Hard components

Polyolefins. In these thermoplastic elastomers the hard component is a crystalline polyolefin, such as polyethylene or polypropylene, and the soft portion is composed of ethylene-propylene rubber. Attractive forces between the rubber and resin phases serve as labile cross-links. Some contain a chemically cross-linked rubber phase that imparts a higher degree of elasticity. 

Iron carbide (3 1), Fe C mol wt 179.56 carbon 6.69 wt % density 7.64 g/cm mp 1650°C is obtained from high carbon iron melts as a dark gray air-sensitive powder by anodic isolation with hydrochloric acid. In the microstmcture of steels, cementite appears in the form of etch-resistant grain borders, needles, or lamellae. Fe C powder cannot be sintered with binder metals to produce cemented carbides because Fe C reacts with the binder phase. The hard components in alloy steels, such as chromium steels, are double carbides of the formulas (Cr,Fe)23Cg, (Fe,Cr)2C3, or (Fe,Cr)3C2, that derive from the binary chromium carbides, and can also contain tungsten or molybdenum. These double carbides are related to Tj-carbides, ternary compounds of the general formula M M C where M = iron metal M = refractory transition metal. 

The complex iron carbonitride is the hard component in steels that have been annealed with ammonia (nitrided steels). Complex carbonitrides with iron metals are also present in superaHoys in the form of precipitates. 

In thermoplastic polyurethanes, polyesters, and polyamides, the crystalline end segments, together with the polar center segments, impart good oil resistance and high upper service temperatures. The hard component in most hard polymer/elastomer combinations is crystalline and imparts resistance to solvents and oils, as well as providing the products with relatively high upper service temperatures. 

Moreover, e e+ pairs are accelerated in the strong magnetic field, which we assume to be present, increasing the hard component of the 7-ray spectrum, that is in coincidence with experimental findings. 

Covalently bonded substructures having compositions distinguishable from their surroundings are formed in multicomponent systems they are called chemical clusters. The adjective chemical defines covalency of bonds between units in the cluster. To be a part of a cluster, the units must have a common property. For example, hard clusters are composed of units yielding Tg domains. Hard chemical clusters are formed in three-component polyurethane systems composed of a macromolecular diol (soft component), a low-molecular-weight triol (hard component) and diisocyanate (hard component). Hard clusters consist of two hard... 

Fractions and functionalities of hard components in initial system. 

Changes in reactivities of functional groups of hard components (substitution effect). 

Any 37/-dimensional Cartesian vector that is associated with a point on the constraint surface may be divided into a soft component, which is locally tangent to the constraint surface and a hard component, which is perpendicular to this surface. The soft subspace is the /-dimensional vector space that contains aU 3N dimensional Cartesian vectors that are locally tangent to the constraint surface. It is spanned by / covariant tangent basis vectors... 

Using distribution (2.112) to evaluate the average of both sides of Eq. (2.110) with respect to fluctuations in the hard coordinates thus yields an effective force balance with soft and hard components... 

The preceding division of the hard components of the total nonhydrodynamic force F n(,x, in Eq. (2.121) between an elastic force F and a sum of constraint forces n(jX, is a matter of convention. Arbitrary shifts in the values of hard components of F could be absorbed into compensating shifts in the... 

The equivalence of Eqs. (2.133) and (2.136) for is a special case of a more general theorem relating inverses of projected tensors, which is stated and proved in the Appendix, Section B. Both Eqs. (2.133) and (2.136) yield tensors that satisfy Eq. (2.135), and that thus have vanishing hard components. The equivalence of the soft components of these tensors may be confirmed by substituting expansion (2.136) into the RHS of Eq. (2.132), expanding on the... 

More generally, contracting the transposed projection tensor with a force to its right (or the projection tensor with a force to its left) produces a constrained force given by the sum of the original force and the constraint force induced by it. Such constrained forces may have nonzero hard components, but, on contraction with induce velocities that do not. 

The biorthogonality and completeness relations presented above do not uniquely define the reciprocal basis vectors and mi a list of (3N) scalar components is required to specify the 3N components of these 3N reciprocal basis vectors, but only (3N) —fK equations involving the reciprocal vectors are provided by Eqs. (2.186-2.188), leaving/K more unknowns than equations. The source of the resulting arbitrariness may be understood by decomposing the reciprocal vectors into soft and hard components. The/ soft components of the / b vectors are completely determined by the equations of Eq. (2.186). Similarly, the hard components of the m vectors are determined by Eq. (2.187). These two restrictions leave undetermined both the fK hard components of the / b vectors and the Kf soft components of the K m vectors. Equation (2.188) provides another fK equations, but still leaves fK more equations than unknowns. Equation (2.189) does not involve the reciprocal vectors, and so is irrelevant for this purpose. We show below that a choice of reciprocal basis vectors may be uniquely specified by specifying arbitrary expressions for either the hard components of the b vectors or the soft components of m vectors (but not both). 

We now show, conversely, that for each projection tensor P j, there exists a unique set of corresponding reciprocal basis vectors that are related to P j, by Eq. (2.195). To show this, we show that the set of arbitrary numbers required to uniquely define such a projection tensor at a point on the constraint surface is linearly related to the set of fK arbitrary numbers required to uniquely specify a system of reciprocal vectors. A total of (3A) coefficients are required to specify a tensor P v- Equation (2.193) yields a set of 3NK scalar equations that require vanishing values of both the hard-hard components, which are given by the quantities n P = 0, and of the fK mixed hard-soft ... 

