Suspensions polydisperse

Rowell and co-workers [62-64] have developed an electrophoretic fingerprint to uniquely characterize the properties of charged colloidal particles. They present contour diagrams of the electrophoretic mobility as a function of the suspension pH and specific conductance, pX. These fingerprints illustrate anomalies and specific characteristics of the charged colloidal surface. A more sophisticated electroacoustic measurement provides the particle size distribution and potential in a polydisperse suspension. Not limited to dilute suspensions, in this experiment, one characterizes the sonic waves generated by the motion of particles in an alternating electric field. O Brien and co-workers have an excellent review of this technique [65]. 

FIG. 2 A polydisperse suspension consists of particles of different sizes. 

FIG. 2.1 Sedimentation field flow fractionation (SdFFF) (a) an illustration of the concentration profile and elutant velocity profile in an FFF chamber and (b) a schematic representation of an SdFFF apparatus and of the separation of particles in the flow channel. A typical fractionation obtained through SdFFF using a polydispersed suspension of polystyrene latex spheres is also shown. (Adapted from Giddings 1991.)... 

For a polydisperse colloid (or for a monodisperse colloid that has evolved to a polydisperse suspension of aggregates of various sizes), we can develop an equation to calculate the concentrations of the aggregates as a function of time. For example, Equation (25) can be generalized to... 

According to this kinetic model the collision efficiency factor p can be evaluated from experimentally determined coagulation rate constants (Equation 2) when the transport parameters, KBT, rj are known (Equation 3). It has been shown recently that more complex rate laws, similarly corresponding to second order reactions, can be derived for the coagulation rate of polydisperse suspensions. When used to describe only the effects in the total number of particles of a heterodisperse suspension, Equations 2 and 3 are valid approximations (4). 

Polydisperse Suspensions. Polydisperse suspensions were approximated by mixtures of monodisperse suspensions. In order not to introduce any bias in the analysis, all the data from three different instruments were analyzed using the same program. The distribution functions were assumed to be a sum of equally spaced histograms for PCS, delta functions for TS and logarithmically spaced histograms for FDPA. The height and position of each parameter was optimized using a nonlinear least squares minimization process (15). 

K (D/Am, np/nm), the extinction coefficient, is in the general case a complicated function of the particle diameter D, the wavelength in the medium m, and the refractive indices np and nm of the particles and the medium, respectively. K can be calculated from the general Mie theory (3). For a polydisperse suspension ... 

Sedimentation in monodisperse and polydisperse suspensions effect of wall inclination. 

This result has been generalized (Batchelor, 1982 Batchelor and Wen, 1982) to a polydisperse suspension of settling spheres, obtaining for the average settling velocity us, of species i. 

The experimental data obtained by us, presented in this survey and compared with literature data show that the free volume is, presumably, a structure parameter characterizing most reliably the properties of highly-loaded suspensions. Using this parameter is most beneficial when we have to deal with a polydisperse suspension and it is difficult to describe it by means of a model. 

Acoustic methods offer several advantages when compared to other comparable techniques (1) applicable to concentrated suspensions (2) less sensitive to particulate contamination (3) better suited to polydisperse suspensions (4) applicable to a wide size range (5) well suited to automated potentiometric titrations and analysis... 

Dilute suspensions Subject to quenching at high colloid loads Little/no sample prep Poor resolution for polydisperse suspensions... 

In the absence of interactions, particles of differing sizes and shapes are statistically independent. For this reason, we can treat the statistical properties of light scattered from a dilute polydisperse suspension as the sum of contributions of many dilute monodisperse suspensions of particles with characteristic shape and size. Suppose that each characteristic shape/size combination is labeled with the index v. Let (V represent the number of particles having a particular shape and size. Clearly, we require N = Ns. All sums over the (V particles in a suspension can be expressed in terms of sums over the shape/size distribution. Thus we have... 

With a suspension containing a distribution of particle sizes, one cannot replace a by the average particle size, since the number of contact points N in a floe is greater than that calculated using an average particle size. With a polydisperse suspension it is difficult to calculate the number of contacts without knowledge of the exact particle size distribution. Since this was not available in the present system, no attempt was made to calculate Xg (and hence compare it with the experimental value). However, the trend in p p correlates very well... 

