Ind. Eng. Chem. Prod. Res. Dev. 1986, 25, 82-87
82
other isomer derived from liquid jojoba wax. The tetraol XV could be obtained from I11 via the diepoxide XIV (Scheme IV). The expoxidation reaction with m-chloroperbenzoic acid causes some isomerization, since up to 15% &-epoxide is detected in the NMR spectrum (see Experimental Section). Hydrolysis of XIV to the tetraol XV brings about the same results, which we noticed earlier in the reaction of partly isomerized jojoba wax (Shani, 1983a). The ratio of threo to erythro tetraols is ca. 1:1, as determined by NMR (2 H at 3.58 for threo and 2 H at 3.78 for erythro). This indicates an SN1type oxirane opening, allowing for isomerization of the trans-epoxide to both diastereoisomers (Shani, 1983a). In contrast, an SN2type reaction of a trans-epoxide, with one-step oxirane opening, should lead to the erythro isomer only. Products Derived from the Acidic (Jojoboyl) and Alcoholic (Jojobyl) Components of the Ester. Several products that are derived from either the acidic component [(E)-jojoboicacid (XVIa)] or the alcoholic one [(E)-jojobyl alcohol (XVII)] have also been prepared. The procedures for their preparation were identical with those employed for the jojoba liquid derivatives studies before. Thus, (E)-jojobamide (XIX) was prepared by high-temperature and high-pressure reaction of (E)-methyl jojoboate (XVIb) (prepared by transesterification of alltrans-jojoba wax with acidic methanol) with NH3 (Shani et al., 1980b) (See Scheme V). (E)-Jojobyl alcohol was obtained by LiAlH, reduction of all-trans-jojoba wax (Shani, 1979), which was then converted to the corresponding chloride, bromide, iodide, mesylate, and several quaternary ammonium salts (Shani and Horowitz, 1980a) (see Scheme VI). As stated in the Introduction, isomerization to the E configuration changes some of the physical properties of the wax and its derivatives. Thus, the wax itself acquires the highest melting point of the isomerized jojoba waxes. Several derivatives exhibit t,he same trend toward a more
crystalline structure, as is expected for the trans configuration and for hydrogenated polyethylene chains. Since jojoba wax is a mixture of homologous long-chain esters: the increase in crystallinity and the consequent higher melting points are remarkable effects. The (E)-diepoxide (XIV) melts at a high temperature (50-52 "C)relative to the viscous liquid products derived from jojoba wax and even from the partly isomerized wax (Shani, 1983a). The (E)-jojoboic acid (XVIa) and (E)-jojobyl alcohol (XVII) and the amide XIX, derived from the all-trans-jojoba wax, have essentially the same melting points as those of the 2 configuration (Shani, 1980b, 1981). It seems, therefore, that the polar groups play a more significant role than the geometrical configuration of the double bonds in determining the strength of packing forces of the molecules in the solid phase. The mesylate (XXII) again shows higher crystallinity since it melts at 34-36 "C, as compared to the liquid (2)-mesylate (Shani, 1979). The other products are either semisolids or viscous liquids. Studies of the technological properties of these many products and others that can be derived from all-transjojoba wax might allow better application in special uses. Literature Cited Brown, J. H.; Oienberg. H. U.S. Patent 4360387; Chem. Abstr. 1983, 9 8 , 364782. Fieser, L. F.; Fieser. M. M. "Reagents for Organic Synthesis"; Wiley: New York. 1967; Vol. 1, pp 1262-1265. Shani, A. J . Chem. Ecol. 1979, 5 , 557. Shani, A. J . Am. OiiChem. SOC.1981, 5 8 , 845. Shani, A. J . A m . OilChem. SOC.1982, 5 9 , 228. Shani, A. Ind. Eng. Chem. Prod. Res. Dev. 1983a, 2 2 , 121. Shani, A. Soap Cosmet. Chem. Spec. l983b, 59(7), 42. Shani, A,; Horowitz, E. J . Am. Oil Chem. SOC.1980a, 5 7 , 161. Shani, A,; Lurie, P.; Wisniak, J. J . A m . Oil Chem. SOC. 1980b, 5 7 , 112. Sonnet, P. E. Tetrahedron Report No. 77 Tetrahedron 1980, 36, 557. Wisniak, J.; Aifandary, P. Ind. Eng. Chem. Process Res. Dev. 1975, 14, 177. Wisniak. J. Proc. Chem. Fats Other Lipids 1977, 15, 167.
