Bulk Modulus, and Cohesion

To predict the equilibrium density, we simply write = 0, using 

Test of the ttiiiiimum of (Hq. 15-16) as a prediction of equilibrium lattice spacing. The first values arc predicted k,. in A, based on i values from the Solid State Table. Values in parentheses arc experimental values of kf from the Solid State Table. 

We may also obtain an immediate estimate of the bulk modulus, 

The first value for each clement is the bulk modulus (in 10 erg/cm ), obtained from B = (l/9)/cf-, Eq, (15-18), evaluated 

SOURCE of experimental values hUenuitional Critical Tables (1928). 

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Table III. Minus the total Si crystal valence electron energy per atom with relaxation energy and pseudopotential corrections included, along with the equilibrium lattice constant, bulk modulus, and cohesive energy calculated with four different exchange-correlation functionals (defined in the caption of Table I) are compared with experimental values. The experimental total energy is the sum of Acoh plus the four-fold ionization energy.

As in the other solid types, the entire range of structural, elastic, and vibrational properties arc determined by the electronic structure. Likewise, as in other systems, the density, bulk modulus, and cohesion arc considcfed together as a separate problem and, for the metals, were treated in Chapter 15. We have given a reasonably simple description of the electronic structure of simple metals in Chapter 16, and can now use it to treat the more detailed aspects of the bonding properties. 

The trends in several ground state properties of transition metals have been shown in Figs. 2, 3 and 15 of Chap. A and Fig. 7 of Chap. C. The parabolic trend in the atomic volume for the 3-6 periods of the periodic table plus the actinides is shown in Fig. 3 of Chap. A. We note that the trend for the actinides is regular only as far as plutonium and that it is also broken by several 3 d metals, all of which are magnetic. Similar anomalies for the actinides would probably be found in Fig. 15 of Chap. A - the bulk modulus - and Fig. 7 of Chap. C - the cohesive energy if more measurements had been made for the heavy actinides. 

Using the relationship between the bulk modulus and the cohesive energy density leads to... 

Timgsten has been of keen theoretical interest for electron band-structure calculations [1.14-1.25], not only because of its important technical use but also because it exhibits many interesting properties. Density functional theory [1.11], based on the at initio (nonempirical) principle, was used to determine the electronic part of the total energy of the metal and its cohesive energy on a strict quantitative level. It provides information on structural and elastic properties of the metal, such as the lattice parameter, the equilibrium volume, the bulk modulus, and the elastic constants. Investigations have been performed for both the stable (bcc) as well as hypothetical lattice configurations (fee, hep, tetragonal distortion). 

Table 1. Experimentally determined lattice constant and calculated properties of two possible configurations of ) i2AION3. Bo and fi are the bulk modulus and its pressure derivative, the cohesive energy per formula unit.

The minimum and curvature of E(V) near the minimum determine the lattice constant and bulk modulus. The cohesive energy can be evaluated by comparing the energy for the solid including a zero-point motion contribution and the isolated pseudoatom ground-state energy. 

Fig. 3. Theoretical (A) and experimental (O) values of (a) equilibrium Wigner-Seitz radii, (b) the bulk modulus and (c) cohesive energies. The experimental value for Pm is not available.

Up to now the best agreement between experimental ground state properties like cohesive energy, lattice constants, and bulk modulus and the corresponding DFT results were obtained with the PBE functional. However, in a recent survey it has been shown that the anisotropy of the bonding in Zn, as reflected by the elastic constants, is not described in a well-balanced manner even with this functional. Indeed, the elastic constants related to distortions in the a-Z)-plane are reproduced better with hybrid functionals with Hartree-Fock and DFT exchange mixed. The drawback of these latter functionals is a worse description of the interlayer interaction, leading with even to a shift of the e value by 0.79A or... 

The cohesive properties which we wish to describe are the atomic volume, the structure, the bulk modulus and the cohesive energy. It is necessary to calculate the total energy of both the free atoms and the solid in order to arrive at the cohesive energy, whereas the former three properties may be obtained from the force theorem (sect. 3.1). Although the total energy could be, and sometimes is, used to compute the volume, structure and bulk modulus, no separation into angular momentum contributions is then possible. 

