THE NON-LOCAL FORMULATION OF ELECTROSTATIC PROBLEMS FOR SENSORS HETEROGENEOUS TWO- OR THREE PHASE MEDIA, THE TWO-SCALE SOLUTIONS AND MEASUREMENT APPLICATIONS. TO BE CONTINUED. SEE ALSO: ISJAEE. 2005. N 3. P. 7-17 Текст научной статьи по специальности «Физика»

ВОДОРОДНАЯ ЭНЕРГЕТИКА И ТРАНСПОРТ

Газоаналитические системы и сенсоры водорода

HYDROGEN ENERGY AND TRANSPORT

Gas analytical systems and hydrogen sensors

THE NON-LOCAL FORMULATION OF ELECTROSTATIC PROBLEMS FOR SENSORS HETEROGENEOUS TWO- OR THREE PHASE MEDIA, THE TWO-SCALE SOLUTIONS AND MEASUREMENT APPLICATIONS

V. S. Travkin \ A. T. Ponomarenko*

1 Member of International Editorial Board 11 Member of International Editorial Advisory Board

Hierarchical Scaled Physics and Technologies (HSPT) 10431 Larwin Ave., Chatsworth, CA 91311, USA E-mail: travkin@iname.com

Enikolopov Institute of Synthetic Polymeric Materials, Russian Academy of Sciences Profsoyuznaya, 70, Moscow, 117393, Russia

Nomenclature

C — mass fraction concentration; B — magnetic flux density [Wb/m2]; cd — mean drag resistance coefficient in the REV; cd — mean skin friction coefficient over the turbulent area of dSw; cp — specific heat, J/(kgK); ds — interface differential area in porous medium, m2;

dS12 — internal surface in the REV, m2; D — molecular diffusion coefficient, m2/s; D — electric flux density, C/m2; E — electric field, V/m;

f ={fi} — VAT intrinsic phase averaged over AO, value f ;

(f)f — VAT phase averaged value f, averaged over AO, in a REV;

f — VAT morpho-fluctuation value of f in a O,;

{f)t — time averaged value f ; j — current density, A/m2;

k1 = kf — fluid phase thermal conductivity, W/(mK);

k2 = ks — homogeneous thermal conductivity of solid

phase, W/(mK);

H — magnetic field, A/m;

m — porosity;

(m) — averaged porosity;

n — refraction index;

p — pressure (Pa) and phase function;

Q — electrical power from heater dissipated through the specimen;

(s2 ) — solid phase fraction;

S12 — specific surface of a porous medium dS12/AO, 1/m;

T — temperature, K;

T2 = Ts — solid phase temperature, K;

Tj — interface surface temperature when i is in

upward direction, K; U — vector component in x-direction; V — vector component in y-direction; W — vector component in z-direction.

Subscripts

c — charge; e — effective; eff — effective; ex — experimental; f = 1 — fluid phase; 1 — first phase;

i — component of vector variable; L — laminar; r — roughness; s = 2 — solid phase; T — turbulent; w — wall.

Superscripts

--value in fluid phase averaged over the

phase REV AOn;

Статья поступила в редакцию 04.01.2005. The artisle has entered in publishing office 04.01.2005.

* — equilibrium values at the assigned surface and complex conjugate variable; — fluctuation value in a phase.

Greek letters

12'

12'

a 21 — averaged heat transfer coefficient over dS W/(m2K);

a C — averaged mass transfer coefficient over dS m/s;

ed, em — dielectric permittivity, Fr/m; |j.m — magnetic permeability, H/m; v — frequency (Hz) and kinematic viscosity, m2/s; pc — electric charge density, C/m3; p — density, kg/m3;

o — medium specific electric conductivity, A/V/m; O — electric scalar potential, V; ra — angular frequency, rad/s; AQ — representative elementary volume (REV), m3; AQ1 = AQf — pore or phase 1 volume in a REV, m3; AQ2 = AQQ — second or phase 2 volume in a REV, m3; Tw — wall shear stress, N/m2.

5. Some applications of the non-local VAT formulated problems for sensor's modeling properties and measuring techniques

5.1. Traditional local and piece-wise distributed coefficients electrostatic problem formulations

In DMM-DNM as, for example, for dielectric medium statements usually can be used equations (see [11])

V(e(r)VO(r)) = 0, reQ, i = 1,2, (76)

where the dielectric coefficient e is the piece-wise function

e(r ) = exy(1) (r ) + e2 y(2) (r )

(77)

and T(i) is the characteristic function of phase i. Interface boundary conditions assumed for these equalities are

«0 (r) = $2 (r), re^, (78)

ei (n VOi (r)) (n VO2 (r)), r e^. (79)

Almost the same traditional notation has the conductivity problem in heterogeneous medium see, for example, the review in [8]

V(o(r )VO(r )) = 0, r eQ, (80)

with the boundary condition on the bounding surface dS (or part of this surface)

n •(j)las = jn (S). (81)

5.2. Bulk effective coefficients modeling for the two-phase medium

Starting, we choose the conductivity problem and first will be treating the example of constant phase conductivity coefficient conventional equations (80) for the heterogenous medium.

As shown above this mathematical statement is incorrect when the equation is applied to the volume containing both phases, even when coefficient o(r) is taken as random scalar or tensorial function.

Conventional theories of treatment of this problem do not specify what is the meaning of the field O, assuming that it is the local variable, or -O = O(r) , where at the point r there is the point value of potential O exists.

Next, the analysis shows that the coefficient

0 = o(r), as long as in each separate point r there is exists the local y with the value either of phase 1 or phase 2, and in each of the phases the value of

01 is constant.

In the DMM-DNM approaches the mathematical statement usually deals with the local fields and as soon as the boundary conditions are taken in some way, the problem became formulated correctly and can be solved exactly as in work by Cheng and Torquato [11], for example.

