The proposal to NSF in 2001 reflected some ideas regarding modeling of the VAT scaled basic electrodynamics problems in few scales heterogeneous media electrodynamics. This kind of mathematical scaling statements as we shown on many occasions had not known in the conventional one scale electrodynamics governing equation statements. As it was demonstrated in many other fields the scaled formulation for the scaled physical problems opens quite new possibilities and just new physical phenomena or makes some phenomena to be observable or assessed in completely different environments. The different scale physics brings up really different scale things than as they generally assumed.
Some of the issues first surfaced in this proposal have been treated, analyzed or solved during the first half of 2000s, others are no more so important (since Maxwell-Heaviside-Lorentz governing equations hardly may be accepted for scalable electrodynamics - and all electrodynamics including school educational volume is actual for polyscale media, etc.). Some information is given at this website, still more have found no website presence and hard copy publications, mostly because of patenting related concerns.
There is a need for a method that enables one to develop general, physically based models of a group of physical objects (for example, molecules, atoms, crystals, phases, etc.) that can be substantiated by data (statistical or analytical in nature). The basis for our proposed treatment of local and non-local electromagnetic wave propagation and thermal transport in heterostructures and photonic crystals is a scaling hierarchical tool called volume averaging theory (VAT). VAT provides the tools for analysis of heterogeneous media problems on the basis of a heterogeneous theory that is based on heterogeneous versions of Gauss--Ostrogradsky theorem, not homogeneous classical mathematical models and equations. The main issue when there is a hierarchical structure is that if at any scale of physical consideration, it can be claimed or one can prove or substantiate that the coefficients are known and/or justified for the medium, then there is still often a need to know the properties of the higher levels of the material's organization. This is especially important when the problem treatment initiates at the lowest possible scale. The problem can be written within the VAT framework allowing a clear connection to be made between structure, morphologies and properties. The problem at present is that closure methods, calculation procedures and solution of VAT integro-differential equations are difficult to develop and they are not known for most problems of interest.
We propose to study heterogeneous media problems with multicomponent (two, three and more components) and three scale morphologies that are constructed in a way that should exhibit the possibility of being a multifunctional (smart) material with controllable behavior using experimental data obtained by others. This behavioral control will be exercised through the EM and temperature fields.
Our proposed technical objectives are to develop a description of the combined physical phenomena of electrodynamics and heat transport phenomena at three adjacent scales of a heterogeneous media, and to describe coupled nanoscale EM and heat transport in an inhomogeneous structure of an interface layer imitating the gradual transition of concentrations from one phase to another, study layered media with different layer morphologies using numerical simulation and scaled experimental results for effective coefficient assessment. From this, we will develop scaled hierarchical models based on VAT for a few problems of coupled heat transport and electrodynamics in layered structures including superlattices and 2D photonic crystal morphologies. Scaled hierarchical mathematical models for transport coefficients including reflectivity and transmittivity in layered media and Si/Si, SiO2/SiO2, Si/Ge superlattices will be developed. We will address the possibilities of controling the photonic medium upper scale EM response with an internal interface coating, using fillers, or by an external fields.
The need for a method that enables one to develop general, physically based models of a group of physical objects (for example, molecules, atoms, crystals, phases, etc.) that can be substantiated by data (statistical or analytical in nature) is well accepted. In modern physics this is usually accomplished using statistical and theoretical methods that are based on the Gauss-Ostrogradsky theorem (developed for homogeneous medium) to obtain governing equations. One of the major drawbacks of this widely used approach is that it does not give a researcher the capability to relate the spacial and morphological parameters of a group of objects to the phenomena of interest when it is described at the upper level of a hierarchy. Often the equations obtained by these methods differ one from another even when describing the same physical phenomena (Nigmatulin, 1987; Nemat-Nasser and Hori, 1999; Papanicolaou, 1998; DeSanto, 1998). The basis for our proposed treatment of local and non-local electromagnetic wave propagation and thermal transport in heterostructures and photonic crystals is a scaling hierarchical tool called volume averaging theory (VAT). VAT provides the tools for analysis of heterogeneous media problems on the basis of a heterogeneous theory that is based on heterogeneous versions of the Gauss--Ostrogradsky theorem, not homogeneous classical mathematical models and equations.
