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Specific Interactions which Need Nonequilibrium Particle/Particle Models E. ST. A. JORDINE 1 Scientific Research Council, P.O. Box 502, Kingston, Jamaica, West Indies Received May 9, 1972; accepted December 12, 1972
1. INTRODUCTION In the accepted methods of calculating the forces between two parallel-plate colloidal particles, the volume density of the ions in the intervening liquid phase is usually expressed with the aid of Boltzmann's theorem. The electrostatic problem generated by such a model is manipulated (1) by the techniques which had been used earlier in the theories of electrolyte solutions (2) and electrochemistry (3, 4). The original classical developments of all these theories are based on the concepts of conducting interfaces, point charges and linear dielectrics. As a consequence, the structural properties of the liquid and the polarization of the solid dielectric interfaces and media, do not enter into the development of the colloidal theories (1) of parallel-plate interaction. In addition the computational difficulties which restrict electrolyte theories to dilute concentrations and 1-1 electrolytes apply to colloidal systems and do not need to be repeated. In the following paragraphs, efforts are made to outline the difficulties which arise and where possible to resolve some of them. 2. STATEMENT OF PROBLEM Although it has been acknowledged in passing in the standard work (1), that it is inappropriate to assume a conducting inter1Present address: Alumina Partners of Jamaica,
Spur Tree, Jamaica, W.I.
face when treating the interaction of nonmetallic colloidal particles, this is not enough, as major discrepancies can arise. It can be stated at the outset that there are systems involving dielectrics, where the conducting model and a Boltzmann distribution of ions, yields results which do not admit experimental support over a range of particleparticle separation. There is also experimental evidence (5, 6) for some systems, that at particle separations of the order of 40/~ a special situation develops in the structural properties of the solvent and counter ions which make conventional solution theories inadmissible. By sheer admission of the physical situation, the colloid micelles have to be regarded as solid heterogenous crystals (7) and not as liquid systems, even though such micelles may be dispersed in a liquid. In addition, the finite thickness of the dielectric particles and the discrete nature of the surface charge cannot be approximated to an infinitely thick conductor-bearing free surface change. Other difficulties such as the finite size of the ions and their specific ion-dipole interactions with the solvent are enough to invalidate any theory based on macroscopic electrostatic parameters. Experimental illustrations of these problems can be found in fields as diverse as surfactant chemistry (8), civil engineering (9), and electrochemistry (10). In the work of Mysels (8), one sees a manifestation of the hydration effects of the ions at work in controlling the
435 Copyright O 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 45, No. 3, December 1973
436
JORDINE
development of second black films, whereas in Arnold's (9) soil consolidation work it is shown that, contrary to the normal repulsive forces expected from the theory of the diffuse double layer for a parallel-plate system, the forces become attractive over a certain particle-particle separation, and an extra increment is contributed to the externally applied pressure. Since the work by Arnold (9) on a Volclay bentonite is an engineering illustration and easily visualized by a wide audience, a discussion follows in order to demonstrate the problem fully. In (9, Fig. 5), it can be observed that at a given voids ratio of a Volclay bentonite, there is a sudden change in slope of the curve for the permeability coefficient, implying a sudden increase in the expulsion of water over the normal value expected. Figure 6 of Ref. (9) shows the corresponding pressure relationships during consolidation. In this figure, the ratio of the externally applied pressure to the theoretical pressure based on double layer theory is termed the activity factor and plotted versus particle-particle spacings estimated from the voids ratio. From the shape of the curve and the decrease shown in the activity factor from the maximum to the minimum, some unaccounted pressure must be operating internally to account for the increased permeability observed in Fig. 5 of Ref. (9) over precisely this region. Since there is nothing in the developments of double layer theory that could explain such a behavior, it must be concluded that as a theory, it is inadequate over this range. Further since the force is the negative of the derivative of the energy with respect to distance, Fig. 6 of Ref. (9) can only be explained by a potential energy curve such as that shown in Fig. 3 of Ref. (7). Other difficulties which arise from the specificity of ions can be best observed from the data in Refs. (5) and (6). Along with Ref. (8) these data demonstrate the specific hydration effects of the ions in a remarkable way. Taking lithium as an example, its
