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A Multivariate Analysis on the Sedimentation of Aqueous Silica Suspensions as Studied by Means of Time Domain Dielectric Spectroscopy
Journal of Colloid and Interface Science 214, 1–7 (1999) Article ID jcis.1999.6142, available online at http://www.idealibrary.com on
A Multivariate Analysis on the Sedimentation of Aqueous Silica Suspensions as Studied by Means of Time Domain Dielectric Spectroscopy Bjørnar Hauknes Pettersen, Egil Nodland, and Johan Sjo¨blom 1 Department of Chemistry, University of Bergen, Allegt. 41, N-5007 Bergen, Norway Received August 4, 1997; accepted February 16, 1999
of all factors influencing the suspension stability, numerous experiments should be performed. A way to reduce the number of experiments required is to use factorial design (11). A full factorial design using two levels of each variable requires 2 n experiments, n being the number of variables. For an increasing number of variables even this method requires many experiments. However, with a large n the desired information can often be obtained by using a reduced factorial design (12). In this article we have investigated the influence on suspension sedimentation of different variables and their interaction parameters. The derivative of static permittivity vs time is introduced as the response variable. In the parameter interval under study the permittivity is linearly dependent on the volume fraction of solid material in suspension.
The sedimentation of aqueous silica suspensions has been studied by means of time domain dielectric spectroscopy (TDDS) in the frequency range 20 MHz to 2 GHz. The experiments were designed with a 2 5–1 reduced factorial design and analyzed in terms of multivariate regression models. Five different variables were varied to investigate the corresponding suspension sedimentation. The influence of the different variables on the sedimentation of the silica suspensions is discussed. © 1999 Academic Press Key Words: suspension stability; sedimentation; dielectric spectroscopy; reduced factorial design; multiple regression; interactions.
INTRODUCTION
In the field of suspensions, stability and the factors influencing stability have been the subject of several studies during the past years (1– 6). The studies reported have mainly been concentrated on variation of one variable at a time. The interactions among the different variables governing the suspension stability have attracted less attention. According to Stokes’ law, the rate of sedimentation depends on gravity, the density difference between the continuous phase and the suspended phase, particle size, and viscosity. The particle size will be affected by particle–particle interactions resulting in flocs or large agglomerates. To follow the sedimentation rate, dielectric spectroscopy has proved to be a convenient measuring technique for investigating suspensions (7–10). The dielectric response of the suspension depends on the dielectric properties of the continuous and disperse phases and their volume fraction. A change in the amount of solid particles in the suspension will give rise to an altered dielectric response, if the static permittivity of the disperse phase differs from that of the continuous phase. The changes in the dielectric response (static permittivity) can be monitored as a function of time after the suspension is mixed. The change in static permittivity as a function of time is thus a measure of the sedimentation rate of the suspension. In order to study the effects 1
EXPERIMENTAL
Chemicals and Sample Preparation Silica particles were prepared using the following chemicals without further purification: absolute ethanol from Vinmonopolet, ammonia (25%) from Merck, sodium chloride from Merck, and tetraethyl orthosilane (TEOS) (98%) from Fluka Chemical. The water was doubly distilled (Seralpure PRO 90CN). Monodisperse silica particles were prepared according to the sol gel method (13). The concentrations of the different chemicals were taken from the work of Lindberg et al. (14) to give a diameter of 1 mm. The size was confirmed by electron microscopy (Fig. 1). The zeta potential of the silica particles was measured using a Zetasizer 4 (Malvern Instruments, UK). The ionic strength was 0.01 M NaCl. The pH was adjusted using NH 3 and acetic acid. The silica suspensions were prepared by weighing the components in glass vials. They were mixed by using an ultrasonic bath before they were introduced to the dielectric cell for sedimentation measurements. Measurements and Data Analysis The dielectric spectra were measured using the time domain spectroscopy technique (TDS) (15). The dielectric cell is
To whom correspondence should be addressed. 1
0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
2
¨ BLOM HAUKNES PETTERSEN, NODLAND, AND SJO
The pulses were observed in a time window of 50 ns and Fourier transformed in the frequency range 20 MHz to 2 GHz. The experimental data were fitted to a Debye function (19):
e *~ v ! 5
FIG. 1. SEM picture of monodisperse silica particles, d ' 1 mm.
placed at the end of a coaxial cable, and the difference in the reflected pulses from a reference liquid and the sample is recorded (16 –18). The complex permittivity of the sample, e*(v), is calculated after a Fourier transform to the frequency domain.
e *~ v ! 5 e 9~ v ! 2 i e 0~ v !
[1]
es 2 e` 1 e `. 1 1 i vt
[2]
Here e s is the static permittivity, e ` is the permittivity at high frequencies, v is the angular frequency, and t is the relaxation time. The experimental setup for the measurements is illustrated in Fig. 2. The dielectric sensor/cell used in this study has a cell length of 0.025 mm. The measuring system consists of a digital sampling oscilloscope (HP 54120A) and a pulse generator (HP 54121A). A computer, which is connected to the oscilloscope, performs all necessary calculations. The dielectric spectra are obtained at different times after the suspension has been introduced into the dielectric cell. The value of static permittivity, e s, is resolved for each spectrum and plotted against the time. A typical plot of e s vs time is shown in Fig. 3. The decline in e s is constant during the first measurements, and this value is used as a variable in the analysis. The data obtained were analyzed using Sirius for Windows (20).
FIG. 2. Experimental setup of time domain dielectric spectroscopy, instrumentation and specifications of sensor.
3
SEDIMENTATION OF AQUEOUS SILICA SUSPENSIONS
FIG. 3. Static permittivity vs time for sample 11 in the design.
