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The deposition of colloidal particles onto the surface of a rotating disk
The Deposition of Colloidal Particles onto the Surface of a Rotating Disk W. I. W N E K Xerox Corporation, Rochester, New York
D, G I D A S P O W Institute of Gas Technology
AND D. T. W A S A N Department of Chemical Engineering, ~llinois Institute of Technology, Chicago, Illinois 60616
Received August 30, 1975; accepted January 29, 1976 The deposition rate of colloidal particles onto a rotating disk was calculated as a function of time by using the convective-diffusion equation subject to the boundary condition of a first order surface reaction which incorporates the effect of surface forces and a charge balance which allows for changes in the charge of the disk due to deposition. Calculations show that such a model accounts for the experimentally observed fact that the deposition rate decreases with time. INTRODUCTION
made for accounting for surface forces which otherwise would not be realized. Since the experimental study was restricted to submicron particles for which Brownian diffusion was the controlling deposition mechanism, this analysis wilI be also so restricted.
I n order to develop design procedures for the removal of colloidal particles from water b y deep bed filters it is necessary to describe mathematically the mechanism b y which deposition of these particles onto much larger particles occurs. However, due to the complicated flow patterns in a filter bed, this is a very difficult problem. Thus, in order to gain some insight to the basic mechanisms involved, it is appropriate to analyze a related but simpler and better defined system. Such a system is the deposition of charged colloidal particles onto the charged surface of a rotating disk immersed in the suspension for which the results of an experimental study are available in literature (5, 8). This system is related to the filter bed in that it exhibits the same basic behavior with respect to surface charge. The results of the analysis to be presented allow simplifying assumptions to be
EXPRESSION FOR THE LOCAL DEPOSITION RATE SpieIman and Friedlander (10) and Ruckenstein and Prieve (14) have shown that the diffusional deposition of charged particles onto a charged surface m a y be formulated as a convective-diffusion problem subject to a first-order surface reaction whose rate constant is a function of the colloidal interaction energy. Clint el al. (2) have performed a similar analysis for the case of a rotating disk. I n an analogous manner, the present authors obtained the following expression for this rate 1
Copyright ~ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 59, No. I, March 15, 1977 ISSN 0021-9797
2
WNEK, GIDASPOW AND WASAN
constant (13)
where
rr~
~[.)o
zc = - -
I = -- expEV(s,,)/kTJ,
EII
P
--0.5 4- (0.25 -- 2.5f)½
where
oem
2f~
zX~ = ai7
eV(")/~'ds.
[-2]
(positive value)
0 (a~j + s) ~
The above equation is a generalization of the results of the above investigators to the case of two arbitrary spheres and the inclusion of viscous interaction. Since ~ occurs at a point where the surface forces are insignificant and A, can be evaluated by the same asymptotic expansion method as for the stability ratio (9), 6 can be replaced by infinity with the result (13) A , = a i j W i j = ai~X(s~)Wi?.
327rata2
-~A + ~ e 2A + 2 ~ 1 ~ nokT X ~
d~V (s~) ds~
e-".
I-4-]
K2
E5-1
the expression for W~;° becomes (13) auI W ~fl =
(s,,, + ,zu) ~
[-6]
2'
-- 2~
An approximate expression for X may be obtained by combining the results of Honig et aI. (4), Happel and Brenner (3), Charles and Mason (1), and Wnek (13): 6u 2 + 13u + 2 x =
6u 2 + 4u
U~
The deposition rate J of colloidal particles onto a rotating disk may now be obtained from Levich's (7) solution for intermediate reaction kinetics on the surface of a rotating disk j0 J = , E8]
1 + (yo/y.)
where j0 = deposition rate in the absence of surface forces ~°Co/AD, A D = 1.61 D Oi/~ul/6/M/~, j , = deposition rate for surface forces controlling KC0.
