Coagulation in processing of ceramic suspensions: Powder size distribution effectsM. Strauss, T. Ring, A. Bleier, and H. K. Bowen Citation: Journal of Applied Physics 58, 3871 (1985); doi: 10.1063/1.335605 View online: http://dx.doi.org/10.1063/1.335605 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/58/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effects of laser sintering processing time and temperature on changes in polyamide 12 powder particle size,shape and distribution AIP Conf. Proc. 1593, 728 (2014); 10.1063/1.4873880 Irreversible volume growth in polymer-bonded powder systems: Effects of crystalline anisotropy, particle sizedistribution, and binder strength J. Appl. Phys. 103, 053504 (2008); 10.1063/1.2838319 Optical effects of fine ceramic powder in solidified gases AIP Conf. Proc. 309, 1515 (1994); 10.1063/1.46269 Effect of particle size distributions on the rheology of concentrated bimodal suspensions J. Rheol. 38, 85 (1994); 10.1122/1.550497 Firstorder diffusive effects in the coagulation of colloidal suspensions J. Chem. Phys. 70, 1129 (1979); 10.1063/1.437612

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CoaguJation in processing of ceramic suspensions: Powder size distribution effects

M. Strauss,a) T. Ring, A. Bleier,b) and H. K. Bowen Materials Processing Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Received 15 May 1985; accepted for publication 2 August 1985)

The origin of agglomerates in slips and slurries of ceramic particles due to particle size distribution effects is considered. Since agglomerates have been related to flaws in sintered ceramic bodies, the parameters which control coagulation of particles into agglomerates need to be defined. Computer calculations, based on Derjaguin, Landau, Verwey, and Overbeek theory, have been used to predict stability curves for particulate suspensions with log normal particle size distributions. Two regimes appear in the calculations: at width parameters (O"z) corresponding to the inherent distribution of a powder the coagulation time decreases rapidly with increasing polydispersity; at larger o"z s corresponding to mixing different sizes of powders to get very broad distributions the coagulation time decreases with increasing polydispersity. This change in behavior occurs at aproximately U z = 1.0.

LIST OF SYMBOLS N?,NJ = Initial number of i and j particles/cm),

(a) = Average particle radius. (a2

) = Second moment of the particle size distribution.

(a3) = Third moment of the particle size

ag

a;o aj

A {3 d (d) Dij D/.Dj

h (tfJ)

distribution. = Geometric mean particle radius. = Particle radii. = Hamaker's constant. = Hydrodynamic interaction parameter. = Particle diameter. = Average particle diameter. = relative diffusion coefficient. = Diffusion coefficient of particles i and j,

respectively. = Fluid dielectric constant. = Fraction of particles agglomerated. = Volume fraction of particles. = Acceleration due to gravity. = Number of all particles colliding with a

particle i per second. = Number of j-type particles colliding with

a particle i per second. = Total number of all particles colliding per

second. = Volume correction term for the settling

time. = Inverse double layer length. = Boltzmann's constant. = relative mobility between particles i and j. = Viscosity. = Total number ofunagglomerated

particles in system per cm3•

= Number of i and j particles/em), respectively.

aJ Part of M. Strauss Ph.D. Thesis. Department of Materials Science and Engineering, MIT. 1985.

b) Now at Oak Ridge National Laboratory.

P(a)

1/10 P PI R

Vr(R)

respectively. = Particle size distribution. = Surface potential on particles. = Particle mass density. = Fluid density. = Separation between particle centers. = Log normal width parameter. = Absolute temperature. = Coagulation time for 2% agglomeration. = Time for a fraction f of the particles to

become agglomerated. = Stokes settling time for 1 cm. = Hydrodynamic separation parameter. = Attractive van der Waals energy between

, particles i and j. = Repulsive double-layer energy between

particles i and j. = Interaction energy for particles i andj

separated by R. = Stability factor for particles a, and aj •

= Total stability factor for polydisperse system.

INTRODUCTiON

Processing ceramics often involves the dispersion of powders as slips and slurries. The use of aqueous and nona­queous solvents combined with electrostatic or steric stabili­zation of the particles against coagulation is convenient for several shaping/forming technologies-e.g., slip casting. tape casting, and electrophoretic deposition-and is used to produce uniform green bodies. Agglomerates that are formed during processing can degrade the properties and performance of sintered ceramics. I This work focuses on the effects of polydispersity on the coagulation of oxide powders suspended in water. The purpose is to determine powder characteristics that minimize the formation of agglomerates as nonhomogeneous centers within the green microstruc­ture. Since homogeneous, unagglomerated slips or slurries are desired for processing, what are the effects of the particle

3871 J. Appl. Phys. 58 (10). 15 November 1985 0021-8979/85/223871-09$02.40 @ 1985 American Institute of PhySiCS 3871

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size and size distribution, volume fraction solids, electrolyte and surface charge characteristics on the stability of the dis­persion relative to coagulation?

