J. CHEM. SOC. FARADAY TRANS., 1991, 87(8), 1163-1168 1163

Evolution of the Cluster-size Distribution in Stirred Suspensions

Ruben D. Cohen Department of Mechanical Engineering and Materials Science, Rice University, Houston, TX 77251, USA

Based on combinatorics, a simple model for predicting the steady-state cluster or drop-size distribution in suspensions undergoing simultaneous coagulation and break-up due to rapid stirring was recently proposed (R. D. Cohen, J. Chem. SOC. Faraday Trans., 1990, 86, 2433). This work extends that model to account for the evolution of the size distribution in stirred systems. The method utilizes the concept of the distribution entropy, and suggests a means for predicting the characteristics of the size distribution as the system approaches steady state. Among the important results is that the distribution functions (size, volume etc.) are path dependent. Furthermore, along the ‘most probable entropy path I , the cluster-size distribution remains closely related to the Poisson distribution up to and including steady state. Another interesting finding is that, along the mentioned path, the distribution functions are concentration specific (intensive property) as long as they are in their tran- sient state. These ultimately become number specific (extensive property) once they reach steady state.

Owing to its important implications in the chemical industry, the behaviour of liquid droplet and colloidal dispersions in stirred vessels arouses great interest within the chemical engineering field. As noted from the amount of relevant papers in the literature, this subject has been studied exten- sively.

It is well known that the mixing of two immiscible liquids forms a dispersion in which coalescence and break-up of the liquid droplets occur simultaneously. Likewise, the same applies when the dispersion comprises colloidal particulates, in which case the end result is the coagulation and break-up of the aggregates. Also known is that if mixing persists over a sufficiently long period of time, a dynamical equilibrium or steady state is established where, beyond this point, the drop- or cluster-size distribution attains a form which remains practically unchanged with time.

Numerous investigations related to this topic have produc- ed measurements of drop-size distributions in stirred vessels. Some of these have concluded that this dynamical equi- librium is accomplished when the rate of break-up equals exactly the rate of coalescence.’ Many others, on the other hand, have conclusively demonstrated that equilibrium can be achieved even when there is no further coalescence.2 This was shown to occur when the turbulence microstructure acts together with the particle-fluid and particle-particle inter- actions in such a manner that, beyond a certain limit, any further break-up of the droplets is completely prevented.

This paper is concerned primarily with the case where both coagulation and break-up occur simultaneously within the agitated system. This, of course, is in contrast to the situation mentioned above which involves only the fragmentation of larger clusters or drops into smaller one^.^.^ The main objec- tive of this work is to extend a recently proposed steady-state model4 to account for the evolution of the cluster-size dis- tribution during rapid mixing. A brief discussion of the steady-state model and the derivation for the present case follow in the next sections.

Problem Formulation Consider a suspension that initially contains a large number of clusters, each composed of a certain number of primary particles or units. Depending on the nature of the primary units, the clusters may be agglomerates of solid colloidal par- ticulates or drops of liquid formed by, respectively, aggre- gation or coalescence.

Related studies have consistently shown that vigorous stir- ring carries the cluster-size distribution through a transient or temporal stage until ultimately steady state is achieved.,~~ Furthermore, if both coagulation and break-up were to occur at random due to stirring, then the steady-state size distribu- tion becomes independent of the input cluster sizes.

As mentioned above, the clusters in the system may be either aggregates or drops. The primary units forming these clusters will be assumed to be monodisperse, each with dia- meter dmin.

Consider now an experiment in which a suspension is being stirred so rapidly that all effects due to Brownian motion and surfaces interactions (such as DLVO) are over- whelmed by the shearing forces and pressure fluctuations of the mixing which cause coagulation and break-up to occur. At some point in time during mixing, there will be a given number of clusters in the suspension. This number shall be denoted by N . We should realize that while N changes with time, the total number of the primary particles in the system remains a constant, equal to N o . Consequently,

N o

N = E N i (1) i = 1

and No

N o = i N i (2) i = 1

where i is the cluster size or the number of primary units in a cluster and Ni is the instantaneous number of clusters with size equal to i.

Following the scheme proposed by C ~ h e n , ~ we assume that the coagulation and break-up processes occur completely at random so that at any stage, the size distribution is deter- minable by combinatorics. This is a result of the vigorous and homogeneous stirring overcoming all effects due to surface interactions. Avoiding the details, we make use of the expression

(3) N o !

