Langmuir 1985,1, 691-696
691
conducted TPD studies for methanol on MgO powders and observed H2C0 and CHI as the principal organic products, these were attributed to decomposition of surface methoxy species. The absence of thermal decomposition of methoxy species in the present study suggests that this reaction may require sites other than those on which the initial proton abstraction takes place. This is not unreasonable since the decomposition of an anionic methoxy species to formaldehyde requires elimination of a hydride ion rather than a proton. Again it appears that a simple acid-base model is inadequate to describe the surface reactivity of magnesium oxide.
3. Acetic acid and acetylene dissociate irreversibly on these oxide films; alcohols and water are reversibly dissociated; propyne, propylene, and formaldehyde exhibit only weak molecular adsorption. The pattern for dissociative adsorption is consistent with previous observations on blocking of anion radical formation by Br~nstedacids on MgO powders. 4. The chemical characteristics of the majority 5-coordinated MgO site pairs appear to be equivalent for bulk MgO powders and for oxide layers formed by oxidation of the bulk metal. Thin oxide films thus represent promising and tractable models for study of the surface reactivity of bulk oxides.
V. Conclusions 1. Thin (one-two atomic layers) oxide layers may be formed on the Mg(001) surface by exposures of less than 20 langmuirs at 300 K. The uptake of oxygen with increasing exposure is sensitive to both temperature and the extent of surface disorder; the oxygen uptake profiles of this study are in agreement with previous results on clean, ordered surfaces. 2. The oxide layers formed are continuous and are inactive for adsorption of CO and C02at temperatures down to 200 K.
Acknowledgment. We gratefully acknowledge the Research Corporation and the National Science Foundation (Grant CPE8311912) for support of the various stages of this work. We wish to thank Professor Dietrich Menzel for providing the magnesium crystal. R.M. thanks the Latin American Scholarship Program of American Universities (LASPAU) for financial support. Registry No. Mg, 7439-95-4; MgO, 1309-48-4;CH3COOH, 64-19-7; CzH2, 74-86-2; HzO, 7732-18-5; H&O, 50-00-0; C3H6, 115-07-1;C3H4,74-99-7; CO, 630-08-0; COz, 124-38-9; MeOH, 67-56-1; EtOH, 64-17-5; i-PrOH, 67-63-0.
Kinetics of Coupled Primary- and Secondary-Minimum Coagulation in Colloidal Dispersions D.L. Feke* and N. D.Prabhu Department of Chemical Engineering, Case Institute of Technology, Case Western Reserve University, Cleveland, Ohio 44106 Received March 22, 1985. In Final Form: July 8, 1985 Binary coagulation rates are often predicted on the basis of a steady-state assumption about the pair distribution function and particle fluxes. On the basis of estimates of characteristic times, this assumption is not always justified for slow coagulation processes. A new type of analysis, appropriate for the initial stages of slow coagulation, is presented. Coagulation is modeled in terms of the rates of rearrangement of particle distribution that occur after the interparticle potential is perturbed at the onset of coagulation. Asymptotic results of the model valid for short times are given. For a colloidal dispersion governed by a DLVO potential, the theory predicts an induction period for the filling of the primary minimum and a coupling between primary- and secondary-minimum coagulation. 1. Introduction Stability of colloidal dispersions is often explained and interpreted in terms of theories of interparticle forces. Coagulation rates are predicted to depend on the magnitude of the energy barriers that particles must overcome on their way to a collision. Colloidal systems that exhibit a high interparticle potential barrier are thought to be stable to coagulation while low potential barriers lead to rapid coagulation. However, for the intermediate case of moderate potential barriers, colloidal systems undergoing slow coagulation may exhibit a more complex behavior. Particularly, coagulation may occur in either the primary or secondary minimum, and the rates of coagulation into the two minima can be inherently coupled. In this paper we present an analytical treatment for the slow coagulation of monodisperse colloidal spheres. In order to better model the coupling between primary- and secondary-minimum coagulation, our theory does not rely on pseudo-steady assumptions about the coagulation 0743-7463/85/2401-0691$01.50/0