DLVO (Derjaguin, Landau, Verwey, and Overbeek) theory2 can describe the interactions between suspended col­loidal particles, and relate interaction energies to coagula­tion rates. In any specific process, the coagulation time (which may be fixed or varied with powder characteristics and processing parameters) should not be shorter than the processing time, e.g., times for electrophoretic deposition, drying, or settling.

This work calculates the coagulation time for various systems with different log normal powder size distributions. The calculations include electrostatic and van der Waals in­teraction energies. The particle diffusion coefficients have been modified to account for hydrodynamic interactions between particles.

In addition, calculations are made which compare co­agulation time to settling time since settling may be impor­tant in many ceramic processes (such as slip casting).

The coagulation time for a polydisperse, interacting, concentrated suspension undergoing shear due to differen­tial settling is too complex for calculation. For systems where the particle size is less than 1 p.m, perikinetic (Brown­ian) coagulation dominates over orthokinetic (shear induced) coagulation. 3 Increasing the volume fraction decreases the settling velocities, which reduces the differential settling. The reduction in shear rate by an increase in volume fraction to one percent is sufficient to make perikinetic coagulation dominate for particle sizes up to 3 p.m.

Work has been done on the coagulation of poly disperse systems evaluating how particle size distributions evolve with time.4-8 However, they have dealt with noninteracting, dilute, coalescing emulsions, and therefore are not applica­ble here.

Recently, Bensely and Hunter9 have proposed a model for concentrated, monodisperse, interacting colloids with

120

100

~ 60

J-

.:::. >-0> 60 :;; c w

~

.02 .04 .06 ,06 ,1 Separation (microns)

FIG. 1. Interaction energy vs separation for different particle size combina· tions. Small particles interacting with either small or large panicles have much reduced energy barriers compared with the large panicle interaction. Thee\ectrolyte concentration isO.OOIM, 1/10 '" 25.0 mV,A = 2.1 X 10-20 J, (a) d. '" d2 = I p.m; (b) d. = 1 p.m, d2 '" 0.1 p.m; (c) d, = d2 = 0.1 p.m.

3872 J. Appl. Phys., Vol, 58, No. 10, 15 November 1985

large potential energy barriers. The extension of their work to polydisperse suspensions has difficulties. In their model, an effective pair potential must be defined taking into ac­count the influence of neighboring particles. Bensley and Hunter do this for a monodisperse system by averaging the Hogg, Healey, Fuerstenau 10 (HHF) pair potentials over pos­sible configurations. For polydisperse systems, the effective pair potential would also depend on the sizes of the neigh­boring particles and additional averaging over the particle size distribution would be necessary. Also, assumptions about how different size particles form clusters would have to be made. In addition, energy barriers between particles for polydisperse systems cannot always be assumed to be large. There will be a distribution of energy barriers as shown in Fig. 1. While the energy barriers for the average particle size may be large, some smaller particle pairs will not have large barriers. For these reasons the Bensley and Hunter approach could not be used for this problem.

The path we have chosen closely follows that of other workers,4,IO,11 who have calculated stability factors for di­lute, interacting, polydisperse colloids. For ceramic systems we need to extrapolate these types of caJ.culations to high volume fraction, and calculate coagulation times in addition to stability factors.

ANALYSIS

The time for a fraction f of the particles to coagulate as shown in the Appendix is

1" = 21T7/WT (a3

) (--.L). (1) c kTrp 1 - f

Here 1J is the viscosity, (0 3) is the third moment of the parti­cle size distribution, k is Boltzmann's constant, and Tis the temperature. Particle size distribution and interactions between particles affect the coagulation time through the stability factor W T :

(J~oo 100 (a j + aj )2 P(aj)P(Oj) )-1 Wr = da; daj , (2)

o 0 OJ OJ W(o;, OJ}

where P (a) is the distribution of particie sizes, W (a;, OJ) is the pair stability factor for particles of radii a; and OJ' and is of the form

2100 eV,)R )/kT dR W(o;. OJ) = (01 + aj) -2,(3)

a,H) [a;lp(uj ) + o/P(U;l} R

where R is the center-to-center separation of the two parti­des, f3 is the hydrodynamic interaction parameter, and U is twice the distance from the surface of a particle to the plane of symmetry between the particles divided by the radius. 12-14 For ease and speed of calculation, the approximation for P (u) derived by Honig et 0/.

