Ind. Eng. Chern. Fundarn. 1986, 25, 561-565
56 1
Analysis of Reaction Processes in Which Microscopic Heterogeneities Appear: Scale-up and Scale-Down of Polymerization Reactions Wllllam E. Ranr Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455
Present-day models of chemical reactions tend to be as specific and complex as the molecules and structures which have been produced by chemical processing. Only venturesome, imaginative approaches to analysis of phenomena on a fine scale of detail will help us in the future to closer understanding of things that industry now knows how to do, or wants to do, empirically. A dearth of generalizations for future scale-up, scale-down, engineering design, and even deductions or proofs of scientific details can be lessened by considering a full range of scales of time and space in the physicochemical model. Two examples of this new viewpoint for analysis, with some generality for applications, are discussed here: formation of smoke and of polymer surface zones between reactants generated by mechanical mixing, externally forced or internally imposed by chemical reactions. The latter models can b e implemented in a "microscopic pilot plant", scaled down from the bench scale of a chemical laboratory.
Introduction Analysis and design of chemical processes classified as heterogeneous and occurring in media with time-varying microscopic structure are much more than a sequence of single-reaction models with rate-limiting steps in continuous time. Participating molecules are spatially segregated in two and three dimensions. Molecular events must be deduced by the evidence of experimental observations at larger space scales and a t longer time scales than those characterizing each event. 0. A. Hougen and K. M. Watson (1947) in Part I11 of Chemical Process Principles first introduced systematic models of physicochemical sequences at various scales of space and time to understand complex heterogeneous reactions of the chemical industry. Modern stochastic experimentation, successful in deducing the atomic details of so complex a molecule as DNA by failure of preconceived models to predict observed X-ray patterns, was represented in this book by determining experimentally the values of molecular scale model parameters which cannot be negative in value. A sufficient version of a necessary model for such deductions depends on its physicochemical validity, not on its mathematical charm and manipulative ability or accidental fit of data. To avoid real-world errors, a general resolution of a complex model must be made at various space and time scales. Portions of the model, which can be verified by experiment, must be separately identified. The model must also be of a form suitable for scale-up and scale-down from the testable portions; that is, it must be a model for design of pilot plants. As an example of such analysis and design, models for studying polymer reactions in a microscopic pilot plant (10-6-10-3 m, 10°-lo" s) are developed here. The class of reactions under consideration are those which do not occur in a solvent, as in reaction injection molding, where relatively fast reactions occur in two-dimensional reaction zones generated by laminar mixing, and interfacial polymerization, where prepolymers or catalysts cannot be dissolved in the same solvent. A sufficient model could also represent, with zero reaction rates, spontaneous formation of emulsions and the kinetics of micelle formation. A microscopic pilot plant permits observation of reaction structures and actions motivated by reaction in the reactor. 0196-43 13/86/ 1025-056 1$01.50/0
It reduces amounts of specialized prepolymers which must be synthesized for study of the effect of different molecular sequences in reactant and product molecules. Along with advances in microanalysis, the microscopic pilot plant can be proposed as a general technique for chemical engineering research and development in cases of heterogeneous reaction. It should have as many applications for engineering as the environmental stage of an electron microscope has had for surface physics and chemistry and be as practical as were the differential catalyst beds of Chemical Process Principles. Since the polymerization models proposed here add only fluid mechanical actions and boundary generation caused by molecular locking of long molecules into fluid, flexible, and elastic structures after snakelike diffusion to reaction zone surfaces, classification of complex reactions is a useful introduction. Recognition of structure and action sequences at a microscopic scale (as well as a microorganism scale) has become as detailed as actions in the solar system (as well as the universe). Chemical engineers must somehow organize and generalize the fascinating detail and not get lost in a scientific forest filled with amateur hunters who will shoot at anything that moves. A descriptive classification of models of practical chemical reaction rates, more or less in the order of their historical complexity, follows.
Reactions in an Apparently Homogeneous Phase Bimolecular encounters would occur in a homogeneous phase, especially in a gas. The frequency of encounters depends on molecular concentration and molecular velocity. Reaction rate is further quantified by the probability of a sufficient kinetic energy associated with the collision event. Macroscopic effects can be observed by concentration and temperature causes. Small time scale effects become intricate with more than two reactants and with different rates for subsequent and parallel reaction steps which reactions can provide. In liquids, a similar picture is seen but with diffusive motion in a dense phase to a target molecule, which reacts on any contact or in a probabilistic manner for sufficient energy of the collision and/or for molecular positions needed for coition. Fast reactions in liquid solvents are 0 1986 American Chemical Society
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so modeled. The model also applies to very fast dehy-
drogenation of aromatic hydrocarbon gases in a heterogeneous formation of spherical smoke pellets which subsequently agglomerate by Brownian diffusion according to a mathematirally similar model.