process as do most other analyses of coagulation phenomena. Rates of slow coagulation are presented in terms of a perturbation series for small changes in the interparticle potential. Our analysis is general and can incorporate DLVO theory to account for interparticle forces and can employ exact mobility functions to describe the hydrodynamic interactions between the particles. In addition, we present a simulation of a typical coagulation process to indicate the nature of the coupling between the two types of coagulation and also to exemplify the induction period for the filling of the primary minimum. 2. Background Coagulation processes can be viewed in terms of the time evolution of the spatial distribution of particles within a suspension. A model for the coagulation of dilute suspensions of monodisperse colloidal spheres is depicted in Figure 1. The analysis is developed in a Lagrangian (particle-centered) reference frame in which the origin 0 1985 American Chemical Society
Feke and Prabhu
692 Langmuir, Vol. I, No. 6, 1985
c-
,
-fl'S , P H ECOAGULATING RE
' E
k-.-
,/
is given as the inner product of a diffusivity tensor with the gradient of the probability function. Thus f = -Pw,-VV - D,.VP (2.4)
'\
In (2.4), w, = w,(r) is the relative particle mobility tensor, D, (=kTw,, k T being the product of Boltzmann's constant and the absolute temperature) is the relative diffusivity tensor, and V the overall interparticle potential. Substitution of (2.4) into (2.5) yields an advective-diffusive equation of the form aP/at - V.P~,.VV = V.D,.VP (2.5) \
\
Figure 1. Definition sketch for the coagulation analysis. Co-
agulating particles penetrate surface S and come to rest between surfaces SI and Sp. Coagulation rates are calculated as the net rate of passage through the various surfaces.
coincides with the center of a reference or test sphere. Coagulation kinetics are determined from the rate at which second particles approach the test sphere. In general, for any surface S that encloses the test particle, the flux of second particles toward the test sphere through surface S can be written (see, e.g.,l for a development of the methodology outlined in this section) as
j , = - l PS( r ) v m ds where v = v(r) is the velocity of the second sphere relative to and away from the reference sphere and P(r) is the distribution function for the number concentration of second particles in space. To calculate coagulation rates within any particular region of space surrounding the test particle, we introduce the auxiliary surfaces S1and S2,both of which enclose the test particle, as shown in Figure 1. The net coagulation to the region of space bound by SI and S2,js1,s2,can be found by repeated application of (2.1): js192=
P(r)v-nd S 1
1P(r)vn dS 82
In order to obtain individual primary- or secondary-minimum coagulation rates, SI and S z must be chosen to correspond to the bounds of the primary or secondary minimum. For the coagulation of spherical particles subject to interparticle forces dependent only on centerto-center separations, S, and S2are selected to be spherical surfaces, concentric with the reference sphere. The suspensions studied here are dilute in the sense that only two-particle interactions are considered. In this case, the distribution function P(r) must satisfy a conservation equation of the form aP/at + v.f = o (2.3) where f (=vP)is the local flux of second particles relative to the test particle. Note that negative fluxes represent the approach of two particles. In the absence of externally imposed flow fields or relative motion driven by external body forces, the relative flux results from interparticle forces and Brownian diffusion. The flux arising from interparticle forces can be formulated as the inner product of a hydrodynamic mobility tensor and the gradient of the interparticle potential, multiplied by the probability function. Similarly, the flux due to Brownian diffusion (1)Feke, D. L.; Schowalter, W. R. J. Fluid Mech. 1983, 133, 17.