12 is used. It is valid to within a few percent over a large range of values. The stability factor is very sensitive to the interaction of energy Vij(R ) through its exponential dependence. The interaction energy in tum is strongly dependent on particle sizes 0; and OJ;

V;j(R) = V~j(R ) + V:/(R ), (4)

V:I(R) = €t/?o (~) lnt 1 + e-K[R-(Q,HjlJ 1. (5) a; +OJ

Strauss sf 81. 3872

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+ In (R 2 - (a j + aj f )] . R2-(a

j _a

j)2

(6)

Equations (5) and (6) are, respectively, the coloumbic repulsion3 and van der Waals attraction 15 between two une­qual spherical particles. The dielectric constant of the fluid medium is E, the surface potential is tPo, and the double-layer thickness is K-

I, which is a function of the electrolyte con­

centration. The Hamaker constant A is a measure of the van der Waals attraction between particles.

The HHFIO electrostatic interaction energy has some implicit assumptions, specifically, the surface potential of the particles should be less than 25 m V (however, the DL VO energy has been used for surface potentials up to 50 or 60 m V), 3 the particle size must be at least five times the charac­teristic length of the electrical double layer, and the particles must be spherical. It is further assumed that the dielectric constant of the fluid is constant throughout, and the electro­lyte ions act as point charges. Figure 1 illustrates the effect of particle size on the interaction energy. Two Jarge particles must overcome a large energy barrier to coagulate while two smaller particles must overcome a smaner barrier. A large and a sman particle together have a greatly reduced barrier. This fact, coupled with the increased probability of an inter­action between small and large particles, explains why small particles preferentiaUy attach to large particles in polydis­perse systems.

A log normal distribution is used to describe the particle sizes, because it best represents actual powders l6:

P( )d - 1 - (In(ologl]'/2"; da a a----e -. uz..{iii a

(7)

The log normal width parameter (jz is related to the geomet­ric standard deviation by

U z = In({jg). (8)

The geometric mean particle size ag differs from the average particle size by

(9)

In general, the nth moment of a log normal distribution is

( n) n n',,;/2 (10) a =age .

Figure 2 illustrates log normal distributions for systems with the same average particle size, but different width param­eters.

For ceramic powders subjected to grinding, a wide dis­tribution corresponds to {jz = 0.75. For l-,um average size particles, 95% ofthe particles, by number, are between 0.17 and 3.28,urn. Ceramic powders made by chemical precipita­tion methods generally have much smaller width param­eters. Silica powders have been produced l

? this way with {jz

= 0.05. Titania powders have been produced l? this way

with {jz = 0.113. 18 For a distribution with {jz = 0.1 and (a) = 1.0,um, 95% of the particles are between 0.82 and 1.1 ,urn.

3873 J. Appl. Phys., Vol. 58, No. 10,15 November 1985

J 'j

1 1.5 2.5 Radius/Average Radius

FIG. 2. Ditrerent log normal distributions, the average particle size is kept constant and the width parameter is varied, (a) (7, = 0.1; (b) (7, = 0.25; (c) (7, = 0.5; (d) (7z = 0.75; (e) (7z = 1.0.

RESULTS AND DISCUSSION

There are two sets of parameters used in the calcula-tions. The first set contains material properties:

(1) Particle density; (2) fluid density; (3) fluid dielectric constant; (4) Hamaker constant. Once a powder and fluid are chosen, these properties are

not variable. The calculations described here are for titania (Ti02) powders suspended in aqueous salt solutions at 25 'C. The density oftitania powder is 3810 kglm3

.19 The Hamaker constant for titania suspended in water is 0.21 X 10- 19

1.18

Water has a dielectric constant of 78.1. 20 The second set of parameters contains variables that can be adjusted while making a ceramic suspension. They are

( I) Average particle size; (2) width parameter of distribution; (3) electrolyte concentration; (4) surface potential; (5) volume fraction of particles. The logarithm of the coagulation time is plotted versus

volume fraction for different degrees of polydispersity in Fig. 3. For most of the range, volume fraction has just a small effect on coagulation time. Significant increases in the co­agulation time occur at very low volume fractions where the particles are far apart, and therefore the diffusion distances and times are very large.

