Matrix models

In the matrix model (Jongschaap, 1990), the global thermodynamic system is composed of two separate physical parts, which are called the environment and the internal variables. For the polymer solutions, for example, the pressure tensor Pv may be the internal variables, and the classical variables density, velocity, and internal energy are the environment variables. 

In the matrix model, the power supplied to the system is characterized by a set of controllable external forces Fex and rate variables and is given by 

When there are no internal variables, the whole supplied power would dissipate. The matrix model is derived fromEq. (14.84) 

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Chemicals Application area Environmental matrix Model inputs REF... 

In chemometrics, the inverse calibration model is also denoted as the P-matrix model (the dimension of P is m x n) ... 

Sun, H. Dalton, L. Chen, A., Systematic design and simulation of polymer microring resonators with the combination of beam propagation method and matrix model, In Digest of the IEEE LEOS Summer Topical Meetings, 2007, 217 218... 

Each column of the matrix Model contains the 3 indices, xyz, representing the stoichiometry for the species shown in the corresponding column header. There are 3 rows for the 3 components. The species and component names in the column and row headers are not part of the matrix Model, they are added for clarity only. 

Genomic analysis integrating APOE + PSl + PSl in a tiigenic matrix model... 

L. F. Cugliandolo, J. Kurchan, G. Parisi, and F. Ritort, Matrix models as solvable glass models. Phys. Rev. Lett. 74, 1012-1015 (1995). 

This leads to die third and final main topic, the use of Heff model u-scaling predictions to detect the changes in the resonance structure that occur near chemically important topographic features of a potential surface. Such features include an isomerization saddle point or a sharp bend in the minimum-energy isomerization path. The key feature of this matrix model is... 

W. H. Miller I would like to ask Prof. Schinke the following question. Regarding the state-specific unimolecular decay rates for HO2 — H + O2, you observe that the average rate (as a function of energy) is well-described by standard statistical theory (as one expects). My question has to do with the distribution of the individual rates about die average since there is no tunneling involved in this reaction, the TST/Random Matrix Model used by Polik, Moore and me predicts this distribution to be x-square, with the number of decay channels being the cumulative reaction probability [the numerator of the TST expression for k(E)] how well does this model fit the results of your calculations ... 

Sefcik M. D., Raucher D. The Matrix Model of Gas Sorption and Diffusion in Glassy Polymers, to be published... 

It was previously noted that no evidence for direct photolysis for PFOS or PFOA has been observed experimentally. In aqueous solutions alone and in the presence of hydrogen peroxide (H2O2), iron oxide (Fe203) or humic material, PFOS has been observed to undergo some indirect photolysis [28] whereas PFOA did not undergo indirect photolysis [29]. Using an iron oxide photo-initiator matrix model, the indirect photolytic half-life for PFOS was estimated to be > 3.7 years at 25 °C. The half-life of PFOA was estimated to be > 349 d. 

Huang, Y., Rumschitzki, D., Chien, S. and Weinbaum, S. (1997) A fiber matrix model for the filtration through fenestral pores in a compressible arterial intima. Am. J. Physiol, 272, H2023-H2039. 

Edwards, A., M.R. Prausnitz. 1998. Fiber matrix model of sclera and corneal stroma for drug delivery to the eye. AIChE Journal 41 214. 

Appendix Comments Concerning The "Matrix" Model For Sorption and Difffusion in Glassy Polymers... 

On a more basic level, since the matrix model implicitly requires a somewhat inconsistent interpretation for the various model parameters in Eq (A-l) and Eq (A-2), it becomes primarily an empirical means of reproducing the observed pure component data with no fundamental basis for generalization to mixtures. One could, of course envision several extensions based on additional a terms in the denominator of Eq (A-l) and additional 8 terms in Eq (A-2). Such an approach to mixture permeation analyses would be completely empirical and mimic the generalization of Eq (2) and Eq (7) however, without any physical justification. The generalizations of Eq (2) and Eq (7) were natural outgrowths of the fundamental physical basis of the Langmuir isotherm. The fact that the mixture data are so consistent with Eq (7) and Eq (9) provides strong support for the physical basis of the dual mode model. 

In the following chapter we present the matrix model of gas sorption and diffusion in glassy polymers which is based on the observation that gas molecules interact with the polymer, thereby altering the solubility and diffusion coefficients of the polymer matrix. 

The gas-polymer-matrix model for sorption and transport of gases in polymers is consistent with the physical evidence that 1) there is only one population of sorbed gas molecules in polymers at any pressure, 2) the physical properties of polymers are perturbed by the presence of sorbed gas, and 3) the perturbation of the polymer matrix arises from gas-polymer interactions. Rather than treating the gas and polymer separately, as in previous theories, the present model treats sorption and transport as occurring through a gas-polymer matrix whose properties change with composition. Simple expressions for sorption, diffusion, permeation and time lag are developed and used to analyze carbon dioxide sorption and transport in polycarbonate. 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

A model for sorption and transport of gases in polymers has to specifically account for the fact that the presence of sorbed gases in the polymer modifies the matrix (7, 9). In developing the matrix model we are guided by the physical evidence relating to the mechanism of sorption and transport. The matrix model is consistent with the following observations and assumptions ... 

A. Matrix Model Expressions for Sorption and Transport 1. Solubility Coefficient... 

II. ANALYSIS OF SORPTION AND TRANSPORT DATA BY THE MATRIX MODEL... 

Both the matrix-model and the dual-model represent the experimental data satisfactory (Fig. 1). After modeling sorption measurements in several gas-polymer systems we have observed no systematic differences between the mathematical descriptions of the two models. 

Figure 1. Sorption isotherm at 35 °C for CO2 in polycarbonate conditioned by exposure to 20 atm CO2. The experimental data are from Ref. 15. The curves, based on the matrix model (solid line) and the dual-mode model (broken line), are calculated using the parameters given in the text.

Comparing the curves in Fig. 2 shows that representing the permeability versus pressure data by either model provides a satisfactory fit to the data over the pressure range of 1 to 20 atm. However, at pressures less than 1 atm. the two models differ in their prediction regarding the behavior of the permeability-pressure curve [Fig. 2]. While the matrix model predicts a strong apparent pressure dependence of the permeability in this range (solid line), the dual-mode model predicts only a weak dependence (broken line). 

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