RESULTS
Table 1 lists the variables chosen and the corresponding values for the high and low levels, as well as for the control points. The complete design matrix is shown in Table 2. The calculated response values (d e s/dt) are also included for the different samples in the same table. Figure 4 displays the corresponding regression coefficients, when Eq. [3] is fitted to the response. The coefficients with positive values will increase the value of d e s/dt, whereas the negative coefficients will decrease the decline, i.e., decrease the sedimentation in the suspension. The full model (Eq. [3]) will explain 100% of the variation in the response, which is evident when 16 variables are used for 16 samples. The full model predicts a value of 7.9 for the control points, while 10.4 was experimentally observed. The observed discrepancy reveals nonlinear effects among one or more of the variables. Predicted vs measured values of d e s/dt for a model based on the main parameters only are shown in Fig. 5. This model explains only 58% of the variation in the response. A model using cross terms must thus be introduced in order to better explain the variation in suspension sedimentation. A model based on the main factors, X i and X 1 X 2 , X 1 X 5 , X 2 X 5 , X 3 X 4 , and X 3 X 5 , is introduced in Fig. 6.
experiments. We have used a reduced factorial design, 2 5–1, thus reducing the required number of experiments to 16. This design is obtained by defining the level of variable 5 as a combination of variables 1, 2, 3, and 4 (X 5 5 X 1 X 2 X 3 X 4 ). Since we use a reduced factorial design, there will be an overlap between main effects and three-factor effects. However, one can assume that three-factor effects are negligible as compared to the main and two-factor effects. In a 2 5–1 reduced factorial design, the variables 1, 2, 3, and 4 are varied to include all the combinations of the levels chosen, while the levels of variable 5 are chosen from the combination of variables 1, 2, 3, and 4. A complete multiple regression model is obtained by using d e /dt as a dependent response variable, the five variables, and their cross terms and independent variables. The complete equation is Yˆ 5 5.33 1 0, 96X 1 1 1, 1X 2 2 1, 24X 3 2 1, 65X 4 1 1, 25X 5 1 1, 46X 1 X 2 1 0, 15X 1 X 3 1 0, 09X 1 X 4 1 0, 79X 1 X 5 1 0, 21X 2 X 3 2 0, 50X 2 X 4 2 0, 80X 2 X 5 1 0, 71X 3 X 4 2 0, 84X 3 X 5 2 0, 43X 4 X 5 .
Here X i corresponds to the different variables in the design as seen in Table 1. The regression coefficients are presented as bars in Fig. 4. For the main effects, a positive regression coefficient means that a change in the level from 21 to 11 increases the response, and a negative coefficient implies a decrease in the response by changing the level from 21 to 11. The positive values cause d e s/dt to increase, and vice versa for the negative values. Equation [3] explains all the variations in the response variable due to the full rank of the design matrix. The calculation of the control points also implies that there is a nonlinearity in the model. It is therefore most interesting to use the model to evaluate the impact on sedimentation by the different variables, not to predict the value of the sedimentation. A model based on the five main effects is not satisfactory to
TABLE 1 Variables and Levels
DISCUSSION
High level (1)
Low level (2)
X1 X2 X3
0.06 11 120
0.02 3 800
0.04 7 400
X4 X5
0.1 25
0.001 4
0.01 15
Factorial Design A two-level factorial design is a convenient way of screening the importance of the different variables in the system under study. With this method, one can extract the information on general trends in variables based on relatively few experiments. Although this cannot fully explore the wide region in the variable space, it will resolve major trends among the variables. The number of experiments required by a full factorial design increases rapidly as the number of variables increases. In this work we have chosen to study five different variables; a complete factorial design would require 2 5 5 32
[3]
Variables Volume fraction disperse phase pH Surface hydroxyl groups (dehydroxylation temperature a (°C)) Salinity [NaCl] Temperature (°C)
Control point
a Reflects the number of 2OH groups at the surface of the silica particles. X 3 5 120°C gives a high number of 2OH groups at the surface; X 3 5 800°C gives a low number of 2OH groups.
2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 20.039 20.039 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 0.0084 0.0084 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 0.0476 0.0476 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 0.0476 0.0476 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 0.998 0.998 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 0 0
2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 0.177 0.177
2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 20.818 20.818
1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 0.0069 0.0069
1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 20.037 20.037
1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 0.0084 0.0084
2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 20.144 20.144
1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 0.1764 0.1764
1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 20.818 20.818
1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 20.144 20.144
Y (de/dt) X4X5 1 X1X2X3 X3X5 1 X1X2X4 X3X4 1 X1X2X5 X2X5 1 X1X3X4 X2X4 1 X1X3X5 X2X3 1 X1X4X5 X1X5 1 X2X3X4 X1X4 1 X2X3X5 X1X3 1 X2X4X5 X1X2 1 X3X4X5 X5 1 X1X2X3X4 X4 X3 X2 X1
TABLE 2 2 5–1 Factorial Design with Cross Terms and Response Values
10.8 2.7 7.2 15 2.8 5.2 3.6 8.5 2.7 6.5 2.3 5.3 2.6 0.5 2.9 6.6 10.4 10.4
¨ BLOM HAUKNES PETTERSEN, NODLAND, AND SJO
4
describe the sedimentation process (see Fig. 5) and only accounts for 58% of the variation in the response. A model that better explains the variation in the sedimentation is obtained by including cross terms. Often there is a clear difference in the significance of the cross terms that can be seen from the magnitude of the regression coefficients. In this work there are no such clear limits concerning the significance of the individual cross effects. We have introduced a model with the cross terms X 1X 2, X 1X 5, X 2X 5, X 3X 4, and X 3X 5. A plot of predicted vs measured values is shown in Fig. 6. This model explains 96% of the variation as compared to the model with only main variables, which explains only 58% of the variation. The residuals from the model are presented in Fig. 7. In a normal distribution plot the residuals are sorted in an increasing order, each accounting for 1/16 of a normal distribution population. If the residuals fall on a straight line the assumption that the residuals are normally distributed holds. This is seen from Fig. 7. Dielectric Measurements on Suspension Sedimentation We have in previous work introduced time domain dielectric spectroscopy as a feasible method to study the sedimentation of suspensions (9, 10). The dielectric response of solid particle suspensions is due to the dielectric parameters of the continuous and disperse phases and the volume fraction solid particles. When sedimentation starts, there will be an increase in volume fraction solid particles at the bottom of the suspension. By placing a dielectric sensor at the bottom of a sedimentation vessel, it is possible to monitor this change in volume fraction of solid particles. We have previously shown that, based on theoretical models for the dielectric properties of heterogeneous mixtures, it is possible to transform the measured static permittivity to a volume fraction of solid particles. In this work we have chosen to study merely the change in static permittivity as a function of time after the sample is introduced into the sedimentation vessel. The response used in this experiment is d e s/dt in the time region where this value is constant. The derivative is proportional to the sedimentation rate of the suspension. A suspension with a high sedimentation rate will have a large d e s/dt due to a fast alteration in the concentration profile of solid particles. More stable suspensions give rise to less pronounced changes in d e s/dt. Sedimentation of Suspensions The stability of suspensions is dependent on a variety of factors. The sedimentation is mainly driven by the density difference between the solid particles and the continuous phase. Interactions between particles and interactions between particles and the continuous phase also play an important role in the observed stability of the suspension. If the particles flocculate, the observed stability will decrease due to the increased sedimentation rate of flocs compared to the sedimentation rate of individual particles. The balance of repulsive and
SEDIMENTATION OF AQUEOUS SILICA SUSPENSIONS
FIG. 4. Regression coefficients. The negative values tend to decrease the value of d e s/dt while the positive values tend to increase d e s/dt.