Heterogeneous Reactions The need for multidimensional as well as varying space scales arises immediately when multistep chemical reactions involve more than one phase and when one of the phases also participates in the overall reaction. Between steps, molecules may have to move from one place to another. Reaction steps may occur on a two-dimensional surface at supermolecular scale. Reaction rate models, where a catalyst specifies a critical step in a possible reaction sequence, have become even more complex than those imagined in Chemical Process Principles. Adsorbed molecules diffuse and react in a two-dimensional patterned surface. Desorbed molecules react in a three-dimensional contact gas or liquid phase. Supported catalyst patches change their dispersed geometry in time. Such surfaces are usually buried in porous pellets of relatively large size. Carbon deposits and unwanted squatter molecules may blanket active surfaces. Useful reaction rate formulas cannot be constructed until all of the steps are determined for the model and their rates and geometry are quantified. Since all of these details cannot be observed directly and separately, they must be deduced from model experiments. Engineering analysis methods are exercised in model experiments a t low pressures in gases and with million-dollar instruments that read the spectroscopic conversation of molecules as they undergo reactions in microscopic supermolecular space scales. Time is resolved by the time scales of the instruments. Polymerization Reactions: Scale-up of Time and Space Polymerization chemistry, as applied by industry, is practiced now far beyond the ability of science to model processes in the total detail scientists desire and beyond the ability of supercomputers to cope with such models, were they available and were they properly resolved in space and time. Polymerization chemistry, as applied by nature, has always been practiced so far beyond the understanding of science or industry that one lets it happen naturally for fear the gods who are said to be in charge of such matters may be angered. Nevertheless, in industry one is quite willing to use whatever one can produce by the wiles of nature and by the trials and errors of experimental chemistry. Polymerization in solvents has been resolved by processes of initiation, chain growth, and chain termination with diffusional contact of reactants in a homogeneous phase. These simultaneous processes can be treated competitively by probabilistic development of a molecular weight distribution in a large enough differential volume. Cross-linking and stoichiometry of monomers add further complications to the models and further variations in the polymers which can be produced. With more concentrated reactants and with less solvent, the time scale of the reaction decreases and reaction sites compete with other reaction sites nearby. Mixing and interdiffusion of reactants, solvent, and catalyst to a uniform initial condition for the model are not achieved, and heterogeneous reaction structures a t larger space scales exist from the time of initial contact and start of reaction. In emulsion polymerization, where the primary reactor is a dispersion of polymer spheres which grow to visibility inside an organic phase solubilized in an aqueous phase
(Ray, 1974), engineers lost much of their innocence about polymer reaction rates and how they should be used in scale-up and process design. We discuss here additional complications of newer practices where the models for reaction rates and polymer properties must consider solventless reactions, mechanical mixing, reactions in twodimensional zones of contact, reaction-generated mixing a t microscopic scale, and heterogeneous separation of product phases (as surf products or precipitated structures). Processes which one eventually hopes to model may be listed as follows: (1)reaction injection molding where liquid reactants become solid structures in minutes, (2) mixed liquid adhesives where viscous liquids become strong adhesives after hours of cure, and (3) interfacial polymerizations where prepolymer units are not soluble in the same solvent and polymer chains grow anchored to an internal interfacial area which can be increased by shear mixing.
