Statistical physics of linear and nonlinear, scalar vector transport processes in disordered media

200

Nuclear Phymcs B (Proc Suppl ) 5A (1988) 200-208 North-Holland, Amsterdam

STATISTICAL MEDIA Muhammad

PHYSICS OF LINEAR AND NONLINEAR,

SCALAR AND VECTOR TRANSPORT PROCESSES

IN DISORDERED

SAHIMI

Department

of Chemical Engineering,

University of Southern California,

Los Angeles,

CA 90089-1211

Recent developments in modelling transport processes in disordered media are reviewed. These include conductivity, diffusivity and elasticity of disordered solids, and failure and yield mechanics of such systems. We focus our attention on transport processes in discrete and continuous percolation systems, and discuss the behavior of transport and mechanical propertles of these close to, and away from the percolation threshold.

shes as g~

i. INTRODUCTION The determination mechanical

of effective

properties

as diffusivity,

and

of disordered media such

conductivity

been of great interest port properties

transport

and elasticity has

for a long time. Trans-

and mechanics

depend on their morphology,

of such systems

i.e., their topolo-

backbone

and the backbone

fractlon X B (the

is made of multiply

IC) vanishes

connected bonds in

as EBB. For length scales L <<

~p,

the IC is a fractal ob3ect with a fractal dimension dlc=d-B/~,

while the backbone has a frac-

tal dimension dBB=d-SB/~, tial dimension.

where d is the spa-

Currently accepted values are,

gy and geometry. While the effect of geometry

~=4/3, ~=5/36 and BB=39/72

has been appreciated

B=0.43 and ~B=0.9 for d=3. For d ~ 6, the mean-

for a long time, until

for d=2, and ~=0.88,

recently I, the effect of topology had been

field exponents,

ignored in most of theoretical

exact. The scaling of transport

efforts.

percolation

works 4'5 have been used extensively disordered media. bond (site)

In a percolation

is the percolation

On the other hand, attention has also been

net-

focussed on the application

to model network each

is present with probability

absent with probability

propertles near

Pc will be dlscussed below.

Since the pioneering works of Last and Thouless 2 and Kirkpatrick 3

V=I/2, B=I and BB=2 become

p and

l-p. For P>Pc' where Pc

threshold of the system,

a

of percolation

theory to modelling of electrzcaland

mechanical

failure in disordered media, which are phenomena of technological

importance.

previous models discussed

Most of the

in the literature

cluster of present bonds is

incorporate

artificial

formed. A bond can represent a channel in a

microscopic

laws of failure, whose contrl-

porous medium through which the flow of fluids

bution may not even be essential,

sample-spanning

takes place,

or it is a resistor that allows the

flow of electrical spring,

current,

representing

or it can be a

a solid phase. Such perco-

features and complex

ication of percolation

and the appl-

theory appears to be

more appealing and promlslng. This paper summarizes

recent advances

in

lation networks have proven to be powerful tools

modelling

of linear and nonlinear

for studying transport processes

in disordered

processes

in disoredred media using percolation

is the behavior

networks.

We focus our attention on, (a) con-

systems.

Of particular

of topological

interest

and transport

As Pc is approached,

properties near Pc"

the correlation

length ~p

diverges as ~-~, where e=p-pc. The fraction of bonds in the infinite cluster

(IC) vani-

0920-5632/88/$03 50 © Elsevier Science Pubhshers B V (North-Holland Physics Pubhshlng D,wslon)

ductivity and diffuslvlty (linear and nonlinear

of percolation

scalar transport),

elastic moduli of dlsoredred by percolation

transport

clusters

systems (b) the

solids represented

(linear and vector tra-

M Sahlml /Dtsorderedmedta

nsport),

and (c) yield and failure phenomena

percolation

systems

in

(linear and nonlinear,

scalar and vector transport).

Particular

201

the details of the system.

Computer slmulations

have shown that, aside from a special case,

atten-

and s are indeed universal.

tion will be paid to the behavior of transport

to this universality

processes near Pc" The interested

conductivity

reader is

of random continuous

referred to Ref. 6, where some other applica-

systems.

tions of percolatlon

so-called ~ i s s - ~ e e s e

theory have been discussed.

