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Langmuir 2000, 16, 9784-9791
Simulation and Experiments on Colloidal Particle Capture in a Shear Field Martin Whittle,* Brent S. Murray, Jianshe Chen, and Eric Dickinson The Procter Department of Food Science, University of Leeds, Leeds LS2 9JT, U.K. Received July 26, 2000. In Final Form: September 25, 2000 Using a colloidal particle scattering apparatus, we have observed shear-driven formation of particle doublets near a wall for latex and emulsion systems. For the latex system, capture is in the expected stable downstream position. For the emulsion system, capture occurs upstream from the fixed particle. Furthermore, the frequency of capture is found to depend on pH in agreement with rheology results. Computer simulations have been performed to examine the criteria for capture in this case, and, of necessity, a novel potential is introduced. Good comparison can be obtained between simulation and experiment when surface charge variations linked to pH are invoked as the key variable.
Introduction The association of colloidal particles into aggregated systems is a feature of great fundamental and practical interest and underlies many important industrial processes.1 In a dispersed colloidal system, aggregation is usually prevented either by electrostatic repulsion or the presence of adsorbed polymers, but relatively small changes in conditions can result in destabilization and subsequent flocculation. Typical perturbations that may influence the strength and range of the colloidal forces include the addition of electrolytes, polymers, or polyelectrolytes or a change in pH or temperature. The stability of a colloidal system is thus dependent on the details of the interaction forces, electrostatic or steric, which in turn are influenced by the fluid environment. In food colloid systems the range of polymeric stabilizers is diverse and includes polysaccharides and proteins.2 Proteins can also frequently act as emulsifiers because of their high surface activity. In homogenized milk, fatty droplets are held in dispersion by a coat of casein micelles. It is the interaction between these entities that ensures the normal stability of milk and also controls the conditions for the formation of products such as cheese and yogurt. The aggregation kinetics and final structure of the colloid may also be affected by a sufficiently large shear field, such as that generated by stirring.3 Also, the presence of a wall can be significant, and here an investigation of colloidal aggregation effectively becomes the study of deposition processes, which is central to an understanding of coating and filtration.4 In this paper, motivated by new experimental observations of particle capture, we discuss the initial stages of aggregation near a wall. The work relates to experimental observations made using a particle-scattering apparatus designed for the determination of interparticle forces from the detailed analysis of trajectories.5-7 In the course of * Corresponding author. (1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (2) Dickinson, E. An Introduction to Food Colloids; Oxford University Press: Oxford, 1992. (3) Axford, S. D. T. J. Chem. Soc., Faraday Trans. 1996, 92, 1007. (4) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: London, 1989. (5) Murray, B. S.; Dickinson, E.; McCarney, J. M.; Nelson, P. V.; Whittle, M. Langmuir 1998, 14, 3466. (6) Wu, X.; Dabros, T.; Czarnecki, J. Langmuir 1999, 15, 8706.
Figure 1. (a) Diagram of the trajectory of a mobile particle around a similar particle fixed at point “P” (shown filled) finally aggregating at the stable downstream position (shown dotted). (b) A particle that has attached at an upstream position. Uy is the shear flow field.
performing these experiments particle deposition may be observed under conditions where the bulk dispersion would be near to destabilization and close to forming flocs or aggregates. In this paper we focus our attention in detail on these sticking events and perform some related simulations to examine what we can learn from them. Experimental Section Our colloidal particle scattering apparatus has been described in detail before.5 Figure 1 serves to illustrate its use in the current experiment. A particle is first attached to a stationary plane glass surface (at the point P in the figure). Couette flow is then applied by moving a parallel plane plate mounted a short distance (∼200 µm) from the fixed plate. In the present case, we are interested only in binary encounters close enough potentially to stick a mobile particle to the fixed particle. To arrange this, a series of mobile particles are positioned with initial surface separations approximately one diameter downstream from the fixed particle, and also with their centers no greater than roughly a particle radius from the center line defined by the fixed particle. When working with the more polydisperse emulsions, care was (7) Whittle, M.; Murray, B. S.; Dickinson, E. J. Colloid Interface Sci. 2000, 225, 367.
10.1021/la001061d CCC: $19.00 © 2000 American Chemical Society Published on Web 11/16/2000
Colloidal Particle Capture in a Shear Field
Figure 2. The experimentally determined capture efficiency ΦE for sodium caseinate stabilized cetyl bromide oil-in-water emulsions at different pH conditions.