WN,, N , 7 N , , . . .) = N o n N , ! [i!]”’ i = 1

where n ( N , , N , , N 3 , ...) is the degeneracy or the total number of ways in which the particles in the dispersion can be distributed into N , clusters of size i = 1, N , clusters of size i = 2, N , clusters of size i = 3 and so on. Upon combining eqn. (3) with the conservation constraint of eqn. (2), we obtain

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1164 J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

the most probable cluster-size distribution at steady state within rapidly mixed system^.^ The end result is a Poisson- type size-distribution probability function, p,(i) , where the subscript co refers to steady state, i.e. t + co. From previous results, this is given by

(4) 2’

p,( i ) = exp( -2) - i !

where 2 satisfies the following relationship

N o = 2 exp Z (5)

Moreover, p, ( i ) is defined by

where N i , is the cluster-size distribution at steady state, and N , , which is the total number of clusters at steady state, is given by4

N , = exp 2 (6b)

Returning now to the present problem concerning the evol- ution of the cluster-size distribution at any time prior to steady state, we implement eqn. (1) as an additional restriction to eqn. (3). Using the method of Lagrange multi- pliers, we introduce the constants Al and A 2 to maximize Q by the following

a -((In f2 + A I N o + A 2 N ) = O d N i

(7)

where, this time, both eqn. (1) and (2) are being used as con- straints. The result of eqn. (7) should, therefore, provide the most probable cluster size distribution in the suspension at any point in time, given N and N o . Note that N and N i change with time so that N ( t --$ a) = N , , as given by eqn. (6b), and Ni( t + co) = N i , .

Upon inserting eqn. (l), (2) and (3) into eqn. (7), we obtain

N i = A - j Pi I .

where for convenience, we have defined A and p as

,u 3 exp Al (94

A = exp A , (94

and

so that they contain the Lagrange multipliers, A1 and A 2 . Substituting eqn. (8) into eqn. (1) and (2) yields the following:

N 1 -exp(-p) _.- - NO P

and

(1 1) N o A = - P exp P

which remain valid for N o >> p (since N o > lo5 for typical suspensions). Thus, for any given N and N o , we can calculate ,u and A using eqn. (10) and (l l) , respectively, and conse- quently obtain the most probable cluster sizes, N i , from eqn. (8).

It is more convenient, however, to deal with the size- distribution probability function, p(i), which, in relation to eqn. (6a), is defined by

Ni p(i) = - N

Therefore, upon combining eqn. (lo), (1 1) and (8), we obtain

p ( i ) = exph) - 1 i ! (13)

which, again, is found to be closely related to the Poisson distribution. In accordance with C ~ h e n , ~ it is useful to note that the steady-state size-distribution function can be derived from the above results when A = 1 and p = 2 [i.e. compare eqn. (11) and (5)]. Since 2 is on the order of In N o and N , 9 1, then eqn. (13) reduces to eqn. (4) because exp 2 9 1.

Other parameters of importance in aggregation and coalescence studies are the volume distribution function, W(i), the cumulative volume function, V(i), and the Sauter mean diameter, d32. In compliance with the parameters used in this work, these are expressed by

and

i N i W( i ) = -

NO

$ idi) d 3 2 i = l --- -

No dmin 1 i2I3p(i)

i = 1

Upon combining eqn. (lo), (12) and (13), we find that the volume-distribution function, W(i), as defined by eqn. (14), is directly related to the size-distribution function through the following expression

W(i ) = [l - exp( -p)]p(i - 1) (17)

It also follows that

V(i) = [I - exp(-p)] p ( j - 1) (18) j = 1

where

1 e x m ) - 1

p(i = 0) =

by virtue of eqn. (13).

Entropic Considerations An interesting aspect of coagulation was discussed previously by Rosen.6 The approach basically utilizes the maximum entropy principle to derive the long-term asymptotic and ‘self-preserving’ drop-size distribution’ in coalescing systems. Briefly, Rosen’s work is based on maximizing the expression Ex - Mx)ln p(x)], which is analogous to maximizing the entropy of the system. Here, Ax) is the size-distribution prob- ability, and x is the ‘reduced’ variable which, in compliance with the nomenclature used in this work, is simply i N / N o . In general, entropy maximization provides the most probable size distribution which satisfies the constraints represented by eqn. (1) and (2) above. Furthermore, it is the dependence of the probability, Ax), on the single parameter, x, that leads to the ‘self-preserving’ nature of the size distribution.

