Langmuir 1993,9, 2836-2843
2836
Prediction of Composite Colloidal Suspension Stability Based upon the Hogg, Healy, and Fuerstenau Interpretation B. A. Wilson and M. J. Crimp' Department of Materials Science and Mechanics, College of Engineering, Michigan State University, East Lansing, Michigan 48824-1226 Received March 17, 1993. I n Final Form: July 6, 199P A computer program was developed to predict the dispersion stability of ceramic composite systems. The program uses the suspension properties of the individual materials such as particle size and zeta potential along with the system processing conditions as the basis for calculation. The program is based upon a model which is a modification of the Hogg, Healy, and Fuerstenau (HHF) method. The homocoagulation and heterocoagulation behaviors of systems are determined in terms of a stability ratio, W, as a function of pH. The particle potential was modified to allow inclusion of electrophoretic and acoustophoretic data. Electrophoresis and acoustophoresis data for Sic and Si3N4 are presented and discussed. The predicted ranges of stability for the SiC/SisNr system were determined. Modeling results were correlated with sedimentation experiments. The predicted effects of variations in the Hamaker constant, relative volume fraction,and temperature were investigated. Variations of *lo% in the Hamaker constant resulted in insignificant alterations in the stability predictions. Differing relative volumefractions of components have little effect on predictions of total stability. However, sedimentation experiments indicated that differing relative volume fractions of Components did have a significant effect on the actual system stability. Although temperature variations of 1 6 'C of room temperature have little effect on the predicted stability, ice bath temperatures significantly alter the stability predictions.
Introduction The majority of ceramic materials are made by processing raw material powders into a green body followed by heat treatment, causing pore elimination, densification, and microstructural development. Uniformity among the composite constituents is important since it governs the microstructures of the final sintered body. This in turn determines the characteristic attributes including mechanical, electrical, and magnetic properties. Areas in which ceramics are presently found include electronics, dentist@, automotive engines, industrial tooling, and biological prostheses. Difficulties with advanced ceramics arise from the inability to reproducibly process ceramics having identical microstructures and properties. The formation of such nonuniform ceramicsand ceramicmatrix composites (CMCs) is caused by inhomogeneities which are a direct result of agglomerates present while the components are in suspension. Agglomerates are one of the main causes of inhomogeneities since they cause an uneven distribution of pores, yielding uneven pore collapse during subsequent consolidation procedures.' The preparation of well-dispersed, stable suspensions helps to alleviate the problems caused by agglomeration and leads to the formation of a uniform green body and consequently results in improved consolidation results. It is important to increase control of the processing in order to reduce microstructural defects and to maximize reliability. Dispersion of the powder consists of two elements: the breakup of agglomerates into primary particles to form a suspension and the stability of this suspension. While it is ideal to have a disperse, stable suspension in order to improve the final microstructure by removing agglomerates, new processing methods contain steps which call for alternating from stable suspensions to unstable coagulated suspensions (containing large, loosely bound agglomerates). Such procedures prevent segregation of the various system components and make it possible to remove excess Abstractpublished in Advance ACS Abstracts, August 15,1993. (1)Dynys, F. and Halloran, J. W. J. Am. Ceram. Soc. 1983,9, 655.