Using for the van der Waals interaction energy (15) vx = -t~/s
2kr
_ _ .31- O~K2e--xSm Sm 3
[31
Further, the need to evaluate 3 has been eliminated. The result for a flat plate is obtained by letting ai go to infinity. Since ao" is then much greater than s, it is seen that the result of the above investigators is obtained for X = 1. In order to obtain an analytical expression for W~fl, the expression for the interaction energy as given by Verwey and Overbeek (11) was extended to the case of two arbitrary spheres with arbitrary surface potentials with the result (13) : V~ = ~ E ~ ? e
P = -\
CALCULATION OF THE AVERAGE DEPOSITION For the case when there are attractive forces acting (large K ) , it has been shown by Clint et al. (2) and Wnek (13) that the deposition rate is for all purposes the value given by Levich (7) in the absence of surface
Journal of Colloid and Interface Science, VoL 59, N'o. I, March 13, 1977
DEPOSITION ONTO A DISK forces, and thus it does not vary with time. However, when surface forces are present, J depends on the charge characteristics of the disk and particles. Due to the deposition of particles onto the disk, the charge characThe change in the surface charge of the disk = dqD
3
teristics of the disk change with time, and thus so does the flux or rate of deposition. In order to take this effect into account, a charge balance on the disk may be performed to yield The rate at which charge is transported to the disk,
Qj,
[9]
dt coul
coul
particles deposited~
unit time/ a result, one may define an effective particle charge as
subject to the initial condition qD = qOD,
I = O,
~,
where
qo -= the charge on the disk, 0r = the effective charge of a suspension particle, t = time. In obtaining Eq. [9] the surface charge has been considered as being smeared over the surface rather than as discrete charges on the surface. The reason that an effective particle charge is used rather than the actual value is to account for the fact that when a surface is completely covered by colloidal particles, it tends to accept their surface properties. One possible way of representing this effect is as follows. Consider a particle approaching another one already on the surface along a path perpendicular to the surface. The charge that the approaching particle effectively sees is the sum of the second particle's charge and the charge due to the surface a distance "2a" away at the second particle's surface. The latter may be expressed as qDe-2~ which is readily obtained from the solution of the linearized Poisson-Boltzmann equation for a flat plate (6). Thus, considering the charge over an area 4a 2, the finite size of the second particle can be thought of as canceling charge on the order of 4a2qD(1- e--2"~). As
= q,-
4@~(t
-
e-2~o).
[1o]
In the limit of very small particles, ~ = qp while for the case of very large ones~ ~ = qp4a2qD. Because J is given as a function of the surface or zeta potential, it is necessary to relate charge and surface potential. For flat surfaces, an appropriate expression is given by (6)
qD =
(2,KTno) ½ se¢oo sinh ~ \ r t 2kT
[11]
for the case of ions of the same valence and for particles q, = e¢@(1 -t- Ka)/4rca.
[12]
The total deposit is given by /
N = ] Jdt J0 or
dN --=
J
El3]
dt
subject to N=0,
t =0.
Equations [9] through [13] define a system
Journal of Colloid and Interface Science, Vol. 59, No. 1, March 13, 1977
4
WNEK, GIDASPOW AND WASAN
of two first-order nonlinear ordinary differential equations in the unknowns q v or ~k0D and N and independent variable t which m a y be solved by any standard numerical technique such as Runge-Kutta. NUMERICAL CALCULATIONS AND COMPARISON OF THEORY WITH DATA Marshall and Kitchener (8) and Hull and Kitchener (5) have carried out experimental studies of the deposition of submicron particles onto a rotating disk and interpreted their results in terms of the Levich theory for mass transfer to a rotating disk. For the case of negatively charged particles and a positive disk, they found that the deposition rate was constant up to times of 10-30 rain whereupon the rate decreased. They also compared the experimental deposition rates with that given by the Levich theory and found them to be in excellent agreement. This confirms the theoretical prediction that in the case of attractive surface forces the deposition rate is not significantly greater than that given by the Levich theory. Numerical calculations of the n u m b e r of particles deposited and the surface charge and zeta potential of the disk as a function of time are presented in Figs. 1 and 2. The values of the various parameters were taken as those given by Marshall and Kitchener (8) :
LEVICH T H E O ~ [
X ;0`5 M ~-
5
~o x
/
h-
4 06M ~x
~ -2
=~
,
L i0
1 20
r 30
r 40
2;i I 50
,o
c~
I 60
TIME (rain)
FiG. 1. Number of particles deposited onto a rotating disk and its surface charge vs time.
