Sum properties

Properties of composites can be more easily adjusted within wide ranges than those of simple solid compounds because the behavior of composites depends on how much each component contributes and how the components are made to cooperate in the composite. These are fixed at the preparation stage. Two types of composite properties are recognized sum properties and product properties. 

Sum properties of composites are a function of the corresponding properties of the components. The relation between the sum property values and the property values of the components is not necessarily simple or linear, and it is definitely morphology-dependent. Four examples of sum properties are  

Ferroelectrics such as BaTi03 usually have a high dielectric constant 8, which decreases with increasing high external electric fields because of saturation. A composite of this ferroelectric with antiferroelectric NaNbOa changes its saturation behavior and with a suitable composition and sintering process. 

Rgure 9.1. Microwave lead-through insulator of porous glass for a vacuum reactor. The dielectric constant n can be made low by making a glass-air composite with closed porosity. Adapted with permission from the Journal of Materials Education. 

The dielectric constant of parallel plates of two dielectrics with different s/s 

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Assume ideal gas at final T P. Sum property changes for the process. For the initial T P ... 

Moreover, a special case of autocorrelation descriptors is the Atom-Type Autocorrelation (ATAC), which is calculated by summing property values only of atoms of given types. The simplest atom-type autocorrelation is given by... 

Thus sum properties are made up of the same properties of the individual components the dielectric constant of the composite is some function of the dielectric constants of the components the velocity of sound in the composite depends on the velocity of sound in the matrix and reinforcement phases. This is not true for product properties. 

Sum properties of disordered materials can be expressed in terms of compound properties and their volume fractions. As an example, the conductivity (electric or thermal) of a dispersion of a phase in a matrix is given in terms of the component conductivities. For inhomogeneous solids a general empirical equation is used to... 

Effective material properties (Newnham 1986) could exhibit values either in the range given by the material coefficients of both phases (sum property), or lie beyond that range (combination property). As an example of the combination property could serve the effective hydrostatic charge coefficient for the parallel cormection of phases. The effective hydrostatic charge coefficient could be calculated as... 

Sum properties are those [4,1632] in which the X-Y effect of the composite is determined by the X-Y effects b pha 1 and 2. As an example, the stiffiiess of a composite is determined by the elakk properties of the oomponent phases and by the mixing rule, which depend on the geometry of the phases. Similarly, permittivity c of the composite consisting of two phases with a, and Cj b determined by the sum... 

A sum property is one where the composite coefficient depends on the corresponding coefficients of the individual phases. For example, the permittivity of a two-phase (or diphasic) composite is given by... 

Calculation of thermophysical properties of gases relies on the principle of corresponding states. Viscosity and conductivity are expressed as the sum of the ideal gas property and a function of the reduced density ... 

The essential property that we use, is the transformation of the product of convolution in a sum. 

The last identity follows from the orthogonality property of eigenfunctions and the assumption of nomralization. The right-hand side in the final result is simply equal to the sum over all eigenvalues of the operator (possible results of the measurement) multiplied by the respective probabilities. Hence, an important corollary to the fiftli postulate is established ... 

These moments are related to many physical properties. The Thomas-Kulm-Reiche sum rule says that. S (0) equals the number of electrons in the molecule. Other sum rules [36] relate S(2),, S (1) and. S (-l) to ground state expectation values. The mean static dipole polarizability is md = e-S(-2)/m,.J Q Cauchy expansion... 

Note that the sums are restricted to the portion of the frill S matrix that describes reaction (or the specific reactive process that is of interest). It is clear from this definition that the CRP is a highly averaged property where there is no infomiation about individual quantum states, so it is of interest to develop methods that detemiine this probability directly from the Scln-ddinger equation rather than indirectly from the scattering matrix. In this section we first show how the CRP is related to the physically measurable rate constant, and then we discuss some rigorous and approximate methods for directly detennining the CRP. Much of this discussion is adapted from Miller and coworkers [44, 45]. 

Shen Y R 1989 Surface-properties probed by second-harmonic and sum-frequency generation Nature 337 519-25... 

Atomistically detailed models account for all atoms. The force field contains additive contributions specified in tenns of bond lengtlis, bond angles, torsional angles and possible crosstenns. It also includes non-bonded contributions as tire sum of van der Waals interactions, often described by Lennard-Jones potentials, and Coulomb interactions. Atomistic simulations are successfully used to predict tire transport properties of small molecules in glassy polymers, to calculate elastic moduli and to study plastic defonnation and local motion in quasi-static simulations [fy7, ( ]. The atomistic models are also useful to interiDret scattering data [fyl] and NMR measurements [70] in tenns of local order. 

An R-matrix has a series of interesting matheinatical properties that directly reflect chemical laws. Thus, the sum of all the entries in an R-matrix must be zero, as no electrons can be generated or annihilated in a chemical reaction. Furthermore, the sum of the entries in each row or column of an R-matrix must also he zero as long as there is not a change in formal charges on the corresponding atom. An elaborate mathematical model of the constitutional aspects of organic chemistry has been built on the basis of BE- and R-matriccs [17. ... 

The law of additivity then says that the sum of the properties of the molecules on the right-hand side is the same as the sum of the properties on the left-hand side of Eq. (1). 

The sum of the properties of the diatomic species X and Y2 is the same as twice the property of XY. This is the zero-order approximation to additivity rules. 

Clearly, there is one molecular property that can be exactly calculated fi om the contributions of its constituent atoms the molecular weight, or, more correctly, the molecular mass, which is exactly the sum of the masses of its constituent atoms. 

Here, the component of the autocorrelation vector a for the distance interval between the boundaries dj (lower) and (upper) is the sum of the products of property p for atoms i and j, respectively, having a Euclidian distance d within this interval. 

The component of the autocorrelation vector for a certain distance interval between the boundaries 4 and du is the sum of the products of the property p x,) at a point Xi on the molecular surface with the same property p Xj) at a point Xj within a certain distance d Xj,Xj) normalized by the number of distance intervals 1. All pairs of points on the surface are considered only once. 

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