salts and surfactants by washing.2 Due to the importance of preparing stable suspensions and the combined use of stable and unstable suspensions, the prediction of the stability of ceramic suspensions would be an important tool to aid in the control of processing advanced ceramics. Pioneering work in this area was done in the 194Os, resulting in the well-known DLVO t h e ~ r y . ~ This ! ~ theory describes the totalinteraction energy between particles in a single-component suspension. From this information, some insight into the stability behavior of the suspension can be inferred. However, with the increasing use of multicomponent systems due to the addition of reinforcements and processing additives (e.g., sintering aids, stabilizers, composites, etc.), it is necessary to develop a theory for the prediction of the stability of multicomponent systems. In the current study, the stability behavior of multicomponent ceramic systems will be examined and predictions of the stability studied by entering system and material data (such as pH, electrolyte concentration, particle size, Hamaker constant, and zeta potential) into a computer program which uses an approach that is an adaptation of a method originally developed by Healy, Hogg, and Fuerstenau.6 Knowledge of the stability conditions of suspensions can be applied in selective flocculation in waste treatment,sg biological/medical (2)Lange, F. F. J. Am. Ceram. SOC.1989, 72,3. (3)Derjaguin, B. V.;Landau, L. D. Acta Physicochim. URSS 1941,14, 633. (4)Verwey, E. J. W.; Overbeek, J. Th. G.Theory of the Stability of the Lyophobic Colloids; Elsevier: Amsterdam, 1948. (5)Hogg, R.;Healy, T. W.; Fuerstenau, D. W. Trans. Faraday SOC. 1966,62,1638. (6) Daniels, S. L. In Theory, Practice, and Processing for Physical Separations; Freeman, M. P., Fitzpatric, J. A., Eds.; Engineering Foundation: New York, 1981;p 1. (7)O'Melina,C. R. In Theory, Practice, and Processing for Physical Separations; Freeman, M. P., Fitzpatric, J. A., Eds.; Engineering Foundation: New York, 1981;p 5. (81 Hahn, H. H.; Eppler, B. Colloid and Interface Science, Volume IV: Hydrosols and Rheology; Academic Press: New York, 1976;p 125. (9)Shaw, D.J. Introduction t o Colloid and Surface Chemistry; 4th ed.; Butterworths: Boston, 1992.
0743-746319312409-2836$04.00/0 0 1993 American Chemical Society
Composite Colloidal Suspension Stability
Langmuir, Vol. 9, No. 11,1993 2837
studies of blood coagulationloand cholesterol stability," paint stability,12paint retention,13deposition and adhesion of material coatings,1p16and oil drilling.17
Literature Review Colloidal Forces. To model the composite interaction forces, it is necessary to describe the forces which effect particle interactions in colloid suspensions. The forces acting upon approaching colloid particles include van der Waals forces, steric forces, and repulsion forces. Repulsion forces are the result of the electrical charges on particule surfaces. Steric forces can be caused by copolymers which are added and adsorbed on the surfaces of particle^.^ The present work will be concerned with aqueous suspensions which do not contain copolymers so that steric repulsion forces will not be considered hereafter. London forces are the dominant attractive force in suspensions unless the materials are highly HamakerlS2O calculated the force due to van der Waals attraction which results from London forcesby using a simple pairwise addition of atomic forces, with the potential decreasing as the inverse of the sixth power of the separation distance.21 Van der Waals attraction without retardation for two different interacting particles of radii ai and aj separated by a distance H can be represented by20
V, = -
--
"[ 12 x
+xy+x
+x
+ x 2 + xy + x ] 2 log x2 + xy + z + y
potential energy of interaction as DLVO theory, namely6~~~ (4) However, HHF uses expanded attraction and repuleion force equations to take into account the differences between the two particle types. In developing an equation to describe the repulsive energy of interaction, the linear form of the Debye-Htickel form of the Poisson-Boltzmann equation is used as in DLV023