specific effects can be seen in two roles: first, as the only inorganic ion capable of activating swelling in the vemiculite crystal and, second, as the only complete inhibitor of the development of the second black film (8). Again these are phenomena which lie outside the scope of conventional theory. 3. PROPOSALS FOR RESOLUTION OF THE PROBLEM As pointed out in the introduction, one of important considerations to be accepted is that at close separations of the order of 40 A, the parallel plate colloidal system has to be considered as a heterogenous crystal and not as a liquid. Such a problem therefore can be defined and set up as in any solid state system involving a crystal lattice (7, 11). This approach is already popular in the so called lattice theories of liquid structure, and in the case of the bentonite and vermiculite systems is physically accurate (5). See also Figs. 1 and 2. In continuation of our discussion, the consequence of equating a solid dielectric colloidal particle to an infinitely thick conductor is that, in calculations of the total energy of the electric field, the energy of the field in the solid phase is omitted. This follows since the energy density in a dielectric is proportional to the square of the electric field, but the field in a conductor is zero and consequently the energy. This oversight of the energy due to the polarization of the solid will become manifest at close separations where the volume of the solid dielectric particle is of the same order of magnitude as the intervening liquid phase. It can be deduced that, with comparable polarizabilities for the two phases, the electrical energy of the two phases may be almost equal and, what is more, may have different signs algebraically (11). It is obvious that we are faced with two errors which do not cancel, but rather reinforce each other. One error ignores terms, while the other presupposes an inadmissible model physically.
Journal oy Colloid and InterJace Science, Vol. 45, No. 3, December 1973
NONEQUILIBRIUM MODELS From the foregoing, it can be seen that, even if for expediency in algebra, it is necessary to assume a conducting particle, there is a critical lower bound of particle separation 2D = L where such a model is inapplicable and does not make sense physically. Some relationship to the particle thickness b, such as L / b = k is to be expected. Of deeper physical significance would be a relationship involving the ratio of the energies of the electric field in the solid and liquid phases. See, for instance, Ref. (11) and the thermodynamic data in Ref. (12), which unfortunately does not extend to large enough spacings. Calculations (7, 11) show that, for a lattice model, the field energy in the solid approaches zero asymptotically from negative values according to the Laplace transform of a Bessel function of the first order. The expression for the energy in the liquid phase is more complex but is shown to be a path of decreasing potential energy, which is in direct contrast to that for the solid. In addition, peculiarities of liquid structure and specificity of ions give rise to immense particle-particle forces at close separations, which cannot be accounted for in a simple way by point charge electrostatics. Such problems need to be treated by techniques such as those developed by Bernal (13) and Debye (14). In these treatments, the classical electric field energy term arises as in any linear dielectric continuum that is electrically saturated but in addition an empirical term which accounts for the microscopic structural features and the interactions of the solvent and ions predominates.
437
however quite general for any charged dielectric plates, hence there is no real specialization in the equations developed in Refs. (7) and (11) for example. A change in the values of the particle thickness, surface charge and dielectric constant is all that is required to apply the equations generally to any plateshaped system. In support of this, it may be recalled that there is a general correspondence in the behavior of diverse systems such as ionized monolayers (8) and clay minerals (5, 6). The correspondence between such widely varying systems indicates that a wide range of extrapolation is possible from conclusions and facts derived from a study of the bentonite and vermiculite systems. Such a procedure of extrapolation is of course a necessary part of any discipline, and mere mathematical insights are pointless without a direct physical model as a reference point. In regard to parallel plate systems, the most suitable candidate that can be visualized as a model is the lamellar crystal vermiculite. From Figs. 1 and 2 we observe that as a model for ideal parallel-plate interaction vermiculite is unsurpassed. Physically also, the crystal is a heterogenous solid (Fig. 1) which after a tenfold expansion still maintains its integrity as a solid in Fig. 2, even though
!ii;!i!;!!!!i:i !i; :)?i
: ?i))iiiiii;¸)!:iil ii iili;i!ii!