•~
o
I x 10-TM
5 X 10-6M
-io 0
I
I
I0
20
~0
I
I
40
50
60
TIME {rain)
Fla. 2. Zeta potential of rotating disk vs time. a = 2.25 X 10- ~ cm, ~ = 8.97 X 10_2 stokes, :Do = 1.08 X 10-~ cm2/sec, ¢0 = 20.5 radians/ sec, AD = 3.58 X 10-4 cm, Co = 6 X 108 particles/cm3, j0 = 1.8 X 104 particles/cm 2 sec. Also ~ was taken as qp. The zeta potentials of the particles and disk were given as --29 and -J-10 mV in distilled water, respectively. However, since the Helmholtz-Smoluchowski equation was used to convert the electrophoretic mobilities to zeta potentials, the method of Wiersema et aI. (12) was used to correct that of the particles to give ~'p = --42.5 inV. The effect of different values of the zeta potentials was investigated by varying the ionic strength and holding the surface charges constant at their values when the ionic strength is 1 X 10-7 moles/liter. I t should be recalled that increasing the ionic strength at constant charge decreases the zeta potential. For the sake of comparison the experimental amount of deposition after 30 rain in distilled water was 0,33 X 107 par-
Journal of Colloid and !r*terfac¢Science, Vo|. 59~ No. !, March ~5. 1977
DEPOSITION ONTO A DISK ticles/cm 2, the predicted value from Fig. 1, 0.75 X 107 particles/cm ~, and the Levich value 3.24 ;< 107 particles/cm 2. Due to the approximate nature of this data, the experimental and theoretical values m a y be considered to be in agreement. As suggested b y Marshall and Kitchener (8), the decrease in the deposition rate could also be due to agglomeration of the colloidal particles. However, calculation of the energy interaction curve showed t h a t the energy b a r r i e r was much too great to allow significant agglomeration. TABLE OF NOMENCLATURE A a a a~ a2
= = = = --
aii
= a i q - a~;
Co
e
= concentration of the suspension far away from the rotating disk; = Brownian diffusivity uncorrected for viscous interaction; = charge of the electron;
f
= 0.5 - e-°.V~;
:Do
J
Hamaker radius of 2ata2/(a~ radius of radius of
constant; a particle; + a~) ; sphere 1; sphere 2;
Sm
t
T
5
separation distance at which the maxim u m interaction energy occurs; time ; = temperature; =
V
s/~; = sum of the interaction
g
.=
energies due to van der Waals and double layer interactions ; attractive energy due to van der VA Waals interaction; VR = repuIsive energy due to electrical double layer interaction; W i i = stability ratio for particles of type i and j ; W i i ° = stability ratio uncorrected for viscous interaction; ze~o~/k T ;
valence.
Greek letters
(32~rn0kT/~')'y-2Eala~ / (al q- a2) ~;
= deposition rate per unit area of rotating disk; j0 = deposition rate in the absence of surface forces; J% = deposition rate for surface forces controlling ; K = X)°/A~ ; k = Boltzmann constant; 21g = molar concentration, moles/liter; no = ionic concentration or strength in the bulk solution; N = number of colloidal particles per unit area deposited onto a rotating disk; q°D = initial surface charge of rotating disk; qD = surface charge of the rotating disk; Op = effective surface charge of a suspension particle; qp = actual surface charge of a suspension particle ; s = shortest distance between two spheres;
3
= (Aa1~2)/6(~1 + ~2);
5"
= kinematic viscosity';
(eYo.,- 1)/(e~'o.2 + 1); 3~ _~ ,yl~e-~n -~ ,y22e2~, -~- 23/13/2; A = 1 in 3'1/3'2; AD = diffusion boundary layer thickness; As a boundary layer thickness over which surface forces are significant; distance where surface forces become insignificant; dielectric constant; zeta potential; K2
8rrnoe~z2/ ek T ;
X
= correction factor for Stokes' law which accounts for the effect of viscous interaction;
~0D
~-
3K/a ; surface potential of rotating disk; surface potential of suspension particles; rotational speed.