Coagulation rates are obtained from the solution of this equation subject to boundary conditions chosen to reflect the appropriate physics of the particular coagulation process being modeled. In the absence of a high interparticle energy barrier and large gradients in the distribution function, corresponding to the case of rapid coagulation, a characteristic time for the evolution of P, t,, is
t,
-
a2/D
(2.6)
where a is a characteristic length of a particle and D is a scalar measure of the particle diffusivity. Using the Stoke-Einstein value for D, (2.6) becomes t,
-
67rpa3 ___
kT which is t, 6 X lov4s for 0.1-pm particles in water at room temperature. Since most coagulation experiments cannot detect transients on this short time scale, many investigators (see, e.g., ref 1-8) have predicted rapid coagulation rates on the basis of a pseudo-steady-state assumption about the distribution function P. For this case, (2.5) is solved with boundary conditions chosen to reflect a pseudo-steady process in which second particles are considered to be removed from the system upon contact with the test particle and are supplied to the system far away from the test particle at a rate equal to their removal rate. For coagulation of identical spheres of radius a, these conditions are expressed by P-Pm r-a (2.8a)
-
P=O
r=2a
(2.8b)
where P, is the bulk number density of spheres. Solution of (2.5) for identical spheres with conditions (2.8) yields 'exp( V / k T )
dr
where D, = D,(r) = r.D,.r/r2. Using (2.9) in evaluating (2.1) for any spherical surface S reveals that
(2) Smoluchowski, M. v. 2. Phys. Chem. 1917, 92, 129. (3) Fuchs, M.2.Phys. 1934, 89, 736. (4) Derjaguin, B. V.; Muller, V. M. Dokl. Akad. Nauk. SSSR 1967,176,
738. (5) (6) 794. (7) (8)
Spielman, L. A. J. Colloid Interface Sci. 1970, 33, 562. van de Ven, T. G. M.; Mason, S. G. Colloid Polym. Sci. 1977,255, Batchelor, G. K. J . Fluid Mech. 1977,83, 97. Melik, D. H.; Fogler, H. S. J. Colloid Interface Sci. 1984, 101, 84.
Kinetics of Coagulation in Colloidal Dispersions
Langmuir, Vol. 1, No. 6, 1985 693
Note that (2.10) shows j s to be independent of S, and, as a consequence, (2.2) indicates that one result of the pseudo-steady assumptio! is that particles do not accumulate in any regions of space. Under these conditions, coagulating particles are pictured to move from the outer regions surrounding the test particle, through the secondary-minimum region (if the interparticle potential is chosen such that one exists), and through the primary minimum at constant rate, eventually disappearing upon collision with the test particle. The result (2.10) has been tested and found adequate in a number of instances. Roeberson and Wiersemaghave investigated the error introduced by neglecting the transient part of (2.5) when the conditions (2.8) are used. They concluded that for most rapid coagulation processes, (2.10) adequately describes experimental results. However, results of slow coagulation experiments show much less agreement with the pseudo-steady theory.'&12 It has been speculated that the failure of the pseudo-steady theory is that it treats coagulation in the primary and secondary minimum as independent p r o c e ~ s e s . ~ To ~ - ~coagulate ~ in the primary minimum, a particle must first enter and then leave the secondary minimum, passing over the potential maximum. In attempting to compare theory to experimental measurements (which cannot easily distinguish between primary and secondary doublets), one must be aware of the coupling between the two coagulation modes. Some attemptsl2JS1' have been made to predict simultaneous primary- and secondary-minimum coagulation. These theories, which extend the pseudo-steady coagulation analysis to include secondary-minimum coagulation, do not adequately explain experimental trends in all instances. Closer examination of the pseudo-steady-state assumptions reveal that they may not be valid for colloidal systems containing large particles or undergoing slow coagulation. For example, (2.7) gives t, 600 s for 10-pm particles in water at room temperature. As a further example, consider the case of a colloidal suspension, initially stable to coagulation, being caused to undergo slow coagulation through the addition of a small amount of salt. If AUK,a difference in the dimensionless Debye length, characterizes the amount of salt added, then a scale for the change in interparticle potential in the vicinity of the potential maximum is ta\kk,2(Aa~)/2a~, where t is the dielectic coefficient of the fluid medium and \ko the surface potential on the particles. By use of this scale factor, (2.5) reveals that a time scale for the evolution of P during slow coagulation, t,, is N
t,
-
3.rrpa2(a~) t\ko2(AaK)
(2.11)
-
Note that in (2.11) the scale factor for the interparticle mobility, w (4/a~)/6apa,was chosen to reflect the hydrodynamic retardation experienced by the particles in the vicinity of the potential maximum. For 10-pm particles (9)Roeberson, G.J.;Wiersema, P. H. J. Colloid Interface Sci. 1979, 49, 98.
