Media Semi-infinite

The diffusion distance concept is best defined for infinite and semi-infinite media diffusion problems. In these cases, C depends on x 4Dt), so if at time ti the concentration is Ci at Xi, then at time tz = 4fi the concentration is Ci at X2 = 2xi (because = Xi/(4Dfi) = 2 = 2/(4T>f2) ). This fact is often referred as the square root of time dependence. That is, the distance of penetration of a diffusing species is proportional to the square root of time. In other words, the concentration profile propagates into the diffusion medium according to square root of time. It can also be shown that the amount of diffusing substance entering the medium per unit area increases with square root of time. The square root dependence is often expressed as... 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

For semi-infinite media, this interface concentration remains constant, but for a particle it changes with time towards. Equations (3-72) and (3-73) are compared with more complete solutions below. 

Schery, S. D., Holford, D. J., Wilson, J. L., and Phillips, F. M. (1988). The flow and diffusion ofradon isotopes in fractured porous media Part 2, Semi-infinite media. Radiation Protection Dosimetry, 24(1/4), 191-197. 

The physical problem to be solved is described in Figure 3.1. Here a stack of m films is contained between two semi-infinite media. The upper medium has refractive index nQ and supports the incident field, E0. The lower medium has refractive index nm + j... 

L2.3.A. Interactions between two semi-infinite media, 182 L2.3.B. Layered systems, 190 L2.3.C. The Derjaguin transform for interactions between oppositely curved surfaces, 204 L2.3.D. Hamaker approximation Hybridization to modern theory, 208 L2.3.E. Point particles in dilute gases and suspensions, 214 L2.3.F. Point particles and a planar substrate, 228 L2.3.G. Line particles in dilute suspension, 232... 

L2.2.A. TABLES OF FORMULAE IN PLANAR GEOMETRY Table P.l.a. Forms of the van der Waals interaction between two semi-infinite media... 

Table P.6.a. Multilayer-coated semi-infinite media...

P.7.C.I. Two semi-infinite media A symmetrically coated with a finite layers a of thickness D with exponential variation ea(z) perpendicular to the interface, retardation neglected... 

The general formula for the electrodynamic free energy per unit area between two semi-infinite media A and B separated by a planar slab of material m, thickness / is... 

TWO SINGLY COATED SEMI-INFINITE MEDIA Exact... 

Following the strategy for extracting small-particle van der Waals interactions from the interaction between semi-infinite media, we can specialize the general expression for ionic-fluctuation forces to derive these forces between particles in salt solutions. Because of the low frequencies at which ions respond, only the n = 0 or zero-frequency terms contribute. In addition to ionic screening of dipolar fluctuations, there are ionic fluctuations that are due to the excess number of ions associated with each particle. 

L3.3. A heuristic derivation of Lifshitz s general result for the interaction between two semi-infinite media across a planar gap, 283... 

Correlated charge fluctuations between anisotropic bodies acting across an anisotropic medium create torques as well as attractions or repulsions. The formulae derived here for semi-infinite media can also be specialized to express the torque and force between anisotropic small particles or between long rodlike molecules. (For example, Table C.4 and Subsection L2.3.G.)... 

Let us consider the laminate system for situation (ii) with a 1 and a very short contact time t = t. This means the initial solute concentration in the vicinity of x=d at t=0 is Cp=cPe and cl.i = 0 (Fig. 7-14a). This illustration is the case of a system with diffusion between two semi-infinite media (Crank, 1975) for which Eq. (7-51) reduces to Eq. (7-54). 

The electrostatic potential satisfies the Laplace equation within each of the two semi-infinite media which lie beyond the surfaces,... 

A basic waveguide structure, which is sketched in Fig. 1, is composed of a guiding layer surrounded by two semi-infinite media of lower refractive indices. The optical properties of the stmcture are described by the waveguiding layer refractive index Hsf, and thickness t, and by the refractive indices of the two surrounding semi-infinite media, here called (for cover) and (for substrate). Application of Maxwell s equations and boundary conditions leads to the well-known waveguide dispersion equation [6] ... 