Received for review April 29, 1985 Accepted September 3, 1985
Polydisperse Suspensions of Spherical Colloidal Particles: Analogies with Multicomponent Molecular Liquid Mixtures Eric Dicklnson Procter Department of Food Science, University of Leeds, Leeds LS2 9JT, England
Polydispersity affects the equilibrium and dynamic properties of suspensions of spherical colloidal particles. Fluid-phase equilibria in polydisperse systems, colloidal or molecular, can be conveniently described by an extended theory of conformal solutions devised originally for binary liquid mixtures. For electrostaticallystabilized polydisperse suspensions, the theory predicts the occurrence of gas-liquid and liquid-liquid phase transitions. There are, however, other important aspects of polydisperse colloidal behavior that do not have any obvious analogue with multicomponent molecular systems. The effects of polydispersity on the order-disorder transition and the settling of particles under gravity both come into this category.
Introduction Suspensions of colloidal particles occur widely in nature and are important industrially. In many colloids of practical interest-food emulsions, for example-the dispersed particles are close to spherical in shape, but their sizes show considerable variations. These colloidal systems
can be regarded as being truly polydisperse, in contrast to a molecular system, which, however complex, is necessarily paucidisperse (from the Latin "paucus" meaning "few"). As well as particle size, there may also be a continuous distribution of other single-particle properties such as density, refractive index, thermal conductivity, or sur-
0196-4321/86/1225-0082$01.50/00 1986 American Chemical Society
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Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986
1
I
’ 0
1.
**
I
I.
^. :.44
0.
Figure 1. Schematic representation of states of a monodisperse colloid: (1)single-phase, stable, gaslike; (2) two-phase, flocculated and dispersed, liquid-gas equilibrium; (3) two-phase, flocculated ordered and flocculated disordered, solid-liquid equilibrium; ( 4 ) single-phase, coagulated, gellike.
face charge. In this paper, however, we shall limit discussion to the polydispersity of particle size in systems of undeformable charged spheres. Some important aspects of polydispersity on equilibrium and dynamic properties will be reviewed, and the effects on sedimentation and the order-disorder transition will be highlighted. Before moving on, a few words are pertinent on the use of the term “equilibrium” in connection with colloids. Thermodynamically, of course, most colloids are not stable but metastable. But, if the rate of bulk phase separation of the dispersed and continuous phases is vanishingly small, as it must be for a “stable” suspension, it is perfectly valid, over the observational time scale, to regard the steady-state metastable behavior as being “equilibrium” in nature, and therefore amenable to the methods of statistical thermodynamics. (In the same way, we routinely apply equilibrium statistical mechanics to systems of molecules that are “metastable”, insofar as the latter may ultimately react to form different “stable” chemical entities.) In this terminology, flocculated colloids can be treated by the methods of equilibrium statistical mechanics but irreversibly coagulated colloids cannot (see Figure 1). Statistical mechanical treatments of colloids (Dickinson, 1983a; van Megen and Snook, 1984) exploit the analogy between a dispersion of particles and an assembly of atoms or molecules. The imaginary transposition from molecular liquid to colloidal dispersion is achieved by replacing the intermolecular pair energy by an effective ”potential of mean force” between pairs of particles. The contribution of solvent and small ions to correlations between particles is contained within the potential of mean force as is the effect of adsorbed macromolecular chains in polymerstabilized colloids. Comparison of theory with experimental data on many colloidal systems is hampered by the presence of an ill-defined particle-size distribution in the experimental samples, and interest in recent years has therefore focused on a few synthetic colloids that can be made to a high degree of monodispersity. In this regard, the most prominent model system consists of spherical polystyrene latex particles (“polyballs”) (Hearn et al., 1981; Pieranski, 1983).