(10)Joeeph-Petit, A.-M.; Dumont, F.; Watillon, A. J.Colloid Interface Sci. 1973, 43,649. (11)Ottewill, R. H.; Shaw, J. N. Discuss Faraday SOC.1966,42, 154. (12)Marmur, A. J. Colloid Interface Sci. 1979, 72, 41. (13)Nir, S.;Bentz, J.; Duzgunes, N. J . Colloid Interface Sci. 1981,84, 266. (14) Nir, S.; Bentz, J. J. Colloid Interface Sci. 1978, 65,399. (15)Richmond, P.;Smith, A. L. J.Chem. SOC.,Faraday Trans. 2 1975, 71, 468. (16)Hogg, R.; Yang, K. C . J . Colloid Interface Sci. 1976, 56, 573. (17)Wiese, G.R.;Healy, T. W. Trans. Faraday SOC.1970, 66, 490.
-
with q0= 10 mV, for example, (2.11) predicts t, 1s for a coagulation process induced by changing the concentration of a 1:l salt from 1.0 X to 2.0 X M. For the coagulation of large particles or slow coagulation processes, there may not be sufficient justification to use the pseudo-steady analysis to interpret experimental results. The analysis presented in the next section better accounts for the physics of slow coagulation processes and allows the prediction of some time-dependent effects that may be observable in coagulation experiments. In addition, it elucidates the physical coupling between coagulation in the primary and secondary minima. 3. Analysis In the analysis presented here, the colloidal dispersion is more realistically considered to be a closed system. The distribution function P remains governed by the conservation equation (2.5), but the boundary conditions reflect the closed nature of the system: f-0 r-w (3.la)
f=O
r=2a
(3.lb)
The first of these conditions indicates that no particles enter or leave the system while the second indicates that particles cannot pass into the reference sphere. An appropriate initial condition for the solution of (2.5) is also required. Before the initiation of coagulation, we assume that P has attained a steady, stable configuration as determined by an interaction potential V,. We shall assume that coagulation commences at t = 0 due to some change in the interparticle potential. This change can be envisioned as, say, the addition of salt to the dispersion. We then model the progress of coagulation after the change in interparticle potential to a different constant value V. Although we have arbitrarily selected a change in interparticle potential as the driving force for coagulation, other mechanisms, such as the imposition of a nonuniform flow field on the dispersion, can similarly be incorporated into the model. To formulate an initial condition for the solution of (2.5), we examine the steady-state solution of (2.5) subject to boundary conditions (3.1) and interparticle potential Vi. The result shows the particles to be Boltzmann distributed according to V, at the onset of coagulation. We thus adopt (3.2) as the initial condition for the
P/P, = exp[-V,/kT] (3.2) solution of (2.5) for slow coagulation processes. In order to examine the salient features of this model, we chose to obtain an asymptotic solution to the governing equations by assuming a series expansion for P in the form P(r,t) = Po(r) + Pl(r,t) + Pz(r,t) + ... (3.3) Similarly, there exists an expansion for the flux of second particles toward the reference sphere given by f = fo(r) + fl(r,t) + f2(r,t) + ... (3.4) in which the f, are related to the P, through (3.5) according to (2.4). Substitution of these series into (2.5) results in the hierarchy of equations (cf. (2.3)) aPo/at = o (3.6a)
Since the chosen boundary conditions apply to f , it is
694 Langmuir, Vol. 1, No. 6, 1985
Feke and Prabhu