Langmuir, Vol. 16, No. 25, 2000 9785 agreement with rheological measurements on sodium caseinatestabilized emulsion gels.8 The isoelectric point for this system is at pH ∼ 4.7, and as it is approached from higher pH values, the small negative charge on the adsorbed casein layer that confers stability to the bulk colloid is reduced and attractive interactions assert their influence.9 Some self-selection of mobile particles may occur near the isoelectric point, since only those particles that have not already aggregated can be chosen for the collision experiment. This is one possible explanation for the failure to reach 100% capture at pH e4.9. The observation of upstream sticking for this system is significant since, as we argue in detail below, it cannot be explained by a radial attractive interaction alone. We must conclude that a tangential force exists between the closely approaching particles. The latex systems (1 and 2) have also been tested at varying pH conditions and salt concentrations. It was found that not every collision results in sticking, even for those conditions where the corresponding bulk colloidal systems were unstable. However, no detailed investigation has, so far, been carried out.
Simulation Model taken to choose mobile particles with diameters close to the fixed one ((10%). A shear rate of 0.28 s-1 was then applied, thereby moving the mobile particle toward the fixed one and creating a collision. This was repeated some 30 times. Some of the collisions resulted in aggregation of the mobile particle with the fixed one and enabled a capture efficiency, ΦE, to be established with an associated measurement error of ∼(1/30. We define this quantity as the ratio of the number of collisions that successfully result in a stuck particle to the total number of trials. To examine shearinduced breakup, the aggregated particle doublets were subsequently exposed to increasing shear rates up to the experimentally available maximum of 12 s-1. We note that other workers have also recently used a similar apparatus to examine doublet breakup.6 Three systems were studied: (1) sodium caseinate-stabilized latex particles (5.1 µm diameter) at different pH conditions; (2) SDS (sodium dodecyl sulfate)-stabilized latex particles (5.1 µm diameter) at different salt (NaCl) concentrations; (3) sodium caseinate-stabilized cetyl bromide in water emulsions (∼5.0 ( 0.5 µm droplet diameter) at different pH conditions. The emulsion system (3) was prepared using a normal blade mixer, and the resulting polydispersed emulsion was highly diluted to give the sample solution. The smooth emulsion droplets were found to be more difficult to immobilize on the glass plate, and another system using SDS/cetyl bromide emulsions proved impossible to study for this reason. The fixed and mobile emulsion droplets were chosen under the microscope by comparison with an image of the known-size latex particle (as reference). The error in chosen emulsion droplet size was estimated to be less than 10%. Results for the protein-stabilized emulsion (system 3) proved to be of most interest, and so we discuss these in most detail. Experimental Observations. For the latex systems stabilized by SDS or sodium caseinate, it was observed that the mobile particle would attach to the fixed particle and then, under influence of the flow, migrate to the down stream stagnation point immediately behind the fixed particle. This is the expected final sticking position for a radially attractive interaction force.7 Once the mobile particle was settled in this stable position, a reversal of the shear field caused the mobile particle to slip around to the new, diametrically opposed, stable point. Subsequent field reversals would repeat this behavior, the particle moving back and forth with the flow. This observation emphasizes that radial forces alone are responsible for the aggregation in this case. By increasing the shear rate, it was usually possible to disrupt the aggregate. Although the emulsion particles stabilized by sodium caseinate are expected to have a smoother surface, we observed that the mobile particle adhered to an upstream surface of the fixed particle on each occasion. Furthermore, it was not subsequently possible to break up these aggregates by the application of high shear rates (up to 12 s-1). Breakup could, however, be achieved by tapping the fixed plate, thus showing that the particles were not irreversibly bound. For the emulsion system we have also found that the capture efficiency, ΦE, varies with pH (Figure 2). This finding is in broad
Computer simulation is necessary to analyze the results of particle scattering experiments,10,11 and we use essentially the same simulation techniques here. The simulation proceeds by computing the generalized (translational and rotational) particle velocity U in the creeping flow limit from a 6 × 6 mobility tensor MFU and the total generalized force:
U ) MFU‚(FH + Fc)
(1)
Here, the hydrodynamic driving force FH in general depends on the relative position of the two particles and the wall, and we suppose that the colloidal interaction force Fc is spherically symmetric. Far from the wall and the fixed particle, FH is simply the Stokes force in the flow direction
Fs ) 6πaηzwGj
(2)
where a is the sphere radius, η is the fluid viscosity, zw is the particle center to wall separation, G is the shear rate, and j is a unit vector in direction y. For particle center-center separations r > 2.5a, the calculation of mobility and hydrodynamic driving force is based on the multisubunit method employing a number of Oseen interaction centers (suitably corrected for the wall) distributed on the particle surfaces. In the near field, r < 2.1a, asymptotic lubrication results are used and in the intermediate regions, where neither approach is strictly valid, values are interpolated. Positions are updated by straightforward Euler integration
r(t + ∆t) ) r(t) + U∆t
(3)
where the time step ∆t is variable and chosen to limit particle movement especially at close approach. For the trajectory simulations a set of eight subunits arranged in a cube were used to calculate the hydrodynamic force. Most of the simulations were performed in the Stokes mode, i.e., without reference to temperature, with the particle radius, fluid viscosity, and shear rate set to unity. In this system, the unit of energy is ηa3G and that of force (8) Chen, J.; Dickinson, E.; Edwards, M. J. Texture Stud. 1999, 30, 377. (9) de Kruif, C. G.; Hoffmann, A. M.; van Marle, M. E.; van Mil, P. J. J. M.; Roefs, S. P. F. M.; Verheul, M.; Zoon, N. Faraday Discuss. 1995, 101, 185. (10) Whittle, M.; Murray, B. S.; Dickinson, E.; Pinfield, V. J. J. Colloid Interface Sci. 2000, 223, 273. (11) van de Ven, T. G. M.; Warszynski, P.; Wu, X.; Dabros, T. Langmuir 1994, 10, 3046.
