The Heterogeneous Multiphysics of Cryogenic Vacuum-Screening Thermal Insulation. Introduction to Design

The project "Hydrogen Detectors" has features those involving many phenomena in multilayer materials making the transport of gaseous and fluid substanses through the insulation the two scales, at least, multiphysics problem. Researchers admit among many physical aspects having place in the Electro-Vacuum Thermal Insulation (EVTI) the few physical mechanisms-processes we have been studying because they have the nature of scaled phenomena.

The general structure of EVTI is the multilayer composite material with at least three distinctive scale of physical nature. There are many layers of permeable metal films on the mostly polymeric substrates, separated in turn each from another one by the low heat conductivity porous layers. Roughly it can be considered as the 3 kind of layers (materials) each with different physical and heterogeneous (morphological) characterisitics located in a periodical lattice. The three also major physical fields acting in the composites are the main concerns for modeling and optimization of the structure. These are the heat- and mass, and momentum transport throughout the structure, with addition of electrical or EM fields distribution and the radiation transport happens to be across the EVTI. Pretty much complicated physics, and as such the 3-phase, triple physics, and three-scale system had never been addressed as the VAT based modeling for itself.

The great number of facts evidence in a favor of interface physical processes and are being the important components of a process or transport. There is also evident the need to understand the physics and to model the local, non-local and multiscale characteristics in heterogeneous media.

It is again worthwhile to remind that 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 transport and effects, etc. to be considered. It is not possible to include all of these characteristics in current models using conventional theoretical approaches based on homogeneous Gauss-Ostrogradsky (GO) theorem.

We are working to make some aspects of physics and fields distribution understood better, and modeled as for the three-scale medium. We've been working with the all three major physics fields more or less for many years to have the VAT scaled physics phenomena developed, understood, and applied.

I will repeat below some of the main introductory factual material to lay down the ground as for the three-scale transport approach for the EVTI modeling.

Accomplished Milestones

In previous years we have been succeeded in developing the 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. Heat, mass and electrical transfer models have been developed for media that take into account at least two scales and three phases of the chosen material.

The new scaled Maxwell's field equations have been developed for description of phenomena in heterogeneous media.

The resulting integro-differential equations posses additional ''morpho-'' integral, and ''morpho-'' diffusive terms that are the main issues in the modeling and optimization method development.

Accurate evaluation of various kinds of medium morphology irregularities results from the modeling methodology once a heterogeneous medium morphology is chosen. An attempt was made to classify morphology irregularities associated with different specific kinds of scale morphology and to quantify the impact of irregularities on the mathematical forms of the electrostatic and magnetization governing equations in addition to the impact on the modeling results.

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.

It has been reported in a number of publications that measured values of superlattice thermal conductivities, for example, GaAs/AlAs, Si/Ge, InAs/AlSb, etc., 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 of the process. One of the used techniques is the 3(omega) measurement of thermal conductivity of superlattices.

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. Substitution of effective coefficients into models of this type is the primary question that must be dealt with.

Effective coefficient models as thermal and electric conductivities, dielectric permittivity, and reflection coefficent are done at present time on the basis of homogeneous medium governing equations. The models for coefficients constructed on the basis of homogeneous medium provisions do not reflect the most influencial and dominant physical phenomena in the heterogeneous media, as - polarizations, microscale heterogeneities, interface microfields, interplay of different effects, etc. Those described features are the part of the VAT physical and mathematical formulations of problem.

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, and dielectric permittivities in superlattices. It is shown that the issues of simulation or measurement of the effective coefficients at the upper scale are essentially the same as simulation of the complete two-scale problem in its complexity. Some of these concerns have been dealt with elsewhere. We have solved few VAT two-scale problems for superlattice transport contributed to understanding of surficial transport and its inclusion into simulation procedures on the upper scale, and the problem of interaction of charge carriers transport and heat transport on both scales.

Has been developed heterogeneous two and three scale volume averaging theory (VAT) Volumetric Heat Transport Device (VHTD) models and mathematical methods for closure of heterogeneous terms in optimization governing equations. Simulated few canonical and test morphology designs on the lower level of heat transport modeling.

The Method of hierarchical optimization of two- and three scale heat transport in a heterogeneous media of a semiconductor heat sink was formulated and developed for the purpose of hierarchical heat sink design. It is shown how traditional governing equations developed using rigorous VAT methods can be used to optimize surface transport processes in support of heat transport technology. Methods for the optimization of transport characteristics were developed and demonstrated by preliminary results for samples of the fundamental morphologies considered.