efficient to rewrite (3.6) in terms o f f by eliminating P through (3.5). First, taking the r derivative of (3.6b) multiplied by the integrating factor eVJkT yields
Note, however, that (3.5) can be arranged to give (3.8) which can be substituted into the first term of (3.7) to yield
a Lf e,V / k T at D ,
a
i a 2
ar
r2
--eV/kT-
dr
r fi-l = 0
(3.9)
4. Results and Discussion To illustrate the use of the theory, we consider 9.G case of spherical colloidal particles governed by a DLV018J9 interparticle potential. Here the total interaction energy is given by the linear sum v = VA + VR + v c (4.1)
where V Ais the attractive potential arising from van der Waals effects, V Ris a short-range repulsion potential due to Born effects, and Vc is a Coulombic repulsive potential owing to the presence of charged double layers. Under the conditions of no retardation, the van der Waals attractive energy for identical spheres is given by the Hamaker expression20
This equation can be rearranged, and, in conjunction with (3.6a), yields the hierarchy of equations
a
-fo
at
(3.loa)
=0
(3.lob) The boundary conditions (3.1) become fi-0 r-a alli fi = 0
r = 2a
all i
(3.11a) (3.11b)
while the initial condition can be obtained from the combination of (3.2) and (3.5) as
D a f o = -P,2e-Vi/kT-(V- Vi), t = 0 kT ar fi=o t = O i > l
2+ - +R2 2+ I n -
VA = -A( 6 R2-4
Note that (3.12a) explicitly shows that the change in the interparticle potential imposed at t = 0 drives the coagulation. The solution to (3.10) subject to conditions (3.11) and (3.12) is given by
2
(4.2)
4,
where R is the distance between the particle centers scaled on the sphere radius, and A, the Hamaker coefficient, is a measure of the strength of the attraction. Several ad hoc formulas have been used to describe the short-range repulsion that acts between colloidal spheres. Since the precise details of the short-range repulsion are not important for the consideration of coagulation processes here, we use a mean-field expressionz1wherein 4! -1 R2 vR=4A - - - 14R + 54 lo! R[ (8- 2)7
(:)
R2
(3.12a) (3.12b)
R2-
+
12R2 + 60 R7
+ 14R + 54 (R
+ 217
+
-
]
(4.3)
Here u represents an atomic collision diameter. This potential is useful for our study because it exhibits some degree of softness yet does not permit particle interpenetration. Many approximate forms are available for the Coulombic repulsion between colloidal spheres. For our demonstration, the approximate expression given by Bell, Levine, and McCartney22was used. For small surface potentials, the expression
(4.4) is known to give an error of less than 3% at all separations in comparison with the exact numerical solution for the interaction of identical spheres. Substituting (4.2), (4.3), and (4.4) into (4.1) and defining a dimensionless potential V = V / k T yields For any interparticle potentials V i and V , (3.13) predicts the flux of second particles toward the reference sphere as a function of position and time after the start of coagulation. Coagulation rates can be found directly from f . Rewriting (2.2) as
+
(
(3.14) shows that for spherical surfaces S1 and Sz, the coagulation rate is given by j r l , r 2 ( t ) = 47dr12f(r1,t)- r22f(r2,t)l
(3.15)
Thus, to find the coagulation rate to the region in space surrounding the reference sphere and bounded by spheres of radius r1 and rz, (3.15) can be applied directly.
A)
Nc 1 - -
In
-2R2 + 60 R2 + 14R + 54 + Rl (R + 2)7
[
1+-
1
R-1
exp(-aK(R
-
2))
1
I
+
(4.5)
(18) Derjaguin, B. V.; Landau, L. D. Acta Physicochem. URSS 1941, 14, 633.