Colloidal Phase Transitions It is now recognized (Dickinson, 198313) that dispersions of spherical particles may exist in colloidal states (ordered, flocculated, and dispersed) equivalent to the three common states of matter (solid, liquid, and gas, respectively). There
83
is, in addition, the possibility of a glasslike colloidal state, and also for nonspherical particles various liquid-crystalline states. With “monodisperse” polystyrene latices, three types of phase transition have been observed in the laboratory: (a) between flocculated (“liquidlike”)and dispersed (“gaslike”) states; (b) between ordered (“crystalline”)and disordered (“fluidlike”) states; and (c) between two different ordered states (body-centered-cubic and face-centered-cubic). Figure 1illustrates schematically the types of aggregation commonly encountered in monodisperse colloidal systems. Situation 1 corresponds to a dilute stable sol of electrostatically stabilized particles at low electrolyte concentration. Situation 2 corresponds to a more concentrated dispersion in which there is an equilibrium between flocculated and dispersed phases. In situation 3, there is an equilibrium between dense ordered and disordered phases, and the whole system may be regarded as being in a flocculated state. Situation 4 represents a coagulated gellike colloid in which the particles are held together by strong short-range forces in a porous network structure. In situations 1-3 all the individual particles are free to move around under the influence of gravity and Brownian motion, since the energies of pair interaction are only of the order of the thermal energy (k7‘). In contrast, situation 4 contains particles that are not free to move, and the whole structure is “frozen” over the normal experimental time scale. There is much interest currently in the kinetics of transformation from stable to coagulated colloids, but as the process is irreversible it is outside the scope of this paper. The maximum number of coexisting phases in a monodisperse system is limited to three by the phase rule. With polydisperse systems, however, there exists the potential for finding a large number of phases in equilibrium (Dickinson, 1983~).But, since entropy considerations imply miscibility, one would not expect a large number of phases to separate out under normal circumstances. The relationship between colloidal and molecular phase diagrams is sometimes obscured by the difference in state variables. With colloids we are concerned with osmotic pressure rather than normal hydrostatic pressure. But, more importantly, we do not use temperature as a state variable because the potential of mean force is often itself very temperature sensitive, making the effective temperature range over which real dispersions are studied extremely limited. In fact, because of the temperature dependence of the potential of mean force, some caution needs to be exercized in expressing thermodynamic quantities as configurational averages (van Megen and Snook, 1984). For the case of electrostatically stabilized dispersions, a convenient variable for labeling phase diagrams is electrolyte concentration. This can be considered as being qualitatively equivalent to reciprocal temperature, since a reduction in the electrolyte concentration normally implies a lowering in the strength of the interparticle attractive energy relative to the thermal energy. We note that in electrostatically stabilized dispersions, the reversible gas-liquid transition is physically quite distinct from irreversible coagulation, which will always occur beyond a critical electrolyte concentration (Victor and Hansen, 1985). As far as transitions between disordered colloidal states are concerned, the influence of polydispersity can in principle be treated by the same sorts of theoretical methods as are applicable to paucidisperse molecular fluids (Bowman, 1949; Salacuse and Stell, 1982; Gualtieri et al. 1982; Briano and Glandt, 1984). In what follows below,
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the statistical thermodynamics in an electrostatically stabilized dispersion is described by a type of van der Waals equation of state in conjunction with conformal solution theory, with compositions of coexisting phases determined by equating chemical potentials as in conventional chemical engineering treatments of fluid-phase equilibria. It follows naturally from this approach that polydisperse systems may exhibit gas-liquid and liquidliquid transitions just like those in binary fluid mixtures. Polydisperse Conformal Solutions Like the interactions between simple molecules, the potentials of mean force between colloidal particles are effectively superimposable after appropriate scaling with respect to energy and distance (Dickinson, 1980). That is, the pair potential can be written as U,]=
QWgJ
(1)
where el, and u , are ~ energy and distance scaling factors for particles of types i and j and F is a universal function. The assumption of superimposability should be valid for all systems whose degree of polydispersity is not too large. A system of particles interacting with potentials of the form of eq 1 is called a conformal solution (Prigogine, 1957). The conformal concept is ideally suited to polydisperse systems, since the continuity in single-particle attributes forces a conformality requirement on all pairs with similar values of attribute z , where y ( z ? dz is the probability of finding a particle with z between z’and z ’ + dz. We are treating the parameter z as a characteristic attribute, real or imaginary, that serves to identify the individual particles in the system. In conformal solution theory, it is convenient to define relative energy and volume parameters (Rowlinson, 1969):
