Aggregation Processes, Particle Interactions, and Colloidal Structure

Aggregation Processes, Particle Interactions, and Colloidal Structure

Aggregation Processes, Particle Interactions, and Colloidal Structure By Eric Dickinson PROCTER DEPARTMENT OF FOOD SCIENCE, UNIVERSITY OF LEEDS, LEEDS...

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Aggregation Processes, Particle Interactions, and Colloidal Structure By Eric Dickinson PROCTER DEPARTMENT OF FOOD SCIENCE, UNIVERSITY OF LEEDS, LEEDS LS2 9JT, UK

1 Introduction The stability and rheological properties of food colloids are dependent on the ways in which the constituent particles and macromolecules are assembled together to form colloidal structures. Different kinds of interparticle interactions lead to differcnt aggregation mechanisms, and different aggregation processes lead to different kinds of colloidal structures. In order to increasc our understanding of the link between the particle interactions and the colloidal properties, it is necessary to consider what are the main factors controlling the mechanisms of colloidal aggregation. What is meant here by the word ‘particle’ is rather wide. At the smallest cxtreme it could refer to single macromolecules (e.g. B-lactoglobulin); at the largest to droplets of emulsified oil or water; it could also be applicable to protein oligomers, protein particles ( e . g . casein micelles) and very small crystals ( e . g . of triglyceride). The only condition is that Brownian motion should be important in determining thc movement of the particlcs and hence the resulting aggregate structure. This restricts the particle size to less than a few micrometres. In categorizing types of colloidal structure, a useful distinction can be made between reversible and irreversible aggregation processes. The former are primarily determined by thermodynamic considerations; the latter by kinetic considerations. Both are influenced by the nature of the interparticle interactions. Net repulsive interactions imply colloidal stability and hence inhibition of aggregation. When the interactions are strongly attractive, aggregated particlcs become permanently bonded together. When the interactions are more weakly attractive, there is opportunity for continuous aggregatc reorganization and the setting up of an equilibrium between coexisting aggregated and unaggregated phases. More complex situations may also occur with particles that are permanently, but flexibly, bonded to other particles; in such systems other repulsive or attractive non-bonded interactions can influence further the 107


Aggregation Processes, Particle Interactions, and Colloidal Structure

detailed colloidal structure. Despite the obvious compositional complexity of real food colloids, it appears that the main structural characteristics of these various classes of multi-particle systems can be understood in general terms using established statistical mechanical theories and computer simulation techniques applied to the behaviour of rather simple models. A crucial aspect of exploring concentrated systems of this sort is that we must learn to think in terms of the structure, not of separate individual aggregates, but of the colloidal system as a whole.

2 Particle Interactions In describing the form of the interparticle interaction, what we are usually concerned with is the change in free energy AG(h) associated with bringing together a pair of colloidal particles with surface-to-surface separation h. It is convenient in many cases to regard this free energy function AG(h) as being equivalent to a pairwise-additive potential energy function U(h)-although strictly speaking the function should be called a ‘potential of mean force’ in recognition of the major entropic contribution to the free energy of interaction.‘ The form of U(h)depends, of course, in a complicated way on the particle surface properties and on the chemical composition of both the particles and the medium in which they are dispersed. For certain idealized cases the general shape of U(h)is known ab initio from analytic t h e ~ r y(and ~,~ occasionally directly from macroscopic surface force experiments). For instance, curve A in Figure 1 shows a plot of U(h) for a pair of electrostatically stabilized spherical particles in a colloidal dispersion as described by classical DLVO theory.2For particles of a micrometre diameter or above in an aqueous

Figure 1

Realistic forms of thepair interaction potential U(h) as a function of surfaceto-surface separation h for (A} electrostatically stabilized particles (DLVO potential) and ( B } sterically stabilized particles

