Elastomer calculations

Fig. 38 Activation energy for thermal oxidation of some butadiene elastomers calculated from different kinetic parameters. The data were taken from [94J3]. (dark grey) 1,4 butadiene rubber (95 wt.% cw-isomer) (pale grey) styrene butadiene rubber (23.5 wt.% st5rrene), (white) butadiene acrylonitrile rubber (38 wt.% acrylonitrile).

The calculation of equilibrium swelling of SPU-1 (after its deformation up to 700 % and subsequent unloading) in a given solvent was carried out using the Flory - Rehner equation. In oiu calculations we used values of the %i parameter of interaction between solvents and elastomer calculated from experimental data of equilibrium swelling of initial sample ( =0) in these solvents. 

The classic model that describes chain scission in elastomers was proposed many years ago by Lake and Thomas [26J. The aim of the model is to calculate the energy dissipated in breaking all the polymer strands that have adjacent cross-links on either side of the crack plane. The basic assumption of this model is that all the main chain bonds in any strand that breaks must be strained to the dissociation... 

The nitrile rubber used for the experimental studies contained 40% acrylonitrile and the values of k, I, m, and n for the elastomer were calculated from spectroscopic studies as 359, 559, 215, and 770, respectively. 

Further to aspects discussed in Section 23.5.1.2, the simplest strategy, if it is practical, is slow decompression. If the pressure can be reduced sufficiently slowly, the gas which is dissolved in the elastomer can diffuse out of the mbber without buUding up sufficient internal pressure to cause damage. The rate of decompression required may be roughly calculated if the diffusion coefficient of the gas and its solubility in the elastomer are known— but in practice it will probably require a laboratory simulation. 

The technique used to study dewetting dynamics on materials consists of making a flat, smooth elastomer surface. A hquid puddle is deposited within a 50-mm-diameter ring of 0.1-mm-thick plasticized adhesive paper adhering to the substrate. The adhesive paper acts as a spacer. A microscope slide is drawn over the liquid to obtain a liquid film of ca. 0.1-mm thickness. At this thickness, the liquid film is unstable, being much less than the equilibrium value, of ca. 1.5 mm calculated from Eq. (29). Nucleation of dry patches... 

The reactor was charged with 37 elastomer which was Insoluble In the chromatographic mobile phase. Thus, the relationship between calculated and measured solids contents should have, and did, differ by at least 37 at low conversions. Expected formation of Insoluble graft polymer would also have Influenced relationship between calculated and measured total solids. 

Because of the improvements in properties exhibited by elastomers having bimodal distributions [5], there have been attempts to prepare and characterize "trimodal" networks [56]. The calculations suggest that adding a small amount of very high molecular weight end-linkable polymer could further improve mechanical properties. 

Additional insights into the dynamics and structure of bimodal elastomers have been obtained by dynamic light-scattering experiments [129], neutron scattering experiments [130] and calculations [131], dual cross-linking system experiments [132], non-affine swelling [133], and the computer simulations already mentioned. 

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

CALCULATED DIMENSIONAL CHANGES IN POLYMER CHAINS IN STRETCHED ELASTOMERS... 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

Preliminary Considerations. Table II gives the mole/kg of the components used in preparing the LHT-240 and Tri-NCO elastomers. The values are based on the assumption that the concentrations of hydroxyl and double-bond groups in the PPG are in fact those obtained from the analyses already discussed and that the NCO/OH ratio is 1.00 instead of 1.02, listed in Table I. In reality, this ratio may have been essentially unity because, as mentioned already, the degree of swelling in benzene and the sol fraction were found to be minima when the nominal ratio was 1.02. The results obtained from the calculations discussed subsequently, and summarized in Table HI, do not differ significantly from those obtained if the NCO/OH ratio is equated to 1.02. 

Equation A1 in the Appendix shows that P is a function of p, and thereby the sol fraction (eq A2) depends solely on p and certain mole and weight fractions given by the a s and w s (Table II). From the determined minima values of the sol fraction (Table I), p and Pxi for the LHT-240 and the Tri-NCO elastomers were first evaluated by an iterative process, and then G ax, vc, and t>e were obtained from eqs 4 and 5. These quantities are given in Table III. Those for the TIPA elastomer are not included because its composition is somewhat questionable, as mentioned earlier. However, if its initial ingredients were in fact 1.00 x 10 4 mole/g of TIPA, PPG, and a stiochiometric amount of TDI, the calculated values of G ax, 104pc, and 104pg are 0.350 MPa, 0.966, and 0.741, respectively. 

Though somewhat invalid, the calculation given under Preliminary Considerations illustrates the sensitive of the modulus to the precise functionality of the PPG. However, if the mole fraction of monohydroxy PPO in the PPG is considered, for illustration, to be 0.040 instead of 0.066, the calculated values of G ax, 104 >c, and 104Re for the LHT-240 elastomer are 0.318 MPa, 0.789, and 0.551, respectively. These values differ only slightly from those in Table III. Such results because, both p and P must become smaller with a decrease in the amount of monohydroxy PPO so that the calculated sol fraction will agree with that observed. 

The conclusions reached are valid qualitatively if the amount of monohydroxy PPO is less than that obtained from the chemical analyses of the PPG. In fact, if no monohydroxy PPO were present, then the calculated values of vcRT and G ax for the LHT-240 elastomer become 0.13 MPa and 0.48 MPa, respectively. Such results because, to obtain the observed sol fraction (0.041), the extent of reaction must be 0.9728 instead of 0.9859 (Table III). Similarly, if the true values of the sol fraction are somewhat larger than those found, calculations then show that vcRT is less and G J13 is larger than those reported in Table III. 

Figure 17 Calculated stress-relaxation behavior at 298 K for five uncross-linked elastomers of M = 200,000 EP, ethylene-propylene (56 44) styrene-butadiene (23.5 76.5), SB natural rubber, N butyl and dimethyl siloxane.

Figure 19 Comparison of calculated (curves) versus experimental (points) modulus results (or the same elastomers shown in Figure 17. M assumed 200,000 in each case / = 298 K. The values of v, x 10 are SBR, 100 butyl, 113 silicone, 38 and hevea, 168.

The ratio of shear stress to shear strain. A property which determines the rate at which elastomers stiffen as the temperature is lowered. The force required to twist the test piece through 90° is measured at each temperature and the modulus calculated from a formula. 

The breaking stress in tension of an elastomer expressed in Mpa, kg/cm2 or lb/in2. It is calculated on the original (unstressed) area of cross-section of the test specimen. 

Table II gives our description for each occurring methylene sequence in terms of conditional probabilities. We take a measured nmr spectrum of an ethylene propylene elastomer and derive a set of six reaction probabilities that fit it best. For possible sets of probability values, a spectrum is calculated and a sum of weighted squares of errors is found. The set of probability values having the smallest sum of weighted squares of error is used as a starting point to minimize the error using the...

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