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Transient Cellular Convection in Electrically Polarized Colloidal Suspensions
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
202, 562–565 (1998)
CS985443
LETTER TO THE EDITOR Transient Cellular Convection in Electrically Polarized Colloidal Suspensions
We present theoretical evidence from computational fluid dynamics for transient isothermal roll-cell convection induced by very strong vertical electric fields in thin horizontal layers of waterin-oil emulsions initially at sedimentation equilibrium under gravity. This convection is driven by the Kelvin body force, which depends on the concentration gradient of droplets and on differences between the isotropic dielectric permittivities of the two phases of a nonionic emulsion. Provided that the applied voltage and hence a nondimensional electric Rayleigh number N is sufficiently large compared to a gravitational Rayleigh number R, rollcell convection sets in, grows, and then decays. If N is sufficiently large this roll-cell convection disturbs the local composition of the suspension and increases the spatial homogeneity of the emulsion. This electroconvective effect, which can accompany dielectrophoresis, is of interest as a novel form of transient pattern formation and as an example of electrically induced stirring. Similar convection patterns should be observable in related colloidal suspensions of fine particles with high dielectric permittivities dispersed in carrier fluids with lower dielectric permittivities and densities. q 1998 Academic Press Key Words: transient electroconvection; isothermal roll-cell convection; water–oil emulsions—electrically induced stirring; dielectrophoresis.
We have modeled the electrohydrodynamic behavior of emulsions somewhat simplistically ( 2 ) using the Navier – Stokes equation for viscous Newtonian flow, a Fickian diffusion equation incorporating gravitational and dielectrophoretic fluxes for the droplets, the hydrodynamic continuity equation for mass conservation, and the Maxwell equation, Çr( e0E / P ) Å 0, consistent with the absence of free charges. The last condition requires the emulsion to be stabilized by nonionic surfactants. Our two-dimensional model employs the barycentric velocity ( 3 ) v , which is a function of the horizontal coordinate x , the vertical coordinate y and the time t . We describe the incompressible flow in terms of a stream function c according to v Å curl( ck),
[1]
where (i, j, k) is the triad of unit vectors along the x, y, and z axes. Thus the vorticity W Å krcurl v Å 0Ç2c. Simple linear constitutive expressions are assumed for the density, r Å r2 (1 / ac), of the suspension and for its relative permittivity, e Å e2 (1 / bc). In these expressions the subscript 2 refers to the continuous phase and 1 refers to the droplets. Thus, a Å r1 / r2 0 1 measures the ratio of the densities of the two phases, b Å e1 / e2 0 1, and c Å c1 is the local mass-fraction of the droplets in the emulsion. The nondimensional governing equations for a dilute emulsion are the Navier–Stokes equation, Ç 2W 0 Wt 0 Rcx Å vrÇW 0 N krÇ c 1 ÇE 2 ,
[2]
(including Boussinesq approximations), the droplet diffusion equation, Consider a water-in-oil (W/O) emulsion containing droplets 0.1 mm in radius sandwiched between two perfectly conducting horizontal plates separated by a small vertical distance d of about 3 mm. It is well known that gravity sets up a significant concentration gradient in a colloidal suspension and eventually leads to a state of sedimentation equilibrium. The water droplets have a dielectric permittivity significantly higher than that of the continuous phase. The impulsive application of a potential difference, V volts, between the plates polarizes the spatially inhomogeneous emulsion and the resulting electric field gradient ÇE produces relative motion of the polarized droplets and the continuous phase. This motion often takes the form of dielectrophoresis (1) of these particles until a new equilibrium state is reached. Here we argue that whenever an electric Rayleigh number N is sufficiently large relative to a gravitational Rayleigh number R (see Eq. [5]) the resulting electrohydrodynamic instability can induce a temporary flow in which some of the electrical energy imparted to the colloidal system is dissipated through the formation of convective roll cells. Such transient flows can be very effective in homogenizing the spatial distribution of colloidal particles. It is remarkable that the roll-cell convection in a thin horizontal layer of polarized suspension occurs under isothermal conditions. For suspensions of this type the gravitational body force is stabilizing but the Kelvin body force, PrÇE, associated with the applied voltage (P is the electric polarization), reinforces any excursions of the droplets vertically upwards and is therefore destabilizing.