REFERENCES 1. CItARLES,G. E. AND N[ASON, S. G., J. Colloid Inlerface Sci. 15, 236 (1960).
Journal of Colloid and Interface Science, VoL 59, No. 1, March 15, 1977
6
WNEK, GIDASPOW AND WASAN
2. CL:N~, G. E., CL:N% ~. H., CO~I~LL, ~[. M., AND WALI:Eg, T., J. Colloid Interface Sci. 44, 121 (1973). 3. HAPPEL, f. AND BRENNER, H., "Low Reynolds Number Hydrodynamics," Prentice-Hall, Englewood Cliffs, New Jersey, 1965. 4. HO~:G, E. P., ROEBERSO:¢, G. J., ANn WIE~SE~tA, P. H., ]. Colloid Interface Sci. 36, 97 (1971). 5. HULL, M. AND KITCHENER, ~. A., Trans. Faraday Soc. 65, 3093 (1969). 6. KRuY% H. R., "Colloid Science," Vol. I, Elsevier, Amsterdam, 1952. 7. LEVlCH,V. G., "Physicochemical Hydrodynamics," Prenficc-HM1, New York, 1962. 8. MARSHALL,J. K. Am) KITCHENER, J. A., J. Colloid Interface Sc¢. 22, 342 (1966).
9. REERINK, H. AND OVERBEEK, j'. TJ-I. G., Disc. Faraday Soc. 18, 74 (1954). 10. SPIELMAN, L. A. AND F:P.IEDLANDEI% S. K., Y. Colloid Interface Sci. 46, 22 (1974). 11. VERWE¥, E. J. W. AND OVERBEEK, J. TII. G., "Theory of the Stability of Lyophobic Colloids," Elsevier, Amsterdam, 1948. 12. WIERSEMA, P. H., LOEB, A. L., AND OVERBEEK, J. TH. G., J. Colloid Interface Sci. 22, 78 (1966). 13. WNEI:, W. J., Ph.D. Thesis, Illinois Institute of Technology, Chicago, 1973. 14. RUCKENSTEIIV, E. AND PRIEVE, D. C., J. Chem. Soc. Faraday Trans.,/Jr 69, 1522 (1973). 15. WIESE, G. R. AND HEALY, T. W., Trans. Faraday Soc. 66, 490 (1970).
Journal of Colloid and !nlerface Science, Vol, 59, No. 1, M~,rch 15, 1977
D, G I D A S P O W Institute of Gas Technology
AND D. T. W A S A N Department of Chemical Engineering, ~llinois Institute of Technology, Chicago, Illinois 60616
Received August 30, 1975; accepted January 29, 1976 The deposition rate of colloidal particles onto a rotating disk was calculated as a function of time by using the convective-diffusion equation subject to the boundary condition of a first order surface reaction which incorporates the effect of surface forces and a charge balance which allows for changes in the charge of the disk due to deposition. Calculations show that such a model accounts for the experimentally observed fact that the deposition rate decreases with time. INTRODUCTION
made for accounting for surface forces which otherwise would not be realized. Since the experimental study was restricted to submicron particles for which Brownian diffusion was the controlling deposition mechanism, this analysis wilI be also so restricted.