The only exception

is when one considers the percolation

The best known of these systems is the model,

in which spherical

holes are randomly placed in a medium having 2. CONDUCTIVITY

AND DIFFUSIVlTY

otherwise uniform transport properties,

Consider a percolation network in which each bond is a resistor whose conductance distributed

quantity.

In the simplest

can take on one of the two values, probabilities

g is a

takes place through the channels

between the nonoverlapping

case, g

a and b , w i t h

p and l-p, respectively.

transport

If a is

spheres. Halperin

et al. 15 have shown that the distributlon conductances f(g) ~ g

of the channels has the form,

Cheese model 15 onto an equivalent

cting and insulating

which the role of the channels

E vanishes as Pc is approached,

conductivity according

~ (p-pc)~

(i) and ~(d=2)=1.3,

one obtains a network of conducting

and superconducting

elements

Z ~ (pc-p) -s

for which

shows that the exponents elasticity

a duality argument 7 s(d=2)=~(d=2); s(d=3)=0.75

one also has,

and s(d ~ 6)=0. The exponent

s

appears in the critical behavior of dielectric constant 8'9, the absorption

exponents 16 discussed below)

models 16. For example,

exponent , and by

coefficient

of metal-

insulator composites I0, in the conduction

~ and s (and the

for the Swiss-Cheese

~(l-a) -I where s is another dynamical

model donor

in the network

models 15. But, the analysis of Halperin et aZ.

different

(2)

in

f(g). The topolo-

differ from their counterparts

and ~(d > 6)=3. If a is infinite and b

is finite,

network,

is played by the

gical exponents of the Swiss-Cheese

where ~ is a dynamical exponent ~(d=3)=2

resistors with a distribution

to the power law 3

of the

, with 0 < ~
finite and b=0, one obtains a network of conduelements, whose effective

and

can be

and related

~ is replaced by ~ +

a result that has been confirmed by

a variety of methods 17-19. This result had actually been predlcted much earlier by approxima~ 20,21 tecnniques . In general, as stated clearly by us 22, the universality is violated

of bi-

if

of transport

r°° f( ) f_l=J0 I---~g -$2- dg

exponen~

(3)

nary metallic alloys 7'II and, possibly, in the 12 dlvergence of the viscosity of gels (see below)°

diverges

The percolation

singular f-i can also cause drastlc changes in

with

conductivity

problems

associated

(i) and (2) are speclal limits of a general

two component

random resistor network of poor

and good conductors

(with a>b). One can

(i.e., the probability

finite). A conductance

sion-controlled

diffusing molecule f-i is divergent,

magnetic

field is played by the ratio

An important

question

and independent

For example,

in

for the dlffu-

and B a statlonary one, if then, the decay with timet

of

the density of A is of the form23,t-X,insteadof

h=b/a

is whether the expo-

nents ~ and s are universal

with a

reaction A+B~B, where A Js a

a magnetic

in which the role of the

distribution

the behavior of other types of phenomena dlsordered media.

develop 7'13 an analogy between this system and system,

that g=O is

of

exp(-t6), Another

where 24 ~=d(d+2) -I important

question is whether ~ and

202

M Sahlml / Dzsordered medm

s are related to topological

exponents

~, B and

BB, i.e., whether dynamical exponents are related to static ones. This question has recently received considerable since Alexander

attention,

especially

and Orbach25(AO)

proposed a

and conjectured

that this is in fact an exact

result. Early plausibility arguments 32'33 and 34~35 numerical simu±ations seemed to support this. But more accurate simulations in both 36 and three dimensions 37, together with an

two

relation between ~, V and B. Before stating the

E-expansion 38 (E=6-d), ds=4[l-E/126

AO conjecture,

indicate that the AO conjecture

percolating

we briefly review diffusion

clusters.

Einstein relation,

Because of the Nernst-

one can replace the conduc-

tivity problem with an equivalent problem.

in

diffusion

implies that

~=[(3d-4)~ -~]/2.

