Langmuir 1991, 7, 1325-1331
1325
The Process of Emulsification: A Computer Model Paul Becher* Paul Becher Associates, Ltd., Wilmington, Delaware 19803
Michael J. McCann McCann Science, Chadds Ford, Pennsylvania 19317 Received August 24, 1990.I n Final Form: November 30,1990 We have derived a computer simulation of the process of emulsification, in which the change in particlesize distribution is calculated with respect to time. Among the assumptions used is that particle rupture is a first-order process, while particle coalescence is second order. However, the particle is not assumed to split into two equally sized smaller particles; rather, the particle is assumed to break into a number of particles, both large and small. These assumptions lead to an explanation of the phenomenon of overemulsification. Introduction The process of emulsification has been studied from the viewpoint of the effect of various experimental conditions on the finished product, e.g., by Walstra-l However, it must be admitted that no satisfactory model of the process has been developed. The simplest approach has been on the basis of the following assumptions: 1. Particle (droplet) rupture or splitting is a first-order process and the split is into two equally sized particles. 2. Particle coalescence is a second-order process. These two assumptions lead to the well-known Bernoulli equation dn/dt = k+z - k,n2
(1)
where n is the concentration of droplets a t time t, and kf and k, are the rate constants for the fracturing (splitting) and coalescence steps, respectively. The Bernoulli equation may be readily integrated, and we find that l / n = k,/kf
+ (l/no - (k,/kf) exp(-k&))
(2)
where no is equal to the number of particles existing a t the beginning of the process and may, for the case in point, be set to unity. As will be noted from eq 2, the particle number or the average particle size (initial volumeln) tends to a steadystate concentration of particles, n,t, proportional to k,/kf. In this paper, we shall understand the “Bernoulli curve” to correspond to this behavior, Le., to an exponential increase in particle number to some steady-state value. As will be shown, our model exhibits the Bernoulli form of response as a particular case. The Bernoulli formulation is limited in that it contains no provision for adjustment of the coefficients kf and kc during the emulsification process. We shall elaborate on this point subsequently. Such behavior is indeed encountered in many practical emulsions, e.g., as reported by RajagopaLz However, this approach does not consider the possibility of overemulsification. Overemulsification describes the phenomenon in which a maximum particle number n (minimum particle size) is reached, followed by a drop in n, i.e., droplet-size (1) Weletra, P. In Encyclopedia ofEmulsion Technology;Becher, P., Ed.;Vol.l,MarcelDekker,Inc.:NewYorkandBasel,1983;Vol.1,Chapter 2. (2) Rajagopal, R. S. Kolloid-2. 1969, 167, 17.
increase, until a new equilibrium is achieved, with n < nat. This has been observed, for example, in ultrasonic emulsification by Rajapopa13and in jet emulsification by Bechera4More recently, similar results have been reported by Djakovi~.~ We have examined the possibility that overemulsification can be explained by calling on the so-called chaos theory.6 Indeed, there does appear to be evidence that overemulsification can be induced under chaotic condit i o n ~but , ~ we remain unconvinced of the validity of this approach. Thus, we have gone back to fundamentals and have devised a computer simulation of the process of emulsification, in which we have included assumptions additional to those stated above. Overview of the Model We have assumed, for purposes of the model, that the emulsification takes place in a stirred reactor vessel. It will be apparent, however, that extension to other types of emulsification equipment is possible. In addition, we add two further assumptions to those listed above: 3. Particle rupture does not necessarily result in two equal-sized particles (as in assumption 1)but may result in the formation of a number of particles, both small and large. 4. The nature of the particle rupture, e.g., the ratio of small to large particles, is related to, among other things, the shear velocity and the properties of the substances involved. Assumption 3 will be recognized as arising from the observations of Rumscheidt and Mason! who first observed the presence of small “satellite” drops on particle rupture. Assumption 4 arises naturally from surface hydrodynamics. Furthermore, the model must meet certain requirements: It must be conservative, in the sense that it observes conservation laws. (3) Rajagopal, R. S. Roc.-Zndian Acad. Sci., Sect. A 1969,49A, 333. (4) Becher, P. J. Colloid Sci. 1967,24, 91. (5) Djakovic, R. Private communication. (6) (a) Hofstedter, D. R. Metamathematical Themas: Questing for theEssence ofMindandPattern; Basic Books: New York, 1985;Chapter 16. (b) Gleick, J. Chaos-Making a New Science; Viking: New York, 1987. (c) Schuster, H. G. Deterministic Chaos: An Introduction, 2nd ed.; VCH Verlagsgesellschaft: Federal Republic of Germany, 1988. (7) Becher, P.; McCann, M. J. Unpublished. (8) Rumscheidt, F. D.; Mason, S. G.J. Colloid Sci. 1961, 16, 238.