d2*,/dx2 = (5) Hogg et al. show that the approximation is valid for fi and * j values of less than 50-60 mV as opposed to 25 mV that is assumed in DLVO. To solve eq 5 in HHF, the two different particles are treated as two plates having a constant potential, separated by a distance 2d with the boundary conditions
*=\koiasH+O
and *-qOjasH-2d
The solution then becomes \k = 'ko, cosh(KH)
+
*o,
(
sinh(KH) (7) The energy of repulsion for HHF was then found using Derjaguin's m e t h ~ d , as ~ ~in?DLVO: ~~ sinh(2nd)
+xy+x+y
(1)
Particle interactions in a medium are less than in a vacuum because of the presence of molecules of the medium between the two interacting particles. Therefore, for two different particles in a medium an effective Hamaker constant is used:9
-
)
- qOi COSh(2rd)
The Hamaker constant, A, must be calculated in order to evaluate the force of attraction due to van der Waals forces. Bleier used a simplified Lifshitz method for calculating the Hamaker constant in a vacuum:22
A,, = (
(6)
~ ~ 1 / 2~ , 1 / 2 ) (JY2 ~
- A m 1/21
(3) HHF Theory. In the mid-19605, Hogg, Healy, and Fuerstenau5 (HHF) built upon theory to develop a quantitative kinetic stability theory for nonidentical particles which was more easily applied to actual systems. HHF theory uses the same approach to describe the total (10) Sprinvasen, S.; Weiss, B. R. Colloidal Dispersions and Miscellar Behavior; American Chemical Society: Washington, DC,1975; p 322. (11) Carrique, F.; Salcedo, J.; Gullardo, V.; Delgado, A. V. J. Colloid Interface Sci. 1991, 146, 573. (12) Parfitt, G. D. Dispersions of Powders in Liquids with Special References to Pigments; Elsevier: New York, 1969. (13) Jaycock, M. J.; Pearson, J. L. J . Colloid Interface Sci. 1976,55, 181. (14) Elimelech, E. M. J. Colloid Interface Sci. 1991, 146, 337. (15) Tamura, H.: Matiievic, - . E.;. Meites. L. J . Colloid Interface Sci. 1983,92,303. (16) Matiievic..E.:. Kuo.. R. J.:. Kolnv. - . H. J. ColloidInterfaceSci. 1981. 80, -94; (17) Everett, D. H. Basic Principles of Colloid Science; Royal Society of Chemistry: London, 1988. (18) Hamaker, H. C. Recl. Trav. Chim. Pays-Bas 1936,55, 1015. (19) Hamaker, H. C. Recl. Trav. Chim. Pays-Bas 1937,56, 727. (20) Hamaker, H. C. Physica 1937, 4, 1058. (21) Tabor, D. Colloidal Dispersions; The Royal Society of Chemistry: London, 1981. .(22) Bleier, A. J. Am. Ceram. SOC.1983, 66, C79.
MI - e x p ( - 2 ~ ~ ) )(8) ] The total potential energy as a function of the interparticle distance for two nonidentical spherical particles is calculated by using eqs l and 8 to solve eq 4 as a function of varying separation distance, H. A graph of total potential energy versus particle separation is used to indicate the stability of single-component and multicomponent systems where a maximum potential energy of 20 kT is often noted to indicate stability.26 However, these plots do not address the kinetic aspect of stability. In order to better describe the effecta of homoand heterocoagulation and to generate a quantitative theory for the overall kinetic stability of the system of nonidentical particles, Hogg et aL5developed a variable, WT,which is similar to an expression by F u ~ hto s describe ~~ the total stability of the system: (1- n)2
+
2n(l-
w,= [&+ly,
q-1
(9)
w12
The factor W is a factor by which rapid coagulation, as described by von S m o l u ~ h o w s k i , 2is~slowed ~ ~ ~ due to a potential barrier to coagulation caused by the repulsive potential energy of particle interaction^.^.^^ This factor (23) Heimenz, P. C. Principles of Colloid and Surface Chemistry,2nd
ed.; Marcel Dekker: New York, 1986. (24) Derjaguin, B. V. Kolloid-Z. 1934,69, 155. (25) Derjaguin, B. V. Acta Physicochim. URSS 1939,10, 33.
(26) Lyklema, J. Colloid andlnterface Science; AcademicPrem: New York, 1976; Vol. I, p 257. (27) Fuchs, N. Z. Phys. 1934,89,736. (28) von Smoluchowski,M. Phys. Z . 1916,17, 557. (29) von Smoluchowski,M. Z. Phys. Chem. 1917,92, 192. (30) Overbeek, J. Th.G. Colloid and Interface Science; Academic Press: New York, 1986; Vol. I, p 431.
