Influence of Dielectric Constant on Sedimentation Rate of Concentrated Suspensions of Aluminum and Magnesium Hydroxides KENNETHs. ALEXANDER*',DAVID DOLLIMORE**, SHAHNAZ s. TATA*,AND A. SAVlTRl MURTHY* Received Au us1 20,1990, from the 'College of Pharmacy and the SDepament of Chemistry, The University of Toledo, 2801 west Banmfl Street, T o l d , OH 43606. Accepted for publication September 12, 1991. Abatract 0 Sedimentationof dilute pharmaceuticalsuspensions obeys Stokes's law, which assumes that there 1s no interaction between particles. The behavior of concentrated pharmaceutical suspensions is generally interpreted by use of modifications of Stokes's law that do not consider chemical Interaction between particles. Properties of the medium itself, such as dielectric constant and surface tension, have not been includedin the established equations. The present work shows that the dielectric constant of the medium has a distinct effect on the rate of sedimentation of the systems investigated.

the hindrance to falling and a term for the initial porosity, Q. Therefore, for concentrated suspensions, Stokes's law becomes the following:

In eq 5, Q is the rate of fall of the interface with respect to time. From his experimental results, Steinour found that the equation could also be expressed as follows:

The forces acting on an immersed body moving relative to a viscous fluid were first studied by Stokes.' He derived an equation to describe the viscous resistance to the motion of a single particle in a n infinite amount of fluid:

In eq 6, A is a constant. The other frequently used equation for the internretation of hindered settling is the one proposed by Richardson and Zaki:4

F = 31~qVd

Q = VsEn


In eq 1,F is the viscous drag on the particle (sphere), d is the diameter of the sphere, 7 is the coefficient of viscosity of the fluid, and V is the velocity of the sphere relative to the fluid. The effective gravitational force acting on the sphere (FJis given by the following equation:

In eq 2, ps and p, are the densities of the solid and the liquid, respectively, and g is the acceleration due to gravity. The sphere achieves a constant terminal velocity (VJ,at which the two forces balance each other. Hence:


As the concentration of the solid in the suspension increases, the particles interact with each other and with the walls of the container. If we assume that the diameter of the container is large compared with that of the sphere, then the interactions between particles become more significant. These interactions are affected by (1)the concentration of the suspension; (2) the tendency of the particles to flocculate; (3) the physical properties of the liquid medium, such as dielectric constant, surface tension, and viscosity; and (4) the chemical properties of the medium.2 S t e i n o d and Richardson and Zaki4 modified Stokes's law to find the extrapolated velocity (VJfor concentrated suspensions. Steinour introduced a shape factor, &Q), to account for 0022-3549/92'~787$02.50/0 Q 7992, American Pharmaceutical Assodation


In eq 7, n is a constant. Both eqs 6 and 7 and also other equations proposed to describe hindered settling do not take into account the dielectricconstant of the medium. Davies and Dollimo+ t i e d to find a relationship between the dielectric constant and the flocculating tendency of a liquid. Dollimore and Karimians examined the sedimentation of concentrated aluminum hydroxide suspensions in Me r e n t alcohols. They showed that for 90 g of aluminum hydroxide, the plot of log Q versus log (dielectric constant) yielded a straight line. They reported that a plot of log Q versus log 7 yielded a straight line. Our data yielded a better fit when log Q versus log 1/q (fluidity)was plotted for s u m . An attempt to correlate parameters resulted in plots of log Q for the sample against viscosity, dielectric constant, density, and surface tension. No simple correlation was found among these parameters. It was noted, however, that viscosity and dielectric constant would provide a reasonable correlation within the limita of the experiments, such that

log Q = log K

+ (nlog q) or Q = Kqn


In eq 8, n and K can be obtained from the slope and the intercept, respectively. Correspondingly, the following relation was obtained in a similar manner:

log Q = logK + (n logE) or Q =KEn


In eq 9, E is the dielectric constant of the medium. The log-log plot was confirmed by the present system. This concept is further examined here with aluminum and magnesium hydroxide suspensions in aqueous solutions of sucrose, glycerin, and sorbitol. Journal of Pharmaceutical Sciences 1 787 Vol. 81, No. 8, August 1992

