Faraday Discuss. Chem. Soc., 1990, 90, 17-29 Electric-field-induced Aggregation in Dilute Colloidal Suspensions Paul M. Adriani and Alice P. Gast Department of Chemical Engineering, Stanford University, Stanford, CA 94305-5025, USA Electric-field-induced chain formation in dilute, non-aqueous suspensions of sterically stabilized, 1 Fm, poly(methy1 methacrylate) (PMMA) latices are investigated. Optical microscopy and digital image analysis provide the chain-length distribution. We find that the particles carry a charge sufficient to inhibit field-induced aggregation. Equilibrium predictions of chain aggre- gation incorporating a screened Coulombic repulsion and field-induced dipole attraction agree well with experimental observations near the onset of aggregation; chain formation becomes diffusion limited above the thresh- old field strength.1. Introduction Colloidal particles subjected to an applied electric field interact via attractive forces due to particle polarization. Dilute suspensions in strong fields typically aggregate into chain-like structures, while concentrated suspensions form a dense complex network. The strength of the attraction is controlled by the magnitude of the applied field and aggregation is reversible upon its removal. Network formation in concentrated sus- pensions can produce large changes in rheological properties; such suspensions are called electrorheological fluids. There is a growing interest in applying electrorheological fluids to a number of devices including electromechanical transducers and vibration isolation elements.''2 Owing to their potential technological importance, electrorheological fluids have been the subject of a number of studies employing a variety of techniques including rheological optical experiment^,^,^ computer simulation^,^'^ and theo- retical models.8-10 Two recent reviews provide an overview of electrorheological fluids and their application.""2 Analogous field-induced flow transitions occurring in ferro- fluids, dispersions of magnetic particles, have also been the subject of much investi- gation.I3 Efforts to model electrorheological fluids are impeded by the concentrated nature of the suspension.The pair-interaction potential provides the basis for fundamental models of suspension behaviour. Strong attractive forces induce particle aggregation, but it is both the interaction forces and the resulting structure that are responsible for the field-induced rheological changes.I4 Determining the precise nature of the particle interactions is difficult in concentrated suspensions where the highly complex structure and the enormous number of particle interactions complicate experimental interpreta- tion.As a step towards understanding the mechanism of the electrorheological effect, we investigate field-induced forces between particles in dilute suspension where one can study isolated particles associating into chains. Electric-field-induced chain forma- tion has been studied in biological cell suspension^^^"^ and in aqueous suspensions of polymer l a t i c e ~ . ~ ~ , ~ ~ In this paper we present a microscopic study of electric-field-induced aggregation in a dilute non-aqueous suspension of polymer latices.We compare aggregate size and 1718 Field-induced Aggregation number distribution to theoretical predictions from a model particle interaction potential and an equilibrium statistical mechanical theory for chain formation in dilute sus- pensions. The interaction potential includes a field-induced dipole attraction; however, the high dipole interaction strength required for aggregation suggested the presence of a repulsive force between the particles. We find that despite the low dielectric environ- ment, the particles carry a significant charge, which we characterize via electrophoresis. The inclusion of an electrostatic repulsion with the dipole attraction yields reasonable agreement between theory and experiment near the onset of aggregation. A transition to transport-limited aggregation occurs at elevated field strengths, indicating the need for additional studies of non-equilibrium chain formation.2. Particle Interactions We first consider particle interactions in the absence of an electric field. A steric layer grafted to the particle surface provides a short-range repulsion sufficient to prevent irreversible particle aggregation due to Van der Waals forces. Since the steric layer repulsion is powerful over only a very short range, we can model it with an effective hard-sphere repulsion. Particle charge creates a Coulombic repulsion screened by ions in the solvent. This electrostatic repulsion can be represented in the linear superposition approximation as a Yukawa potential," where k,T is the thermal energy, a is the particle radius, and r is the centre-to-centre distance between particles