Mathematical Models for Electrostatics of Heterogeneous Media

Since most of twenty years intensive investigations, in Physical Chemistry and applied Physics, have been directed to various basic processes for complexly structured material systems, with a strong accent on surface phenomena problems. Such systems, known as heterogeneous, consist typically of different bulk phases, separated by specific interfaces, which can be also realized as containing distinct but near by sub-phases. Additionally, the common line contours, of 2D sub-phases, are treated as materially autonomous (1D) phases as well. One of the main directions in said topics consider surface nucleation phenomena in gas-liquid systems. The interest here has been provoked mainly from the open questions for the mechanism of the surface nucleation – in particular in lipid systems. Said questions essentially concern basic topics of Physical Chemistry, related also to ecological applications. As a second class, note the problems on structure building of semiconductor films of aircrystal media, via an actual technological interest: it primarily concerns the main factors governing the growth and roughness of semiconducting surface films. Another (third) class of related topics is shown by the recent studies on cell biology problems (e.g. [4]). This class includes also mathematical models for detecting of anomalies in the human organic systems – for instance, the blood circulatory system and that of the white liver oxygen transfer.


Introduction
Since most of twenty years intensive investigations, in Physical Chemistry and applied Physics, have been directed to various basic processes for complexly structured material systems, with a strong accent on surface phenomena problems. Such systems, known as heterogeneous, consist typically of different bulk phases, separated by specific interfaces, which can be also realized as containing distinct but near by sub-phases. Additionally, the common line contours, of 2D sub-phases, are treated as materially autonomous (1D) phases as well. One of the main directions in said topics consider surface nucleation phenomena in gas-liquid systems. The interest here has been provoked mainly from the open questions for the mechanism of the surface nucleation -in particular in lipid systems. Said questions essentially concern basic topics of Physical Chemistry, related also to ecological applications. As a second class, note the problems on structure building of semiconductor films of aircrystal media, via an actual technological interest: it primarily concerns the main factors governing the growth and roughness of semiconducting surface films. Another (third) class of related topics is shown by the recent studies on cell biology problems (e.g. [4]). This class includes also mathematical models for detecting of anomalies in the human organic systems -for instance, the blood circulatory system and that of the white liver oxygen transfer.
The electrostatic properties of matter have been taken as the basic framework for investigations of surface phenomena problems. Especially, adequate expressions have been sought for the electric potential -as the key quantity, integrating the basic electrostatic parameters of medium. Our main goal here consists in finding such expressions, primarily concerning the interface potential of complex (3-2-1D) heterogeneous systems. Said aim yields the key question how to construct a proper mathematical model of the matter electrostatics, which introduces a correct problem for the electric potential. Secondly, it is necessary to do the main steps in the mathematical analysis of the relevant problem. Here we propose an answer of the above question, taking into account two required basic steps. The first one consists in introducing the object called heterogeneous media, when in reality we have given a material system of different bulk (3D) phase, for instance -gas and liquid, with a relatively thin transition layer (say emulsion). In our treatment an additional stage of heterogeneity is presumed: the bulk transition layer consists also of two near-by sub-phases (of differing matter). According to the Gibbs idealizing approach ( [6]), we have to consider www.intechopen.com  takes the form of the known Boltzmann distribution, for instance in case of electrolytes. Because the Gibbs idealization assumes a step transition across the interface (consequently again such transition, but of lower dimension, is assumed across the phase contour l ), the next appearing problem reads: how to formalize said step transitions, in order to use effectively the Maxwell system. A useful suggestion for the first transition stage (across the interface) can be found in the monograph of D. Bedeaux and J. Vlieger ( [2]). According to Bedeaux [3]). In the above generality of formulation, problem (0.1) -(0.3) remains however as open one.
In this chapter we give the main steps of deriving and solve two sub-cases of the general problem (0.1) -(0.3), related to heterogeneous systems with flat interfaces, respectively of semiconductor and organic nature. A straight line contour l enters in both the models as 1D phase of anomalies. Note that the model, with a defect line on the semiconductor interface, is closely related to real experimental data, found by scanning tunneling microscopy. On the other hand, in the case of organic interface (considered in our second model as the known lamina basale), the lamina folio is supposed cleft in two sub-phases by a (straight) line of functionally anomalous intercellular spaces (holes enter nonlinearly in the cases, respectively of semiconductor and organic interfaces. By transforming relations (0.2), (0.3), we derive and solve the relevant integral equations for the surface potential S u . We obtain also effective formulas for certain approximations of S u (in the case of S -a semiconductor folio), and -for the exact potential (in the case of organic S ). Recall here that the contemporary problems primarily focus in detecting the surface values of the electric potential. As a consequence of finding the surface potential, we express in addition, by the classical Dirichlet problem, the bulk potentials u  as well. Thus we get expressions for potential u , valid far from the interface, which ensure in particular important diagnostic analyses (made, for instance, by parametric identification inverse problems).
In Sect. 2 we give phenomenology comments and a common derivation of the two considered models. The basic results on the surface potential are presented in Sect. 3. The question for determining of explicit approximations to (,,) uxyz is discussed in Sect. 4, in case of semiconductor interface.