Wilson and Crimp
2838 Langmuir, Vol. 9, No. 11,1993 is called the stability ratio and is essentially the ratio of particle collisions t o collisions resulting in coagulation.26 For two nonidentical particles of radii ai and aj and a distance of separation r (measured from particle center to particle center) the stability ratio is31 (10) Wiese and Healy later derived an energy of repulsion solution similar to the HHF solution with the same form, but changed the boundary conditions for systems with particle charges which remain constant:32
ln(1- exp(-2m)
1
(11)
For systems in which both components are not considered to have constant potential or constant charge, Kar, Chander, and Mika33used an approach similar to the one used by Hogg e t al. to develop a solution for systems with boundary conditions in which one of the components has constant charge and the other has constant potential. Barouch and MatijeviE have worked with various other researchersm7 to develop a complex interaction potential for nonidentical particles that uses an approximate solution of the Poisson-Boltzmann equation in its two-dimensional form with boundary conditions of constant potential. Barouch and MatijeviE note there is only one axis of symmetry for spherical particles, and conclude that a twodimensional solution yields more accurate results. The results of their solution for the interaction potential are cornpared with those of Hogg, Healy, and F~erstenau.~' The two models agree fairly well for unlike particles with potentials of opposite sign and similar magnitudes. For particles with the same sign and different magnitudes, the HHF equation overestimates the interparticle repulsion according to Barouch and M a t i j e ~ i E .However, ~~ the model developed by Barouch, MatijeviE, et al. is mathematically complex, and its solution is complicated, involving intricate use of several algorithms as the authors a~knowledge,3~ and there is some question as to the accuracy of their method.3741 Therefore, the computer program developed by the authors will be based on HHF's original paper.5
Experimental Procedure The two-component systemused for the stability investigation was the SiC/SisN4 system. The colloidal and surface characteristics of the two types of powder were known due to extensive (31) Overbeek, J. Th. G. In Colloid Science Volume I: Irreversible Systems; Elaevier: New York, 1952. (32) Wiese, G. R.; Healy, T. W. Trans. Faraday SOC. 1970, 66, 490. (33) Kar, G.; Chander, S.; Mika, T. S. J. Colloid Interface Sci. 1973, 44,347. (34) Barouch, E.; Matijevic, E.; Ring, T. A.; Finlan, J. M. J. Colloid Interface Sci. 1978, 67, 1. (35) Barouch, E.; Matijevic, E. J. Chem. SOC.,Faraday Trans. 1 1986, 81. 1797. (36) Barouch, E.; Matijevic, E.; Wright, T. H. J. Chem. SOC., Faraday Trans. 1 1985,81,1819. (37) Barouch, E.; Matijevic, E. J. Colloid Interface Sci. 1985,106,505. (38) Chan, D. Y.C.; White, L. R. J. Colloidlnterface Sci. 1980,74,303. (39) Chan, B. K. C.; Chan, D. Y. C. J. Colloid Interface Sci. 1983,92, 2&1. (40) Overbeek, J. Th. G. J. Chem. SOC.,Faraday Trans. 1 1988,84, 3079. (41) Barouch, E. J. Chem. SOC.,Faraday Tram. 1 1988,84,3093.
investigations by The Sic was an a-Sic powder (UF10,Lo-) with a manufacturer-reportedparticle size of 1.8 wm. The ShN4 powder was an a-SisN4 (SN-E10, UBE) with a manufacturer-reported particle size of 0.5 pm. The Hamaker constants used for this systemwere determinedby Bleiern using the Lifshitz method. The values for the Hamaker constantawere 3.0 X and 1.6 X WeJ for Sic and Si& respectively.n Zeta Potential. The zeta potential data wed in this investigation were of two types. The F i t was electrophoresis zeta potential data previously collected by Crimpu using a PENKEM SYSTEM3000automated electrokineticeanalyzer. These data were collectedfor both typea of powder. Each typeof powder was suspended in deionized water which had an indifferent electrolyte concentration of 10-9M KNOs. The pH was varied for this system by the addition of HNOa and KOH. The syetems were dispersed using an ultrasonic probe to eliminate previous particle agglomeration. The other type of data used was electrokineticsonicamplitude (ESA) zeta potential data collected in this investigation using a Matec ESA-8000 system. The electrokinetic phenomena of acoustophoresis (ESA)has been describedin detail by O'Brien&# and Oja." An acoustic wave is generated by particles as they oscillate in an alternating electric field due to their charge and a densitydifferenceexistingbetween the particlesand the liquid. The formation of this acoustic wave is termed the electrokinetic sonic amplitude (ESA) and is measured as a pressure amplitude per unit electric field applied. The ESA dynamic mobility was found by using an equation derived by O'Brien:&# = ESA(W)/&APC (12) where ESA(w)is the pressure amplitude per unit electric field, & is the volume fraction of solids, Ap is the density difference of particles and liquid, and c is the velocity of sound in the PJW)