Smoke Formation: A Model of Sequential Time Scales Carbon black is composed of round carbon pellets 10-100 nm in diameter. The distribution of sizes is remarkably narrow, and the primary particles are seen by an electron microscope only as agglomerates. A recent report of industrial research (Harris, 1984) supports the following model developed in 1979. Consider the possibility of a very rapid heterogeneous dehydrogenation of some intermediate (perhaps C2H2)on growing carbon black particles kept spherical by surface tension. These particles begin on some sort of nucleus (perhaps a radical) and continue growth until the intermediate is depleted in the gas phase. This initial process occurs in a time so short, 10-7-10-6 s, that the primary particles finish growth before they collide with another particle and start agglomerating. One speculates that the rate process can be simplified to diffusion of the intermediate from a relatively large spherical reservoir associated with each nucleus to zero concentration a t the particle surface. Here a very fast reaction allows no significant concentration of the intermediate in the gas phase at the surface. The instantaneous rate is taken to be the steady-state rate for diffusion to a sphere from infinite surroundings. However, the infinite surroundings are made finite with regard to the concentration of the intermediate in the reservoir. At some time, t , after an initial appearance of n nucleii per unit volume and the formation of a particle of diameter D, on each nucleus, the mole fraction, x, of diffusing species in the reservoir can be approximated by where f is moles of carbon black particle formed per mole of diffusing species reacted cf = 2 if reaction is C2H, = 2C + H2),c = p / R T , the mole concentration in the gase phase (c = 8.1 X mol/cm3 a t 1500 K, 1atm), xo is the mole fraction of diffusing species in reservoir a t time t = 0, pp is the density of carbon black particle material (specific gravity is approximately 2.0), and Mp is the molecular weight of carbon black material (M,, = 12 g/g-mol i f f = 2, a C2 compound is the diffusing component, and the particle is nearly all C). The quantity expressing our basic ignorance is n, which is assumed to be time independent. Furthermore, its value is not independently specified. We are forced to deduce its value from observations of final particle size. It is convenient to define for x = 0 at t =
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
where Dp,max is the maximum possible particle diameter for n concentration of nucleii. We obtain a value of n by substituting observed sizes in this definition. For 10-nm particles, n would have to be of order 10'3-1014nucleii/cm3; for 100-nm particles, n would have to be of order 101o-lO1l nucleii/cm3. The rate of particle growth is given by d z ( ~ p n D p ~ / 6 M p=) fNr=Dp/z"Dp2 (3) is the mole flux of diffusing species to the where NrEDPl2 growing particle surface. For spherical symmetry and pseudosteady diffusion from an infinite reservoir Nr=Dp/zDp/C(X- Xr=D,/z)D = 2
(4)
where x , = ~ , , ~= 0 is the mole fraction of diffusing species in the gas phase a t the particle surface and D is the diffusivity of diffusing species (the diffusivity of CzHzin air or nitrogen is about 1.6 cm'/s at 1500 K, 1 atm). A differential equation for D, vs. t can now be constructed. The solution for an initial condition, D, = 0, takes the form
s,""""""~ dr/(l
- r3)=
t/tc
(5)
such that D, is about 95% of DpSmax when t = t,, a characteristic time. The characteristic time, t,, for diffusive growth can be specified in two ways:
t , E (9p,/16n2D3Mpfcxon2)1~3 if n is known is observed = ppDp,maxz/4DMpfC~0 if Dp,max The time t , for a 10-nm particle is of order 10-8-10-7 s; for a 100-nm particle, 104-10-5 s. Thus, the proposed mechanism gives sufficiently short times to be a possible mechanism. The time scale for subsequent agglomeration (development of carbon black structure) is much larger by a factor of lo3. The very rapid rates of 104-10-5 s for the diffusive growth of primary particles suggest that time scales of Brownian agglomeration should be treated with a similar model. While growing, a nucleated particle wanders in time t a distance r given by d(r2)/dt = 2DB
(6)
563
particles in times of longer scale. While this second process is a series of encounters, it can still be averaged by the same diffusional analysis as that used for primary particle growth. Now the concentration in question is number of agglomerates per unit volume, the diffusivity is a Brownian diffusivity of the agglomerates, and the "diameter" of the agglomerate is related to the number of primary particles collected in the agglomerate. Let n, be the number of agglomerates per unit volume in the suspension at time t,, where n, is n when t , is zero. At time t, the average agglomerate will have n/n, primary units and, if the agglomerate is globular, will have a diameter for Brownian motion of order (n/n,)1'3Dp,max and a diameter for collection on encounter of twice this value. At an instant of time, considering one agglomerate as collector and doubling the diffusivity to account for equal kinetic movement of the collector, the equations for spherically symmetrical diffusion given for the n, vs. t , equation d
dt,(nlns)
=
rate of collection of primary units by o n e target agglomerate
collecting area of target agglomerate
no. flux of agglomerates t o collecting agglomerate (Nsh = 2)
no. of primary particles per agglomerate
where D, is of order (n/ns)1/3Dp,max. Note that D, cancels from the equation and that the configuration of the agglomerate will probably have only a minor effect on the rate of agglomeration. The agglomeration rate equation reduces to d(n;')/dt,
= 8g,~T/3p
(8)
Integration with initial condition n, = n when t, = 0 gives
S
n / n , = 1 + (t,/tsc)
(9)
where S is the average number of primary particles per agglomerate, a quantitative measure of structure, and t,, = 3p/8g,~Tnis the coagulation time, a characteristic time for the agglomeration process. This is the famous Smoluchowski equation for Brownian coagulation. For a carbon black n of 10" ~ r r -t,, ~ ,is of order s. Thus, S is of order of 10 particles per agglomerate after about s of hot flow after black formation and before quench. These times scales make good sense when compared with the time scales of the operating process.
