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Transfer of particles over a randomly fluctuating energy barrier
Transfer of Particles over a Randomly Fluctuating Energy Barrier PIOTR WARSZYNSKI AND JAN CZARNECKI l Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek, 30-239 Krakow, Poland Received November 30, 1987; revised March 30, 1988 The effect of fluctuations in the interaction energy on particle transport to the interface is considered by solving the one-dimensional stochastic transport equation both analytically (approximately) and numerically. The results show that in the presence of fluctuations a considerable increase of particle transfer is predicted, provided that the correlation time of the fluctuations is longer than 0. l-1 ms for l ~m particles in an aqueous solution. © 1989AcademicPress,Inc.
INTRODUCTION
According to the DLVO theory the energy between interacting bodies, i.e., between particles in coagulation or a particle and a collector in deposition, is expressed as the sum of the electric double layer and dispersive contributions. The stability of colloidal systems is related to the form of the relationship of the interaction energy vs the distance between the interacting bodies. If the double-layer interactions are repulsive, a characteristic maxim u m in the total energy may exist (the socalled energetic barrier). The rate of particle transfer over the barrier, i.e., the rate of coagulation or deposition, depends strongly on the barrier height and thickness. The DLVO theory assumes that the form of the energy vs distance relationship remains constant during a particle-particle or a particle-collector encounter, and that the interaction energy is a function of the distance between the interacting bodies only. It is usually assumed that the considered system is described by the mean values of the respective parameters. However, these parameters may fluctuate around their mean
1 On sabbatical leave with the Department of Chemistry, McGiU University, Montreal, Canada.
values for thermodynamic reasons and this may produce temporary changes in the interaction energy. There are fluctuations in the concentration of ions forming the electrical double layers, both in the diffuse layers and in the Stern layers. The amounts of adsorbed ions and dissociated groups responsible for the surface charge also fluctuate, leading to fluctuations in the interaction energy. In every real system the interaction energy, even for spherical particles, is not dependent only upon the coordinates normal to their surfaces (i.e., the distance between the interacting bodies), but also depends on the mutual particle orientation, due to surface roughness or heterogeneity in the surface charge distribution. Also, in particle deposition the interaction energy for a given particle is a function of its position above the collector surface, as was pointed out in (1). Thus, in connection with the translation or rotation of a particle due to convection or Brownian motion, surface irregularities produce temporal, random changes of the interaction energy. The randomness of these changes is due to the fact that the surface shape and surface charge are statistically distributed. An attempt at evaluating the rate of particle transport in systems where the interaction en-
137 0021-9797/89 $3.00 Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989
Copyright © 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
138
WARSZYNSKI A N D C Z A R N E C K I
ergy is a stochastic function of time is given below. The flux of particles over the energetic barrier is calculated using a one-dimensional equation of transport. The value obtained is then compared with the particle flux through a static barrier whose height is equal to the mean height of the fluctuating one. An analytical solution restricted to small fluctuations of the barrier height is discussed. Next, the numerical solution of the transport equation is presented. Then, on the basis of the results of both solutions, we evaluate the characteristic value of the correlation time of energy barrier fluctuations. Fluctuations with a correlation time longer than the characteristic one can significantly affect particle transport. The results are compared with those of a previous "deterministic" attempt at evaluating the influence of fluctuations on particle transfer over the energetic barrier (2). A N A L Y T I C A L SOLUTION FOR SMALL F L U C T U A T I O N S
We shall consider the one-dimensional equation of transport. This kind of equation describes the deposition of particles onto a rotaring disk collector (3) or deposition on other collectors near the forward stagnation point (4): Off 0 = -z-y..J± - P e F 3 ( H ) ( H + 1)f O-7 O H
+ Q ( f , H, r)
[1]
Do, H i s the dimensionless surface-to-surface distance scaled with respect to a, ~(H, r) is the interaction energy in kT units, F I ( H ) , F2(H), F3(H) are the universal correction functions accounting for hydrodynamic interactions between a particle and a fiat collector (5, 6), Ex is the normal component of the resultant external force scaled with respect to kT / a, Q( f , H, ~') is a dimensionless source term, and Pe is the Prclet number characterizing the significance of convection effect relative to diffusion. For a rotating disk, 0)3/2(23
Pe-- 1.02
vl/2vo~ '
where o:~ is the angular velocity of the disk and v is the fluid kinematic viscosity. In the formula above the interaction energy is, due to fluctuations, a stochastic function of time, like the one illustrated in Fig. 1. Following the surface boundary layer approximation (7, 8), we can assume that within the region close to the collector surface (region I in Fig. 2a) where the energetic barrier exists, the only mechanism of particle transport is diffusion and rewrite Eq. [ l ] as 0--~=ffH0ff 0 F 1(H )[~--~+~-~f r 0ff 0~ ] •
[2]
If the changes of the height of energy barrier are faster than 1/rD, where rD is the dimensionless time for formation of a panicle concentration profile in the vicinity of the collector (2), the particles in the region where surface
where Off O~b(H,7") _ j± = FI(H) - ~ + OH n
1
+ ~ P e F 2 ( H ) ( H + l)2ff + Exff
]
is the normal component of the dimensionless panicle flux j± = ja/Do~n~, no~ is the bulk concentration, a is the particle radius, Do is its bulk diffusion coefficient, f is the dimensionless concentration scaled by n ~, r is the dimensionless time scaled with respect to a2/ Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989
FIG. 1. Fluctuations of the interaction energy between a panicle and a collector surface. The D L V O theory concerns the mean value o f energy ( ~ ) .
