appears isotropic and therefore settles vertically. Thus there will be no translational diffusivity except for the Brownian translational diffusivity. The results reported so far are applicable only for particles subjected to no external torque. An external torque would tend to produce preferred orientations. It is relatively easy to compute the effective diffusivity if the external torque is given by Hooke's law, i.e., if the restoring torque is proportional to the angle between Fex and the particle's symmetry axis, and if the coefficient of proportionality K is sufficiently large compared to kT. Then only slight deviations from the vertical orienta-
TABLEI Comparison of the Effective Translational Diffusivity with the Brownian Diffusivity for Cylinders of Density 2 g/cm a Sedimenting Through Water in Response to Gravity. L
l/~rt
(~m)
(sec)
Nzrv/'N
~
~EFF
(cmVsec)
(cmVsec)
1
1.36 × 10 '
1.06 × 10 - r
1.17 x i 0 -s
1.18 × 10 - i s
5
1.70 × 10
4 . 1 4 × 10 -2
2 . 3 4 × 10 -a
9 . 6 9 × 10 - n
7.44
5 . 6 0 x 10
1
1.57 x 10 -9
1.57 x 10 -9
1.36 x 10 z 1.09 x 10a
1.06 x 10 2.71 x 103
1.17 x 10 ~ 5 . 8 5 x 10 -1°
1.24 x 10 8 1.59 x 10 -6
10 20
tion are likely and sin 0 cos 0 in [3] and subsequent equations can be replaced by 0. Analysis along the lines presented here then shows ~EFF,x =l_ . { F e x ] 2 30e71 - f ~ l ) ~ e R
2 ~K-)
( f l 1 + 2f~-1)
[291
provided K/kT >> 1. It might be of interest to work out the transition from [29] for K/kT >> 1 to [25] for K/kT ~ 1 as a function of K/kT.
REFERENCES 1, Brenner, H., and Condiff, D. L W., J. Colloid Interface Sci. 41, 228 (1972). 2, Gallily, I., and Cohen, A-H., J. Colloid Interface Sci. 56, 443 (1976). 3, Happel, J., and Brenner, H., " L o w Reynolds Number Hydrodynamics." Prentice-Hall, Englewood Cliffs, N. J., 1965. 4. Fuchs, N. A., "The Mechanics of Aerosols." Macmillan Co., New York, 1964. 5. Broersma, S., J. Chem. Phys. 32, 1626 (1960). 6. Batchelor, G., J. Fluid Mech. 44, 419 (1970). 7. Uhlenbeck, G. E., and Ornstein, L. S., in "Selected Papers on Noise and Stochastic Processes" (Nelson Wax, Ed.), pp. 93ff. Dover, New York, 1964. 8. Goldstein, H., "Classical Mechanics," pp. 107f, 158. Addison-Wesley, Reading, Mass., 1959. 9. Gans, R., Ann. Phys. (Leipzig) 86, 628 (1928). 10. O'Neill, M. E., and Goren, S. L., to be published. 11. Goldman, A. J., Cox, R. G., and Brenner, H., Chem. Eng. Sci. 21, 1151 (1966). 12. Szu, S. C., and Hermans, J. J., J. Polym. Sci. 11, 1941 (1973).
Journal of Colloid and lnterJace Science, V o L 71, N o . 2, S e p t e m b e r 1979
@Evv/~ = ( FexL/32kT) 2. INTRODUCTION
This paper calls attention to the fact that the coupling of Brownian rotation with the sideways drift of sedimenting, nonspherical particles results in effective vertical and lateral diffusion of the particles over and above that due to Brownian translational diffusion. The importance of coupling of rotational and translational effects on the transport properties of colloidal particles was recognized by Brenner and Condiff (1). They presented a general and rigorous framework for the discussion of these effects, but this was not specialized to the case considered in the present paper. Such coupling was also recognized by Gallily and Cohen (2) who concluded that even a monodisperse collection of nonspherical particles would give rise to a distribution of aerodynamic diameters as measured by sedimentation times. The calculations of 1This paper was presented at the European Mechanics Colloquium on Low Reynolds Number Flow held in Jablonna, Poland, March 29, 1978.