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[3]
where J Å 0S(Ç c / Kcj 0 LcÇE 2 ), and the Maxwell equation, Çr(1 / bc)E Å 0,
[4]
for the vanishing of the divergence of the electric displacement in the absence of free charges. Subscripts t and x denote partial derivatives. The diffusive mass flux J of the droplets is defined in terms of a parameter S, which is the nondimensional ratio, D/ n, of the droplet diffusion coefficient D to the kinematic viscosity, n, of the emulsion. In these equations spatial coordinates have been scaled with respect to d, time with respect to d 2 / n, and the standards for velocity, vorticity, and electric field strength E Å 0Ç F are £s Å n /d, Ws Å n /d 2 and Es Å V/d, respectively. The gravitational and electric Rayleigh numbers in the vorticity equation [2] are defined by
R Å ar2 gd 3 / nh
and
NÅ
1 e0e2bV 2 / nh, 2
and the corresponding parameters in the diffusion equation are
562
0021-9797/98 $25.00 Copyright q 1998 by Academic Press All rights of reproduction in any form reserved.
AID
ct / vrÇc / Çr J Å 0,
coidas
[5]
LETTER TO THE EDITOR
TABLE 1 Values of Physical Parameters at 207C Used in the Calculations c0 d g n V R
Å Å Å Å Å Å
r1 r2 e1 e2 D N
0.050 3.0 mm 9.8 m s02 8.33 1 1007 m2 s01 450 V 1.59 1 104
K Å M1 gd(1 0 r2 / r1 )/RT
and
Å Å Å Å Å Å
LÅ
1000 kg m03 960 kg m03 78.5 2.0 2.68 1 10012 m2 s01 1.03 1 105
1 e0e2bV 2 M1 / r0 RTd 2 , [6] 2
where g is the acceleration due to gravity, e0 is the permittivity of free space, h the shear viscosity of the emulsion, M1 the molar mass of the droplets and RT their thermal energy at the absolute temperature T . To complete the system of governing equations we require the velocity and the diffusive flux to vanish at the walls of the rectangular domain. The nondimensional electric potential takes the value 1 at the top plate, 0 on the bottom plate, and the normal component of the electric field vanishes on the side walls. The initial conditions are that the vorticity has zero amplitude throughout the fluid domain and that the initial sedimentation profile is gc0 exp ( 0 Ky ) , where c0 is the spatially uniform mass fraction of colloidal droplets prior to gravitational sedimentation and g Å K / [1 0 exp ( 0 K ) ] . An extremely small homogeneous velocity field of 3 1 10 011 m s 01 was applied in the positive y direction to initiate flow of the unstable system. Using the physical parameters listed in Table 1 we have solved the governing equations on a rectangular domain of length 3 d and height d using Fastflo finite-element software. This
563
domain was divided into 2838 six-node triangular elements and Crank – Nicolson time stepping was used to follow the evolution of the solutions. Although the quantitative values of these solutions at a particular time after the voltage is applied depend on the initial velocity field, these solutions seem structurally independent of the velocity perturbation used to trigger the flow. Thus the qualitative appearance of the roll cells, the maximum value of the mass flux, and the maximum velocities of the roll cells appear to be insensitive to the magnitude of this initial velocity. To test these conclusions experimentally, at a semiquantitative level, it may be desirable to impose a voltage oscillating at several kHz to avoid complications associated with finite conductivities. The voltage V can then be interpreted as a root mean square value. A model W / O emulsion ( with c0 Å 0.05 ) at sedimentation equilibrium was subjected to the spatially inhomogeneous electric field created by a potential difference of 450 V applied across electrodes separated by 3 mm. Shortly after the voltage is applied incipient roll cells are evident ( see Fig. 1a ) . The rapidly growing cellular convection soon affects the early vertically stratified concentration profile ( see Fig.1b ) with significant undulations evident ( see Fig. 2b ) only 5.4 s after the electric field is applied. After 5.7 s the spiral motions associated with the roll cells are already causing major upwelling and sinking in the distribution of water droplets over the interior of the fluid domain ( see Fig. 3b ) . Dielectrophoresis in the absence of the roll-cell convection leads to a small mass flux vertically upward until a new state of equilibrium is reached. This mass flux is increased by many orders of magnitude by electrohydrodynamic roll-cell convection. Finally we remark that we have also studied initiation of convection by dielectrophoresis from the quiescent state of sedimentation equilibrium. Here the solutions to the governing equations appear to be less structurally stable. This subject, together with a description of the decay of the transient convection as the suspension becomes more spatially homogeneous, will be described in an expanded account.