Narrowing the width of the particle size distribution has a much larger effect on increasing the coagulation time. This is seen more directly in Fig. 4. Along each curve (a) is kept constant and the logarithm of the coagulation time is plotted versus {jz.

For example, monodisperse 0.6-,um particles take roughly a year to coagulate, while those with a {jz of 0.4 coagulate in about an hour. Larger particles are inherently more stable because of their high-energy barriers, therefore, the rapid coagulation in polydisperse systems is due to the small particles in the distribution. This is observed in Eq. (2).

Strauss et a/. 3873

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'" o -'

6

o~ ___ _

-2+---------.-------~~------_.,-o .2 .4 .6 .8

Volume Fraction

FIG. 3. Logarithm of the coagulation time in seconds vs volume fraction. Increasing the width parameter reduces the coagulation time by orders of magnitude. The electrolyte concentration is O.OIM, 1/10 = 25.0 mY, and A = 2.1 X 10-20 J, (a) Uz = 0; (b) Uz = 0.25; (c) Uz = 0.5; (d) Uz = 0.75.

The total stability factor for a polydisperse system is found by summing the reciprocals of the individual pair stability factors, much like resistances in parallel. In this way pairs of particles which have large stability factors contribute little toward the total, while pairs which have small stability fac­tors contribute greatly.

Monodisperse slips which have sman or no energy bar­riers are affected only slightly by polydispersity because they are already coagulating rapidly such as for O.2-llm particles in Fig. 4, which have a coagulation time ofless than a second. Increasing U z does not decrease the coagulation time appre­ciably for these particles. The calculations for weakly inter­acting systems are similar to the results of Cooper, II who calculated stability factors, but not coagulation times, for very small particles with a Gaussian distribution of particles.

'" o ....J

The energy barrier between particles is enlarged by in-

8,---------------------------------------,

2,

o

-d

-4~1----~,------~, ----~,----~,----~ o .2 .4 .6 .8

Width Parameter

FIG. 4. Logarithm of the coagulation time in seconds vs width parameters for different average particle sizes. Polydispersity has a large eft'ect on sys­tems where the energy barrier for the average size particle is large. The elec­trode concentration is 0.01M, 1/10 = 25.0 mY, A = 2.1 X 10-20 J, (a) (d) = 0.6 pm; (b) (d) = 0.5 pm; (c) (d) = 0.4 pm; (d) (d) = 0.2 pm.

3874 J. Appl. Phys., Vol. 58, No.1 0, 15 November 1985

:J I I

~ I

N i 0, I 0 -'

0

-21 -4

0 .2 .4 .6 .8 Width Parameter

FIG. 5. Logarithm of the coagulation time in seconds vs width parameter for dilferent surface potentials. The electrolyte concentration is O.OIM, A = 2.1 X 10-20 J, (d> = 0.1 pm, (a) 1/10 = 40.0 mY; (b) 1/10 = 36.0 mY; (c) 1/10 = 33.0 mY; (d) 1/10 = 30.0 mY.

creasing the surface potential or decreasing the electrolyte concentration, which in tum increases the coagulation time. The effects of surface potential and electrolyte concentration are shown in Figs 5 and 6, respectively. Even though the energy barrier for the average particle size may be high enough to hinder coaguJ.ation, polydispersity will cause some of the energy barriers to be small or nonexistent, there­by reducing the coagulation time.

For the range of width parameters used. in Figs. 3-6 (O"uz " 1), the coagulation time decreases as U z is increased. This range corresponds to the inherent size distribution in most powders used to produce parts which need a homogen­eous fine-grain microstructure.

At very high levels ofpolydispersity (uz > 1), the coagu­lation time increases as U z is increased. These broad distribu­tions are generally obtained. by mixing different size ranges

3

-3~i-------r------~------~,------~,------~ 0.2 .6.8

Width Parameter

FIG. 6. Logarithm of the coagulation time in seconds vs width parameter ror different electrolyte concentrations. 1/10 = 25.0 mY, A = 2.1 X 10- 20 J, (d> = 0.2 pm, (a) Ce = O.OOIM; (b) Ce = 0.OO3M; (e) Ce = O.OO6M; (d) Ce=O.OIM.