scaled on the diameter, 2a.The Debye length, K - ' , reflects the effective range of the Coulombic repulsion screened by the ions in solution. The magnitude of the electrostatic repulsion, scaled by the parameter A, takes two forms, 1 a W: constant potential A =i (7) [ Q2/(l + Ka)' constant charge where the Bjerrum length I = e2/4mo&,kB T is the characteristic range of interaction between two charges e in a medium of relative permittivity E , ; E~ is the vacuum permittivity. The first expression applies to particles maintaining a constant surface potential, qo = +oe/ k , T, as they approach one another.We approximate the surface potential, +b0, with the zeta potential measured from the particle electrophoretic mobility. The second form of the electrostatic repulsion applies to particles with constant surface charge Q = ZZ/a, where 2 is the number of charges on the particle. When the surface potential is low, the charge is related to the potential as Q = (1 + Ka)Wo. Note that the Bjerrum length is greatly extended in a low relative permittivity solvent, thus the relative magnitudes of charge, zeta potential and interaction strength typical of aqueous sus- pensions are quite different in non-aqueous suspensions. The range of the particle electrostatic repulsion depends on the ion concentration, No = 4v2a2no, through the Debye screening length K - ~ , as14 2No 3Q4 (Kal2=-+- 1-4 1 - 4 (3) where 4 is the particle volume fraction.The second term is the contribution from the Q counter-ions associated with each particle to maintain electroneutrality in the sus- pension. In non-polar solvents the ionic species are unknown and their concentrations are extremely low. We approximate the ionic concentration from the suspension conduc- tivity, a, as a = 2e20no where we assume univalent ions with mobility, o, dominated by the fastest species and related to the ion diffusivity by D = wk,T."P. M. Adriani and A. P. Gast 19 The linear superposition approximation (LSA) in eqn ( 1 ) becomes a poor approxima- tion to the exact numerical solution of the non-linear Poisson-Boltzmann equation as the particles approach c ~ n t a c t .' ~ The magnitude of error depends on the degree of charge regulation on the particle surface; in the limit of constant surface charge the exact repulsion exceeds the LSA force at small separations by a factor of four for Ka = 2, while for constant surface potential the exact repulsion near contact is half that of the LSA force. An alternative to the LSA is the Derjaguin approximation, which agrees well with the exact solution for particles with constant potential but diverges for constant charge.lg Since the degree of charge regulation is unknown in our system and to retain a potential amenable to statistical mechanical analysis, we choose the LSA. The field-induced particle interactions are highly complex for d.c.or low-frequency a.c. owing to the mobility of the charged particles and counter-ions. Particle interactions simplify considerably for high-frequency a.c. fields. The particles and the counter-ion cloud can no longer follow an applied a.c. field at frequencies higher than 1- 10 ~ H z , ~ ~ ~ ~ ~ thus particle electrophoresis is suppressed and the counter-ion cloud retains its zero-field distribution. The field-induced interaction then simplifies to a dipole attraction arising from the difference in polarizability (or relative permittivity) between the particles and the solvent. The dipole moment of an isolated sphere induced by an electric field E ( t), po( t ) = 4fl~,&,pu~E( t), depends on the difference between the relative permittivities of the particle, E ~ , and the .solvent, E , , through /3 = ( E~ - E , ) / ( cp + 2 ~ , ) where a is the sphere radius.The upper limit, p = 1, represents conductive particles, whereas the lower limit, p = -1/2, applies to spheres much less polarizable than the solvent as in an aqueous suspension of polystyrene latices.17918 The dipole moment is modified by the interaction between a dipole and the field produced by a nearby second dipole.' Higher-order multipoles are induced as however, both of these effects contribute to a higher order in /3 and can be neglected for a pair of spheres if lpl<0.5 and for the extreme case of an infinite chain of spheres if < 0.2.22 We assume the relative permittivity of the PMMA latices in this study to be the same as bulk PMMA, 3.0 in the kHz range, while that of hexane is 1.9.These relative permittivities yield p =0.16, a value small enough to render higher-order interaction terms negligible. The dipole strength for two spheres at contact with dipoles oriented head to tail is p = po[l +p/4+ O(p')]; neglect- ing the higher-order terms constitutes a 4% error in p for this system. The time-averaged potential energy of interaction udip( r, 8) between two hard spheres with