f,, = c l l / ~ o
h,, = glJ3/uo3
(2)
The scaling factors eo and uo refer to a monodisperse reference substance with z = zo. The most obvious choice of reference substance is that for which zo takes the mean value 2 = s z y ( z ) dz
(3)
A polydisperse system in this nomenclature is described by two functions, f(z’,z’’) and h(z’,z’q, which represent the distributions of relative energy and volume parameters for all pairs with z = z’and z = z”. In the well-known yonefluid” van der Waals model, which works well for simple liquid mixtures (Leland et al., 1968), it is assumed that the multicomponent system can be replaced by an equivalent monodisperse substance with parameters (f) and ( h ) . These are obtained by summing over all interacting pairs in a paucidisperse system and integrating over all interacting pairs in a polydisperse system: ( f ) = (h!-’SSy(z?y(z’~f(z’,z’?h(*’,z’? dz’“’’
( h )=
s
sy(z’)y(z’?h(z’,z’? dz’“’’
(4)
(5)
The configurational Gibbs free energy G, of the equivalent monodisperse substance at temperature T and pressure p has the form G,[T,p; ~ ( ~ =1 (1 f ) G , ( T / ( f ) , ( h ) p / ( f )-) N k T In ( h ) (6) where N is the number of particles, k is Boltzmann’s constant, and Go is the free energy of the reference substance. In a mixture of M discrete components, the condition for equilibrium between two phases a and fi is
F~“(T,= P )pl’(TIP)
(i = 1, 2,
M)
(7)
-
where pi is the chemical potential of species i. In a polydisperse system ( M a),eq 7 is replaced by A p k ) = rc.”(T,p;2) - pLP(T,p; 2) = 0
(8)
where p(T,p; z ) is a chemical potential function that depends on the value of the attribute z. For a pair of phases with distributions y,(z) and y&), the chemical potential difference A p can be written as a generalization of the binary expression of Rowlinson and Watson (1969)
.b = NkT(h’[z,y,(z)](Z,“ - 1) - hTz,yfi(z)l X (Z,B - 1) + In [y,(z)] - In [ ~ , & ) l l +G,” - G,P + f T Z , Y a k ) I U e “ - fTz,y&)lU,8 (9) where 2,and U, are the compressibility factor and internal energy of the equivalent monodisperse substance. The quantities f’and h’are given by (Rowlinson and Watson, 1969) f’[z,u(z)l = 2 ( [ f ( z ) h ( z ) / ( f ) ( h- )Eh(z)/(h)lJ l (10) hTz,y(z)I = 2([h(z)/(h)I - 11
(11)
where the functions f(z) and h(z)are the so-called “twofluid” averages (Leland et al., 1969):
f k )= [ h ( z ) I - ’ S ~ ( z ? f ( z , z ? h ( z , tdz‘ ?
(12)
h(z) = I y ( z q h ( z , z ’ )dz’
(13)
In order to use the conformal solutions approach, it is necessary to assign an equation of state to the reference substance. A convenient choice is an equation of state that combines the hard-sphere expression of Carnahan and Starling (1970) with the well-known van der Waals attractive term. Dickinson (1980) has solved eq 8 and 9 simultaneously for M discrete, equally-spaced values of z. For a rectangular distribution of particle radii in the range 0.51-0.69 pm, the same gas and liquid compositions were found with M = 10 as with M = 20, from which it can be inferred that an adequate representation of a polydisperse colloidal system is possible with about 10 effective components. The calculations show that in colloidal transitions between flocculated and dispersed states, the range of possible interparticle energies is sufficient to cause an order-of-magnitude difference in the concentrations of largest and smallest particles in the flocculated phase (see Figure 2).
Liquid-Liquid Phase Equilibrium Some binary liquid mixtures exhibit liquid-liquid phase separation below an upper critical solution point (Rowlinson, 1969). The phenomenon is interpreted statistically in terms of a relative weakness in the pair interaction between unlike species. In a polydisperse system, all species are to some extent different, and we can designate the degree of “unlikeness” in terms of two functions, [ ( z ’ , ~ ’ ? and { ( z ’,z ’9, defined by f ( z ’,2’9 = E(z ’,z ’9 [f(z’,z ?f(z
”,z ’911’2
(14)
[ h ( ~ ’ , 2 ’ 9 ] l=/ ~{ ( Z ’ , Z ’ ? ~ [ ~ ( Z ’ , Z ? ] ~+/ ~[ h ( ~ ” , 2 ’ 9 ] ~ / ~ } / 2 (15)
Let us suppose that in a particular system z runs from z1 to z 2 . By analogy with the binary case, a polydisperse system is susceptible to liquid-liquid immiscibility if [ ( z ’ , ~ ’ ’ ) is much less than unity for z ’ = z1 and z ” = z 2 or vice versa. Entropy considerations favor miscibility in polydisperse systems much more than they do in binary
Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986 85
theoretical predictions based on an analytic mean-field approach (Smith and Rowlinson, 1980). That is, if the unlike interactions are energetically weak enough, then indeed, a polydisperse system can show genuine liquidliquid phase separation. Whether such interactions do exist in practice remains an open question. Apparently, no phase separation of this type has yet been reported for a polydisperse colloidal suspension.