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medium of low ionic strength, the theoretical potential U(h)is characterized by a large energy barrier. This so-called ‘primary maximum’ prevents pairs from irreversibly jumping together at very close separations (in the ‘primary minimum’) over the normal experimental timescale. At larger separations (beyond the energy barrier in the DLVO potential), there is a shallow attractive region (the ‘secondary minimum’) which induces pairs to associate reversibly into what, in principle at least, should be freely rotating doublets (though, confusingly, some recent experiments4 are apparently consistent with such flocs behaving as rigid doublets!). Under different conditions, the energy barrier could be of reduced or zero height (e.g. at medium or high ionic strength), or the secondary minimum could be missing altogether ( e . g . with small colloidal particles). Curve B in Figure 1 represents the equivalent theoretical potential .~ the for particles sterically stabilized by an adsorbed layer of p o l y m ~ rWhile functional form of U(h) resembles curve A at largc h, the short-range steric repulsive region of curve B is distinctly steeper at small pair separations, and the DLVO primary minimum is absent altogether because the presence of the (irrcversibly) adsorbed polymer prevents the particles from getting so close. The ‘pseudo-secondary minimum’ of the steric potential may also be missing in many systems (e.g. with small particles, long polymer chains, a n d o r good solvent conditions). In food colloids the precise form of U(h)is never known. What is sometimes known is the likely mechanism of stabilization and the approximate magnitude of some energy barrier or attractive well depth. The latter are typically inferred from experimental observations of coagulation kinetics, phase transitions or rheological behaviour. This is done using some established theory which relates the measured quantity (turbidity, viscosity, erc.) to the interparticle interactions. As the validity of the theory normally relies on a number of assumptions that are of questionable validity when applied to food systems, the potential parameters inferred in this way can be regarded only as a very rough indication of relative interaction strengths, and not as numbers of absolute significance. The limited aim of this paper is to highlight the key aspects of interparticle interactions that are responsible for determining the main features of structure, stability and rheology in aggregated colloidal systems. So, rather than discussing realistic potentials whose forms vary considerably from system to system, it is here more useful to consider sets of idealized potentials such as the ones illustrated in Figure 2. In case (a) the potential is strongly attractive at very close separations and zero at larger separations. Particles of this hypothetical type are susceptible to fast irreversible coagulation leading to diffusioncontrolled fractal-type aggregates with little relative particle movement or aggregate rearrangement. In case (b) the potential is repulsive at close separations and attractive at intermediate separations. Particles of this type are susceptible to rapid reversible flocculation, with the resultant aggregate structure and mobility dependent on the potential well depth Umin(relative to the thermal energy kT, where k is Boltzmann’s constant and Tis the temperature). For very deep potentials of type (b), the aggregation becomes effectively


Aggregation Processes, Particle Interactions, and Colloidal Structure




Figure 2





Four idealized forms of the pair interaction potential U(h) as a function of surface-to-surface separation h. See text for full discussion of the cases (a) through (d)

irreversible [like case (a)], but extensive aggregate rearrangements may still take place [unlike case (a)]. Potential (c) combines the key features of potentials (a) and (b). Particles of this type are susceptible both to irreversible coagulation at very close separations and reversible flocculation at intermediate separations. Unlike case (a), however, the coagulation is reactioncontrolled, with the value of the rate constant decreasing exponentially with the potential barrier height U,,,, and the resulting aggregate structure being sensitively dependent on floc rearrangements prior to coagulation. Type (c) systems with large U,,, ( 2 15k7') are stable to coagulation and therefore behave like type (b) systems. A further degree of complexity occurs in case (d) with the presence of a second potential maximum at large separations. Particles of this type experience a kinetic barrier to reversible flocculation, with the flocculation rate constant dependent on the secondary barrier height Ukar So far it has been assumed that the interparticle potential U ( h ) is a simple time-invariant function characteristic of the particular colloidal system. However, this is oftcn not the case in food systems where the aggregation process is triggered gradually by heating, pH change or enzyme action. In such cases, for instance, there may be occurring a continuous change in either the well depth for a type (b) system or the barrier height for a type (c) system. These effects


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Figure 3

h 3 Illustration of ways in which the pair interaction U(h) may change with time: (i) the depth of the secondary minimum of type (b) potential (see Figure 2 ) increases with time; (ii) the height of the primary maximum of type (c) potential (see Figure 2) decreases with time

are illustrated schematically in Figure 3. Time-dependent changes in U ( h )of this sort are manifest experimentally as an apparent 'lag phase' in the kinetics prior to the initial detection of flocculation or coagulation. Alternatively they may be indicated by structural changes associated with complex reorganizational events during or following the onset of aggregation.