suspension. The zeta potential is calculated using the mobility (asis the case for electrophoresis)with the additionof a correction for the inertia of the particle in an alternating field. This correctionreduces the velocityamplitudeof particle motion. The equation to calculate the zeta potential was derived by O'Brien using the Helmholtz-Smoluchowski equation:&,&
l =(rddcoc,)G(a)-' where G(a)is the inertia correction term:
(13)
ESA zeta potential measurementa were performed on both powders with the suspensions prepared at an electrolyte concentration of 1W M KNOs and a solid content of 0.5 vol % (Figure 2). The suspensions were initially dispersed by an ultrasonic probe. The pH was then decreased by addition of HNOa to pH 4.0, and the suspensions were again ultrasonically dispersed before loading into the ESA testing equipment. Once testing began, the system automatically increased the pH to 11 with KOH and then decreased to pH 4 with HNOs. Experimental Stability. Sedimentation experimenta were performed using suspensions of 0.5 vol % solids in electrolyte (1VM KNOs). The suspensions were initially dispersed by an ultrasonic probe. The pH was then decreased by addition of HNOs to pH 4.0, and the suspensions were again ultrasonically dispersed before the pH was varied from 4 to 11 in 0.5 pH incrementa by addition of KOH. Photographs were taken at varying time intervals to record the sedimentationheight versus time. Stability Predictions. The computer prediction program was originally used to verify the HHF predictions in order to be (42) Crimp, M. J. Colloidal Characterization of Silicon Carbide and Silicon Nitride. M.S. Thesis, Case Western RBserve Univemity, 1986. (43) Crimp, M. J.Surface Charaderization of SiliconNitride and Silicon Carbide. Ph.D. Dissertation, Case Western Reserve University, ID88. (44)Crimp, M. J.; Johnson, R. E.; Halloran, J. W.; Feke, D. L. Science of Ceramic Chemical Processing; Wiles New York, 1988, p 539. (45) OBrien, R. W. J. Fluid Mech. 1986, 190, 71. (46)OBrien, R. W. J . Fluid Mech. 1990,212, 81. (47)Oja, T. Matec Applied Sciences Inc., New York. Personal communications.
Composite Colloidal Suspension Stability
Langmuir, Vol. 9, No. 11, 1993 2839
'i
4 0 1 20
11
Material
20
Material +SIC
* S4N.
I
-60
-80 2
3
4
5
6
7
8
9 1 0 1 1
PH
4
5
6
7
8
9 1 0 1 1
PH
Figure 1. Zeta potential vs pH for Sic and SisNI from electrophoresis measurements at an electrolyte concentration of 10-9 M KNOS and solids content of 0.01% .
Figure 2. Zeta potential vs pH for Sic and SiaN, from ESA electroacousticmeasurements at an electrolyte concentration of 10-9 M KNOs and solids content of 0.5%.
certain that the program was executing properly. Input data identical to those from HHF's original papeld were used for a series of computer runs in order to duplicate their data. The output data were found to replicate the stabilitydata from HHF's
contribution G(a) is overestimated, resulting in an underestimation of the zeta potential. A factor responsible for the differencesin particle radius and the need for an effective radius is the agglomeration of particles into larger effective particles during ESA testing.& Electrophoresis does not experience a similar problem with particle size determination since it does not %ee" an effective particle radius because the electrophoresis measurementsare taken at lower volume percents of solids to liquids in the suspension. Predictive Program. With all of the different models for the repulsive interaction for unlike particles, the HHF method is the only model which incorporates the kinetic aspect of the system stability into quantitative expressions for the coagulation behavior of a system of nonidentical particles. This method is therefore best suited for use in the prediction of ceramic composite suspension stability. The present study developed a computer program which allows prediction of the stability for ceramic composite suspensions and is based upon a method originally developedby Hogg, Healy, and F~erstenau.~ An advantage of this method is that various adjustments can be made within the model to improve the method's stability predictions. The authors' prediction program includes several changes or adjustments to the original HHF method. The first change is the use of zeta potential, 5; versus pH data to replace the surface potential, $PO.This eliminates the need for a calculated surface potential taken from models for surface charge generation and experimentally determined, point of zero charge data. These surface charge models require considerable knowledge of ion groups present on the particle surfaces as well as an understanding of how these groups react with the medium to generate ~harge.~=2The zeta potential is a value calculated from experimentallymeasured mobility values which requires little knowledge of the particles other than size and density."
originalpaper.
