Volume fraction dependence

Figure 19. ( ) Rate coefficient /(<(>) as a function of the square root of the volume fraction O for a = 3 and kgT =. The solid fine is determined using Eq. (92), while the dashed line is obtained using a higher-order approximation to the volume fraction dependence. 

MPC dynamics follows the motions of all of the reacting species and their interactions with the catalytic spheres therefore collective effects are naturally incorporated in the dynamics. The results of MPC dynamics simulations of the volume fraction dependence of the rate constant are shown in Fig. 19 [17]. The MPC simulation results confirm the existence of a 4> 2 dependence on the volume fraction for small volume fractions. For larger volume fractions the results deviate from the predictions of Eq. (92) and the rate constant depends strongly on the volume fraction. An expression for rate constant that includes higher-order corrections has been derived [95], The dashed line in Fig. 19 is the value of /. / ( < )j given by this higher-order approximation and this formula describes the departure from the cf)1/2 behavior that is seen in Fig. 19. The deviation from the <[)11/2 form occurs at smaller values than indicated by the simulation results and is not quantitatively accurate. The MPC results are difficult to obtain by other means. 

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 volume fraction dependence of /cq/(4>) is plotted in Fig. 21b and shows that it increases strongly with 4>- Recall that this rate coefficient is independent of 4> if simple binary collision dynamics is assumed to govern the boundary layer region. The observed increase arises from the obstacle distribution in the vicinity of the catalytic sphere surface. When obstacles are present, a reactive... 

However some problems must be addressed prior to the use of Equation (3.66) from the data generated above. The first problem is that the ions in the diffuse layers must be included in the estimation of k. This becomes essential as soon as the particle-particle interactions become significant and so Equations (3.62) to (3.65) should contain a volume fraction dependence of k of the form given by Russel,30 for example for a symmetrical electrolyte ... 

The fiber continuity requirement for the taper follows from the assumption that all fibers are infinitely long and their number is constant. A mass balance gives the fiber-volume-fraction dependency along the x-axis as... 

Here volume fraction of component i in phase j at tie line k. The calculated volume fractions depend on the parameters, p. 

The Flory-Huggins parameter Xi was initially derived as an experimentally determined interaction enthalpy parameter which was supposed to be independent of poljnner concentration. However, experiments have shown that it is dependent on pol5mier concentration. Evans and Napper [40] have shown that the polymer volume fraction dependence of X can be given by... 

However, the model over predicted the shear rate for thickening and also failed to capture the volume fraction dependence of the critical shear rate (Chow and Zukoski, 1995a, 1995b). 

When 8 < Zf, two different modes of failure can occur depending on (jy (see Fig. 15.13). As noted, the strength of the composite a for small fiber volume fractions depends essentially on the matrix. When the matrix fails, all the load is transferred to the fibers, but, as there are not enough fibers to take the load, the composite will fail. The strength of the lamina can be written as... 

Figure 5.22 shows that the critical filler volume fraction depends on the structure of carbon black which is here characterized by DBF absorption. 

The critical volume fraction of the filler has a different application in the case of conductive materials. As the amount of conductive filler is increased, the material reaches a percolation threshold. Below the percolation threshold concentration, the electric conductivity is similar to that of matrix. Above the percolation threshold conductivity rapidly increases. Above the critical volume fraction of filler which is, in turn, a concentration above the percolation threshold, there is a rapid increase in conductivity. " The critical volume fraction depends on the type of filler and its particles size. For example, for silver powder, it ranges from 5 to 20 vol% for... 

Flocculated Systems. The viscoelastic responses of flocculated systems are strongly dependent on the suspension structure. The suspension starts to show an elastic response at a critical solid volume fraction of 0ct = 0.05 — 0.07, at which the particles form a continuous three-dimensional network (211-213). The magnitude of the elastic response for flocculated suspensions above 0ct depends on several parameters, such as the suspension structure, interparticle attraction forces and particle size, and shape and volume fraction. Buscall et al. (10) found that the volume fraction dependence of the storage modulus follows a power-law behavior. 

It has been shown (Evans and Napper, 1977) that the interaction between polymer chains, which display a concentration dependent interaction parameter, vanishes when x, = /( +1) for all positive integral values of i. This relationship sug ts that near to the 0-point Xt=h Z2 = l/3=(2/3)xi and X3 = U4=(i)xi- In these circumstances, we mi t guess that the simplest volume fraction dependence might be... 

If only a rough idea of the temperature dependence is required, the series expansion in equation (12.91) can be truncated after the first x i) tenn (i.e. any polymer volume fraction dependence is ignored). Under this assumption... 

One of the first studies of PNC physical aging was published by Lee andLichtenhan [1998] for epoxy containing w = 0 to 9wt% of polyhedral oligomeric silsesquiox-ane (POSS). The presence of POSS increased Tg and the relaxation time thus, the nanoflller slowed down the molecular dynamics. For amorphous polymers at Tpstructural cluster model. The cluster volume fraction depends on temperature ... 

The ratio of the critical volume fractions depends on the frequency. For instance, for neoprene latex, the critical thermal volume fraction is 10 times higher than the critical viscous volume fraction for 1 Mhz and only three times higher for 100 Mhz. 

Doherty RD, Srolovitz DJ, Rollett AD, Anderson MP (1987) On the volume fraction dependence of particle limited grain-growth. Scr Metall 21 675-679... 

Figure 7.31. Volume-fraction dependence of relative viscosity for a suspension of calcium carbonate dispersed with NaPA...

When the particles in suspension are non-spherical, the rotation of the particles due to Brownian motion results in an excluded volume, which is higher than the volume-fraction of the particles. As the degree of anisotropy increases, the effects become more dramatic. Figure 9.6 shows the volume-fraction dependence of the viscosity of silicon nitride, alumina and silicon carbide whisker (SiCw) suspensions. The experimental points were fitted to a modified Krieger-Dougherty equation ... 

The fibre volume fraction depends heavily on the method of manufacture. A uni directional composite may have a fibre volume fraction as high as 75%. However, this can only be achieved if all the fibres are highly aligned and closely packed. A more typical fibre volume fraction for uni directional composites is 65%. If the fibre configuration is changed to put fibres in other directions, then the maximum fibre packing is reduced further. A typical fibre volume fraction for bi-directional reinforcement (woven fibre) is 40% and a typical volume fraction for random in-plane reinforcement (chopped strand mat) is 20%. 

Figure 5.9 The CB volume fraction dependence of the electrical conductivity in a CB-polymer composite in which the CB particles are essentially spherical. The curves...

The free energy of the lamellar structure does not depend on the volume fraction of the f) phase, only on the lamellar spacing A. The free energy of the rodlike structure depends both on the rod spacing A and the volume fraction of the P phase. We can explicitly incorporate this volume fraction dependence by expressing how depends on A and r ... 

M. Watzlawek, G. Niigele, Self-diffusion coefficients of charged particles Prediction of nonlinear volume fraction dependence. Phys. Rev. E 56(1), 1258-1261 (1997). doi 10.1103/PhysRevE. 56.1258... 

For composites with well-bonded filler particles the variation of failnre stress on filler volume fraction depends greatly on the system concerned. For example stress may be independent of filler volume fraction, increase slightly with filler volume fraction or, more usually, exhibit a slight or moderate decrease followed by a recovery. 

As expected for a three dimensional swelling, the volume fraction depends linearly on and decreases to zero as... 

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