Particle interactions

Since most colloidal dispersions are stabilized by particle interactions, the use of equation (10.51) may lead to biased estimates of particle size that are often concentration dependent. The effect may be taken into account by expanding the diffusion coefficient to a concentration power series that, at low concentrations, gives  

The equation reduces to the Stokes-Einstein equation for spherical particles. Since the friction coefficient for a non-spherical partiele always exceeds the friction coefficient for a spherical particle, over estimation of particle size will occur if equation (10.41) is applied. 

The virial coefficient kD is positive for repulsive particle interaction and negative for attractive interaction. Thus if particle interaction is neglected the apparent size will be concentration dependent, increasing with increasing concentration for attractive interactions and decreasing with repulsive interactions. In such cases, the diffusion coefficient should be determined at a range of concentrations and Dq determined by extrapolating to zero concentration. 

The effect of particle interaction is proportional to the average Interparticle distance that, for a fixed volume concentration, decreases with particle size. Hence, the effect of interaction reduces as particle size increases. However, small particles scatter much less light than large particles and it is necessary to use a higher concentration for reliable PCS measurements. In these cases the concentration needs to be increased to volume fractions up to 0.1% and, again, particle sizes can only be determined from extrapolations to zero concentration. 

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Energetic particles interacting can also modify the structure and/or stimulate chemical processes on a surface. Absorbed particles excite electronic and/or vibrational (phonon) states in the near-surface region. Some surface scientists investigate the fiindamental details of particle-surface interactions, while others are concerned about monitormg the changes to the surface induced by such interactions. Because of the importance of these interactions, the physics involved in both surface analysis and surface modification are discussed in this section. 

For systems in which the constituent particles interact via short-range pair potentials, W = (F. 

The solution detennines c(r) inside the hard core from which g(r) outside this core is obtained via the Omstein-Zemike relation. For hard spheres, the approximation is identical to tire PY approximation. Analytic solutions have been obtained for hard spheres, charged hard spheres, dipolar hard spheres and for particles interacting witli the Yukawa potential. The MS approximation for point charges (charged hard spheres in the limit of zero size) yields the Debye-Fluckel limiting law distribution fiinction. 

Assuming that additive pair potentials are sufficient to describe the inter-particle interactions in solution, the local equilibrium solvent shell structure can be described using the pair correlation fiinction g r, r2). If the potential only depends on inter-particle distance, reduces to the radial distribution fiinction g(r) = g... 

Figure B3.3.3. Periodic boundary conditions. As a particle moves out of the simulation box, an image particle moves in to replace it. In calculating particle interactions within the cutoff range, both real and image neighbours are included.

Particles can be manipulated in suspension using strongly focused laser beams ( optical tweezers ) [25] or magnetic fields [26] and by collecting statistics on tire particle movements using video microscopy, infonnation on the particle interactions can be obtained. 

At finite concentration, tire settling rate is influenced by hydrodynamic interactions between tire particles. For purely repulsive particle interactions, settling is hindered. Attractive interactions encourage particles to settle as a group, which increases tire settling rate. For hard spheres, tire first-order correction to tire Stokes settling rate is given by [33]... 

In particular, in polar solvents, the surface of a colloidal particle tends to be charged. As will be discussed in section C2.6.4.2, this has a large influence on particle interactions. A few key concepts are introduced here. For more details, see [32] (eh 13), [33] (eh 7), [36] (eh 4) and [34] (eh 12). The presence of these surface charges gives rise to a number of electrokinetic phenomena, in particular electrophoresis. 

Particularly in polar solvents, electrostatic charges usually have an important contribution to tire particle interactions. We will first discuss tire ion distribution near a single surface, and tlien tire effect on interactions between two colloidal particles. 