PARTICLE TRANSFER OVER AN ENERGY BARRIER
forces may be neglected will not feel the relatively fast changes in the barrier height. Thus, we can assume that the particle concentration at the outer boundary of the region I (cf. Fig. 2a) is practically independent of time. Moreover, if the mean energy barrier is high, the transport of particles over the barrier will be the rate-determining step of the process. The case when the barrier is high is the most interesting since for such a barrier the influence of the fluctuations on particle flux is expected to be significant. In such a case we can assume that the convective diffusion is fast in comparison with the barrier penetration, so that ff = 1 outside the barrier. Thus, the boundary conditions to Eq. [ 2 ] are
-I
I
Z
I
a
I I I
\
q~s
![
I
t7 = 0 t7= I
and
and
139
q~ ~ - o o
for
~=0 for
H>t~0+~B,
for
0
F1 02~ ~
0~
-- F I F B ( r) ~-~ = 0
~i(0, r) = 0 t7(oo, r) = 1.
b
.\', /oe \\N ,
[4]
in semi-infinite space with boundary conditions: H
g
[31
For simplicity we take ~o = 0, and ~B is a stochastic function of time only. If the average barrier height fulfills the condition that (~B(Z)) ~> 1, the solution of Eqs. [ 2 ] and [3 ] is practically the same as the solution of 0~
~,~
[2a]
where ]B is the width of the barrier. Here the perfect sink approximation was used at the separation $0 to collector surface (4). To obtain an approximate analytical solution of Eq. [2] let us consider the simplest case, where the energetic barrier has a triangular shape:
0---~ --
i
It II
H < ~o
-
[4a]
In Eq. [4 ] we disregarded the dependence of F~ on H (F1 is the value of F1 (H) calculated somewhere between 0 and ~B) and FB = 4~B/~Bis the interaction force. We now divide the interaction force into two parts, FB(z) = 2 F + 2ef(r),
." ] :'l
H FIG. 2. The shapes of the interaction energy profiles introduced in (a) the approximate analytical solution, and (b) the numerical solution of the stochastic transport equation.
[51
where 2 F is a nonrandom part of the force equal to the average force (FB(r)), 2~f(r) is the fluctuating part of the force acting on a particle whose mean ( f ( r ) ) is zero, and E is a small expansion parameter which assured the condition
'l/(f2(r)) ,~ 1. F Journal of Colloid and Interface Science, Vol.
128, No. 1, March 1, 1989
140
WARSZYNSKI AND CZARNECKI
The factor 2 is included to simplify the calculations. Substituting Eq. [ 5 ] into Eq. [ 4 ] we get On
d2ff
Or
OHz
Off Off 2F-z-,,o,1- 2cf(r)~-~ = 0.
ON &
02N OH 2
2F ON OH
- 4~2 ( f ( r) -~H £" £~° dr' dti [6]
× Go(H,~,r-r')f(r')~}
Here we have rescaled the dimensionless time r as r/F~. There are two operators in Eq. [6 ]: the deterministic one, O
Zo = 07"
02 0H 2
0
2 F ~OH'
and the random operator, l?(r) = 2f(r)(O/ OH). Introducing these operators in Eq. [6] we obtain
[Lo + W(r)]fi(H, r) = 0.
[71
To calculate the mean rate of particle transfer over the energy barrier, we have to find the equation for the mean concentration profile N(H, r) based on the equation for the instantaneous concentration (Eq. [ 6 ] or [ 7 ] ). Then differentiating N(H, r) with respect to H we obtain the mean particle flux. To find the equation for the mean concentration, we use the Bourret approximation (9, I0) for Eq. [7], [Lo - ~2(I?LolI?}]N(H, r) = O(e2),
[8]
where Lo I is the operator inverse to L0 (11) given by
X N(~, r ' ) = O.
Using the commutation rule for integration and averaging in Eq. [11 ], and assuming that the fluctuations of the interaction force are represented by a stochastic process with the correlation function (13 ) given by
( f ( r ) f ( r M ) > = a2e -e=l*-''l,
[121
where f12 = 1/rcF~, re is the correlation time of the interaction force fluctuations, and a 2 is the mean square fluctuation, we get
ON (Or
OeN
2F O___N OH
OH 2 2
2
0
r
- 4 e a ~-~£ d r ' e - a " - " l £
oo
dli
0
X Go(H, ~, r - r') -~ N(~, v') = 0
N(0, r) = 0 and N ( ~ , r) = 1.
drd~Go(x, ~, t - r)f(~, r),
[9]
where G(x, ~, t - r) is Green's function for the diffusion into the semi-infinite space (12): e-F(x-O-F 2(t-r)
Go(x, ~, t - r) =
1/41r(t - r)
X [e -((x-~)2/*('-~)) - e-"X+o~/4"-*)q. [10] Substituting Eq. [9] into Eq. [8 ] we obtain the integrodifferential equation for the mean particle concentration: Journal of Colloid and lnterface Science,
Vol. 128,No. 1, March 1, 1989
[131
with the boundary conditions
L~lf(x, t) =
[Ill
[13a]
We solve Eq. [13] by the successive approximation method with the restriction to times much greater than the fluctuation correlation time re. In the first approximation, the dimensionless mean particle flux - F1 ON/ OH for H = ~B is given by ( j l > = Jo(]s)
r8e, a2e-2r" r 2 F ~ ,
+&t ~7~
8F(~2 °'2 +
1 F]
L'v+F ]
e
,
[141
141
PARTICLE TRANSFER OVER AN ENERGY BARRIER (l±> Jo
A 3.0
2.0
1.0 100
10 4
10-2
10-3
10-4
10-5
Tc
FIG. 3. The normalized mean flux of particles to the collector surface as a function of the correlation time of fluctuations of the interaction energy. Curve A: a = 2.5, curve B: a = 2.0, curve C: ~ = 1.5. The mean barrier height ( ~ ) = 8 and its width ~B = 0.03 are c o m m o n to all curves.