('the

ant') which performs an unbiased and nearestneighbour

random walk on a percolation

('the labyrinth').

cluster

The root mean-squared

displ-

(RMSD) R of the walk grows with time as

R ~ tk~ where d =k -I is the fractal dimension of w the walk. If R >> ~p, then dw=2 for all d, but if R << ~p, then, d w depends on d. This problem 27 was first studied by Mitescu and Roussenq and Straley 28. Gefen et al. 29 and Havlin et al. 30 showed that if one averages R over all percolation clusters, whereas

Note that the AO conjecture

(5)

Thus, de Gennes 26 suggested that one

should consider the motion of a partlcle

acement

+O(~2)]/3,

is not exact, . 739,40 although the matter is still controversla±

then, d =2(2~+~-B)(2~-8) -I, w if R is calculated for only the IC~ then

Aharony and Stauffer41(AS) the AO conjecture

argued that even if

is exact,

it might be so only

above a lower critiaal dimensionality

dl, and

suggested that at d=d/ one must have df=2. Sahimi 42 provided further support

for the AS

argument by using a relation between the conductivity problem and the Heisenberg

ferromagnets

at low temperatures. The AO conjecture means that dw=3df/2 , whereas AS suggested dw= df+l for d ~ d 1. These conjectures

have also been

tested for other fractal systems 43-46, and their validity

is still an open questlon;

see

the review of Havlin and Ben-Avraham 47. Note that if one averages N(~) over all clusters,

d =2 + (~ w

~)Iv,

where both of these relations

(4) for d

for R << ~p (e.g., at pc ) . Thus, random walks on percolation a means of estimating

one obtains N(~) ~

are valid

w simulating

an effective-medium

clusters provides

~30,31.

Such random walks

also provide 25 a simple way of measuring

the

power law behavior of harmonic excitations density of states N(0J) at low frequencies

and ~0:

N (~) ~ ~ ds-l. For ~0>0~ , where ~0 is a critical d e frequency and ~0c ~~-P w~2, one has the usual relation N(~) ~ d - z .

Here d s is the fracton or

~q-1, where48 q=2d/dw,whic h

does not agree with the calculation of N(~) by approximation 49'50 (EMA).

Note also that d

governs all properties of $ random walks on fractals; mean number of sltes

visited scales as 32 t X, the probability

of

return to the origin as 25~32 t -X, and mean number of visits to the origin scales as 51 t ° , where, X=~ +l=d /2. It has also been shown s that 47'52'53 for most diffusion processes on fractal systems,

the probabillty

of finding the

spectral dimension 25, ds=2df/dw, where df is the

random walker at a point X at time t is no~

fractal dimension of the system~ and for perco-

Gaussian.

lation d

is given by (4) (if a single cluster

de Gennes

54

also suggested

that one can use

w

is considered). diffusion

The crossover between normal

and that characterized

by d takes w such that t =~0-2. Moreover, c e c that for the IC, d s =4/3 for 2~_d <--6,

random walks to study superconducting tion clusters.

The properties

percola-

of such random

place at a time t

walks have been studied recently 13'55-58.

AO observed

is the so-called termite problem.

Thls

[t appears

M Sahtmt /Dtsordered rnedla

203

that, in contrast with the ant limit, one

if some of the springs are already stretched in

cannot define a fractal dimension d

equilibrium

termltes. s=~-B/2,

for the

w It has also been suggested that 59

to some rubbers and gels),

data 60 .

More recently, and voltages

dominant, the distributions

in percolation

studied 61-63.

of currents

clusters have been

It has been shown that each mom-

or use

with a distinct

and an infinite set

models,

is needed to fully characterize

such distributions.

The distribution

has also been used to study resistance

G vanishes

fluctua-

tions and i/f noise in metal-insulator mixtures.

rigidity percolation,

Mechanical

IN ELASTIC NETWORKS

properties,

data 73'74

It has been argued 70 that f>l+~d. No estimate is available yet, but Sahimi 75 and

Roux 76 have proposed

e.g., the elastic

that

f=~+2~,

moduli G, of percolation networks were first

to that of

and in fact72,f(d=2)=3.96

which is consistent with experimental

of f(d=3) 3. PERCOLATION

as (p-pc)f , in which f is

neither equal to ~, nor is it equal

of currents

to Gare

and the analogy 65 between elasticity

that can bend and stretch (the bending model), 71 beams instead of springs. For such

is characterized

of exponents

it has been

and conductivity may hold. Otherwise, in order 7O to shift Pce to Pc' one has to use springs

ent of such distributions exponent,

then,

argued 69 that the scalar contributions

but this does not agree with the

available

(a situation which may be relevant

(7)

studied by Jerauld 64. He studied 2d percolation

which agrees with the estimate of f(d=2), and 77 appears also to be exact within a Flory-like