0 1991 American Chemical Society
Becher and McCann
1326 Langmuir, Vol. 7,No. 7, 1991 It must be able to represent a wide range of particle sizes. It has to account for the physical properties of the materials. It has to relate to the actual conditions existing in, e.g., a stirred vessel. It is dynamic and must be able to cope with the range of time scales covering both the initial emulsification and the subsequent reagglomeration. Finally, in order to keep within useful bounds, it must be simple enough to run on a personal computer. We shall discuss the elements of the program under a number of headings, which will illuminate the overall plan. LogarithmicSize Sequence. The key design element is the choice of a logarithmic sequence of particle size categories, each of which has some mass of material associated with it. Thus, if, as in this demonstration of the model, there are ten size categories and we choose to make each size twice the size of the preceding, then the range of sizes is 512:l. The choice of ten size categories, although arbitrary, is good enough to show at least a bimodal distribution with reasonable exactness. Obviously, however, making the size ratio a ratio of masses is not the only option; in fact, masses may be converted to volume or number concentration terms, if one knows the density of the dispersed particles, or if one can assign a diameter to each size category. The distribution of particle sizes will be represented by a set of material masses, each set being made up of particles of the same size, and where each size will differ from the next by a factor of 2 in the volume (or mass). The corresponding ratio of the raddi is 21J3= 1.2599. Well-StirredVessel. The next key feature is the way in which the rates of fragmentation and coalescence are determined. We postulate a well-mixed vessel, so that the size categories are not spatially segregated, i.e., all sizes are completely mixed together. We assume that as particles become smaller they become more difficult to split into smaller fragments. In fact, we assume that there will be a smallest possible size in a particular vessel (or with a particular procedure), below which no particles will appear, since there is nowhere in the vessel sufficient shearing action to cleave them. This is consistent with the observations of Becher4 for the case of jet emulsification. Fragmentation and Coalescence. The process of emulsification can then be represented by material being transferred from a large-size category to a smaller. In so doing, the number of particles is greater, but the model tracks the mass and calculates only the particle numbers from the mass and size in each category. The converse coalescence process can be represented as particles getting together to make bigger particles. In the model, this appears as a process whereby mass is moved from one or more smaller size categories into a larger-size category. The model is a set of first-order, ordinary differential equations, one for the mass in each size category. It is an initial value problem with time as the independentvariable and is therefore amenable to solution by numerical integration. Given this framework, the behavior of the system is determined by the rules for transferring mass from one size category to another. Consider first the breakdown to smaller sizes. When a large particle fragments, it can break down into several smaller particles (cf. above, assumptions 1 and 3). The simplest model is one in which it is assumed to always split into two equal-sized parts. Thus, if the rate of splitting is 1 kg/s, then that is both the rate of increase
h
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1
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0.2
V 0 1
7
4
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Particle Size Number (Log Volume)
Figure 1. Cumulative probability P of finding a particle in a region of the vessel where V, > Vp, Le., where the shear forces are sufficient to fragment the particle, as a function of q (eq 7).
V
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--+
Figure 2. Course of emulsification for q = 0, R = 0, pseudoBernoulli case.
of the mass in the smaller size category and the rate of reduction of the larger. The number of particles in the system increases because twice as many smaller particles appear. Likewise,the simplest model of the coalescence process allowstwo particles of the same size to join to form a single larger droplet. This appears as a reduction of the mass in the smaller size category and a growth in the larger. It is a limitation of this type of model that only certain combinations of particle sizes can be allowed to combine and still make recognizably sized larger particles. Nevertheless, migration up the size scale is'possible, and given a sufficiently small size ratio between categories, it seems to work. We postulate that the fragmentation rate for particles of a particular size is determined by the probability that a particle of that size will find a region of the vessel where the shearing action is sufficient to cleave it. We further assume that this probability is the product of the mass fraction (represented by their size category in the whole contents of the vessel)and the cumulative volumetric shear stress distribution calculated by integrating down from maximum shear to the level that will just split the size particle in question.