4. MODELS AND REVIEW OF SOME CALCULATIONS So far the discussion has been modelled on bentonite systems as though they were the only plate-shaped particles. Indeed, they lend themselves to experimental study of equilibrium interparticle spacings by X-ray diffraction, more than any other known systems and hence represent the best model systems. The electrostatics applied and discussed is
FIG. 1. Vermiculitecrystal at equilibrium position.
Journal of Colloid and Interface Science, Vol. 45, No. 3, D e c e m b e r 1973
438
JORDINE 5. DISCUSSION
FIo. 2. Vermiculite crystal in swollen state. gel-like. The stability phenomena at various particle separations are shown in Ref. (6). Examination of the data in Ref. (6) reveal that, taking lithium vermiculite as an example, the potential energy curve must display a minimum at a particle-particle separation of 2D - 5 A and a saddle point between 5 A and 72 A. Similar arguments apply for lithium and sodium bentonites. It can be seen, therefore, that the potential energy curves for these systems are all of the form shown in Ref. (7). Bearing these data in mind, if we seek the appropriate model based on what X-ray studies (6) have shown for vermiculite and calculations of the potential energy which will satisfy the above, the suggested model will be slabs of dielectric interspersed regularly by counterions. For simplicity, this is reduced to a cell model in which each ion is assumed to have a region of influence over two solid particle surfaces which demarcate a liquid region. Proceeding on this model for the vermiculite system by the methods of Refs. (7) and (11), Fig. 3 was obtained. If we estimate the location of the minimum from this curve, it will be seen to be very close to the experimental value of 2 D = 5 A determined in Ref. (6) by X-ray diffraction.
It cannot be denied that simple electrostatic models can yield the correct type of energy curves and give minima which are close to those found experimentally. See also the Figures in Ref. (II). The depth of the minima may not be exact but they seem to have the correct multiple of kT. It also seems that a minimum which shows a binding energy of the order of 6-8 kT resists expansion, and the crystals do not swell beyond this spacing. This is the case for all the inorganic ions listed except lithium and sodium. In passing, it may be observed that all the potential energy curves may be recognized (15) as curves which depict the potential energy of a two-body system under central forces. This suggests a critical energy of escape for the particles to separate beyond the minimum. The soil mechanics data of Arnold (9) on Volclay bentonite now find a logical explanation in terms of the family of potential energy curves cited, which show that the force between bentonite particles change from one of repulsion to one of attraction as the hump in the curve is approached. The same feature holds for vermiculite Fig. 3 and is supported by the X-ray and swelling data in Ref. (6). The simple model we have chosen gives a faithful representation of the particle-particle spacings of charged particles as well as the observed behavior in soil mechanics. The implications and need for such specific binding models to explain many coagulation phenomena especially with polyvalent ions need no assertion. Further the activation and stability of biological, industrial, and mineralogical surfaces can, in the author's opinion, only be explained by a spectrum of potential energy curves for the various ions. The specific behavior of the inorganic ions in Mysels work (8) on black films also fits this spectrum of energy curves. Returning in closing to the old controversy due to Langmuir (16), it must be concluded as he suggested, that stable biparticle aggre-
Journal of Colloid and Interface Science, Vo]. 45, No. 3, December 1973
NONEQUILIBRIUM MODELS
439
SO /,0 30 20 10 0 10"~ En==I,=~ -10 "20 -30
-40 -50 -60
-70 -80 -90
-100
2 a ~ s s 7 8 9 ~o~,2,a~s~s~7~8~20
0(i) FIG. 3. Energy of interaction between two faces of vermiculite in relation to one-half distance of separation. gates are i n d e e d achieved to a large extent, b y " h y d r a t i o n " forces c o u n t e r a c t i n g Coulombic forces. L a n g m u i r ' s general idea c a n n o t therefore be easily dismissed (1). REFERENCES 1. VERWEY, E. J. W., AND OVERBEEK, J. T. G., "Theory of Stability of Lyophobic Colloids" Elsevier, New York 1948, 2. DEBYE, P., AND HUCKEL, E., Physik. Z. 24, 185 (1923). 3. CHAPMAN,D. L., Phil. Mag. 25, 475 (1913). 4. GouY, G., Ann. Phys. 7, 129 (1917). 5. NogRIsrI, K., Discuss. Faraday Soc. 18, 120 (1954). 6. NORRISH, K., AND RAUSELL-COLOM,J. A., Clays Clay Minerals 10, 123 (1963).