Difficulties arise when the result of this solution needs to be interpreted — and this is in the majority of problem statements in heterogeneous media, in terms of non-local fields, but averaged in some way. The averaging procedure usually is proclaimed in one of the fashions — either doing 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.

Secondly, as soon as the boundary surfaces are intersecting both of the phase subvolumes, the conventional boundary conditions become invalid as well as the DMM-DNM solution. In the event when boundary surfaces are chosen to belong to only one of the phases in a random or irregular morphology medium the problem becomes formulated as in a special kind of periodic medium and its solution has peculiarities and useful value for the problem stated only with one phase boundary condition. When a problem's physical model includes phenomena of wave propagation through the boundary of heterogeneous body as, for example, in most of electromagnetic problems, the one phase boundary condition has no value for that problem.

Further, a more complicated situation arises when the intention is to formulate and find effective transport coefficients in heterogeneous medium. Let us consider the conductivity problem in a two-phase medium. According to most accepted mathematical statements this problem is given on the lower (local) scale as (80) and (81).

Meanwhile, if the coefficient of conductivity in the phase one is a constant value then the upper scale VAT equation for potential field in this phase is

/

v2(fa)фi)+V- -1- J ФА

+ — J Ф1 - dsi = 0,

AßdS 1 1

which has one more term in the case of the inho-mogeneous or nonlinear conductivity in phase one

/

V-^V^i)Ф1 ))+V- Ö!A^J Ф^

dS,2

+V - (ô1V<p}1 + aQ J G1V01 - ds1 = 0. (83)

'dSv,

The constant current flow equation in the phase one in terms of potential flux (o1V^) when o1 is a constant value looks like

V-(01V01 )) =

= V o^V0^1 +J V<-dsx =0, (84)

and in the second phase the same kind of potential equation

(V • (o2V02 )2 = V- o2 (V0^2 + ^ J V<0 • ds2 = 0.

Observing these equations one can assume that the effective conductivity coefficient in a medium should come from the equation of sum of the previous equations for each phase which can be reduced consequtively accounting for the conservation of current j at the interface dS12

as to

n - j1 las12 = n - j2 las12, П -C1V°1 k = П -G2V°2 ldS12

V- [o^ VO) 1 (VO) 2 ] = 0,

(85)

(86)

or

V- [o^fo)ф )] + V- AQ J

+V-[o2V(m^ Ф 2 )) +V

—— J Ф2 ds2

AQj 2 2 dS12

= 0,

which results in

V-[oiV((i) <01 ) + 02V(m^ <5 2) + ,

+ (°1 -°2)m J ^i = 0, (87)

because O = 0, and ds-, =-ds,, also the same

1 z I ooi2 2 1

kind of expression can be obtained

V-[oiV((mi) <01 ) + 02V(m^ <5 2) + ,

= 0,

+ (o2 -o1 )aq J ф2ds,

AQ 3S12

(88)

assuming that the phase two is the globular inclusions phase. Standard definition of effective (mac-

roscopic) conductivity tensor determines from the following equation

j = 0* (VO), (89)

in which assumed that the current vector can be presented as

j = >(r)V< = - [01 <V°) 1 + 02 (V°)2 ] =

= -o* (V<i) = -o* V(«&)

as soon as

VФ (r )) = ^Ф) 1 + ^Ф) 2 =

= V(( 1 +(Ф> 2 )=^Ф(Г ) ,

because

(то) 1 = V(m)ф 1 )+AQ J<Ms

(90)

(91)

(92)

3S12

V«&)2 =V(m^Ф2) + J Ф2ds2, ds1 =-ds2, (93)

AQ3S19

L J фД J ф2ds

° J AQ-

AQ

(94)

Further, we assume that

j = j 1 + j 2 =-°1 \ -o 2 W2 =

= -o„

(vФ)=-o;. [(vФ) 1 +<то) 2 ]=

= -o * ^Ф X -o * <TO> 2,

(95)

and, for the usually assumed at the interface dS12 physics the effective coefficient determines as

O* (V^ = [Cl (TO)! +02 <V«) 2 ] = = o1V((m1) « 1 ) + O2V(m^ « 2 ) +

or

oj =

(o1-o2)AQdl °lds1,

aSn

^(Ц) Ф1 ) + o2V(m^ Ф 2 )

(96)

+(o1 -o2 )aQ J фА

AQ 3S19

(97)

or

oj =

[o1 V ((m^ Ф 1 ) + o2V(m^ Ф 2 )

+ (o2 -o1 )aQ J Ф2ds

AQ3S19

/[V(( )1 + <ф)2 )] (98)

involving knowledge of the three different potential functions ««1, Ô2, and «|3 in the upper scale space consideration in the volume W and at the interface. This formula for the steady state

-1

effective conductivity can be shown is equal to the known expression

o*. = + -o2 J VOdœ, (99)

12

which means, when applied the WSAM theorem for averaging the operator V that O* (VO) is equal

O* ( VO) = O2 VO + (Oi -O2 ))VO(100)

That is one of the evidences that for some (very few) instances there can be drawn the direct analytical comparison of homogeneous and heterogeneous formulae. For this particular above formula the comparison shows the coincidence. We will demonstrate below for the layered medium one more example of a complete inheritance and conjugation for homogeneous and heterogeneous descriptions.

It is worth to note here that the known formulae for the effective dielectric conductivity (or permittivity) of the layered medium

c E (mi

i=1

i = 1,2,

(101)

for electric field applied in parallel to interface of layers, and

Ce =

E

i =1

m

\-1

(102)

when the electric field is perpendicular to the interface, can be derived from the general expression (??), see [9, 46], using the assumption that displacement fields expressed in terms of gradients of averaged potentials are equal

OlVi°l}i =O2V{° 2 >2, (103)

which is incorrect. We discuss these implications and solve exactly the two scale electrostatic problem for layered medium with high level of detail below in 5.3.