In this work, we shall also investigate the cluster-size dis- tribution in a stirred vessel using the entropy concept. We should mention, however, that Rosen’s approach,6 along with others that followed later, have dealt with purely coagulating systems while the present work concerns simultaneous coagu- lation and break-up. For this matter, we return to our expres- sion for the degeneracy, R, available from eqn. (3), and in

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J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87 1165

*O.

0 s h

10 -

111 ’v,

0 10-l0 I O - ~ l o o

NINO

Fig. 1 S us. N / N o at different typical values of N o : (a) lo9, (b) lo8, (c) 107, (4 106

accordance with Cohen,8 define the ‘distribution entropy,’ S, of the system using Boltzmann’s relation

( 20)

where k , is Boltzmann’s constant (note that the above is commonly used in statistical mechanics). Moreover, we intro- duce the ‘specific dimensionless entropy,’ s, and define it as

S = k , In R

by means of eqn. (20). The quantity s, however, can be easily evaluated upon inserting eqn. (3) into eqn. (21). This yields

N o In N o ! - c (1n N, ! + Ni In i!)

Since we have readily available the most probable cluster sizes, N i [from eqn. (8)] for any given N and N o , we can obtain the ‘most probable specific dimensionless entropy,’ denoted here by S, for the same N and N o simply by substi- tuting eqn. (8) into eqn. (22). After some manipulation, this leads to

A + In A - - (1 + [expb) - llln A } (23) NO

where p satisfies eqn. (10) for any given 0 < N / N o < 1, and A comes from eqn. (1 1). This result is illustrated in Fig. 1 as S us. N / N , for variable N o in the range lo6 6 N o < 10’. This range of N o is a fair representative of typical suspensions.

In summary, the following conclusions can be derived from Fig. 1 : (i) Every S us. N / N , curve (for each different N o ) has a distinctive maximum which occurs at a particular value of N / N o . We denote this maximum S,,,. (ii) The value of S,,, corresponds exactly to the steady-state conditions discussed briefly in the beginning of this paper, and in more detail in C ~ h e n . ~ Hence, S,,, , occurs at N / N o = N,/No, where A = 1, p = Z [recall that 2 satisfies eqn. (5)], and N , = exp 2 from eqn. (6b). Inserting these into eqn. (23) and con- sidering that N o is several orders in magnitude, we obtain

(24) 1 z S,,, = 2 - 1 + -

as formulated in ref. 8.

Discussion It is important now to focus our attention on Fig. 2 which represents a typical S us. N / N o curve. Here, for convenience, the point of maximum entropy, S,,, , is indicated by 0.

path I (A -+ 0) -1

Fig. 2 Typical curve of S us. N / N o depicting the point of maximum entropy, S,,, occurring at 0, and the two most probable paths for the development of the cluster-size distribution during mixing

Consider now that the suspension is initially at some state A located on the curve. At this point we begin to stir the suspension vigorously and homogeneously. As a result, the suspended clusters undergo simultaneous break-up and coag- ulation at random, until after some period of time, the cluster- size distribution reaches steady state. The size-distribution function [which is obtainable from the result of eqn. (13) in which p is related to N / N o through eqn. (lo)] can, therefore, be conveniently traced through the S us. N / N o curve of Fig. 2. As an example, the most probable temporal evolution must follow path I, moving along states A -+ B -+ C -+ 0, where eventually at point 0, when S cannot increase any further, steady state is achieved. Of course, the entropy here is S,,, .

Similarly, the size distribution can be traced as it evolves along path I1 in Fig. 2. Here, for instance, state A’ is the initial condition, and the most probable path must follow A’ -, B’ + C’ -+ D’ -+ 0, until steady state is reached at S,,, . It is worth noting that N / N o = 1 signifies a suspension that is fully dispersed, with all particles being the primary units of size dmin . This may very well serve as the initial condition For path 11.

In order to illustrate how the clusters evolve along the two paths of interest, we have included Fig. 3(a) and 3(b) where the volume-distribution function, W(i) from eqn. (17), is plotted against cluster size, i, at the different stages indicated in Fig. 2. Fig. 3(a) displays the behaviour of W(i) along path I, from A + 0, whereas Fig. 3(b) depicts W(i) along path 11, going from A’ + 0. Obviously, the dominant action along path I [Fig. 3(a)] is break-up, whereby it is coagulation along path I1 [Fig. 3(b)].

