Parinov I.A. Microstructure and Properties of High-Temperature Superconductors

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C.2 Heat Conduction of Heterogeneous Systems 525

(b)

(c)

(a)

Fig. C.4. Deﬁnition of averaged geometrical parameters of elements of the ﬂux

tube in the frame: (a) disposition of contacts and through pores in cubic package of

balls; (b) mean cross-section through pores; (c) element of chaotic structure with

averaged parameters

Using (C.12) and (C.22), the dependence y

= y

)maybestated.

The heat resistance of the ﬁrst type element is equal to the sum of straight

) and truncated (AA

B) cylinders (Fig. C.3b). Heat resistance

of the truncated cylinder is equal to

= h

/(λ

) , (C.23)

where h

and S

is the mean height and base square of the truncated cylinder,

respectively. Moreover

= S

sin α, (C.24)

where S

is the base square of the straight cylinder;

=(r − h

)sinα, (C.25)

where r is the ball radius and h

is the height of the straight cylinder. Then,

we have from (C.23) to (C.25)

=(r − h

)/(λ

) . (C.26)

Because geometrical parameters of the straight cylinder depend on coor-

dination number only, then the heat resistance of the ﬁrst type element does

526 C Macrostructure Modeling of Heat Conduction

not depend on contact location. Therefore, during the study of heat transfer

through elements of ﬁrst type, it may be restricted by only considering the

central element.

In the second type elements, contact points are disposed near interface

plane and heat transfer depends signiﬁcantly on fraction of through pores. Due

to the existence of ﬁrst, second and mixed type elements in chaotic structure,

it is assumed that eﬀective heat conduction of ﬂux tube is equal to the heat

conduction of combined element with averaged parameters, that is, the ﬁrst

type element with mean fraction of the through pores belonging to the second

type element. In order to deﬁne mean fraction of the through pores, tangential

planes, forming spatial polyhedron, are drawn about any particle in the points

of contact with neighbors (Fig. C.2a). Average porosity of these polyhedrons

and all systems are coincided.

Assumption 3. Irregular shape of particles (in the form of polyhedron) can

be substituted by system of equal volume concentric balls with the same porosity

(Fig. C.4b).

It is followed from Fig. C.4b that

/r =(1−m

)

−1/3

= r

/r , (C.27)

where m

=(V

−V

)/V

; V

and V

is the volume of external ball and particle,

respectively and m

is the porosity. Quantitative estimation of the through

pores is obtained by using the ratio of the through pore square per particle

to the mean cross-section square of the particle.

Assumption 4. Through pores may be presented in the form of cylinders

with annular base, enclosing central element of the tube (Fig. C.4c). In this

case, relative square of the cylinder bases is the same as relative square of the

through pores: π(r

− r

)/πr

= π(r

− r

)/πr

We have from (C.22) and (C.27)

= r

/r = y

(1 − m

)

−1/3

. (C.28)

Equations (C.22), (C.27) and (C.28) deﬁne ﬁnally the averaged geometrical

parameters of the considered system (Fig. C.4). Presented model takes into

account the existence of continuous contacts of particles in any direction (the

stability condition) and isotropy of the granular system with chaotic structure.

One is applied for porosity in the range of 0 ≤ m

≤ 0.4.

C.2.4 Heat Flux Through Averaged Element

After estimation of the averaged geometrical parameters of granular system,

heat ﬂux through an averaged element is investigated. As this element, we con-

sider an element with bases presenting isothermal planes, and lateral surfaces

C.2 Heat Conduction of Heterogeneous Systems 527

(b)

(a)

1’

Fig. C.5. Deﬁnition of heat resistance of chaotic structure element of the granular

system: (a) distribution of heat ﬂuxes (actual ﬂux spreading is shown at right hand,

and idealized picture at left hand), (b) schematic junction of the heat resistances for

single sections

are adiabatic (Fig. C.5). A division by adiabatic surfaces parallel to the heat

ﬂux gives overstated value for heat resistance, but a division by isothermal

surfaces leads to understated value. A division of mixed type leads to more

accurate results with error about 10%.