They were strictly derived for canonical morphologies. A common occurrence in porous media modeling is the inverse problem requiring the calculation of many coefficients. To obtain the coefficients for a fundamental morphology class, exact closure procedures were derived for layered, capillary, and globular type morphology descriptions at the lowest level scale.

These technical tools, developed and demonstrated, can be used to derive optimization methods for vastly more complex morphologies for VHTDs than those considered before (semiconductor heat sinks).

For the first time, the potential to control transport properties by means of recognizing and manipulating the strict dependencies between the control functions, transport properties and medium morphology (surface system as well) was made available just for heterogneous transport processes.

The same methodology we have been applying to the transport optimization for such devices as the EVTIs. Experience in hierarchical upscale modeling and experimental data reduction techniques for heat transfer and flow in porous and heterogeneous media will be a critical component in the suggested nano- microscale phenomena modeling.

This goal will be achieved using the techniques developed for describing the connection and intercommunication of the phase - interface - phase media properties between the neighboring phases and scales and will include the thermal, electrical phase and effective three-scale coefficients.

2. Few samples of the Governing Equations Used.

We are using one of the generalized forms of two-temperature second upper scale non-local VAT governing equations in heterogeneous medium of EVTI. Based on these forms as will be shown later the whole class of governing equations on micro- and nanoscale levels can be the subject of investigation and treated accordingly.

The solid phase of one of the EVTI layers, for example, averaged equation of heat transfer can be written for the second scale in the form for constant conductivity coefficient


At the same time nonlinear parabolic VAT heat conduction equation in one of the phases of the superlattice EVTI structure is



At the same time, as this area is the region of highly diverse methods and techniques the VAT presents itself the indispensable tool for purposes of evaluation and bringing together different methods and quite useful as the most valid basis for comparative validation of techniques using theory and experiment (Travkin et al., 2001a). Equation of second scale heat transport in the second, and third phase would be the same and one can obtains the three-phase temperature model which means actually the hyperbolic types of governing equations (Travkin et al., 2001a).

In our work we will include the interphase boundary thermal resistance into the VAT heat conductivity model in media with three phase morphology. We intend to simulate interface resistance using the integral additional terms in the upper scale VAT governing equations. The influence of these interphase additional terms can be sufficiently high (Catton and Travkin, 1997; Travkin and Catton, 1998).

The main idea in considering the interface zone as the physical object with a finite but very thin thickness helps in understanding why it is different from mathematically assigned surface. This construction allows to manipulate with the transport processes in grain boundaries media as in surficial entities. Meantime, it is the entity which still can be considered as the 3rd medium between known and considered two phases of the composite which can be described as


If physical evidencies are good enough to consider the interface in solid composite medium as the primarily 2D entity (with negligeble thickness ) - then equation of property transport $\Phi $ is


where $\Gamma $ is the surface density or mass per unit area (MATH, $\QTR{bf}{j}^{s}$ is the property flux within the surface $\partial S_{12}$ (along this surface). We've developed few surficial models for transport of mass, energy and EM wave propagation in interfaces (some of the results were presented at conferences - Travkin et al., 1998a,b; Travkin et al., 1999b,c). We would like to extend the surficial transport modeling capability in the current EVTI research applied toward electrodynamics and energy transport models.

The VAT based 1-D momentum equation for laminar or turbulent flow in void media of each kind of layer's material number 1 is depicted as following

where $\nu $ is the gas kinematic viscosity.

For laminar porous medium flow, the two-temperature energy equation can be simplified to MATHwhile in the neighboring solid phase of the same layer, the corresponding equation is


Meanwhile, when coefficient of conductivity (or dielectric permittivity) in the phase one is a constant value then the VAT second scale equation for potential in this phase has the form


while in the second phase of the same layer


With account for the conservation of current $\QTR{bf}{j}$ at the interface $\partial S_{12}$


the effective conductivity coefficient determined as



It is worth to note here that the known formula for the effective dielectric permittivity and conductivity of the layered medium is equal


when the electrical field is perpendicular to the interface, is easily (but wrongly) can be derived from the general expression above.

Let's just make an example of the three-phase layered morphology effective conductivity (permittivity) coefficient (assuming that all three phases are conductive (dielectric), although it is not really restrictive assumpltion), then to find out the effective MATH for perpendicular electric field, one needs to have at least the lower scale (second) potential fields solved that allows finally having





MATHwhich seems points out on the quality which first makes impact on the coefficient value, and that is the relative locations of the layers. Here is the direct connection between the relative location of the layer and its impact onto the effective coefficient via the temperature distribution within the layer and with its bounding surfaces temperatures.