Experimental Section Materials-The suepended solids were magnesium hydroxide, N.F., Po. (lot KHMG Sherman Research Laboratories, Toledo, OH), and aluminum hydroxide (lot 901900A; Fisher Scientific, Fair Lawn, NJ). The different media were solutions of glycerin, USP (lot KLNZ; Mallinckrodt,Paris, KY),sucrose, N.F. (lot H23407P16; Amend Drug and Chemical, Irvington, NJ), and sorbitol (lot 022582; Sherman) in distilled water. Methods-The hindered-settling experiments were conducted in 250-ml Pyrex measuring cylinders with an internal diameter of 3.6 cm. A scale marked in O.lcm increments was attached to the side of each cylinder. The suspension was prepared by placing different weighed quantities of the solid into the cylinders. The medium was added up to the 100-ml mark and leR overnight to saturate the solid. On the next day, the suspension was brought to 150 ml with the medium and dispersed. Each suspension was redispersed before observation by inverting each cylinder 20 times. The cylinders were set against a dark background, and a 100-W lamp was used to measure the position of the interface at different timee. The experiments were performed with different concentrations of glycerin, sorbitol, and sucrose solutions as a convenient way to change the dielectric constant without altering the chemical properties of the medium. Ambient temperature (25 2 1"C) was used throughout the study. The viscosities of the sucrose-water and glycerin-water systems were determined by use of an Ostwald viscometer and a thermostated water bath kept at 25 2 0.5 "C. The materials used in these experiments are produced by the suppliers via a specific set of manufacturing procedures, and lot-blot variances result. These variances caw the materials to exhibit performancedifferencesso it is essential for all the experiments to be carried out with the same lot of ingredient. Comparison of results from different lots may not be possible; however, the calculated parameters may be similar. Initial experiments must establish the minimum and maximum concentrations at which hindered settling begins and terminates for any given material. Thus, for the aluminum hydroxide lot, the concentration range was 3343% (w/v), and for the magnesium hydroxide lot, the concentration range was 13430% (w/v).




_i 4.8

-12 ' l l

. 0.04



. 0.W














, 0.98


Figure 2-Plot of log (0'2) versus E for magnesium hydroxide suspensions in water.



-2.0+ -0.08

. , . , . , . , .0.07





, 4.03

. , . , . .0.02





Flgun &Plot of log 0 versus log E for magnesium hydroxide suspensions in water.

Results and Discussion The height of the interface was plotted against time (Figure 1).Three distinct regions were observed in the plot an initial

region, or the time taken by the suspension to achieve stability; a linear region, or the hindered-settling zone; and a compressive region, representing the approach to saturation and the final settled volume. The slope of the linear region is considered to be the rate of sedimentation (Q). The extrapolated Stokes's velocity can be determined in two ways: by Steinour's equation3 and by Richardson and M i ' s equation.' In Steinour's method, log (QlP)is plotted against E. From the intercept, V,can be determined (Figure 2). In Richardson and Zaki's method, log Q is plotted against log E, and the intercept is equal to log V, (Figure 3). Particle size was



determined by microscopy, with the limitation that only particles between 0.8 and 150 pm could be measured. Both low-power ( x 100) and high-power ( ~ 4 5 0 )magnifications were used. The average particle sizes for magnesium hydroxide were 21.1 pm ( ~ 1 0 0 and ) 2.36 pm (x460), and those for aluminum hydroxide were 8.8 pm ( x 100) and 22 pm ( x 450). The apparent discrepancy is explained by the fact that, under low power ( X 1001, particles larger than 7 pm could be seen, and it was impossible to determine whether the smaller shapes were aggregates or individual particles. The particle size of the aggregate of the suspended solid was calculated from hindered-settling measurements as described by Steinour.3 The particle size was observed to be between 6 and 20 times larger than that measured by the high-power ( ~ 4 5 0 ) resolution of the microscope. The "ultimate" particle (the smallest state of subdivision that retains all the physical and chemical properties of that substance) was probably not measured. When unidirectional microscopy is used to determine the sizes of irregularly shaped particles, the arithmetic nonequivalent diameter is elucidated. Sedimentation, on the other hand, measures the geometric average. For glycerol and sorbitol, the particle size decreased. As the concentration of water in the medium increases, the dielectric constant of the medium changes. The dielectric constants for different dilutions of glycerin, sorbitol, and sucrose were taken from the literature7.8 and are shown in Table I. When alcohols were used by Dollimore and Karimian,6 the variable dielectric constant was contingent on the carbon length of the alcohol. The carbon length of the alcohol added another variable to the sedimentation phenomenon by