Fig. 27.14 Size dependence of a the elastic modulus [147] and b the Raman shifts [144, 146], c temperature [141, 148], and d pressure [143, 148] dependence of the bulk modulus and Raman shift Aig mode (inset Eg—639 cm ) of Ti02[144, 151]. The paradox in the size-induced shift trends arises from the involvement of the different numbers of atomic CN of the specific atom. Theoretical matching gives rise to the mode cohesive energy and Debye temperature as listed in Table 27.5 (Reprinted with permission from [151])...

Therefore, the macroscopically measurable B and Aco connect directly to the bond length and energy of the specimen and their response to the intrinsic coordination imperfection and the applied stimuli through the BOLS correlation and the LBA approach. Exercises lead to derived information of the cohesive energy and Debye temperature, energy density, compressibility, the bulk modulus and their first-order derivatives, and the analytical correlation between the B and Aco. 

Applications of the quantum methods Although the quantum methods require a much larger computational effort than classical approaches, they have now been applied to as wide a range of systems as the latter. For example, they have been used to describe not only the cohesive properties, such as the bulk structural parameters, the bulk modulus and the electronic structure (Binggeli et al, 1991 Dovesi et al, 1992 Mackrodt... 

Material properties can be further classified into fundamental properties and derived properties. Fundamental properties are a direct consequence of the molecular structure, such as van der Waals volume, cohesive energy, and heat capacity. Derived properties are not readily identified with a certain aspect of molecular structure. Glass transition temperature, density, solubility, and bulk modulus would be considered derived properties. The way in which fundamental properties are obtained from a simulation is often readily apparent. The way in which derived properties are computed is often an empirically determined combination of fundamental properties. Such empirical methods can give more erratic results, reliable for one class of compounds but not for another. 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

In (8.1), is the specific cohesive energy, v = 1/p is the specific volume and the reciprocal of the density p. Vq is the specific volume at zero pressure as shown in Fig. 8.1. The final parameter a is constrained by the relation for the bulk modulus... 

Table4.6 Lattice constants a, volume V, cohesive energy and bulk modulus 6 for fee gold from nonrelativistic and relativistic pseudopotential DFT calculations (from Ref [402]).

Other authors have studied other correlations. Two are Povarennykh (1964), and Goble and Scott (1985). The latter emphasized compressibility (inverse bulk modulus) as did Beckmann (1971). The bulk modulus is not a reliable measure for the same reason as the cohesive energy. It is volume dependent rather than shear dependent. Still another attempt to correlate hardness and compressibility was that of Yang et al. (1987). This was followed by a proposal by Liu and Cohen (1990) that hardness and bulk moduli are related. This proposal was refuted by Teter (1998) who showed that hardness values correlate better with shear moduli than with bulk moduli. 

The Cohesive Energy and the Bulk Modulus of Light Actinides in Friedel s 101... 

Indices are dimensionless parameters derived from various mechanical and physical properties of the tablet blend and resulting compacts. Mechanical properties typically measured include indentation hardness (kinetic and static), elastic modulus, and tensile strength (10,11). Physical properties include particle size, shape, and size distribution, density (true, bulk, and tapped), flow properties and cohesive properties. 

The cohesive energy, equilibrium atomic volume, and bulk modulus across a transition metal series may now be evaluated by choosing the following simple exponential forms for ( and h(R), namely... 

Fig. 7.12 The theoretical ( ) and experimental (x) values of the equilibrium band width, Wigner-Seitz radius, cohesive energy, and bulk modulus of the 4d transition metals. (From Pettifbr (1987).)...

Table 10.5 Hildebrand s solubility parameter and cohesive energy density determined from this cohesive energy density from bulk modulus Hydroxyl concentration for some networks, (a) Molar ratio dimethacrylate/ methacrylate = 5 x 10 4 (500 ppm) (b) aromatic poly(bismaleimide) from BASF. 

To summarize, the bulk modulus of thermosets is proportional to the cohesive energy density and does not depend practically on temperature in the 200 K - (Tg — 30 K) temperature range. There is no significant effect of crosslink density on K, which can be predicted (in the temperature interval under consideration) using K=ll (CED), with an incertitude of about... 

The bulk modulus depends practically only on cohesion. It does not exhibit viscoelastic effects and depends only slightly on temperature, except at the glass transition where it varies by a factor of about 2. 

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