aligned dipoles of moment p = ( p ( t))r,,,s. is where 8 is the angle between the dipole moment vector and the centre-to-centre vector r, r is scaled on the particle diameter 2a, P2(x) = (3x2-1)/2 is the second Legendre polynomial, and A = p2/ T T T T E ~ E , ~ ~ ~ k , T is the dimensionless dipole interaction strength. The attraction between aligned dipole moments is maximized when the dipoles are positioned head to tail with the contact potential energy, 2A, scaling as the square of the applied electric field.A peak-to-peak applied potential of 800V yields E,,,,,= 0.50 V pm-' and A = 4.7, or a maximum energy at contact of 9.4kBT. Combining the dipole attraction with the electrostatic repulsion yields r < l k,T A/rexp[2~a(r-l)]-(2A/r~)P,(cos 8) r 2 1 a potential energy whose minimum is no longer at hard-sphere contact with a magnitude significantly lower than the dipole contact energy, -2A. Typically with A = 10, Ka = 3, and A ranging from 4 to 10, the potential-energy minimum varies from -2.4 to -10kB7', while the dipole contact energy is -8 to -20kBT. ( 5 ) -=[ ub-9 6 ) a20 Field-induced Aggregation 3.Equilibrium Model for Chain Formation Particles having aligned dipole moments will aggregate into chains. We are interested in chain formation by particles having induced dipole moments aligned in an applied electric field. A similar problem occurs in chain formation among particles with per- manent dipole moments fully aligned in the limit of strong applied fields. An equilibrium theory of chain formation for hard spheres with permanent dipoles has been developed previo~sly.~~-*~ In the limit of strong fields, this theory can be applied to the induced dipole problem. In this section we extend these results to include electrostatic repulsion and we derive modified expressions for the chain-length distribution. At very low concentrations, only the equilibrium between singlets and doublets needs be considered.The singlet and doublet number densities p1 and p2 are governed by the partition function of each state q1 and q2 via the relation and by particle conservation P1+2P2= P (7) where p is the total number density of particles. Solving eqn (6) and (7) for p1 and p2 to O(p2) yields The osmotic pressure of the particles, neglecting chain-chain interactions, is II = (2pi)kBT; therefore, to O ( p 2 ) we have n -- - P1 +P2 = P -B,P2 k B T (9) where we can now identify -B, as the second virial coefficient. The singlet partition function accounts for the translational degrees of freedom with q1 = V/A3, where A = (h2/2.rrrnkBT)”2 is the thermal wavelength, h is Planck’s constant, and rn is the particle mass.The doublet partition function is a sum over all bound states of a particle pair. In previous studies on strong dipolar interaction^,^'-^^ the relevant particle configurations are highly localized, and the pair interaction potential was expanded in a Taylor series about the potential-energy minimum. This expansion is not useful for the present interaction potential, since the energy minimum moves and becomes quite broad as A is varied. The potential-energy minimum is given implicitly by 6h - = d i m ( 1 + 2 ~ a d , ~ , ) exp [ - 2 ~ a ( dmin - I ) ] A if dmin > 1 otherwise The shape of the interaction potential near the energy minimum varies with the relative magnitude of the electrostatic repulsion to the dipole attraction rendering simple approxi- mations unreliable.We therefore evaluate the doublet partition function numerically employing the full potential. The doublet partition function is proportional to the configuration integral over all bonded states of a particle pair. The integral is dominated by two potential-energy minima, at r = dmin , 8 = 0 and T. By symmetry we need only integrate over a hemisphereP. M. Adriani and A. P. Gast 21 to give exp [ - u ( r ) / k , T ] d+ sin 8 d8r2 dr. (11) Substitution of the pair potential, eqn ( 5 ) , and recognizing that sin 8 = 8 and P,(cos 8) == 1 - 302/2, in the region near 8 = 0, permits the angular integrals to be evaluated analyti- cally. The choice of upper limit on 8 is irrelevant for large dipole strengths. We choose the upper limit of d 2 to preserve the correct form in the absence of a dipole. We thus obtain VB, 42 =-yT I 1 (12) r3 I:' 6A ~ , = 2 7 ~ ( 2 a ) ~ exp [-Urep(r)/kBT] exp ( 2 ~ / ~ ~ ) - [ i - e x p (-6A/r3)]r2dr where ureP(r) is the electrostatic repulsion energy, and r is dimensionless on 2a.The form of eqn (12) holds for any potential having an isotropic repulsion, and reduces to previous results for B, in the limit of large A and a hard-sphere The choice of integration limits, rl and r 2 , corresponds to the definition of a bonded state. The doublet partition coefficient is only sensitive to these limits when the attractions become very weak. A reasonable definition of a doublet includes those particles falling within the nearest-neighbour shell 1 < r < 2. This choice applies well to both weak and strong particle interactions and corresponds to our criterion for counting doublets in our video images as described below.This theory of singlet-doublet equilibrium can be extended to consider the equili- brium of chains of all length^.