Similar verificationsuccessfullyoccurred,showing
the repulsion potentials calculated were identical to other published HHF repulsion p~tentials.~~ Extensive stability prediction program runs were performed using the adapted method for the Sic powderISisN4 powder system.
Results and Discussion Zeta Potential. The electrophoresiszeta potential data for both S i c and Si3N4 are plotted as zeta potential versus pH and shown in Figure 1. The curves in these and all other subsequent graphs are computer drawn best fit curves for the given data points. The data for Sic from pH 2 to pH 5 (Figure 1)are actual measurements which cease at pH 5 since the zeta potential values level off at this point. A logarithmic regression was done in order to extend this data to higher pHs using a standard logarithmic regression, and these are the data points shown for pH > 5. Figure 2 shows the acoustophoresis zeta potential data for both the Sic and Si3N4powders plotted as zeta potential versus pH. Examination of the Si3N4 data shows the magnitude of the acoustophoretic ESA zeta potential data (Figure 2) as lower in magnitude than the values of the electrophoresis data (Figure 1) for the same type of Si3N4 powder. A similar discrepancy between measured acoustophoretic ESA data and electrophoresisdata was discussed by James, Hunter, and O'Brien& in testing UBE SN-E10 Si3N4. For Si3N4, these authors showed that if an effective radius at each pH was used in place of the average particle size to calculate the zeta potential with the acoustophoretic ESA technique, the results were found to approximately match the electrophoresis data. The reasons are found in eqs 13 and 14 and the definition of a (a= wa2p/v). The inertial correction to the dynamic mobility, G(a), is dependent upon both the frequency of the acoustic wave and the particle size. Generally, G(a)is an inverse function of the particle radius. If the radius is underestimated,the inertial (48)James, M.;Hunter, R. J.; O'Brien, R. W. Langmuir 1992,8,420.
(49)James, R.0.; Parks, G. A. Surface and Colloid Science; Plenum Press: New York, 1982;Vol. 12,p 119. (50) Johnson, R. E., Jr. J. Colloid Interface Sci. 1984,100, 540. (51)James, R. 0.In Adsorption of Inorganics at Solid-Liquid Interfaces; Ann Arbor Science: Ann Arbor, 1981;p 219. (52)Healy,T.W.;White, L. R. Advances in Colloid and Interface Science; Elsevier: Amsterdam, 1978;Vol. 9,p 303.
Wilson and Crimp
2&40 Langmuir, Vol. 9, No. 11, 1993
The second change is the use of an effective Hamaker constant (eq 3) in place of an in vacuo Hamaker constant. Thirdly, the Lifshitz method for calculating Hamaker constants (eq 2) is incorporated since a method for this calculation was not addressed by the original theory. The fourth change is the use of an expanded expression for the potential of interparticle attraction (eq 1)in place of the equation used in the original HHF method (which is a simplification of eq 1with an approximation that assumes small particle separations (H/a 10, for the Indicated Figures SiC/SiC SiC/SisN4 SisNJSisNA total 7-11 allpH 4-5.5,7-11 7-11 Figure3 Figure4 6-11 nopH nopH no pH allpH 6.5-11 4-5.5,7.5-11 7-11 Figure5 Figure6 6-11 7.5-11 4-5.5,7-11 7.6-11 ~
Table 111. pH Ranges of Suspension Stability Predicted for log W > 40, for the Indicated Figures SiC/SiC SiCfSisN4 SisNJSi& total 4-5.5,8-11 6-11 allpH 7.5-11 Figure3 Figure4 6-11 nopH nopH no pH 6.5-11 4-5.5,7-11 7-11 allpH Figure5 Figure6 6-11 8-1 1 4-4.5,8-11 8-11 4
5
6
7
8
9
1011
PH
Interactions
'0. SIC/SlC
+ SIC/SI,N, * S\N,/S\N, *TOTAL
Figure 6. Stability ratio vs pH using eq 8 and acousto- and electrophoresis data for SiC/SirN4 where the temperature is 25 OC, the RVFC is 0.5,and the concentration of KNOSelectrolyte is 10-9M.