The Process of Emulsification
Langmuir, Vol. 7, No. 7, 1991 1327 loo
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a8
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aG
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Figure 4. Course of emulsification for q = 0, Yi = 1.0, the true Bernoulli case. 100 80
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\ 6
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+ = 2.0, R = 0. Note
Size Category
Figure 6. Course of emulsification for q overemulsification.
5
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Figure 6. overemulsificationand shift of maximum to higher size category.
s
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--+ Course of emulsification for q = 2.0,R = 0.5. Note
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Size Category
m
1
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11 I',.,\ \
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*
Figure 3. Course of emulsification for q = 0, 93 = 0.5, pseudoBernoulli case.
x0
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: E
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We allow for the possibility that particles may fragment into several, Le., more than two, smaller particles, and that some of these will be significantly less than half the
60
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Figure 7. Course of emulsification for q = 2.0, R = 1.0. Note overemulsification. Distribution is unimodal. size of the fragmenting particle. In fact, this assumption is crucial to the explanation of overemulsification (cf. above, assumption 3). We further postulate that the coalescence rate depends on the probability that two (or more) particles of appropriate size will meet. This probability is assumed to be defined by the product of their respective mass fractions in the whole contents of the vessel. This necessarily makes the model nonlinear and is therefore a major determinant of the choice of numerical integration procedure and, hence, of restricting the number of size categories. In addition to the probabilities, there are rate constants to be considered. We do not know the nature of the cumulative shear distribution, nor do we really know what are the minimal shear levels for particles of a particular size. However, we can construct mathematical relationships to reflect what we understand of the underlying physical processes and then explore the extent to which choices of rate constants, assumed distribution shapes, and allowablefragmentation patterns influence the overall behavior of the model emulsification apparatus. In order to relate the critical shear required for fragmentation to the size of a given particle, we consider the particle to be sheared by forces proportional to shear stress and the particle area and resisting fragmentation
Becher and McCann
1328 Langmuir, Vol. 7, No. 7, 1991 100
.-2
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Figure 8. Course of emulsification for q = 6.0, R = 0.0. Note overemulsification, with considerable broadening of the distribution.
i2
60
d
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,
8
9
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+
Figure 9. Course of emulsification for q = 6.0,R = 0.50. Note persistence of bimodal distribution. through surface tension forces proportional to the particle circumference. As the relative masses change, so also does the level of shear which balances the surface tension. Hence, it is possible to assign a critical shear to any given particle size. Our model assumes that the regions of high shear occupy a small fraction of the total volume, since the edges of stirrer blades and the high shear regions between blades and vessel walls are but a very small fraction of the total volume. The model includes a function that permits us to adjust the shape of this distribution in order to determine its effect. For convenience, the numbers have been normalized so that the highest shear is the level that will just allow the smallest particles to be generated by splitting a larger particle, but not allow them to be further divided. This prevents useless computation of masses of nonexistent particles. Whenever a particle (or more accurately in terms of this model, a mass of material) fragments, we have it split both into particles of the next smaller and some of the very smallest particles (cf. ref 8). The model has been kept simple by looking at only two categories of resultant particles. The investigation of the relative proportions of
2 3
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8