where J0(]B) = 2Fl Fe -2eg~ is the steady-state flux for the mean barrier and 3' = ~ +/32. In Fig. 3 the mean flux calculated from Eq. [14 ], normalized by the steady-state flux for the mean barrier, is shown as a function of the correlation time of fluctuations of the interaction force. Curves A, B, and C are cal-
culated for the same average barrier height but for three different magnitudes of fluctuations. For all cases presented in Fig. 4 the dispersion o f fluctuations is exactly the same and the curves differ from each other in mean barrier height. The conclusions are presented in the next section.
( i±> Jo
30 A 2£
C
1.0 .
.
.
.
.
.
.
.
.
i
100
I0-I
10-2
10-3
I0-4
10"5
,re
FIG. 4. The normalized flux of particles as a function of the correlation time of interaction energy fluctuations where the mean square fluctuation ~2 = 4 and the mean energetic barrier height: Curve A: ( ~ ) = 10, curve B: ( ~ ) = 8, and curve C: ( ~ = 6. Barrier width ~B = 0.03 is c o m m o n to all curves. Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989
142
WARSZYNSKI AND CZARNECK/
10 9 8
7 6
otT
FiG. 5. An example of the numerical model o f the random function $ ( r ) used in the calculations. NUMERICAL SOLUTION
The numerical solution of the stochastic transport Eq. [1 ] was obtained as follows: we took 20 possible realizations of the stochastic function $ ( H , r) and solved the deterministic transport equation using a Cranck-Nicholson method (14). Next, we averaged the solutions for all realizations to get a single solution of the stochastic equation. The interaction energy was expressed as
~(H, 7-)
=
t~m(n)[1
-4- t~r(T)] ,
where ~m(H) is the mean interaction energy profile illustrated in Fig. 2a and ~,(r) is the fluctuating term. A single realization of the stochastic function ~r(r) was obtained in the following way. The value ~r was selected randomly from the previously assumed distribution of interaction energy with mean value ~"and variance (mean
1012I
A
B
{
Jo
10-3
104
10-2
10-3
10-4
10-5 TC
Fie. 6. The mean flux of particles to the collector surface as a function of the correlation time of fluctuations o f the interaction energy obtained from the numerical solution of the stochastic transport equation where the mean barrier height (~_~ = 8. Curve A: cr/<~) = 0.31, curve B: ~ / < ~ ) = 0.19, and curve C: a/<~) = 0.125. The barrier width t~B = 0.03 and Pe = 0.02. ]ournal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989
[15]
143
PARTICLE TRANSFER OVER AN ENERGY BARRIER
square fluctuation) 0"2, simultaneously with changes in the Poisson process with the rate a. The probability that at least one change of the Poisson process occurs between r = 0 and r = T is equal to P =
1 -
[16]
e -'~T.
This procedure was applied until r = 6 0 0 / a , and then the average flux was calculated. In Fig. 5 an example of a single realization of the stochastic function ~r(r) obtained in such a way is presented. Figure 6 presents the results of the numerical solution of the stochastic transport Eq. [ 1] for the fixed mean energy profile and various mean square fluctuations of the interaction energy. The correlation function for the stochastic process used in the numerical solution of the transport equation to simulate fluctuations of the interaction energy is given by R ( r ) = ~-2 + ¢2e-~1,1
[17]
As the same correlation function (with ~"= 0) was previously assumed in the approximate analytical solution of the transport equation, we are able to compare the results of both methods directly. In Figure 7 the mean flux of particles (normalized with respect to the steady-state flux through the mean barrier) obtained from the numerical solution of the transport Eq. [ 1] is shown together with those calculated using an approximate analytical method. It can be seen that the analytical solution is valid only for small fluctuations where the analytical and numerical results are in good agreement. For large fluctuations only the numerical solution can be used. Based on the results presented above, one can draw the conclusion that the fluctuations of the interaction energy with short correlation time (to < 10 -4) cannot significantly affect the rate of the transport process. However, sufficiently large and long-correlated (long-lasting) fluctuations (re > 10 -3) may strongly increase the rate of the transport process by as much as one order of magnitude.
