Kieffer model

The bias observed between experimental measurements and Kieffer s model predictions is due to the relative paucity of experimental data concerning cutoff frequencies of acoustic branches, and also to the assumption that the frequencies of the lower optical branches are constant with K and equivalent to those detected by Raman and IR spectra (corresponding only to vibrational modes at K = 0). Indeed, several of these vibrational modes, and often the most important ones, are inactive under Raman and IR radiation (Gramaccioli, personal communication). The limits of the Kieffer model and other hybrid models with respect to nonempirical computational procedures based on the equation of motion of the Born-Von Karman approach have been discussed by Ghose et al. (1992). 

Table 5.24 lists selected data concerning entropy and isobaric heat capacity, covering andradite, grossular, pyrope, and almandine terms, compared with results of calculations based on the Kieffer model (Ottonello et al., 1996). 

In the Kieffer model (Kieflfer 1979a, b, and c cf section 3.3), the isochoric heat capacity is given by... 

The Kieffer model correctly predicts the systematic change of the reduced partition functions of various minerals with structure, as indicated by Taylor and Epstein (1962). For anhydrous sihcates, the decrease in the sequence framework-chain-orthosilicate reflects the decreasing frequency of antisymmetric Si-O stretching modes. The internal frequencies of the carbonate ion give a high reduced partition function at all T. The value for rutile is low because of the low frequencies of the Ti-0 modes (Kieffer, 1982). 

Kieffer has estimated the heat capacity of a large number of minerals from readily available data [8], The model, which may be used for many kinds of materials, consists of three parts. There are three acoustic branches whose maximum cut-off frequencies are determined from speed of sound data or from elastic constants. The corresponding heat capacity contributions are calculated using a modified Debye model where dispersion is taken into account. High-frequency optic modes are determined from specific localized internal vibrations (Si-O, C-0 and O-H stretches in different groups of atoms) as observed by IR and Raman spectroscopy. The heat capacity contributions are here calculated using the Einstein model. The remaining modes are ascribed to an optic continuum, where the density of states is constant in an interval from vl to vp and where the frequency limits Vy and Vp are estimated from Raman and IR spectra. 

The Kieffer approach uses a harmonic description of the lattice dynamics in which the phonon frequencies are independent of temperature and pressure. A further improvement of the accuracy of the model is achieved by taking the effect of temperature and pressure on the vibrational frequencies explicitly into account. This gives better agreement with experimental heat capacity data that usually are collected at constant pressure [9],... 

Crystals lack some of the dynamic complexity of solutions, but are still a challenging subject for theoretical modeling. Long-range order and forces in crystals cause their spectrum of vibrational frequencies to appear more like a continuum than a series of discrete modes. Reduced partition function ratios for a continuous vibrational spectrum can be calculated using an integral, rather than the hnite product used in Equation (3) (Kieffer 1982),... 

For crystals with molecule-like constituents, like the BO, " and BO4 " groups in some borates, semi-quantitative models of the molecular component as a gas-phase entity have been proposed (Oi et al. 1989). This is conceptually similar to the approximation made for species in solution, although in practice most studies of crystals consider additional frequencies that reflect inter-molecular vibrations. The spectroscopic data on these vibrations (which typically have lower frequencies than the intra-molecular vibrations) are often available, at least approximately, from infrared and Raman spectroscopy and elastic properties. This type of hybrid molecule-in-crystal model has been applied to many minerals in theoretical studies of carbon and oxygen isotope fractionation, the most noteworthy being studies of calcite (Bottinga 1968 Chacko et al. 1991) and sihcates (Kieffer 1982). Because specfroscopic dafa are always incomplete (especially for subsfances substifufed wifh rare isolopes), some amounl of vibralional modeling is necessary. 

Therefore, in the first model, the specific surface area increases with decreasing porosity, while in the second, the opposite relationship is specified. While some attempts have been made to experimentally verify these models in individual rock types (Kieffer et al. 1999 Jove Colon et al. 2004), the data concerning a wide range of rocks and precipitation-dissolution reactions remain limited. 