where DB = g,~T/3npD, is the Brownian diffusivity, g, is a dimensional constant (g, = 1 (g/dyn)(cm/s2)), K is the Boltzmann constant (K = 1.38 x dyn cm/K), and p is the viscosity of gas ( p = 5.6 X g/(cm s) for air at 1500 K). DB is at a maximum value of the order of D when Polymerization Reaction Rates i n t h e t = 0 and decreases to near the final value, g , ~ T / 3 n p D ~ , ~ ~Rapid , Absence of Solvents when t It,. If D, had the maximum value, D, during the A logical extension of models for fast reaction in liquids time, t,, r would be equal to (2Dt,)1/2. Thus, r has to be to fast polymerization reactions is to assume that a viscous less than 10-3-10-4 cm while the primary particle is growing polymer layer forms on contact between a liquid preby diffusion. If this maximum possible distance of polymer A and its companion prepolymer B plus catalyst. movement during growth is compared with the average Rate of reaction is controlled by the rate of diffusion of distance between nucleated particles, which is of order A through a growing pdymer layer. Because the inter(lo-" ~ m ~ / n u c l e u s )=' / ~ cm, then it is clear that the material area per unit volume, a,, increases by fluid meproposed mechanism can explain formation of primary chanical mixing, the layer thickens by reaction while it particles before agglomeration begins. thins by stretching (Ranz, 1979). The physical picture of the proposed mechanism should Following a microscopic model in Lagrangian frame and be kept in mind. The growing particle is rapidly slowed assuming pseudosteady diffusion of A, the thickness, s,(t), in its kinetic motion while many speedy intermediate gas of the polymer lamina at time t for a,(t) is given by molecules encounter it, react, and deposit C. The averaged process is described by the diffusion analysis. However, d(app) avDApc.4 -the complete primary particle continues its Brownian (10) dt PPSP motion and eventually joins others to form aggregates of
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where DApcAis the diffusivity of A through polymer P multiplied by the appropriate partial density of A in polymer P at the A-P phase interfaces and where pp is polymer density. This rate equation, in terms of a fluid mechanical stretching parameter, a ( t ) = -d(ln a,)/dt = d(ln s)/dt, where s is the striation thickness of an unreacted prepolymer A “ghost” lamina, becomes d(s,2)/dt - 201sP2= 2DApcA/pp
(11)
An initial condition is sp = 0 and a, = 0, whan A and B contact at t = 0. If a = 0, that is, if diffusion has a very much longer time scale than mixing, sp = (2DAp~At/pp)1/2 Imaximum size of sp permitted by the ghost, s, of the limiting reactant. If CY = -C, where C is some apparent average in time scale t sp2 =
(DApCA/CPP)(l- exp(-2Ct))
(1
-)
+ So2CP,
= &In DA~CA
(-)
SO2CPP DA~CA
B
Initial polymer layer grows in thickness.
Drop forms appear and move into phase-B
Volcanic ruptures spray drops into phase-B
(12)
At the end of reaction, when sp(t)= so(pp/p,) exp(-Ct) and t = tf
t f = -In 1 2c
A
Tubule formation; stalk and bulb growth.