The main issue when there is a hierarchical structure is that if at any scale of physical consideration, it can be claimed, or one can prove or substantiate, that the coefficients are known and/or justified for the medium, then there is still a need to know the properties of the higher levels of the material's organization. This is especially important when the problem treatment initiates at the lowest possible scale. The problem can be written within the VAT framework allowing a clear connection to be made between structure, morphologies and properties. The problem at present is that closure methods, calculation procedures and solution of VAT integro-differential equations are difficult to develop and they are not known for most problems of interest.
Travkin et al. (1998b,1999b,2000b), Ponomarenko et al. (1999,2001), and Travkin and Catton (2001a) have carried out VAT based analysis of several electrodynamics problems and shown that what is calculated as effective transport coefficients in many cases are not the conventional effective coefficients. They are effective coefficients, but in a form not conventionally formulated when transient, nonlinear, interface dependent, with non-constant characteristics on the lower level components and dependence on temperature or level of EM propagation must be considered in a problem.
Some aspects of the development of the needed theory are now well understood and have seen substantial progress in thermal physics and in fluid mechanics sciences. The basis for this progress is volume averaging theory as first proposed in the sixties by Anderson and Jackson (1967), Slattery (1967, 1980), Whitaker (1967, 1977, 1997), Marle (1967), and Zolotarev and Radushkevich (1968). This approach was later advanced for application to nonlinear physical phenomena in thermal physics and fluid mechanics by Primak et al. (1986), Shcherban et al. (1986), Travkin and Catton, (1992,1995,1998a), Catton and Travkin (1997).
It is worth noting that the non-local mathematical modeling we have been developing is very different from modeling based on homogenization assumptions and other methods used in most current research efforts (Rytov, 1955; Landauer, 1978; Gubernatis, 1978; Levy and Stroud, 1997; Shalaev and Sarychev, 1998; McLachlan et al., 1990; Lindell, 1992; etc.). There is an extensive literature and many books on propagation of electromagnetic and acoustic waves in homogeneous and inhomogeneous media. All of them describe the propagation of waves based on governing equations derived from application of the homogeneous Gauss-Ostrogradsky theorem.
Solving the new VAT integro-differential transport statements that result from the application of the theory to heterogeneous media is a current issue in studies of transport phenomena. The theory allows one to take into consideration characteristics of a multi-component multi-phase media with perfect as well as imperfect morphologies and inter-phases.
Among the important features of VAT are that it allows specific medium types and morphologies, lower-scale fluctuations of variables, cross-effects of different variable fluctuations, and interface variable fluctuations effects to be considered. It is not possible to include all of these characteristics in current models using conventional theoretical approaches based on the homogeneous Gauss-Ostrogradsky theorem. Experimental measurement methods and analysis based on VAT tools of thermal physics, fluid mechanics and electrodynamics in heterogeneous media and are just beginning (Ponomarenko et al., 1999,2001; Ryvkina et al., 1998,1999; Rizzi et al., 2001; Travkin et al., 2001a,b). We propose to study heterogeneous media problems with multicomponent (two, three and more components) and three scale morphologies that are constructed in a way that should exhibit the possibility of being a multifunctional (smart) material with controllable behavior using our experimental data obtained previously and obtained by others. This behavioral control will be exercised through the EM and temperature fields.
Our proposed technical objectives are the following:
1) Develop a description of the combined physical phenomena of electrodynamics and heat transport phenomena at 3 adjacent scales. The modeling will include lower to upper scale and upper to lower scale (symbolized by ) simulation of interdependent phenomena. These scales and their interactions are:
2) Describe coupled nanoscale EM and heat transport in an inhomogeneous structure of an interface layer imitating the gradual transition of concentrations from one phase to another. The 1D problem will consider the nanoscale features of materials like Si, SiC, SiO_{2} and Si/Ge.
3) Study interface physics phenomena that are important at a scale of .................................................
..............................subcrystalline level (Fig. 1).
4) Study layered media with different layer morphologies using numerical simulation and scaled experimental results for effective coefficient assessment. Some electrophysical properties for heterostructures and possible photonic crystals like reflectance, , transmittance, effective permittivity coefficient, and layer by layer permittivity coefficients , and will be compared with experiment.