Jo 22
2.O
-
t
"t8
16 B 1.4
1.2
=
1.0
10 -~
10 -2
10 `3
10 -~
10 "~
Tc
FIG. 7. The normalized mean flux of particles as a function of the correlation time of interaction energy fluctuations obtained from the numerical solution of the stochastic transport equation (points) shown together with one obtained from the approximate analytical solution: Curve A: a ( $ ) = 0.19, curve B: a / ( ( o ) = 0.125. The barrier width ~B = 0.03 and Pe = 0.02. Journal of Colloid and Interface Science, Vol. 128,No. 1, March 1, 1989
144
WARSZYNSKI AND CZARNECK/
In a previous analysis of the effect o f fluctuations a "deterministic" model of fluctuations was used (2); i.e., the interaction energy was expressed as ~ ( H , r) = ~ m ( H ) ( 1 + B cos ¢or), where B and o~ = 2~r/rv were dimensionless constants characterizing the amplitude and frequency of energy oscillations (rF is an oscillation period). If we calculate the mean recurrence time of the mean value ~'m~ (15) for the process used in the numerical solution of the stochastic transport equation and assume that rmr = rF/2, then we can compare the results of these two methods. This is done in Fig. 8 where the values of the mean flux, obtained from the numerical solution of the transport equation and from the "oscillation" model, are shown together as functions of r~r = rF/2. It is seen that such a deterministic model of fluctuations may be useful as a first approximation. CONCLUDING REMARKS It was shown that fluctuations of the interaction energy may appreciably increase the average flux of particles penetrating the energy barrier, providing the dimensionless correlation time of fluctuations is longer than about
10-4 for a typical barrier width of 10 -2 to 5 X 10 -2 (normalized with respect to particle radius). If the correlation time is shorter, the changes of particle flux are damped and the flux approaches its steady-state value for the mean barrier height. In dimensional units, for a 1-#m particle in an aqueous solution, the critical correlation time is about 10 -4 s. It should be noted that for very fast changes of the forces acting on particles, Eq. [ 1] is no longer valid. This is because Eq. [ 1] neglects the inertia of the particles and the surrounding fluid, which becomes significant for rapid fluctuations, which are much faster than those corresponding to our critical correlation time. However, the inertia effects cause averaging of the forces acting on the particles and the resuiting flux may be expected to attain the steady-state value for the mean barrier. Thus, neglecting inertia effects in Eq. [ 1] should not affect the fast fluctuation limit o f our predictions. Crude estimates show that the fluctuations of the concentration of ions in the electrical double layer within the volume between interacting particles have too short a correlation time (re -~ 10 -6 s) to significantly affect the rate of particle transport. On the other hand,
10 "2
A
+
Jr fl-
+
4-
4-
4-
10-3
10"~
10-2
10-3
-e.l_
10-4
10-5
"/'mr
FIG. 8. Comparisonof the results of the numerical solutionof the stochasticequation (points) with those obtained using a "deterministic"approximationof fluctuations. Curve A: ~r/(~ > = 0.31, curve B: ~r/(~) = 0.19 and B = ~r~/(~). The averagebarrier height (~) = 8. Journal of Colloid and InteoeaeeScience, Vol. ! 28, No. 1, March 1, 1989
PARTICLE TRANSFER OVER AN ENERGY BARRIER the fluctuations o f the local surface charge d e n s i t y a n d fluctuation d u e to surface roughness a n d heterogeneity o f surface charge distribution, p r o v o k e d b y the m o t i o n o f particles, have sufficiently long c o r r e l a t i o n t i m e s to increase particle transfer rate over the energetic barrier. It is w o r t h y to n o t e t h a t such a n increase has i n d e e d b e e n o b s e r v e d in particle d e p o s i t i o n e x p e r i m e n t s (16, 17). Also, o u r p r e d i c t i o n s agree, at least qualitatively, with those o f Prieve a n d L i n (18), w h o s t u d i e d enh a n c e m e n t in the initial c o a g u l a t i o n rate for a n e n s e m b l e o f particles with a d i s t r i b u t i o n o f surface properties, r a t h e r t h a n particles with fluctuating surface properties.
5. 6. 7. 8. 9. 10. 11. 12.
13.
REFERENCES
14.
1. Czarnecki, J., and Warszynski, P., Colloids Surf. 22, 207 (1987). 2. Adamczyk, Z., Czarnecki, J., and Warszynski, P., J. Colloid Interface Sci. 106, 299 (1985 ); Czarnecki, J., Adv. Colloid Interface Sci. 24, 283 (1986). 3. Adarnczyk, Z., Dabros, T., Czarnecki, J., and van de Ven, T. G. M., J. Colloid Interface Sci. 97, 91 (1984). 4. Adamczyk, Z., Dabros, T., Czarnecki, J., and van de
15. 16. 17. 18.
145
Ven, T. G. M.,Adv. ColloidlnterfaceSci. 19, 183 (1983). Brenner, H., Chem. Eng. Sci. 16, 325 (1983). Goldman, J. H., Cox, R. G., and Brenner, H., Chem. Eng. Sci. 22, 637 (1967). Ruckenstein, E., and Prieve, D. C., J. Chem. Soc. Faraday Trans. 2 69, 1522 (1973). Spielman, L. A., and Friedlander, S. K., J. Colloid Interface Sci. 46, 22 (1974). Bourret, R. C., Nuovo Cim. 26, 1 (1962). Van Kampen, N. G., Phys. Rep. C24(3), 171 (1976). Byron, F. W., and Fuller, R. W., "Mathematics of Classicaland Quantum Physics." PWN, Warszawa, 1975. [ Polish edition ] Carslow, H. S., and Jaeger, J. C., "Conduction of Heat in Solids." Oxford (Clarendon) Univ. Press, Oxford, 1959. Landau, L., and Lifshitz, E. M., "Statistical Physics." PWN, Warszawa, 1959. [ Polish edition ] Adamczyk, Z., and van de Ven, T. G. M., J. Colloid Interface Sci. 80, 340 (1981). Wong, S., "Stochastic Processes." WNT, Warszawa, 1974. [ Polish edition ] van de Ven, T. G. M., Dabros, T., and Czarnecki, J., J. Colloid Interface Sci. 93, 580 (1983); Dabros, T., thesis, Jagiellonian University, Krakow, 1986. Hull, M., and Kitchener, J. A., Trans. Faraday Soc. 65, 3093 (1969). Prieve, D. C., and Lin, M. M. J,, J. Colloid Interface Sci. 86, 17 (1982).
Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989
INTRODUCTION
According to the DLVO theory the energy between interacting bodies, i.e., between particles in coagulation or a particle and a collector in deposition, is expressed as the sum of the electric double layer and dispersive contributions. The stability of colloidal systems is related to the form of the relationship of the interaction energy vs the distance between the interacting bodies. If the double-layer interactions are repulsive, a characteristic maxim u m in the total energy may exist (the socalled energetic barrier). The rate of particle transfer over the barrier, i.e., the rate of coagulation or deposition, depends strongly on the barrier height and thickness. The DLVO theory assumes that the form of the energy vs distance relationship remains constant during a particle-particle or a particle-collector encounter, and that the interaction energy is a function of the distance between the interacting bodies only. It is usually assumed that the considered system is described by the mean values of the respective parameters. However, these parameters may fluctuate around their mean
1 On sabbatical leave with the Department of Chemistry, McGiU University, Montreal, Canada.
values for thermodynamic reasons and this may produce temporary changes in the interaction energy. There are fluctuations in the concentration of ions forming the electrical double layers, both in the diffuse layers and in the Stern layers. The amounts of adsorbed ions and dissociated groups responsible for the surface charge also fluctuate, leading to fluctuations in the interaction energy. In every real system the interaction energy, even for spherical particles, is not dependent only upon the coordinates normal to their surfaces (i.e., the distance between the interacting bodies), but also depends on the mutual particle orientation, due to surface roughness or heterogeneity in the surface charge distribution. Also, in particle deposition the interaction energy for a given particle is a function of its position above the collector surface, as was pointed out in (1). Thus, in connection with the translation or rotation of a particle due to convection or Brownian motion, surface irregularities produce temporal, random changes of the interaction energy. The randomness of these changes is due to the fact that the surface shape and surface charge are statistically distributed. An attempt at evaluating the rate of particle transport in systems where the interaction en-
137 0021-9797/89 $3.00 Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989
Copyright © 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
138
WARSZYNSKI A N D C Z A R N E C K I
ergy is a stochastic function of time is given below. The flux of particles over the energetic barrier is calculated using a one-dimensional equation of transport. The value obtained is then compared with the particle flux through a static barrier whose height is equal to the mean height of the fluctuating one. An analytical solution restricted to small fluctuations of the barrier height is discussed. Next, the numerical solution of the transport equation is presented. Then, on the basis of the results of both solutions, we evaluate the characteristic value of the correlation time of energy barrier fluctuations. Fluctuations with a correlation time longer than the characteristic one can significantly affect particle transport. The results are compared with those of a previous "deterministic" attempt at evaluating the influence of fluctuations on particle transfer over the energetic barrier (2). A N A L Y T I C A L SOLUTION FOR SMALL F L U C T U A T I O N S
We shall consider the one-dimensional equation of transport. This kind of equation describes the deposition of particles onto a rotaring disk collector (3) or deposition on other collectors near the forward stagnation point (4): Off 0 = -z-y..J± - P e F 3 ( H ) ( H + 1)f O-7 O H
+ Q ( f , H, r)
[1]
Do, H i s the dimensionless surface-to-surface distance scaled with respect to a, ~(H, r) is the interaction energy in kT units, F I ( H ) , F2(H), F3(H) are the universal correction functions accounting for hydrodynamic interactions between a particle and a fiat collector (5, 6), Ex is the normal component of the resultant external force scaled with respect to kT / a, Q( f , H, ~') is a dimensionless source term, and Pe is the Prclet number characterizing the significance of convection effect relative to diffusion. For a rotating disk, 0)3/2(23
Pe-- 1.02
vl/2vo~ '
where o:~ is the angular velocity of the disk and v is the fluid kinematic viscosity. In the formula above the interaction energy is, due to fluctuations, a stochastic function of time, like the one illustrated in Fig. 1. Following the surface boundary layer approximation (7, 8), we can assume that within the region close to the collector surface (region I in Fig. 2a) where the energetic barrier exists, the only mechanism of particle transport is diffusion and rewrite Eq. [ l ] as 0--~=ffH0ff 0 F 1(H )[~--~+~-~f r 0ff 0~ ] •
[2]
If the changes of the height of energy barrier are faster than 1/rD, where rD is the dimensionless time for formation of a panicle concentration profile in the vicinity of the collector (2), the particles in the region where surface
where Off O~b(H,7") _ j± = FI(H) - ~ + OH n
1
+ ~ P e F 2 ( H ) ( H + l)2ff + Exff
]
is the normal component of the dimensionless panicle flux j± = ja/Do~n~, no~ is the bulk concentration, a is the particle radius, Do is its bulk diffusion coefficient, f is the dimensionless concentration scaled by n ~, r is the dimensionless time scaled with respect to a2/ Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989
FIG. 1. Fluctuations of the interaction energy between a panicle and a collector surface. The D L V O theory concerns the mean value o f energy ( ~ ) .