FIG. 1. (a) Streamlines of roll-cells produced in the W/O emulsion 2.8 s after 450 V is applied across the two horizontal electrodes. Adjacent rollcells rotate in opposite directions. The equally spaced contours of the streamfunction range from 05.9 1 10 011 m2 s 01 at the center of the second rollcell from the left (with clockwise rotation) to /5.9 1 10 011 m2 s 01 at the center of the fifth roll-cell. (b) Equally spaced contours of the water droplet mass-fraction, c, for the W/O emulsion. At the bottom plate c Å 0.087 while at the top plate c Å 0.026.
AID
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LETTER TO THE EDITOR
FIG. 2. (a) Streamlines of roll-cells produced in the W/O emulsion 5.4 s after 450 V is applied across the electrodes. The equally spaced contours of the stream function range from /2.3 1 10 06 m2 s 01 at the center of the third roll-cell from the left (with counterclockwise rotation) to 02.3 1 10 06 m2 s 01 at the center of the fourth roll-cell. (b) Equally spaced contours of the water droplet mass-fraction, c, for the W/O emulsion. At the bottom plate c Å 0.087 while at the top plate c Å 0.026. These concentration contours, which were effectively horizontal 2.6 s earlier, exhibit pronounced undulations due to the rapidly increasing hydrodynamic convection.
FIG. 3. (a) Streamlines of roll-cells produced in the W/O emulsion 5.7 s after 450 V is applied across the electrodes. The equally spaced contours of the stream function range from /4.6 1 10 06 m2 s 01 at the center of the third roll-cell from the left with counterclockwise rotation to 04.5 1 10 06 m2 s 01 at the center of the fourth roll-cell. (b) Equally spaced contours of the water droplet mass-fraction, c, for the same emulsion. Note the three pronounced upwellings of the water droplets at this epoch where the cellular convection has reached its maximum amplitude and the vertical mass flux is also close to its maximum value.
AID
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LETTER TO THE EDITOR Similar electrically induced stirring should be observable in related colloidal suspensions of fine particles with high dielectric permittivities ( such as barium titanate ( 4 ) and semiconductors ( 5 ) ) dispersed in organic liquids with lower dielectric permittivities and densities. We predict that electrically induced convection should also occur in creamed suspensions ( such as oil-in-water ( O / W ) emulsions ) where the particles have much lower permittivities and are less dense than the carrier fluid. However, in the case of O / W emulsions it is difficult to use very large electric fields and in many such emulsions well-known electrokinetic effects associated with double layers will normally take precedence over the flows described in this letter.
ACKNOWLEDGMENTS We thank Dr. R. J. Hunter, Mr. M. Kagan, Dr. C. L. Russell, and Dr. A. N. Stokes for valuable contributions and the Australian Research Council for financial support.
1. Pohl, H. A., ‘‘Dielectrophoresis: The Behaviour of Neutral Matter in Non-Uniform Electric Fields.’’ Cambridge Univ. Press, Cambridge, UK, 1978.
JCIS 5443
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6g42$$$341
Peter John Stiles 1 Helen May Regan School of Chemistry Macquarie University New South Wales 2109, Australia
Received October 17, 1997; accepted January 23, 1998
REFERENCES
AID
2. Lhuillier, D., in ‘‘Flow of Particles in Suspensions’’ (U. Schaflinger, Ed.) pp. 39–91. Springer-Verlag, New York, 1996. 3. de Groot, S. R., and Mazur, P., ‘‘Non-Equilibrium Thermodynamics.’’ North-Holland, Amsterdam, 1962. 4. Trau, M., Sankaran, S., Saville, D. A., and Aksay, I. A., Nature 374, 437 (1995). 5. O’Brien, R. W., J. Colloid Interface Sci. 177, 280 (1996).