Further analysis gives us few known dependencies for effective coefficients still using the VAT formulations. Thus, assuming that the values of o1 and o2 differ slightly one from another then the known approximate formula (see [9, 10], for example)

Cy = C

V ((1 ) Ф1 ) + °- V((m2) Ф 2 ) +

(C2 - C ) 1 л r

+--^ J Ф2dS2

C1 A- 3*2

yo)-1 =

= C

V((m^ Ф1 ) + C V((m2) Ф 2 ) +

, , (C2 - C ) 1 Nk Г r

O A- k=1 a*.,

VO-1 ;

; C1 + (m2)(o2-C1 )

{^20 }2

= C1 + ( ml) (C2 -C1 )

{E20 }2

(104)

E

can be obtained for a dilute mixture of spherical particles (phase two) in a matrix of phase one, as long as the surface integrals over the second phase separate finite number Nk of dilute globulars

Nk

AQ2 = E(AQ2k) (k = 1, 2, ..., Nk) in the REV are

k=1

equal to the sum of volume integrals over the separate particles AQ2k and it is assumed that when the homogeneous Ostrogradsky-Gauss theorem of gradient is applied to these integrals

Nk

A^ J ф2ds2 = -NT^-E J °2Äk ;

7-2 dS12 E (A-2k )k=1 ^

E

O2 + 2O1

into (104) gives also the known formula [27]

Cy =о+m

301 (02 -p1 ) c2 + 2o1

;J VO20dro = (VO20}2 =-{E20(105)

2 ao2

then the substitution of the solution for a single sphere in a homogeneous field (E) effective medium [9, 27]

{E20 }2 = 30i

(106)

(107)

From this comparison one can see how the

difference between the exact VAT formulation (96)*

(98) for the effective conductivity coefficient oij in heterogeneous medium and widely known approximate formula (107) can be shown. The approximation (107) simplifies the expression for effective conductivity in two ways, one is through using the condition of a small difference between phase conductivities, while another source of approximation stems from the substitution of heterogeneous WSAM heterogeneous theorem equation by the homogeneous Ostrogradsky-Gauss theorem of gradient.

In section 5.6 we are solving this problem for globular (spherical particles) medium with high accuracy for the difficult statement when the media's morphological properties are changing throughout the space.

One of the simplified methods of closure of mathematical models for processes in a heterogeneous media is the quasihomogeneous method. In this case, the transfer process is modeled as an ideal continuum with homogeneous effective transport characteristics instead of the real heterogeneous characteristics of a porous medium. This

method of closure of the diffusive terms in diffusion equations results in certain limitations: (a) the two-phase medium components are taken without fluctuations in each of the phases; and (b) the transfer coefficients being constant in each of the phases results in simplification of phase transport. These equations relate the unknown averaged diffusion flows in each of the phases in the following form for mass transfer, for example, as

=-K;(y)(VC), (108)

Vo = v[( щ {C}f +(1 - m ){C} ]-+— J K+- C-)dS,

A— V '

(109)

' dSw

(fYjf))-1

j) = -vc . (110)

VT)f + (VT)s = (VT

-KTf (VT) -Ksr {VT) =-K* (VT

\ If \ is

Applying the closure relation, for example

Kf {VT)f = KS (VT)s,

which is generally incorrect (might be correct for only (m) = 0.5), but

Kf {VT} = KT {VT}s, {VT} =

KT {VT}s

KTf

(112)

m {vt }f=(V^f

(m)KT(s){VT}s щKT {VT

(s)Kf

{s)K{

(113)

is correct for some morphologies-as when the multilayer has a cross heat flux or current, and substituting

VT) =

/

(m)KT {VT

(s)Kf

we can get

and

VT

VT

ys)KT + (m)KT

Kj Kf

T

(s) + (m

Kf / kT

(kt / Kf )

Kf

s)Kf + (m) KS

(114)

and this is the known formula also. Returning to the usually applied incorrect formula

VT) =-

f

KS (VT

KTf

we yield the effective stagnant coefficient 2Kf Kf

Kr = -

( +Kt )'

(115)

Here C + and C~ are the values of the concentrations (or temperatures) at both sides of the phase transition surface dSw (they do not have to be

necessarily equal), K^), Ksc(i}-) are the transfer

coefficient tensors in each of the phases, and K*(j)

is the effective diffusion coefficient. The problem

of closure has been reduced to finding K*^) and

integrals across the interface of the difference of the values of limits of admixture concentrations (or temperature) at both its sides. For temperature fields, the above relationships will be similar (neglecting at this time the heat resistance of the interface boundary)

(111)

that represents the lower bound of the effective stagnant conductivity for a two-phase material.

There are other closure equations for calculating the stagnant effective conductivity.

The quasi-homogeneous approach has several defects: a) the basis for the quasi-homogeneous equations is in question, b) the local fluctuation values, as well as in homogeneity and dispersivity of the medium, are neglected, and c) the interdependence of the correlated coefficients and arbitrary adjustment to fit the measured data significantly reduce the generality of the results.

Considering the effective coefficient problem as the cornerstone issue in the heterogeneous media transport, one needs to accommodate the reasoning that the absolute majority of problem stated and studies include the following assumptions:

1) a composite is a two phase media consisting of a continuous matrix phase and embedded inclusions of disperse phase;

2) phase materials are homogeneous and iso-tropic, their properties are temperature independent;

3) disperse phase consists of the equally sized spherical particles, uniformly distributed within a matrix phase. As a result, the composite is assumed to be macroscopically isotropic;

4) interfaces have the conditions of conventional boundary transport laws, for example, perfect thermal contact is supposed to be maintained;

5) the external (heat) flux is supposed to be a time-independent and macroscopically uniform, etc.

The methodology of heterogeneous VAT applied toward the effective coefficient problem allows to formulating the task strictly and with mathematics that is solvable even using desktop computers.

5. 3. Single phase effective coefficients in the two-phase sensor's medium

Starting again with the second phase potential field upper scale equation

s

s

V2 (ы ф 2 )+V.