1 600


1 600

Tlme (rrcondr)

Figure 1-Typical plot of height of interface versus time. Key: (A) initial region; (B) linear region; (C) compressive region. 788 I Journal of Pharmaceutical Sciences Vol. 81, No. 8, August 1992

Tabb CDklectrlc Constat. at C d Dif?wenlConcmtntlona of Qlvcerln, Sorbltol, and Sucrom In W.1.P

Medium and Concentration, % (wh)

Dieiectrk Constant

Water Glycerin


20 40 60

74.2 68.9 62.6 52.4 43.0

80 95

Sucrose 10

20 30 40 50 60


76.2 73.6 70.9 67.7 64.2 60.2





100 (dlelactric

I .a0

I .M



Flgure 4-Plots of log 0 versus log (dielectric constent) for different weights of magnesium hydroxide in aqueous glycerin sdutions. Key: (0) 20 8; (0)25 g; (W 35 g; (0)45 g.

Sorbitol 20 40 70

72.7 68.8 60.9

’Data are from refs 7 and 8. also altering chemical properties. The use of sucrose-water, sorbitol-water, and glycerin-water systems maintains chemical properties but changes the dielectric constant. The dissacharide sucrose acts as a flocculant because of its structural and chemical properties, whereas the monosaccharide sorbito1 and the polyhydric alcohol glycerin do not contribute to chemical properties. The dielectric constant is a measure of the effectiveness of the medium in separating two oppositely charged ions.9 The higher the dielectric constant, the greater the solubility of an ionic substance. The dielectric constant of the medium can be related to the rate of fall of the interface (8) of an insoluble substance. Changing the dielectric constant alters the forces of interactions between the particles. In a previous study, Dollimore and Karimiane noted that the logarithm of the dielectric constant of the medium has a linear relationship to log Q.Unfortunately, only one weight of the solid was used. The rate of sedimentation increased with an increase in the dielectric constant of the medium, perhaps because of a n increase in the agglomeration of the particles. The chemical effect of the dielectric constant of the medium is to cause particles having like charges to agglomerate and, thus, increase the sedimentation rate. An increase in the dielectric constant causes a concomitant increase in medium polarity 80 that sedimentation will be faster in a more polar medium than in a less polar medium. To determine the effect of the dielectric constant on the rate of sedimentation, we plotted log Q against log (dielectric constant) for different solid concentrations in different dilutions of glycerin and sucrose. Straight lines with positive slopes were obtained (Figures 4 and 5).Similar effects were observed with the other systems studied (Figure 6). An interesting observation is that the slopes decreased with a n increase in weight and that the lines appeared to meet. This observation was verified by plotting the intercepts of the lines against the corresponding slopes. Figures 7 and 8 show that straight-line plots were obtained. Most real systems deviate from ideality. Solution deviation has been extensively studied by Hildebrand et al.”JJ1Investigators have been attempting to establish a similar relationship for dispersed systems. The current investigation has established a means by which the ideal system can be determined from real system data. This effect is called “compensation,” and it shows that all the experimental lines intersect at a single point. The coordinates







, 1.84






I 100

100 (dlbibclrk COnaIanI)

Figure G P l o t s of log 0 versus log (dielectric constant) for dtfferent weights of aluminum hydroxide in aqueous sucro88 sdutions. Key: (0) 70 g; (0)80 g. 50 g; (0)60 g; .1.2.I4


1 .53

1 .a5


1 .a7

I .an

iog (dieteetric conitant)

Flgun -lots of log 0 versus log (dielectric constant) for dtfferent weights of magneslum hydroxide in aqueous sorMtd solutions. Key: (0) 20 g; (0)25 g;I.( 35 g; (0)45 g.

of this point of intersection can be determined as follows. The equations of n lines, all of which pass through a point (xl, yl), are as follows:

y1 = (mlrl)+ c1 Y1 = ( m s d + c2

. .