5) Simulate graded 2D media composed of .................................................. scaled models.
6) Develop scaled hierarchical models based on VAT for a few problems of coupled heat transport and electrodynamics in layered structures including superlattices.
7) Develop 2D and 3D photonic medium ...........................
..................... in these media.
8) Develop hierarchical model closure ................................................................
...............................
9) Develop mathematical methods and software for a two scale integrated solution of VAT governing equations for superlattices and 2D photonic crystal morphologies.
10) Develop scaled hierarchical mathematical models for transport coefficients of ..................................................................... Si/Si, SiO_{2}, Si/Ge superlattices.
11) Develop two scale models and suggest improvements for heat conductivity measuring techniques - .............................................................coupling of electrical-temperature fields.
12) Study 2D and simple 3D photonic medium ................................................Address the possibilities of controling the photonic medium EM ............................................
13) Study the morphology of multilayered structures and superlattice samples using optical and electron microscopy with consequent simulation of VAT morphological parameters.
14) Study the scaled problem in the 3rd ......................................................................................... ............................. globular fillers.
In our analysis particular attention will be paid to widely used electronic materials and components as Si, SiO_{2}, SiC, Si/Ge, etc. because of their importance as candidate material components for many applications. Experimental data and analysis show that the performance of these material is the result of highly coupled thermal and electrical effects. The coupled physical processes are also dependent on the material microstructural physical characteristics. The governing energy transport equations used in most past work have treated heterogeneous (and subcrystalline) heat transport, whether they are of differential or integro-differential type, as if it were in a homogeneous media. This type of idealization significantly reduces the robustness of the physical description loosing the relationship between microscale parameters and the macroscopic behavior.
The basic elements of a hierarchical medium description must account for the possibility that the mathematical representation of physical phenomena at the different scales can be very different, even if the phenomena are identical, and that the lower scale features must be transported to the upper level of description in a mode that useful information is communicated to the upper level.
The volume average value of one phase in a two phase composite medium in a representative elementary volume (REV) and its fluctuations in various directions are defined Five types of two-phase medium averaging over the REV function are defined by the following averaging operators (Whitaker, 1977, 1986; Primak et al., 1986, Travkin and Catton 1998a) where the phase and intrinsic phase averages are given by and
A full description of the derivation of the VAT non-local electrodynamics governing equations is given in Travkin et al. (1999a, 2000b), and Travkin and Catton (1999c;2001a). These equations and some of their variations are a basis for modeling of electromagnetic and coupled energy related fields at the microscale level in heterostructures. It can be seen that the most advantageous feature of the heterogenous media scaled electrodynamics equations is the inclusion of terms reflecting phenomena at the interface surfaces, and that this allows one to incorporate morphologically precisely multiple effects occurring at the interfaces.
A possible application to be addressed with scaled VAT electrodynamics approach is the formulation of models of electromagnetic waves in a heterogeneous medium of photonic crystals, for example, see Soukoulis (1993,1996), Joannopoulos et al. (1995), Ho et al. (1990). The problem of photonic band-gap in composite materials has received a great deal of attention since 1987 (Yablonovitch, 1987; John, 1987) due to its exciting potential. The most interesting applications are the design of such materials exhibiting selective, at least in some wave bands, propagation of electromagnetic energy (Soukoulis, 1993; Leonard et al., 2000).
At present two-scale photonic crystal band gap problems are formulated on the basis of single scale homogeneous Maxwell equations. The most common way to treat such problems has been numerical experiments over more or less the exact morphology of interest; often called Detailed Micro-Modeling (DMM) and often based on Direct Numerical Modeling (DNM) (for example, see Figotin and Kuchment, 1996; Figotin and Godin, 1997; Busch and John, 1998). Questions often arise about the differences between DMM-DNM and Heterogeneous Media Modeling (HMM) where the media is averaged.
There are three compelling reasons for wanting to modeled each physical scale and the connections between them with correct transformations from the lower and upper levels (if only two scales considered):
1) the ability to measure and compare field variables with those that are physically meaningful at the both scales is limited; for example, when the wave length is greater then the inclusion mean size,
2) proper theoretical models of upper scale (heterogeneous scale) phenomena are required because neither measurements nor theoretical analysis is complete at the upper scale of a hierarchy, and
3) it is difficult to suggest a technically correct way to improve the characteristics or functionality of a heterogeneous media without proper theoretical models and understanding of the phenomena at all of its important scales.
