Power-law dependence

The JKR model predicts that the contact radius varies with the reciprocal of the cube root of the Young s modulus. As previously discussed, the 2/3 and — 1/3 power-law dependencies of the zero-load contact radius on particle radius and Young s modulus are characteristics of adhesion theories that assume elastic behavior. 

Upon comparison of Eqs. 29 and 36, it is readily apparent that both theories predict the same power law dependence of the contact radius on particle radius and elastic moduli. However, the actual value of the contact radius predicted by the JKR theory is that predicted by the DMT model. This implies that, for a given contact radius, the work of adhesion would have to be six times as great in the DMT theory than in the JKR model. It should be apparent that it is both necessary and important to establish which theory correctly describes a system. 

Rimai et al. [57] determined the power-law dependence of the contact radius on the substrate s Young s modulus for another quintessential JKR system that of a soda-lime glass particles on polyurethane substrates. They reported that the contact radius varied as with iua calculated to be 0.12 J/m. The results... 

There have been several theories proposed to explain the anomalous 3/4 power-law dependence of the contact radius on particle radius in what should be simple JKR systems. Maugis [60], proposed that the problem with using the JKR model, per se, is that the JKR model assumes small deformations in order to approximate the shape of the contact as a parabola. In his model, Maugis re-solved the JKR problem using the exact shape of the contact. According to his calculations, o should vary as / , where 2/3 < y < 1, depending on the ratio a/R. 

Other ideas proposed to explain the 3/4 power-law dependence include effects due to viscoelasticity, non-linear elasticity, partial plasticity or yielding, and additional interactions beyond simply surface forces. However, none of these ideas have been sufficiently developed to enable predictions to be made at this time. An understanding of this anomalous power-law dependence is not yet present. 

As is evident, there are several distinctive characteristics of adhesion-induced plastic deformations, compared to elastic ones. Perhaps the most obvious distinction is the power-law dependence of the contact radius on particle radius. Specifically, the MP model predicts an exponent of 1/2, compared to the 2/3 predicted by either the JKR or DMT models. 

The total van der Waals interaction potential is obtained by simply adding the individual contributions arising from the Keesom, Debye, and London interactions. Because the radial power-law dependencies of all these interactions vary as 1 /r, the total van der Waals interaction can be expressed simply as... 

The k p scheme has been used also for the study of transport across junctions connecting tubes with different diameters through a region sandwiched by a pentagon-heptagon pair [25]. In Junctions systems, the conductance was predicted to exhibit a universal power-law dependence on the ratio of the circumference of two CNTs [26]. An intriguing dependence on the magnetic-field direction was predicted also [27]. These newer topics will be discussed elsewhere. 

While at high densities we observe perfect exponential scaling of p x), at lower dilute densities (with sufficiently long chains ) one observes results consistent with Eq. (16b). The insert in Fig. 5 shows that the MWD at dilute densities agrees with the additional power-law dependence p x) oc in the limit of small x, confirming the theoretical predictions [33,34]. 

Here, is an effective overlap parameter that characterizes the tunneling of chaiges from one site to the other (it has the same meaning as a in Eq. (14.60)). T0 is the characteristic temperature of the exponential distribution and a0 and Be are adjustable parameters connected to the percolation theory. Bc is the critical number of bonds reached at percolation onset. For a three-dimensional amorphous system, Bc rs 2.8. Note that the model predicts a power law dependence of the mobility with gate voltage. 

We see that the bridge contribution also changes the power-law dependence (L a A T y), making it closer to experimental observation. In fact the resulting value of the theoretical exponent y is 0.69, to be compared with the experimental value 0.70 we point out that bridge becomes the main contribution to n(new) for small values of AT. 

Dynamic rheological measurements have recently been used to accurately determine the gel point (79). Winter and Chambon (20) have determined that at the gel point, where a macromolecule spans the entire sample size, the elastic modulus (G ) and the viscous modulus (G") both exhibit the same power law dependence with respect to the frequency of oscillation. These expressions for the dynamic moduli at the gel point are as follows ... 