Gallily and Cohen are based on Monte Carlo techniques and so the results, while illustrative of the effect being considered, are not of general applicability. In the present work the long-time behavior of sedimenting nonspherical particles is examined analytically and relatively simple formulas for the effective diffusion coefficient are deduced. Knowing the effective diffusion coefficients, one could easily calculate the apparent distribution of aerodynamic particle size for a sedimenting cloud of particles or the spatial distribution of such particles. ANALYSIS
A small particle in creeping motion experiences a drag force proportional to the particle's velocity. The proportionality factor (more generally the translation tensor (3) multiplied by the viscosity /x of the suspending medium) is called the friction coefficient and is denoted by the symbolf.
209 Journal of Colloid and Interface Science, Vol. 71, No. 2, September 1979
0021-9797/79/11020%07502.00/0 Copyright© 1979by AcademicPress, Inc. All rights of reproduction in any form reserved.
210
SIMON
L. GOREN
For nonspherical particles, f depends on the shape and orientation of the particle relative to the direction of motion. In this paper we consider only particles that possess both an axis of symmetry and a plane of symmetry perpendicular to this axis. These particles are characterized by two generally different friction coefficients, fl and f~, for translation parallel to and translation perpendicular to the axis of symmetry, respectively. Examples for elongated particles are prolate spheroids and long circular cylinders (rods). With L denoting the major axis and D the minor axis, theory (4-6) shows that fi = 2~r~L/ln f~ = 47r/xL/ln
(2L/D), (2L/D)
[1] [2]
for both types of particles provided L/D is very much greater than unity. Examples of squat particles are oblate spheroids and short circular cylinders (disks). A particle of the type considered here when sedimenting at low Reynolds number in response to an external force Fex generally does not move totally in the direction of the applied force. Instead it also acquires a horizontal component of velocity that depends on the angle 0 between the direction of the applied force and the symmetry axis of the particle. In the absence of Brownian and fluid inertial effects, the particle maintains its initial orientation. It is easy to show by a force balance (3) that the sideways velocity is given by
Squaring [4] gives 1)2
y~(t)=F~x(~ x cos
f2
t I o I i sin 0(s)
O(s) sin O(r) cos O(r)dsdr. [5]
If the particle is sufficiently small it will also be subjected to Brownian diffusion and Brownian rotation; these motions are assumed to be uncorrelated. Here we are primarily interested in the interaction of the Brownian rotation with the sideways drift. The particle's orientation gradually changes due to the Brownian rotation so that the particle executes a complicated zig-zag path as it sediments. As this motion has all the characteristics of a random walk, a cloud of sedimenting particles is expected to diffuse laterally. The effective lateral diffusion coefficient may be found from the long-time behavior of y2(t) according to the well-known (7) formula
~E~v = lim E{y~(t)}/2t. [6] t--,~ E{ } represents the expected value or ensemble average of the quantity in braces. Moreover, since the vertical velocity fluctuates with fluctuations in orientation, there will be a distribution of vertical displacements about the average vertical position; this gives rise to an effective vertical diffusion coefficient. To calculate these effective diffusion coefficients the stochastic properties of the orientation angle O(t) must be known so that the long-time behavior of y2(t) can be deduced from -~- - F e x f2 s i n 0 c o s 0 . [3] Eq. [5]. The orientation of a particle can be speciAllowing for a time-dependent orientation fied by the three Eulerian angles 0, ~b, ~b. 0 but neglecting the particle's inertia, we can is the angle between a vertical axis fixed in integrate [3] to find the instantaneous dis- space and the instantaneous axis of symmetry of the moving particle and is the placement y(t) for a particle starting from angle appearing in the discussion so far. y=0. 4) is the angle in a fixed horizontal plane 1 between a fixed horizontal line and the "line y(t) = F e x ( ~ f 2 ) of nodes," this being the intersection of the horizontal plane and a plane moving with x sin O(s) cos O(s)ds. [41 the particle and normal to its axis of symmetry. The angular motion of the particle is
f2
Journal of Colloid and Interface Science, Vol. 71, No. 2, September 1979
SEDIMENTING PARTICLES governed by the well-known Euler equations of motion (Goldstein (8)) which express the law of conservation of angular momentum about three orthogonal axes moving with the particle. Using Euler's equations, it is straightforward to express the law of conservation of angular momentum about the line of nodes. For the symmetrical particles considered here subjected to no external torques other than the hydrodynamic resistance to rotation and the random bombardment of solvent molecules, the result is I
d~O dt 2
dgp d O dt dt
+ I~ - - -
-
sin 0
(I - Is) - - - - dd~b t h sin 0 cos 0 dt dt = -fR
dO dt
+ B(t).