Elements of phenomenology and mathematical modeling
Let us give firstly some phenomenology comment on said two classes of heterogeneous media. Beginning with the case of organic interface, we should note the following. The interest of tools for biomedical detections of anomalies in the human circulatory system, via the walls-structure of the blood vessels, is directly motivated from the quite specific ruling function of the wall-layers. To recall and clarify the main (simplified) viewpoints here (cf. e.g. [11]), we assume a stretched location of the wall, as the flat surface ( 0 z  ) on Fig. 1, below. Said construction is introduced as an admissible version of the real situation: a 3D localization is made to a capillary (practically cylindrical) vessel in the human white liver and the vessel-wall is (functionally) identified with its middle layer (called lamina basale). In reality the wall is a 3D organic threefold layer, deep not less than 120-180 nm and the midmost (just the lamina basale) is of corpulence about 40-60 nm. The upper (external) and lower (internal) layers, built -as a short description -respectively of endothelian and adventitale cells, are neglected. They are considered with a secondary role (compared to lamina basale). As known, acting as a typical bio-membrane of polysaccharide-matter, with a fine fibers structure, lamina basale is the main factor for the oxygen transfer to the blood. Via the Gibbs approach (and taking into account the ratio of the wall corpulence to the radius of the capillary vessel, which is 1  ), we consider lamina basale layer as infinitely thin; thus we get an interface film of organic matter in an air-blood (vacuum-blood) heterogeneous 3D media. On the lamina (2D) film it is uniformly distributed a set of pointspresenting the holes (tunnels, in reality radial to the vessel axis and known as intercellular spaces), which provide the oxygen contact to the blood; they are assumed however of certain functional anomaly, extremely activated on a relatively narrow (cylindrical, in reality) strip, interpreted, following Gibbs, as the middle circumference of the strip, across to the vessel-axis. Thus a specific homogeneous 1D matter phase (of the extreme anomalies) has appeared. Stretching (locally) the curved anomaly-line (and the surrounding cylindrical surface, together), we get the above mentioned (flat-interfaced) construction. Now the organic lamina-film can be presented as the plain 0 z  (regarding a Cartesian (,,) xyzcoordinate system), where said 1D contour, of defective air permeability, has already shaped as a straight line. We shall take this line as the Oy -axis. This manner the laminafilm is cleft in two electrostatic equivalent 2D (sub-) phases by the anomaly contour and we have given a typical case of 3-2-1 D heterogeneous system, schematically shown on Fig. 1, below. The system consists in upper and lower 3D (bulk) phases, respectively of air and liquid, and a complex-structured organic interface (with a special role of a line phase). The bulk phases B  , B  fill the subspaces 0 z  (B  ), 0 z  (B  ) and their common 2D boundary -the organic interface S -is given (as already noted) by the equation 0 z  ; S consists in the two neighbouring surface phases It is possible however a sharp variant of anomalies: an air volume can leave involved between the blood and the surface, shaping an internal (lower) air bulk phase. Such presence of two-side bulk air phases is due to the anomalous air transfer: the outgoing stage gets blocked, after a previous air invasion. We would have then a heterogeneous system with upper and lower air (vacuum) bulk phases, a two-phased lamina-interface (as a 2D film) and a separating the surface phases homogeneous 1D material (straight line) phase.
