Equilibrium Mechanical Measurements

A typical evolution of equilibrium mechanical properties during reaction is shown in Fig. 6.1. The initial reactive system has a steady shear viscosity that grows with reaction time as the mass-average molar mass, M , increases and it reaches to infinity at the gel point. Elastic properties, characterized by nonzero values of the equilibrium modulus, appear beyond the gel point. These quantities describe only either the liquid (pregel) or the solid (postgel) state of the ma terial. Determination of the gel point requires extrapolation of viscosity to infinity or of the equilibrium modulus to zero. 

Accurate measurement of the equilibrium modulus is extremely difficult because its value remains below the detection limit for a considerable time and, theoretically, an infinite time is required to perform the measurement. 

Steady shear viscosity measurements are very simple and are often used in practice. Very often a viscosity of 10 Pas is arbitrarily identified with the gel point. But the determined gelation time, tgei, depends on the shear rate, and extrapolation to zero shear rate meets the following difficulties  

tgei may depend on the shear rate due to shear thinning at high rates. 

The network structure near the gel point is very fragile and can be broken by the shear flow experiment. 

---

The swelling and equilibrium mechanical measurements were the basic methods used for determination of the appearance and magnitude of the phase transition. Swelling of the gels (prepared in the form of cylinders) was characterized by the swelling ratio [11]... 

In summary, therefore, solution and fiber biochemistry have provided some idea about how ATP is used by actomyosin to generate force. Currently, it seems most likely that phosphate release, and also an isomerization between two AM.ADP.Pj states, are closely linked to force generation in muscle. ATP binds rapidly to actomyosin (A.M.) and is subsequently rapidly hydrolyzed by myosin/actomyosin. There is also a rapid equilibrium between M. ADP.Pj and A.M.ADP.Pj (this can also be seen in fibers from mechanical measurements at low ionic strength). The rate limiting step in the ATPase cycle is therefore likely to be release of Pj from A.M.ADP.Pj, in fibers as well as in solution, and this supports the idea that phosphate release is associated with force generation in muscle. 

When determining the solubility and dissolution rate of amorphous or partially crystalline solids, the metastability of these phases with respect to the highly crystalline solid must be considered. While the low diffusivity of the molecules in the solid state can kinetically stabilize these metastable forms, contact with the solution, for example during measurements of solubility and dissolution rate, or with the vapor, if the solid has an appreciable vapor pressure, may provide a mechanism for mass transfer and crystallization. Less crystalline material dissolves or sublimes whereas more crystalline material crystallizes out. The equilibrium solubility measured will therefore approach that of the highly crystalline solid. The initial dissolution rate of the metastable form tends to reflect its higher... 

Hersey and Robinson also foundthat many guest species that show kinetic behavior apparently explicable in terms of a single-step binding, give a discrepancy between the values of the equilibrium constant determined kinetically and those determined from equilibrium studies. It was found that the equilibrium constant, deterrmned spectrophotometrically, was usually greater than the ratio of the forward and backward rate-constants, determined kinetically. They therefore suggested that this discrepancy could be adequately explained if the two-step mechanism just described was used to interpret the results. A similar proposal has also been made by Hall and coworkers, who observed a large discrepancy between AV° values for the inclusion of 1-butanol and 1-pentanol by alpha cyclodextrin, calculated from equilibrium-density measurements and kinetic, ultrasonic-absorption data. 

This chapter describes recent work in our laboratories examining density modification of DNAPLs through a combination of batch non-equilibrium rate measurements and DNAPL displacement experiments in 2D aquifer cells. The objective of this work was to evaluate the applicability of nonionic surfactants as a delivery mechanism for introducing hydrophobic alcohols to convert the DNAPL to an LNAPL prior to mobilizing the NAPL. Three different nonionic surfactants were examined in combination with n-butanol and a range of DNAPLs. Overall, it was found that different surfactants can produce dramatically different rates of alcohol partitioning and density modification. However, for some systems interfacial tension reduction was found to be a problem, leading to unwanted downward... 

This brings up the question of how this scheme has to be modified when equilibrium is not attained. The answer is that the identity of thermod3mamical and mechanical measurement persists, but that the value obtained for y differs from y (eq.) in fact, often y (non-eq.) > y (eq.). Suppose a given interface is created very rapidly and then starts to relax with a time scale r. At any time t equilibrium state, characterized by the pertaining y (non-eq.). Even then, this non-equilibrium surface tension can be measured, provided the time-scale of our measurement is short in comparison with t. When different methods of measurement, either thermodynamical or mechanical, do not yield the same y this may either mean that there have been errors in the measurement or that they apply to different moments (or time intervals) of the relaxation period. The downward tendency of y(f) reflects the general trend of F(V,T,n) and G(p,T, n) to become minimal at equilibrium (sec. 1.2.12). When only relaxation of the interface takes place, y must decrease. However, when the bulk phases also relax slowly or when the relaxation is determined by adsorption-desorption processes, y may also increase. For instance, this would be observed if... 