attractive interparticle forces dictates the degree of flocculation. The variables chosen in this study will all influence these interparticle forces to some extent. Variables 1– 4 listed in Table 1 influence the surface properties of the solid particles while variable 5 (temperature) should influence both particles (Brownian motion) and the continuous phase (viscosity). We have chosen to study monodisperse silica particles in order to more-or-less eliminate effects like particle polydispersity and gradients in gravity. For monodisperse systems like ours, the main effect will be interparticle interactions (flocculation/agglomeration) and their influence on the sedimentation rate. Evaluation of the Different Variables Influencing the Observed Suspension Sedimentation
5
FIG. 6. Measured vs predicted values of d e s/dt for model based on main variables and cross terms X 1 X 2 , X 1 X 5 , X 2 X 5 , X 3 X 4 , and X 3 X 5 . Explains 96% of the variance in the dependent variable.
suspensions. The five variables chosen are all known to influence the sedimentation. Volume fraction disperse phase (X 1). The value of d e s/dt increases as the volume fraction of solid particles increases. Due to volumetric reasons, the particle–particle interaction will be higher at higher f. This can result in, for instance, a higher degree of flocculation, which will accelerate the sedimentation. pH (X 2). A higher pH tends to increase de s/dt, indicating a less stable suspension against sedimentation. This is not in agreement, according to classical DLVO theory concerning zeta potential and stability. Silica has a high negative zeta potential at high pH and a low value at low pH (Fig. 8). One would therefore expect an increase in stability at high pH values. Allen and Matijevic (21) found that Ludox HS precipitated silica does not
In the following section, we will briefly examine the different variables and explain their influence on the stability of the
FIG. 5. Measured vs predicted values of d e s/dt for model based on main variables only. Explains 58% of the variance in the dependent variable.
FIG. 7. Fig. 6.
Normal plot of the residuals in model with cross terms, from
¨ BLOM HAUKNES PETTERSEN, NODLAND, AND SJO
6
FIG. 8. Zeta potential of silica particles [NaCl] 5 0.01 M.
coagulate in acidic pH, where the electrophoretic mobility is minimum. Wattilon and Ge`rard (22) attributed the exceptional stability of silica suspension to the presence of an immobilized surface water layer. The first evidence of this hydration layer was given by Derjaguin and Zorin (23), and further evidence of the existence of this hydration force has been given by several authors (24, 25). Hence, the pH dependence is not entirely correlated with an electrostatic stabilization of the silica particles and zeta potential. Level of salinity [NaCl] (X 3). A higher level of salinity in the continuous phase decreases the value of d e s/dt. For electrostatically stabilized systems, one would expect that an increase in salinity would decrease the stability of the suspensions. Iler (26) stated that for dilute silica sols around pH 2, where there is little ionic charge on the particles, no coagulation by adding electrolyte is observed, presumably because of the hydration layer. The mechanism of coagulation with salts has been investigated for half a century, and the mechanism is not yet fully understood. Dumont (27) calculated the Hamaker constant for a series of oxide particles, based on an approximate expression derived by Israelachvili (28), A 13 5
F
e1 2 e3 3 kT 4 e1 1 e3
G
2
1
3h n e ~n 21 2 n 23 ! 2 2 2 3/ 2 , 16 Î2 ~n 1 1 n 3 !
SiO 2 suspended in water an electrolyte addition will mainly alter the solvent properties (dielectric constant and refractive index). For the Hamaker constant this will result in a substantial decrease in the static part and a smaller decrease in the dispersive part. All-in-all, the electrolyte addition will decrease the attraction energy and hence stabilize the system. Qualitatively, our findings are in accordance with these results from the literature. It should also be noted that we are limited in the concentration of electrolyte due to the experimental setup. Using a higher salinity will cause problems in resolving the dielectric spectra. Level of OH groups (dehydroxylation temperature, X 4). A higher level of this variable tends to reduce the value of d e s/dt, i.e., increase the stability against sedimentation. This is explained on the basis of the increased hydrophilicity of the silica particles with increasing numbers of OH groups at the solid surface (29). Figure 9 shows the FT-IR spectra of the silica particles “dried” at 120 and 800°C, respectively. It is seen that the intensity in the region of OH groups (30), 3400 –3750 cm 21, is reduced at higher temperatures. Studies of dehydroxylation and OH coverage of silica particles have been conducted by different authors, who have reported that the coverage of OH groups is almost independent of silica type and is dependent solely on the dehydroxylation temperature (31–33). The degree of rehydroxylation is small for silica dehydroxylated at 800°C when the particles are dispersed in water at ambient temperature and below (21). The number of OH groups will dictate the level of hydration of the particles and the strength of
[4]
where A 13 is the Hamaker constant for particles (medium 1) dispersed in medium 3, e 1 and e 3 are the static dielectric constants for the media, n 1 and n 3 are the refractive indexes in the visible range, n e is the adsorption frequency of the media, assumed to be the same for both of them, k is the Boltzmann constant, T is the temperature, and h is Planck’s constant. In this expression one can identify a static term (depending on the dielectric constant) and a dispersive term (depending on the refractive index and the frequency.) For SiO 2 the total Hamaker is of the order 0.65 3 10 220 J while TiO 2 has a value of 23 3 10 220 J. As a consequence the attraction between two SiO 2 particles is about 35 times smaller than the similar attraction between TiO 2 particles. Due to the small value of the attraction force, the total interaction energy between two SiO 2 particles will depend on structural forces like hydration forces to a higher extent than does, for instance, the TiO 2 system. For
FIG. 9. FT-IR spectra of silica particles, dehydroxylated at 120 and 800°C.