^^-^' In general, eqn (6) becomes and the conservation equation is Detailed calculations of the chain partition function, q,, have been carried out for hard spheres with dipole^.'^,^^ All relevant short-range effects are incorporated in the doublet equilibrium constant, B,, and in the equilibrium particle spacing, dmin . The screened Coulombic repulsion decays exponentially making the electrostatic interactions negli- gible beyond nearest neighbours. Since there are no new long-range interactions, the chain partition function remains the same as for hard spheres [eqn (3.7) of ref.(24)] with eqn (12) for B, and the substitution A + A' = A/dLin, where dmin is determined from eqn (10). The calculated partition functions yield the number densities of all chain lengths. From this chain-length distribution one can calculate associated quantities such as the number fraction of singlets or the average chain length. Equilibration produces an average chain length near unity for small dipole strengths increasing rapidly above a critical threshold. In fig. 1 we compare the predictions for singlet-doublet equilibrium with full equilibrium among chains of all lengths employing parameters appropriate to our experimental system. The average chain length increases more abruptly for the full equilibrium calculations, as the equilibrium shifts toward long chains.The onset of aggregation in the singlet-doublet equilibrium occurs at the same field strength as the full equilibrium, but the average chain length increases more gradually. The full equilibrium model for chain length distribution is applicable when the potential-energy minimum is on the order of a few kBT and there is sufficient time for equilibration. In dilute suspensions, when equilibration time is limited, the singlet- doublet model is more appropriate. When attractions due to large applied fields exceed 6 k,T, aggregation becomes irreversible on timescales comparable to our experiments Cnp, = p . (14)22 Field-induced Aggregation n 1.6 1.4 1.2 1 .o ._ 0 250 500 750 1000 1250 1500 V,-,P Fig.1. Number average chain length predicted from eqn (8) and (12) against the peak-to-peak applied voltage for particle interaction parameters A = 10, K a = 3, and volume fractions 4 = 0.0067 and 0.0041, respectively, for singlet-doublet equilibrium (- and - - -), and equilibrium of all chain lengths ( * + - and - - - - -). and a non-equilibrium description of chain formation such as the Smoluchowski equation is more appr~priate.~ 4. Experimental We study aggregation of 0.98 pm diameter poly(methy1 methacrylate) (PMMA) latices suspended in hexane at volume fractions in the range 0.001-0.01. The particle synthesis and characterization are described e l s e ~ h e r e . ~ ~ - * ~ A covalently bound poly( 12-hydroxy- stearic acid) steric layer, ca. 0.01 pm thick, provides a strong, short-range repulsion to ensure stabilization against aggregation due to van der Waals attractions.We observe aggregation between two thin electrodes through a 20x objective on a Zeiss Axioplan optical microscope in transmission. The objective working distance allows us to focus at any level within the cell, while the depth of field is comparable to one particle diameter. The electrode cell, comprising a machined acrylic substrate, flat copper electrodes, and a glass cover slip is sealed with silicone grease and compressed together by small screws. A 560 pm gap between the 420 pm thick electrodes provides a reasonably homogeneous electric field near the middle of the cell. Segments of 610 pm outer diameter polyethylene tubing squeezed into the gap permit injection of the suspension while minimizing solvent evaporation. We maintain a slight convection parallel to the electrodes to provide a continuous supply of particles to observe.Our observation time is limited by a slow particle drift toward the cell walls due to dielectrophoresis. The dielectrophoretic force, Fdiel = p, - V E == pa3V E 2, induces the motion of a dipole in a non-uniform electric field. The dielectrophoretic velocity increases with the square of the field gradient. At field strengths near the onset of aggregation, dielectrophoresis is negligibly slow compared to aggregation, convection, and observation times. Doubling the field strength increases the dielectrophoretic velocity by a factor of four, reducing the dielectrophoresis timescale to a few minutes.Since the dielectrophoretic force is independent of the sign of the applied field, it