[7]
[8]
Is is the moment of inertia of the particle about its symmetry axis. B(t) is a random, Gaussian torque of zero mean exerted on the particle about the line of nodes by the molecules of the suspending medium. Because of the symmetry of the particles, this torque does not depend on orientation. About the only important characteristic of B needed for our purposes is that the autocorrelation of B, E { B ( t ) B ( t + s)}, at two different times, t and t + s, is a very rapidly decreasing function of the time difference s. The integral I ~ E { B ( t ) B ( t + s)}ds = fl
is a constant /3, as yet unknown, and the great preponderance of the integral defining /3 is contributed near s = 0. Equation [7] is complicated by terms quadratic in the angular velocities. However, as the following estimate shows, these terms will be negligible compared to the viscous term under many applications. Since the average kinetic energy of rotation approaches that given by the equipartition of energy at long times, for order of magnitude purposes we estimate the angular velocities by (kT/I) 1/2. Then the order of magnitude of the ratio of the nonlinear inertial terms to the viscous term is kT/ fR(kT/I) l/z= (kTI)l/2/fR. Using Eq. [8] to calculate the rotary friction coefficient for a long circular cylinder whose moment of inertia about a transverse axis is 7rpD2L3/48, we write the ratio as QUADRATIC TERMS
I is the moment of inertia of the particle about a line perpendicular to its axis of symmetry and through its midsection, fR is the rotary friction coefficient which is the proportionality factor between the torque and the angular velocity about the line just described. For both prolate ellipsoids (9) and circular cylinders (5, 6) of very large aspect ratio fR = ½rclxL3/ln (2L/D).
211
[9]
VISCOUS TERM 3kTio
- (1-6~tz2/
/ 1/2 I n
(2L/D)
(L/D)
[10]
For a circular cylinder of density 2 g/cm 3, length 10 /xm, and aspect ratio L/D of 10 immersed in water at 25°C, this ratio is computed to be 6 x 10-5, implying that to a very good approximation the quadratic terms can be omitted from Eq. [7]. Since the above ratio varies as only the inverse square root of particle size L for fixed aspect ratio L/D, the quadratic terms are negligible down to particles of molecular dimensions suspended in water. With the neglect of the quadratic terms, Eq. [7] reduces to d20 I -- dt 2
dO -JR - - + B(t). dt
[11]
The mathematical treatment of Eq. [11] from which the rotary diffusion coefficient DR can be deduced is well known (7). Key points required for the present development are summarized below. The formal solution of [11] for the angular velocity O(t) and angular position O(t) of a particle starting from rest with 0 -- 0 is Journal of Colloid and Interface Science, VoI. 71, No. 2, September 1979
212
SIMON L. GOREN
o(t)- dOd~- exp(-fRt/I)
Ii B(O I exp(fR~/I)d~
[121
and
O(t) = fot B(~) fR dE _ exp(--fRt/I)
-I Ii -B(O
exp(fR~/I)d~.
[131
If the particle is free of external torques as implied by [11], then the expected values of the angular velocity and angular position are both zero at any time. The autocorrelations of these quantities h o w e v e r are not zero, but are related to the autocorrelations of the random torque. Following (7) we find
E{O(t)b(t)} = exp(--2fRt/I) f l I o E { B ( ' ) B12( ~ ) } e x p ( f R [ , + l x ] / I ) d t z d , -
E{O(t)O(s)} =
[3 2if.