PARTICLE TRANSFER OVER AN ENERGY BARRIER
forces may be neglected will not feel the relatively fast changes in the barrier height. Thus, we can assume that the particle concentration at the outer boundary of the region I (cf. Fig. 2a) is practically independent of time. Moreover, if the mean energy barrier is high, the transport of particles over the barrier will be the rate-determining step of the process. The case when the barrier is high is the most interesting since for such a barrier the influence of the fluctuations on particle flux is expected to be significant. In such a case we can assume that the convective diffusion is fast in comparison with the barrier penetration, so that ff = 1 outside the barrier. Thus, the boundary conditions to Eq. [ 2 ] are
-I
I
Z
I
a
I I I
\
q~s
![
I
t7 = 0 t7= I
and
and
139
q~ ~ - o o
for
~=0 for
H>t~0+~B,
for
0
F1 02~ ~
0~
-- F I F B ( r) ~-~ = 0
~i(0, r) = 0 t7(oo, r) = 1.
b
.\', /oe \\N ,
[4]
in semi-infinite space with boundary conditions: H
g
[31
For simplicity we take ~o = 0, and ~B is a stochastic function of time only. If the average barrier height fulfills the condition that (~B(Z)) ~> 1, the solution of Eqs. [ 2 ] and [3 ] is practically the same as the solution of 0~
~,~
[2a]
where ]B is the width of the barrier. Here the perfect sink approximation was used at the separation $0 to collector surface (4). To obtain an approximate analytical solution of Eq. [2] let us consider the simplest case, where the energetic barrier has a triangular shape:
0---~ --
i
It II
H < ~o
-
[4a]
In Eq. [4 ] we disregarded the dependence of F~ on H (F1 is the value of F1 (H) calculated somewhere between 0 and ~B) and FB = 4~B/~Bis the interaction force. We now divide the interaction force into two parts, FB(z) = 2 F + 2ef(r),
." ] :'l
H FIG. 2. The shapes of the interaction energy profiles introduced in (a) the approximate analytical solution, and (b) the numerical solution of the stochastic transport equation.
[51
where 2 F is a nonrandom part of the force equal to the average force (FB(r)), 2~f(r) is the fluctuating part of the force acting on a particle whose mean ( f ( r ) ) is zero, and E is a small expansion parameter which assured the condition
'l/(f2(r)) ,~ 1. F Journal of Colloid and Interface Science, Vol.
128, No. 1, March 1, 1989
140
WARSZYNSKI AND CZARNECKI
The factor 2 is included to simplify the calculations. Substituting Eq. [ 5 ] into Eq. [ 4 ] we get On
d2ff
Or
OHz
Off Off 2F-z-,,o,1- 2cf(r)~-~ = 0.
ON &
02N OH 2
2F ON OH
- 4~2 ( f ( r) -~H £" £~° dr' dti [6]
× Go(H,~,r-r')f(r')~}
Here we have rescaled the dimensionless time r as r/F~. There are two operators in Eq. [6 ]: the deterministic one, O
Zo = 07"
02 0H 2
0
2 F ~OH'
and the random operator, l?(r) = 2f(r)(O/ OH). Introducing these operators in Eq. [6] we obtain
[Lo + W(r)]fi(H, r) = 0.
[71
To calculate the mean rate of particle transfer over the energy barrier, we have to find the equation for the mean concentration profile N(H, r) based on the equation for the instantaneous concentration (Eq. [ 6 ] or [ 7 ] ). Then differentiating N(H, r) with respect to H we obtain the mean particle flux. To find the equation for the mean concentration, we use the Bourret approximation (9, I0) for Eq. [7], [Lo - ~2(I?LolI?}]N(H, r) = O(e2),
[8]
where Lo I is the operator inverse to L0 (11) given by
X N(~, r ' ) = O.
Using the commutation rule for integration and averaging in Eq. [11 ], and assuming that the fluctuations of the interaction force are represented by a stochastic process with the correlation function (13 ) given by
( f ( r ) f ( r M ) > = a2e -e=l*-''l,
[121
where f12 = 1/rcF~, re is the correlation time of the interaction force fluctuations, and a 2 is the mean square fluctuation, we get
ON (Or
OeN
2F O___N OH
OH 2 2
2
0
r
- 4 e a ~-~£ d r ' e - a " - " l £
oo
dli
0
X Go(H, ~, r - r') -~ N(~, v') = 0
N(0, r) = 0 and N ( ~ , r) = 1.
drd~Go(x, ~, t - r)f(~, r),
[9]
where G(x, ~, t - r) is Green's function for the diffusion into the semi-infinite space (12): e-F(x-O-F 2(t-r)
Go(x, ~, t - r) =
1/41r(t - r)
X [e -((x-~)2/*('-~)) - e-"X+o~/4"-*)q. [10] Substituting Eq. [9] into Eq. [8 ] we obtain the integrodifferential equation for the mean particle concentration: Journal of Colloid and lnterface Science,
Vol. 128,No. 1, March 1, 1989
[131
with the boundary conditions
L~lf(x, t) =
[Ill
[13a]
We solve Eq. [13] by the successive approximation method with the restriction to times much greater than the fluctuation correlation time re. In the first approximation, the dimensionless mean particle flux - F1 ON/ OH for H = ~B is given by ( j l > = Jo(]s)
r8e, a2e-2r" r 2 F ~ ,
+&t ~7~
8F(~2 °'2 +
1 F]
L'v+F ]
e
,
[141
141
PARTICLE TRANSFER OVER AN ENERGY BARRIER (l±> Jo
A 3.0
2.0
1.0 100
10 4
10-2
10-3
10-4
10-5
Tc
FIG. 3. The normalized mean flux of particles to the collector surface as a function of the correlation time of fluctuations of the interaction energy. Curve A: a = 2.5, curve B: a = 2.0, curve C: ~ = 1.5. The mean barrier height ( ~ ) = 8 and its width ~B = 0.03 are c o m m o n to all curves.
where J0(]B) = 2Fl Fe -2eg~ is the steady-state flux for the mean barrier and 3' = ~ +/32. In Fig. 3 the mean flux calculated from Eq. [14 ], normalized by the steady-state flux for the mean barrier, is shown as a function of the correlation time of fluctuations of the interaction force. Curves A, B, and C are cal-
culated for the same average barrier height but for three different magnitudes of fluctuations. For all cases presented in Fig. 4 the dispersion o f fluctuations is exactly the same and the curves differ from each other in mean barrier height. The conclusions are presented in the next section.