is not suppressed by increasing the frequency of the applied field. This force is proportionalP. M. Adriani and A. P. Gas? 23 to the number of dipoles in a chain, thus long chains drift more rapidly to the cell walls. This effect is minimized by making the applied field as homogeneous as possible. In our electrode geometry, the field is most homogeneous at depths near the middle of our cell; we therefore focus in this region for our microscopic observations. To prevent particle accumulation at the electrode surface due to electrophoresis, dielectrophoresis, or image forces, we wrap each electrode with a 100 pm layer of Teflon tape to act as a particle barrier. The dielectric constant of Teflon, 2.1, is similar to that of hexane, 1.9, and the tape is uniform in thickness, so the field between the electrodes should not be significantly perturbed by the Teflon layer.To induce aggregation, we subject the suspension to a sinusoidal ax. field at a frequency of 3 kHz and a peak-to-peak potential, Vp-p, in the range 0-1200 V from a Tektronix model FG504 40 MHz function generator with a Trek model 610B high-voltage amplifier. The potential is monitored at the generator, amplifier, and electrodes on a Hewlett-Packard model 54501A digitizing oscilloscope. We record suspension behaviour on a Sony model VO-5800H video cassette recorder digitizing individual frames on an Imaging Technology Series 100 real-time digital image processing board in an IBM PC AT.We analyse images with IMAGELAB and IMAGETOOL image processing software by Werner Frei Associates. We typically record aggregation experiments for one minute at a given applied field with approximately 100 particles in the field of view. Particles move across the field of view on a timescale of 5-10 s owing to slow convection parallel to the electrodes produced in order to observe independent sets of particles. From each minute of recorded observations, we digitize and average five frames spaced 10 s apart. On longer timescales particle sedimentation and dielec- trophoresis become significant, so the cell is occasionally flushed with fresh suspension. We measure suspension conductivity, a, in a stainless steel Couette cell of a 13 mm cylinder with a 12 mm radius and a gap of ca.0.5 mm. We need large electrodes and a small gap to measure the low suspension conductivity. The conductance, C, across the gap is measured with a Yellow Springs Instrument Co. Model 35 Conductance Meter. Conductivity is proportional to conductance, a = k , C, with the cell constant, k , = 0.046 cm-', determined by the cell geometry and calibrated with a salt solution in a YSI model 3403 conductivity cell of a known cell constant. We found that suspension conductivity depended on sample age owing to the accumulation of ionic impurities. In order to maintain reproducible solution ionic strengths, we saturated our hexane with sodium acetate. We then found the conductivity of the PMMA suspensions in saturated hexane, a = 5 x R-' m-', to reflect an ionic concentration of no ==: 5 x 1013 cm-3 (= 1 x mol dmP3) assuming univalent ions of diffusivity D = (0.5-1) x lop9 m2 s-'.This ionic strength corresponds to a screening length of 2.6 < Ka < 3.7; the screening due to counter-ions alone would contribute Ka = 0.2 for a particle volume fraction 4 = 0.01. We assume K a = 3 for our theoretical predictions of suspension aggregation, except for one suspension with conductivity a = 6.4 x We measure the particle electrophoretic mobility in the same electrode cell for the aggregation experiments. Applying a weak, low-frequency (f = 0.5 Hz) sinusoidal a.c. field R-' m-', and 3.1 < K a < 4.3, where we assume Ka = 3.7. E ( t ) = (Ep-p/2) cos (2Tft) where Ep-p is the peak-to-peak amplitude of the field, typically Ep-p = 20 Vp-,/560 pm = 0.036 V pm-'.We follow particle oscillations z ( t ) = zO+(zp-,/2) sin (2r-t) where z is in the direction of the applied field and zP-, is the amplitude of oscillation, to determine the particle velocity U ( t ) = z'(?) = (2Tf)(ZP-,/2) cos ( 2 T f f )24 Field- induced Aggregation and the electrophoretic mobility We then calculate the zeta potential from the particle m~bility,~' assuming monovalent ions and molar ionic conductances of 70 cm2 K' mol-', equivalent to an ion diffusivity of 1 x m2 s-l. Again, to prepare suspensions with a controlled zeta potential, we saturate the solution with sodium acetate. After such contact, the measured particle mobility is 1.6 x lop9 m2 V-' s-' and the zeta potential is 40 mV, corresponding to a coefficient for the electrostatic repulsion [eqn (2)] of A = 10.This zeta potential corre- sponds to 100 charges per particle and an extremely low surface charge density of 3 x lop5 nm-*. The suspension with cp = 0.0028 has a lower electrophoretic mobility, 0.9 x m2 V-' s-', reflecting a weaker electrostatic repulsion of A = 3. This is the suspension having a higher conductivity, owing to a larger concentration of ionic impurities in the solution. 