Journal of Phannaceutlcal sciencesI 789 Vd. 81, No. 8, August 1992

Glvcerin. % Iw/wl

Dielectric Constant

Stokes’s V e W , cm/s

10 20


76.4 74.2 71.6 67.4 66.0

0.0678 0.0423 0.0280 0.0098 0.0072

SorMtol, % (w/w)

Dielectric Constant

Stokes’sVelocity, a n / s

10 20 30

74.8 72.7 70.4 68.8

0.0436 0.0188 0.0144 0.0028




.n .?II 10






Figure 7-Plot of the intercepts versus the slopes of the plots of log 0 versus log (dielectric constant) for different weights of. magnesium hydroxide in different concentrations of glycerin:




f\ we












1A 4




log (dlolectrlc conatant)

Figure S-Plot of log V, versus log (dielectric constant) for magnesium hydroxide suspensions in aqueous glycerin solutions.

Figure &Plot of the intercepts versus the slopes of the plots of log 0 versus log (dielectricconstant) for different weights of aluminum hydroxide in different concentrations of sucrw.

Eq 10 indicates a linear relationship between intercept c and slope m. When the slopes are plotted against the corresponding intercepts, the x coordinate of the point of intereaction is the negative slope of the line, and the y coordinate is the intercept of the line. The x intercept establishes the slope of the ideal line for the ideal system, and the y intercept establishes log Q.This method provides investigators a means of establishing ideal conditions for systems that have been considered too difficult to describe or characterize. The pharmaceutical systems examined were magnesium hydmxide and aluminum hydroxide suspensions in Merent concentrationsof glycerin, sorbitol,and BUC~OBB.All the systems showed compensation when treated as described earlier. Becam the rate of sedimentation of the system determines the extrapolated Stokee’s velocity, as shown by eqs 6 and 7, an attempt was made to determine whether there is a relationship between V, and the dielectric constant. The velocities determined by eq 6 for magnesium hydroxide suspensions in glycerin and sorbitol solutions are shown in Tables II and III, respectively. Log V, in glycerin was plotted against log (dielectric constant) to yield a straight line (Figure 9). This plot led to the conclusion that the dielectric constant a!€& the extrapolated Stokes’s velocity. Suspensions in sumw did not demonstrate thisrelationship, because sucrose is a flooculatingagent and has a point of minimum flocculating action.

790 I Journal of Phannaceutkal Sciences Vol. 87, No. 8, August 7992

Conclusions The radius of the aggregate and, therefore, the extrapolated Stokes’s velocity CVJ depend on the dielectric constant. We suggest that the dielectric constant of the medium should be included as one of the variables in Steinour‘s modifications of the Stokes’s law equation. The exact manner in which the dielectric constant should appear in the equation has not been determined. However, this study has laid the basis for some serious thought on the effect that the dielectric constant has on the sedimentation of concentrated suspensions.

References and Notes 1. Stokes, G.G.Math. Phys. Pap. 1901,3,1. 2. Bhatty, J. I.; Davies, L.; Dollimore, D.; Zahedi, A. H. Surf. Technol. 1982,16,323-344. 3. Steinour, H.H.Znd. Eng. Chem.1944’36,818424. 4. Richardeon, J. F.;Zaki, W. N. Trans. Znst. Chem.Eng. 1954,32, 36-53. 5. Daviea, L.; Dollimore, D. Powder Technol. 1978.19,l. 6. Dollimore, D.; Karimian, R. Surf. Technol. 1982,17,239-250. 7. Sorby, D. L.;Bitter, R. G.; Webb, J. G. J . Phurm. Sci. 1963,52, 1149-1163. 8. Malmbera, C.G.;Maryott, A. A. J . Res. 1950,46,299-303. Sciences, 17th ed.; Gennar~,A.R., 9. Remm@&’s Phar&ut&l Ed.; ack Easton, PA, 1987;p 218. 10. Hildebrand, J. H.;Prausnitz, J. M.;. Scott, R. L. R ulur and Related Solutwns; Van Noatrand Reinhold New Yo%, 1970;p 22. 11. Hildebrand, J.H.;Scott, R. L. SolubiliCy of Non-Electrolytes; Dover: New York, 1968; p 274.

Influence of dielectric constant on sedimentation rate of concentrated suspensions of aluminum and magnesium hydroxides.

Sedimentation of dilute pharmaceutical suspensions obeys Stokes's law, which assumes that there is no interaction between particles. The behavior of c...
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