Aggregation Processes, Particle Interactions, and Colloidal Structure

3 Irreversible Cluster-Cluster Aggregation While the experimental observation of the formation of dendritic and chainlike aggregate structures by irreversible coagulation had been recorded on many occasions in the colloid science l i t e r a t ~ r e it, ~was ~ only through the application of computer simulation that the origin of this type of aggregate structure was definitively as being simply the result of random irreversible collisions. The major step forward over the past decade or so has been the re'~ognition"-'~that colloidal aggregates are examples of statistical fractal objectsI4 which are on average self-similar over certain length scales. More precisely, they can be characterized as muss fractals since the mass M scales with the average size ra according to


- r$,

where df is the fractal dimensionality. In practice, the main factor determining df is the form of the interparticle potential U(h). Experimental studies of irreversible coagulation of particles in dilute dispersions typically give values of df = 1.8 under fast aggregation conditions and df = 2.1 under very slow aggregation conditions. These two limiting types of behaviour are well described in terms of two distinct cluster-cluster aggregation models-so-called 'diffusion-limited cluster aggregation' (DLCA) and 'reaction-limited cluster aggregation' (RLCA). In a computer simulation of cluster-cluster aggregation, the particles and clusters move by Brownian diffusion in a continuum fluid (or by random walks on a lattice), and the cluster diffusion coefficient is typically scaled in inverse proportion to the cluster radius. The DLCA model is applicable to interacting particles with U(h)like that in Figure 2(a): as soon as any two particles on different clusters get within a certain minimum distance (or, equivalently, they occupy neighbouring lattice sites), the original clusters become permanently joined together and thereafter diffuse as a new single cluster. In the RLCA model, however, a repulsive barrier must be crossed [see U ( h ) in Figure 2(c)] before the colliding clusters can contact each other to become irreversibly joined. The barrier height U,,, cannot be too large, of course, because otherwise the dispersion will be stable over the experimental (or simulation) timescale. On the other hand, it cannot be too small either, because the process then becomes diffusion-limited for large (slowly moving) clusters, implying a gradual 'crossover' from RLCA kinetics to DLCA kinetics (and hence a reduction in df) as the aggregation proceeds. A widely recognized process of irreversible aggregation in food science is the coagulation of casein particles induced by lowering of pH or enzymic hydrolysis with chymosin. It has been demonstrated experimentally's-17 that the growing aggregates which appear during the formation of an acid casein gel do indeed have a fractal character, and that the structure and rheological properties of the resulting casein gel can be understood in terms of scaling laws based on the effective characteristic fractal dimensionality. However, the df values

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fitted15-" for casein gels (d, -- 2.3 k 0.2) are significantly higher than the values quoted above for simulations based on the simple DLCA or RLCA model. The question arises, then, as to what are the important features missing from these simple aggregation models which need to be added before the computer simulations can be regarded as giving reliable representations of network structures formed from aggregating casein micelles (or other particles such as protein-stabilized emulsion droplets). There are two main factors that need to be addressed when considering the fractal properties of particle gels-(i) the effect of the finite particle concentration on aggregate growth and interpenetration, and (ii) the effect of interparticle interactions on aggregate restructuring before, during and after gelation. While the two issues are somewhat interrelated, let us first consider them separately, beginning with the issue of particle volume fraction $p. In the simplest cluster-cluster gelation m ~ d e l , ' ~ the - ' ~individual aggregates are assumed to grow exactly as they would at infinite dilution ($,, + 0) with a single fractal dimension df in the range 1.8-2.1 depending on whethcr the mechanism is DLCA, RLCA, or something in between. The number of particles of radius R in a fractal aggregate of radius ra and dimensionality d f is Nd

- (ra/R)df,

as compared with the number of particles in the equivalent close-pack:d aggregate:


- (r,/R)3.