(19) Verwey, E. J.; Overbeek, J. Th. G . "Theory of the Stability of Lyophobic Colloids";Elsevier Publishing Co.: Amsterdam, 1948. (20) Hamaker, H. C. Physica (Amsterdam) 1937.4, 1058. (21) Feke, D. L.; Prabhu, N. D.; Mann, J. A., Jr.; Mann, J. A., I11 J . Phys. Chem. 1984,88, 5735. (22) Bell, G . H.; Levine, S.;McCartney, L. N. J. Colloid Interface Sci. 1970, 33, 365.
Langmuir, Vol. 1, No. 6, 1985 695
Kinetics of Coagulation in Colloidal Dispersions
R
Table I. Dimensionless Parameters Governing the Interparticle Potential range for typical nOUD disDersions NA = A/GkT 1 to 10 10-18 to 10-23 NR= ( 4 A / k T ) (4!/10!)(u/al6 Nc = eaPo2/kT 1 to 104 0.1 to 104 aK
2. I
0
Four dimensionless parameters govern the shape of the interaction potential. Table I summarizes the groups and their numerical values for typical dispersions. The parameters N A is the ratio of the van der Waals attractive energy to the thermal energy in the dispersion, NR is a measure of the Born repulsion energy to the thermal energy, and N c relates the strength of the Coulombic repulsion energy to the thermal energy. The fourth parameter, aK, is the ratio of the particle radius to the diffuse double-layer thickness. Presumably, for coagulation processes induced by the addition of salt to the suspension, the values of NA, NR, and NC will be the same before and after the salt is added. The assumption, which implies that A , u, e, and \koare not dependent on the fluid environment, should be valid for small added quantities of salt. Thus, coagulation is assumed to come about through a change in aK only. It is also convenient to define a dimensionless flux, f = 6.rrpa2f/kTP,,and a dimensionless coagulation rate, j = 3pj,f2kTPm.With these definitions, (3.13) becomes
E L c
i -0.020 T R ,
-0.025
1
'
.
'
2.2
I PARTICLE
2.3
2.4
2.5
DEPLETION-,
ACCUMULATION
Figure 2. Plot of the zeroth-order asymptotic solution of the relative particle flux for the case NA = 1, NR = lo-", Nc = 525, ( u K )= ~ 50, and ( U K ) = 25. Note that regions of positive slope roughly correspond to particle depletion, and negative slopes indicate particle accumulation. For the particular coagulation example illustrated in this paper, the dimensional coagulation rate can be calculated from fo = (5.32 X lo8)@fo m-2 s-l, where 9 is the volume fraction of particles.
2.50 -
2.00 A
h
fl/t
1.50-
(4.6a)
0-
- - a
f i = D -"-evre dR
i -(R2L'?i-1(3) a
R2 dR
di)
)
I
2 ~ 2 - E
' 2.4DEPLETION-, 2.5
R
(4.6~)
where 8, = 6spaDr/kT and t^ = kTt/6aha3, and (3.15) becomes
~ R ~ , R=~ R12?(Ri,0 (O - R22?(R2,0
(4.7)
Thus, if R1 and R2 correspond to the inner and outer boundaries of an interparticle minimum, (4.7) would give the coagulation rate in that minimum. If R1 = 2 (the collision surface), (4.7) reduces to (4.8) In this instance, if Rz is larger than the distance corresponding to the secondary minimum, (4.8) gives the total of the primary and secondary coagulation rates. As an example, consider the coagulation process for a suspension of 5-pm spheres in water with \ko 10 mV, A 2.5 X J, and coagulation induced by changing the to 4.0 X M. concentration of a 1:l salt from 1.0 X For these conditions we use N A = 1, NR = (arbitrarily), Nc = 525, ( u K )=~ 25, and ( U K ) = 50. Shown i,n Figuzes 2 and 3 are plots of the spatial dependence of f o and fl/f for this combination of parameters. These-figures were obtained by direct evaluation of (4.6) with V given by (4.5). Consider first the results shown in Figure 2, in which 3, is seen to be negative for all R . These negative values
which indicates that wherever
N
N
particle depletion occurs at those locations. This condition is satisfied for the regions to the right of the negative peak in Figure 2 , R 2 2.22. Similar reasoning leads to the conclusion that particles accumulate in the region 2.13 5 R 5 2.22 which corresponds to the bounds of the secondary minimum. For the region 2.00 6 R 5 2.13, which corresponds to the vicinity of the primary minimum, there is virtually no particle accumulation. This stems from the choice of the initial interparticle potential Vi, which dic-