General heat ﬂux, Q, is presented by the sum: Q = Q

+ Q

, where ﬂuxes

and Q

pass through particle and through pore in the averaged element, re-

spectively. Then, heat resistances R

, R

and R

are found. The resistance R

is related to spherical section of solid particle down to intermediate isotherm



–1



(Fig. C.5a). Then, the pointed region is divided into annular adiabatic

planes. Conductivity of single layer, dσ

, with thickness, dx,iscalculatedas

(see Fig. C.6) [234]

dσ



∗

∗∗



−1

=dS



r −

√

− x

√

− x



− r



−1

, (C.29)

where dS

= x dx dθ.

Because the radiant component into gap between spheres is signiﬁcantly

smaller than the molecular one (λ

<< λ

), then in integration of (C.29),

we take into account only the molecular heat transfer. Hence, we have from

(C.18)

≈ λ

= λ



∗

(x)H



−1

. (C.30)

528 C Macrostructure Modeling of Heat Conduction

dθ

1’

Fig. C.6. Deﬁnition of heat resistance of the element’s sections with averaged

parameters

Total conductivity of the gap between spheres, taking into account only

molecular transfer is

2π



dσ

2πλ

1 − ν



D − 1+ω

− D

− 1



, (C.31)

where

D =



1 − y

; y

= r

/r; ω

=[1− ν

D + B/(Hd)]/(1 − ν

);

= λ

/λ

. (C.32)

Now, we take into account a contribution of the radiant component of the

heat conduction factor into gap between spheres, summarizing it with the

molecular component. We obtain from (C.31) and (C.32)

2πλ

1 − ν



D − 1+ω

− D

− 1



, (C.33)

where

=[1− ν

D+B/(Hd)]/(1−ν

); ν

= λ

/λ

; λ

= λ

+λ

. (C.34)

The value of λ

into gap between spheres is estimated approximately,

replacing a complex shape of the gap through buﬀer layer with thickness δ

which is equal to the gap thickness between spheres, to being average-integral

value on square, πr

(see Fig. C.6), that is,

≈ 4ε

S–B

, (C.35)

C.2 Heat Conduction of Heterogeneous Systems 529

here

πr

2π



, (C.36)

where

= l

∗

=(r −

√

− x

)x dx dθ; ε

= ε/(2 − ε); and σ

S–B

Stefan–Boltzmann constant.

At integration of (C.36), it is taken into account that dependence r

f(N

) is deﬁned by relation (C.22). Then, we have from (C.36)

= d/N

. (C.37)

(a) Finally, the heat resistances are found as

=1/σ

; (C.38)

(b) the heat resistance, R

, of the section between intermediate isotherm 1



−1



and upper base of the element (see Fig. C.5a) is

πr



1 − y

πry

; (C.39)

,is

π(r

− r

)

πr(y

− y

)

; y

= r

/r , (C.40)

where λ

is the heat conduction of gas into through pores with height

equal to the particle radius; one is computed by using (C.20).

By using heat conduction, R

, of single sections (C.33)–(C.40) and taking

into account their schematic junction (see Fig C.5b), eﬀective heat resistance,

R, of element with averaged parameters is calculated as

R = f(R

) . (C.41)

On other hand, we obtain by assuming that all volume of the element

is ﬁlled by homogeneous matter with eﬀective heat conduction, λ

,whichis

equal to eﬀective heat conduction of the granular system frame:

R = r/(λ

πr

) . (C.42)

530 C Macrostructure Modeling of Heat Conduction

Finally, we ﬁnd eﬀective heat conduction of the frame, λ

, from (C.41) and

(C.42):

⎧

⎨

⎩



(1 − ν

)

2ν



D − 1+ω

− D

− 1



−1



−1

⎫

⎬

⎭

;

(C.43)

where

D =(1− y

)

1/2

; E = y

− y

; ω

1 − ν

(1 − y

)

1/2

+ B/(Hd)

1 − ν

;

(C.44)

; ν

; B =

4γ(2 − a)Λ

(γ +1)aPr

. (C.45)

This formula could be used to calculate eﬀective heat conduction system

at m

< 0.5.

Eden Model

The Eden’s model is a sequential model for the stochastic growth of compact

clusters [244]. In this model, each new element of a cluster is added at growth

site that is chosen with equal probability, from a set of all possible growth sites.