7. HURST, C. A., AND JORDINE, E. ST. A., J. Chem. Phys. 41, 2735 (1964). 8. JONES, M. N., MYSELS, K. J., AND SCHOLTEN, P. C., Trans. Faraday Soc. 62, 1336 (1966). 9. ARNOLD, M., "Int. Soc. Soil Mech. and Found. Eng. Conf. 6th," p. 12, 1965. 10. MAcDoNALD, J. R., AND BAgLOW, C. A., J. Chem. Phys. 36, 3062 (1962). 11. JORDINE, E. ST. A., STEEL, B. J., AND WOLFE, J. D., Bull. Chem. Soc. Japan 38, 199 (1965). 12. BARSHAD,I., Clays Clay Minerals 8, 84 (1960). 13. BERNAL, J. D., AND FOWLER, R. H., J. Chem. Phys. 1, 515 (1933). 14. DEBYE, P., "Polar Molecules." Dover, New York, 1929. 15. GOLDSTEm, H., "Classical Mechanics." AddisonWesley, Reading, MA., 1959. 16. LANGMUIR, I., J. Chem. Phys. 6, 873 (1938).
Journal of Colloid and InterfaceScience, Vol. 45, No. 3. December 1973
1. INTRODUCTION In the accepted methods of calculating the forces between two parallel-plate colloidal particles, the volume density of the ions in the intervening liquid phase is usually expressed with the aid of Boltzmann's theorem. The electrostatic problem generated by such a model is manipulated (1) by the techniques which had been used earlier in the theories of electrolyte solutions (2) and electrochemistry (3, 4). The original classical developments of all these theories are based on the concepts of conducting interfaces, point charges and linear dielectrics. As a consequence, the structural properties of the liquid and the polarization of the solid dielectric interfaces and media, do not enter into the development of the colloidal theories (1) of parallel-plate interaction. In addition the computational difficulties which restrict electrolyte theories to dilute concentrations and 1-1 electrolytes apply to colloidal systems and do not need to be repeated. In the following paragraphs, efforts are made to outline the difficulties which arise and where possible to resolve some of them. 2. STATEMENT OF PROBLEM Although it has been acknowledged in passing in the standard work (1), that it is inappropriate to assume a conducting inter1Present address: Alumina Partners of Jamaica,
Spur Tree, Jamaica, W.I.