The one part of the problem reviewed here with only the two-phase superlattice composite has been evaluated and discussed in this website another subsection -

  • - "Scaling Design Goals for 1D Heat Resistance Composites"

    with the thorough solution given for the electric fields (potentials) (which is valid and for the temperature steady-state distribution) and explanations layed out with the smallest detail in -

  • - "When the 2x2 is not going to be 4 - What to do?".


    $\QTR{bf}{B}$ - magnetic flux density [Wb/m$^{2}$]
    $c_{p}$ - specific heat [$J/(kg\cdot K)$]
    $ds$ - interface differential area in porous medium [$m^{2}$]
    $\partial S_{12}$ - internal surface in the REV [$m^{2}$]

    $\QTR{bf}{D\ }$- electric flux density [C/m$^{2}$]

    $\QTR{bf}{E\ }$- electric field [V/m]

    $\widetilde{f_{i}}$ $\equiv $ MATH- VAT intrinsic phase averaged over $\Delta \Omega _{i}$ value $f$

    $<f>_{f,1}$ - VAT phase averaged value $f$, averaged over $\Delta \Omega _{i}$ in a REV

    MATH - VAT morpho-fluctuation value of $f$ in a $\Omega _{i}$

    $I_{\nu }$ - radiation intensity [W/(m$^{2}$ sr Hz)]

    $I_{\nu b}$ - spectral blackbody intensity [W/(m$^{2}$ sr Hz)]

    $I_{b}$ - total blackbody radiation intensity [W/(m$^{2}$ sr)]

    $I_{b21}$ - blackbody specific surface radiation intensity [W/(m$^{2}$)]

    $\QTR{bf}{j}$ - current density [A/m$^{2}$]

    $<f>_{t}$ - time averaged value $f$

    $k_{1}=k_{f}$ - phase 1 or fluid phase thermal conductivity [$W/(mK)$]

    $k_{2}=$ homogeneous thermal conductivity of one of the solid phases [$W/(mK)$]

    $\QTR{bf}{H}$ - magnetic field [A/m]

    $m$ - porosity [-]

    MATH - averaged porosity [-]

    $n$ - refraction index [-]

    $\QTR{bf}{q}^{r}$ - radiation flux [$W/m^{2}$]

    $p$ - phase function [-]

    MATH - solid phase 2 volumetric fraction [-]

    $S_{12}$ - specific surface of a porous medium in the REV MATH [$1/m$]

    $T$ - temperature $[K]$

    $T_{2}=T_{s}$ - solid phase temperature $[K]$

    MATH - interface surface temperature when $i$ is in upward direction $[K]$


    $f$ $\equiv 1$ - phase

    $s$ $\equiv 2,3$ - phase's number


    $\thicksim $- value in phase averaged over the $\Delta \Omega _{1}$

    $\ast $ - complex conjugate variable

    Greek letters

    MATH - averaged heat transfer coefficient over $\partial S_{12}$ $[W/(m^{2}K)]$

    $\beta $ - total extinction coefficient

    $\beta _{\nu }$ - extinction coefficient [1/m]

    $\varepsilon _{d}$ $-$ dielectric permittivity [Fr/m]

    $\varepsilon _{ij}$ - radiative hemispherical emissivity from phase $i$ to phase $j$ with phase $i$ being up into the direction $\QTR{bf}{\Omega }$

    MATH - total radiative hemispherical emissivity from phase $2$ to phase $1$ in the REV

    $\lambda _{j}$ - prescribed Markovian transition length in medium $j$ [m]

    $\mu _{m}$ - magnetic permeability [H/m]

    $\nu $ - frequency [Hz]

    $\rho $ - electric charge density [C/m$^{3}$]

    $\sigma =k_{B}$ - Stephan-Boltzmann constant [W/(m$^{2}$ K$^{4}$)]

    $\sigma _{e}$ - medium specific electric conductivity [A/V/m]

    $\Phi $ - electric scalar potential [V]

    $\psi $ - particle intensity per unit energy (frequency)

    MATH - interface ensemble-averaged value of $\psi ,$ with phase $j$ being to the left

    MATH - ensemble-averaged value of $\psi $

    $\omega $ - angular frequency [rad/s]

    $\varkappa _{\nu a}$ = $\varkappa _{a}$ - absorption coefficient [1/m]

    $\varkappa _{\nu s}$ = $\varkappa _{s}$ - scattering coefficient [1/m]

    $\Delta \Omega $- representative elementary volume (REV) $[m^{3}]$

    MATH $\Delta \Omega _{f}$ - phase 1 or pore volume in a REV $[m_{3}]$

    MATH- solid phase 2 volume in a REV $[m_{3}]$


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