— J Ф2 ds2

AQal? 2 2

+— J VФ2 - ds2 = 0,

AQa!

(116)

or

V-(02V«2 )) =V-02 (V<0^2 J V<2 -dS2 =

Ai2 dS12

= V-(o2 PV« 2 )=0,

one can write generally that the effective phase conductivity being assessed via

V-(o2 pVO 2 )=

= V-

2V((m^ Ф2 )+o2"^ J Ф2dS2 +o2C2

AQ 3^12

= 0,

o

o2V-C2 = -Q AQ

J VФ 2 - ds2 = o2 B.

bSn

Now we can have for the steady-state effective conductivity problem in the single phase as

o2 p =

[o2 ^Ф2 (X)2 +o2C2 ]

2 (X )2

2V(m^ф2 ) + o2 aQ J Ф2ds2 +o2C2

3S12

V((({ф2 (x)>2 ) [o2 V ((2 ) Ф2 ) + o2 A + o2C2 ] V((({Ф2 (X)} )

A = — J Ф2 ds2,

AQ34 2 2

where we can write that

V C2 =AQ J VФ2 -^2 = B (x ).

(117)

(118)

(119)

Now, for the absence of the current exchange in the medium — B _ 0 (for some media with the homogeneous morphology), we can write the formula for the phase two effective conductivity o2p as

* _[°2V((m2)02 ) + 02a]

V(({ф2 (X)>2 )

But then, sometimes this coefficient is substituted into the inapplicable for this coefficient equation

V-[o2V(m^ Ф 2 )] = ^^Ф2 ] = 0,

(120)

which must be

V • [0*2V ((«2 ><02 )] _ V- [o2 2 ] _ 0. (121)

The last equation when B ^ 0, must be substituted by

V-[o2V(

m2)ф2))

AQ

J

Э£19

- ds~, = 0,

or

V-(o2 р^Ф 2 )=0.

(m2) = const,

(122)

5. 4. Layered medium effective dielectric permittivity exact simulation

There is the system combined of multiple k layers with posible different thicknesses lt, of layers. We take the two-component system, when layers organised in a periodical structure with the period of two different layers with the constant e1 and e2 permittivities.

Fig. 2. Two phase layered structure composite — superlattice of 100 layers with 50 layers of each phase

For the dielectric static problem the boundary conditions can be taken with the assigned electrical field values on both ends of the stack

El _-(v<)|l _ fL (<),

ER _-(v<)|R _ fR (<R ) or just the potential values

<L _ fL (<L ),

<R _ fR (<R ) This case as shown to be suggesting the ef-fecfive permittivity coefficient as

e. _[£1 (V<(* ) 1 +£2 (V°(X ) 2 ]_

V<(x)

(123)

(124)

= [e1V(w^) Ф ! ) + e2V((m2) Ф 2 )-

+ (e2 -ei 1 Ф2ds2

3S12

/(( V^ (x) + (m2)VÖ2 (x)).

(125)

As one can see in this morphology the only one coordinate argument is used.

In this morphology the key element to obtain the simplified formula is the recognition that the traversing potential field at steady state conditions is the same for both phases ejVOj = e 2VO 2. That formula may be used for substitution as

VO2 = fe/ e2 )VO 1» and the specific asumption for this morphology is that the averaged variables can be used in this equality (meaning, if take that eiVÔ 1 = e2 VÔ 2), then following the above precaution remarks one can get for this morphology (meaning ((mj),( m2) ) = const )

e!2 =

[(Ы + (m2) )УФ 1 ] [(Ц) + (e /e2))m2))Ф 1 ]

[( + fa) )e1]

m

(m2 )

Ы

S e

, i = 1,2, (126)

because if the values ®2(dS12) on both surfaces of a one separate layer are very close then the following term can be taken as

(e2 -e1 1 Ф2ds2 =

(127)

scales — on the lower, conventional adopted spatial scale; and for the Upper averaged fields scale.

The upper (the second) scale solutions are not achievable in brackets of homogeneous physics. There are no definitions and mathematics (at least correct ones) for that in homogeneous physics. That is why there is continuous search for experimental methods to measure the superlattice conductivity and dielectric permittivity. Researchers calculate often the different — the Transient bulk coefficients. That kind of coefficient differs from a steady-state sought effective coefficient. Also, with these techniques there is no way to find out the Phase Effective Coefficients of conductivity or permittivity in each phase. We show this with our exact solution.

Because the solution available analytically for linear and with the help of numerical simulation for both scales nonlinear problems for this morphology of 1D layers, this is the unique problem to test and compare the both scales solutions. Up to this day it is known as only the first kind of such a problem for layered medium with the analytical solution. There are few more the two-scale exact solutions have been achieved for other problems [32, 33, 37, 46-48].

The dependencies outlined in homogeneous statement solutions are incorrect if being applied with the exact fields for the lower scale solutions as, for example, the homogeneous formula gives the displacement field in the system through the averaged electric fields (potentials)

D

hom

Meanwhile, the assumption £jVOy = £2VO2 is generally not correct for heterogeneous media, but the equality

£1 (x )}1 =£2 {O (x )}2,

is correct for this morphology linear problems.

Problem Statement and Simulation. There are the exact solutions available for the lower (1st) scale of this 2-scale two-phase problem for the steady-state Laplacian equation fields

V201 (r) = 0, V2O2 (r) = 0, r eQ, (128)

applied in-cross for the 1D layered (superlattice) media. These solutions are given in textbooks on few physical disciplines (thermal physics, electrostatics, civil engineering, materials science, etc.) and often cited as to be the complete and final result.

We have found that the linear problems have the 2-scale Exact Analytical Solutions for the 3 kinds of Boundary Conditions: 1) field values of potentials; 2) displacement or electric fields applied; 3) Neumann's conditions of the 3rd kind. The two-scale solution here meaning the solutions on both

= (( {Ei}i + (m2) {E2 } ) = = (-£1 (my) V{Oi}i-£2 (m2) V{O 2 }),

and this value can be tens and hundreds times different from the correct one. We are assuming here as usual that according to homogeneous Ostrograd-sky-Gauss theorem the following is true — the averaged ( )2 over the each phase 1, 2 gradient operator acting on the potential function ® is equal to

(129)

which is not correct for heterogeneous media.