[1 -
exp(--2fRt/I],
I[ E{B(~)B(~)}dtzd( -
f l
exp(--fRt/I)
× exp(fR~/I)dtzd~ - exp(-frts/I) + exp(-frt[s +
[14]
Jl
Io' E{B(~)B(Ix)} f~
s E{B(()B(tz)} exp(fRtz/I)dtxd~
f~
t]/I) Ii Irs E{B(~)B(tz)} exp(fR[~: + tx]/I)dlxd~ .o f~
fll - - - [fRt/1 - 1 - ½ exp(-fR[s - t]/I) + exp(-fRs/I)
f~
+ exp(-fnt/I) - ½ exp(--fR(s + t)/I)], for s > t with an equation analogous to [15] for s < t. In obtaining these results use has been made of the very rapid decay of the autocorrelation of the r a n d o m torque and of exchanging the order of averaging and integrating. Since the particle eventually achieves thermal equilibrium with the suspending medium, its expected rotary kinetic energy at infinite time, ½IE{O2(~)}, must be given by the equipartition of energy ½kT which requires fl = 2kTfR. [ 16] The rotary diffusion coefficient @R' is the limit of E{O2(t)}/2t as t approaches infinity; setting s = t in [15] and taking the indicated limit results in ~r~ = lira t--,~
E{O2(t)} - kTfR.
[17]
2t
The dimensions of Or are reciprocal time. Journal of Colloid and Interface Science, Vol. 71, No. 2, September 1979
[15]
Inserting [13] into [5] gives the time dependence ofy2(t) for an individual particle. The e x p e c t e d value ofy2(t) for an ensembie of particles is
1 )2
E{y2(t)} =F~x(~ × cos
f2
t Ii Ii E{sin O(s)
O(s) sin O(r) cos O(r)}dsdr.
[18]
A theorem from statistics states that if X and Y are random Gaussian variables with zero means, then
E{exp(i[X + Y])} = e x p ( - ½ E { X 2} -
½E{
i12} _ E { X Y } ) .
[19]
I am grateful to Professor N o r m a n Kaplan for calling my attention to this theorem. Since the trigonometric functions can be represented in terms of exponentials with
SEDIMENTING PARTICLES imaginary argument and since O(t) is a random Gaussian variable with zero mean, use of [19] reduces [18] to
E{yZ(t)} = x
VsF~x( 1_ 1) ~ \fl A
f f2
These two effective diffusivities are over and above those due to Brownian translational diffusion. For comparison, the Brownian translational diffusion coefficient @ for a particle free to assume all orientations randomly is
exp(-2[E{O2(s)}
= kT(f-f 1 + 2f~-1)/3.
[20] ~Erv,x
The three functions E{O2(s)}, E{O2(r)}, and E{O(s)O(r)} appearing in the integrand are all known from Eq. [15]. We expect the coupling of Brownian rotation with sideways drift to result in lateral diffusion, so E{y2(t)} is expected to increase linearly with t for sufficiently long times. In accordance with [6], the effective diffusivity is found from the limit of E{y2(t)}/2t as t becomes infinite. Application of L'Hospital's rule twice to the indefinite quotient E{y2(t)}/2t as t ~ oo with E{y2(t)} given by [20] does indeed yield the following "constant" as the limit: ~ E F F ~- F 2 x ( f l
1 --
f~1)2/32~R.
[24]
On dividing [22] by [24] and replacing DR by [17] we find
+ E{O2(r)})[exp(4E{O(s)O(r)}) - exp(-4E{O(s)O(r)})]dsdr.