The present design of photonic media uses experimental studies and numerous investigations are based mostly on the method of plane wave expansion (PWEM), see, for example, Busch and John (1998, 1999). While these methods can be used for one cell periodic medium structures....................... .............................................................................................
These methods can, however, be used for the VAT lower scale simulation. From a pure mathematical point of view, the insufficiency of a homogeneous wave propagation description in a heterogeneous medium was addressed by Figotin and Kuchment (1998) by searching via "heuristic arguments" for another type of governing operator that could better explain the behavior of the frequency spectrum eigenmodes.
The general band-gap formulation in which the features of the whole volume of a photonic crystal are sought could be treated using two-scale HMM statements; use the VAT equations at the upper scale accompanied by use of homogeneous lower scale analysis like that of Busch and John (1998, 1999). Straightforward description of a band-gap problems has been discussed by Travkin and Catton (2001a), and Travkin et al. (2000b).
As an example, when the dielectric permittivity function is homogeneous in each of two phases, then the VAT upper scale photonic band-gap equations can be reduced to one equation in each phase and written in a more simple form for phase 1:
For the third scale, ...........................................................
.....................................................................we propose to first complete the development of....... ................................................ models with exact closure for physical phenomena like heat conductivity in disperse media, convective conjugate heat transport in capillary morphologies and in semiconductor heat sinks, electrostatic fields in layered and capillary morphologies and EM wave propagation and reflection in regular graded surface layer that are under development for theoretical and experimental purposes (Travkin and Catton, 1995, 1998a; Catton and Travkin, 1997; Travkin et al., 1999a 2001a, 2001b, 2001c). .........
......................................................... There are existing VAT methods to connect the transport models for the 1st, 2nd, and 3rd scales independently. Using these techniques it is possible to simulate 3-scale transport of charge, heat and electromagnetic waves in superstructures.
Recent discussions of mechanisms of high level EM transmission through subwavelength gratings (Ghaemi et al., 1998; Sönnichsen et al., 2000; Treacy, 1999; Porto et al., 1999) brought out the importance of processes at interface surfaces in a heterogeneous medium. We have developed several surficial models for transport of mass, energy and EM wave propagation at interfaces (some of the results were presented at conferences, see Travkin et al., 1998b,c;1999b; Travkin and Catton, 1999d) for cases where the property transport equation for a field .....................................
..............................................................................................
....................................................................We would like to extend the third (surficial) VAT modeling capability developed in our current research to superstructure electrodynamics and energy transport.
It has been reported in a number of publications that measured values of superlattice thermal conductivities, for example, GaAs/AlAs, Si/Ge, InAs/AlSb do not compare well with expected or modeled values. There are questions about measurement techniques that are used and some improvements have been suggested for simulation. It is well known that the scale of measurements and of the modeling must correspond one to another. This obvious and simple principle is violated when what is clearly a two scale physical problem is described on the upper (measurement) scale with the same kind homogeneous mathematics as is used for the lower scale. The substitution of effective coefficients into models of this type is the primary question that must be dealt with.
In our efforts to relate the scaled volume average theory (VAT) description and simulation of semiconductor heat sinks heat transfer (heat--sink-to-air) to experimental measurements, we developed a process of coupling two scale DMM-DNM and their corresponding experimental results for a semiconductor heat sink design (Travkin et al., 2001a; Rizzi et al., 2001). At present, efforts to advance the theory and models of heterogeneous media transport based on a VAT description into experimental practice is in its initial stages. Only a few studies exist in this area. In their recent studies, Ryvkina et al.(1998,1999) and Ponomarenko et al.(1999a,b) outlined a few of the more frequently addressed issues of VAT application to heterogeneous ferrite media experimental measurements of effective conductivities and permittivities in single and composite media. Studies of optimization of semiconductor chip heat sinks by Travkin et al.(2000a, 2001b) and Rizzi et al. (2001) are based on a scaled approach that allows the connections between the homogeneous lower scale heat transfer and the convection inside the heat sink and the upper scale bulk (actually averaged) fields of momentum, energy transport and general characteristics of effectiveness of the device to be made.