( i±> Jo
30 A 2£
C
1.0 .
.
.
.
.
.
.
.
.
i
100
I0-I
10-2
10-3
I0-4
10"5
,re
FIG. 4. The normalized flux of particles as a function of the correlation time of interaction energy fluctuations where the mean square fluctuation ~2 = 4 and the mean energetic barrier height: Curve A: ( ~ ) = 10, curve B: ( ~ ) = 8, and curve C: ( ~ = 6. Barrier width ~B = 0.03 is c o m m o n to all curves. Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989
142
WARSZYNSKI AND CZARNECK/
10 9 8
7 6
otT
FiG. 5. An example of the numerical model o f the random function $ ( r ) used in the calculations. NUMERICAL SOLUTION
The numerical solution of the stochastic transport Eq. [1 ] was obtained as follows: we took 20 possible realizations of the stochastic function $ ( H , r) and solved the deterministic transport equation using a Cranck-Nicholson method (14). Next, we averaged the solutions for all realizations to get a single solution of the stochastic equation. The interaction energy was expressed as
~(H, 7-)
=
t~m(n)[1
-4- t~r(T)] ,
where ~m(H) is the mean interaction energy profile illustrated in Fig. 2a and ~,(r) is the fluctuating term. A single realization of the stochastic function ~r(r) was obtained in the following way. The value ~r was selected randomly from the previously assumed distribution of interaction energy with mean value ~"and variance (mean
1012I
A
B
{
Jo
10-3
104
10-2
10-3
10-4
10-5 TC
Fie. 6. The mean flux of particles to the collector surface as a function of the correlation time of fluctuations o f the interaction energy obtained from the numerical solution of the stochastic transport equation where the mean barrier height (~_~ = 8. Curve A: cr/<~) = 0.31, curve B: ~ / < ~ ) = 0.19, and curve C: a/<~) = 0.125. The barrier width t~B = 0.03 and Pe = 0.02. ]ournal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989
[15]
143
PARTICLE TRANSFER OVER AN ENERGY BARRIER
square fluctuation) 0"2, simultaneously with changes in the Poisson process with the rate a. The probability that at least one change of the Poisson process occurs between r = 0 and r = T is equal to P =
1 -
[16]
e -'~T.
This procedure was applied until r = 6 0 0 / a , and then the average flux was calculated. In Fig. 5 an example of a single realization of the stochastic function ~r(r) obtained in such a way is presented. Figure 6 presents the results of the numerical solution of the stochastic transport Eq. [ 1] for the fixed mean energy profile and various mean square fluctuations of the interaction energy. The correlation function for the stochastic process used in the numerical solution of the transport equation to simulate fluctuations of the interaction energy is given by R ( r ) = ~-2 + ¢2e-~1,1
[17]
As the same correlation function (with ~"= 0) was previously assumed in the approximate analytical solution of the transport equation, we are able to compare the results of both methods directly. In Figure 7 the mean flux of particles (normalized with respect to the steady-state flux through the mean barrier) obtained from the numerical solution of the transport Eq. [ 1] is shown together with those calculated using an approximate analytical method. It can be seen that the analytical solution is valid only for small fluctuations where the analytical and numerical results are in good agreement. For large fluctuations only the numerical solution can be used. Based on the results presented above, one can draw the conclusion that the fluctuations of the interaction energy with short correlation time (to < 10 -4) cannot significantly affect the rate of the transport process. However, sufficiently large and long-correlated (long-lasting) fluctuations (re > 10 -3) may strongly increase the rate of the transport process by as much as one order of magnitude.