Because appears contracted with in the equation of motion, the hard components of have no dynamical effect, and are arbitrary. The values of the soft components of F depend on the form chosen for the generalized projection tensor, and reduce to the metric pseudoforce found by Fixman and Hinch in the case of geometric projection. 

Note that the form of the projection tensor P depends on the form chosen for the hard components of Z v Specifically, values of the mixed soft-hard components of P, , which are not specified by the definition of a generalized projection tensor given in Section VIII, are determined in this context by the values chosen for the mixed components of Z v, which specify correlations between hard and soft components of the random forces that are not specified by Eq. (2.295) for Z v... 

The results given above are essentially identical to those obtained by Hinch [10] by a similar method, except for the fact that Hinch did not retain any of the terms involving the force bias (tIv)o which he presumably assumed to vanish. An apparent contradiction in Hinch s results may be resolved by correcting his neglect of this bias. In a traditional interpretation of the Langevin equation as a limit of an underlying ODE, the bead velocities are rigorously independent of the hard components of the random forces, since the random forces in Eq. (2.291) appear contracted with K , which has nonzero components only in the soft subspace. Physically, the hard components of the random forces are instantaneously canceled by the constraint forces, and thus can have no effect... 

In either interpretation of the Langevin equation, the form of the required pseudoforce depends on the values of the mixed components of Zpy, and thus on the statistical properties of the hard components of the random forces. The definition of a pseudoforce given here is a generalization of the metric force found by both Fixman [9] and Hinch [10]. An apparent discrepancy between the results of Fixman, who considered the case of unprojected random forces, and those of Hinch, who was able to reproducd Fixman s expression for the pseudoforce only in the case of projected random forces, is traced here to an error in Fixman s use of differential geometry. 

In the traditional interpretation of the Fangevin equation for a constrained system, the overall drift velocity is insensitive to the presence or absence of hard components of the random forces, since these components are instantaneously canceled in the underlying ODF by constraint forces. This insensitivity to the presence of hard forces is obtained, however, only if both the projected divergence of the mobility and the force bias are retained in the expression for the drift velocity. The drift velocity for a kinetic interpretation of a constrained Langevin equation does not contain a force bias, and does depend on statistical properties of the hard random force components. Both Fixman and Hinch nominally considered the traditional interpretation of the Langevin equation for the Cartesian bead coordinates as a limit of an ordinary differential equation. Both authors, however, neglected the possible existence of a bias in the Cartesian random forces. As a result, both obtained a drift velocity that (after correcting the error in Fixman s expression for the pseudoforce) is actually the appropriate expression for a kinetic interpretation. 

We now show that the same drift velocity is obtained from a Stratonovich interpretation of the Langevin equation with unprojected and projected random forces that have the same soft components but different hard components. We consider an unprojected random force ri (t) and a corresponding projected random force Tj (f) that are related by a generalization of Eq. (2.300), in which... 

Equation (A.99) may, however, also be obtained by explicitly using projected random forces, as Hinch did in order to reproduce Frxman s result. The use of the projected friction tensor for Z v in Eq. (2.306), rather than the unprojected tensor has the same effect in that equation as did Fixman s neglect of the hard components of because the RHS of Eq. (2.306) depends only on the dot... 

The shape of the spectra A and B would hardly be explicable in terms of a single component. In addition, while these two spectra are significantly different from each other below 15 keV, they are essentially the same above 15 keV. These facts suggest that there are two separate components, a soft component and a hard component, of which the hard component remains essentially constant independent of the soft component. This explains the little intensity change observed in the hard X-ray band. Besides, a flux minimum in the 10 15 keV range in the spectrum A suggests that the hard component is cut off below 20 keV. 

We attempt to determine the spectral shape of the hard component by assuming the form E-1 x expf-nN, ). However, the assumed power-law form is not critical for the present energy range limited below 40 keV. The power-law E-1 was simply chosen for a qualitative representation of the result of the Kvant observations in the higher... 

In order to determine the absorption column Nn, the above model was fitted to the average spectra. As the result, the spectra A and B gave Nu-values both approximately equal to 102K H atoms/cm2 for the cosmic abundances of element. The spectrum C is insensitive to determine the N -value uniquely. However, we assume the same form of the hard component also during the January flare, because the hard component is likely to be independent of the soft component. We therefore fixed the Nn at 102S H atoms/cm2, and performed fitting to the spectrum C. The intensity of the hard component was dealt with as a free parameter. 

Satisfactory fit was obtained for all cases. As the result, the intensity of the hard component turned out to be the same for all three spectra within the errors. This implies that the hard component remained essentially constant for more than 200 days. The temperature kT of the soft component is found to be about 10 keV for the spectra A and B but is higher than 50 keV for the spectrum C. The count rate increase in the hard X-ray band (16 - 28 keV) during the January flare (see Fig. 3) is thus interpreted as due to an enhanced contribution of the hardened soft component. The luminosity of the soft component reached 1038 ergs/sec at the flare peak, for the assumed distance of 50 kpc. 

---