face when treating the interaction of nonmetallic colloidal particles, this is not enough, as major discrepancies can arise. It can be stated at the outset that there are systems involving dielectrics, where the conducting model and a Boltzmann distribution of ions, yields results which do not admit experimental support over a range of particleparticle separation. There is also experimental evidence (5, 6) for some systems, that at particle separations of the order of 40/~ a special situation develops in the structural properties of the solvent and counter ions which make conventional solution theories inadmissible. By sheer admission of the physical situation, the colloid micelles have to be regarded as solid heterogenous crystals (7) and not as liquid systems, even though such micelles may be dispersed in a liquid. In addition, the finite thickness of the dielectric particles and the discrete nature of the surface charge cannot be approximated to an infinitely thick conductor-bearing free surface change. Other difficulties such as the finite size of the ions and their specific ion-dipole interactions with the solvent are enough to invalidate any theory based on macroscopic electrostatic parameters. Experimental illustrations of these problems can be found in fields as diverse as surfactant chemistry (8), civil engineering (9), and electrochemistry (10). In the work of Mysels (8), one sees a manifestation of the hydration effects of the ions at work in controlling the
435 Copyright O 1973 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vol. 45, No. 3, December 1973
436
JORDINE
development of second black films, whereas in Arnold's (9) soil consolidation work it is shown that, contrary to the normal repulsive forces expected from the theory of the diffuse double layer for a parallel-plate system, the forces become attractive over a certain particle-particle separation, and an extra increment is contributed to the externally applied pressure. Since the work by Arnold (9) on a Volclay bentonite is an engineering illustration and easily visualized by a wide audience, a discussion follows in order to demonstrate the problem fully. In (9, Fig. 5), it can be observed that at a given voids ratio of a Volclay bentonite, there is a sudden change in slope of the curve for the permeability coefficient, implying a sudden increase in the expulsion of water over the normal value expected. Figure 6 of Ref. (9) shows the corresponding pressure relationships during consolidation. In this figure, the ratio of the externally applied pressure to the theoretical pressure based on double layer theory is termed the activity factor and plotted versus particle-particle spacings estimated from the voids ratio. From the shape of the curve and the decrease shown in the activity factor from the maximum to the minimum, some unaccounted pressure must be operating internally to account for the increased permeability observed in Fig. 5 of Ref. (9) over precisely this region. Since there is nothing in the developments of double layer theory that could explain such a behavior, it must be concluded that as a theory, it is inadequate over this range. Further since the force is the negative of the derivative of the energy with respect to distance, Fig. 6 of Ref. (9) can only be explained by a potential energy curve such as that shown in Fig. 3 of Ref. (7). Other difficulties which arise from the specificity of ions can be best observed from the data in Refs. (5) and (6). Along with Ref. (8) these data demonstrate the specific hydration effects of the ions in a remarkable way. Taking lithium as an example, its
specific effects can be seen in two roles: first, as the only inorganic ion capable of activating swelling in the vemiculite crystal and, second, as the only complete inhibitor of the development of the second black film (8). Again these are phenomena which lie outside the scope of conventional theory. 3. PROPOSALS FOR RESOLUTION OF THE PROBLEM As pointed out in the introduction, one of important considerations to be accepted is that at close separations of the order of 40 A, the parallel plate colloidal system has to be considered as a heterogenous crystal and not as a liquid. Such a problem therefore can be defined and set up as in any solid state system involving a crystal lattice (7, 11). This approach is already popular in the so called lattice theories of liquid structure, and in the case of the bentonite and vermiculite systems is physically accurate (5). See also Figs. 1 and 2. In continuation of our discussion, the consequence of equating a solid dielectric colloidal particle to an infinitely thick conductor is that, in calculations of the total energy of the electric field, the energy of the field in the solid phase is omitted. This follows since the energy density in a dielectric is proportional to the square of the electric field, but the field in a conductor is zero and consequently the energy. This oversight of the energy due to the polarization of the solid will become manifest at close separations where the volume of the solid dielectric particle is of the same order of magnitude as the intervening liquid phase. It can be deduced that, with comparable polarizabilities for the two phases, the electrical energy of the two phases may be almost equal and, what is more, may have different signs algebraically (11). It is obvious that we are faced with two errors which do not cancel, but rather reinforce each other. One error ignores terms, while the other presupposes an inadmissible model physically.