(13)
showing in such a case the simultaneous effect of mechanical mixing. In reaction injection molding, RIM, polymerization begins with impact jet mixing in a small prechamber, motionless mixing in a runner, and laminar mixing during mold filling. The mold-filling time is of the order of a few seconds. The total reaction time at demolding is a matter of minutes. Back-calculations from measurable and otherwise deducible dependent quantities, say tf of order lo2 s and snakelike diffusivities of order 10-lo-lO-ll m2/s, indicate the need for an s,, of the order loo pm. The experimental evidence and theoretical prediction evidence are that an so of order lo2 pm is obtained by mechanical mixing in RIM practice (Kolodziej, 1982). The model obviously fails for urethane systems which are molded successfully. It may explain, however, why kinetically faster chemical systems show evidence of insufficient mixing in the same practice. Logic suggests that, in the microscopic workings of RIM polymerizations, certain chemical systems during reaction, by mechanical chemical action, increase the apparent a, by factors of 10’-102. Some do not. The phenomenon suspected is the same as that which occurs in spontaneous emulsification and can also occur in interfacial polymerization.
Mechanical Chemical Actions Which Generate Intermaterial Area Microscopic observations of the initial contact surfaces between reactant liquids and reactant liquid solutions (Extrand, 1985; Wickert, 1985) show many loo- and 101-pm structures which are formed and outlined by encapsulating product layers or surf products. The structures formed by spontaneous emulsification, micellation, and single molecular membranes or coatings are probably the result of the same kinds of directed forces and physicochemical actions. In surf-product cases, the action can be more vigorous than in surfactant cases where the necessary third component is initially dissolved in one of the two phases. In surf-product cases, catalysts and initiators can be primarily actors in the additional formation of a,. Considering the variety of chemical systems, despite contraints of three-dimensional geometry, the sufficient model expands immediately in complexity of a,(t) even for a specific
Figure 1. Intermaterial polymer structures forming a t initial contact surfaces between reactant liquids and reactant liquid solutions.
case where experimental observations can be made of some of the details at small time and space scales. Some of the structures seen are diagramed in Figure 1: The vertical line represents a product layer that merely grows in thickness, as described in the simple model developed above. Even if the polymer was ultimately soluble in one or both of the reactant phases, a three-phase system has been formed for short times. The line can be observed to thin by mechanical stretching (if it has the properties of a liquid) or, perhaps to fracture like a sheet of ice (if it has the properties of a solid). Droplets (1-5 km in diameter) moving away from the line into one of the two reactant phases, somewhat like a random rain in one direction, are moving at a decelerating rate and will stop farther out in a pileup of polymer capsules. The formation of each drop is so fast and the drop is shot at such a high velocity into the receiving phase that it does not appear to the eye until the polymer layer around the drop has grown thick enough to be visible and the drop has slowed down enough to be followed. Stopping-distance models for Stokes resistance to motion extrapolated back to the original intermaterial interface can be used to deduce the initial velocity and the surface forces acting impulsively. This must occur at some defect in the original polymer layer, perhaps an impurity, a solvent molecule, or a catalyst in the initial polymer layer. A spray of droplets in slow motion may issue from a volcanic rupture in the polymer layer, again in one direction with the drop forms piling up against some stopping distance. In the case of RIM chemical systems the stopping distance and the distance between volcanoes appear to be fortuitously of order lo2 pM = so. In the microscopic observations, the ejection activity slows with time, probably because the original polymer layer continues to thicken. However, a bit of shear restores the dispersion action. There is evidence of tubes being formed and pushed into the second phase. One elaborate structure seen was a bulb or large drop on the end of a tube which continued to feed the bulb reactant and propel it into the second phase. Meanwhile, the bulb was surrounding itself with a cloud of smaller droplets, each of which may have had a tube
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feeding it until closed by polymer growth or broken by drop inertia. Tubular structures are also observed in spontaneous emulsification and micelle formation. It is apparent that the first diffusion model for rapid polymerization rates and mechanical mixing is not sufficient when the limitations to mechanical mixing can be circumvented by physicochemical means. Further, the physicochemical means are not independently related to the molecular structure for controlled sequencing of the polymer chain and for particular polymer properties. Successful trial-and-error experimentation for scale-up would depend on improbable luck without the scale-down experiments discussed here. a,(t) is complex enough in fluid mechanical mixing. In mechanical chemical actions, av(t)is even more complex in the geometric configurations of boundary conditions that grow between reactant phases.