Strauss et al. 3874

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ofpowders21 and produce sintered bodies which have either heterogeneous or large-grain microstructures. Figure 7 shows this coagulation behavior. It is the same as Fig. 4 except the calculations are extended to larger U z •

The reason for this behavior is that polydispersity af­fects coagulation in two ways. It has already been shown that it can decrease the stability factor by orders of magnitude, but it also decreases the total number of particles in a system, as seen in Eq. (A26). Since the kinetics of coagulation is sec­ond order, as shown in Eq. (A19), decreasing the number of particles in the system increases the time it takes for a frac­tion of them to coagulate.

These effects may be separated by taking the logarithm of the coagulation time:

(21T7J W T (aP)

10g(t ) = log + 1.30-;. c 49kTt/J

(11)

The first term on the right-hand side ofEq. (11) contains the effect of polydispersity on the interactions through the stability factor. The second term, a simple quadratic in U z ' is the effect of poJydispersity on the total number of particles. These two terms are graphed separately in Fig. 8.

The quadratic in U z is relatively flat for U z < 1, the re­gion where the stability factor decreases very rapidly. For U z > 1, the quadratic term grows at a much greater rate and overtakes the effect of the diminishing stability factor, which is also decreasing but more slowly because of the large num­ber of particles in the system.

In many ceramic processes, the processing time is essen­tiaUy the time it taKes for the particles to settle into their greenbody positions. For these situations the coagulation time may be scaled to the settling time. As an example, we will assume a slip remains stable if it takes at least 10 times as long to coagulate 2% of the particles as it does for the parti­cles to settle 0.01 m. The time a particle takes to settle a distance I in a fluid is given by the Stokes relationships. 22

(12)

----------- .--.- ---'-

o~~

::l~ o .4

W,dth Parcmeter

, 1 2

~---J 1.6

FlG. 7. Logarithm of the coagulation time in seconds vs width parameters for different average particle sizes. Same as Fig. 4, but extended to larger U,s. The electrolyte concentration is O.OIM, 1/10 = 25.0 mV, A = 2.1 X 10-201, (a) (d) = 0.61lm; (b) (d > = 0.51lm; (c) (d > = O.4llm; (d) (d> = 0.21lm.

3875 J. Appl. Phys., Vol. 58, No.1 0, 15 November 1985

-2 ~ I

-4 L-~------'---::~'----------==--==9 o .5 1.5

Width Parameter

FlG. 8, This graph splits the coagulation time into two parts. Curves a-d are the parts only due to the stability factor and are the logarithms of coagula­tion times minus 1.3 0;. Curve e is the part due to fewer particles in the system. The electrolyte concentration is O.OIM, "'0 = 25.0 mV, A = 2.1 X 10-20 1, (a) (d > = 0.6 11m; (b) (d > = 0.5 11m; (c) (d > = 0.4 11m; (d) (d) = 0.2 11m; (e) \.3 0;.

(a2) is the second moment of the particle size distribution, p

and P f are the densities of the particles and fluid, respective­ly, g is the acceleration due to gravity, t/J is the volume frac­tion, and h (t/J) is the volume fraction correction term due to the shear layer which develops in the fluid between moving particles and impedes their motion. There are many expres­sions for h (t/J) reported in the literature,2J but for simplicity we will use the one reported by Happel and Brenner2:

h _ 3 + 4.5W13 - t/J1/3) - 3t/J2

(t/Jl - 3 + 2tfJ5 / 3 (13)

Taking Eqs. (1) and (12) for the coagulation and settling times, and applying our criterion for stability, we get the expression

h (t/J) _ 2205 kTl T - 2WT (a3

) (a 2 )1T{p - P f)g (14)

For a given set of dispersion parameters the right-hand side of Eq. (24) can be calculated. The corresponding volume fraction is then numerically evaluated using Eq. (13). This volume fraction is an upper stability limit; any larger volume fraction yields an unstable system using our stability crite­rion-the time to coagulate 2 % of the particles is 10 times the time to settle 0.01 m,

Plots of volume fraction versus average particle size are then made for a given set of parameters. The plots represent boundaries between regions of stability and instability. It must be noted that processing can still be done in the unsta­ble region, but the amount of agglomerates may cause the final fired piece to be rejected, depending on the number of inhomogeneities that can be tolerated in its application.