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Table 1. Conversion Factors for Systems with a ) 2.5 × 10-6 M, η ) 10-3 Pa s, and a Nominal Temperature T ) 298 K Giving kBT ) 4.114 × 10-21 J G/s-1
ηa3G/J
ηa3G/kBT
ηa2G/N
3.33 0.28
5.2031× 10-20 4.3750 × 10-21
12.65 1.064
2.0813 × 10-14 1.7500 × 10-15
ηa2G. We have performed simulations with forces scaled to shear rates of G ) 0.28 and 3.33 s-1. For these settings, and for other parameters appropriate to our experimental variables, we choose the following: particle radius a ) 2.5 × 10-6 m, viscosity (water) η ) 10-3 Pa s, and a nominal temperature T ) 298 K; some conversion factors are given in Table 1. Colloidal Interaction Force. The radial interaction potential for particles of radius a is considered to be made up of three components each dependent on the surfacesurface separation h ) r - 2a
φ(h) ) -
AHa + E0 exp(-κh) + φst(h) 12h
(4)
The first term in eq 4 is the unretarded dispersion force with AH being the Hamaker constant. The second term is an electrostatic repulsion where the simplest identification is
E0 ) 2πar0ψ2
(5)
and, for the range parameter, we have
κ2 )
2z2e2n0 r0kBT
(6)
In these expressions, r is the relative permittivity, 0 is the permittivity of free space, ψ is the surface potential, e is the charge on the electron, z is the valency of the ions, n0 is the number density of ions, and kB is Boltzmann’s constant. The third term in eq 4 represents a steric interaction which we discuss below. Differentiating eq 4 gives a radial force where Fst is the steric interaction term.
F(h) ) -
AHa 12h2
+ E0κ exp(-κh) + Fst(h)
(7)
De Gennes Steric Interaction. The emulsion droplets are considered to be stabilized by a brushlike forest of caseinate molecules.12 Among several possible forms for the steric term, the de Gennes interaction13 seems quite appropriate, since this was designed on the basis of two brushlike surfaces with a thickness hc/2 and grafting separation s (gap between the bristles). In this model, the bristles are anchored in fixed positions on the surface. The energy of interaction is obtained as the sum of osmotic and elastic contributions. Braithwaite et al.14 have integrated this to obtain a force which we write as
Fst )
kBT s
3
ahc
[( ) ( ) 4 hc 5 h
5/4
+
4 h 7 hc
7/4
-
48 (35 )]
(8)
h e hc The sum of the three terms in eq 7 produces an overall (12) de Kruif, C. G.; Zhulina, E. B. Colloids Surf. 1996, 117, 151. (13) de Gennes, P. G. Adv. Colloid Interface Sci. 1987, 27, 189. (14) Braithwaite, G. J. C.; Luckham, P. F.; Howe, A. M. J. Colloid Interface Sci. 1999, 213, 525.