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Figure 10. Course of emulsification for q = 6.0,R = 1.0. Note overemulsification. Distribution is unimodal,broad, and showing no smaller particle categories. these two daughter categories is another aspect of our investigation. For overemulsification to be observed, there must some period during which there is a greater abundance of smaller particles than in the final equilibrium distribution. However, if particles split only into the next smaller size, the distribution remains unimodal, and the peak migrates slowly from large particles to a final steady state without ever passing beyond it. The behavior is simply that predicted by the Bernoulli equation (eq 1). To get the temporary superabundance of smaller particles, we assume that they appear by some other route, which bypasses (in some sense) the average size, hence the reason for assumption 3, that is, allowing large particles to break up into some very small ones as well as some the next size down. Let us now consider a small spherical particle subject to splitting because it is in a region where avelocity gradient exists. It is supposed that the particle can be fragmented if the splitting forces are greater than the binding forces. Thus for any given particle size there is a critical velocity gradient level required to tear it apart. The critical velocity is estimated by balancing the shear forceagainst the surface tension force. If the velocitygradient is Vg(&),the surface tension is y (N m-l), the dynamic viscosity is u (Pas), and the particle radius is R (m), we may assume the shearing force to be
Vg X
uX
?rR2
and the surface tension force to be y X 2uR
Equating the two forces
R = 2y/ V,a (3) We now consider the consequences of its fragmentation for each size of particle. The simplest model would, of course, be the Bernoulli model, in which each particle is assumed to break into particles of a smaller size, thus yieldingtwo half-size particles. For the coalescence process the simple model is one in which the reverse effect applies. To relate these processes to the particle size distribution existing in the model a t any given time, the rate of fragmentation for particles in the size range i is
Langmuir, Vol. 7, No. 7, 1991 1329
The Process of Emulsification
k; x m [ i ] x P [ i ]
(4)
where k/ is a rate constant, m [ i ] is the mass fraction in the size category i, and P [ i ]is the probability that a particle in that size category is in a region of high enough velocity gradient to be fragmented? In order to describe coalescence, the rate of merging of particles of any given size is given by
k,'
X
m [ i ]X m [ i ]
(5)
where k,' is the rate constant for coalescence. As we have pointed out above in connection with eq 1, the Bernoulli model is limited. The need to adjust the coefficients kfand k , of eq 1to represent the changes due to shifting particle size is a consequence of that model being a low-order, aggregate representation. In our model, the overall coefficients, were they to be calculated explicitly, would be seen to change continuously in response to the shifting size distribution. Our coefficients relate to the individual particles (droplets). The probabilities of drops splitting are dealt with explicitly by relating the size of the drops to the available shearing forces. The effect, in the aggregate, is then felt by the different behavior of different sizes of droplets in proportion to their populations and to the probability that there will be some region in the vessel where the shearing forces are big enough to tear them apart. Coalescence is dealt with by assessing the probability that the droplets will come together based on their populations by size category within the whole well-stirred vessel. Thus, the coefficients in eqs 4 and 5, arenot thesameas thoseofeq l,exceptinaformalmanner, and are distinguished by the prime ('). At this time, the dependence of the rate constant k, on the stirring, Brownian motion, the shape or size of the vessel, or other sources of interparticle interaction have not been investigated. In order to extend the model to allow for variation in stirring (down to zero agitation), this will be necessary. This last relation may be interpreted as stating that the coalescence process takes place between like-sized particles and that the rate of coalescence is determined by the probability that two similar particles wilI find themselves in proximity. On the assumption that, in a well-stirred vessel, the probability that particle of size i is at a particular place is proportional to the mass m [ i ] of such particles; thus, the joint probability that two particles will be in "proximity" is seen to be proportional to the square of the mass. Shear Distribution Equation. If V , is the greatest velocity gradient to be found in the stirred vessel, the corresponding radius R m is the least radius for a particle that can be split, or, alternatively, the maximum radius of a particle that cannot be fragmented. The same balance of shearing and surface (i.e., forces opposing splitting) forces is used in defining the capillary number.lOJ1 This critical V , might correspond in reality to the smallest clearance between moving parts and their corresponding relative velocities in that region. Given the dimensions of the vessel and its stirrers and their speeds of stirring, the critical particle size R, is, for our analysis, presumed to be defined. There will be particles smaller than R m in the vessel owing to the splitting of larger ones, (9) The product m[i] X P[i] is,of course, the compound probability not only that there is a region in which particle may be fragmented but also that there is a particle in that region. (10) vandenVen,T. G. M. Colloida1Hydrodynamics;AcademicPrees: New York, 1989; p 189 et seq. (11) Probstein, R. F. Physicochemical Hydrodynumics: An Introduction; Butterworths: Boston, MA, 198%p 279.