networks

theory•

in which each bond represents

that can be stretched. lation threshold

a spring

He found that the perco-

estimates

Pce of the system is much

higher than the connectivity

threshold

Equation

Pc' the

(7) also predicts

that f=3.76

at d=3, which agrees with the experimental

rials 74

of f for gels 78 and sintered mateEquation

(7) is also exact for d> 6

reason for this being the fact that the defor-

where f=4. Continuum

mation of many configurations

been studied 14'16, and have been shown to be

of the network

can be done at no cost to the elastic energy of

G ~ (p-Pce)f

de Gennes's conjecture

that 65 f=~. Feng and Sen 66 independently

system, usually called the are different lation.

properties

of this

rigidity percolation,

from those of connectivity

For example,

reach-

It has been shown 67'68

that even the topological

the correlation

perco-

length of

the system diverges as P+Pce with an exponent e where ~e~l.l;

dBB=I.9 and f=1.45 for d=2. No

reliable estimates

of these are available at

d=3. Because of these pecularities, percolation

is said to be

large for the Swiss-Cheese model. An EMA for G

developed 64'79 for all d, and for the bending

(6)

ed the same conclusion.

to f have also

in the rigidity percolation model has been

the system. He also found that near Pce

where f>~, contradicting

corrections

the rigidity

underconstrained. But,

model 80 for d=2. Other mean-field

like argu-

ments have also been advanced 81 for estimating Pce for various

systems.

More recently,

the distribution

elastic percolation

of forces in

networks has been studiedS~~

It appears that one needs an infinite exponents

set of

to describe this distribution,

the determination distribution of estimating

and

of the second moment of this

provides a highly accurate method f and other quantities.

In addi-

tion, the concept of nolse in electrical net82 works has been extended to elastic networks .

since Pce>Pc , this is not a very realistic model

Finally,

for disordered

changes the values of the two Lame'coefficients

solids.

If the random system is pre-constrained,

i.e.,

it has been suggested

in the rigidity percolation,

that 83 if one

f would vary

204

M Sahlml / Disordered medta

continuously

with the ratio of the two

percolation

Similar to superconducting works,

percolation

net-

one may also consider 67 a superelastic

network

cluster.

Ra(L)

coefficients.

and as L -~°, equation familiar form R

in which a fraction p of the springs

Then,

~ L ~(a)/v,

exponent

(9)

(9) crosses over to the

Ip-pc I-~(~). The conductivity

is then given by N=(d-2)~+[~-v]/e.

It

are totally rigid. For such a network one has -T G ~ (Pce-P) , (8)

has been shown 87 that at d=2 one has s(a-l)=

as P+P~e"

nent of the nonlinear network.

It has been argued 67 that such a net-

work might be a better model for explaining divergence

of viscosity

the superconducting percolation,

the

of gels near Pce' than

networks 12. For the rigidity

it appears that 67 T
the bending model 84'85 T=s. No reliable estimate of T at d=3 is currently available, relation between

T and other exponents has

linearities

on transport

in relatively

the number of bonds on the shortest path ^ through the cluster, Lmi n - L ~min, w h e r e 89

~ . ~l.l mln

and 1 . 6 3 f o r

^

d=2 and 3. B l u m e n f e l d

unless the network is fully symmetrLc

the effects of non-

the number of bonds in the Zongest self]avoiding

processes

few papers.

have been

walks between the two terminals,

Most of

Moreover,

L ~L ~max max they proved that as a+ -~, one has

~(a) ~ zlal, where z describes

of elements

maximal

involving both reversible and

irreversible nonlinearities.

The first type is

by nonliear resistors or springs

for which the Ohm's law or the Hook's law has been generalized,

whereas the second type invo-

lves burntout or breakage of such elements. first type of models such as nonNewtonian porous media. models

is

relevant

The

to phenomena

and turbulent flows in

of mech-

anical and electrical yielding and fracture

in

materials.