The known "classical" formulae for effective coefficients can be calculated as bulk values only for linear problems. It is important to say that the formula for the linear effective bulk coefficient

Уф1 (x ) 1 =V(®1 (x) 1, Уф2 (x)2 =V(®2 (x))2,

e!2 =

-1

I

i=1

m

i = 1,2,

(130)

is correct, but is not valid for the derivation. We show why is that. The final result and the formula is correct for linear statement, but the derivation is not correct. It can not be derived even for linear problem when applied only the one scale homogeneous statememts.

The critical issue is that ejVjOj} * e2V{O2}2, in spite that the homogeneous fields formula e1VO1 = e 2 VO 2 is correct and the local lower scale

V<O = (e2 / e )VO 2 is correct also. We supply below the direct calculations supporting this statement based on the exact solution.

As we will see, this particular layered morphology of the two-scale problem dictates the peculiarities of the physical phenomena involved. Thus, for the linear problem there are the two physical mechanisms involved in field's transport: 1) is the INTRAPHASE, in each phase; and 2) is the INTERFACE transport. In most common situations those are the three different components for the transport characteristics. The later properties described and openly accounted for as to the physical phenomena seems only in chemistry, chemical technology. In electrodynamics it is still considered as the 3D phenomena of a one scale.

The calculation of two components of the in-terscale transport — the Intraphase and the Interfacial one gives the understanding and values of new phenomena in superlattice transport. The mathematical formulation and exact calculation of the surficial transport components give an expected explanation of decreased overall bulk permittivities and what is more important of their magnitudes. It also allows to compare transport in each separate phase along with values for their components.

Needless to say, that the unknown component of interface transport can reach values of hundreds and even thousands percent of effective bulk values — see Figs. 3, 4. This correction can be directed toward the explanation of some known physical phenomena as, for example, like the interface resistance and polarization among others.

There are new phenomena verified by direct simulation for the "bottom-up" as well as for the "top-down" scaling connection of physical entities as, for example, it can be found out the lower scale complete solution without solving it as it is known to be solved. All is based on the knowledge of the upper scale characteristics. This can be chracterized as the kind of Interscale Mathematical Transform (IMT), where the upper scale properties determine directly the lower scale properties. This is the strict and direct evidence of communication of physical properties on both scales, and this is one of new discoveries.

We supply here few details and the numerical simulation results using the exact solution for the lower scale superlattice in-cross electrostatics problem. It is given in terms formulated for the electric fields strength (displacement fields), but is valid perfectly for thermal physics conductive problems as well.

One of the variants presented here was calculated for the number of layers k = 100, with the layers left boundary potential = 46.8292682927

[V], while layer's stack right boundary is at = 34.6341463415 [V], thicknesses are l1 = = 0.001 [m], l1 = 0.001 [m], the coordinate of the left boundary is Z1 = 0.001 [m].

The effective dielectric permittivity coefficient

" F ]

for such a combined heterogeneous system is

m

=

m.

m,

-i

where

m о =

l

m

m

-i

(131) ^

— average volume frac-

1,2 ! (4) + (4)

tions of the phase 1, 2. Meanwhile, the bulk medium electric flux density (displacement field) can be found as

Dbulk = -£ (VO) =

= -£ ((V{01}1 + {m2)V{®2}2). (132)

Tedious VAT derivation brings that for the only linear phase coefficients the VAT derived effective coefficient is also equal

* _pi (V^1 (x) 1 +£2 2 (x)2 ] = £l2 = (V®(x))

(133)

while the complications as, for example, the internal charge sources or additional interface resistance or/and the phase's nonlinear data of input will get much more complicated formulae for the effective coefficients. It worth to note that in this way was obtained the nice analytical result — that the VAT gives the same results for the same aspects as the homogeneous physics, when the statement of the problem is clear, and the morphology can be tracked down precisely as in this case with the cross-layers direction of electrical field or electrical potential those can be calculated exactly for this morphology and constant phase coefficients.

It is no secret, that for the inhomogeneous, nonlinear problems of this kind there is no solution for effective coefficient in the homogeneous physics. More of that, the Direct Numerical Modeling (DNM) results if applied straight toward the derivation of the effective variables using formulae for homogeneous media gives an incorrect output as, for example, when calculating the derivative content

(D) = e (x, °1,{°1 (x)} )V(( {O (x)})+

+e2 (x,O2,{O2 (x)}2)V(({O2 (x)}2), (134) will give incorrect average displacement field, even when the source fields {O (x)}, {O2 (x)}2 are known with high accuracy.

Taking the dielectric permittivity coefficients as rather different (£1 = 1, £2 = 9) we get the following values for chosen boundary conditions

(m2) = 0.5; Ц) = 0.5;

D 219.51219512

* ее 1.8

* е_1_2 1.8

ё while having

Фх 46.8292682927

Фд 34.6341463415

D2 (x) = -е^Ф2 (x) 219.5121951

D1 (x) = -V (x) 219.5121951

D* bulk 219.5121951

( m) v{Mi-£2 Ы v{<£ 2 }2 )

-£

-е1УФ1 (x) = -е2УФ2 (x), and {-е^Ф1}={-^Ф2 }2,also -£2 {VФ2 }2 =£2V {Ф2 }2, so

-eiVMi =-е2V{Ф2 }

but we have

(137)

Using the exact solution values one can calculate that

gradhom = (V°) =

= (( V{Ox}x + (m2) V{( 2 }2 ) =-121.9512195,

while the corresponding effective coefficient following the homogeneous definition is

£sim (V(} = ((D)! + (D)2 ) =

= - (m) v{o1}1 -£2 fa) v{( }2,

which gives

= ( (m)V{(^1}1 -£2 {mùV {(2}2 ) = 609.7560975.