SEDIMENTATION OF AQUEOUS SILICA SUSPENSIONS
the hydration forces, which is the source of the deviation from classical DLVO theory. Hence our result, that an increase in the number of OH groups will stabilize the system, is logical. Level of T, temperature (X 5). A higher level of temperature in the suspensions increases de s/dt. An increase in T will most likely influence the diffusion properties (Brownian motions) of the particles and the viscosity of water. A higher Brownian motion will increase the collision frequency between the particles and enhance the possibility of a floc formation. This is especially so when the repulsive hydration force between the SiO 2 colloids declines as a function of temperature. The viscosity of water at 5°C is slightly higher than at 25°C and may thus contribute to the increased value of de s/dt at higher temperature. Interaction terms. As seen from Fig. 5, interaction terms must be included in a model of the sedimentation process. Omitting the interaction terms gives a model that explains only 58% of the variation in the response whereas in the model with the interaction terms 96% of the variation is explained. Further experiments should be done to study the interactions terms and their effects in more detail. The largest interaction term is X 1 X 2 , i.e., an interaction between volume fraction silica and pH. This effect is also the second most dominating effect of all the regression coefficients. At high volume fractions and pH the probability of flocculation is increased. The other interaction effects included in the reduced model example all have approximately the same magnitude.
7
ACKNOWLEDGMENTS Bjørnar Hauknes Pettersen would like to acknowledge the Norwegian Research Council (NFR) for a Ph.D. grant. Financial support from Elf Aquitaine, Saga Petroleum, and Statoil is also acknowledged. Financial assistance from the technology program Flucha, financed by NFR, and the oil industry is also acknowledged.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
12. 13. 14.
CONCLUSION
15.
We have in this work introduced time domain dielectric spectroscopy in combination with reduced factorial design as a screening method for investigating the SiO 2 suspension sedimentation. The method allows us to view the influence of the different variables on the sedimentation stability of aqueous silica suspensions. Of the variables chosen, the number of surface OH groups (X 4 ), temperature (X 5 ), and salinity (X 3 ) are the most important main variables. It has been demonstrated that it is insufficient to consider only isolated variables, i.e., not taking into account the interactions between the variables. An improved model is obtained by introducing cross terms (interaction effects). The model with cross terms explains 96% of the variation in the observed sedimentation rate whereas the model with only main variables explains only 58% of the observed variation. The interaction term of volume fraction and pH (X 1 X 2 ) is of considerable importance regarding the stability of the aqueous silica suspensions. The influence of the main variables is in good agreement with previously reported data. The regression model is unfortunately not linear. The model is therefore not suitable for prediction of stability for values other than the levels chosen for the variables. The procedure of multivariate analysis is, however, useful to obtain a qualitative evaluation of the different variables and their interaction terms affecting the suspension stability.
16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
Dollimore, D., and Kariman, R., Surf. Technol. 17, 239 –250 (1982). Hunter, R. J., and Ekdawi, N., Colloids Surf. 18, 325–340 (1986). Abel, J. S., and Stangle, G. C., J. Mater. Res. 9, 451– 460 (1994). Shih, Y. T., Gidaspow, D., and Wasan, D. T., Powder Technol. 50, 201–215 (1987). Okubo, T., J. Phys. Chem. 98, 1472–1474 (1994). Yotsumoto, H., and Yoon, R.-H., J. Colloid Interface Sci. 157, 434 (1993). Gaigalas, A. K., and Whetstone, J. R., Chem. Eng. Commun. 40, 85–95 (1986). Yuan, F., and Pal, R., Chem. Eng. Sci. 50, 3525–3533 (1995). Pettersen, B., and Sjo¨blom, J., Colloids Surf. 113, 175–189 (1996). Pettersen, B., Bergflødt, L., and Sjo¨blom, J., Colloids Surf. 127, 175– 86 (1997). Box, G. E. P., Hunter, W. G., and Hunter, J. S., in “Statistics for Experiments, An Introduction to Designs, Data Analysis and Model Building,” Chap. 10. Wiley, New York, 1978. Førdedal, H., Nodland, E., Sjo¨blom, J., and Kvalheim, O. M., J. Colloid Interface Sci. 173, 396 (1995). Sto¨ber, W., Fink, A., and Bohn, E., J. Colloid Interface Sci. 26, 2 (1968). Lindberg, R., Sundholm, G., Pettersen, B., Sjo¨blom, J., and Friberg, S. E., Colloids Surf. 123–124, 549 –560 (1997). Sjo¨blom, J., and Gestblom, B., “Organized Solutions.” Dekker, New York, 1992. Cole, R. H., Mashimo, S., and Winsor, P., J. Phys. Chem. 84, 786 (1980). Cole, R. H., Berberian, J. G., Mashimo, S., Chryssikos, G., Burns, A., and Tombari, E., J. Appl. Phys. 66, 793 (1989). Dawkins, A. W. J., Grant, E. H., and Sheppard, R. J. O., J. Phys. E 14, 1429 (1981). Debye, P., “Polar Molecules.” Reinhold, New York, 1929. Sirius for Windows, version 1.1, PRS A/S, Bergen High Technology Center, Tormøhlensgt 55 N-5008 Bergen, Norway. Allen, L. H., and Matijevic, E., J. Colloid Interface Sci. 31, 287 (1969). Watillon, A., and Ge´rard, P., Proc. Int. Congr. Surface Activity 4, 1261 (1964). Derjaguin, B. V., and Zorin, Z. M., “Proceedings of the 2nd International Congress of Surface Activity,” Vol. 2, p. 14. Butterworths, London, 1957. Pashley, R. M., and Kitchener, J. A., J. Colloid Interface Sci. 71, 491 (1979). Rabinovich, Ya. I., Derjaguin, B. V., and Churaev, N. V., Adv. Colloid Interface Sci. 16, 63 (1982). Iler, R. K., “The Chemistry of Silica.” Wiley, New York, 1979. Dumont, F., “Stability of Sols, The Colloid Chemistry of Silica, pp. 143–146. Am. Chem. Soc., Washington, DC, 1994. Israelachvili, J. N., “Intermolecular and Surface Forces.” Academic Press, New York, 1979. In “The Colloid Chemistry of Silica” (H. E. Bergna, Ed.), Chap. 1. Am. Chem. Soc., Washington, DC, 1994. Hair, M. L., “Infrared Spectroscopy in Surface Chemistry.” Dekker, New York, 1967. Zhuravlev, L. T., Langmuir 3, 316 –318 (1987). Zhuravlev, L. T., Colloids Surf. 74, 71 (1993). Hair, M. L., and Bell, A. T., “ACS Symposium Series, 137.” Am. Chem. Soc., Washington, DC, 1980.
A Multivariate Analysis on the Sedimentation of Aqueous Silica Suspensions as Studied by Means of Time Domain Dielectric Spectroscopy Bjørnar Hauknes Pettersen, Egil Nodland, and Johan Sjo¨blom 1 Department of Chemistry, University of Bergen, Allegt. 41, N-5007 Bergen, Norway Received August 4, 1997; accepted February 16, 1999
of all factors influencing the suspension stability, numerous experiments should be performed. A way to reduce the number of experiments required is to use factorial design (11). A full factorial design using two levels of each variable requires 2 n experiments, n being the number of variables. For an increasing number of variables even this method requires many experiments. However, with a large n the desired information can often be obtained by using a reduced factorial design (12). In this article we have investigated the influence on suspension sedimentation of different variables and their interaction parameters. The derivative of static permittivity vs time is introduced as the response variable. In the parameter interval under study the permittivity is linearly dependent on the volume fraction of solid material in suspension.
The sedimentation of aqueous silica suspensions has been studied by means of time domain dielectric spectroscopy (TDDS) in the frequency range 20 MHz to 2 GHz. The experiments were designed with a 2 5–1 reduced factorial design and analyzed in terms of multivariate regression models. Five different variables were varied to investigate the corresponding suspension sedimentation. The influence of the different variables on the sedimentation of the silica suspensions is discussed. © 1999 Academic Press Key Words: suspension stability; sedimentation; dielectric spectroscopy; reduced factorial design; multiple regression; interactions.
INTRODUCTION
In the field of suspensions, stability and the factors influencing stability have been the subject of several studies during the past years (1– 6). The studies reported have mainly been concentrated on variation of one variable at a time. The interactions among the different variables governing the suspension stability have attracted less attention. According to Stokes’ law, the rate of sedimentation depends on gravity, the density difference between the continuous phase and the suspended phase, particle size, and viscosity. The particle size will be affected by particle–particle interactions resulting in flocs or large agglomerates. To follow the sedimentation rate, dielectric spectroscopy has proved to be a convenient measuring technique for investigating suspensions (7–10). The dielectric response of the suspension depends on the dielectric properties of the continuous and disperse phases and their volume fraction. A change in the amount of solid particles in the suspension will give rise to an altered dielectric response, if the static permittivity of the disperse phase differs from that of the continuous phase. The changes in the dielectric response (static permittivity) can be monitored as a function of time after the suspension is mixed. The change in static permittivity as a function of time is thus a measure of the sedimentation rate of the suspension. In order to study the effects 1
EXPERIMENTAL
Chemicals and Sample Preparation Silica particles were prepared using the following chemicals without further purification: absolute ethanol from Vinmonopolet, ammonia (25%) from Merck, sodium chloride from Merck, and tetraethyl orthosilane (TEOS) (98%) from Fluka Chemical. The water was doubly distilled (Seralpure PRO 90CN). Monodisperse silica particles were prepared according to the sol gel method (13). The concentrations of the different chemicals were taken from the work of Lindberg et al. (14) to give a diameter of 1 mm. The size was confirmed by electron microscopy (Fig. 1). The zeta potential of the silica particles was measured using a Zetasizer 4 (Malvern Instruments, UK). The ionic strength was 0.01 M NaCl. The pH was adjusted using NH 3 and acetic acid. The silica suspensions were prepared by weighing the components in glass vials. They were mixed by using an ultrasonic bath before they were introduced to the dielectric cell for sedimentation measurements. Measurements and Data Analysis The dielectric spectra were measured using the time domain spectroscopy technique (TDS) (15). The dielectric cell is
To whom correspondence should be addressed. 1
0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
2
¨ BLOM HAUKNES PETTERSEN, NODLAND, AND SJO
The pulses were observed in a time window of 50 ns and Fourier transformed in the frequency range 20 MHz to 2 GHz. The experimental data were fitted to a Debye function (19):
e *~ v ! 5
FIG. 1. SEM picture of monodisperse silica particles, d ' 1 mm.
placed at the end of a coaxial cable, and the difference in the reflected pulses from a reference liquid and the sample is recorded (16 –18). The complex permittivity of the sample, e*(v), is calculated after a Fourier transform to the frequency domain.
e *~ v ! 5 e 9~ v ! 2 i e 0~ v !
[1]
es 2 e` 1 e `. 1 1 i vt
[2]
Here e s is the static permittivity, e ` is the permittivity at high frequencies, v is the angular frequency, and t is the relaxation time. The experimental setup for the measurements is illustrated in Fig. 2. The dielectric sensor/cell used in this study has a cell length of 0.025 mm. The measuring system consists of a digital sampling oscilloscope (HP 54120A) and a pulse generator (HP 54121A). A computer, which is connected to the oscilloscope, performs all necessary calculations. The dielectric spectra are obtained at different times after the suspension has been introduced into the dielectric cell. The value of static permittivity, e s, is resolved for each spectrum and plotted against the time. A typical plot of e s vs time is shown in Fig. 3. The decline in e s is constant during the first measurements, and this value is used as a variable in the analysis. The data obtained were analyzed using Sirius for Windows (20).
FIG. 2. Experimental setup of time domain dielectric spectroscopy, instrumentation and specifications of sensor.
3
SEDIMENTATION OF AQUEOUS SILICA SUSPENSIONS
FIG. 3. Static permittivity vs time for sample 11 in the design.