Figure 9 shows the maximum volume fraction possible as a function of average particle size. The only difference between the four curves shown is the width parameter of the size distribution U z ' For narrow distributions (i.e., small uz )

high volume fractions of particles can be kept in suspension without agglomeration. For example, using curve d in Fig, 9,

Strauss et al. 3875

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75

.6

I c 2 U 45 0

t " i f

i ::J

3 i '0

! I /

>

.15 J / ! '

~ o I ,1) ,~ .5 1 1.5 2.5

Average Diome~er microns

FIG. 9. Maximum volume fraction vs average particle size. Higher volume fractions are achieved with narrower size distribution. The electrolyte con­centration is O.OIM, ,po = 25.0 mV, A = 2.1 X 10-20 J, (a) CT, = 0; (b) CT,

= 0.25; (c) CT, = 0.5; (d) CT, = 0.75.

a dispersion ofpartic1es with an average diameter of 1.5 /-lm and a CTz of 0.75 will not be stable, no matter what the volume fraction. For the same average particle size but CTz = 0.5, up to 30% particles, by volume, can be suspended, as shown by curve c. If the distribution is narrow with CTz = 0.25, essen­tially any amount of powder can be kept in suspension with­out agglomeration, as shown in curve b, as long as the mean size is greater than 0.75 /-lm. If the distribution is monodis­perse, CTz = 0.0, any amount of powder may be kept in sus­pension if the average particle size is greater than 0.6 /-lm as shown in curve a.

The effect of changing the salt concentration is iHustrat­ed in Figs. 10 and 11. For a given surface potential and width parameter, a higher electrolyte concentration decreases the volume fraction of particles that can be suspended because ions shield the coulombic repulsion and reduce the energy

75 1

,6 J

c .2

45 i U 0

t " E ::J

.3 ~ 0 >

'5 ~

D~ 0 .5 1.5 2.5

Average Diameter "T1icrons

FIG. 10. Maximum volume fraction vs average particle size. Lower salt concentration reduces the screening of repulsive forces, making coagulation time longer. Higher volume fractions are achieved with lower salt concen­trations. ,po = 30.0 mV, CT, = I, A = 2.1 X 10- 20 J, (a) Ce = O.OOOIM; (b)

Ce = O.OOIM; (c) Ce = O.OIM; (d) Ce = 0.02M.

3876 J. Appl. PhyS., Vol. 58, No. 10,15 November 1985

.75

.6

c .2 U 0

t " E ::J

0 >

, 1 5 2 2.5 Average Diameter "T1icrans

FIG. II. Maximum volume fraction vs average particle sizes. Same as Fig. 3 but with a narrower size distribution, with a width parameter of 0.5. ,po = 30.0 mY, CT, = 0.5; A = 2.1 X 10- 20 J, (a) Ce = O.OOOIM; (b) Ce = O.OOIM; (c) Ce = O.OIM; (d) Ce = 0.02M.

barrier to coagulation. This decreases the coagulation timell

and lowers the maximum attainable volume fraction of pow­der. Figure 11 is identical to Fig. 10, except it has a smaller value of CTz (i.e., 0.5 compared to 1.0). The curves with a U z of 0.5 are shifted to a higher volume fraction compared to those with a width parameter of 1.0.

Increased surface potential of the particles increases the amount of particles that can be suspended as shown in Figs. 12 and 13. The surface potential. of Ti02 and most oxide ceramics are changed by adjusting the solution pH and elec­trolyte concentration. Larger surface potentials result in larger repulsion between interacting particles. This increases the coagulation time. I I Figure 13 is similar to Fig. 12, but with a narrower size distribution. Comparison of these fig­ures once again shows that a narrow particle size distribu­tion is more stable than a wide distribution, for all surface potentials .

. 75 -

.5

c

~ u

.'5 0

t " E ::J

0 .3 >

.15

0 0 .5 1 1.5 2 2.5

Average Diameter microns

FIG. 12. Maximum volume fraction vs average particle size. Increasing the surface potential heightens the energy barrier between particles, increasing the coagulation time. Therefore, higher volume fractions of particles can be suspended. The electrolyte concentration is 0.01 M, CT, = I, A "",2. 1 X 10-20

J, (a) "'0 = 40.0 mY; (b~,po = 35.0 mY; (c) "'0 = 30.0 mY; (d~,po = 25.0 mY.

Strauss et al. 3876

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·75 -

I

i 6 -i

I c I 0 I

~ i 0 .45 •

~

" E 3 -I ~

a >

I

151

0 0 15 2 2.5

Average Diameter microns

FIG. 13. Maximum volume fraction vs average particle size. Same as Fig. 5 but with a narrower size distribution, O'z = 0.5, the electrolyte concentra­tion is O.OIM, A = 2.1 X 1O~2() J, (a) tPo = 40.0 mY; (b) t/lo = 35.0 mY; (e) ,po = 30.0 mY; (d) t/lo = 25.0 mY.