5. Results We quantify suspension images by counting the number and length of chains to determine a chain-length distribution and average chain length. The image analysis procedure begins with a 464 x 512 pixel grey scale image; each pixel has an intensity value ranging from 0 for black to 255 for white.The pixels are wider than they are tall with an aspect ratio of 1.2. Calibration of the video image with a microscope grid yields a scale of 2-06 vertical pixels pm-'. Two sets of particles are visible in the original image. The particles within the focal plane appear bright while dark particles appear from other planes. Since the depth of field is comparable to the particle diameter we can calculate the apparent particle volume fraction from the number of bright particles. We employ this relation to cut the image intensity histogram at a value appropriate to include only the bright particles. Variations in illumination are corrected by subtracting a smoothed image from the original to equalize the intensity histogram. We count particle chains by searching for vertical strings of bright pixels.Requiring a continuous line of bright pixels is too strict; permitting breaks of one or two pixels captures the chains visible in the original image. A chain of 1-3 pixels corresponds to a singlet, 4-6 to a doublet, and so on. The physical particle size is 2.06 pixels, but since the image is somewhat larger, a length criterion of 3 pixels is best for distinguishing singlets and doublets; this may lead to a slight underestimate of the length of longer chains. We show the singlet and doublet number fractions for suspensions of volume fraction 0.0067 and 0.0041 in fig. 2 ( a ) and ( b ) . Below 750 V, the number fractions of singlets and doublets correspond well to the theoretical prediction for nearest neighbours in a Yukawa fluid.The expected number of doublets would be about eight times greater for hard spheres without electrostatic repulsion. Doublet formation begins near 750 VP+ for the 0.0067 volume fraction suspension and around 850 VP+ for cp = 0.0041. The singlet and doublet fractions change with applied field in reasonable agreement with the theoretical predictions of the singlet-doublet equilibrium model until the highest field strengths where the experimental points begin to saturate. The saturation occurs at a lower field strength for more dilute suspensions suggesting that the aggregation becomes diffusion limited. The theoretical predictions of singlet and doublet number fraction show the effect of 10% uncertainty in the inverse Debye screening length, K ; a 10% uncertainty in the electrostatic repulsion parameter, A, is of comparable con- sequence.The experimental number fractions of singlets and doublets sum to unity near the onset of aggregation, but triplets and higher n-mers account for 10% of the aggregates at the highest field strength. The presence of longer chains should be more pronouncedP. M. Adriani and A. P. Gast 25 250 500 750 1000 1250 1500 0 Fig. 2. Theoretical and experimental chain number fractions of (a) singlets and ( b ) doublets against the peak-to-peak applied potential for particle interaction parameters A = 10, KU = 3 * 0.3, volume fraction 4 = 0.0067 (- and 0), and volume fraction 4 = 0.0941 (- - - and 0). at higher particle volume fractions. The presence of triplets and higher n-mers does not explain the slight saturation in singlet number fraction for 0.0041 since the effect of longer chains is to reduce the number fraction of singlets not to increase it.The singlet and doublet number fractions for volume fractions of 0.0028 and 0.0014 are displayed in fig. 3(a) and (b). As before, the distributions agree well with those predicted theoretically up to a field strength where doublets comprise ca. 10% of the suspension. Again we observe a saturation reflecting a transition to diffusion-limited kinetics, a process naturally occurring at lower fleld strengths in more dilute suspensions. The profound influence of the electrostatic repulsion is evident in our results. While our addition of sodium acetate facilitated controi over the suspension ionic strengths, the suspension having a volume fraction, cp = 0.0028, had a slightly higher conductivity.This results in a lower particle surface charge, a smaller electrostatic repulsion, A = 3 , and enhanced screening, Ka = 3.7. The aggregation threshold predicted for these condi- tions is substantially reduced to about 800 Vp-p(A = 4.7) from the 1000 Vp-p ( A = 7.4) required for aggregation when A = 10 and KU = 3. If the electrostatic repulsion were negligible ( A = 0), the required potential would be only 700 Vp-p (A = 3.6). Thus the