RESULTS
Table 1 lists the variables chosen and the corresponding values for the high and low levels, as well as for the control points. The complete design matrix is shown in Table 2. The calculated response values (d e s/dt) are also included for the different samples in the same table. Figure 4 displays the corresponding regression coefficients, when Eq. [3] is fitted to the response. The coefficients with positive values will increase the value of d e s/dt, whereas the negative coefficients will decrease the decline, i.e., decrease the sedimentation in the suspension. The full model (Eq. [3]) will explain 100% of the variation in the response, which is evident when 16 variables are used for 16 samples. The full model predicts a value of 7.9 for the control points, while 10.4 was experimentally observed. The observed discrepancy reveals nonlinear effects among one or more of the variables. Predicted vs measured values of d e s/dt for a model based on the main parameters only are shown in Fig. 5. This model explains only 58% of the variation in the response. A model using cross terms must thus be introduced in order to better explain the variation in suspension sedimentation. A model based on the main factors, X i and X 1 X 2 , X 1 X 5 , X 2 X 5 , X 3 X 4 , and X 3 X 5 , is introduced in Fig. 6.
experiments. We have used a reduced factorial design, 2 5–1, thus reducing the required number of experiments to 16. This design is obtained by defining the level of variable 5 as a combination of variables 1, 2, 3, and 4 (X 5 5 X 1 X 2 X 3 X 4 ). Since we use a reduced factorial design, there will be an overlap between main effects and three-factor effects. However, one can assume that three-factor effects are negligible as compared to the main and two-factor effects. In a 2 5–1 reduced factorial design, the variables 1, 2, 3, and 4 are varied to include all the combinations of the levels chosen, while the levels of variable 5 are chosen from the combination of variables 1, 2, 3, and 4. A complete multiple regression model is obtained by using d e /dt as a dependent response variable, the five variables, and their cross terms and independent variables. The complete equation is Yˆ 5 5.33 1 0, 96X 1 1 1, 1X 2 2 1, 24X 3 2 1, 65X 4 1 1, 25X 5 1 1, 46X 1 X 2 1 0, 15X 1 X 3 1 0, 09X 1 X 4 1 0, 79X 1 X 5 1 0, 21X 2 X 3 2 0, 50X 2 X 4 2 0, 80X 2 X 5 1 0, 71X 3 X 4 2 0, 84X 3 X 5 2 0, 43X 4 X 5 .
Here X i corresponds to the different variables in the design as seen in Table 1. The regression coefficients are presented as bars in Fig. 4. For the main effects, a positive regression coefficient means that a change in the level from 21 to 11 increases the response, and a negative coefficient implies a decrease in the response by changing the level from 21 to 11. The positive values cause d e s/dt to increase, and vice versa for the negative values. Equation [3] explains all the variations in the response variable due to the full rank of the design matrix. The calculation of the control points also implies that there is a nonlinearity in the model. It is therefore most interesting to use the model to evaluate the impact on sedimentation by the different variables, not to predict the value of the sedimentation. A model based on the five main effects is not satisfactory to
TABLE 1 Variables and Levels
DISCUSSION
High level (1)
Low level (2)
X1 X2 X3
0.06 11 120
0.02 3 800
0.04 7 400
X4 X5
0.1 25
0.001 4
0.01 15
Factorial Design A two-level factorial design is a convenient way of screening the importance of the different variables in the system under study. With this method, one can extract the information on general trends in variables based on relatively few experiments. Although this cannot fully explore the wide region in the variable space, it will resolve major trends among the variables. The number of experiments required by a full factorial design increases rapidly as the number of variables increases. In this work we have chosen to study five different variables; a complete factorial design would require 2 5 5 32
[3]
Variables Volume fraction disperse phase pH Surface hydroxyl groups (dehydroxylation temperature a (°C)) Salinity [NaCl] Temperature (°C)
Control point
a Reflects the number of 2OH groups at the surface of the silica particles. X 3 5 120°C gives a high number of 2OH groups at the surface; X 3 5 800°C gives a low number of 2OH groups.
2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 20.039 20.039 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 0.0084 0.0084 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 0.0476 0.0476 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 0.0476 0.0476 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 0.998 0.998 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 0 0
2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 0.177 0.177
2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 20.818 20.818
1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 0.0069 0.0069
1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 20.037 20.037
1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 0.0084 0.0084
2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 20.144 20.144
1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 0.1764 0.1764
1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 20.818 20.818
1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 20.144 20.144
Y (de/dt) X4X5 1 X1X2X3 X3X5 1 X1X2X4 X3X4 1 X1X2X5 X2X5 1 X1X3X4 X2X4 1 X1X3X5 X2X3 1 X1X4X5 X1X5 1 X2X3X4 X1X4 1 X2X3X5 X1X3 1 X2X4X5 X1X2 1 X3X4X5 X5 1 X1X2X3X4 X4 X3 X2 X1
TABLE 2 2 5–1 Factorial Design with Cross Terms and Response Values
10.8 2.7 7.2 15 2.8 5.2 3.6 8.5 2.7 6.5 2.3 5.3 2.6 0.5 2.9 6.6 10.4 10.4
¨ BLOM HAUKNES PETTERSEN, NODLAND, AND SJO
4
describe the sedimentation process (see Fig. 5) and only accounts for 58% of the variation in the response. A model that better explains the variation in the sedimentation is obtained by including cross terms. Often there is a clear difference in the significance of the cross terms that can be seen from the magnitude of the regression coefficients. In this work there are no such clear limits concerning the significance of the individual cross effects. We have introduced a model with the cross terms X 1X 2, X 1X 5, X 2X 5, X 3X 4, and X 3X 5. A plot of predicted vs measured values is shown in Fig. 6. This model explains 96% of the variation as compared to the model with only main variables, which explains only 58% of the variation. The residuals from the model are presented in Fig. 7. In a normal distribution plot the residuals are sorted in an increasing order, each accounting for 1/16 of a normal distribution population. If the residuals fall on a straight line the assumption that the residuals are normally distributed holds. This is seen from Fig. 7. Dielectric Measurements on Suspension Sedimentation We have in previous work introduced time domain dielectric spectroscopy as a feasible method to study the sedimentation of suspensions (9, 10). The dielectric response of solid particle suspensions is due to the dielectric parameters of the continuous and disperse phases and the volume fraction solid particles. When sedimentation starts, there will be an increase in volume fraction solid particles at the bottom of the suspension. By placing a dielectric sensor at the bottom of a sedimentation vessel, it is possible to monitor this change in volume fraction of solid particles. We have previously shown that, based on theoretical models for the dielectric properties of heterogeneous mixtures, it is possible to transform the measured static permittivity to a volume fraction of solid particles. In this work we have chosen to study merely the change in static permittivity as a function of time after the sample is introduced into the sedimentation vessel. The response used in this experiment is d e s/dt in the time region where this value is constant. The derivative is proportional to the sedimentation rate of the suspension. A suspension with a high sedimentation rate will have a large d e s/dt due to a fast alteration in the concentration profile of solid particles. More stable suspensions give rise to less pronounced changes in d e s/dt. Sedimentation of Suspensions The stability of suspensions is dependent on a variety of factors. The sedimentation is mainly driven by the density difference between the solid particles and the continuous phase. Interactions between particles and interactions between particles and the continuous phase also play an important role in the observed stability of the suspension. If the particles flocculate, the observed stability will decrease due to the increased sedimentation rate of flocs compared to the sedimentation rate of individual particles. The balance of repulsive and
SEDIMENTATION OF AQUEOUS SILICA SUSPENSIONS
FIG. 4. Regression coefficients. The negative values tend to decrease the value of d e s/dt while the positive values tend to increase d e s/dt.
