Crowding volume fractions

Crowding Volume Fractions vc, Crossover Molecular Weights Mbt and Crossover End-to-End Distances lb as Functions of the Salt Concentration c and Pure Aqueous d-Value d0... 

One characteristic of shear banded flow is the presence of fluctuations in the flow field. Such fluctuations also occur in some glassy colloidal materials at colloid volume fractions close to the glass transition. One such system is the soft gel formed by crowded monodisperse multiarm (122) star 1,4-polybutadienes in decane. Using NMR velocimetry Holmes et al. [23] found evidence for fluctuations in the flow behavior across the gap of a wide gap concentric cylindrical Couette device, in association with a degree of apparent slip at the inner wall. The timescale of these fluctuations appeared to be rapid (with respect to the measurement time per shear rate in the flow curve), in the order of tens to hundreds of milliseconds. As a result, the velocity distributions, measured at different points across the cell, exhibited bimodal behavior, as apparent in Figure 2.8.13. These workers interpreted their data... 

The interior of the living cell is occupied by structural elements such as microtubules and filaments, organelles, and a variety of other macromolecular species making it an environment with special characteristics [97], These systems are crowded since collectively the macromolecular species occupy a large volume fraction of the cell [98, 99]. Crowding can influence both the... 

We focus on the effects of crowding on small molecule reactive dynamics and consider again the irreversible catalytic reaction A + C B + C asin the previous subsection, except now a volume fraction < )0 of the total volume is occupied by obstacles (see Fig. 20). The A and B particles diffuse in this crowded environment before encountering the catalytic sphere where reaction takes place. Crowding influences both the diffusion and reaction dynamics, leading to nontrivial volume fraction dependence of the rate coefficient fy (4>) for a single catalytic sphere. This dependence is shown in Fig. 21a. The rate constant has the form discussed earlier,... 

The mean-field theory has a number of shortcomings, including the approximations of a mean concentration around all particles and the establishment of spherically symmetric diffusion fields around every particle, similar to those that would exist around a single particle in a large medium. The larger the particles total volume fraction and the more closely they are crowded, the less realistic these approximations are. No account is taken in the classical model of such volume-fraction effects. Ratke and Voorhees provide a review of this topic and discuss extensions to the classical coarsening theory [8]. 

If the internal phase in an emulsion has a sufficiently high volume fraction (typically anywhere from 10 to 50%) the emulsion viscosity increases due to droplet crowding, or structural viscosity, and becomes non-Newtonian. The maximum volume fraction possible for an internal phase made up of uniform, incompressible spheres is 74%, although emulsions with an internal volume fraction of 99% have... 

Since wet foams contain approximately spherical bubbles, their viscosities can be estimated by the same means that are used to predict emulsion viscosities. In this case the foam viscosity is described in terms of the viscosity of the continuous liquid phase (tjo) and the amount of dispersed gas (4>). In dry foams, where the internal phase has a high volume fraction the foam viscosity increases strongly due to bubble crowding, or structural viscosity, becomes non-Newtonian, and frequently exhibits a yield stress. As is the case for emulsions, the maximum volume fraction possible for an internal phase made up of uniform, incompressible spheres is 74%, but since the gas bubbles are very deformable and compressible, foams with an internal vol-... 

In describing the viscosity of an emulsion, the volume fraction of the dispersed phase is the most important parameter. A model suggested by M. Mooney (15) in 1951 for solid suspensions and emulsions with highly viscous dispersed phase described the relative viscosity as a function of volume fraction and a coefficient, k, called the self-crowding factor. 

In dry foams, where the internal phase has a high volume fraction, the foam viscosity increases strongly, because of bubble crowding or... 

The foam of everyday experience, though different in several ways from the foam that occurs and is used in reservoirs, is worthy of some examination. Such everyday foam is a two-phase mixture of gas and liquid, in which the liquid is the continuous fluid and the gas is held in separate cells. See also the discussions in Chapters 1—3 of this book.) To display the distinctive foamlike characteristics, the volume fraction of the discontinuous phase must be greater than about 70%. At this high gas volume fraction (the so-called quality), the bubbles of gas are closely crowded together so that they cannot move independently. They also change in shape, and the walls of the cells become approximately planer, polygonal surfaces that are called lamellae or bubble-films. 

There is, however, an excluded volume effect since when the particle fraction was added, not all the fluid volume was available, part being already taken up by the particle fraction ,. Krieger and Dougherty therefore assumed that the volume fraction available to the new particles was only 1 — k(f>, where is a constant termed a crowding factor. It follows that Eq. (9.3.2) should be written... 

The HCP model implies that in diluted systems ( < 0.005, where exfoliated clay platelets may freely rotate), individual HCPs are dispersed in a polymeric matrix and values of the interaction parameters are constant. As the concentration increases, the domains of reduced mobility around HCPs begin to overlap, macromolecules with bulk properties disappear, and the interactions change with clay content. Above the encompassed clay platelet volume fraction, rot = Q.99 p 0.005, there is a second critical concentration, Wmax 3.6 wt% or (/>max 0.015, at which the clay platelets with adsorbed solidified organic phase begin to overlap. Due to platelet crowding, CPNC approaching this concentration forms stacks thus, the assumption that individual exfoliated platelets are present is no longer valid. 

As a result, this equation is usually the only one needed for liquid or solid aerosols. Figure 6.18 shows several sets of experimental data compared with the Einstein equation. In practice once cp reaches between 0.1 and 0.5, dispersion viscosity increases significantly and can also become non-Newtonian (due to particle/droplet/bubble crowding or structural viscosity). The maximum volume fraction possible for an internal phase made up of uniform, incompressible spheres is 0.74, although emulsions and foams with an internal volume fraction of over 0.99 can exist as a consequence of droplet/bubble distortion. Figure 6.18 and Equation 6.33 illustrate why volume fraction is such a theoretically and experimentally favoured concentration unit in rheology. In the simplest case, a colloidal system can be considered Einsteinian, but in most cases the viscosity dependence is more complicated. 

The latter form is a good approximation for any 0> Oq and h/R 1. In most foams, the effect is expected to be minimal, as the bubbles tend to be relatively large. For emulsions of small drop size, however, the effect may be considerable and the peculiar properties resulting from extreme crowding may commence at an apparent volume fraction that is considerably smaller than one would expect for zero film thickness. For example, in an emulsion with droplets of 2/ - 1 um and A = 50 nm, the effective volume fraction already reaches a value of 0.74 at an apparent volume fraction of only about 0.64 The finite film thickness... 

Tanner et al. (2010a) have extended the above results to concentrated regimes by using the Roscoe procedure (Roscoe 1952, also see Phan-Thien and Pham 2000). In concentrated suspensions, some of the fluid is trapped between particles, and hence Roscoe (1952) suggested that the increment of small amount of volume fraction d(p results in an effective increase of concentration of d(p/ — (p/(p ), which is called the crowding function, where (j is the maximum volume fraction. We use N ) as an example to describe the procedure. From Eq. 5.61, one has... 

Table 10.1 shows reasonable values for the crowding factor k. However, a is much larger than the theoretical value of 2.5. This discrepancy is due to the presence of the surfactant layer, which causes an increase in the core volume fraction. 

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