I n order to develop design procedures for the removal of colloidal particles from water b y deep bed filters it is necessary to describe mathematically the mechanism b y which deposition of these particles onto much larger particles occurs. However, due to the complicated flow patterns in a filter bed, this is a very difficult problem. Thus, in order to gain some insight to the basic mechanisms involved, it is appropriate to analyze a related but simpler and better defined system. Such a system is the deposition of charged colloidal particles onto the charged surface of a rotating disk immersed in the suspension for which the results of an experimental study are available in literature (5, 8). This system is related to the filter bed in that it exhibits the same basic behavior with respect to surface charge. The results of the analysis to be presented allow simplifying assumptions to be
EXPRESSION FOR THE LOCAL DEPOSITION RATE SpieIman and Friedlander (10) and Ruckenstein and Prieve (14) have shown that the diffusional deposition of charged particles onto a charged surface m a y be formulated as a convective-diffusion problem subject to a first-order surface reaction whose rate constant is a function of the colloidal interaction energy. Clint el al. (2) have performed a similar analysis for the case of a rotating disk. I n an analogous manner, the present authors obtained the following expression for this rate 1
Copyright ~ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 59, No. I, March 15, 1977 ISSN 0021-9797
2
WNEK, GIDASPOW AND WASAN
constant (13)
where
rr~
~[.)o
zc = - -
I = -- expEV(s,,)/kTJ,
EII
P
--0.5 4- (0.25 -- 2.5f)½
where
oem
2f~
zX~ = ai7
eV(")/~'ds.
[-2]
(positive value)
0 (a~j + s) ~
The above equation is a generalization of the results of the above investigators to the case of two arbitrary spheres and the inclusion of viscous interaction. Since ~ occurs at a point where the surface forces are insignificant and A, can be evaluated by the same asymptotic expansion method as for the stability ratio (9), 6 can be replaced by infinity with the result (13) A , = a i j W i j = ai~X(s~)Wi?.
327rata2
-~A + ~ e 2A + 2 ~ 1 ~ nokT X ~
d~V (s~) ds~
e-".
I-4-]
K2
E5-1
the expression for W~;° becomes (13) auI W ~fl =
(s,,, + ,zu) ~
[-6]
2'
-- 2~
An approximate expression for X may be obtained by combining the results of Honig et aI. (4), Happel and Brenner (3), Charles and Mason (1), and Wnek (13): 6u 2 + 13u + 2 x =
6u 2 + 4u
U~
The deposition rate J of colloidal particles onto a rotating disk may now be obtained from Levich's (7) solution for intermediate reaction kinetics on the surface of a rotating disk j0 J = , E8]
1 + (yo/y.)
where j0 = deposition rate in the absence of surface forces ~°Co/AD, A D = 1.61 D Oi/~ul/6/M/~, j , = deposition rate for surface forces controlling KC0.
Using for the van der Waals interaction energy (15) vx = -t~/s
2kr
_ _ .31- O~K2e--xSm Sm 3
[31
Further, the need to evaluate 3 has been eliminated. The result for a flat plate is obtained by letting ai go to infinity. Since ao" is then much greater than s, it is seen that the result of the above investigators is obtained for X = 1. In order to obtain an analytical expression for W~fl, the expression for the interaction energy as given by Verwey and Overbeek (11) was extended to the case of two arbitrary spheres with arbitrary surface potentials with the result (13) : V~ = ~ E ~ ? e
P = -\
CALCULATION OF THE AVERAGE DEPOSITION For the case when there are attractive forces acting (large K ) , it has been shown by Clint et al. (2) and Wnek (13) that the deposition rate is for all purposes the value given by Levich (7) in the absence of surface
Journal of Colloid and Interface Science, VoL 59, N'o. I, March 13, 1977
DEPOSITION ONTO A DISK forces, and thus it does not vary with time. However, when surface forces are present, J depends on the charge characteristics of the disk and particles. Due to the deposition of particles onto the disk, the charge characThe change in the surface charge of the disk = dqD
3
teristics of the disk change with time, and thus so does the flux or rate of deposition. In order to take this effect into account, a charge balance on the disk may be performed to yield The rate at which charge is transported to the disk,
Qj,
[9]
dt coul
coul
particles deposited~
unit time/ a result, one may define an effective particle charge as
subject to the initial condition qD = qOD,
I = O,
~,
where
qo -= the charge on the disk, 0r = the effective charge of a suspension particle, t = time. In obtaining Eq. [9] the surface charge has been considered as being smeared over the surface rather than as discrete charges on the surface. The reason that an effective particle charge is used rather than the actual value is to account for the fact that when a surface is completely covered by colloidal particles, it tends to accept their surface properties. One possible way of representing this effect is as follows. Consider a particle approaching another one already on the surface along a path perpendicular to the surface. The charge that the approaching particle effectively sees is the sum of the second particle's charge and the charge due to the surface a distance "2a" away at the second particle's surface. The latter may be expressed as qDe-2~ which is readily obtained from the solution of the linearized Poisson-Boltzmann equation for a flat plate (6). Thus, considering the charge over an area 4a 2, the finite size of the second particle can be thought of as canceling charge on the order of 4a2qD(1- e--2"~). As
= q,-
4@~(t
-
e-2~o).