Journal oy Colloid and InterJace Science, Vol. 45, No. 3, December 1973
NONEQUILIBRIUM MODELS From the foregoing, it can be seen that, even if for expediency in algebra, it is necessary to assume a conducting particle, there is a critical lower bound of particle separation 2D = L where such a model is inapplicable and does not make sense physically. Some relationship to the particle thickness b, such as L / b = k is to be expected. Of deeper physical significance would be a relationship involving the ratio of the energies of the electric field in the solid and liquid phases. See, for instance, Ref. (11) and the thermodynamic data in Ref. (12), which unfortunately does not extend to large enough spacings. Calculations (7, 11) show that, for a lattice model, the field energy in the solid approaches zero asymptotically from negative values according to the Laplace transform of a Bessel function of the first order. The expression for the energy in the liquid phase is more complex but is shown to be a path of decreasing potential energy, which is in direct contrast to that for the solid. In addition, peculiarities of liquid structure and specificity of ions give rise to immense particle-particle forces at close separations, which cannot be accounted for in a simple way by point charge electrostatics. Such problems need to be treated by techniques such as those developed by Bernal (13) and Debye (14). In these treatments, the classical electric field energy term arises as in any linear dielectric continuum that is electrically saturated but in addition an empirical term which accounts for the microscopic structural features and the interactions of the solvent and ions predominates.
437
however quite general for any charged dielectric plates, hence there is no real specialization in the equations developed in Refs. (7) and (11) for example. A change in the values of the particle thickness, surface charge and dielectric constant is all that is required to apply the equations generally to any plateshaped system. In support of this, it may be recalled that there is a general correspondence in the behavior of diverse systems such as ionized monolayers (8) and clay minerals (5, 6). The correspondence between such widely varying systems indicates that a wide range of extrapolation is possible from conclusions and facts derived from a study of the bentonite and vermiculite systems. Such a procedure of extrapolation is of course a necessary part of any discipline, and mere mathematical insights are pointless without a direct physical model as a reference point. In regard to parallel plate systems, the most suitable candidate that can be visualized as a model is the lamellar crystal vermiculite. From Figs. 1 and 2 we observe that as a model for ideal parallel-plate interaction vermiculite is unsurpassed. Physically also, the crystal is a heterogenous solid (Fig. 1) which after a tenfold expansion still maintains its integrity as a solid in Fig. 2, even though
!ii;!i!;!!!!i:i !i; :)?i
: ?i))iiiiii;¸)!:iil ii iili;i!ii!
4. MODELS AND REVIEW OF SOME CALCULATIONS So far the discussion has been modelled on bentonite systems as though they were the only plate-shaped particles. Indeed, they lend themselves to experimental study of equilibrium interparticle spacings by X-ray diffraction, more than any other known systems and hence represent the best model systems. The electrostatics applied and discussed is
FIG. 1. Vermiculitecrystal at equilibrium position.
Journal of Colloid and Interface Science, Vol. 45, No. 3, D e c e m b e r 1973
438
JORDINE 5. DISCUSSION
FIo. 2. Vermiculite crystal in swollen state. gel-like. The stability phenomena at various particle separations are shown in Ref. (6). Examination of the data in Ref. (6) reveal that, taking lithium vermiculite as an example, the potential energy curve must display a minimum at a particle-particle separation of 2D - 5 A and a saddle point between 5 A and 72 A. Similar arguments apply for lithium and sodium bentonites. It can be seen, therefore, that the potential energy curves for these systems are all of the form shown in Ref. (7). Bearing these data in mind, if we seek the appropriate model based on what X-ray studies (6) have shown for vermiculite and calculations of the potential energy which will satisfy the above, the suggested model will be slabs of dielectric interspersed regularly by counterions. For simplicity, this is reduced to a cell model in which each ion is assumed to have a region of influence over two solid particle surfaces which demarcate a liquid region. Proceeding on this model for the vermiculite system by the methods of Refs. (7) and (11), Fig. 3 was obtained. If we estimate the location of the minimum from this curve, it will be seen to be very close to the experimental value of 2 D = 5 A determined in Ref. (6) by X-ray diffraction.