The effect of po]ydispersity can be evaluated by plotting the maximum allowable volume fraction versus the width parameter, holding the average particle size constant. This is done in Fig. 14 for a surface potential of 25.0 m V and an electrolyte concentration ofO.OIM. For most average parti­cle sizes, narrowing the particle size distribution causes a dramatic increase in the maximum volume fraction. Figure 14, curve a, shows that a 0.5-,um average particle size system is unstable even if it is monosized. For this specific set of conditions that is true; it is not generally the case. For 0.5-,urn particles with 25.0-m V surface potential and O.OIM elec­trolyte concentration, the energy barrier is roughly 20.0 kT. This is a fairly large energy barrier, but not enough to pre­vent coagulation on the 2% coagulation time scale. Larger energy barriers are required to prevent coagulation. Increas­ing the surface potential to 30.0 m V increases the energy

.75 -

\ .6 \

\ c \ a

\ ~ ~

45 ", \ I u. \ '" E \ :J

.3 ~: \ 0 >

\ \

, 5-j \ \ \

0+--= . '\ 0 4 6

Width Parameter

FIG. 14. Maximum volume fraction vs width parameter. The surface poten­tial is 25.0 mV and the electrolyte concentration is O.OIM for all except curve b, which has an electrolyte concentration of 0.00 1M, ,po = 25 mY, A = 2.1 X 1O~20 J, (a) (d) = 0.5 p.m, Ce = O.OIM; (b) (d) = 0.5 p.m, Ce = O.OOIM; (c) (d) = \.5 p.m, Ce = O.OIM; (d) (d) = 2.0 I'm, Ce =O.OIM.

3877 J. Appl. Phys., Vol. 58, No. 10, 15 November 1985

barrier to 50.0 kT. Decreasing the electrolyte concentration to O.OOIM increases the energy barrier to 60.0 kT. The effect of increasing the energy barrier by reducing the electrolyte concentration is shown in the dotted curve b in Fig. 14.

Keeping everything else constant, smaller particles have smaller energy barriers, which lead to higher coagulation rates. But this is only part of the picture. Smaller particles also have longer settling times: 0.5-,um particles take about 10.0 h to settle 0.01 m, while l-,um particles take only 2.5 h. Therefore, to meet our criterion of stability, small particles must have longer coagulation times relative to large parti­cles. This is an example of where the scaling of the coagula­tion time to settling time breaks down, in which case a differ­ent processing time should be used for comparison.

CONCLUSIONS

The small particles in a polydisperse system cause rapid coagulation because they have low-energy barriers. The en­ergy barriers between aU the particles should be large to pre­vent coagulation. Polydispersity can decrease the stability factor, and in turn, coagulation time by orders of magnitude even if the surface potential is high and the electrolyte con­centration is low, causing very rapid coagulation for a poly­disperse suspension.

In systems that have a very wide particle size distribu­tion, polydispersity still reduces the stability factor, but also reduces the total number of particles in the system, keeping volume fraction constant. Because the kinetics are second order, the coagulation time for a given volume fraction of particles increases.

APPENDIX: COAGULATION TIME

Following Verwey and Overbeek,2 the total number of particles of type j colliding with a particle of type i per sec­ond is

(AI)

D·· is the relative diffusion coefficient of the two particles, I)

N, is the concentration of particles of type j, ,uij is the ) . . .

mobility of particles of type j, and Vij IS the mteractlOn ener-gy between the particles. The first term in brackets is a flux due to a particle concentration gradient and the second one is the flux due to motion in a potential gradient.

Using the Nernst-Einstein relation24

,uij = Di/kT, (A2)

Eq. (ll) becomes

_ 2 (dNj ~ dVij(R l) (A3) Gij - 41TR Dij dR + kT dR .