This is not a close match to the correct Dj = =219.5121951

609.75609756 * 219.5121951, and one can calculate that effective coefficient obtained via the homogeneous formulae above

= -5.0. (135)

Meaning, that they are not equal

eSim = 5.0 * 1.8 = e; (136)

and that the difference is 177.78 %. What does it mean we will see while doing the two-scale solution for this problem.

Heterogeneous VAT Simulation of the Data in Each of the Phases. Now let we find the displacements fields of the averaged functions in both phases

1v{°1(x" )}1 *-e2V{{(x")}

121.9512195 * 1097.5609755,

meanwhile, according to homogeneous one scale statement for this problem should be satisfied

V{ (x" )} = 121.9512195 * е2V{Ф2 (x" )} = 1097.5609755,

(138)

the difference is in whole 7-8 times.

That means also that the specific assumption for this morphology — is that the averaged variables can be used in this equality (meaning that

£1V{(1}1 = £2V{(2 }2) to derive the effective coefficient formula when using V{( 2 }2 = (£ / £2 ) V{(1}1

£* = [(( + (m^)£1V{°1}1 ] = £±2 [(( + (£,/£2 Km^)V{(1}1]

[(( + (m) )е1 ]

m

■ц + ^2)

m

-1

=1 е

, /' = 1,2, (139)

is not valid for the derivation. The final result and the formula is correct, but the derivation is

not correct. Because e1V{O1}1 *e2V{O 2 }2, while the homogeneous e1VO1 =e2VO2 is correct.

Calculating the VAT based integrals using the lower scale exactly calculated fields we get the values

An (x" )=дк J ^ A'2 (x" )=лк J Ф2^

3S12

Ap2 = 48.780487797,

(140)

and

D2 (x) = -e2VO2 (x ) = = -9.0 * (-24.3902439) = 219.5121951. (141) Symmetrically we can get the Ap1 equal Ap1 = -48.780487803,

while

D1 (x ) = -e1VO1 (x ) =

= -1.0 * (-219.5121951) = 219.5121951.

The whole phase 1 heterogeneous upper scale displacement field is

{D( )}1={e:V°:}1 ={D(x" )}i, + {D(x" )}

e1V( 1 ) + e1AT 1 ô1ds1

((-^1)1 )

as.

= 219.512195106,

m

which means that the upper scale electric density flux is exactly coincides with the lower scale displacement.

Now — calculating the intraphase transport phase one only related part

{D (" }

D

m

1P

= 6°.9756°975/°.5 = 121.9512195,

(142)

and at the same time, the surface transport part occurring only in this phase which is

{D (x")}

D (x" )

m

(143)

= 48.780487803/0.5 = 97.560975606, one gets

{(x" )} ={ (x" )} ^ +{d (x" )} =

= 121.9512195 + 97.560975606 = 219.512195106. (144) Calculations show that the homogeneous lower scale displacement value

D2 (x) = -£2 VO2 (x) = 219.5121951, (145)

and the heterogeneous internal phase two displacement field are the same

{D (x" )}2 ={-e2V02} 1

2 V (Ф 2 )

AO

2 dsv

1 Ф2 ds2

((2^2)2 )

(146)

=219.512195154.

m

Now, calculating the intraphase transport part

(D (x" ))

{D (x" )}22 p

2 P

m

= 548.78048775/0.5 = 1097.5609755, and the surface transport part as

(147)

, \D (x")

{d (x")} =^m-4

L v 'hi (m2)

= -439.024390173/0.5 = -878.048780346, (148) giving the balance of fields as the perfect one

{D(x" )}2 ={D(x" )}2, +{D(x" »2 =

= 1097.609755 - 878.048780346 =

= 219.512195154,

while the lower scale homogeneous flux equality is perfect also — (the exact solution)

D2 (x) = -e2V®2 (x) = = -9.0 * (-24.3902439) = 219.5121951,

D1 (x ) = -e1V01 (x ) = = -1.0 * (-219.5121951) = 219.5121951.

The internal stack's homogeneous formula left hand electric displacement field is

D =-e* (VO) = -eI ((Vj®} + (m2)V{®2}2) =

= 219.5121951. That means the balances of displacement field in the two scale VAT approach are perfect for all the scales and for the effective coefficient statement problem.

Heterogeneous Upper Scale Displacement Fields in Terms of VAT. Now, let we do the calculation of the right hand side for the equation for effective coefficients modeling and simulation in this medium

D =-e! (VO) = = -e (( V{01} + mj V{02}2 ) =

= (D (x" )) + (D (x" ) =

= (-e1V$ ^ +(-e2V^ 2 =

v

= -e -e

e1V((m^ ф 1 ) + e1 AO 1 ф1^1

dS12

— f Ф 2 ds2 AOdS

dSn

e2V(m^ Ф 2 ) + e2

m,)V{01 (x" )}1 + Ap1 (x" )

m2)V{ (x" )}2 + Ap2 (x" ) = 219.51219513.

(149)

To combine the components of the homogeneous formula we get

(D, (x" )) = -£ * m) V{ (x" )}1 -

-£2 *(m2)v{ (x" )}2 =

= -1.0 * (-121.9512195)* 0.5 --9.0 * (-121.9512195)* 0.5 = 609.7560975,

which is grossly incorrect and needs some explanation.

But when we use the surface components as the VAT indicates to us we got the complete satisfaction with numbers

Di (x" )) = - *Ap1 (x" )-e2 * Ap2 (x" ) =

= -1.0 * (-48.780487803)--9.0 * (48.780487797 ) = -390.24390237, (150)

^ then, the right hand side balans is

D(x" ) = 609.7560975 - 390.243902424 =

= 219.512195076, (151)

is exactly the same number. This shows us that the displacement fields are equal on any of two scales, as they should be.