attractive interparticle forces dictates the degree of flocculation. The variables chosen in this study will all influence these interparticle forces to some extent. Variables 1– 4 listed in Table 1 influence the surface properties of the solid particles while variable 5 (temperature) should influence both particles (Brownian motion) and the continuous phase (viscosity). We have chosen to study monodisperse silica particles in order to more-or-less eliminate effects like particle polydispersity and gradients in gravity. For monodisperse systems like ours, the main effect will be interparticle interactions (flocculation/agglomeration) and their influence on the sedimentation rate. Evaluation of the Different Variables Influencing the Observed Suspension Sedimentation
5
FIG. 6. Measured vs predicted values of d e s/dt for model based on main variables and cross terms X 1 X 2 , X 1 X 5 , X 2 X 5 , X 3 X 4 , and X 3 X 5 . Explains 96% of the variance in the dependent variable.
suspensions. The five variables chosen are all known to influence the sedimentation. Volume fraction disperse phase (X 1). The value of d e s/dt increases as the volume fraction of solid particles increases. Due to volumetric reasons, the particle–particle interaction will be higher at higher f. This can result in, for instance, a higher degree of flocculation, which will accelerate the sedimentation. pH (X 2). A higher pH tends to increase de s/dt, indicating a less stable suspension against sedimentation. This is not in agreement, according to classical DLVO theory concerning zeta potential and stability. Silica has a high negative zeta potential at high pH and a low value at low pH (Fig. 8). One would therefore expect an increase in stability at high pH values. Allen and Matijevic (21) found that Ludox HS precipitated silica does not
In the following section, we will briefly examine the different variables and explain their influence on the stability of the
FIG. 5. Measured vs predicted values of d e s/dt for model based on main variables only. Explains 58% of the variance in the dependent variable.
FIG. 7. Fig. 6.
Normal plot of the residuals in model with cross terms, from
¨ BLOM HAUKNES PETTERSEN, NODLAND, AND SJO
6
FIG. 8. Zeta potential of silica particles [NaCl] 5 0.01 M.
coagulate in acidic pH, where the electrophoretic mobility is minimum. Wattilon and Ge`rard (22) attributed the exceptional stability of silica suspension to the presence of an immobilized surface water layer. The first evidence of this hydration layer was given by Derjaguin and Zorin (23), and further evidence of the existence of this hydration force has been given by several authors (24, 25). Hence, the pH dependence is not entirely correlated with an electrostatic stabilization of the silica particles and zeta potential. Level of salinity [NaCl] (X 3). A higher level of salinity in the continuous phase decreases the value of d e s/dt. For electrostatically stabilized systems, one would expect that an increase in salinity would decrease the stability of the suspensions. Iler (26) stated that for dilute silica sols around pH 2, where there is little ionic charge on the particles, no coagulation by adding electrolyte is observed, presumably because of the hydration layer. The mechanism of coagulation with salts has been investigated for half a century, and the mechanism is not yet fully understood. Dumont (27) calculated the Hamaker constant for a series of oxide particles, based on an approximate expression derived by Israelachvili (28), A 13 5
F
e1 2 e3 3 kT 4 e1 1 e3
G
2
1
3h n e ~n 21 2 n 23 ! 2 2 2 3/ 2 , 16 Î2 ~n 1 1 n 3 !
SiO 2 suspended in water an electrolyte addition will mainly alter the solvent properties (dielectric constant and refractive index). For the Hamaker constant this will result in a substantial decrease in the static part and a smaller decrease in the dispersive part. All-in-all, the electrolyte addition will decrease the attraction energy and hence stabilize the system. Qualitatively, our findings are in accordance with these results from the literature. It should also be noted that we are limited in the concentration of electrolyte due to the experimental setup. Using a higher salinity will cause problems in resolving the dielectric spectra. Level of OH groups (dehydroxylation temperature, X 4). A higher level of this variable tends to reduce the value of d e s/dt, i.e., increase the stability against sedimentation. This is explained on the basis of the increased hydrophilicity of the silica particles with increasing numbers of OH groups at the solid surface (29). Figure 9 shows the FT-IR spectra of the silica particles “dried” at 120 and 800°C, respectively. It is seen that the intensity in the region of OH groups (30), 3400 –3750 cm 21, is reduced at higher temperatures. Studies of dehydroxylation and OH coverage of silica particles have been conducted by different authors, who have reported that the coverage of OH groups is almost independent of silica type and is dependent solely on the dehydroxylation temperature (31–33). The degree of rehydroxylation is small for silica dehydroxylated at 800°C when the particles are dispersed in water at ambient temperature and below (21). The number of OH groups will dictate the level of hydration of the particles and the strength of
[4]
where A 13 is the Hamaker constant for particles (medium 1) dispersed in medium 3, e 1 and e 3 are the static dielectric constants for the media, n 1 and n 3 are the refractive indexes in the visible range, n e is the adsorption frequency of the media, assumed to be the same for both of them, k is the Boltzmann constant, T is the temperature, and h is Planck’s constant. In this expression one can identify a static term (depending on the dielectric constant) and a dispersive term (depending on the refractive index and the frequency.) For SiO 2 the total Hamaker is of the order 0.65 3 10 220 J while TiO 2 has a value of 23 3 10 220 J. As a consequence the attraction between two SiO 2 particles is about 35 times smaller than the similar attraction between TiO 2 particles. Due to the small value of the attraction force, the total interaction energy between two SiO 2 particles will depend on structural forces like hydration forces to a higher extent than does, for instance, the TiO 2 system. For
FIG. 9. FT-IR spectra of silica particles, dehydroxylated at 120 and 800°C.