05-13-98 07:38:39
1
To whom correspondence should be addressed.
coidas
202, 562–565 (1998)
CS985443
LETTER TO THE EDITOR Transient Cellular Convection in Electrically Polarized Colloidal Suspensions
We present theoretical evidence from computational fluid dynamics for transient isothermal roll-cell convection induced by very strong vertical electric fields in thin horizontal layers of waterin-oil emulsions initially at sedimentation equilibrium under gravity. This convection is driven by the Kelvin body force, which depends on the concentration gradient of droplets and on differences between the isotropic dielectric permittivities of the two phases of a nonionic emulsion. Provided that the applied voltage and hence a nondimensional electric Rayleigh number N is sufficiently large compared to a gravitational Rayleigh number R, rollcell convection sets in, grows, and then decays. If N is sufficiently large this roll-cell convection disturbs the local composition of the suspension and increases the spatial homogeneity of the emulsion. This electroconvective effect, which can accompany dielectrophoresis, is of interest as a novel form of transient pattern formation and as an example of electrically induced stirring. Similar convection patterns should be observable in related colloidal suspensions of fine particles with high dielectric permittivities dispersed in carrier fluids with lower dielectric permittivities and densities. q 1998 Academic Press Key Words: transient electroconvection; isothermal roll-cell convection; water–oil emulsions—electrically induced stirring; dielectrophoresis.
We have modeled the electrohydrodynamic behavior of emulsions somewhat simplistically ( 2 ) using the Navier – Stokes equation for viscous Newtonian flow, a Fickian diffusion equation incorporating gravitational and dielectrophoretic fluxes for the droplets, the hydrodynamic continuity equation for mass conservation, and the Maxwell equation, Çr( e0E / P ) Å 0, consistent with the absence of free charges. The last condition requires the emulsion to be stabilized by nonionic surfactants. Our two-dimensional model employs the barycentric velocity ( 3 ) v , which is a function of the horizontal coordinate x , the vertical coordinate y and the time t . We describe the incompressible flow in terms of a stream function c according to v Å curl( ck),
[1]
where (i, j, k) is the triad of unit vectors along the x, y, and z axes. Thus the vorticity W Å krcurl v Å 0Ç2c. Simple linear constitutive expressions are assumed for the density, r Å r2 (1 / ac), of the suspension and for its relative permittivity, e Å e2 (1 / bc). In these expressions the subscript 2 refers to the continuous phase and 1 refers to the droplets. Thus, a Å r1 / r2 0 1 measures the ratio of the densities of the two phases, b Å e1 / e2 0 1, and c Å c1 is the local mass-fraction of the droplets in the emulsion. The nondimensional governing equations for a dilute emulsion are the Navier–Stokes equation, Ç 2W 0 Wt 0 Rcx Å vrÇW 0 N krÇ c 1 ÇE 2 ,
[2]
(including Boussinesq approximations), the droplet diffusion equation, Consider a water-in-oil (W/O) emulsion containing droplets 0.1 mm in radius sandwiched between two perfectly conducting horizontal plates separated by a small vertical distance d of about 3 mm. It is well known that gravity sets up a significant concentration gradient in a colloidal suspension and eventually leads to a state of sedimentation equilibrium. The water droplets have a dielectric permittivity significantly higher than that of the continuous phase. The impulsive application of a potential difference, V volts, between the plates polarizes the spatially inhomogeneous emulsion and the resulting electric field gradient ÇE produces relative motion of the polarized droplets and the continuous phase. This motion often takes the form of dielectrophoresis (1) of these particles until a new equilibrium state is reached. Here we argue that whenever an electric Rayleigh number N is sufficiently large relative to a gravitational Rayleigh number R (see Eq. [5]) the resulting electrohydrodynamic instability can induce a temporary flow in which some of the electrical energy imparted to the colloidal system is dissipated through the formation of convective roll cells. Such transient flows can be very effective in homogenizing the spatial distribution of colloidal particles. It is remarkable that the roll-cell convection in a thin horizontal layer of polarized suspension occurs under isothermal conditions. For suspensions of this type the gravitational body force is stabilizing but the Kelvin body force, PrÇE, associated with the applied voltage (P is the electric polarization), reinforces any excursions of the droplets vertically upwards and is therefore destabilizing.
