S. Torquato and G. Stell. Microstructure of two-phase random media.I. The n-point probability functions. Journal of Chemical Physics, 77(4):2071.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... # 2# 2 # 2 # # h(x 3 x 4 )m(x 1 x 3 )m(x 2 x 3 )dx 3 dx 4 , 2.35) where# u (x 1 x 2 ) 2# # int (x 1 x 2 ) is the union volume (area) of two particles with centers located at points x 1 and x 2 . This relation is equivalent to formulas obtained by Torquato in [Torquato and Stell, 1985] using slightly di#erent procedure. To make Eq. 2.35) as simple as possible, let us consider statistically isotropic twodimensional composite formed by identical circles with radius R. For this particular case, Eq. 2.35) yields Smm (r 12 ) 1 ## u (r 12 ) # 2# 2 # 2 # # h(r 34 ....

Torquato, S. and Stell, G. (1985). Microstructure of two-phase random media. V. The n-point matrix probability functions for impenetrable spheres. Journal of Chemical Physics, 82(2):980--987.

.... x 2 ) # 2# 2 # 2 # # h(x 3 x 4 )m(x 1 x 3 )m(x 2 x 3 )dx 3 dx 4 , 2.35) 18 where# u (x 1 x 2 ) 2# # int (x 1 x 2 ) is the union volume (area) of two particles with centers located at points x 1 and x 2 . This relation is equivalent to formulas derived by [Torquato and Stell, 1985] using a slightly di#erent procedure. To further simplify Eq. 2.35) we consider a statistically isotropic two dimensional composite formed by identical circles with radius R. In such a case, the two point matrix probability function is provided by Smm (r 12 ) 1 ## u (r 12 ) # 2# 2 ....

Torquato, S. and Stell, G. (1985). Microstructure of two-phase random media. V. The n-point matrix probability functions for impenetrable spheres. Journal of Chemical Physics, 82(2):980--987.

....rule is the variational bound of Weissberg and Prager [109] derived for a particularly simple model of a random composite called the penetrable sphere model. The penetrable sphere model is a theoretical construct for which exact information is available about the statistics of the microgeometry [96]. The model is constructed by throwing points randomly in a box and then letting spheres grow around the points until the desired porosity is reached. The result of Weissberg and Prager [109] is Gamma 2 9 OER 2 ln OE ; 96) where R is the radius of the spheres and OE is the porosity. 7.2. ....

....1 2 135 64 (1 Gamma OE) ln (1 Gamma OE) 16:5(1 Gamma OE) 98) where the exact result for Stokes flow through a dilute assemblage of spheres of radius R is Stokes = 2R 2 9(1 Gamma OE) 99) 7.3. Examples 7.3.1. Penetrable sphere model. Results for the penetrable sphere model [96] are shown in Figure 11. The solid volume fraction is 1 Gamma OE and =R 2 is the normalized permeability, where R is the radius of the spheres in the model. The Kozeny Carman empirical relation used in the plot is Stokes =KC = 10(1 Gamma OE) OE 3 ; 100) where the Stokes permeability in a ....

Torquato, S., and G. Stell, Microstructure of two-phase random media. III. The n-point matrix probability functions for fully penetrable spheres, J. Chem. Phys., 79, 1505--1510, 1983.

....of (13) for large arguments solves the problem of obtaining accurate values of S 3 for large arguments and also eliminates the storage problem, since S 2 is much easier to compute and store than S 3 . Finally, note that the approximation (13) is superior to other approximations that have been used [19] such as S 3 (r; s; S 2 (r)S 2 (s) OE which has no angular dependence (and therefore would produce nothing if used in the integrals) or such as S 3 (r; s; S 2 (r) S 2 (s) S 2 (t) 2OE which does not satisfy all of the limiting conditions. We will show an example later that illustrates ....

....in Figure 2 were chosen to illustrate the characteristics of the correlation function, but higher resolution is required to obtain accurate values of the parameters of microgeometry if this integration scheme is to be used. Other integration schemes have been used to produce more accurate results [2, 19, 20]. 5. Discussion. We have shown that it is possible to obtain useful measured values of three point correlation functions for real materials, to interpolate and integrate those values to find the parameters of the composite microgeometry, and to use new visualization techniques to gain insight ....

S. TORQUATO and G. STELL, Microstructure of two-phase random media. III. The n-point matrix probability functions for fully penetrable spheres, J. Chem. Phys., 79 (1983), pp. 1505-1510.

No context found.

S. Torquato and G. Stell. Microstructure of two-phase random media.I. The n-point probability functions. Journal of Chemical Physics, 77(4):2071.

No context found.

S. Torquato and G. Stell. Microstructure of two-phase random media, i: The n{point probability functions. J. Chem. Phys., 77:2071.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC