Fractal aggregation in relation to formation and properties of particle gels

The purpose of this study is to gain insight into the conditions determining whether small particles in a liquid are able to jointly occupy the total volume thus forming a gel network. In order to build a network the colloidal particles have to be 'sticky', unstable. In the unassociated state the particles move at random through the liquid due to collisions with solvent molecules. This movement, called thermal or Brownian motion or diffusion, depends on the temperature and on the size of the particles. By thermal motion particles may meet and subsequently stick, thus forming clusters or floes. This process is called perikinetic aggregation. Other transport mechanisms that lead to aggregation involve velocity gradients in the dispersion (orthokinetic aggregation) or sedimentation of the flocs.A model describing the formation of a gel out of aggregating floes is derived in chapter II. Floes that are formed by aggregation have in general a fractal geometry. This implies that repetitive levels of detail exist on all length scales between the size of the primary particles and the size of the floe. A floe is built up of smaller floes that, on their turn, are built of still smaller floes, etc. Each separate fractal floe has its own geometry, different from that of any other floe. However, all floes share a similar average structure characterized by a stochastic fractal nature, and are in this respect scale invariant. The efficiency with which floes fill the available space is expressed by the fractal dimensionality, D, which is the exponent in the power-law relation between the number of particles in a floe and the size of the floe. A low value of D implies a small number of particles needed to build up a floe of certain size, and thus a high spacefilling efficiency. In a three- dimensional system, D may attain values between 1 and 3. Computer simulations of the aggregation process yield D = 1.8 if all collisions lead to attachment (diffusion - limited cluster aggregation) and D = 2.1 if the sticking probability is very low (reactionlimited cluster aggregation). Rearrangement of the floes during aggregation results in higher fractal dimensionalities.Since D is generally smaller than 3 the volume fraction of the particles in a fractal floc decreases as the size of the floc increases. When the flocs jointly become space-filling a gel is formed; now the overall volume fraction of particles in the system equals the average volume fraction of particles in the flocs. This implies that fractal flocs have the ability to become space-filling at any volume fraction of primary particles if the flocs have enough space to grow large enough, i.e., if the container is large enough. In section 2.3 a model is derived that describes the relation between the average size of clusters in a gel and the overall volume fraction of the particles.Gels formed at high and low volume fractions will have similar geometric stuctures, be it on different length scales, if the relative size distribution of the clusters remains constant during the aggregation process. This is shown experimentally in section 4.3 where micro-graphs of gels at various volume fractions and magnifications are compared. Scale invariance enables the derivation of scaling relations between gel properties and the volume fraction of the particles or the length scale on which the gel is studied. Relations between the permeability and the correlation length versus the volume fraction, and the correlation function and the turbidity versus length scale (wavelength), are derived in section 2.4. These relations can be used to obtain the fractal dimensionality from experimental results.In chapter III materials and methods are described that have been used to make and study the gels. For the preparation of the gels it is essential that aggregation occurs under quiescent conditions, because velocity gradients may lead to rearrangement of the flocs into more compact clusters with a lower space-filling efficiency. Different methods of destabilisation that have been used to obtain quiescent aggregation, are described: 1) slowly warming up a dispersion that is stable at low, but unstable at higher temperatures, 2) acidification of a dispersion that is unstable at low pH by a slowly hydrolyzing acid precursor, and 3) addition of an enzyme that removes stabilizing compounds from the surface of the particles. The permeability of these gels was studied by measuring the flow rate, caused by a certain pressure gradient, through tubes in which a gel was constrained. The geometry of the network was studied by confocal scanning laser microscopy. A small spot inside the gel, that is provided with a suitable fluorescent label, is illuminated by a focused laser beam and the fluoresced light that stems from the spot is detected via a microscope in a photon multiplier. Many positions in the gel are scanned in this way and optical sections and three- dimensional images are obtained.In chapter IV results of the gelation and coagulation studies are described. At quiescent conditions dispersions of small particles may gel at extremely low volume fractions. Spherical, palmitate covered polystyrene particles (α= 35 nm) formed space-filling networks even at volume fractions below 0.1 %! Aggregation caused by electrolyte addition to polystyrene or haematite sols (coagulation) may also result in continuous networks. The mixing of the electrolyte and the sol leads presumably to rearrangement of the floes into more compact clusters. The volume fraction of particles that was necessary for gel formation turned out to be roughly 5 % at high NaCl concentrations. A relative measure for the coagulation rate is the initial rate of change of the turbidity after electrolyte addition. It is found that at some critical salt concentration, the ccc, this rate attains a maximum; here, the