In addition, Strauss s work developed the means to determine the osmotic pressure for a polydisperse suspension corresponding to a lognormal size distribution. With this S5/stem a particular osmotic pressure, II(= 4cksT sinh [ezt/ (j8)/2A BT]), is assumed specifying a value of i/ (j8). The volume fraction is then calculated. For a particular particle size, a, the volume of fluid associated with the particle is determined by the specified value of t//(j8) and the boimdaiy conditions. With the outer boundary condition, dt/ /dr r=(3 = 0 and //(j8) = constant for a specific value of the osmotic pressure, the total volume firaction, can be determined by summation of the volume fraction associated with... 

For a monodisperse suspension the decay rate can be described by a first order rate equation. For a polydisperse suspension the decay rate is a sum of exponentials. Measurement of the decay rate permits computation of particle size [338]. 

Other problems in deriving a priori equations result from the polydisperse namre of pharmaceutical suspensions. The particle size distribution will determine rj. A polydisperse suspension of spheres has a lower viscosity than a similar monodisperse suspension. 

The maximum packing fraction cp. can be easily calculated for monodisperse rigid spheres. For an hexagonal packing

random packing (Pp = 0.64. The maximum packing fraction increases with polydisperse suspensions for example, for a bimodal particle size distribution (with a ratio of 10 1), 0.8. 

For polydisperse suspensions, the first-order autocorrelation function is an intensity-weighted sum of autocorrelation function of particles contributing to the scattering. 

Petkanchin I, Bruckner R, Sokerov S, Radeva Ts. Comparison of electric light scattering and birefringence for polydisperse suspensions. Colloid Polymer Sci 1979 257 160-165. 

Although the discussion of the high shear limit viscosity relations has centered on monomodal spherical suspensions, Probstein et al. (1994) have shown the applicability of Eq. (9.3.8) to bidisperse and polydisperse suspensions experimentally and on theoretical grounds. 

In this section we turn our attention to bidisperse suspensions, following which we will briefly discuss polydisperse suspensions. It is known that for a bidisperse suspension the relative viscosity decreases significantly in comparison to that of a monodisperse suspension of the same material and with the same solids volume fraction. 

It should be noted that Farris (1968) developed a bimodal model for polydisperse suspensions in which the fine and the coarse fractions were also assumed to behave independently of each other. However, the arguments were purely geometric and the issues related to the non-Newtonian character of the viscosity were not treated. 

The bimodal model has also been applied to polydisperse suspensions (Probstein et al. 1994), which in practice generally have particle sizes ranging from the submicrometer to hundreds of micrometers. In order to apply the bimodal model to a suspension with a continuous size distribution, a rational procedure is required for the separation of the distribution into fine and coarse fractions. Such a procedure has not been developed so that an inverse method had to be used wherein the separating size was selected which resulted in the best agreement with the measured viscosity. Again, however, the relatively small fraction of colloidal size particles was identified as the principal agent that acts independently of the rest of the system and characterizes the shear thinning nature of the suspension viscosity. 

Using the formulas (8.147) and (8.148), it is possible to determine experimentally the properties of infinite diluted suspensions containing same-sized particles (a monodisperse suspension), for example, the mass concentration and size of particles. If the suspension contains particles of different sizes (a polydisperse suspension), then dividing the entire spectrum of particle sizes from amin to amax into a finite number of fractions, it is possible to carry out the argumentation stated above for each fraction, and to determine the laws of motion for the corresponding discontinuity surfaces. Measuring the velocities of discontinuity surfaces in an experiment, it is possible to determine the characteristics of each fraction and thereby the size distribution of particles. 

It is probably the most widespread method it is based on the direct observation, with a suitable magnifying optics, of individual particles in their electrophoretic motion. In fact, it is not the particle what is seen but its scattering pattern when illuminated in a dark background field. It allows direct observation of particles in their medium and the observer can in principle select a range of sizes to be tracked in case of polydispersed suspensions [29,30], As the observations are possible only if the suspensions are dilute enough, even moderately unstable systems can be measured, as aggregation times are expectedly large for such dilute systems. 

Diameter, diameter of ith generation of particles in polydisperse suspensions... 

The experiment of Kumar et al (2000) consists of continuously feeding the polydisperse suspension through a vertical column in the well-mixed state and allowing the relative motion of particles to exit at an outlet located at a suitable distance from the point of entry. The relative motion of particles will have established a steady state, spatially uniform distribution of particles with an exit number density that can be measured by a device such as a Coulter counter. The population density, / (z, v) in vertical coordinate z and particle size described by volume v, satisfies the population balance equation... 

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