Feke and Prabhu
696 Langmuir, Vol. 1, No. 6, 1985
0,5[ I
AP 0
t 20.1-
h
I
\
,0.00
stages of this coagulation process in Figure 4. Note that particles continuously are depleted from the region R 5 2.27 which is a direct result of ongoing coagulation. Consider also the region at R = 2.18. For very short times after the onset of coagulation, the particle concentration may increase above its initial value, but with the progression of time, particles may begin to vacate this region. This is seen in Figure 4 as the shift in the position of the AP = 0 crossing points to higher values of R at longer times. For the short times depicted in Figure 4, the filling of the primary minimum is insignificant. The analysis presented here is valid only for the initial stages of coagulation. For longer times, more terms in the asymptotic series are needed to give precise estimates of coagulation rates, especially in the vicinity of the two minima. An additional limitation is that the analysis only accounts for singlet-singlet particle interactions. Beyond the point of doublet formation, a comprehensive model of coagulation behavior must account for singlet-multiplet and multiplet-multiplet interactions. As a rough estimate of the duration of validity of our analysis, we calculate the time for one particle to coagulate through
L.5
Figure 4. Variation in particle number density with time after the onset of coagulation. Note that in some regions in space, particles may first tend to accumulate and then vacate. For the coagulation example illustrated throughout this paper, real time can be calculated through the formula t = 72f 6.
tates that no particles be in the vicinity of the primary minimum and implies an induction period for primaryminimum coagulation. Figure 3 shows the first-order correction to the coagulation f l u given by Figure 2. From (4.9), it can be reasoned that particles tend to continuously deplete from the region R 2 2.27 but accumulate in the region 2.18 5 R 5 2.27, which indicates continuing secondary-minimum coagulation. Note, however, the region 2.10 ,< R 5 2.18, the inner portion of the secondary minimum, wherein particle depletion is predicted to begin as coagulation proceeds. This illustrates that particles vacate part of the secondary minimum to begin to fill the primary minimum and exemplifies the physical coupling between the two types of coagulation. The relative particle fluxes depicted be Figures 2 and 3 indicate a rearrangement of the spatial distribution function of particles surrounding the reference sphere. This can be seen more explicitly by integrating (2.3) to yield
The temporal and spatial dependence of the increment to the probability distribution function is shown for the initial
(4.11)
In (4.11),,@is the volume fraction of spheres and R1 corresponds to a separation outside the secondary qinimum well, defined here arbitrarily as the point where V = -1.0. Hence, (4.11) is used to predict the average time for which one particle leaves the outer space surrounding the test sphere and becomes associated with it. For the particular coagulation example illustrated throughout, (4.11) gives t^* 20, which corresponds to t* 24 min, when @ = 0.01. For less concentrated dispersions, the validity of the theory will be correspondinglylonger. Clearly, experiments of this duration can be performed with conventional techniques. For times longer than the value predicted by (4.11) the actual coagulation rate is expected to decrease with time, eventually slowing to zero. In order to test our model, it would be necessary to measure time-dependent effects for both types of coagulation processes. Since published coagulation data do not include this information at present, direct verification of this theory cannot be made at this time.
-
-
Acknowledgment. Partial support of this work was provided by the Research Corporation.