4.1. Transport

Nonlinear Elements

obeys the generalized

i.e., the largest number of bonds

N which one can cut in order to break the backbone into two pieces;

resistors,

percolation

each of which

Ohm's law, V=rlll~sgnl,

N~L z. It was also shown 90

that d~/d~ < 0, and that ~(e)=l for aTl a and d ~ 6. For a < 0 there exists 90 a fami7 U of

solutions, corresponding

to different

the network.

directions

through some bonds, and a local

extremum in the dissipated

electrical

power of

Meir et al. 91 studied the critical

behavior of such nonlinear networks using series while Harris 92 developed

sion for ~(~). It appears that 90'93, the linear networks,

Kenkel and Straley 86'87 proposeda network of nonlinear

the terminals,

expansions,

in Networks with Reversibly-

the scaling of the

cutting surface of the backbone between

of the currents

The study of the second type of

is relevant to the description

heterogeneous

(which is

clusters near pc ) ,

~(~) has a singularity at a=0 and, in particular ^ ~(0-)=~max, where ~max describes the scaling of

these studies are based on percolation networks

characterized

and

NETWORKS

IN PERCOLATION

Depite their importance,

Blumenfeld

A h a r o n y 88 p r o v e d t h a t ~ ( ~ ) = ~ ( ~ ) / v = - 1 , a n d ^ ~ ^ ^ ~(0 )=~min' where ~min describes the scaling of

not the case for percolation

4. NONLINEAR TRANSPORT

expo-

et a l . 90 proved that ~(a=-l)=dBB and that,

and no

been proposed.

explored

a~(a), where s(~) is the superconducting

of exponents

similar to

one needs an infinite

to fully characterize

bution of currents and voltages networks.

an e-expan-

Noise and resistance

set

the distri-

in the nonlinear fluctuations

have

between the voltage drop V, current I and the

also been studied 91'g3 in the nonlinear networks,

nonlinear

which are relevant to charge-density-wave

resistance r, where ~ is a constant.

Consider the mean resistance terminals,a

distance L apart,

R (L) between two on the same

ems and noise in metal-insulator

mixtures

syst93 94 ' .

M Sahlml / Disordered medta

4.2. Transport in Networks with Irreversibly-

205

all springs whose lengths have exceeded I

Nonlinear Elements

are c broken. The shape of the macroscopic fracture

Such models have been essentially developed

depends crucially on the form of the probability

for studying yield and failure phenomena in disordered solids. Earlier models did not use percolation concepts.

Molecular-Dynamic

They were either 95'96 based on

simulations

(MDS) based on

density function

(PDF) of 1

or k . If the first c e inverse moment of this PDF is finite (i.e., if

1 and k do not assume zero values), one always c e obtains a single fracture spanning the system

Newtonian dynamics and a Lennard-Jones potential,

in which,

or used 97 models that were based on diffusion-

fraction of springs have broken. However,

limited aggregates which are nonequilibrium

this moment is divergent

structures.

(i.e., ~

or k

if do

c e assume zero values with a finite probability),

Chakrabarti and co-workers 98 used

MDS and studied the fracture behavior of disord-

then, one obtains a branched, fractal-like fracture, and the breakdown of the system is

ered solids near Pc.a/?9 De Arcangelis et

in the limit L -~°, a vanishingly small

and Takayasu

I00

were the

more gradual. The first case is similar to a

first to use percolation networks to study

first order phase transition, whereas the second

failure phenomena.

one resembles a second order phase transition.

In the model of de Arcangelis

et al., each bond of a network is a fuse with

Of particular interest is the distribution of

probability p and an insulator with probability

breakdown strengths.

l-p. Each fuse has a unit conductance and it

that, on application of an external voltage

burns out and becomes an insulator if a voltage

gradient VB/L (or a strain S), the system will

drop of more than unity is imposed on it.

breakdown.

Fracture occurs when enough fuses have burnt out

ture phenomena I04, this distribution is in the

such that the entire network has broken down. We

form of a Weibull distribution,

In the classical literature on frac-

call this the fuse model. A closely related model which is called I01'I02 the dielectric model,

P=l-exp[-c I

is

This is the probability P

Ld

(V/L)

ml

],

(i0)

the one in which each vacant bond can withstand

where c I and m I are constant. Duxbury et al. I05

a voltage drop of unity, beyond which it becomes

showed that for the fuse model near p=l,

a conductor.

The network suffers dielectric P=l-exp[-c2Ldexp(-m2L/V)],

breakdown if enough of the vacant bonds breakdown. In Takayasu's model, the resistance of the bonds are uniformly distributed,

and if the

(ii)

where c and m 2 are also constant. Stephens and I~6 Sahimi have shown that equation (ii) is valid

voltage drop along a given bond exceeds a pre-

even if the system does not have the topology of

assigned value,

a percolation cluster, and if the conductances

its resistance is reduced by a

large amount. The damaged resistors are subse-

of the fuses are distributed quantities,

quently not altered.

first inverse moment of the PDF is finite.

