Squared Gradient Method

Scheutjens-Fleer theory 5.31, 5.5 square gradient method 5.33 statistics 5.29... 

Once the free energy of an inhomogeneous system is given, one can calculate by standard methods the properties of the interface—for example, the interfacial tension or the density profile perpendicular the interface [285]. Weiss and Schroer compared the various approximations within square-gradient theory discussed earlier in Section IV.F for studying the interfacial properties for pure DH and FL theory [241, 242], In theories based on local density approximations the interfacial thickness and the interfacial tension were found to differ by up to a factor of four in the various approximations. This contrasts with nonionic fluids, where the density profiles and interfacial... 

The least-squares methods may generally be applied using one of two possible approaches. One of them requires a computation of the function gradients whilst the other does not. Gradient methods use iterative computation of corrections to consecutive approximate solutions g[Pg.265]

R. W. H. Sargent The method used is given in the Results section of the paper, page 160. The errors between the model and experimental results at 29 points were computed and the sum of squares of the errors minimized by adjusting Dp and P using Powells conjugate gradient method Computer Journal, 1964). 

Nonlinear least-squares inversion by the conjugate gradient method This method uses the same ideas as the regularized conjugate gradient method ... 

Using equations (5.102), (5.103), (5.105), and (5.106), we can obtain m iteratively. 5.3.6 The numerical scheme of the regularized conjugate gradient method for nonlinear least-squares vnversion... 

Figure 1, Composition of the critical cluster (bubble of critical size), Xgas, thermodynamic driving force of critical bubble formation, CJ,radius of the critical bubble, Rc, and work of critical bubble formation, [J Gc, computed for the case of boiling in binary liquid-gas solutions in dependence on supersturation here expressed via the density of the liquid puq (for the details see Ref 21). By the number (1), the results are shown computed via the classical Gibbs approach employing the capillarity approximation, number (2) refers to computations via the generalized Gibbs approach and number (3) to computations via the van der Waals square gradient density functional method.

It is obvious that the number of equations can be taken either equal to the unknown parameters or greater. In the latter the least squares method can be applied if the system is linear or in a more general case gradient methods can be used. Function Gi La0) can be either the geometric factor or a very thin cylindrical shell (the differential response) or that of a relatively thin cylindrical layer. In such a case this function is expressed through the integral response of a two-coil induction probe Gi(La0)-... 

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