After multiplying both sides of Eq. (A3) by eV,j.R)lkT dR I 41TR 2Dij , and rearranging terms, we find

G .. eV/)RVkT( dR ) = (eV,)R)/kTdN. I) 41TR 2D. )

I)

N.eV/j.RllkTdVij(R )) + J kT

= d (NjeV,j.R)lkT). (A4)

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Since Gjj is considered to be constant for short times and small depletion, Eq. (16) can be integrated over the limits R = aj + aj to R = 00. The interaction energy for large sep­arations is equal to zero and the concentration of j-type par­ticles assumes the bulk concentration. At the surface of the i­type particle the concentration of j-type particles is initially zero. Therefore, at the initiation of coagulation,

Gij roo eV,)R VRr dR 2 = NJ. 41r Ja,+ aj DijR

(A5)

The relative diffusion coefficient can be related to the particle size by ll,14.24

Dij =Dj +Dj , (A6)

kT kT D; = , Dj = , (A 7)

6rrqaJ3 (ui ) 6rrqa/3 (uj )

where fJ (u) is the correction term due to the hydrodynamic interactions between particles and U j is twice the distance from the surface of particle i to the plane of symmetry between the two particles divided by a;. The plane of symme­try is defined using bispherica1 coordinates-the natural co­ordinate system for two interacting particles. The parameter U; can be shown to bel4

(AS)

The function {3(u j ) has been solved as a series expan­sion,12.14,15 but is too time consuming to use repeatedly in computer calculations. Instead, we used an approximation derived by Honig et al. 12 which is good to a few percent over a large range of U values:

{3(u) = 6u2 + l3u + 2 . (A9) 6u2 + 4u

A dimensionless parameter, the pair stability factor may be defined as W(a/> aj). This term contains the hydrodynamic interactions

W(a j , aj) = (a j + aj )2

X r"" ( eV,)R)/kT dR) (AW) )a/Hj [a;/.B(uj ) + a;lfJ(u;)] R 2 •

It is written in this form to keep it consistant with the defini­tion of stability factors which assume no hydrodynamic in­teractions. Inserting the pair stability factor into Eq. (A5) yields

2kT(aj +aj )2NJ G j = (All)

• 31/a j aj W(a;,aj )

If the number distribution of particle sizes is P (a), with a bulk concentration of particles No. then the concentration of j-type particles is

NJ = Nc!'(a/'tdaj • (AU)

Therefore, the total number of particles of all types imping­ing on a particle i per second is

G - 2kTN, 50"" (a j + ajf P(aj ) -1 1--- 0 "aj'

3q 0 a/aj W(a;, aj) (Al3)

3878 J. AppI. Phys .. Vol. 58, No. 10, 15 November 1985

The total number of all types of particle sticking together per second must be

(AI4)

(A16)

(AI7)

W T is the total stability factor for a polydisperse colloidal system.

Since Gr is also the total number of particles coagulat­ing per second, Gr is the rate of depletion of the number of particles

Gr = (dNoIdt). (Al8)

dNoidt = (- 2kT /3qWT)N~. (A19)

By integrating Eq. (A19) we get

1 I 2kTt No(t) - No(O) = 3qWr'

(A20)

This equation is only valid for small times, when few parti­cles have agglomerated, because integrating Eq. (A19) as­sumes that the particle size distribution in Eq. (A12), Pta). does not change with time.

At time t f. if a fraction I of the particles have agglom­erated, then

No(t 1) = (l - I)No(O) (A2I)

and

I 1 2kT --=---t1 •

(1 - IINo No 3qWT (A22)

3qWr ( I ) t 1 = 2kTNo 1 -I .

(A23)

In our criterion for stability, 2% agglomerated particles is the limit, therefore

te = 3qWr/98kTNo. (A24)

No is the number of particles per unit volume. The volume fraction t/J, is the volume occupied by the No particles:

t/J = So'"' No(41r/3)a3P(a)da,

t/J = (4/3)1T{a3 )No.

Using Eq. (A261 the coagulation time is

te = 21T7JWr {a3 }/49kTt/J.

(A2S1

(A26)

(A27)

To calculate the stability factor. Eq. (10) is used for the particle size distribution and the following substitutions are made:

[i.a,;< a{ =a,e •

[i.,,'y aj = age .

The stability factor then becomes

Strauss et al.

(A2S1

(A29)

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rr(f"" f"" e-x2e-y'cosh2[(O"z/~)(x-y)] )-1 WT =- dxdy

4 -"" -00 W(ai,aj )

(A30)

The integrals in Eq. (A30) are evaluated using Gauss-Hermite quadrature26 which converges for 16 weights.

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3879 J. Appl. Phys., Vol. 58, No. 10,15 November 1985

!,OG. Herdan, Small Particle Statistics (Butterworths, London, 1960), pp. 80-90.

17T. C. Huynh, B. Bleier, and H. K. Bowen, Presentation at Convention of the American Ceramics Society, Cincinnati, 1982 (unpublished).

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