Applications to the Models

The mathematics underlying transformation of the data from different experiments can be applied to simple models. In the case of the relationship between G (a ) and G(t) it is straightforward. To give an example, consider a Maxwell model. It has an exponentially decaying modulus with time. We have indicated that the relationship between the complex modulus and the relaxation function is given by Equation (4.117). So if we substitute the relaxation function into this expression we get 

We should not be intimidated by the presence of i = ( —1)1/2, as the integral can be performed in a straightforward manner with i treated as a constant  

This expression is the transform of the relaxation function. However, it is not in a readily recognisable form. If we multiply top and bottom of the quotient by rr we get 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give 

So suppose that we apply this property to our relaxation integral (Equation 4.47) such that the relaxation spectrum is replaced by a Dirac delta function at time rm  

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For application to the model of Dodelet and Freeman (1975a,b), the joint density of n ordered distances should be... 

G.M. Come, V. Warth, P.A. Glaude, R. Foumet, F. Battin-Leclerc, and G. Scacchi. Computer-Aided Design of Gas-Phase Oxidation Mechanisms—Application to the Modeling of n-Heptane and Iso-Octane Oxidation. Proc. Combust. Inst., 26 755-762,1996. 

In 1969 Gray and Yang [70] formulated an extremely simple scheme which could reproduce these phenomena and it is described in detail in Chapter 5. Its importance lay, not so much in its application to the modelling of practical systems, but in its provision of a conceptual base for further development. It incorporated the essential features of the science, particularly thermokinetic feedback - the interaction between a branched radical chain and the reaction-generated temperature rise. 

Several application of oxide models have been presented. These included the importance of within-die characterization and prediction for process optimization, and the use of the pattern dependent model in run by ran feedback control. Work has also been done to apply the density model to STI polish. We believe that the generalized framework presented for copper polish is also applicable to the modeling of dishing and erosion in STI CMP. 

J. Higo, V. Collura, J. Gamier, Development of an extended simulated annealing method application to the modeling of complementary determining regions of immunoglobulins. Biopolymers, 32 (1992) 33. 

Duan, Z., and Hu, J. (2004) A new cubic equation of state and its applications to the modeling of vapor-liquid equilibria and volumetric properties of natural fluids, Geochimica et Cosmochimica Acta 14, 2997-3009. 

Obviously, the solutions of Eqs. 1.74-1.77 depend on the coefficients that appear in these equations. Solutions of Eqs. 1.74-1.77 are equally applicable to the model and prototype (where the model and prototype are geometrically similar systems of different linear dimensions in streams of different velocities, temperatures, and concentration), if the coefficients in these equations are the same for both model and prototype. These coefficients, Pr, Re, Sc, and Ec (called dimensionless parameters or similarity parameters), are defined in Table 1.10. Focusing attention now on heat transfer, from Eq. 1.14, using the dimensionless quantities, the heat transfer coefficient is given as ... 

WARTH V., STEF N., GLAUDE P.A., BATTIN-LECLERC F., SCACCHI G., COME G.M., Computer-Aided Derivation of Gas-Phase Oxidation Mechanisms Application to the Modelling of the Oxidation of n-Butane, Comb. Flame, 114. 81-102 (1998). 

Boundary Element Method and Its Applications to the Modeling of MEMS Devices... 

Boundary Element Method and Its Applications to the Modeling of MEMS Devices, Fig. 4 A convergence plot of the simulated drag force... 

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