In many colloidal systems, both in practice and in model studies, soluble polymers are used to control the particle interactions and the suspension stability. Here we distinguish tliree scenarios interactions between particles bearing a grafted polymer layer, forces due to the presence of non-adsorbing polymers in solution, and finally the interactions due to adsorbing polymer chains. Although these cases are discussed separately here, in practice more than one mechanism may be in operation for a given sample. 

In the previous section, non-equilibrium behaviour was discussed, which is observed for particles with a deep minimum in the particle interactions at contact. In this final section, some examples of equilibrium phase behaviour in concentrated colloidal suspensions will be presented. Here we are concerned with purely repulsive particles (hard or soft spheres), or with particles with attractions of moderate strength and range (colloid-polymer and colloid-colloid mixtures). Although we shall focus mainly on equilibrium aspects, a few comments will be made about the associated kinetics as well [69, 70]. 

We will focus on one experimental study here. Monovoukas and Cast studied polystyrene particles witli a = 61 nm in potassium chloride solutions [86]. They obtained a very good agreement between tlieir observations and tire predicted Yukawa phase diagram (see figure C2.6.9). In order to make tire comparison tliey rescaled the particle charges according to Alexander et al [43] (see also [82]). At high electrolyte concentrations, tire particle interactions tend to hard-sphere behaviour (see section C2.6.4) and tire phase transition shifts to volume fractions around 0.5 [88]. 

If the gas particles interact through a pairwise potential, then the contribution to the viriai from the intermolecular forces can be derived as follows. Consider two atoms i and j separated by a distcmce r. ... 

This example shows the round particle in cell B,B with two possible nonbonded cutoffs. With the outer cutoff, the round particle interacts with both the rectangle and its periodic image. By reducing the nonbonded cutoff to an appropriate radius (the inner circle), the round particle can interact with only one rectangle—in this case, the rectangle also in cell B,B. ... 

Rheology. Flow properties of latices are important during processing and in many latex appHcations such as dipped goods, paint, inks (qv), and fabric coatings. For dilute, nonionic latices, the relative latex viscosity is a power—law expansion of the particle volume fraction. The terms in the expansion account for flow around the particles and particle—particle interactions. For ionic latices, electrostatic contributions to the flow around the diffuse double layer and enhanced particle—particle interactions must be considered (92). A relative viscosity relationship for concentrated latices was first presented in 1972 (93). A review of empirical relative viscosity models is available (92). In practice, latex viscosity measurements are carried out with rotational viscometers (see Rpleologicalmeasurement). 

Dispersion of a soHd or Hquid in a Hquid affects the viscosity. In many cases Newtonian flow behavior is transformed into non-Newtonian flow behavior. Shear thinning results from the abiHty of the soHd particles or Hquid droplets to come together to form network stmctures when at rest or under low shear. With increasing shear the interlinked stmcture gradually breaks down, and the resistance to flow decreases. The viscosity of a dispersed system depends on hydrodynamic interactions between particles or droplets and the Hquid, particle—particle interactions (bumping), and interparticle attractions that promote the formation of aggregates, floes, and networks. 

For higher (0 > 0.05) concentrations where particle—particle interactions are noticeable, the viscosity is higher than predicted by the Einstein equation. The viscosity—concentration equation becomes equation 10, where b and c are additional constants (87). 

If there is particle—particle interaction, as is the case for flocculated systems, the viscosity is higher than in the absence of flocculation. Furthermore, a flocculated dispersion is shear thinning and possibly thixotropic because the floccules break down to the individual particles when shear stress is appHed. Considered in terms of the Mooney equation, at low shear rates in a flocculated system some continuous phase is trapped between the particles in the floccules. This effectively increases the internal phase volume and hence the viscosity of the system. Under sufficiently high stress, the floccules break up, reducing the effective internal phase volume and the viscosity. If, as is commonly the case, the extent of floccule separation increases with shearing time, the system is thixotropic as well as shear thinning. 

The classical motion of a particle interacting with its environment can be phenomenologically described by the Langevin equation... 

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