Heterogeneous Upper Scale Dielectric Permittivity Coefficient Components in Terms of VAT. Taking the equality

-e

12 p *

УФ> = (Dp (x" )),

(152)

where the left hand side is assumed to be written in homogeneous definitions, while the right hand side in heterogeneous ones we can get the intraphase component of effective permittivity as it understood in homogeneous physics as following

el2 p =

Dp (x" )) (УФ)

(-61 * (m) v{ (x" )}1 -e2 *(m2) v{ (x")} (( ^Ф^ + (m2) у{ф 2 }2 ) = -5.0,

or it appeared that

ei2 p =esim = 5A

at the same time

6i2p = 50 * 1.8 = 6I2.

(153)

(154)

(155)

Now, the second component of the effective bulk permittivity — the interface component is

e —

Di (x")

m =

(уФ)

= (( * Ap1 (x" )-e2 * Ap2 (x" )) = = ((У{Ф1}1 + (m2)У{Ф2}2) = (-1.0 * (-48.78°4878°3) - 9.0 * (48.780487797)) -121.9512195 = 3.199999999754.

(156)

We see that their sum is exactly equal to the composite's effective permittivity coefficient

e12 =e12 p + e12i = 5.0 - 3.199999999754 = = 1.8°°°°°°°°246.

(157)

Calculation of the Each Phase Effective Coefficients in Terms of VAT. We know that

D2 (x) = -e2V®2 (x) =

= -9.0 * (-24.3902439) = 219.5121951, making equal the both parts — the left one, which is in homogeneous notations and definitions, and the right one which is based on heterogeneous medium determined mathematics, for purposes of finding the phase effective coefficient

Db2 (x) = (D2 (x)2 = = -e2 Ы У{Ф2 (x )}2 =-e2V0^2 =

2V(({ф2 (x)}2 )+e2 AO.1 ф2ds

From this equality here one can get the phase two effective coefficient

e, =-

e2V

m

){Ф2 (x)}2 +e2 AO 1 Ф2

AO 3S12

(m2)V{®2 (x)}2 e2 (xu ) + A^ (xu ) _

K>V{ (x)} =

(m2) V{®2 (xu )}2 + A,2 (xu )

(m2) V{®2 (x)}2 = 1.80000000044.

Now let we present the phase effective coefficient as of the two parts — intraphase e*p part and the interface surface influenced transport component e*;

e 2 = e 2 p + e 2i =

e^m^у{ф2 (x" )}2 e2Ap2 (x" )

(m2)У{Ф2 (x)}2 (m2)У{Ф2 (x)}2 )у{ф2 (x" )}2

(158)

e2 \m2l

e2 p = "

= 9.° = e

2

while

e2 Ap2

m2)У{Ф2 (x)}

= 9.° * (48.780487797)

m

)У{Ф2 (x)} 0.5 * (-121.9512195)

= -7.19999999956.

(159)

The interface input to the phase 2 effective coefficient of permittivity is equal to

e2 Ap2

m

>V{®2 }

= e~

((, f (У{Ф2}2 )),

(160)

and for this problem it will be the component not dependent on spacial argument in this particular statement.

Doing the same for the phase 1 effective permittivity coefficient we get the similar results (see more features in [49]).

Some data obtained during simulation of this problem for the both scales depicted in Figs. 3-4.

or

100

£r = £2 / £1

Fig. 3. The effective coefficient of permittivity (-■-) for the superlattice of 100 layers and the 2nd phase interface conductivity (-—-)component relative value

2500

2000

1500

1000

(V • (c^VO) = V o2 (VO)2 + J VФ2 • ds2 = 0,

V2 ((rn2) Ф 2)

+ V-

АЛ

J Ф2 ds2

dS 2

= 0 (161)

because neglecting the really 3rd — the smaller scale phenomena in the conductive phase two near the interface dS12 is rather the general assumption leading to the situation when inside of the conductive particle and at its internal boundary surface the field (E2 = -VO2 ) =0.

Meanwhile, the solution of the separate conductive problem (161) within the only medium of conductive system phase is possible without solving the another phase — the dielectric phase — problem, if the boundary conditions on the interface are assigned as for non-conjugate problem

in2 =°2 En2 L , =0, En2 =

дФ2 (x)

dn

3S, 2

^ f VO

2 • ds2 = 0.

2V(m^ Ф2 ) + Ö2 J Ф2ds.

Э! -

(( V(O 2 ))

- 500

-2500

80 100

£r = £2 / £

Fig. 4. Medium's nondimensional interface effective coefficient of permittivity for the phase 2 of superlattice with 100 layers — the real and the absolute relative values: -■- —medium effective interphase permittivity; n — interphase effective medium permittivity

5.5. Dielectric-conductive morphology medium

Another case appeared when one of the phases (the second, for example) is conductive with o2 = const, £2 = const, while the first phase is not o1 = 0, £1 = const, then the conductivity equation for the composite could be deduced as for the bare second phase averaged potential equation

= 0. (162)

That means the surface integral in the VAT upper scale equation (161) should be

(163)

This determines the effective conductivity coefficient in the phase two, keeping in mind that the boundary conditions are set separately for each phase

*o2. (164)

Э! 2

The formula (164) is the result of the nature of defined in this way the one-phase medium effective coefficient of conductivity and of assumptions which have been made. This is not a surprising result, as soon as the outcome obtained for the singled second phase effective coefficient of conductivity becomes inadequate, if considering only conductivity problemwithout the interface surface charge and smaller scale jump of potential

gradient of the second phase since the (VO) 1 is not determined via the conductivity statement, and so the E1.

This new VAT potential equation in the second conductive phase directly depends and controlled by the morphology of interface and connectivity of the conductive phase throughout the entire medium morphology of the dielectric phase and it's model solution. The tangential boundary condition at the interface is set-up as usual

e (r )=e* (*t12 =(n x ^ (f )))= 0 (165)

where index c means in the conductive phase, Edtg is the tangential component of electric field on the dielectric side in the interface.