With chemical relaxation methods, the equilibrium of a reaction mixture is rapidly perturbed by some external factor such as pressure, temperature, or electric-field strength. Rate information can then be obtained by following the approach to a new equilibrium by measuring the relaxation time. The perturbation is small and thus the final equilibrium state is close to the initial equilibrium state. Because of this, all rate expressions are reduced to first-order equations regardless of reaction order or molecularity. Therefore, the rate equations are linearized, simplifying determination of complex reaction mechanisms (Bernasconi, 1986 Sparks, 1989),... 

The DTA curves show no evidence for a Tp mechanism in the case of liquid selenium. The specific heat at constant pressure, Cp, vs. T curve of liquid selenium shows a minimum near 500°C . The increase after this minimum is ascribed to depolymerization, which is endothermic. However, care must be exercized in determining the polymer-ring equilibrium mechanism from the heat capacity measurements. The following contribute to the specific heat ... 

The upper-part of Table 3 shows the rapid equilibrium mechanisms and the lower-part the steady-state mechanisms. At the end of the table is the Steady-State Random Bi Bi mechanism that is included for comparison. The steady-state random case in practice gives the same patterns as rapid equilibrium ones one can usually teU the difference only by differential rates of isotopic exchange or the measurement of stickiness. 

The Epons 828,1001,1002,1004, and 1007 fully cured with stoichiometeric amounts of DDS are examples of well-characterized networks. Therefore, mechanical measurements on them offer insight into the viscoelastic properties of rubber networks. The shear creep compliance J t) of these Epons were measured above their glass temperatures [11, 12, 14]. From the statistical theory of rubber elasticity [1-5, 29-33] the equilibrium modulus Ge is proportional to the product Tp, where p is the density at temperature T, and hence the equilibrium compliance is proportional to (Tpy Thus J t) is expected to be proportional to and J(t)Tp is the quantity which should be compared at different temperatures. Actually the reduced creep compliance... 

Results of dynamic mechanical measurements are shown in Fig. 42 none of the gels show a real equilibrium plateau for G, so a real permanent network is not obtained. Moreover, the gels show thixotropic (two phase) behaviour, which follows from the observations that the gels are liable to a continuous stress relaxation upon applying a constant deformation and that the linearity of the dynamic moduli already disappears at strains of less than 1% the non-linear response, however, is almost instantaneous and, by turning back into the linear region, the initial values are quickly regained. These phenomena are typical for two-phase systems and especially for thixotropic ones. 

A mechanism is a step-by-step map that shows the presence or absence of intermediates and the total number of steps required to convert the string materials to the products 13, 36, 37, 38. Reversible chemical reactions have an E t (reverse) that is equal to or less than E ct (forward), for the forward and reverse reactions. The position of the equilibrium is measured by the equilibrium constant K, defined as products divided by reactants 14, 33. In ionic reactions, an atom with an excess of electrons (- or 8-) donates two electrons to an atom that is electron deficient (+ or 8+) to form a new bond. When this occurs, a bond in the original molecule must break, with transfer of the two electrons in that bond to one of the atoms in the starting material 40, 41. 

The first step is a disrotatory cyclohexadiene-hexatriene isomerization. Its product, cf5-dihydrobenzocyclooctatetraene, is less stable than the trans dimer and is known to isomerize to it, [27] the isomerization presumably taking place via an a" displacement that reduces symmetry to Ci, in which no reaction is forbidden. At the higher temperatures at which fragmentation occurs, the first product should be in equilibrium with the reactant, and its eight-membered ring is sufficiently flexible that a similar desymmetrization would allow it to serve as an unstable intermediate. The activation parameters cited above, which - for the postulated mechanism - measure the enthalpy and entropy differences between the reactant and the transition state of the second step, are not inconsistent with concerted electrocyclic rupture of both bonds via a relatively unconstrained transition state. 

An unfamihar aspect of the mechanical properties of gels is that they will tend to lose water under compression and take up water under tension. As a result, the mechanical properties will be different depending on the test speed. Thus the fast modulus of cartilage is 2.5 MPa while the equilibrium modulus, measured as water is displaced from the structure, is about 0.7 MPa [8, 12]. The transition between these two values will depend on sample size, as the water has to flow out of the gel. Likewise, testing under water will result in properties that differ from those measured in air. 

---