213
[21]
1 {Fex ]2 3(f11 _ f~-x)~fR - -~\'-~] ( f l x T 2f~-1)
[25]
For the special case of prolate spheroids or circular cylinders of very large aspect ratio L/D, use of [1], [2], and [8] reduces [25] to
~EFr,x _ 1._}__(F¢xL / 2 " @ 1024\ kT ]
[26]
For the special case of oblate spheroids or circular cylinders of very small aspect ratio, inserting known values (2) offl,f2, andfa for circular disks into [25] yields
@EVr,x __
l___~(FexD] z 2048\ kT ] "
[27]
For intermediate values of the aspect ratio, @~vv,xl~ depends on L/D and the specific geometry of the particle. This dependence is illustrated in Fig. 1 for several geometries. These results were calculated from [25] for oblate and prolate spheroids using the friction coefficients derived by Overbeck as reported in Fuchs (4) and Gans (9), for circular cylinders using the friction coefficients derived by Batchelor (6), and for toroids using the friction coefficients ~EFF,x = ~EFF,y derived by O'Neill and Goren (10). It is also = F~x(fi-' --f~l)2/64~R. [22] possible to calculate the effective diffusivity for dumbbells (i.e., two spheres of each of By an analysis following the same lines as above, it is possible to show that the effec- diameter D connected by an infinitely thin tive diffusivity in the vertical direction aris- rigid rod of length (L - 2D) using the exact ing from fluctuations of the settling velocity translational frictional coefficients of Goldwith fluctuations in orientation is just twice man et al. (11) and an approximate rotary friction coefficient of Szu and Hermans (12). the lateral diffusivity, i.e., Here we find the surprising result that for ~EFF,z = F 2 x ( f ~ -1 - f~-l)2/32~R. [23] large separations the effective diffusion Because the angle 0 always lies in a plane containing the vertical axis but this plane is free to rotate about the vertical axis, the lateral displacement computed above is a radial displacement &. Since &2= x = + y2 and we have no reason to distinguish between diffusion in the x and y directions, the effective diffusivity in the lateral direction is one-half that given by [21] or
Journal of Colloid and Interface Science, Vol. 71, No. 2, September 1979
214
SIMON L. GOREN 0.05 ~
_0.05
I 0.04~o
I I
,.,
~-c~|
CirculorCylinders ,, L . _ " J ~
/ for discs - -
_
-
I, ~
,I
J J /JProlate
/"
~-
\Ob,o~e Spheroids
Toroids
/
\
f
,.
e, Spheroids L
o.,
-
o.o3
%-~'.%
=1"
//
/~/
O.OI ~
o.o,
-0.04
,.o
-
,o
,oo
0.01
,ooo
L/D Fro. 1. Plot of a dimensionless ratio proportional to the effective diffusivity against aspect ratio for circular cylinders, prolate spheroids, oblate spheroids, and toroids. coefficient becomes independent separation and is given by @~vv,x _
9
(FexD] 2
of the
[28]
with @ = kT/6~rpJ) as L / D --+ ~. F o r L / D = 4.7622, the numerical value of the coefficient in [28] is reduced to 1.87 × 10 -~ as c o m p a r e d to 2.20 × 10-8 for L/D--+ ~; thereafter, the coefficient reduces more rapidly as the spheres approach and for touching spheres it is estimated to be 2.6 × 10 -4. DISCUSSION In order to characterize the dispersion of nonspherical particles due to the interaction of Brownian rotation and sideways drift by diffusivities independent of time, it is necessary for sufficient time to elapse so that the particles execute many changes of orientation. Thus the elapsed time should significantly e x c e e d the time 1/~R characteristic of rotary diffusion. H o w e v e r , if one is dealing with a large n u m b e r of particles whose initial orientations are random, then the slurry probably can be described by a time-independent diffusivity much more quickly than if all orientations Journal of Colloid and Interface Science,
Vol.71, No. 2, September1979
were initially the same. F o r particles of a specified geometry, 1/@a varies as the volume of the particle or the third p o w e r of particle size so that the time before these results b e c o m e applicable increases rapidly as particle size increases. The Brownian diffusivity decreases with the inverse first p o w e r of particle size. H o w e v e r , Eq. [25] for the ratio of effective diffusivity to Brownian diffusivity is extremely sensitive to particle size. If the external force Fex is a body force, then @ErF/@ varies as the eighth p o w e r of particle size! F o r sufficiently small particles the effect discussed here will be negligible c o m p a r e d to Brownian diffusion, but for sufficiently large particles this effect will dominate Brownian diffusion. This is illustrated in the following table for circular cylinders of aspect ratio L / D = 10. The cylinders are taken to have a density of 2 g/cm ~ and to be sedimenting through water (density = 1 g/cm 3, viscosity = 10-z P, absolute temperature = 300°K) in response to gravity. F o r L = 1 /~m, @ = 1.17 × 10-s cm2/sec and ~ E F F / ~ = 1.06 × 10 -7, showing that @EFt is completely negligible. H o w e v e r at L = 10 ~m, @ is reduced to 1.17 x 10-9 cm2/sec but @Ev~/@ is increased to 10.6, giving @Err = 1.24 x 10-s cm2/sec; now the Brownian