The values plotted in Figure 4 were calculated using the constant potential repulsive equation along with ESA zeta potential data. The constant charge form of the repulsive potential equation was used along with electrophoretic data for the calculations shown in Figure 5. The Sic ESA zeta potential and the Si3N4 electrophoresisdata were used along with the constant potential repulsive equation to calculate the values in Figure 6. HunteP notes that, for a system with a potential barrier peak of 25 kT,an approximate stability value can be estimated to be loB, since the peak is dominant in determining the value of W. Since a potential barrier peak of greater than 20 kT is often used to indicate a ~~~~
(55) Hunter, R.J. Foundations of Colloid Science; Clarendon Press: Oxford, 1987; Vol. 1.
stable system, a system with a stability ratio of 10'0 or greater can initially be used to indicate a system which would be stable with respect to agglomeration. Using this initial estimation of system stability for the computergenerated stability values in Figures 3-6, some initial estimates of system stability were made for the different particle interaction types and the total interaction which is the combined effect of the other interactions (Table
11). The discrepanciesbetween the predicted stability ranges (Table 11) and the experimental stability ranges (Table I) highlight a discrepancy between the stability ranges initiallyestimated from Figures 3-6 and the stabilityranges seen in actual experimentation (Figure 7). In Figure 3, the Si3N4 experimental stability ranges (Table I) for both the lower and higher pH ranges of stability correspond to log W > 40. Examining Figure 4, the experimental stability occurs at pH > 5.5 which corresponds to a log W > 40. Using log W values of greater than 40 as an indication of stability, the stability ranges for Figures 3-6 are shown in Table 111. Comparing Tables I and I11 indicates that the program can indicate ranges of stability. Using the predictive program along with experimental observations leads to stability ranges indicated by pHs which have log W values greater than 40. Hamaker Constant. The Hamaker constant can be a
2842 Langmuir, Vol. 9, No. 11, 1993
Wilson and Crimp 120-
P #
4
5
6
8
7
*
*
+
*' '
9 1 0 1 1
PH e - 3 . 0 , 1.6 + 3.3, 1.44 "2.7.
1.76
Figure 8. Stability ratio vs pH using eq 8 and acousto- and electrophoresis data for SiC/SisN, where the Hamaker constant is varied by &lo%, the temperature is 25 O C , the RVFC is 0.5, and the concentration of KN03 electrolyte is 1(Fs M. difficultvalue to ascertain. HornMfinds dielectricconstant data to be enigmatic for some materials. Problems with obtaining Hamaker constant values for materials raise questions about how much of an effect changes in Hamaker constant will have on stability ratios predicted by the program. Therefore, additional predictive computer runs were carried out where the Hamaker constant was varied. In the first of these runs, the Hamaker constant for Sic was increased by 10%and the Hamaker constant for Si3N4 was decreased by 10%. In the second of these runs, the Hamaker constant for Sic was decreased by 10% and the Hamaker constant for Si3N4was increased by 10%. These results are shown in Figure 8 along with the original, unchanged Hamaker constant predictive runs. The figure illustrates at most a 0.1 pH shift in the pH range for stability. This is a fairly insignificant shift in the pH range since, for practical purposes, intervals of 0.5 pH are only experimentallypractical. This result is encouraging since it allows for more leeway in determiningdielectric constant data and calculating Hamaker constants. Temperature. The effect of normal laboratory temperature fluctuations on the predicted stability ranges was also investigated. Two predictive computer runs were performed where the temperature was increased to 30 "C and decreased to 20 "C from the ambient laboratory temperature of 25 OC. When these data were plotted, no measurable change in the stability was seen. The predicted effect of larger changes in temperature on stability were also investigated. Predictive computer runs were performed with temperature values of 0,25, 50, and 75 OC. These predictive values are shown in Figure 9. The pH of stability deviated slightly from room temperature for 50 and 75 O C . For 75 OC, Figure 9 shows less than a 0.2 pH shift in predicted stability from the stability predicted forroom temperature. Ashiftof0.5pHlessforthestability versus the 0 O C (ice bath) case was predicted. This is an interesting result since it indicates that, by processing at lower temperatures, stability could be achieved up to 0.5 pH lower than at room temperature. This effect should also be magnified since, at the lower temperature, particle (56)Horn, R. G.J . Am. Ceram. SOC.1990,73, 1117.