paDp2uo/18pB,giving also a uo which cannot be observed under the microscope. Apparent uo values so calculated are huge, of order m/s, which is of the same order as the thermal velocity of A molecules, ( ~ R T / T M ~ ) ~This /'! shepherding of molecules suggests that the action at time and space too small to observe is more fantastic than one can yet imagine. Certainly the simple model described here is not sufficient unless it proves to be so accidentally. To carry the above model to a practical conclusion, the generation rate of intermaterial area av(chem) by interfacial energy would be written dav(chem) 6 9.rr ~ A ~ V ~ O %-I -- flAUvQA- = 2-
dt
DP
DP
3Tgc
nAavrO3UAB
(16)
DpkASp
Fluid Mechanical Actions and Intermaterial Area Generation Caused by Interfacial Energies Wickert (1985) has suggested capillary motion in tubes due to surface tension forces as an appropriate model for the phenomena he observed. This idea merits more development than he was able to include in his M.S. thesis. A resolution of surface energies of orienting reacting molecules as tangential intermolecular forces for A, P, and B interface zones would be an extension which could model a rupture hole or leak in the polymer layer as a drop A cannon. The forces themselves might also be measured separately in nonreacting cases and/or related to solubility of the terminated polymer in the different reactants. As one example of many which could be considered for experimental tests, a fluid mechanical model for physicochemical dispersion of A through the a,(t) of mechanical mixing is proposed here. Assume that nAruptures per unit area of av(t)are dispersing prepolymer A into prepolymer B through tubular holes in polymer P. The holes have a radius, ro, and a length, s,(t), as previously modeled. The picture is one of pseudosteady laminar flow drawn through the hole by interphase tension. A stream of A is being projected into B, but it is encapsulated simultaneously by a very thin layer of P and collapses into droplet form
projecting force
viscous resistance force t o laminar flow
average inertial force of liquid A projected into B in terms of initial projection velocity u o
where CAB is the interphase tension, pA is the viscosity of prepolymer A, and &A is the volumetric flow rate through one hole. uo is the average inertial velocity of liquid A being projected into liquid B, as droplets of eventually observable diameter Dp. While the initial formation of a single drop and the initial velocity have not been seen, the movement positions, x ( t ) , of the observable droplet can be used to back-calculate an apparent uo assuming Stokes drag
where pB is the viscosity of prepolymer B. The farthest distance reached, a Stoke's law stopping distance, is
where dt is of order 1 s and assumes a critical role in rate of reaction in RIM after the mold is filled. sp(t) is the polymer layer thickness of the a, contact areas created by mechanical mixing. As av(chem) increases with the molded liquid volume, Sp(chem) increases within the molded liquid volume, Sp(chem), at an instant of time, has a distribution of values because of differences in the lifetimes of the drop forms. Adiabatic temperature rise would be one way to measure the overall reaction rate for confirming or falsifying the model; infrared analysis of a double striation of microscopic area scale might be another.
Models for Heterogeneous Reactions in the Chemical Industry One should not apologize for the complexities of nature or try to ignore laws governing actions at microscopic scales, especially if one is trying to use nature for a preconceived advantage. Development of new chemical products, of new methods of producing new products in quantity, and of improvements in the manufacturing methods which will be used requires the simultaneous construction of a sufficient analytical model and the performance of scale-up and scale-down experiments. The experiments should be limited to those which measure parameters of the model, or falsify the model, and suggest more meaningful experiments for the same purposes. In the same way, one can deduce the scientific secrets beyond the reach of present comprehension. Thoughtful and sagacious engineering is also good science. Acknowledgment Material concerning polymerization in reaction injection molding (Wickert, 1985) was obtained in graduate research supported by NSF/CPE-8118232. Undergraduate research studies (Extrand, 1985), concerning interfacial polymerization, were supported by a undergraduate summer research grant from the 3M Co. Literature Cited Extrand, C. W. Undergraduate Student Research Project Reports, Interfacial Polymerization, University of Minnesota, 1985. Harris, S.J. Chem. Eng. News 1984, 62, 13. Hougen, 0. A,; Watson, K. M. Chemical Process Principles; Wiiey: New York. 1947; Part 111. Kolodziej, P.; Macosko, C. W.; Ranz, W. E. Polym. Eng. Sci. 1982, 22, 388. Ranz, W. E. AIChE J. 1979, 25, 41. Ray, H. J . Macromol. Scl., Rev. Macromol. Chem. 1974, C 1 1 , 177. Wickert, P. D. M.S. Thesis, University of Minnesota, Minneapolis, MN, 1985.
Received for review June 16, 1986