JCIS 5443
/
6g42$$$341
05-13-98 07:38:39
[3]
where J Å 0S(Ç c / Kcj 0 LcÇE 2 ), and the Maxwell equation, Çr(1 / bc)E Å 0,
[4]
for the vanishing of the divergence of the electric displacement in the absence of free charges. Subscripts t and x denote partial derivatives. The diffusive mass flux J of the droplets is defined in terms of a parameter S, which is the nondimensional ratio, D/ n, of the droplet diffusion coefficient D to the kinematic viscosity, n, of the emulsion. In these equations spatial coordinates have been scaled with respect to d, time with respect to d 2 / n, and the standards for velocity, vorticity, and electric field strength E Å 0Ç F are £s Å n /d, Ws Å n /d 2 and Es Å V/d, respectively. The gravitational and electric Rayleigh numbers in the vorticity equation [2] are defined by
R Å ar2 gd 3 / nh
and
NÅ
1 e0e2bV 2 / nh, 2
and the corresponding parameters in the diffusion equation are
562
0021-9797/98 $25.00 Copyright q 1998 by Academic Press All rights of reproduction in any form reserved.
AID
ct / vrÇc / Çr J Å 0,
coidas
[5]
LETTER TO THE EDITOR
TABLE 1 Values of Physical Parameters at 207C Used in the Calculations c0 d g n V R
Å Å Å Å Å Å
r1 r2 e1 e2 D N
0.050 3.0 mm 9.8 m s02 8.33 1 1007 m2 s01 450 V 1.59 1 104
K Å M1 gd(1 0 r2 / r1 )/RT
and
Å Å Å Å Å Å
LÅ
1000 kg m03 960 kg m03 78.5 2.0 2.68 1 10012 m2 s01 1.03 1 105
1 e0e2bV 2 M1 / r0 RTd 2 , [6] 2
where g is the acceleration due to gravity, e0 is the permittivity of free space, h the shear viscosity of the emulsion, M1 the molar mass of the droplets and RT their thermal energy at the absolute temperature T . To complete the system of governing equations we require the velocity and the diffusive flux to vanish at the walls of the rectangular domain. The nondimensional electric potential takes the value 1 at the top plate, 0 on the bottom plate, and the normal component of the electric field vanishes on the side walls. The initial conditions are that the vorticity has zero amplitude throughout the fluid domain and that the initial sedimentation profile is gc0 exp ( 0 Ky ) , where c0 is the spatially uniform mass fraction of colloidal droplets prior to gravitational sedimentation and g Å K / [1 0 exp ( 0 K ) ] . An extremely small homogeneous velocity field of 3 1 10 011 m s 01 was applied in the positive y direction to initiate flow of the unstable system. Using the physical parameters listed in Table 1 we have solved the governing equations on a rectangular domain of length 3 d and height d using Fastflo finite-element software. This
563
domain was divided into 2838 six-node triangular elements and Crank – Nicolson time stepping was used to follow the evolution of the solutions. Although the quantitative values of these solutions at a particular time after the voltage is applied depend on the initial velocity field, these solutions seem structurally independent of the velocity perturbation used to trigger the flow. Thus the qualitative appearance of the roll cells, the maximum value of the mass flux, and the maximum velocities of the roll cells appear to be insensitive to the magnitude of this initial velocity. To test these conclusions experimentally, at a semiquantitative level, it may be desirable to impose a voltage oscillating at several kHz to avoid complications associated with finite conductivities. The voltage V can then be interpreted as a root mean square value. A model W / O emulsion ( with c0 Å 0.05 ) at sedimentation equilibrium was subjected to the spatially inhomogeneous electric field created by a potential difference of 450 V applied across electrodes separated by 3 mm. Shortly after the voltage is applied incipient roll cells are evident ( see Fig. 1a ) . The rapidly growing cellular convection soon affects the early vertically stratified concentration profile ( see Fig.1b ) with significant undulations evident ( see Fig. 2b ) only 5.4 s after the electric field is applied. After 5.7 s the spiral motions associated with the roll cells are already causing major upwelling and sinking in the distribution of water droplets over the interior of the fluid domain ( see Fig. 3b ) . Dielectrophoresis in the absence of the roll-cell convection leads to a small mass flux vertically upward until a new state of equilibrium is reached. This mass flux is increased by many orders of magnitude by electrohydrodynamic roll-cell convection. Finally we remark that we have also studied initiation of convection by dielectrophoresis from the quiescent state of sedimentation equilibrium. Here the solutions to the governing equations appear to be less structurally stable. This subject, together with a description of the decay of the transient convection as the suspension becomes more spatially homogeneous, will be described in an expanded account.