~~
(57) Ohehima, H.; Healy, T. W.; White, L. R.J. ColloidInterface Sci. 1982,89,484.
(58)Ohehima,H.;Chan, D. Y. C.;Healy, T. W.; White,L. R. J.Colloid Interface Sci. 1983,92, 232. (59) Wang, Q. J. Colloid Interface Sci. 1991, 145, 99. (60)Weaver, D. W.; Feke, D. L. J. Colloid Interface Sci. 1986,103, 267.
Composite Colloidal Suspension Stability
dispersive X-ray detector to help differentiate homocoagulation and heterocoagulation.
Conclusions A computer program was developed to predict the dispersion stability of ceramic composite systems. The program uses the suspension properties of the individual materials such as particle size and zeta potential along with the system processing conditions as the basis for calculation. The program is based upon a model which is a modification of the Hogg, Healy, and Fuerstenau (HHF) method. The homocoagulation and heterocoagdation behavior of systems is determined in terms of a stability ratio, W, as a function of pH. Modifying theoretical parameters allowed inclusion of electrophoretic and acoustophoretic data for particle potential. Constant potential and constant charge expressions for the interparticle repulsive potential were used in the program. The authors developed an expression which replaces the number fraction of components with the volume fraction of components (eq 15). Electrophoresis and acoustophoresis particle potentials of S i c and Si3N4 were measured (Figures 1 and 2). The measured acoustophoresis (ESA) data for S4N4 were much lower in magnitude than the electrophoresis data for the same material. The discrepancy was attributed to problems in characterizing the particle size. A similar discrepancy for t potential measurements of SisN4 was observed by James et al." Modeling results were correlated with sedimentation experiments for the SiC/Si3N4 system. Stability ranges were found to be indicated by pHs which correspond to log W values greater than 40. The predicted range for total stability of the SiC/SisN4 system was pH 7-11 or 8-11 (Table 111), while the experimental total stability range was pH 7.5-11 (Table I). The predicted range for the stability of S i c with respect to homocoagulation was pH 6-11 (Table 111) which was the same range found experimentally (Table I). The predicted effects of variations in the Hamaker constant, relative volume fraction, and temperature were investigated. Variationsof &lo% in theHamakerconstant resulted in insignificant alterations in the stability predictions. Differing relative volume fractions of components have little effect on predictions of total stability; however, experimentation did not confirm this. Temperature variations of k5 "C of room temperature were
Langmuir, Vol. 9, No. 11, 1993 2843
found to have little effect on the predicted stability. Ice bath temperatures significantly alter the predicted stability, decreasing the predicted stability range by 0.5 pH compared to the room temperature stability range; however, sedimentation experiments were not performed to c o n f i i this predicted stability range shift.
List of Symbols A = Hamaker constant Aeff = effective Hamaker constant (accounting for the medium) A,,, = Hamaker constant of the medium a = particle radius c = velocity of sound in a suspension e = electron charge (1.6023-19 (C)) ESA(o) = pressure amplitude per unit electric field G(a).= correction factor for particle inertia H = mterparticle separation distance i = d-1 k = Boltzmaun's constant (1.3813-23 (J/K)) n = overall proportion of the number of particles of . type 1 ni" = number of ions of type i in the bulk r = diatance from the particle center RVFC = relative volume fraction of components T = temperature (K) VA = attractive energy VR = repulsive energy VT = total energy W = stability ratio Wij = stability ratio for stability between particles i and j WT = total stability ratio x = H/(ai + aj) y = aJaj zi = valence of ion i u = qaZp/v t = dielectric constant = €06, t o = permitivity in a vacuum f8.8543-12 [C2/(Jm)ll tr = relative permitivity q = viscosity of the liquid medium K = Debye-Htickel parameter (2 = e2Cni0zi/kT)23 l' = zeta potential g d = dynamic mobility p = particle density Ap = density difference of particles and liquid Y = kinematic viscosity (VIP) $C = volume fraction of particles 9 = particle potential 90= particle surface potential o = angular frequency of the applied field