but those below R m in size may not split further. They may, however, coalesce into larger particles. With this rationale, the design of the stirred vessel and the speed of stirring in the vessel will determinethe amount of the vessel volume that is capable of splitting the particles of any given size. There is a distribution of velocity gradient in the vessel ranging from the highest V to the lowest or zero. We now consider the vessel as f & n g a volumetric velocity gradient distribution. The issue for any given particle is the probability that it will be found in a region where the velocity gradient is sufficient to split it, i.e., where the velocity gradient is greater than the particle's own critical V,. For the case of the stirred reactor, it can be assumed that the vessel is divided (in a manner defined by the stirring patterns) into regions of varying shear gradients. These regions can be defined as turbulent or nonturbulent if one wishes, but this is not necessary for the operation of the model. The probability P of finding a region with a velocity gradient greater than any particular value varies ,. In from unity (certainty) at V, = 0 to zero at V , = V such a function the steepness of the decline between the extreme values will obviously depend on the design of the reactor and, since this may be important to the emulsification process, the model provides for adjustment of the shape of this cumulative probability curve. We know of no data on such a probability distribution, but the expression must be primarily a function of the design and operation of the mixing vessel. For example, it would be expected that where spinning blades come close to static components or to counterrotating blades, shear forces will be high, but these regions will represent only a small fraction of the total volume of fluid. An expression of the proper sort would have the following properties: It can be adjusted to shift the balance between high and low shear force regions (q in eq 6). It can be adjusted to have a maximum achieved shear force by changing the maximum velocity gradient (V, in eq 6). It is in cumulative probability form so that the probability of the shear force exceeding a certain level is given directly. It must be easy to compute. A simple way to generate such a curve is to assume a form of the volumetric shear force distribution, such as
P = (1- v,/vP,y where q is any positive exponent. As q becomes larger, the decline becomes more rapid, and, hence, the regions where splitting of the smaller particles is possible become smaller, and thus P can be read as the probability that the shear force will exceed that represented by the velocity gradient V,. Needless to say, the form of this distribution is not sacrosanct, and it may be replaced by better data if such are available. We use it simply to explore the effects of these design factors on overemulsification. The radius R m corresponding to V , is taken to be that of the smallest size particles in the model, Le., those which are not fragmentable anywhere in the system, but which themselves arise from the fragmentationof larger particles. To be absolutely precise, R m could be taken as lying between the smallest particles and the next smallest, but this adds nothing to our understanding and introduces extra coefficients. We define the critical velocity gradient V,[i] for the size category i as equal to K / R [ i ] ,where the constant K involves such properties of the system as the densities
1330 Langmuir, Vol. 7, No. 7,1991 and interfacial tension of the liquids involved,etc. Because of the way in which the system is normalized, these quantities do not have to be explicitly stated. We now may express the probability factor (eq 6) in terms of the individual size categories as
P = (1- V,[i]/v,[l])q (7) where the critical velocity gradients are normalized with respect to that corresponding to the smallest (nonsplittable) size, i.e., i = 1. The probability distribution of eq 7 (for a number of values of q ) is shown in Figure 1,where it is plotted against i, i.e., the particle size number. This is, in effect, a logarithmic scale appropriate to the wide range of sizes considered in this simple version of the model. All the curves of Figure 1have the property that as size increases, the probability of there being a shear region with shear forces large enough to split the droplets increases. Eventually, all the curves come to unit probability. Increasing values of the exponent q have the effect of altering the shape of the distribution so that less of the volume of the vessel is able to shear the smaller droplets. In other words, the probability of splitting a particle in one of the larger size categories decreases with increasing q. Thus, for high q the probability of a small particle (low i) being in a region where it can split is vanishingly small. Hence, the small particles will either simply do nothing or they will coalesce to larger sizes, i.e., they will be subject to overemulsification. Normalization. This choice is the link between the “real” emulsifying apparatus and our model system. The particle size distribution in the model is normalized by the choice of this radius. If the stirring speed in the real apparatus changes, or some design adjustment is made to the clearances, for example, then R m will also change, and the particle size distribution in the model will have to be rescaled so that the smallest size corresponds to the new conditions. An alternative approach would be to allow the model to have a large range of permitted sizes, but only use those above R,. Since the purpose of this study was to determine the type of behavior capable of explaining overemulsification, we have chosen the normalization approach. Changing the speed will not be reflected in the model itself but only in the interpretation of the particle size distribution. Unfortunately, this makes this version of the model incapable of demonstrating the effects of transients in stirring speed. Since the critical V, for any size of particle is inversely proportional to its radius, the probability P of being in a region where the velocity gradient is greater than V, may be related to the radius R. To keep matters simple, the set of particle sizes will be indexed by an integer i, with the values of i increasing with particle size, Le., i = 1corresponding to the smallest size, and so on. Split Ratio. For the model to show the oueremulsification effect, the consequences of allowing the fragmentation to take two routes must be investigated. One way is to allocate the fragmented mass to the next size down, as above. The other route is to allocate the fragmented mass to a much smaller size. Since the model is normalized, this is taken to be the smallest allowable size (i = 1). A fraction R of the fragmented maas is assigned to the next smaller size; the remainder (1- R ) is assigned to the smallest size. The Equations for Emulsification The rates of the fragmentation and coalescence reactions for each size category may be written (with reference to
Becher and McCann P
0 2 6
Table I. Test of Emulsification Model R = 0.00 R = 0.50 R = 1.00 2 3 4 5 6 I 8 9 10
eqs 5 and 6) as
f [ i ]= kd
X
(8)
P[i] X m[i]
c[i] = k,’
X
m[iI2
(9)
where f[i] and c[i] are the fragmentation and coalescence rates, respectively,for size category i, P[i] is the probability defined by eq 7, and m[i]is the mass (volume) concentration in size category i, while ki and k,‘ are the rate constants defined earlier. The overallrate of change in concentration for all particle size categories from i = 0 to i = n is then given by dm[l]/dt = f[2]
+
c(1-R) n
X
fu] - c[i]
(10)
j=3
fori = 2 t o n - 1 dm[i]/dt = ~ [- 1 i1 - ~ [ + i ]R
X
f[i + 11-f[i] (11)
dm[n]/dt = c[i- 11 - f[n] (12) where t is the time and R is the split ratio defined earlier. R may take on values between zero and 1; the larger the value of R,the smaller the fraction of smaller particles.