SEDIMENTATION OF AQUEOUS SILICA SUSPENSIONS
the hydration forces, which is the source of the deviation from classical DLVO theory. Hence our result, that an increase in the number of OH groups will stabilize the system, is logical. Level of T, temperature (X 5). A higher level of temperature in the suspensions increases de s/dt. An increase in T will most likely influence the diffusion properties (Brownian motions) of the particles and the viscosity of water. A higher Brownian motion will increase the collision frequency between the particles and enhance the possibility of a floc formation. This is especially so when the repulsive hydration force between the SiO 2 colloids declines as a function of temperature. The viscosity of water at 5°C is slightly higher than at 25°C and may thus contribute to the increased value of de s/dt at higher temperature. Interaction terms. As seen from Fig. 5, interaction terms must be included in a model of the sedimentation process. Omitting the interaction terms gives a model that explains only 58% of the variation in the response whereas in the model with the interaction terms 96% of the variation is explained. Further experiments should be done to study the interactions terms and their effects in more detail. The largest interaction term is X 1 X 2 , i.e., an interaction between volume fraction silica and pH. This effect is also the second most dominating effect of all the regression coefficients. At high volume fractions and pH the probability of flocculation is increased. The other interaction effects included in the reduced model example all have approximately the same magnitude.
7
ACKNOWLEDGMENTS Bjørnar Hauknes Pettersen would like to acknowledge the Norwegian Research Council (NFR) for a Ph.D. grant. Financial support from Elf Aquitaine, Saga Petroleum, and Statoil is also acknowledged. Financial assistance from the technology program Flucha, financed by NFR, and the oil industry is also acknowledged.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
12. 13. 14.
CONCLUSION
15.
We have in this work introduced time domain dielectric spectroscopy in combination with reduced factorial design as a screening method for investigating the SiO 2 suspension sedimentation. The method allows us to view the influence of the different variables on the sedimentation stability of aqueous silica suspensions. Of the variables chosen, the number of surface OH groups (X 4 ), temperature (X 5 ), and salinity (X 3 ) are the most important main variables. It has been demonstrated that it is insufficient to consider only isolated variables, i.e., not taking into account the interactions between the variables. An improved model is obtained by introducing cross terms (interaction effects). The model with cross terms explains 96% of the variation in the observed sedimentation rate whereas the model with only main variables explains only 58% of the observed variation. The interaction term of volume fraction and pH (X 1 X 2 ) is of considerable importance regarding the stability of the aqueous silica suspensions. The influence of the main variables is in good agreement with previously reported data. The regression model is unfortunately not linear. The model is therefore not suitable for prediction of stability for values other than the levels chosen for the variables. The procedure of multivariate analysis is, however, useful to obtain a qualitative evaluation of the different variables and their interaction terms affecting the suspension stability.
16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
Dollimore, D., and Kariman, R., Surf. Technol. 17, 239 –250 (1982). Hunter, R. J., and Ekdawi, N., Colloids Surf. 18, 325–340 (1986). Abel, J. S., and Stangle, G. C., J. Mater. Res. 9, 451– 460 (1994). Shih, Y. T., Gidaspow, D., and Wasan, D. T., Powder Technol. 50, 201–215 (1987). Okubo, T., J. Phys. Chem. 98, 1472–1474 (1994). Yotsumoto, H., and Yoon, R.-H., J. Colloid Interface Sci. 157, 434 (1993). Gaigalas, A. K., and Whetstone, J. R., Chem. Eng. Commun. 40, 85–95 (1986). Yuan, F., and Pal, R., Chem. Eng. Sci. 50, 3525–3533 (1995). Pettersen, B., and Sjo¨blom, J., Colloids Surf. 113, 175–189 (1996). Pettersen, B., Bergflødt, L., and Sjo¨blom, J., Colloids Surf. 127, 175– 86 (1997). Box, G. E. P., Hunter, W. G., and Hunter, J. S., in “Statistics for Experiments, An Introduction to Designs, Data Analysis and Model Building,” Chap. 10. Wiley, New York, 1978. Førdedal, H., Nodland, E., Sjo¨blom, J., and Kvalheim, O. M., J. Colloid Interface Sci. 173, 396 (1995). Sto¨ber, W., Fink, A., and Bohn, E., J. Colloid Interface Sci. 26, 2 (1968). Lindberg, R., Sundholm, G., Pettersen, B., Sjo¨blom, J., and Friberg, S. E., Colloids Surf. 123–124, 549 –560 (1997). Sjo¨blom, J., and Gestblom, B., “Organized Solutions.” Dekker, New York, 1992. Cole, R. H., Mashimo, S., and Winsor, P., J. Phys. Chem. 84, 786 (1980). Cole, R. H., Berberian, J. G., Mashimo, S., Chryssikos, G., Burns, A., and Tombari, E., J. Appl. Phys. 66, 793 (1989). Dawkins, A. W. J., Grant, E. H., and Sheppard, R. J. O., J. Phys. E 14, 1429 (1981). Debye, P., “Polar Molecules.” Reinhold, New York, 1929. Sirius for Windows, version 1.1, PRS A/S, Bergen High Technology Center, Tormøhlensgt 55 N-5008 Bergen, Norway. Allen, L. H., and Matijevic, E., J. Colloid Interface Sci. 31, 287 (1969). Watillon, A., and Ge´rard, P., Proc. Int. Congr. Surface Activity 4, 1261 (1964). Derjaguin, B. V., and Zorin, Z. M., “Proceedings of the 2nd International Congress of Surface Activity,” Vol. 2, p. 14. Butterworths, London, 1957. Pashley, R. M., and Kitchener, J. A., J. Colloid Interface Sci. 71, 491 (1979). Rabinovich, Ya. I., Derjaguin, B. V., and Churaev, N. V., Adv. Colloid Interface Sci. 16, 63 (1982). Iler, R. K., “The Chemistry of Silica.” Wiley, New York, 1979. Dumont, F., “Stability of Sols, The Colloid Chemistry of Silica, pp. 143–146. Am. Chem. Soc., Washington, DC, 1994. Israelachvili, J. N., “Intermolecular and Surface Forces.” Academic Press, New York, 1979. In “The Colloid Chemistry of Silica” (H. E. Bergna, Ed.), Chap. 1. Am. Chem. Soc., Washington, DC, 1994. Hair, M. L., “Infrared Spectroscopy in Surface Chemistry.” Dekker, New York, 1967. Zhuravlev, L. T., Langmuir 3, 316 –318 (1987). Zhuravlev, L. T., Colloids Surf. 74, 71 (1993). Hair, M. L., and Bell, A. T., “ACS Symposium Series, 137.” Am. Chem. Soc., Washington, DC, 1980.