The volume fraction of particles in a fractal aggregate of radius ra is

The gel point is assumed to be reached at time t = t , when Gf becomes equal to $p, as illustrated in Figure 4. The critical aggregate radius r,* at the onset of gelation is then given by

There are two main assumptions in the above. (1) All aggregates have the same size at the gel point. This is obviously not correct because it is known" that fractal colloidal aggregates of finite extent produced by DLCA or RCLA have a wide time-dependent cluster-mass distribution N ( M , t ) . In reality, in a fairly concentrated dispersion, when the biggest clusters are just beginning to join together to form a system-spanning network, a significant proportion of the particles will still exist as monomers (or very small clusters) at t = t,; these 'free' particles will only become fully


Aggregation Processes, Particle Interactions, and Colloidal Structure


Figure 4

Illustration of gelation according to the simple cluster-cluster aggregation model: (i) single fractal aggregate of particle volume fraction $f;(ii) gel point is reached when Gfequals the overall particle volume fraction @,

incorporated into the developing network structure at times well beyond the gel point (t >> t g ) . ( 2 ) Cluster-cluster interactions are neglected. This assumption must be strictly invalid for any system in the immediate pre-gelation regime ( t + tg) for which the average cluster size is approaching the critical size r z , and it is effectively invalid at all stages of the aggregation process (t > 0)for moderately concentrated systems. In the latter case, even in the very initial stages of coagulation, the local cluster growth is determined by the local particle

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concentration gradient, which in turn depends on the growth of neighbouring clusters. The evolving aggregate fractal structure in a concentrated system is therefore never at any stage the same as that for cluster-cluster aggregation at high dilution. Irrespective of whether the system is concentrated or dilute, however, once the mean cluster radius has approached r t , the increasing crowding and interpenetration of the largest fractal clusters must inevitably change the structure of aggregates in the gelling system from that in the nongelling DLCA or RLCA system at infinite dilution. Intuitively, this effect of cluster4uster 'excluded-volume' interactions would be expected to increase the compactness of the largest clusters, i.e. to increase the effective value of df. Hence, for these two important reasons, the simple cluster-cluster gelation model dcfined by equations (2) to ( 5 ) does not constitute a proper description of a particle gel system formed by diffusion-limited or reaction-limited mechanisms at finite particle volume fraction. The problem lies in part with the nonflcxibility of the aggregates formed by DLCA or RLCA. Both models assume that, once two particles on different clusters actually become joined togetherhowever little or much time this takes-the bond between them thereafter remains permanent and rigid. For a computer simulation of irreversible coagulation in a concentrated dispersion of N particles with periodic boundary conditions, this leads to the curious result'' that the final configuration is a single fractal-type cluster which does not form a percolating network. Another anomaly arising with aggregating fractal clusters which stick together rigidly and irreversibly is that, in principle, there is no lower theoretical particle concentration below which gelation cannot take place, since the volume fraction of a single fractal aggregate, &, can always become equal to the particle volumc fraction of the whole system, @p, if it is allowed to grow indefinitely (@f-+ 0 as r t - + CQ and tg-+ C Q ) .This contrasts with the behaviour in real experimental situations, where it is usually found that there is a welldcfined reproducible critical gelation concentration qbg.

4 More Realistic Particle Gel Models What is crucially missing from the DLCA model is the opportunity for clusters to deform or rearrange during aggregation. In principle, there are various ways in which this severe constraint of the DLCA model can be removed in both lattice and non-lattice simulations: (a) by allowing for bonds to break as well as to form;2"'21 (b) by introducing some flexibility into all bonds (continuum models ~ n l y ) ; ~or l - (c) ~ ~by incorporating internal cluster flexibility through more complex algorithms (bond f l u c t ~ a t i o nor~ ~aggregate shakin8'). The allowance for aggregate flexibility and/or restructuring by one of these methods does lead to behaviour that lies much closer to what is observed experimentally. True gelation occurs only above a certain critical particle volume fraction qbg, and computer simulation in a periodic cell results in a percolating network structure (@p > @.J instead of the single final cluster of the DLCA or RLCA models.