The challenge at the present moment is for experimentalists to begin to realize the difference in data obtained when medium is, in essence, heterogeneous. Methods and procedures for reduction of experimental data in heterogeneous media through application of scaled VAT will be addressed.
In DMM-DNM, the mathematical statements usually deal with local fields and as soon as the boundary conditions are stated in some way, the problem is formulated correctly and can be solved exactly as, for example, is done by Cheng and Torquato (1997). Difficulties arise, however, when the result of this solution needs to be interpreted, as is the case for a majority of problem statements in heterogeneous media, in terms of non-local fields that are averaged in some way. The averaging procedure is usually either stochastic or spacial volumetric integration. Almost all of these averaging developments are done incorrectly due to disregard of averaging theorems for differential operators in heterogeneous medium (see Travkin and Catton, 1998a). A standard definition of an effective (macroscopic) conductivity tensor is determined from the following equation
The effective permittivity coefficient field, for example, for a composite or a nanocrystalline material with constant phase coefficients is given by
Even though both above formulae are based on constant phase coefficients, they still need to incorporate the fields of averaged variables of temperature fluxes or electric fields in both phases These functions are not merely statistically (ensemble) or volumetrically averaged or and. They should be found through properly constructed sets of models for the upper fields, for example, for (confinement features included) and often lower and scales (Travkin and Catton, 1998a,1999a,b, 2001a; Travkin et al., 1999b). Numerous studies have been devoted to these findings. Further, the effective coefficients will be different for stationary and time-dependent fields. It is obvious from this why more attention needs to be placed on obtaining more exact evaluations of these fields , as was done for thermal fields.
We will discuss two possible techniques to measure and model multilayer films as scaled hierarchical objects. One is the Scanning Laser Thermoelectric Microscope (SLTM) technique (see, for example, the recent experiments performed by Borca-Tasciuc and Chen, 1997, 1998; Borca-Tasciuc, 2000) used for measurement of thermal conductivity and diffusivity of thin films. The equation of heat conduction in a homogeneous film used for data reduction for the superlattice,
Another method used for assessment of heat conductivity coefficient in superlattice structures is the 3 technique (Cahill et al. 1989; Cahill, 1990; Cahill et al., 1992).
Equations of heat transfer in each i-th layer was used in the form
The electric field equation in each of the layers of a superlattice can be described by two forms (with the constant coefficients ),
As the number of layers in the superlattice can be substantial, the actual response of the superlattice and it's temperature and heat conductivity coefficients become bulk (averaged) quantities. The volume of averaging can reach a proportion of a superlattice thickness in a cross-section, see Figs. 1-2, and a number of mathematical consequences and non-local models can be derived with the simplest set of governing equations for the two-component superlattice.
The full two-scale heat transport and electrodynamics governing equations were used to achieve understanding of the possible mechanisms that play a role in shaping the effective (measured) coefficients of thermal and electrical conductivities in superlattices (Travkin and Catton, 2001b,c). It was shown that the complexity of simulation or measurement of the effective coefficients at the upper scale are essentially the same as simulation of the complete two-scale problem.
Analysis shows that averaging of known linear solutions at the lower scale do not correspond to the stated problem at the upper scale. This finding changes an approach for treatment of this problem (Travkin and Catton, 2001c). Further simulation of this problem, including coupled heat transport and electrodynamics in heterogeneous media will be addressed. Some of these problems have been dealt with elsewhere. We will contribute to the understanding of surficial transport and its inclusion into simulation procedures at the upper scale, and the problem of interaction of charge carriers transport and heat transport at both scales.
We will use our past bulk experimental data and experiments by Chen and co-authors (Borca-Tasciuc and Chen, 1997, 1998; Borca-Tasciuc, 2000) to develop methods to investigate structural, electrodynamic, polarization and thermal properties of amorphous, nanocrystalline and polycrystalline heterostructures in single and laminated samples of different materials (like Si, SiO_{2}, SiC, Si/Ge,) using the VAT. Interrelation of structural and electrodynamic properties are expected to be established. Special attention will be paid to the effect of different fields on the physical properties, and their relationship to the actual hierarchical structure of the material.
