Fractional linear solid

A pseudo solid-like behavior of the T2 relaxation is also observed in i) high Mn fractionated linear polydimethylsiloxanes (PDMS), ii) crosslinked PDMS networks, with a single FID and the line shape follows the Weibull function (p = 1.5)88> and iii) in uncrosslinked c/.s-polyisoprenes with Mn > 30000, when the presence of entanglements produces a transient network structure. Irradiation crosslinking of polyisoprenes having smaller Mn leads to a similar effect91 . The non-Lorentzian free-induction decay can be a consequence of a) anisotropic molecular motion or b) residual dipolar interactions in the viscoelastic state. 

Use of tie lines and the lever rule enable one to determine those compositions, as illustrated in Figure 4.6a for a binary system. The composition of the solid (Cs) as a function of the fraction of solid transformed (fs), assuming linear solidus and liquidus lines, is given by... 

Thus, the strain at limit of linearity decreases as a function of increasing volume fraction of solids for networks in the strong-link regime, while it increases for networks in the weak-link regime. 

Although experimental values of E can be obtained from the test data via the grade efficiency curve (from which X50 is determined—see chapter 3, Efficiency of Separation ), industrialists prefer to measure the fraction of solids unsedimented (1 — Ei) and plot this against the ratio of the measured flow rate and the calculated E value (which can be varied by changing the speed of rotation). This curve, which often comes out as a straight line on log-probability paper, is naturally a function of the size distribution of the feed but can be used to find the ratio of Q/IL for acceptable efficiency with the given feed material. Extrapolation of the data over the linear parts of the graph can be made with caution. 

Consider a molecular liquid with Newtonian behaviour (see Chapter 4) such as water, benzene, alcohol, decane, etc. The addition of a spherical particle to the liquid will increase its viscosity due to the additional energy dissipation related to the hydrodynamic interaction between the liquid and the sphere. Further addition of spherical particles increases the viscosity of the suspension linearly. Einstein developed the relationship between the viscosity of a dilute suspension and the volume fraction of solid spherical particles as follows (Einstein, 1906) ... 

Initial Specific Volume of Condensed Explosive Mass Fraction of Undecomposed Explosive Mass Fraction of Solid or Gaseous Component Linear Coefficient of Thermal Expansion... 

It is interesting that there is a qualitatively similar character in the plots of the dependence on Tg, and also on the fraction of polymer in the surface layer, v, on the filler concentration. The absence of a linear relation between Tg, in the filled system, and the amount of filler is usually explained by the fact that with an increase in the fraction of solid phase there is an aggregation of the solid particles and reduction in the effective surface of contact with the polymer phase. At the same time, we may expect there is a certain linear dependence of the properties of filled polymer (e.g., Tg and Vh/vi, o) on the fraction of polymer in the surface layer. Plots of these dependencies are shown in Figure 3.12. 

The measured electrical conductivity of the probe volume will vary as the solids concentration in the probe volume changes. Generally, the measured conductivity is linearly related to the volume fraction of solids over a wide range of solids concentrations ... 

Figure 9.2 Zero-shear viscosity versus weight-average molecular weight of linear (solid line) and long-chain-branched fractions of LDPE samples E and P (symbols dashed lines) at 150°C. From Mendelson etal. [2],...

The increase in viscosity with the volume fraction of solids in the suspension does not in itself imply any non-Newtonian behavior, as the stress can remain strictly linear in the shear rate (Guazzelli and Morris 2012). In fact, when we consider the hard-sphere suspension in a Newtonian liquid under Stokes flow conditions, the expectation is that the rheology should be Newtonian. This is a basic result of dimensional analysis since there is no intrinsic rate associated with either the fluid or the particles, the only rate is that set by the flow shear rate. 

C, is the concentration of impurity or minor component in the solid phase, and Cf is the impurity concentration in the hquid phase. The distribution coefficient generally varies with composition. The value of k is greater than I when the solute raises the melting point and less than I when the melting point is depressed. In the regions near pure A or B the hquidus and solidus hues become linear i.e., the distribution coefficient becomes constant. This is the basis for the common assumption of constant k in many mathematical treatments of fractional solidification in which ultrapure materials are obtained. 

Some negligible amounts of linear oligomers, obtained as a non-volatile, solid brown residue, were also formed on reaction of Illb with o-phenylenediamine these were almost completely soluble in toluene. No high molecular fraction was detected among the reaction products. 

Sorption. Capture of neutral organics by non-living particulates depends on the organic carbon content of the solids (9). Equilibrium sorption of such "hydrophobic" compounds can be described by a carbon-normalized partition coefficient on both a whole-sediment basis and by particle size classes. The success of the whole-sediment approach derives from the fact that most natural sediment organic matter falls in the "silt" or "fine" particle size fractions. So long as dissolved concentrations do not exceed 0.01 mM, linear isotherms (partition coefficients) can be used. At higher concentrations, the sorptive capacity of the solid can be exceeded, and a nonlinear Freundlich or Langmuir isotherm must be invoked. 

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