Aggregation Processes, Particle Interactions, and Colloidal Structure

Cluster reorganization prior to gelation ( r < tg) and restructuring of the aggregated network after gelation ( t B tg) are both dependent on the nature of the non-bonded particle-particle interaction potential U(h).Recent Brownian dynamics simulations in two and three dimensions of a model involving flexible irreversible bond formation and non-bonded particle interactions have shown21.22that the particle gel structure at constant q5p is sensitive to two key parameters: the rate of bonding as measured by the probability of bonding during a single simulation time-step, and an interaction parameter measuring the strength of the medium-range interparticle force (attractive or repulsive). For a periodic system of 1000 particles at volume fraction q5p = 0.1, Figure 5 illustrates the effect on the three-dimensional network structure of changing from (1) net repulsive non-bonded interparticle force to (2) net attractive nonbonded interparticle force. In both cases, particle pairs are allowed to stick together moderately slowly via irreversible bond formation, so that the final gel structure becomes held together by a percolating network of permancnt flexible bonds.26However, due to differences in the nature of the non-bonded pair interactions during gelation, the particle gel shown in picture 1 is characterized by a smaller average pore size and a spatially more homogeneous distribution of particle positions than that in picture 2. A general requirement for the formation of a homogeneous gel of uniform pore size is the presence of a repulsive interparticle interaction and a relatively low bonding probability. With particles having attractive non-bonded forces, the simulated gels are heterogeneous, and the network can have fine or coarse pore-size distributions depending on whether the bonding is fast or slow. In systems with non-bonded attractive interactions which are strong enough to induce phase separation [as in Figure 5 ( 2 ) ] , the emerging domains grow to some characteristic average size before they get ‘pinned’ by the irreversible cross-linking. The larger the bonding probability, the smaller the domains can grow before they become incorporated into the connected network structure. Some qualitative analogies can easily be identified between these simulation results and experimental data for particle gels made from food proteins. For instance, the known dependence of the degree of heterogeneity of globular protein gels on pH and heating conditions27328 can be readily explained in terms of the balance between two types of aggregation processes-(i) the generation of intermolecular cross-links (covalent and non-covalent) bctween denatured protein molecules which leads to permanent aggregation and network formation, and (ii) the thermodynamically reversible flocculation which leads to protein precipitation under poor solvent conditions. Let us return now to the factors affecting the fractal dimensionality of particle gels formed from aggregating spherical particles. In terms of increasing distance r between particle centres, three spatial scales of structure can be. expected:w (i) short-range liquid-like order from packing and cxcluded volume effects, (ii) medium-range disorder associated with the fractal character of pregclation clusters, and (iii) long-range homogeneity associated with a uniform bulk material. The short-range structure is reflected in the characteristic

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Figure 5


Snapshots of three-dimensional simulated structure of bonded networks formed from Id spherical particles aggregating with ( I ) repulsive nonbonded interactions and (2) attractive non-bonded interactions26

Aggregation Processes, Particle Interactions, and Colloidal Structure


oscillatory behaviour of the radial distribution function g ( r ) of a concentrated monodisperse and beyond some characteristic correlation length 5 the long-range structure is uniform [g(r) = 11. In between may lie a fractal scaling regime of necessarily limited spatial range. Irrespective of the form of U(h), with increasing overall particle concentration, the upper and lower boundaries of the medium-range fractal scaling regime will tend gradually to converge, eventually to disappear altogether for, say, &, =: 0.3. The effective value of dfcan be inferred” from the set of particle coordinates in a simulated gel by calculating the slope of the linear region of a plot of log n(r) against log r , where n(r) is the average number of particle centres lying within a distance r from a given particle. For large separations, we must have

where R is the particle radius. In the intermediate fractal scaling regime, we have

where the pre-factor no is the average number of particles in the primary clusters from which the fractal scaling regime is built. A low value of the prefactor (no 1) implies a fine-grained microstructure, whereas a high value (no >> 1) implies a coarse microstructure. The correlation length of a gel is given by”