Test of the Model The model may be tested initially by determining if it gives results in accord with practical experience. In order to test the model, eqs 10-12 must be integrated. As suggested earlier, a satisfactory value for n is 10 (Le., allowingfor 10 size categories in the distribution), and for the values of q and R shown in Table I, where the numbers refer to the corresponding figures. Numerical integration by the Runge-Kutta method was used, and the results are shown in the three-dimensional histograms of Figures 2-10. In these diagrams, the time and size category axes are in arbitrary units, while the mass percent axis represents the percentage of the total mass of material in each size category; thus at t = 0,100 5% of the mass is in size category 10. In Figures 2-4 q = 0, hence P = 1, i.e., the probability that a particle is in a place in the vessel where it may be sheared is unity. This obviously corresponds to a very well-stirred vessel. The classical Bernoulli curve is exhibited by Figure 4 (q = 0, R = 1.0). It will be noted that the distribution is invariably unimodal, but sharpens with time. Figures 2 and 3 exhibit what might be called “pseudoBernoulli” behavior, in that the mass percent increases in which might pass for a logarithmic manner. Note, however, that the process is initially bimodal, and that the distribution is sharper than that demonstrated by the Bernoulli case. Figures 5-7 and 8-10 demonstrate the effect of increasing q, i.e., decreasing the probability that a particle will be in a region where it may be sheared. The most obvious result is that the phenomenon of overemulsification is now seen to be occurring, as evidenced by the upward shift in the distribution maximum after a short time. The effect of increasing 59 is to broaden the distribution and increase
The Process of Emulsification
the average particle size. Note also that for R = 1.0,the distributions remain unimodal throughout the process. The following conclusions may be drawn from study of Figures 2-9: For q = 0 (where a drop may be split anywhere in the stirred vessel) the Bernoulli curve results when R = 0; at lower values of R,a pseudo-Bernoulli curve is observed. Decreasing the probability of a particle finding itself in a region of shear sufficiently great to permit particle splitting (increased q ) increases overemulsification and broadens the final distribution. Decreasing the split ratio R,i.e., increasingthe fraction of small droplets formed on particle rupture, favors the formation of smaller particle sizes but broadens the final distribution, as well as increasing the time to equilibrium. It is probable that variations in q and R (hence, in P) are not totally independent in a given stirring environment. However, it may be assumed that the probability of being in a high shear region is a function of the stirring mode (propeller shape and placement, rpm, etc.), as well as the presence or absence of turbulence. The split ratio will, of course, be more strongly influenced by the surface and other physical properties of the 1iquids.l0J1 One final word may be required to clarify our approach.
Langmuir, Vol. 7, No. 7, 1991 1331
In our model, for the purposes of dealing with overemulsification (which, it must be said, was the motivation for the study), we have normalized the particle sizes to be multiples of smallest, not-quite-splittable, droplet size. As a result, the computational load is reduced and fewer parameters are needed. For a real case,the range of droplet sizes would be extended (and not normalized), and the emulsificationmodel allowed to generate such droplet sizes as were from time to time appropriate to the prevailing conditions. Thus, for example, a period of running at high mixer speed would create a population of small droplets. Slowing down would allow them to coalescebut not permit generation of any new droplets. Finally, we expect that, without stirring, the representation of the coalescence coefficient would require modification, since the wellmixed assumption can no longer be relied upon. Clearly,however, the model gives results that correspond to emulsificationsituations encountered in practice. Study of real systems in terms of the factors outlined here should ultimately enable us to calculate q and R for a given set of emulsification conditions and thus to predict the shape of the droplet-size distribution curve and the presence or absence of overemulsification.