FromEqs. (2.60), (2.61), one can deduce an approximate power-law dependence of specific reaction rates on temperature, Tv, since... 

Verify Eq. (2.62) for the power-law dependence of reaction rate on temperature. 

Dependence on Base Concentration. The dissolution rates of substituted PHHPs at different alkali concentrations are displayed 1n Figure 1 for seven different novolac resins. In each case, there appears to be a limiting concentration C0 below which the rate of dissolution 1s too slow to be measured 1n the experimental time scale. The ascending portion of the curve can be represented by a power law dependence of the rate on concentration C, eq.(1),... 

The time scale tG and the amplitudes hq from Eq. [56] are predicted by MCT to show a power law dependence on T — Tc. When one plots fCT and the amplitudes hq taken to the inverse of the predicted exponent versus temperature, one can directly find the critical temperature of MCT, T = 0.45, as shown in Figure 11. From the MCT analysis in the (3-regime, one also obtains the von Schweidler exponent, b = 0.75, and therefore all other exponents through Eqs. [52], [55], and [57]. Another test of MCT, which is suggested by the form of Eq. (56), is to plot the ratio130,131... 

HF (ethanol 50% HF, 1 1), in contrast, a power law dependence on current density J (in mA cm-2) over three orders of magnitude has been observed [Le5] ... 

It should also be noted that ternary and higher order polymer-polymer interactions persist in the theta condition. In fact, the three-parameter theoretical treatment of flexible chains in the theta state shows that in real polymers with finite units, the theta point corresponds to the cancellation of effective binary interactions which include both two body and fundamentally repulsive three body terms [26]. This causes a shift of the theta point and an increase of the chain mean size, with respect to Eq. (2). However, the power-law dependence, Eq. (3), is still valid. The RG calculations in the theta (tricritical) state [26] show that size effect deviations from this law are only manifested in linear chains through logarithmic corrections, in agreement with the previous arguments sketched by de Gennes [16]. The presence of these corrections in the macroscopic properties of experimental samples of linear chains is very difficult to detect. 

Now the functions for doing simple power law-dependent simulations are developed. The zero-shear viscosity, //o. is 1.268 x 10 Pa-s as shown by Fig. 3.22 and the viscosity data in Table 3.6. This holds for all shear rates in the plateau range. For the power law fit, the last six entries in Table 3.6 are used to develop a regression fit, and then the line is extrapolated back to lower shear rates. The regression fit is as follows ... 

The rheological properties change behavior, relative to more dilute solutions, above cp = 0.2, where non-Newtonian behavior is then exhibited. The power law dependence of rj on cp is in harmony with the Zimm rather than the Rouse model, which suggests that hydrodynamic interactions between these polymers, in a mean field sense, are important. Electrical properties also begin to deviate for Nafion solutions above cp = 0.2, and mechanical percolation is essentially the same for the sodium and acid forms. 

For diffusion-controlled gelation and low droplet fraction, qg follows a power-law dependence [27] ... 

The results for the equilibrium foulant film thickness are shown plotted on a log-log scale in Figures 4 to 6. The experimental points were reduced using Eqs. (18b) and (21). Absolute thicknesses are shown for Sf only to give some idea of the magnitudes involved. The use of absolute units does not, of course, affect the slope of the data, which within the experimental accuracy is seen to follow a straight line in these plots, indicating a power-law dependence on the independent variables. In order to reduce the data by means of Eq. (18b) it is necessary to know the behavior of the unfouled membrane flux V. The experiments showed... 

Figure 1. Per cent degradation (%D) as a function of Molecular Weight (MW) to determine onset of degradation (critical MW) and power law dependence of degradation on MW.

As a consequence of this model it is qualitatively easy to anticipate when degradation will occur if (M .)vx3c is known. That is, (Me)GPC,is (Mj.) Vise rough dependence of degradation on Mexp or m >j q will not be far from the correct result for PS or PIB if (M< )gPC i estimated correctly. At high shear stresses the (M(.)gPC ior u-Styragel is lower and the power law dependence in M is lower ( M ). A more exact description of these phenomena is currently under investigation, theoretically and experimentallj(15). 

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