electrostatic repulsion has enormous influence on the field required to produce aggregates. The electrostatic repulsion is also evident in observations of aggregate disruption after removal of electric field. Studying particle chains we find that they dissociate immediately upon removal of the applied field.Particles move apart initially via ballistic motions and then continue on Brownian trajectories beyond one diameter.26 Field- induced Agg reg a t io n " 0 250 500 750 1000 1250 1500 V,-,/V Fig. 3. Theoretical and experimental chain number fractions of ( a ) singlets and ( b ) doublets against the peak-to-peak applied potential for particle interaction parameters A = 3, ~a = 3.7 * 0.3, and volume fraction qh = 0.0028 (- and 0); and A = 10, KU = 3 k0.3, and qh = 0.0014 (- - - and 0). The average chain lengths in fig. 4 ( a ) and (b) are near unity at low field strengths for all particle volume fractions. At the onset of aggregation the increasing number of doublets causes the average chain length to rise.The average chain length saturates at fi = 1.2 for the dilute volume fractions 0.0028 and 0.0014. Average chain-length predic- tions agree well at low fields and near the onset of aggregation but diverge as the experimental data saturate. We discuss this saturation in terms of equilibration and diffusion times below. 6. Discussion Our theoretical predictions of chain aggregation incorporate electrostatic repulsion and dipole attraction into the pair-interaction potential. The agreement with experimental measurements of chain formation near the onset of aggregation suggest that we have a reasonable representation of the particle interaction potential. We show in fig. 5 the particle interaction potentials at the theoretical onset of aggregation for our four experimental volume fractions.The magnitude of the energy minimum required for aggregation varies from 3 to 6k,T as volume fraction decreases from 0.0067 to 0.0014. The 0.0028 volume fraction suspension conforms to this trend despite its lower electro- static repulsion and higher ionic screening compared to the other suspensions. Chain formation is an equilibrium process provided the magnitude of the interaction potential remains moderate. At elevated field strengths, the magnitude of the potential-P. M. Adriani and A. P. Gust 27 0 250 500 750 1000 1250 1500 V,-*IV Fig. 4. Theoretical and experimental number average chain length against the peak-to-peak applied potential for the same particle interaction parameters as in fig. 2 and 3 and particle volume fractions ( a ) 4 = 0.0067 (- and 0) and 4 = 0.0041 (--- and 0 ) and ( b ) 4 = 0.0028 ( - * - and A) and 4 = 0.0014 (- - - - - and 0).energy minimum I urnin( exceeds 6- 10kB T rendering equilibration slow. The characteristic equilibration time, t,, = 67r77a3/ kB T exp ( - urnin/ kB T ) , where 77 is the solvent vis~osity,~' increases from ca. 5 s near the onset of doublet formation to 500 s where saturation occurs. Thus we find that doublet formation becomes a diffusion-limited, non-equili- brium process above ca. 1000 V for A = 10 and K a = 3. Diffusion-limited chain formation is a slow process in dilute suspensions. The characteristic time for doublet formation with no repulsive energy barrier and a short- range attraction is tdblt = 7r77a3/ 4kB T3' Under our most dilute conditions, 9 = 0.0014, and the characteristic time for initial doublet formation is 20s.This time increases as the aggregation proceeds and the population of single particles diminishes. Convection of chains to the cell walls due to electric-field gradients prevents studies of chain formation over long periods; this limitation motivates future improvements in our experimental design to minimize electric-field gradients. We note that the electrostatic repulsion evident in these suspensions is somewhat surprising since the relative permittivity of the solvent is so low. The possibility of significant electrostatic repulsions in otherwise nearly hard-sphere suspensions can have important implications for interpretation of order-disorder transition^^^ as well as rheological behaviour. In a previous study, Ottewill and co-workers28 measured the scattering structure factor of similar 30 nm diameter PMMA particles suspended in dodecane. By comparing the structure factor with that derived from the mean spherical approximation for particles interacting via a Yukawa potential, the authors determined28 Field-induced Aggregation 2l r Fig.5. Theoretical particle interaction potential for aligned dipoles (6=0) at the onset of aggregation for the same interaction parameters as in fig. 2 and 3 and particle volume fractions 4 = 0.0014 (-), 4 = 0.0028 (- - -), 4 = 0.0041 ( - - . ), and 4 = 0.0067 (- - - - -). that a repulsion acting over 10nm best described their particles. This repulsion was attributed to the PHS steric stabilization layer; however, the range of the repulsion varied substantially with particle volume fraction, and it was supposed that this reflected changes in the thickness of the steric layer.In light of the results of this investigation, we suggest that a moderate electrostatic repulsion may play a role in these suspensions. Further studies of PMMA latices coated with PHS should address this possibility. 