[1o]
In the limit of very small particles, ~ = qp while for the case of very large ones~ ~ = qp4a2qD. Because J is given as a function of the surface or zeta potential, it is necessary to relate charge and surface potential. For flat surfaces, an appropriate expression is given by (6)
qD =
(2,KTno) ½ se¢oo sinh ~ \ r t 2kT
[11]
for the case of ions of the same valence and for particles q, = e¢@(1 -t- Ka)/4rca.
[12]
The total deposit is given by /
N = ] Jdt J0 or
dN --=
J
El3]
dt
subject to N=0,
t =0.
Equations [9] through [13] define a system
Journal of Colloid and Interface Science, Vol. 59, No. 1, March 13, 1977
4
WNEK, GIDASPOW AND WASAN
of two first-order nonlinear ordinary differential equations in the unknowns q v or ~k0D and N and independent variable t which m a y be solved by any standard numerical technique such as Runge-Kutta. NUMERICAL CALCULATIONS AND COMPARISON OF THEORY WITH DATA Marshall and Kitchener (8) and Hull and Kitchener (5) have carried out experimental studies of the deposition of submicron particles onto a rotating disk and interpreted their results in terms of the Levich theory for mass transfer to a rotating disk. For the case of negatively charged particles and a positive disk, they found that the deposition rate was constant up to times of 10-30 rain whereupon the rate decreased. They also compared the experimental deposition rates with that given by the Levich theory and found them to be in excellent agreement. This confirms the theoretical prediction that in the case of attractive surface forces the deposition rate is not significantly greater than that given by the Levich theory. Numerical calculations of the n u m b e r of particles deposited and the surface charge and zeta potential of the disk as a function of time are presented in Figs. 1 and 2. The values of the various parameters were taken as those given by Marshall and Kitchener (8) :
LEVICH T H E O ~ [
X ;0`5 M ~-
5
~o x
/
h-
4 06M ~x
~ -2
=~
,
L i0
1 20
r 30
r 40
2;i I 50
,o
c~
I 60
TIME (rain)
FiG. 1. Number of particles deposited onto a rotating disk and its surface charge vs time.
•~
o
I x 10-TM
5 X 10-6M
-io 0
I
I
I0
20
~0
I
I
40
50
60
TIME {rain)
Fla. 2. Zeta potential of rotating disk vs time. a = 2.25 X 10- ~ cm, ~ = 8.97 X 10_2 stokes, :Do = 1.08 X 10-~ cm2/sec, ¢0 = 20.5 radians/ sec, AD = 3.58 X 10-4 cm, Co = 6 X 108 particles/cm3, j0 = 1.8 X 104 particles/cm 2 sec. Also ~ was taken as qp. The zeta potentials of the particles and disk were given as --29 and -J-10 mV in distilled water, respectively. However, since the Helmholtz-Smoluchowski equation was used to convert the electrophoretic mobilities to zeta potentials, the method of Wiersema et aI. (12) was used to correct that of the particles to give ~'p = --42.5 inV. The effect of different values of the zeta potentials was investigated by varying the ionic strength and holding the surface charges constant at their values when the ionic strength is 1 X 10-7 moles/liter. I t should be recalled that increasing the ionic strength at constant charge decreases the zeta potential. For the sake of comparison the experimental amount of deposition after 30 rain in distilled water was 0,33 X 107 par-
Journal of Colloid and !r*terfac¢Science, Vo|. 59~ No. !, March ~5. 1977
