TRANSPORT PROCESSES IN GLASS

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TRANSPORT PROCESSES IN GLASS

A. Cooper, Jr

To cite this version:

A. Cooper, Jr. TRANSPORT PROCESSES IN GLASS. Journal de Physique Colloques, 1982, 43

(C9), pp.C9-369-C9-380. �10.1051/jphyscol:1982971�. �jpa-00222500�

JOURNAL DE PHYSIQUE

Colloque C9, supplément au n°12, Tome 43, décembre 1982 page C9-369

TRANSPORT PROCESSES IN GLASS

A.R. Cooper, J r .

Department of Metallurgy and Materials Soienae 3Case Western Reserve University, Cleveland, Ohio, U.S.A.

Résumé. - La vitesse de liquéfaction d'un verre dépend du transport combiné de plusieurs grandeurs: chaleur, force, masse, et charge électrique. Chacun de ces phénomènes obéit un ensemble similaire de lois et.il existe une variété de porteurs et divers modes de transport pour chacune de ces grandeurs. Des coefficients et matrices d'interdiffusion sont en outre reliés et dépendent des coefficients d'autodiffusion de masse, des phonons. On discute sous cet angle la conductivitê thermique, les coefficients et matrices d'interdif- fusion, la conductivitê électrique et la viscosité des liquides formateurs de verres.

Abstract. - The rate of glass melting depends on the combined influence of the transport of several quantities: heat, momentum, species, and electric charge. The phenomenology of each of these processes obey a similar set of laws and there are a variety of carriers and several modes of transport for each. The.transport coefficients are likewise related and depend on the self diffusion coefficients of species, phonons and photons. Thermal con- ductivity and viscosity of glass forming liquids are discussed from this point of view.

1. Introduction. - Most of the energy required to heat glass batch to the "melting temperature" is utilized in supplying the specific heat of the products and react - ants and only a small proportion is required for batch and fusion reactions them- selves. This places the glass melting process in contrast to the smelting of iron, aluminum and other metals where the majority of the energy is expended in overcoming the free energy of formation of the oxide ore.

The actual "melting temperature" of glass is well above the liquidus tempera- ture. It is not determined thermodynamically, but kinetically from the rates of two processes, fining and homogenization, each of which is determined by transport:

momentum transport governs buoyant rise of bubbles through the melt and hence affects fining. Homogenization occurs by what is sometimes termed mass transport but which we will refer to as species transport. Viscosity decreases and diffusiv- ity increases with temperature. Hence, increased melting rates are achieved by higher temperatures. Temperatures actually used in glass melting are limited by the increased volatilization and refractory and electrode corrosion associated with higher temperatures and by the excess cost necessary to increase flame temperatures for example with oxygen enrichment of air [1].

Since refining and homogenization take place throughout the melter it is evident that heat transport, from flame to glass or from a hotter to a colder

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982971

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DE PHYSIQUE

r e g i o n w i t h i n t h e m e l t , i s a l s o an e s s e n t i a l component of g l a s s melting. When ohmic h e a t i n g of t h e melt i s used t r a n s p o r t of e l e c t r i c charge through t h e g l a s s i s r e - q u i r e d . Understanding of t h e s e t r a n s p o r t p r o c e s s e s : h e a t , s p e c i e s , momentum

( f l u i d flow) and e l e c t r i c a l charge,

i s

t h u s fundamental t o a n a l y s i s of t h e g l a s s m e l t i n g process and, a s we s h a l l s e e , t h e t o p i c becomes c h a l l e n g i n g and d i f f i c u l t because of t h e c o n s i d e r a b l e i n t e r a c t i o n between t h e p r o c e s s e s .

Glass forming i s v i s c o u s deformation, i . e . momentum t r a n s p o r t , complicated by t h e f a c t t h a t t h e r e i s a f r e e s u r f a c e which i s changing with time. The importance of h e a t t r a n s p o r t i n t h e a n a l y s i s and understanding o f g l a s s forming a r i s e s because of t h e s t r o n g dependence of v i s c o s i t y on temperature. Hence t h e d e t a i l s of t h e deformation p r o c e s s depend on h e a t flow a s well a s f l u i d flow.