Another class of heterogeneous media is that including a semiconductor interface. The model under consideration relates to electrostatics for the specific case of air-gas matter, with a semiconductor separating surface (interface), which includes moreover a defect straight line (considered below as the Oy -axis, see Fig. 2). In said case of systems the importance of the interface electrostatics is motivated, as already noted, from actual technological questions (e.g. [5], [7]).The structure of such a system can be explained as a space location of given electronic device. For a short description we will take into account the following. By a teen boundary wall of semiconductor-matter it is closed a volume of gas, and the external medium is of air. This boundary, generally curved, will be treated here as flat (observing a small part of it). Thus the system possesses two bulk phases -of internal (gas) and external (air) media, and a flat semiconducting interface, with a fine surface www.intechopen.com www.intechopen.com roughness, as a straight line defect. The bulk phases are considered as materially equivalent (3D) sub-domains of vacuum. The separating boundary is of indium-phosphorus, InP(110), semiconductor: the real corpulence of the InP(110)-wall is neglected and the wall is identified with its external surface film. The defect line, playing the role of a homogeneous detachment, i.e. of an electrostatic autonomous material 1D phase, separates said interface in two surface (2D) phases, denoted as before by S  ( 0, 0 xz   ), S  ( 0, 0 xz  ). It is posed again a typical case of 3-2-1 D heterogeneity -by a material system, interpreted as vacuum-semiconductor-vacuum (Fig. 2). The InP(110) surface film is presumed as the plain 0 z  as well (see Fig. 2) and the Oy -axis is oriented on the defect straight line. The vacuum bulk phases fill the upper and lower semi-spaces, 0 z  and 0 z  , respectively. Each 2D phase (on 0 z  ) is characterized by an essentially dominating distribution of positively charged phosphorus vacancies, while, as a key anomaly, the line phase l ( lO y  ) enters in the surface electrostatics symmetrically surrounded by an extremely narrow strip of width 2d (0 d  ). The whole this band is denuded of phosphorus vacancies ( [5], [7]). The above construction is essentially supported by real experimentally found data. A credible visual result of [5] and [7] (see Fig. 2, [7]) has been found by the so-called scanning tunneling microscopy. The picture (Fig. 2, [7]) shows the surface structure, fixed after annealing of InP(110) samples at temperatures up to 480 K, followed by heat normalizing. The scanned image includes two near by surface domains (let us denote them by , consider T as an equipollent (say to , PP   ) 2D surface component. Moreover, the two relations -that of T -wide (10 nm) to the above density unit (5.5 nm), and the other -of possible (averaged) electrostatic impact of T to the influence of its middle axis l , allow some identifying of the strip T with the axis l ; thus l takes the role of an intrinsic 1D phase. Let us note that, via the Gibbs approach, the semi-zones ( ( 0), ( 0 phases however enter also negligibly in the surface electrostatics: across , dd ll   the surface electric field stays continuous, under equal permitivity-values s  . So-described picture (of a smooth, flat 2D film) represents a real surface layer with certain nanoscopic roughness, due to step defects. In reality l is actually the edge of a step, deep about 4 nm ( [7], Fig. 1), P  and P  are the terraces (lower and upper, say) of the step, and T is a space construction, divided by l in two halves, T  (lower) and T  (upper), marking off the edge from the relevant terraces.
The next stage of this section is to sketch the basic step of modeling. Via the introduced framework (see Sect. 1) the key tool for description of electrostatic phenomena in complex media relates to the Maxwell system (in case of dielectrics, e.g. [10], [12]): Here  is the nabla operator, D is the vector of the electric induction ( [12]), called also (in Electrochemistry, e.g. [9]) electric displacement, .  D is the formal scalar product of the vectors nabla and D, i.e. .  D div  D;  is the charge density;  is the relative dielectric permitivity for the relevant part of the medium (in particular , where () u  represents the electric field, propagated in the whole 3D material system. Equations (2.1) hold for the total (3D) system and, as known, potential u is a continuous function of (,,) xyz , in spite of the various material phases; the heterogeneity of the system is indicated however mainly by the quantities D and  (note that the permitivity  enters in these quantities). Next, from the singular decompositions, mentioned in Sect.1, applied below for quantities D and  , we get the following problem. Find the (admissibly regular) solutions (D, u) to (2.1), corresponding to the said singular decompositions.