Fig. 4(a) and q b ) , respectively, show the cumulative volume, V(i), available from eqn. (18), plotted against size, i, for the cluster-size evolution along paths I and 11. In addi- tion, Fig. 5 traces the development of the Sauter mean dia- meter ratio, d3Jdmin [eqn. (16)], along paths I and 11. Interestingly, along the S us. N / N o curve, d3Jdmin is found to be related to N / N o by the following simple expression

Moreover, it is important to note that along the S us. N / N o curve, the transient distribution functions, i.e. before they reach steady state, are dependent only on p, by virtue of eqn. (13), (17), (18) and (25), and therefore only on the ratio

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0 . 1 5

0.8-

0.6-

: 0 . 4 -

0.2-

0 . 0 - 1

0.10

: zi 0.05

0 . 0 0 . 0

1 .o

0.8

0.6 f? 1 v

0.4

0.2

0.0 5 0 . 100 150

1 .o

/ *’

\ / Bcl

10 20 30 40 i

Fig. 3 (a) Volume distribution function, W(i), 0s. cluster size, i, during the evolution of the cluster-size distribution following path I (A + 0) shown in Fig. 2. (b) Volume distribution function, W(i), us. cluster size, i, during the evolution of the cluster-size distribution fol- lowing path I1 (A’ -, 0) shown in Fig. 2

N / N o , owing to eqn. (10). Upon defining the ‘number con- centration,’ c, as

N V

c = -

where V is the total volume of the system, we can write

where co is the concentration based on the number of the primary particles, N o . Therefore, it follows from eqn. (27) that at any transitional or time dependent point prior to steady state, the distribution functions, p(i), W(i) and V(i), depend only on the concentration ratio, and are explicitly independent of the total volume, V, of the system.

At steady state, however, when p = 2, where 2 is given by eqn. (5) , the distribution functions, pm(i), Wm(i) and Vm(i), are governed by the total number, N o (instead of the concentra- tion ratio, c/co), and therefore on the total volume, V, of the system (recalling that N o = coV). This simply implies that during its developmental stage, the cluster-size distribution is

0.8

f? 0.6 z

0.4

0.2

0.0

1. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87

0 10 20 30 40 1

Fig. 4 (a) Cumulative volume, V(i), us. cluster size, i, during the evolution of the cluster-size distribution following path I (A -+ 0) shown in Fig. 2. (b) Cumulative volume, V(i), us, cluster size, i, during the evolution of the cluster-size distribution following path I1 (A’ -+ 0) shown in Fig. 2

-4 -3 -2 -1 0

W N / N , )

Fig. 5 log(d,,/d,,,) us. log(N/N,), showing the development of the Sauter mean diameter ratio along the two paths, I and 11, depicted in Fig. 2

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J. CHEM. SOC. FARADAY TRANS., 1991, VOL. 87 1167

maximum entropy at Doint o \

possible paths v A " (A*+ 0)

1 0 - l ~ l o - * 1 0 - 4 10-2 100

NIN,

Fig. 6 Some possible paths on the S us. N / N , diagram when the initial condition, A*, does not correspond to a point on the curve

concentration dependent, thus making it an intensive pro- perty. At steady state (t + a), however, the cluster-size dis- tribution function is dependent on the total volume of the system, thereby making it an extensive property or 'system specific.'

In view of the above, we refer to Smoluchowski's simplified version of the perikinetic (Brownian) coagulation expression for an i-fold aggregate (in a monodisperse system) which is given by9

ci N i (t/t,)'- _ - -- - - co N o (1 + t / tJi+'

where ci (= N J V ) is the number concentration of the aggre- gates of size i, t, is the 'flocculation time' expressed by 3p/4kB Tc,, and T and p are the absolute temperature and the continuous phase viscosity, respectively. Interestingly, we observe that N J N , in eqn. (28) is governed by the primary particle number concentration, c,, and not on the total number of the primary particles, N o , in the system. In rela- tion to the present work, therefore, we can only speculate that the reason for this is perhaps because Smoluchowski's size distribution maintains its unsteady nature at all times.

Additional Comments

We have until now discussed the transition of the cluster-size distribution as it starts at one point on the S us. N / N , curve, and evolves along it until steady state. An interesting ques- tion is what if the starting point or initial condition is not located on the S us. N / N , curve but below it, as signified by point A* in Fig. 6. The state, A*, for example, could represent a suspension of equal-sized clusters. It should be noted that it is impossible for A* to be arbitrarily chosen above the curve since S cannot be exceeded at that particular value of N / N o .