aggregation rate is diffusion limited. In the case of coagulation by NaCl, floes of polystyrene or haematite particles were relatively compact at the ccc. This resulted in high values of the turbidity plateau that an aggregating system approaches, and in small sediment volumes. At higher NaCl concentration the floes become more ramified. Presumably, high salt concentrations cause stronger inter-particle junctions and less rearrangement, leading to more ramified floes.The fractal dimensionalities obtained from permeability studies were 2.34 and 2.21 for acid casein and palmitate covered polystyrene gels, respectively (section 4.2). From the absolute value of the permeability the ratio between the radius of the clusters in the gel and their effective hydrodynamic radius was estimated to be 1.13. This value is substantially lower than expected from calculations on fractal floes obtained by computer simulation. The permeability of most gels did not change during ageing, indicating that no large scale rearrangements occur. However, the permeability of rennet-induced casein gels increased during ageing, because microsyneresis causes a coarsening of the gel.Microsyneresis occurs if a gel tends to synerese whereas shrinkage is impossible. The process leads to local condensation of the network and the formation of large pores elsewhere.Changes in rennet-induced casein gels during ageing are shown on micrographs in section 4.3. The mesh-sizes of fresh rennet induced casein gels and acid casein gels with the same casein concentration are similar, but after ageing the mesh size of rennet-induced gels increases dramatically whereas that of acid gels is constant. Close to the glass surface the density of the gel is relatively high because particles stick to the surface during aggregation. Sections were taken at a depth larger than the diameter of the clusters to ensure the observation of a bulk gel. The fractal dimensionality obtained from the relation between the correlation length in micrographs of acid casein gels and the volume fraction of the particles was 2.35. A similar value was obtained from the relation between the correlation function and the length scale. Results of turbidity measurements as a function of wavelength yielded a value of D for acid casein gels of roughly 2.3.Various models may be derived to relate rheological properties of particle gels to the volume fraction of the particles. In addition to the geometric structure of the network, the interactions between and rheological properties of the particles are important. In chapter V two models are derived for gels built up of fractal clusters. One model applies to type 1 gels in which the stress carrying strands in a gel gradually get stretched, e.g. due to microsyneresis. The other model applies to type 2 gels in which no large scale rearrangements occur after the gel point. Rennet induced casein gels and acid casein gels made by slowly warming up a cold (4 °C) casein dispersion of pH 4.6 to 30 °C) turned out to be type I gels, whereas uninduced casein and palmitate covered polystyrene gels were type 2 gels. At the same volume fraction, type I casein gels were much stiffer than type 2 gels but the strength, i.e. the stress at which fracture occurs, was roughly similar. The strain at which the gels fracture was larger for type 2 gels. The values of D obtained by applying the models to results on the shear modulus versus the volume fraction, were 2.24 and 2.36 for type I and 2 casein gels, respectively, and 2.26 for palmitate-covered polystyrene gels. The stiffness of all gels studied increased during ageing. Since permeability studies show that no large scale rearrangements occurred during ageing (except in rennet gels), the increase in stiffness must be due to an increase of the strength of the inter-particle junctions, e.g. due to some kind of sintering.The aggregation time of an unstable colloidal dispersion, defined as the time after which aggregation becomes visible, depends both on the bond formation rate and on the way the structure of the aggregates develops. The latter is not taken into account in the traditional theory of aggregation kinetics where only the bond formation rate is considered. In chapter VI it is shown that the structure of colloid aggregates has a large effect on the aggregation rate and an even larger one (up to several orders of magnitude) on the aggregation time. Approximate expressions for the aggregation time at different conditions are derived. Due to the intricacy of the subject it is mostly impossible to derive exact expressions. For many situations it is possible, though, to roughly predict the aggregation time. Complications, due to deviations from the ideal case of spherical, smooth, monodisperse particles at low volume fractions, are compiled and their effects on the aggregation rate and time are estimated. It is shown that small velocity gradients often cause a huge decrease of the aggregation time. Small velocity gradients occur also in 'quiescent' systems, e.g. due to convection.Fractal aggregates will always fill the entire volume if they are allowed to grow without being disturbed. Factors that can disturb the gelation. or change the gel structure, are described in chapter VII. It is argued that small velocity gradients are essential for the formation of networks at very low volume fractions because the floes forming the network are so large that the time needed for diffusion over a distance equal to the radius of the floe is much longer than the time needed to settle over that distance. Larger velocity gradients may cause compaction or breakup of the floes, thus hindering gelation. All systems studied in this thesis that gel at low volume fraction show some sintering or fusion, which may be ascribed to an increase in the number of bonds per junction. This may be a prerequisite for aggregation of colloidal particles leading to a gel at low volume fraction. n