To appreciate the predictive properties of Kieffer s model, it is sufficient to compare calculated and experimental entropy values for several phases of geochemical interest in table 3.1, which also lists entropy values obtained through apphcation of Debye s and Einstein s models. One advantage of Kieffer s model with respect to the two preceding formulations is its wider T range of applicability (Debye s model is appropriate to low frequencies and hence to low T, whereas Einstein s model is appropriate to high frequencies and hence to high T). 

Figure 5.16 Heat capacity of pyrope at constant P, as determined from the Kieffer vibrational model, compared with low-P (upper part of figure) and high-P (lower part of figure) experimental evidences. From Ottonello et al. (1996). Reprinted with permission of The Mineralogical Society of America.

Combining the results of Kieffer s model and of laboratory experiments, Clayton and Kieffer (1991) obtained a set of third-order equations relating the reduced partition functions of various minerals to the inverse of the squared absolute temperature (table 11.9) according to... 

Figure 11.31 shows the results of equations 11.138 to 11.141 applied to the calcite-diopside couple compared with experimental evidence and with the indications of Kieffer s (1982) model. 

Figure 1 Common model for the p-, k- and 5-opioid receptor (modified from Gaveriaux-Ruff and Kieffer, 1999).

Benech-Kieffer, F., P. Wegrich, R. Schwarzenbach, G. Klecak, T. Weber, J. Leclaire and H. Schaefer (2000). Percutaneous absorption of sunscreens in vitro interspecies comparison, skin models and reproducibility aspects. Skin Pharmacol Appl Skin... 

Santoni, V., S. Kieffer, D. Desclaux, F. Masson, and T. Rabilloud. 2000, Membrane proteomics use of additive main effects with multiplicative interaction model to classify plasma membrane proteins according to their solubility and electrophoretic properties. Electrophoresis 2 3329-3344. 

Figure 11. Density of vibrational frequencies of calcite, computed by the RIM (rigid-ion-model) potential (above), by the SM (shell model) potential and by Kieffer s model (below, thick and thin lines, respectively).

It is also interesting to examine the Cy results obtained for calcite by using the empirical and purely harmonic Kieffer s model[55, 56, 57]. A density of states function was built up by considering three acoustic modes with sin-like dispersion and cut off fi equencies of 76, 102 and 173 cm ... 

Fig. 12. Curves of Cy versus T for experimental (crosses, from Cp results of Stavely and Linford[35] and Jacobs et al.[51]) and computed (dots RIM circles SM triangles Kieffer s model) data of calcite.

Figure 14. Curves of experimental Cp (crosses, from results of Stavely and Linford [35] and computed Cv (dots RIM squares RIMl triangles Kieffer s model) data of aragonite versus T.

The densities of vibrational states for the two potentials (Fig. 15) are actually very similar, exeept for a somewhat wider gap for RIMl between the two peaks at higher frequencies, corresponding to inner modes of CO3. As for the Kieffer s model, the cut off frequencies of acoustic modes were derived from elastic constants by the Voigt-Reuss-Hill approximation[38], amounting to 51, 66 and 90 cm"i. An optic continuum ranging from 113 to 287 cm was used, and four Einstein oscillators at 708,867, 1042 and 1470 cm with appropriate weights represented the internal optical modes. The... 

Dreher, F, Patouillet, C., Fouchard, R, Zanini, M., Messager, A., Roguet, R., Cottin, M., Leclaire, J., Benech-Kieffer, F. (2002) Improvement of the experimental setup to assess cutaneous bioavailability on human skin models dynamic protocol. Skin Pharmacol Appl Skin Physiol, 15 (Suppl 1), 31-39. 

Kendrew JC, Bodo G, Dintzis HM, Parrish RG, Wyckoff H, PhilUps DC (1958) A three-dimensional model of the myoglobin molecnle obtained by X-ray analysis. Nature 181 662-666 Kieffer SW (1980) Thermodynamics and lattice vibrations of minerals 4. Application to chain and sheet silicates and orthosilicates. Rev Geophys 18 862-886... 

---