FIG. 1. (a) Streamlines of roll-cells produced in the W/O emulsion 2.8 s after 450 V is applied across the two horizontal electrodes. Adjacent rollcells rotate in opposite directions. The equally spaced contours of the streamfunction range from 05.9 1 10 011 m2 s 01 at the center of the second rollcell from the left (with clockwise rotation) to /5.9 1 10 011 m2 s 01 at the center of the fifth roll-cell. (b) Equally spaced contours of the water droplet mass-fraction, c, for the W/O emulsion. At the bottom plate c Å 0.087 while at the top plate c Å 0.026.
AID
JCIS 5443
/
6g42$$$341
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coidas
564
LETTER TO THE EDITOR
FIG. 2. (a) Streamlines of roll-cells produced in the W/O emulsion 5.4 s after 450 V is applied across the electrodes. The equally spaced contours of the stream function range from /2.3 1 10 06 m2 s 01 at the center of the third roll-cell from the left (with counterclockwise rotation) to 02.3 1 10 06 m2 s 01 at the center of the fourth roll-cell. (b) Equally spaced contours of the water droplet mass-fraction, c, for the W/O emulsion. At the bottom plate c Å 0.087 while at the top plate c Å 0.026. These concentration contours, which were effectively horizontal 2.6 s earlier, exhibit pronounced undulations due to the rapidly increasing hydrodynamic convection.
FIG. 3. (a) Streamlines of roll-cells produced in the W/O emulsion 5.7 s after 450 V is applied across the electrodes. The equally spaced contours of the stream function range from /4.6 1 10 06 m2 s 01 at the center of the third roll-cell from the left with counterclockwise rotation to 04.5 1 10 06 m2 s 01 at the center of the fourth roll-cell. (b) Equally spaced contours of the water droplet mass-fraction, c, for the same emulsion. Note the three pronounced upwellings of the water droplets at this epoch where the cellular convection has reached its maximum amplitude and the vertical mass flux is also close to its maximum value.
AID
JCIS 5443
/
6g42$$5443
05-13-98 07:38:39
coidas
565
LETTER TO THE EDITOR Similar electrically induced stirring should be observable in related colloidal suspensions of fine particles with high dielectric permittivities ( such as barium titanate ( 4 ) and semiconductors ( 5 ) ) dispersed in organic liquids with lower dielectric permittivities and densities. We predict that electrically induced convection should also occur in creamed suspensions ( such as oil-in-water ( O / W ) emulsions ) where the particles have much lower permittivities and are less dense than the carrier fluid. However, in the case of O / W emulsions it is difficult to use very large electric fields and in many such emulsions well-known electrokinetic effects associated with double layers will normally take precedence over the flows described in this letter.
ACKNOWLEDGMENTS We thank Dr. R. J. Hunter, Mr. M. Kagan, Dr. C. L. Russell, and Dr. A. N. Stokes for valuable contributions and the Australian Research Council for financial support.
1. Pohl, H. A., ‘‘Dielectrophoresis: The Behaviour of Neutral Matter in Non-Uniform Electric Fields.’’ Cambridge Univ. Press, Cambridge, UK, 1978.
JCIS 5443
/
6g42$$$341
Peter John Stiles 1 Helen May Regan School of Chemistry Macquarie University New South Wales 2109, Australia
Received October 17, 1997; accepted January 23, 1998
REFERENCES
AID
2. Lhuillier, D., in ‘‘Flow of Particles in Suspensions’’ (U. Schaflinger, Ed.) pp. 39–91. Springer-Verlag, New York, 1996. 3. de Groot, S. R., and Mazur, P., ‘‘Non-Equilibrium Thermodynamics.’’ North-Holland, Amsterdam, 1962. 4. Trau, M., Sankaran, S., Saville, D. A., and Aksay, I. A., Nature 374, 437 (1995). 5. O’Brien, R. W., J. Colloid Interface Sci. 177, 280 (1996).
05-13-98 07:38:39
1
To whom correspondence should be addressed.
coidas