215
SEDIMENTING PARTICLES
diffusion is negligible. For the example chosen, a fiber 10 /zm long and 1 /zm in diameter has an effective diffusivity comparable to the Brownian diffusivity of a fiber 1/xm in length and 0.1/zm in diameter, provided the time for sedimentation exceeds about 136 sec or provided one is dealing with large numbers of particles whose initial orientations are largely random. It is worth pointing out the complete parallel of this diffusive process with the well-known phenomenon of Taylor dispersion. For laminar flow in a pipe as an example, the coupling of axial convection with radial diffusion gives rise to an effective axial diffusivity (after sufficient time) in addition to the molecular diffusivity 9. This is given by V2D2/192~, where D is the pipe diameter and V is the fluid velocity. A molecular species of very high @ samples all radial positions sufficiently rapidly so that it is convected down the pipe with the average fluid velocity; the only axial dispersion would be due to molecular diffusion. Compare this with Eq. [22] or [23] which shows an effective diffusivity proportional to the square of a characteristic particle velocity, Fex(fi -1 - f~-l), and inversely proportional to the Brownian rotary diffusivity @R. A particle with very high @a samples all orientations sufficiently rapidly that it appears isotropic and therefore settles vertically. Thus there will be no translational diffusivity except for the Brownian translational diffusivity. The results reported so far are applicable only for particles subjected to no external torque. An external torque would tend to produce preferred orientations. It is relatively easy to compute the effective diffusivity if the external torque is given by Hooke's law, i.e., if the restoring torque is proportional to the angle between Fex and the particle's symmetry axis, and if the coefficient of proportionality K is sufficiently large compared to kT. Then only slight deviations from the vertical orienta-
TABLEI Comparison of the Effective Translational Diffusivity with the Brownian Diffusivity for Cylinders of Density 2 g/cm a Sedimenting Through Water in Response to Gravity. L
l/~rt
(~m)
(sec)
Nzrv/'N
~
~EFF
(cmVsec)
(cmVsec)
1
1.36 × 10 '
1.06 × 10 - r
1.17 x i 0 -s
1.18 × 10 - i s
5
1.70 × 10
4 . 1 4 × 10 -2
2 . 3 4 × 10 -a
9 . 6 9 × 10 - n
7.44
5 . 6 0 x 10
1
1.57 x 10 -9
1.57 x 10 -9
1.36 x 10 z 1.09 x 10a
1.06 x 10 2.71 x 103
1.17 x 10 ~ 5 . 8 5 x 10 -1°
1.24 x 10 8 1.59 x 10 -6
10 20
tion are likely and sin 0 cos 0 in [3] and subsequent equations can be replaced by 0. Analysis along the lines presented here then shows ~EFF,x =l_ . { F e x ] 2 30e71 - f ~ l ) ~ e R
2 ~K-)
( f l 1 + 2f~-1)
[291
provided K/kT >> 1. It might be of interest to work out the transition from [29] for K/kT >> 1 to [25] for K/kT ~ 1 as a function of K/kT.
REFERENCES 1, Brenner, H., and Condiff, D. L W., J. Colloid Interface Sci. 41, 228 (1972). 2, Gallily, I., and Cohen, A-H., J. Colloid Interface Sci. 56, 443 (1976). 3, Happel, J., and Brenner, H., " L o w Reynolds Number Hydrodynamics." Prentice-Hall, Englewood Cliffs, N. J., 1965. 4. Fuchs, N. A., "The Mechanics of Aerosols." Macmillan Co., New York, 1964. 5. Broersma, S., J. Chem. Phys. 32, 1626 (1960). 6. Batchelor, G., J. Fluid Mech. 44, 419 (1970). 7. Uhlenbeck, G. E., and Ornstein, L. S., in "Selected Papers on Noise and Stochastic Processes" (Nelson Wax, Ed.), pp. 93ff. Dover, New York, 1964. 8. Goldstein, H., "Classical Mechanics," pp. 107f, 158. Addison-Wesley, Reading, Mass., 1959. 9. Gans, R., Ann. Phys. (Leipzig) 86, 628 (1928). 10. O'Neill, M. E., and Goren, S. L., to be published. 11. Goldman, A. J., Cox, R. G., and Brenner, H., Chem. Eng. Sci. 21, 1151 (1966). 12. Szu, S. C., and Hermans, J. J., J. Polym. Sci. 11, 1941 (1973).
Journal of Colloid and lnterJace Science, V o L 71, N o . 2, S e p t e m b e r 1979