7. Conclusions We have presented a study of electric-field-induced aggregation in a non-aqueous colloidal suspension. We observed 1 pm PMMA particles in dilute suspension in hexane subjected to fields on the order of 500 V mm-I. The extreme fields required to effect chain formation and the rapid disruption of chains upon removal of the field both reflect the electrostatic repulsion operative in these suspensions.This electrostatic repulsion in otherwise model non-aqueous suspensions may have profound implications for systematic studies. We focused our attention on dilute suspensions to provide a test of our proposed interaction potential energy. In very dilute suspensions, chain formation can be approxi- mated by singlet-doublet equilibrium. A model of equilibrium doublet formation accurately reflects the population distribution and average chain length around the aggregation threshold. This suggests that we may model the interparticle interactions in these suspensions via an electrostatic repulsion added to a field-induced dipole attraction. This provides the basis for further studies of more concentrated suspensions.In elevated fields the aggregation process is no longer represented by an equilibrium population distribution and, at low particle concentrations, chain formation becomes diffusion limited. These indications provide motivation for future investigation of non-equilibrium field-induced aggregation and structure formation at higher particle concentrations.P. M. Adriani and A. P. Cast 29 This work was supported in part by Lord Corp., Ford Motor Co. and by IBM. We thank R. Ottewill for his generous gift of the polymer latices used in this study, C. Steinmetz for his donation of a digitizing oscilloscope, G. Homsy for his loan of image analysis equipment, and K. Smith for her help in measuring suspension conductivity. References 1 W.M. Winslow, J. Appl. Phys., 1949, 20, 1137. 2 Z. P. Shulman, R. G. Gorodkin, E. V. Korobko and V. K. Gleb, J. Non-Newtonian Fluid Mech., 1981, 3 D. L. Klass and T. W. Martinek, J. Appl. Phys., 1967, 38, 67, 75. 4 G. G. Fuller, Annu. Rev. Fluid Mech., 1990, 22, 387. 5 G. G. Fuller, K. Smith and W. R. Burghardt, submitted. 6 P. Bailey, D. G. Gillies, D. M. Heyes and L. H. Sutcliffe, Mol. Sim., 1989, 4, 137. 7 S. Miyazima, P. Meakin and F. Family, Phys. Rev. A, 1987, 36, 1421. 8 P. M. Adriani and A. P. Gast, Phys. Fluids, 1988, 31, 2757. 9 D. J. Klingenberg and C. F. Zukoski IV, Langmuir, 1990, 6, 15. 8, 29. 10 R. Tao, J. T. Woestman and N. K. Jaggi, Appl. Phys. Lett., 1989, 55, 1844. 11 A. P. Gast and C. F. Zukoski, Adv. Colloid Interface Sci., 1989, 30, 153. 12 H. Block and J. P. Kelly, J. Phys. 0, 1988, 21, 1661. 13 For a bibliography on ferrofluids, see S. W. Charles and R. E. Rosensweig, J. Magn. Magn. Mater., 14 W. B. Russel, in Theory of Dispersed Multiphase Flow, ed. R. E. Meyer (Academic Press, New York, 15 H. P. Schwan and L. D. Sher, J. Electrochem., 1969, 116, 22C. 16 S. Takashima and H. P. Schwan, Biophys. J., 1985,47, 513. 17 F. Richetti, J. Prost and N. A. Clark, in Physics of Complex and Macromolecular Fluids, ed. S . A. Safran and N. A. Clark (John Wiley & Sons, New York, 1987), pp. 387-411. 18 S. Fraden, A. J. Hurd and R. B. Meyer, Phys. Rev. Lett., 1989, 63, 2373. 19 A. B. Glendinning and W. B. Russel, J. Colloid Interface Sci., 1983, 93, 95. 20 K. H. Lim and E. I. Franses, J. Colloid Interface Sci., 1986, 110, 201. 21 D. J. Jeffrey, Proc. R. SOC. London, Ser. A, 1973, 335, 355. 22 T. B. Jones and R. D. Miller, J. Elecrrostar., 1990, in press. 23 P. C. Jordan, Mol. Phys., 1979, 38, 769. 24 P. C. Jordan, Mol. Phys., 1973, 25, 961. 25 P. G. de Gennes and P. Pincus, Phys. Kondens. Mater., 1970, 11, 189. 26 R. H. Ottewill, Langmuir, 1989, 5, 4. 27 L. Antl, J. W. Goodwin, R. D. Hill, R. H. Ottewill, S. W. Owens and S. Papworth, Colloid SurJ, 1986, 28 I. Markovic, R. H. Ottewill and S. M. Underwood, Langmuir, 1986, 2, 625. 29 D. J. Cebula, J. W. Goodwin, R. H. Ottewill, G. Jenkin and J. Tabony, Colloid Polym. Sci., 1983, 261, 30 R. J. Hunter, The Zeta Potential in Colloid Science (Academic Press, New York, 1981), pp. 100-112. 31 W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions, (Cambridge University Press, 32 W. van Megen and P. N. Pusey, Nature, 1986,320, 340. 1983, 39, 190. 1983), pp. 1-34. 17, 67. 555. New York, 1989), pp. 270, 329. Paper 01022666; Received 17th May, 1990