Glass annealing and tempering depend on t h e v i s c o u s r e l a x a t i o n of s t r e s s e s t h a t a r e produced e l a s t i c a l l y due t o d i f f e r e n t l o c a l r a t e s of change of temperature.

Again, momentum t r a n s p o r t and h e a t t r a n s p o r t a r e t h e dominant p r o c e s s e s . In view of t h e i r importance i n g l a s s technology, it i s t h e purpose of t h i s paper t o d i s c u s s t r a n s p o r t p r o c e s s e s i n g l a s s forming systems from a g e n e r a l p o i n t of view making comparison where

i t

seems a p p r o p r i a t e t o o t h e r m a t e r i a l s .

2 .

T r a n s p o r t C a r r i e r s and Modes.

-

The mathematical s i m i l a r i t i e s between h e a t t r a n s - p o r t , s p e c i e s t r a n s p o r t and momentum t r a n s p o r t make

it

convenient t o look a t t h e s e v a r i o u s p r o c e s s e s i n a u n i f i e d manner and t h e r e a r e s e v e r a l e x c e l l e n t t r e a t m e n t s

[ 2 ]

which a r e p e r t i n e n t t o problems i n g l a s s making.

A

v a r i e t y of " c a r r i e r s " t r a n s p o r t momentum, h e a t , chemical s p e c i e s , and e l e c t r i c charge. These i n c l u d e molecules, i o n s , e l e c t r o n s , phonons, photons and chemical bonds. Many c a r r y more t h a n one t y p e of charge, e.g. molecules i n a g a s c a r r y h e a t , momentum and s p e c i e s .

Likewise t h e r e a r e d i s t i n c t i v e l y d i f f e r e n t modes of t r a n s p o r t : conduction due t o t h e random motion o f charge c a r r i e r s r e l a t i v e t o t h e m a t e r i a l a s a whole, convec- t i o n due t o t h e motion of t h e m a t e r i a l a s a whole, r a d i a t i o n due t o t h e d e t e r m i n i s -

tic

motion of charge c a r r i e r s through t h e m a t e r i a l without i n t e r a c t i o n . Conduction d e s c r i b e s t h e c a s e where t h e mean f r e e p a t h of t h e c a r r i e r i s much s h o r t e r than a c h a r a c t e r i s t i c dimension of t h e specimen while r a d i a t i o n a p p l i e s t o t h e case where mean f r e e p a t h i s much longer than such a c h a r a c t e r i s t i c d i s t a n c e .

3 .

Flux D e n s i t i e s .

-

The amount of charge t r a n s p o r t e d p e r u n i t a r e a p e r u n i t time i s termed f l u x d e n s i t y , j . As such i f q r e p r e s e n t s t h e amount of charge

I f q i s a s c a l a r l i k e h e a t , chemical s p e c i e s o r e l e c t r i c a l charge then j

i s

a vector*

i n o r d i n a r y t h r e e dimensional space.

^On

t h e o t h e r hand, i f q i t s e l f h a s d i r e c t i o n as i n t h e case of momentum,

j

i s a

3 x 3

m a t r i x .

Flux d e n s i t y may have d i f f u s i v e ,

jD, c o n v e c t i v e , j C , and r a d i a t i v e , j R ,

*We do not u s e a s p e c i a l symbol f o r v e c t o r s i n t h r e e dimensional space.

components. Thus

j = j D + j c + j R . ( 2 )

The phenomenological

equations

f o r t h e d i f f u s i v e components a r e w r i t t e n :

j D = -D Vp =

-k V$

9 9 4 9

^{( 3 )}

where

p q , Dq,

4q, and k a r e r e s p e c t i v e l y t h e charge d e n s i t y , d i f f u s i o n c o e f f i c i e n t , 9

potential,

and c o n d u c t i v i t y f o r q. For example i n h e a t t r a n s f e r

p

i s t h e enthalpy d e n s i t y , D 1 s t h e thermal d i f f u s i v i t y , 4

^E T,

t h e temperature, and k i s t h e 9

9 9 9

thermal c o n d u c t i v i t y .

The convective p o r t i o n of t h e f l u x d e n s i t y i s g i v e n by

j , = , v 9

where v i s t h e o v e r a l l v e l o c i t y v e c t o r .