This compares with 5 = r*, [equation (5)j for the DLCA model (no= 1). For an aggregating system with an extremely low bonding probability and a moderately strong attractive pair interaction [ e . g . , in Figure 2, the type (b) potential with Umin>> k T ] , a gel-like network formed in the early stages will tend gradually to change its fractal character due to restructuring and coarsening during spinodal d e c o m p o ~ i t i o nIf. ~cross-linking ~ is absent altogether, the fractal character associated with the interconnected structure will eventually disappear altogether since the system is ultimately thermodynamically unstable with respect to phase separation. This type of behaviour has recently been simulated for systems of aggregating Lennard-Jones particles using both molecular dynamics in two dimensions34 and Brownian dynamics in three dimensions.2’ The evolving microstructure is a percolating network structure composed of finely dispersed clusters arranged as filaments which gradually thicken with increasing simulating time. This is reflected in a reduction in df and a simultaneous increase in no. During the relentless drive towards phase separation, when some existing fine filament branches are becoming elongated and broken, the process of short-range densification gradually produces a coarser overall blob-like structure (i.e. increasing no) with larger voids. The

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decrease in df is explicable2’ in terms of a net movement of particles (to ‘feed’ the thickening of the filaments) from the intermediate fractal scaling regime whose structure therefore becomes more stringy. It is important to note that thc fractal dimensionality deduced from the scaling behaviour of n(r) for a particle gel simulated at moderate volume fractions (say, &, 0.1) is to be regarded as a property of the whole interconnected network and not a property of individual clusters. When the particle density is moderately high, and there is a reasonably strong attraction between pairs of closely spaced neighbouring particles, small clusters join together simultaneously at a very early stage in the aggregation process. So, when a convincing fractal scaling regime first becomes evident in the statistical analysis of the simulated n(r) data, the system already exists as a percolating gel-like network. Subsequent changes in microstructure of the ageing gel depend in a complicated way on the nature of the interparticle interactions, including both the weak reversible interactions as well as any strong irrevcrsible bonding interactions. While it is expected that many gels formed by particle aggregation do have a convincing fractal scaling regime, it seems clear also that many do not. Most gels made at particle concentrations above q!~ = 0.3, both simulated and real (e.g. adhesive oil-in-water emulsion gels j), appear to have insignificant fractal character. Homogeneous polymer-like 0.1) gel networks formed at moderately low concentrations (say, @p from bonding particles with repulsive non-bonded interactions are also non-fractal. 21,22




5 Reversible Aggregation: Depletion Flocculation Reversible aggregation processes are controlled primarily by thermodynamic factors, and the transition between dispersed and flocculated colloidal states can usefully be regarded as analogous to the gas-liquid transition of simple fluid systems. 31336 This thermodynamic emphasis can bc justified by the fact that strong corrclations have been identified37,38between the solvent conditions required to reversibly flocculate dispersions and those rcquired to induce liquid-liquid phase separation in the absence of colloidal particles. One of the most important mechanisms of reversible aggregation is that duc to the attractivc ‘depletion’ interaction which exists between pairs of large spherical particles disperscd in a dilute solution of non-adsorbing polymer molecules or small solid particles. That depletion flocculation is reversible to dilution was first demonstrated by B ~ n d for y ~the ~ creaming of natural rubber in the presence of water-soluble polymers. In more recent times, qualitativcly similar behaviour was observed when non-adsorbing polymers (hydroxyethyl cellulose or sodium carboxymethyl ccllulose) were added to electrostatically stabilized latices,40 silica particles4’ or emulsion droplets,42 or when microbial polysaccharides (xanthan or dextran) were added to oil-in-water emulsions stabilized with small-molecule emulsifier^^^' or milk proteins.46 Recently, it