These sites are deﬁned each time by a microscopic rule for cluster expansion,

which, in general, identiﬁes sites that are on the cluster edge. The set of

possible growth sites reﬂects instantaneous (i.e., in the present time) shape of

the cluster (see Fig. D.1).

Eden’s clusters are compact in all space dimensions [886], but their sur-

faces demonstrate self-aﬃne fractal geometry [662, 917]. Computer simulation

methods are used extensively to examine the interfacial properties of lattice-

based Eden’s clusters [502, 503, 852]. In [503], three microscopically distinct

versions of the Eden’s growth process are considered and it is shown that they

all have similar scaling behaviors with a ﬁnite size (or roughening) exponent,

α =0.50 ±0.02, and a dynamic exponent, β =0.30 ±0.03. The Eden’s model

describes a large class of irreversible, interfacial growth processes, which in-

cludes processes described by continuum equations.

The relaxation processes, which occur concurrently with the growth and

disposition of one cell in a group of cells, are substantially unquantiﬁed. For

example, in the extended Eden’s model [52], relaxation is restricted to the

synchronous motion of a train of neighboring cells. Computer simulation is

performed on a square lattice in 2D strip with size L.Planesaty =0and

y = L are periodic boundaries, and the strip is inﬁnite in the x-direction.

Initially, all the lattice sites with x ≤ 0 are occupied, and all those with x>0

are empty. At each time step, an occupied lattice site r is chosen randomly.

Then, if there are some unoccupied sites, r

, such that r − r

= sk,where

k is a lattice vector and s ≤ q (q is the maximum train length), the site r

becomes, momentarily, doubly occupied. This double occupancy is relaxed by

the occupation of a site chosen randomly from the set of unoccupied sites r

for

which |r − r

| is the minimum. In most cases, this set has only one member.

The relaxation is constructed from the simultaneous motions of a train of

532 D Eden Model

L = 16

Fig. D.1. Typical cluster obtained, using Eden model. Circles denote sites located

along the cluster perimeter

neighboring cells in the direction of least resistance with the maximum train

length, q. Note that for q = 1 this growth process has been modeled in [503].

The growth process leads to a compact cluster with an irregular, rough

surface. At any time, the surface of the cluster may be deﬁned by a set of

heights, h(y

), 1 ≤ i ≤ L. The heights, h(y

), represent the extension of the

cluster in the x-direction at a lateral point y

. For a cluster with constant

density, the mean height is found as

= L

−1

i=1

h(y

) . (D.1)

For a particular value of h

, the standard deviation of the heights

σ(L, h

−1

i=1

[h(y

) − h

(L)]

(D.2)

represents the surface roughness and has a scaling form [52]

σ(L, h

)=L

f(h

);

∝ h

at h

<< L

;(D.3)

∝ L

at h

>> L

where f(x) is a scaling function and z = α/β. These equations describe a

surface with a strip width that initially increases and then, after a time, which

depends on the strip width L, reaches a saturated value.

D Eden Model 533

The growth from beneath the surface of an expanding cluster leads to an

interface that roughens slowly compared to one that is driven by a simple sur-

face addition process. For the extended model z ≈ 2, and the temporal spread

of height ﬂuctuations is diﬀusive [52]. This is in contrast to the super-diﬀusive

behavior of pure Eden’s model where z<2. Moreover, in the steady state of

the extended Eden’s model, the ﬂuctuations of the heights are less correlated

along the surface than in the case of pure “deposition”, that is, in the ex-

tended model the surface is less rough. The results are currently insuﬃcient

to establish the precise dependence of the exponents on the parameter q, but

they establish a qualitative change, in an observable property of the cluster

morphology, that arises as a result of the subsurface growth.

The averaged Green’s function, g(r − r

), for the steady-state growth in

the extended Eden’s model is largely independent of the system size and

geometry. It represents, quite generally, the probability that an unoccupied

site at r

becomes occupied at the same time as a new particle is introduced

at site r. The probability of growth increases with the distance beneath the

surface because the number of equally weighted growth sites increases with

depth through the surface proﬁle. Simulations with q>σ(L, ∞) conﬁrm that

g(r) → const for r>σ(L, ∞) [52]. Thus, the averaged Green’s function may

be used in a numerical scheme to construct, directly, clusters in diﬀerent

geometry.

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