0.69
a/ym
0.60
0.51 0
1.0
2.0
k/h Figure 2. Coexisting compositions of gaslike and liquidlike colloidal phases calculated from conformal solution theory (Dickinson, 1980). The particle radius a runs from 0.51 to 0.69 fim, and the flocculated phase has a particle-size distribution such that yL(a)= constant; i.e., all particle sizes are equally probable. The ratio yc/yL of the compositions of the gaslike and liquidlike phases is plotted against particle size. Note that the concentration of smallest particles in the dispersed phase is an order of magnitude larger than that of the largest particles.
mixtures, so the simple question arises as to whether any actual polydisperse system could ever satisfy the thermodynamic conditions for separation into two liquidlike phases. The thermodynamics is greatly simplified if the phase diagram is symmetrical about z = zo. This will happen if particle interactions are symmetrical about z = z,; i.e., if f(z’,z’?
= f(z”,Z’?
h(z’,z’? = h(Z’,z”?
(16)
where z‘ = 22, - z , and if the compositions are symmetrically distributed: (17)
342) = 342)
If immiscibility occurs at all, a critical point will appear at z = zo, subject to the thermodynamic condition [ a ~ l ( z ) / a y ( ~ ) I ~ y ( z=’ =0 ~ ~ ,(21 ~,~ -< z
-< 22)
(18)
Equation 18 implies that y,(z) # y&z) for z # zo, so that if any one particle in the polydisperse system has a higher probability of being found in one phase than in the other, the same is true for all other particles (except for the unique case z = zo). This intimate coupling of interparticle and macroscopic functions is a particular feature of polydisperse systems. Some calculations have been done (Dickinson, 1980) for a simple symmetrical polydisperse system by using the modified van der Waals equation of state. Unlike energy interactions are distributed according to
- z,)l”] (19) where l* and m are constants and y ( z 9,f(z ’,z 9 and h ( z ‘,z 9 E(z’,z’? = 1 - [(l- E*)I(z’- z’?/(zp
are set equal to unity for all values of z’ between z1and E* = 0.5, it is found that the system does separate into two liquidlike phases and that the distribution of species between phases is sensitive to the degree of coupling amongst the interparticle interactions, as reflected in the value of the parameter m. The results from the conformal solution approach are qualitatively consistent with evidence from a molecular dynamics simulation of polydisperse hard spheres (Dickinson, 1979a) and with z2. With
Polydispersity and the Order-Disorder Transition The order-disorder transition found experimentally in dispersions of charged polystyrene particles (Hachisu and Takano, 1982) is formally equivalent to the freezing transition of a simple fluid. Despite its fludity under low shearing stresses, the ordered colloid is a real crystal (Pieranski, 1983): in dilute latex dispersions that have been treated with ion-exchange resin, the interparticle spacing is of the order of the wavelength of visible light, and Bragg reflections from colloidal crystallites lead to brilliant opalescence. Body-centered-cubic and face-centered-cubic lattice types are favored at low and high volume fractions, respectively. The order-disorder transformation of a perfectly monodisperse colloid is thermodynamically first order, and it can be treated theoretically either as a Kirkwood-Alder transition or as a melting transition of a Wigner lattice (Dickinson, 1983a). Polydispersity becomes important when the range of the screened interparticle electrostatic forces is much less than the mean particle diameter (Dickinson, 1979b). Experimentally, this means that the electrolyte concentration is moderately high, but not so high as to induce irreversible coagulation. Under these conditions, the repulsive pair potential is steep, and a t high-volume fractions the dispersion prefers to exist as an assembly of close-packed hard spheres. But particles with diameters greater than the mean are not accommodated on the crystal lattice without increasing the overall free energy of the ordered phase. So, one might intuitively expect there to be a certain degree of polydispersity beyond which the disordered phase is thermodynamically the more stable. For ordered solidlike states, it is found (Dickinson, 197913) that the osmotic pressure increases with the degree of polydispersity, in conformity with predictions from a simple cell model. If it were possible to increase the degree of polydispersity continuously for a stable dispersion just on the ordered side of the phase transition, one might expect to reach a critical extent of polydispersity above which ordered phase would melt spontaneously (Dickinson and Parker, 1985). Such an experiment is, of course, impossible in the laboratory, but it can be done on a computer. The phase diagram of a model colloid has been mapped out as a function of polydispersity by using the numerical method of molecular dynamics (Dickinson et al., 1981). The form of the indiscriminate radial distribution function does not easily distinguish between ordered and disordered states in a polydisperse system. But, from the magnitude of computed single-particle diffusion coefficients, it is possible to assign state points as being unambiguously either “crystalline” or “fluidlike”, as shown in Figure 3. The simulated free energy and osmotic pressure of the disordered colloid are insensitive to the degree of polydispersity, in agreement with hard-sphere theory (Dickinson, 1978). By way of contrast, the osmotic pressure of the ordered phase increases strongly as the square of the degree of polydispersity, in agreement with cell theory (Dickinson, 1979b). The net effect of polydispersity on the phase diagram is to shift the position of the melting transition to higher volume fractions at constant electrolyte concentration.