In this case, the distribution could gain access to the steady state, S,,, at point 0, by following any of the suggested routes. For instance, one possible path could be A* + 1 + 0, where, from a thermodynamical standpoint, A* -+ 1 occurs isentropically. Another probable path is A* + 2 + 0, where along A* + 2, N / N , remains constant. Some other possible paths, as shown in Fig. 6, include A* + 3 -+ 0 and A* + 0, with the latter being the most direct route to steady state. At this time, unfortunately, the model is not yet developed to a stage that we are able to determine the most probable path to steady state should the initial condition of the suspension lie below the S us. N / N o curve. Therefore, we shall avoid any further discussions regarding this matter.

Conclusion Upon extending the combinatorial model proposed by Cohen4 to account for the development of the cluster-size dis- tribution functions in rapidly mixed dispersions, we find that, in general, the evolution is strongly dependent on the path. Nevertheless, it is physically conceivable to define an optimal path which falls directly on the S us. N / N , curves shown in Fig. 1. The size distribution function along this path, as given by eqn. (13), is found to be similar to the Poisson distribu- tion. It is interesting to note that during the developmental stage, if it happens to follow this path, the distribution func- tions are concentration dependent, whereas at steady state, they are governed by the total number of primary particles. This is obvious from the fact that prior to steady-state condi- tions, p(i) depends only on p and therefore only on N / N , , while at steady state, p,(i) is governed by 2, and therefore on N o alone.

At present, however, the results of this work do not let us predict the rates simply because there is no time parameter involved in the model. However, in reference to the numerous relevant experimental s t u d i e ~ , ~ , ~ . ' ~ there is reason to believe that the rate dependence is strongly affected by the stirring conditions. Finally, besides the considerable amount of addi- tional and more focused experimental data needed to back up this work, its extension to account for the rate dependence appears to be very likely far from trivial.

Glossary A c c, d,, dmin i N N ,

Ni Ni, N o

p(i)

p,(i)

S distributional entropy [eqn. (20)] s specific dimensionless entropy [eqn. (21) and (22)] S most probable specific dimensionless entropy [eqn.

S,,, maximum most probable specific dimensionless

t , flocculation time [eqn. (28)] V V(i) W(i) 2

i,, i, Lagrange multipliers [eqn. (7)] Q degeneracy [eqn. (3)] p distribution characteristic during the evolutionary

defined in eqn. (9b) [satisfies eqn. (1 l)] number concentration based on N [eqn. (26)] number concentration based on N o [eqn. (27)] Sauter mean diameter [eqn. (16)] diameter of the primary particles cluster size or number of primary particles in a cluster total number of clusters in the system [eqn. (l)] total number of clusters in the system at steady state

number of clusters of size i number of clusters of size i at steady state total number of primary particles or units in the

size-distribution probability function [eqn. (1 2) and

size-distribution probability function at steady state

Ceqn. (6b)l

system [eqn. (2)]

(1311

[eqn. (4) and (6a)l

( 2 3 ~

entropy [eqn. (24)]

total volume of the system cumulative volume [eqn. (15) and (IS)] volume-distribution function [eqn. (14) and (17)] distribution characteristic at steady state [satisfies eqn.

(5)1

stage [satisfies eqn. (lo)]

Referewes 1 S. L. Ross, F. H. Verhoff and R. L. Curl, Znd. Eng. Chem.

2 Fundam., 1978,17, 101. R. Shinnar, J. Fluid Mech., 1961, 10, 259.

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3

4 5

6

G. Narsimhan, D. Ramkrishna and J. P. Gupta, AZChE J. , 1980, 26,991. 22, 126. R. D. Cohen, J. Chem. SOC., Faraday Trans., 1990,86,2133. G. B. De Boer, C. De Weerd and D. Thoenes, Chem. Eng. Res. Des., 1989,67, 308. J. M. Rosen, J. Colloid Interface Sci., 1984,99, 9.

7 S. K. Friedlander and C. S. Wang, J. Colloid Interface Sci., 1966,

8 R. D. Cohen, Powder Tech., 1990,63,261. 9 J. Gregory, Crit. Reo. Enoiron. Control, 1989,19(3), 185.

10 B. J. McCoy and A. J. Madden, Chem. Eng. Sci., 1969,24,416.

Paper 0/03858J; Received 28th August, 1990

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