Figure 3 illustrates the geometry. There are a total of N layers, where the semi-infinite media to the left and to the right of the layer stack have the indices 0 and N + 1. Later on, the crystal will usually be layer 1 and the index 1 will be replaced by q. Each layer j is characterized by the thickness, dj, an acoustic impedance, Zj, and a speed of sound, Cj. Both the impedance Zj and the speed of sound cj are complex. They are given by ... 

Equation 62 is very attractive for the study of adhesion between polymers and sohd surfaces, since it allows for the determination of the viscoelastic constants of the adhesive in the immediate vicinity of the contact. Unfortunately, the QCM does not work well with semi-infinite media when the viscosity, r], is larger than about 50 cP. The bandwidth in this case is too large. Most polymers exceed this limit. If, however, the contact area can be confined to a small spot in the center of the crystal the measurement becomes feasible [74]. Such a small contact area can, for instance, be estabhshed with a JKR tester [75]. The area of contact can be determined by optical microscopy. Of course, this kind of sample is laterally heterogeneous and the apphcabihty of simple models may be questioned. Experiment shows that the finite contact area can be reasonably well accounted for by modifying as ... 

Consider a ray of light incident on a thin film of top antireflection coatings sandwiched between two semi-infinite media air (or water) and the photoresist (Fig. 9.4). Assuming normal incidence, at the boundary between two media, say, medium 1 (air or water) and medium 2 (top antireflection coatings), the... 

Note that for semi-infinite media in case of conservative scattering, a = 1, we find from Eqs. (33a), (33) and (30) that the energy flux integral for the surface Green s function matrix will be equal to zero. 

For semi-infinite media, the a-transformation greatly simplifies. In this case, dne to Eq. (33a), we find fromEq. (36) that... 

From the computational point of view, the (o-transformation is most advantageous for multiple scattering in optically thick media with small tme absorption. In particular, for semi-infinite media, the simple transformation formulae, Eqs. (44) and (48), offer the possibihties for highly efficient computational schemes. The ( -transformation also may be useful for avoiding problems of convergence, which may occur for adding and doubling computational schemes. 

Early modeling efforts focused on equilibrium and near equilibrium conditions. Olander (J 3) developed analytical solutions for a variety of reaction shemes under reaction equilibrium conditions. Friedlander and Keller (44) used an affinity function to obtain analytical solutions for systems near reaction equilibrium. Secor and Beutler (45) used penetration theory to calculate transient mass transfer by numerical methods. Their results for semi-infinite media can be used for short time results. 

One finds that u = —Wonn / A%D ). The interaction energy per unit area between two membranes of thickness, 2, separated by a distance, 2D is proportional to d /D when d D and is attractive. For small separations relative to the membrane thickness, D d, the interaction decays as l/D. In this case, the interaction decays with the same power law as in the case of two semi-infinite media separated by a gap of distance D, since when d D, the membranes appear to be infinitely thick compared to the gap size. Another important quantity is the self energy of a slab of thickness D. For this calculation, a cutoff on the short distance part of the energy is needed, so one can write the dispersion interaction as [r -t- where... 

The samples were freestanding films with thickness 40-100 p,m. This thickness is greater than the electromagnetic penetration depth [5 = c/ 2tt xwo-(w), where c is the speed of light and 10 S/cm. Samples with or cTdc < 10 S/cm were first checked to determine whether they had transmitted far-IR radiation. Therefore, the reflectance can be analyzed using the Fresnel reflection coefficients for semi-infinite media [103]. 

If an isotropic layer with a thickness d.2 is located at the planar interface of two semi-infinite media (Fig. 1.12), the incident wave gives rise to reflected and refracted waves in all the media except for the ontput halfspace, where only the refracted wave exists. For such an optical configuration, the Fresnel coefficients (1.4.5°) can be rewritten in the Drude (exact) form [9] as... 

TRANSMISSION OF LAYER LOCATED AT INTERFACE BETWEEN TWO ISOTROPIC SEMI-INFINITE MEDIA... 

On the basis of Eqs. (3.7) and (3.9), the dispersion relation for electromagnetic waves at the interface of two isotropic nonconducting semi-infinite media can be written as... 

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