86
Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986 I
I
I
0.24
0.28
0.32
0.36
$
Figure 3. Phase diagram of simulated polydisperse colloidal system near the order-disorder transition. The compressibility factor p V j N k T is plotted against volume fraction Cp for five degrees of polydispersity: (*) 0; (0)0.1; (A)0.141;(m) 0.173; (V)0.2. The open symbols refer to the disordered liquidlike state.
Polydispersity and Sedimentation Polydispersity also has an important effect on the dynamic properties of colloidal suspensions. This is manifest in the diffusion coefficients measured in dynamic lightscattering experiments (Pusey and Tough, 1982). Of more practical significance to colloid technologists, however, is the related problem of the particle settling rate in polydisperse suspensions under the influence of gravity or a centrifugal field. On average, the relative velocity of two neighboring particles falling under gravity in a monodisperse suspension is 0 (the relative PBclet number is 0). It follows that the structure of a sedimenting monodisperse system is therefore unaffected by the presence of the sedimenting field. Only in a monodisperse system, however, are particle encounters driven just by Brownian motion. In a polydisperse system, there is a complex coupling between the effects of sedimentation and Brownian motion on the relative particle trajectories. In the dilute case, where three or more particles have negligible chance of coming close together, expressions are now available (Batchelor, 1982) for the mean settling speed as a function of the relative PBclet number. A t small absolute PBclet numbers (small settling speed compared with Brownian diffusion), the effect of sedimentation on dispersion structure can be treated as a perturbation about the equilibrium. At high PBclet numbers, structure is controlled almost entirely by the deterministic hydrodynamic motion. In a polydisperse colloid, there is a continuous distribution of relative PBclet numbers. The motion of individual particles on a time scale much longer than that for solvent motion depends on an interplay amongst the various types of forces acting on the particles: Brownian, van der Waals, electrostatic, hydrodynamic, and gravitational. One approach to the problem that seems promising is the numerical simulation technique of Brownian dynamics (Bacon et al., 1983a,b). Results have been obtained (Dickinson and Parker, 1984) for the effect of electrolyte concentration and sedimenting field strength on the rate of dissociation of a pair of floc-
culated particles of diameters of 1 and 2 pm. At low field strengths, Brownian motion predominates and floc stability depends mainly on the interparticle potential; at higher field strengths, the dynamics depends on a delicate balance between gravity, Brownian motion, and interparticle forces (both hydrodynamic and nonhydrodynamic). A simulation of a single sphere falling in a concentrated system of neutrally buoyant, but otherwise identical, spheres shows that the local structure around a sedimenting particle becomes nonsymmetrical at high sedimenting field strengths (Bacon et al., 1983b). A well-known phenomenon in colloid and emulsion science is the enhancement of particle aggregation by sedimentation or creaming. The process is a property of polydisperse systems and occurs when large particles “catch up” smaller ones. Once aggregation has begun to take place, the system is effectively more polydisperse than it was initially, since flocs settle faster than isolated particles. This positive feedback mechanism makes sedimentation behavior especially sensitive to small degrees of polydispersity. In the absence of Brownian motion (large particles), one can use the concept of a “critical trajectory” to estimate the number of binary encounters that will lead to capture. Where Brownian motion cannot be neglected, computer simulation shows that the probability of particle capture is a sensitive function of field strength, electrolyte concentration, and relative particle separation (Dickinson and Parker, 1984). In fact, deviations from the critical trajectory may be so large that macroscopic concepts are not useful at all for describing gravity-induced flocculation of colloidal dispersions.
Summarizing Remarks Some progress is being made in predicting the equilibrium and dynamic properties of polydisperse colloidal suspensions. The statistical mechanics of polydisperse systems of spherical particles clearly has much in common with the statistical theory of multicomponent liquid mixtures. In particular, conformal solution theory provides a sound and convenient way of describing polydisperse fluidlike colloidal states. The calculated phase behavior resembles that found in some paucidisperse molecular mixtures (e.g., petroleum fractions) with the occurrence of liquid-liquid type equilibrium not being ruled out theoretically. By way of contrast, the effects of polydispersity on the crystalline melting transition and the kinetics of sedimentation are particular to the colloidal regime. One important problem not touched on here is the effect of polydispersity on the viscosity of colloidal dispersions. Some progress is being made now with computer simulation (Ansell et al., 1985) and simple theory (KovgF and Fortelnjr, 19841, but a full solution probably awaits further developments in the field of fluid mechanics. It is generally agreed that a polydisperse suspension has a lower viscosity than the equivalent monodisperse suspension with the same volume fraction, and the normal intuitive explanation is that spheres of different sizes pack together more efficiently than those of the same size and that this leads to less structural resistance to flow and therefore to a lower dispersion viscosity. But how more complex features like shear-thinning, dilatancy, or thixotropy are affected by polydispersity is not at all clear. In monodisperse systems in the vicinity of the solid-fluid transition, the imposition of shear flow causes the development of structures in which two-dimensionally ordered layers slide over one another with a greater fluidity than would occur if the system were totally disordered and liquidlike (Woodcock, 1984). One would expect such flow-induced ordering to occur less
Ind. Eng. Chem. Prod. Res. Dev. 1986,
readily in polydisperse colloids, with obvious implications for the rheological behavior. Acknowledgment
Part of the material in this paper was presented as an oral contribution t o an AIChE symposium on :Thermophysical properties of systems with very many components" in Atlanta, GA, in March 1984. Financial support from the Science and Engineering Research Council (UK) is gratefully acknowledged. Literature Cited Ansell, G. C.; Dickinson, E.; Ludvigsen, M, J . Chem. SOC., Faraday Trans. 2 1985,81, 1269. Bacon, J.; Dickinson, E.; Parker, R.; Anastasiou, N.; Lal, M. J . Chem. SOC., Faraday Trans. 2 I983a, 79,91. Bacon, J.; Dickinson, E.; Parker, R. Faraday Discuss. Chem. SOC. 1983b, 76,165. Batchelor, G. K. J . Fluid Mech. 1982, 119,379. Bowman, J. R. Ind. Eng. Chem. 1949,41,2004. Briano, J. G.; Glandt, E. D. J . Chem. Phys. 1984,80,3336. Carnahan, N. F.; Starling, K. E. J . Chem. Phys. 1970, 5 3 , 600. Dickinson, E. Chem. Phys. Len. 1978, 57, 148. Dickinson, E. Chem. Phys. Lett. 1979,66,500. Dickinson, E. J . Chem. SOC., Faraday Trans. 2 1979, 75,466. Dickinson, E. J. Chem. SOC.,Faraday Trans. 2 1980, 76,1458. Dickinson, E. I n "Colloid Science"; Everett, D. H.,Ed.; Royal Society of Chemistry: London, 1983a; Specialist Periodical Reports, Vol. 4, p 150.