DEPOSITION ONTO A DISK ticles/cm 2, the predicted value from Fig. 1, 0.75 X 107 particles/cm ~, and the Levich value 3.24 ;< 107 particles/cm 2. Due to the approximate nature of this data, the experimental and theoretical values m a y be considered to be in agreement. As suggested b y Marshall and Kitchener (8), the decrease in the deposition rate could also be due to agglomeration of the colloidal particles. However, calculation of the energy interaction curve showed t h a t the energy b a r r i e r was much too great to allow significant agglomeration. TABLE OF NOMENCLATURE A a a a~ a2
= = = = --
aii
= a i q - a~;
Co
e
= concentration of the suspension far away from the rotating disk; = Brownian diffusivity uncorrected for viscous interaction; = charge of the electron;
f
= 0.5 - e-°.V~;
:Do
J
Hamaker radius of 2ata2/(a~ radius of radius of
constant; a particle; + a~) ; sphere 1; sphere 2;
Sm
t
T
5
separation distance at which the maxim u m interaction energy occurs; time ; = temperature; =
V
s/~; = sum of the interaction
g
.=
energies due to van der Waals and double layer interactions ; attractive energy due to van der VA Waals interaction; VR = repuIsive energy due to electrical double layer interaction; W i i = stability ratio for particles of type i and j ; W i i ° = stability ratio uncorrected for viscous interaction; ze~o~/k T ;
valence.
Greek letters
(32~rn0kT/~')'y-2Eala~ / (al q- a2) ~;
= deposition rate per unit area of rotating disk; j0 = deposition rate in the absence of surface forces; J% = deposition rate for surface forces controlling ; K = X)°/A~ ; k = Boltzmann constant; 21g = molar concentration, moles/liter; no = ionic concentration or strength in the bulk solution; N = number of colloidal particles per unit area deposited onto a rotating disk; q°D = initial surface charge of rotating disk; qD = surface charge of the rotating disk; Op = effective surface charge of a suspension particle; qp = actual surface charge of a suspension particle ; s = shortest distance between two spheres;
3
= (Aa1~2)/6(~1 + ~2);
5"
= kinematic viscosity';
(eYo.,- 1)/(e~'o.2 + 1); 3~ _~ ,yl~e-~n -~ ,y22e2~, -~- 23/13/2; A = 1 in 3'1/3'2; AD = diffusion boundary layer thickness; As a boundary layer thickness over which surface forces are significant; distance where surface forces become insignificant; dielectric constant; zeta potential; K2
8rrnoe~z2/ ek T ;
X
= correction factor for Stokes' law which accounts for the effect of viscous interaction;
~0D
~-
3K/a ; surface potential of rotating disk; surface potential of suspension particles; rotational speed.
REFERENCES 1. CItARLES,G. E. AND N[ASON, S. G., J. Colloid Inlerface Sci. 15, 236 (1960).
Journal of Colloid and Interface Science, VoL 59, No. 1, March 15, 1977
6
WNEK, GIDASPOW AND WASAN
2. CL:N~, G. E., CL:N% ~. H., CO~I~LL, ~[. M., AND WALI:Eg, T., J. Colloid Interface Sci. 44, 121 (1973). 3. HAPPEL, f. AND BRENNER, H., "Low Reynolds Number Hydrodynamics," Prentice-Hall, Englewood Cliffs, New Jersey, 1965. 4. HO~:G, E. P., ROEBERSO:¢, G. J., ANn WIE~SE~tA, P. H., ]. Colloid Interface Sci. 36, 97 (1971). 5. HULL, M. AND KITCHENER, ~. A., Trans. Faraday Soc. 65, 3093 (1969). 6. KRuY% H. R., "Colloid Science," Vol. I, Elsevier, Amsterdam, 1952. 7. LEVlCH,V. G., "Physicochemical Hydrodynamics," Prenficc-HM1, New York, 1962. 8. MARSHALL,J. K. Am) KITCHENER, J. A., J. Colloid Interface Sc¢. 22, 342 (1966).
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Journal of Colloid and !nlerface Science, Vol, 59, No. 1, M~,rch 15, 1977