Quite another situation arises if the interface related surface charges are having included into the physical and mathematical models.

Now we may adopt the second phase as a par-ticulate phase with the globular spherical conductive inclusions. The reason why the phase average

of the potential O 2 in conductive particulate phase two is not equal to zero in spite that inside of the AQ2 domain in each separate conductive inclusion bk, k = 1, 2, ..., Nk and without connection to the boundary and while conductive inclusions are surrounded by insulator the electrical field should be equal to zero E2 = V®2 = 0, because the actual jump of value of field E near an interface occurs still inside of the closed volume AQ2k of each conductive particle bk and that the value of the potential on the interface of each particle is different one from another for the conjugate problem, which implicates that

or

ф22 (""' )=Ж" J

¿ЛЬЛ") л гл

ф d ю =

2 ДП

-X J ф 2kdЮ =

K-J2k )k=1 ^

k=1

1 Nk _

Nk-ХФ2k > 0

K-J2k )

k=1

(166)

k=1

having the charge p

sc ldS12

on the surface of con-

ductor's inclusions in dielectric — matrix phase. This one is usually found after the solution of the potential problem for the dielectric phase via

psc dS12 ps2

dSy

= -е

ЭФ1 (x )

дп

(167)

dSy,

where the normal n to the dS12 is taken as an external to the conductor — a conducting inclusion.

Exploring the media electrostatics in both dielectric and conductive phases a little bit more one gets after summation of the equation of conductivity in phase two and the potential in a dielectric phase one the equation for the averaged function of the potential in the whole medium

V-[«è (V<&) 1 +02 ^Ф)2 ] + C1 = 0,

(168)

V^

where

1 Nk

—X

-J k=1

^(Ы C°2 ) + тП J Ф2 dS2

ДП dS12

е1V ((1 ) c°1 ) + -- J Ф1^1

ДП dS12 -Scp (е1, dS12 )= 0,

с = J VФ1 • ds1 =

(169)

J (s^k)ds]

dSk

•nk

1 Nk

= TJI(-Psk ) = ДП k=1 (170)

where O 2k is the distributed function of the mean value of the potential on the interface surface of each particle and that in the inhomogeneous matrix ®1 field the values O2 (x(2) ) ^ O2 (x(22) ), where

x(2) and x(22) neighboring points in the second level "averaged" coordinates, in which the all averaged functions are assumed to be defined and sought.

The boundary conjugate condition in the interface between the conductive-dielectric paire is

= AQ ^cP (e1' ^^12 ) = ^P (e1' ^^12 ^

Pit = J Psc • ds2,

dSk

is the surface averaged density of surface charges Scc on the surfaces of conductive particles in the REV. Of coarse, we do not know the charge distribution up-front at the time of the problem's formulation.

After reducing this equation it shows after derivation that the effective conductivity coefficient in this medium if determined using the both phases field distributions as they are intimately connected

=

Ö2V(m^ Ф2 ) + -J J Ф2 dS2 +

+ 81V(m^) «Ф1 ) + - J J ФД + F1

ДП dS12

X[(VФl}l +^Ф 2)2 ^

(171)

V-F1 =C„

where the coefficients o2 and £1 may be substantially different in magnitudes (up to 18 orders of magnitude for good conductors). This expression implicitly includes the polarization of the conductive particles's surface (170). Following this derivation the effective conductive-dielectric heterogeneous medium effective conductivity coefficient VAT equation appears to be satisfying the homogeneous equation

V ' (( (V®)) = = V'(( [V^) ((1 ) + V(m^ O 2 )])=0. (172)

Thire is no our further intension to go deeply into the interface dielectric-conductor physics for-

mulation problem (boundary surface for each of the phases) because this would change our subject of general description. In this case we need to introduce at least the third (smaller then the first initial lower scale) scale of physical consideration. We intend here just to depict explicitly the appearance and implications of the two scale formulation for the obviously heterogeneous media when the characteristics and properties sought on the upper scale — means on the scale of averaged piece of the material large enough to include many hundreds or thousands heterogeneous elements.

To determine the single phase effective conductivity coefficient in general situation for not a perfect conductor (semiconductor, for example) with the presense of the first phase as in a heterogeneous system, the way to formulate the effective conductivity problem is to go through the accounting of all the terms — including the interface exchange term C2

eff ,2

2/2 =

V-

°2У((т2) Ф 2 )+°2 J Ф2

3Ä,

2 + C2

= 0, (173)

V C2 = f VO2 -ds2,

which would bring the effects of sharp local gradients of electric field into the model.

The effective conductivity coefficient in the conductive phase (one of the two) will result in

2V(m^ ф2 )+Ö2 -1- J ф2dS2 + C2

3S12

(174)

This value must be determined through the solution of the coupled (conjugate) twophase potential problem, as we show in the next section using the example of the globular inclusions dielectric-dielectric composite. Thus, the both phase equations need to be solved.

References

46. Travkin V. S., Catton I. Heat and Charge Conductivities in Superlattices — Two-Scale Measuring and Modeling // Proc. Int. Mech. Engin. Cong. Expo (IMECE'2001). New York, 2001. P. 1-12.

47. Travkin V. S., Hu K., Catton I. Exact Closure of Hierarchical VAT Capillary Thermo-Convective Problem for Turbulent and Laminar Regimes // Ibid.

48. Travkin V. S., Catton I., Hu K., Ponoma-renko A. T., Shevchenko V. G. Transport Phenomena in Heterogeneous Media: Experimental Data Reduction and Analysis // Proc. ASME, AMD-233. 1999. Vol. 233. P. 21-31.

49. http://travkin-hspt.com/eldyn/ WhatToDo2.htm.

Продолжение. Начало см.: ISJAEE. 2005. № 3. С. 7-17. To be continued. See also: ISJAEE. 2005. No. 3. P. 7-17.