Figure 3. Colloidal interaction force from eq 7. AH/ηa3G ) 0.02, hc ) 0.006a, s ) 0.003a, κa ) 80. From bottom to top: E0/ηa3G ) 0.0, 0.25, 0.5, 0.75, 1.0, 1.25, and 1.5. Fc(max) is indicated for E0/ηa3G ) 1.5.
interaction force that may include a close-range attractive region and a force barrier at longer range, depending on the relative values of the electrostatic constants. Some representative forms are depicted in Figure 3. We label the height of the force barrier by Fc(max). In some simulations for particles with attractive interactions, the mobile particle is first attracted into the well while upstream of the fixed particle, and then, under the influence of the shear field, it slides around the fixed particle to take up residence in a stable downstream position (Figure 1a). In these cases aggregation can be recognized by the vanishing particle velocity and capture can be understood entirely in terms of radial interactions. We have discussed the criteria for this behavior in an earlier paper.7 As in that study a short-ranged particlewall repulsion with the form
φw(h) )
( )
B a 6 hw
6
(9)
is also included to prevent particle-wall collisions. Here, hw is the particle-wall surface separation and B is a parameter with units of energy. In this work we have used B ) 10-21. De Kruif et al.9 has described the physical properties of milk with some success using the Baxter adhesive hard sphere (AHS) potential15 to represent the interaction between native casein micelles. By use of this model, the AHS well depth was related to pH through the empirical expression
1 ) kBT pC - pH
(10)
where pC is the pH at the point of flocculation (at or near the isoelectric point for the surface polyelectrolyte). In a shear field, the hydrodynamic force (FH) on the mobile particle can always be split into components radial (FHR) and tangential (FHT) with respect to the fixed particle surface. Consequently, if the mobile particle attaches to the fixed particle by means of a radially attractive force, slippage to the downstream stable position (Figure 1a) behind the fixed particle will always occur in the absence of any impediment to tangential movement. It is, therefore, impossible to model the phenomenon of upstream sticking (Figure 1b), as observed in our experiments on proteinstabilized emulsion droplets, using a purely radial interaction force. Simulation of the unexpected observation of (15) Baxter, R. J. J. Chem. Phys. 1968, 49, 2770.
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Langmuir, Vol. 16, No. 25, 2000 9787
Figure 4. Representation of spherical polar coordinates (r, ξ, φ) centered on the fixed particle.
Figure 5. The hydrodynamic force as a function of the azimuthal angle ξ, for φ ) (π/2 and a surface-to-surface separation of h ) 0.01a. The y-component, FH‚j, s; tangential component, FHT, O; radial component FHR, b.
radial hydrodynamic force is upstream sticking must therefore involve the inclusion of a new interactive component. We model this as a frictional term, Fstick, acting tangentially to the radial force and thus capable of preventing slip. We can only speculate at this stage whether a physical interaction between the surface brushes or a chemical interaction is responsible for this tangential force. We note that tangential forces have recently been reported for heterodoublets of silica/ polystyrene spheres using the new technique of differential electrophoresis.16 Hydrodynamic Forces. The hydrodynamic force, FH, that drives the mobile particle is computed during the course of simulations to an accuracy of ∼3%, which we consider is adequate for this particular application.10 In the vicinity of the fixed particle or the wall, the Stokes force from eq 2 is only a reasonable approximation to this force. The magnitude of the tangential hydrodynamic force is given by
FHT )
1 |F × r| r H
(11)
By replacing FH with the Stokes force Fs, we can obtain a rough analytical estimate for particles in contact (r ) 2a), expressed in spherical polar coordinates (see Figure 4) centered on the fixed particle, i.e.
FST ) 6πa2ηG(2 cos ξ + 1)(cos2 ξ + sin2 ξ cos2 φ)1/2 (12) For this coordinate system, the Cartesian mobile particle position is x ) r sin ξ cos φ, y ) r sin ξ sin φ, and z ) r cos ξ. This estimate has a maximum value of 56.55/ηa2G at the azimuthal position (ξ ) 0). Close to the wall the minimum value that must be encountered by a particle passing close to the fixed particle is 18.85/ηa2G, while immediately in front and behind the fixed particle the tangential force drops to zero. We have obtained accurate values of the hydrodynamic force using the multi-subunit model using 60 subunits arranged in a “buckyball” (truncated icosahedron) configuration. These are shown in Figure 5 for a surface-to-surface separation of h ) 0.01a and for positions in the plane x ) 0 (φ ) (π/2). The true maximum tangential hydrodynamic force estimated by this method for this separation (Figure 5) is some 1.135 times larger (FHT(max) ) 64.2/ηa2G) than the Stokes force at this point. The corresponding Stokes estimate of the (16) Velegol, D.; Catana, S.; Anderson, J. L.; Garoff, S. Phys. Rev. Lett. 1999, 83, 1243.
FSR ) 6πa2ηG(2 cos ξ + 1)sin ξ
(13)
We find that the maximal radial hydrodynamic force is FHR(max) ) 36.2/ηa2G; this is ca. 1.09 times the Stokes estimate. The true hydrodynamic force also varies slightly in direction from the Stokes force. Hydrodynamic forces may distort the shape of emulsion droplets, but in this case the capillary number, Ca, defined by
Ca ) ηGa/σ
(14)
where σ is the interfacial tension, is