Aggregation Processes, Particle Interactions, and Colloidal Structure

has been recognized474‘ that depletion flocculation of oil-in-water emulsions can also be induced by small surfactant micelles. The simplest working theory of depletion flocculation is that due to Asakura and Oosawas” for the case of rigid spherical molecules in the gap between a pair of particle surfaces. They showed that, because solute molecules are excluded from the region between the surfaces when the gap width h is less than the molecular diameter d , there is a net attraction between the particles arising from the lower osmotic pressure in the gap than in the bulk medium of solute volume fraction &. The Asakura-Oosawa depletion potential has the form

(-3kTR$,/d3)(d - h)’,

(h < d ) (h 2 d)


where R is the particle radius. For two particles of radius R = 0.5 p m in a solution containing solute molecules of diameter d = 10 nm at concentration qjs = 0.02 (2 vol%), equation (9) predicts a maximum depletion attraction energy at contact ( h = 0) of u d = -3 kT. This sort of value is probably large enough to be detectable as enhanced flocculation in creaming or rheology experiments. The depletion interaction for flexible polymer solute molecules is much greater than for spherical solute molecules because the configurational entropy of the chains is substantially reduced near the particle s ~ r f a c c . ~A’ -theoreti~~ cal expression by Vincent et for the depletion potential in this case has thc form Ud(h) = - hRTI(A - th)[l + (2A/3R)

+ (h/6R)],


where I7 is the osmotic pressure of the polymer solution, and A is the so-called ‘depletion layer thickness’, which is approximately equal to the radius of gyration of the unadsorbed polymer in the dilute polymer concentration regime. Further refinements of t h e depletion theory have allowed for the particle surface to be coated with a sterically stabilizing layer of adsorbing polymer,s4 and for the inclusion of electrostatic polymer-polymer and particle-polymer interaction^.^"^^ In all these variants, however, the essential physical origin of the depletion attraction remains the same as that originally proposed by Asakura and 00sawa.’~ Whilst emphasizing the predominantly destabilizing effect of the depletion mechanism, Napper also suggested56 the intriguing possibility that, for somc systems of high polymer concentration, a kinetic stabilization effcct could possibly arise due to the presence of a maximum in the depletion potential Ud(h) at larger values of h. For the case of rigid spheres in the gap between a pair of surfaces, the origin of this putative depletion repulsion is the wellknown damped oscillatory potential of mean force associated with layered particle packing at liquid-like den~ities.”,’~Recent calculations of the hardspherc depletion interaction to second order in the solute volume fraction & have shownss*s9that the depletion potential U d ( h )has a positive maximum value of

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[email protected]?/d




d ( l - [email protected],/2)


and a minimum value at contact (h = 0) of

Ud(hmin)= - [email protected]$d - [email protected],2/5d.