Methods for controlling electromagnetic wave propagation in composites by varying the temperature fields and the scale of order-disorder in hierarchical levels will be investigated. We will recommend improvements for experimental procedures and data reduction will be made for the SLTM and 3 experimental techniques. We will use these improvements to perform some test data reduction using available experimental data.
Models for effective coefficients like dielectric permittivity, conductivities, magnetic permeability, and reflection coefficient constructed on the basis of homogeneous medium provisions do not reflect the most influential and dominant physical phenomena in the heterogeneous media. Some of these physical phenomena are polarization, microscale heterogeneity, interface demagnetization microfields, and interplay of different effect. VAT leads to fundamentally different governing equations that present an opportunity for formulation of new modeling and simulation procedures that include these important phenomena as well as for the design optimization. A rigorous theory for addressing multi-scale, multi-phase transport phenomena problems in electrodynamics has been proposed on the basis of heterogeneous volume-surface Gauss-Ostrogradsky theorems to treat various aspects of electrical along with thermophysical transport in these media. VAT physical and mathematical formulations of a problem are rigorous and allow greater inclusion of important phenomena enabling accurate evaluation of various kinds of medium morphology irregularities once a heterogeneous medium morphology is chosen.
- specific heat []
- internal surface in the REV [m^{2}]
- electric flux density [C/m^{2}]
- electric field [V/m]
- VAT intrinsic phase averaged over value
- VAT phase averaged value , averaged over in a REV
- VAT morpho-fluctuation value of in a
- magnetic field [A/m]
- phase fraction [-]
- specific surface of a porous medium [1/m]
- temperature
Subscripts |
Superscripts |
- fluid or phase 1 | - value in phase 1 or 2 averaged over the or |
- solid phase | - complex conjugate variable |
- dielectric permittivity [Fr/m];
- magnetic permeability [H/m];
- electric charge density [C/m^{3}];
- medium specific electric conductivity [A/V/m];
- angular frequency [rad/s];
- representative elementary volume (REV) [m^{3}];
- phase 1 volume in a REV [m^{3}].
We have had some success in our recent proposal related studies (funded by the DOE Office of Basic Energy Sciences) that support the current proposal:
1) We developed physical and mathematical models of multiscale heterogeneous electrodynamics (Ponomarenko et al., 1999,2001; Travkin et al., 1999a,2000b; Travkin and Catton, 1999c,2001a). The physics of polarization were introduced into the upper scale electrodynamics-heat transfer governing equations. This presents an opportunity to simulate exact polarization processes in and around a matrix of submerged objects together with local-nonlocal electrodynamic fields.
2) We developed scaled concepts to address the issues of nanoscale multiphysics heat conductivity measurement techniques in electronic materials. The two methods usually applied toward these tasks are made in terms of hierarchical scaled theory of VAT (Travkin and Catton, 1998a, 2000a, 2001b,c). Some of these issues will be addressed in our proposed work.
3) We developed the fundamentals of a two-scale heterogeneous experimental technique for parameter measurements and design. We used this technique for evaluation of devices of practical importance like semiconductor heat sinks. We obtained exact relationships between improvements in performance and widely used industry parameters of performance and showed their flaws in terms of assessment and comparison with physical laws (Travkin and Catton, 1998a, 2000a, 2001b,c; Rizzi et al., 2001). In our efforts to relate the scaled volume averaging theory (VAT) description and simulation of heat transfer devices (heat sinks) to experimental measurements, we developed a process of coupling two scale detailed micro-modeling - direct numerical modeling (DMM-DNM) and their corresponding experimental results for a devices as heat sink. We described in detail how, and for what reasons, the measured data are to be simulated or measured and represented in a way that allows design goals to be formulated primarily with averaged (or bulk) physical characteristics. We demonstrate why studies of only averaged local integrated variables are not enough.
4) We developed the governing equations for thermal physics and fluid mechanics in hierarchical materials with exact closure and solutions for the VAT specific globular and capillary morphologies for a two-scale problem (Gratton et al., 1995,1996; Travkin et al., 1994;2001b; Hu et al., 2001; etc.). We demonstrated their value for important applications like membrane transport.