25,87-89
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Dickinson, E. I n "Annual Reports C"; Symons, M. C. R., Ed.; Royal Society of Chemistry: London, 1983b; p 3. Dickinson, E. Chem. Phys. Lett. 1983, 101,562. Dickinson, E.; Parker, R. J . Colloid Interface Sci. 1984,97,220. Dickinson, E.;Parker, R. J . Phys. Len. 1985, 46,L229. Dickinson, E.; Parker, R.; Lal, M. Chem. Phys. Len. 1981,79,578. Gualtieri, J. A.; Kincaid, J. M.; Morrison, G. J . Chem. Phys. 1982, 77,521. Hachisu, S.;Takano, K. Adv. Colloid Interface Sci. 1982. 16,233. Hearn, J.; Wilkinson, M. C.; Goodall, A. R. Adv. Colloid Interface Sci. 1981, 14, 173. KovS?, J.; Forteinq, I. Rheol. Acta 1984,23,454. Leland, T. W.; Rowlinson, J. S.; Sather, G. A. Trans. Faraday SOC.1968,64, 1447. Leland, T. W.; Rowlinson, J. S.;Sather, G. A,; Watson, I . D. Trans. Faraday SOC. 1969, 65,2034. Pieranski, P. Contemp. Phys. 1983,24, 25. Prigogine, I.; Bellemans, A.; Mathot, V. "The Molecular Theory of Solutions"; North-Holland: Amsterdam, 1957. Pusey, P. N.; Tough, R. J. A. I n "Dynamic Light Scattering and Velocimetry: Applications of Photon Correlation Spectroscopy"; Pecora. R., Ed.; Plenum: New York, 1982. Rowiinson, J. S."Liquids and Liquid Mixtures", 2nd ed.; Butterworths; London, 1969. Rowlinson, J. S.; Watson, I. D. Chem. Eng. Sci. 1969, 24, 1565. Saiacuse, J. J.; Stell, G. J . Chem. Phys. 1982,77,3714. Smith, E. R.; Rowlinson, J. S. J . Chem. SOC.,Faraday Trans. 2 1980, 76,
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Received for review June 4, 1985 Accepted September 17, 1985
Method for Reducing Elution of Therapeutic Agents from Teeth Robert W. H. Chang Personal Care Products Laboratory, 3M Center, Building
230-2s-06,St. Paul, Minnesota 55 144-1000
A simple method to prevent and control dental disease is to maintain the tooth enamel at maximum resistance to acid dissolution and prevent plaque formation and bacterial colonization by treating the tooth with a therapeutic agent, such as fluoride, and then encapsulatingthe treated tooth with a suitable hydrophobic or polymeric membrane. The fluoride in the tooth enamel will not leach readily and will slowly release to the surface of the membrane or into the plaque, if attached, thus interfering with bacterial colonization. I f the membrane has a low surface energy, it will be difficult for the bacteria to colonize and for the pellicle and plaque to adhere, and it will be much easier to remove bacteria, pellicle, and plaque from the surface.
Introduction
Dental caries and periodontitis, not ordinarily considered to be endangering to life, are among the most troublesome afflictions of mankind. Dental plaque results when cariogenic bacteria (e.g., Streptococcus mutans) collect in colonies and form deposits on tooth surfaces (Fitzgerald, 1968; Fitzgerald and Jordan, 1968; Gibbons, 1968, 1969; Gibbons and Nygaard, 1968; Newburn, 1967; Guggenheim and Schroeder, 1967). The presence of the bacteria and deposits, if left unchecked, may result in infected gingival tissue, the formation of dental caries, and possibly periodontal disease. Plaque formation has also been reported to involve the adhesion of bacteria to the acquired pellicle in the absence of sucrose metabolite products (Hay, 1967; Meckel, 1965; Sonju and Rolla, 1973). Thus, the development of caries requires interaction among three factors: a microbial agent, a susceptible tooth, and the presence of a suitable dietary substance (Keys, 1962, 1969). The logical approach to control them, therefore, is to modify one or more of the factors in this host-diet-bacteria complex. Fluorides have been used for a long time, and their potential to prevent and control dental disease is far
greater than that presently attained by most of the preparations and methods of application in use today (Keys and Englander, 1975). The anticaries and antiplaque potential of fluorides comes from their ability to harden the mineral of the enamel, and thus increase its resistance to demineralization, and their ability to interfere with the growth of microorganisms in plaque. Fluorides are a logical choice for a therapeutic agent; however, fluorides can also be easily leached into the mouth. One simple way to keep a fluoride on tooth enamel longer is to encapsulate the treated tooth with a hydrophobic or polymeric membrane. The fluoride will slowly release to the surface of the membrane and then into the plaque, thus interfering with bacterial colonization. If the membrane has a low surface energy, it will be difficult for the bacteria to colonize and for the pellicle and plaque to adhere, and it will be much easier to remove pellicle and plaque, if already attached, from the surface. Chemical R e a c t i v i t y of T o o t h E n a m e l
The selection of a membrane material is based on chemisorption on tooth enamel of model organic compounds having different functional groups. A novel
0 1986 American Chemical Society 0196-4321/86/1225-0087$01.50/0