The first term of equation (12) (to order &) is the Asakura-Oosawa result [equation (9) at h = 01. For the set of parameter values referred to above ( R = 0.5 pm, d = 10 nm, & = 0.02), equation (11) predicts Ud(hmaX) = 0.05 k T , which is negligible. However, if the solute volume fraction were to be increased to GS = 0.2 (i.e. 20 vol%), then the putative maximum in the depletion potential would become ca. 5 kT. This value would certainly be significant in retarding the aggregation, although it would still represent a barrier height insufficient to confer full kinetic stability. The depletion flocculation of large spheres by small spheres can be used to explain the flocculation of casein-coated oil-in-water emulsion droplets induced by unadsorbed sodium caseinate.m Figure 6 shows light micrographs of four undiluted groundnut oil-in-water emulsions (10 vol% oil, average droplet diameter d32= 0.56 f 0.02pm, p H 7) containing different amounts of sodium caseinate as the sole emulsifier. A t just 2 wt% protein content [picture (a)] the emulsion shows only a slight indication of aggregation. But, when the protein content is increased to 2.4 wt% [picture (b)], we can begin to see numerous individual aggregates in the size range 10-20 pm. At 2.8 wt% protcin, many of the aggregates are joined together into a percolating network structure, and at 3.2 wt% a particle gel structure is clearly evident with considerable close-range densification and possibly some medium-range fractal character. The presence of emulsion flocculation is reflected experimentallym in a greatly enhanced rate of creaming and a substantial change in the low-stress rheological behaviour. As expected for a depletion flocculation mechanism, the aggregation process is completely reversible to dilution.m Assuming that the sodium caseinate is composed of particles of diameter 8-10 nm and that the average emulsion droplet size is 0.6pm, we can estimate from equation (9) that the maximum average depletion interaction strength (h = 0) for a caseinate particle volume fraction of @s = 0.03 (3 ~ 0 1 %is) Ud(h) = 3 k T . This theoretical value is certainly sufficiently large to explain the substantially changed creaming behaviour observed experimentally in conccntrated emulsions containing excess caseinate.m There is also separate experimental evidence that ordered layers of casein particles in thin liquid films between droplets may provide a significant stabilizing mechanism against coalescence in caseinate-containing emulsions.61

6 Concluding Remarks This article has considcred the relationship between particle interactions and the structure and stability in concentrated aggregating colloidal systems. For


Aggregation Processes, Particle Interactions, and Colloidal Structure

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the case of irreversible aggregation, the emphasis here has been on the effect of particle volume fraction and the form of the interparticle potential on the fractal structure of simulated particle gels formed by computer simulation. A critical analysis of the predictions of the infinite-dilution cluster-cluster aggregation models (DLCA and RLCA) has been presented. For the case of reversible aggregation, emphasis has been on the conditions inducing depletion flocculation (and possibly depletion stabilization) in systems of large particles due to the presence of small particles (or polymers). In addition to the mechanisms referred to above, of course, there are various other kinds of particle interactions that can lead to aggregated particle gel structures in food colloids. For instance, small spheres (or polymers) which adsorb onto the surface of large spheres may form network structures due to a bridging flocculation mechanism if the particle concentration is sufficiently high. Such flocculation can be modelled statistically as a simple binary mixture of spheres with a sticky-sphere unlike interaction The implication of the theoretical [email protected] is that the rheology of concentrated colloids should be extremely sensitive to the addition of weakly adsorbing particles (or polymers). This has recently been verified e ~ p e r i r n e n t a l l yfor ~~~~ the case of some concentrated protein-stabilized emulsions containing small amounts of weakly adsorbing polysaccharide. Most of the visible manifestations of colloidal aggregation in the laboratory arise as a result of the influence of gravity in inducing phase separation or density gradients in aggregating emulsions or dispersions. There is dill much to be done to understand properly the effect of interparticle interactions on the structure of aggregating systems undergoing creaming or sedimentation. While some progress can be made in simulating the settling of aggregating systems using simple lattice models,[email protected] this research needs to be extended to nonlattice models and to a wide range of interparticle interactions. A final point to note is that, whereas the current theories typically assume monodispersity, most real colloids are substantially polydisperse. This is particularly true of food emulsions. Polydispersity complicates the interpretation of the origin of colloidal structures because the strength of dropletdroplet interactions and the effect of gravity on droplet motion are both very dependent on the droplet size. Of particular interest, therefore, are recent experimental development^^^^'^ in the formulation of nearly monodisperse emulsions. Measurements on these surfactant-stabilized systems caii be expected to allow more systematic comparisons to be made with analytical theories and computer simulations, e.g. in studies of the fractal structure of concentrated emulsion gels.71 The widespread availability of equivalent monodisperse emulsions stabilized by milk proteins would provide exciting new experimental opportunities for food colloid scientists.

References 1. E. Dickinson, ‘An Introduction to Food Colloids’, Oxford University Press, Oxford, 1992, p. 14.


Aggregation Processes, Particle Interactions, and Colloidal Structure

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