Jo 22
2.O
-
t
"t8
16 B 1.4
1.2
=
1.0
10 -~
10 -2
10 `3
10 -~
10 "~
Tc
FIG. 7. The normalized mean flux of particles as a function of the correlation time of interaction energy fluctuations obtained from the numerical solution of the stochastic transport equation (points) shown together with one obtained from the approximate analytical solution: Curve A: a ( $ ) = 0.19, curve B: a / ( ( o ) = 0.125. The barrier width ~B = 0.03 and Pe = 0.02. Journal of Colloid and Interface Science, Vol. 128,No. 1, March 1, 1989
144
WARSZYNSKI AND CZARNECK/
In a previous analysis of the effect o f fluctuations a "deterministic" model of fluctuations was used (2); i.e., the interaction energy was expressed as ~ ( H , r) = ~ m ( H ) ( 1 + B cos ¢or), where B and o~ = 2~r/rv were dimensionless constants characterizing the amplitude and frequency of energy oscillations (rF is an oscillation period). If we calculate the mean recurrence time of the mean value ~'m~ (15) for the process used in the numerical solution of the stochastic transport equation and assume that rmr = rF/2, then we can compare the results of these two methods. This is done in Fig. 8 where the values of the mean flux, obtained from the numerical solution of the transport equation and from the "oscillation" model, are shown together as functions of r~r = rF/2. It is seen that such a deterministic model of fluctuations may be useful as a first approximation. CONCLUDING REMARKS It was shown that fluctuations of the interaction energy may appreciably increase the average flux of particles penetrating the energy barrier, providing the dimensionless correlation time of fluctuations is longer than about
10-4 for a typical barrier width of 10 -2 to 5 X 10 -2 (normalized with respect to particle radius). If the correlation time is shorter, the changes of particle flux are damped and the flux approaches its steady-state value for the mean barrier height. In dimensional units, for a 1-#m particle in an aqueous solution, the critical correlation time is about 10 -4 s. It should be noted that for very fast changes of the forces acting on particles, Eq. [ 1] is no longer valid. This is because Eq. [ 1] neglects the inertia of the particles and the surrounding fluid, which becomes significant for rapid fluctuations, which are much faster than those corresponding to our critical correlation time. However, the inertia effects cause averaging of the forces acting on the particles and the resuiting flux may be expected to attain the steady-state value for the mean barrier. Thus, neglecting inertia effects in Eq. [ 1] should not affect the fast fluctuation limit o f our predictions. Crude estimates show that the fluctuations of the concentration of ions in the electrical double layer within the volume between interacting particles have too short a correlation time (re -~ 10 -6 s) to significantly affect the rate of particle transport. On the other hand,
10 "2
A
+
Jr fl-
+
4-
4-
4-
10-3
10"~
10-2
10-3
-e.l_
10-4
10-5
"/'mr
FIG. 8. Comparisonof the results of the numerical solutionof the stochasticequation (points) with those obtained using a "deterministic"approximationof fluctuations. Curve A: ~r/(~ > = 0.31, curve B: ~r/(~) = 0.19 and B = ~r~/(~). The averagebarrier height (~) = 8. Journal of Colloid and InteoeaeeScience, Vol. ! 28, No. 1, March 1, 1989
PARTICLE TRANSFER OVER AN ENERGY BARRIER the fluctuations o f the local surface charge d e n s i t y a n d fluctuation d u e to surface roughness a n d heterogeneity o f surface charge distribution, p r o v o k e d b y the m o t i o n o f particles, have sufficiently long c o r r e l a t i o n t i m e s to increase particle transfer rate over the energetic barrier. It is w o r t h y to n o t e t h a t such a n increase has i n d e e d b e e n o b s e r v e d in particle d e p o s i t i o n e x p e r i m e n t s (16, 17). Also, o u r p r e d i c t i o n s agree, at least qualitatively, with those o f Prieve a n d L i n (18), w h o s t u d i e d enh a n c e m e n t in the initial c o a g u l a t i o n rate for a n e n s e m b l e o f particles with a d i s t r i b u t i o n o f surface properties, r a t h e r t h a n particles with fluctuating surface properties.
5. 6. 7. 8. 9. 10. 11. 12.
13.
REFERENCES
14.
1. Czarnecki, J., and Warszynski, P., Colloids Surf. 22, 207 (1987). 2. Adamczyk, Z., Czarnecki, J., and Warszynski, P., J. Colloid Interface Sci. 106, 299 (1985 ); Czarnecki, J., Adv. Colloid Interface Sci. 24, 283 (1986). 3. Adarnczyk, Z., Dabros, T., Czarnecki, J., and van de Ven, T. G. M., J. Colloid Interface Sci. 97, 91 (1984). 4. Adamczyk, Z., Dabros, T., Czarnecki, J., and van de
15. 16. 17. 18.
145
Ven, T. G. M.,Adv. ColloidlnterfaceSci. 19, 183 (1983). Brenner, H., Chem. Eng. Sci. 16, 325 (1983). Goldman, J. H., Cox, R. G., and Brenner, H., Chem. Eng. Sci. 22, 637 (1967). Ruckenstein, E., and Prieve, D. C., J. Chem. Soc. Faraday Trans. 2 69, 1522 (1973). Spielman, L. A., and Friedlander, S. K., J. Colloid Interface Sci. 46, 22 (1974). Bourret, R. C., Nuovo Cim. 26, 1 (1962). Van Kampen, N. G., Phys. Rep. C24(3), 171 (1976). Byron, F. W., and Fuller, R. W., "Mathematics of Classicaland Quantum Physics." PWN, Warszawa, 1975. [ Polish edition ] Carslow, H. S., and Jaeger, J. C., "Conduction of Heat in Solids." Oxford (Clarendon) Univ. Press, Oxford, 1959. Landau, L., and Lifshitz, E. M., "Statistical Physics." PWN, Warszawa, 1959. [ Polish edition ] Adamczyk, Z., and van de Ven, T. G. M., J. Colloid Interface Sci. 80, 340 (1981). Wong, S., "Stochastic Processes." WNT, Warszawa, 1974. [ Polish edition ] van de Ven, T. G. M., Dabros, T., and Czarnecki, J., J. Colloid Interface Sci. 93, 580 (1983); Dabros, T., thesis, Jagiellonian University, Krakow, 1986. Hull, M., and Kitchener, J. A., Trans. Faraday Soc. 65, 3093 (1969). Prieve, D. C., and Lin, M. M. J,, J. Colloid Interface Sci. 86, 17 (1982).
Journal of Colloid and Interface Science, Vol. 128, No. 1, March 1, 1989