It cannot be denied that simple electrostatic models can yield the correct type of energy curves and give minima which are close to those found experimentally. See also the Figures in Ref. (II). The depth of the minima may not be exact but they seem to have the correct multiple of kT. It also seems that a minimum which shows a binding energy of the order of 6-8 kT resists expansion, and the crystals do not swell beyond this spacing. This is the case for all the inorganic ions listed except lithium and sodium. In passing, it may be observed that all the potential energy curves may be recognized (15) as curves which depict the potential energy of a two-body system under central forces. This suggests a critical energy of escape for the particles to separate beyond the minimum. The soil mechanics data of Arnold (9) on Volclay bentonite now find a logical explanation in terms of the family of potential energy curves cited, which show that the force between bentonite particles change from one of repulsion to one of attraction as the hump in the curve is approached. The same feature holds for vermiculite Fig. 3 and is supported by the X-ray and swelling data in Ref. (6). The simple model we have chosen gives a faithful representation of the particle-particle spacings of charged particles as well as the observed behavior in soil mechanics. The implications and need for such specific binding models to explain many coagulation phenomena especially with polyvalent ions need no assertion. Further the activation and stability of biological, industrial, and mineralogical surfaces can, in the author's opinion, only be explained by a spectrum of potential energy curves for the various ions. The specific behavior of the inorganic ions in Mysels work (8) on black films also fits this spectrum of energy curves. Returning in closing to the old controversy due to Langmuir (16), it must be concluded as he suggested, that stable biparticle aggre-
Journal of Colloid and Interface Science, Vo]. 45, No. 3, December 1973
NONEQUILIBRIUM MODELS
439
SO /,0 30 20 10 0 10"~ En==I,=~ -10 "20 -30
-40 -50 -60
-70 -80 -90
-100
2 a ~ s s 7 8 9 ~o~,2,a~s~s~7~8~20
0(i) FIG. 3. Energy of interaction between two faces of vermiculite in relation to one-half distance of separation. gates are i n d e e d achieved to a large extent, b y " h y d r a t i o n " forces c o u n t e r a c t i n g Coulombic forces. L a n g m u i r ' s general idea c a n n o t therefore be easily dismissed (1). REFERENCES 1. VERWEY, E. J. W., AND OVERBEEK, J. T. G., "Theory of Stability of Lyophobic Colloids" Elsevier, New York 1948, 2. DEBYE, P., AND HUCKEL, E., Physik. Z. 24, 185 (1923). 3. CHAPMAN,D. L., Phil. Mag. 25, 475 (1913). 4. GouY, G., Ann. Phys. 7, 129 (1917). 5. NogRIsrI, K., Discuss. Faraday Soc. 18, 120 (1954). 6. NORRISH, K., AND RAUSELL-COLOM,J. A., Clays Clay Minerals 10, 123 (1963).
7. HURST, C. A., AND JORDINE, E. ST. A., J. Chem. Phys. 41, 2735 (1964). 8. JONES, M. N., MYSELS, K. J., AND SCHOLTEN, P. C., Trans. Faraday Soc. 62, 1336 (1966). 9. ARNOLD, M., "Int. Soc. Soil Mech. and Found. Eng. Conf. 6th," p. 12, 1965. 10. MAcDoNALD, J. R., AND BAgLOW, C. A., J. Chem. Phys. 36, 3062 (1962). 11. JORDINE, E. ST. A., STEEL, B. J., AND WOLFE, J. D., Bull. Chem. Soc. Japan 38, 199 (1965). 12. BARSHAD,I., Clays Clay Minerals 8, 84 (1960). 13. BERNAL, J. D., AND FOWLER, R. H., J. Chem. Phys. 1, 515 (1933). 14. DEBYE, P., "Polar Molecules." Dover, New York, 1929. 15. GOLDSTEm, H., "Classical Mechanics." AddisonWesley, Reading, MA., 1959. 16. LANGMUIR, I., J. Chem. Phys. 6, 873 (1938).
Journal of Colloid and InterfaceScience, Vol. 45, No. 3. December 1973