Both the considered cases of heterogeneous systems are however homogeneous on the ydirection, due to assumed homogeneity of the 1D phase l , and the electric potential (,,) uu x y z  will actually depend on , Applying systematically a double decomposition scheme in reworking of the Maxwell system (see below), we shall establish the following final formulation to the sought mathematical models: .) As known, parameters b k , b  and , ss k  are the main factors of the system-electrostatic nature; they are given step constants: , while a linear approximation (regarding potential u) is assumed adequate to reality out of the strip.
Via the phenomenology-essence potential u will be searched for a bounded function (condition (2.3)), continuous in 3 R , classically regular in the sets 0 z  and 0( 0) xz   , with continuous From the vacuum assumption for the upper (external) air phase ( 0 z  ) we suppose from now on: The central results below relate to determination of the surface potential (possibly by an explicit approximation) -as the key first step in solving the full (2.2) -(2.6) -problem.
Let us now sketch the main steps for derivation of the final mathematical models, starting from the Maxwell system (2.1). Via the presumed complex heterogeneity, we shall seek solutions (D, u) of system (2.1) by decompositions in two levels (bulk and surface), of the following type: (2.10) In the above relations ( ) z are respectively the Heaviside forward/backward functions (i.e.
( ) 1 z . . A preliminary motivation to do that follows from the argument that the polysaharide matter of the interface admits to consider it as a lipid medium, where the potential-magnitude can be assumed relatively smaller than the basic ratio (RT 0 )/F, which yields that linear approximations become acceptable (F, R, T 0 are -as follows -the so-called Faraday and gas constants, and the absolute temperature , forecasting possible unknown complications, close to the line contour. The next main step of modeling consists in some reworking to system (2.11) -(2.13). By the right hand sides from (2.11.b) we firstly express ( )  D b , ( )  D b in (2.11.a) and come to the Helmholtz-Laplace equations from (2.2). As noted, condition (2.3) corresponds to the physical nature of the potential (to be a space-bounded and continuous quantity). Going to the next relations, (2.4.a), (2.6.a), they show that potential stays continuous across the transition surfaces and lines. On the other hand condition (2.5) introduces the asymptotic value of the surface potential (, 0 ) ux -they are considered as experimentally known (gauged) data. Afterwards we replace D s in (2.12.a) by the right hand side of (2.12.b) and use that This completes the sketch of derivation to final form (2.2) -(2.6) of our mathematical models.

The basic integral equation and finding of surface potentials
We shall reduce in this section problem (2.2) -(2.6) to a corresponding (nonlinear) integral equation -as a background for finding of explicit type presentations to the surface electric potential. Recall firstly the supposed electrostatic equivalence of the surface phases, which yields that sss    , sss kkk    .
For the needed technical reworks the x -Fourier transformation is systematically taken into account below -by well known conventional expressions (e.g. as in [8]). By the x -Fourier transformation to the relations in (2.2) we find ordinary differential equations (regarding z), which yield the following presentation (for a classical solution (,) uu x z  to problem (2.2) -(2.6)): It is denoted here by ˆ(,) uz  the (partial) Fourier transformation of (,) uxz -with respect to x .
In ( We have denoted by  and  respectively the first and second derivative of () x  , and by ()

Proposition
For arbitrary non zero asymptotic mean value * Thus we have given the proof of the following basic result.

Proposition
For arbitrary values of positive parameters s  , s k , each asymptotic data    of the surface potential and line charges

Explicit approximations in the case of semiconductor interface
Via the possible applications, it is important to ask for a suitable approximation 0 () x