Elastic percolation models of fracture phenomena were introduced by Sahimi and Goddar~0~ In their model,

each bond of a fully-connected

However, neither

if this moment is divergent,

if the

then,

(i0) nor (ii) can provide a good fit to

P. An equation similar to (ii) also holds for

network is a spring that breaks if stretched

the dielectric model I05.

beyond a critlcal length 1 . Both 1 and the c c elastic constant k of the spring are distribue ted quantities. At each step of the simulations,

voltage (strain) VB, i.e., the voltage

Another quantity of interest is the breakdown

at which the network becomes conducting

(strain)

M Sahzml /Dtsordered media

206

(dielectric model),

or insulating

For the dielectric model,

(fuse model).

VB/L=I at p=0 and

VB/L=O for p> Pc" However,

studied I08'III

V B by the

gap gm of the network,

the minimum number of conducting

Here a(p) and b(p) are some

of p. Continuum corrections

and have been found to be

large for the Swiss-Cheese model.

i.e.,

bonds which

models of failure phenomena, principle

have also been developed I12

order to get a conducting

properties

according

sample-spanning Xl Then, gm ~ (Pc -p) , where Xl=X to Stinchcombe e~ a~., who also argued

that x=v. This is supported by the numerlcal

ACKNOWLEDGEMENTS I would like to thank S. Arbabi,J.D. B.D. Hughes,

fuse model one has

M.D. Stephens

where I08 y=~-(d-l)~.

,

G.R. Jerauld,

failure

model I03 S diverges as

for their collaborations

st~-nulating discussions. were supported

,

Goddard,

H. Siddlqui and on

portions of the work revlewed here, and for

(13)

For the mechanical

S ~ (p-pc)-ym

above,

, and their

have been studied.

results of Manna and Chakrabarti I02. For the

VB/L ~ (p-pc)-y

Several other

which are in

similar to the ones discussed

are to be added to the insulating network in

cluster.

to the

exponents x, y, Ym and fm have also been

(12)

Stinchcombe c~ ~7. I07 approximated minimum insulating

functions

as p+pc one has

VB/L ~ (pc-p) x.

failure models.

Portlons of this work

in part by the NSF (;rant

CBT 8615160 and the Air Force Office of Scien-

(14)

where simulations of Sah~ni and Goddard I03 and 109 experiments on perforated metal fomls both

tific Research Grant 87-0284.

indicate that Ym=l.4 at d=2. No relatlon between

REFERENCES

Ym and the other exponents

I. J.M. Ziman, Models of Disorder (Cambridge University Press, Cambridge, 1979).

yet. If mechanical applying

an external

force F, then

F - (p-pc)fm

,

which is conslstent

2. B.J. Last and D.J. Thouless, 27 (1971) 1719.

(15)

and it has been suggested fm=[f+(d-dBB)V]/2.

has been suggested

failure occurs because of

3. S. Kirkpatrlck,

that 98'I0g,

Thls yields fm=2"25 at d=2, with the experimental

Phys. Rev. Lett.

Rev. Mod. Phys. 45(1973)574.

4. M. Sahimi, in The Mathematics and Physics of Disordered Media, eds. B.D. Hughes and B.W. Ninham (Springer, Berlin, 1983) pp 314-346.

result I09, f =2.4+ 0.4. The relation m -Ii0 fm=[f+(2-dic)V]/2 has also been proposed

5. D. Stauffer, Introduction to Percolation Theory (Taylor and Francis, London, 1985).

which yields f

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=2.05, which is still consistent m with the experimental result and, thus, the precise relatlon between f is yet to be determined. are supposed

m

and other exponents

Equations

to be valid for P=Pc" However,

p=l (or p=0 for the dielectric model) a different

7. J.P. Straley,

(12)-(15) for

one has

result I01'I05

VB/L ~ [a(p) +b(p)(in L)~] -I,

Phys. Rev. BI5 (1977)

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5733.

Phys. Status

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where 1/2 < y < i for the dielectric model, [2(d-l)] -I < y < 1 for fuse and mechanical

and

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