Unlike convection and conduction f l u x d e n s i t i e s t h e r a d i a t i v e f l u x d e n s i t y ,

j R '

does n o t depend on t h e l o c a l v a l u e s b u t r a t h e r on t h e v a l u e s of t h e i n t e n s i t y , I , o f r a d i a t i o n a t t h e bounding s u r f a c e s .

A s

such

The f a c t o r s i n t h e i n t e g r a n d of e q u a t i o n 5 a r e b e s t understood by c o n s i d e r i n g a vec- t o r , r , j o i n i n g a n i n c r e m e n t a l a r e a dA on t h e s u r f a c e with t h e incremental a r e a , dA, where t h e f l u x d e n s i t y i s t o be determined; i s t h e a n g l e between t h e normal t o t h e a r e a where t h e f l u x d e n s i t y i s t o be determined and t h e v e c t o r ;

O 2

i s t h e angle between t h e inward normal t o t h e bounding a r e a and t h e v e c t o r . The r a d i a t i o n i n t e n - s i t y , I , a t t h e bounding s u r f a c e i s t h e amount of charge per u n i t a r e a normal t o

^r

p e r u n i t s o l i d angle emitted o r r e f l e c t e d from t h e s u r f a c e . I f t h e m a t e r i a l i s i s o - t r o p i c , I does n o t depend on d i r e c t i o n . I t does depend on p o s i t i o n on t h e bounding s u r f a c e .

4.

Conservation Equations.

-

The q u a n t i t i e s ( h e a t , momentum, s p e c i e s and e l e c t r i c charge) t r a n s p o r t e d i n glassmaking a r e conserved, allowing t h e following r e l a t i o n t o a p p l y i n a l l c a s e s

where

S

i s t h e s t r e n g t h of a l o c a l source of q , t h e amount of q p e r u n i t volume p e r 9

u n i t time. Ignoring r a d i a t i o n and s u b s t i t u t i n g from e q u a t i o n s

3

and

4

g i v e s

3 _{a t}

⁼

^VD _{q q}

Vp -

vVp _q

+

5 _q (71

A

number o f s o l u t i o n s t o equation

7

f o r simple c a s e s can be found i n s t a n d a r d r e f e r - ences

[ 3 ] .

For complex shapes a n a l y t i c a l s o l u t i o n s a r e r a r e l y a v a i l a b l e and r e - c o u r s e t o numerical s o l u t i o n s i s o f t e n r e q u i r e d . When

D =

D

(p )

t h e e q ~ ~ a t i o n

9 9 9

becomes non l i n e a r and hence more d i f f i c u l t .

5 .

I n t e r a c t i o n s Between Transport Processes.

-

Both t h e complexity and t h e c h a l l e n g e

C9-372 JOURNAL D E PHYSIQUE

a r e i n c r e a s e d by t h e f a c t t h a t i n glassmaking t h e r e i s c o n s i d e r a b l e i n t e r a c t i o n be- tween t h e t r a n s p o r t p r o c e s s e s . Sometimes two, t h r e e o r even f o u r p r o c e s s e s i n t e r a c t .

According t o t h e Curle P r i n c i p l e [4] t r a n s p o r t p r o c e s s e s of t h e same rank* a r e coupled i n t h e sense t h a t t h e d i f f u s i v e f l u x d e n s i t y f o r each p r o c e s s i s a f f e c t e d by t h e p o t e n t i a l g r a d i e n t of a l l of t h e independent p r o c e s s e s . I n a multicomponent sys- tem t h e f l u x d e n s i t y of each of t h e independent s p e c i e s i s a f f e c t e d by t h e p o t e n t i a l g r a d i e n t of each of t h e o t h e r independent s p e c i e s a s w e l l a s by t h e temperature g r a d i e n t . Llkewise t h e h e a t f l u x 1 s a f f e c t e d by t h e chemical p o t e n t i a l g r a d i e n t s o f a l l t h e s p e c i e s . For a system with two independent s p e c i e s

A

and B, a dependent s p e c i e s

C,

and non-uniform temperature [ 5 ] .

When independent f l u x d e n s i t i e s and p o t e n t i a l s a r e u t i l i z e d , t h e

L

m a t r i x , above, i s r e q u l r e d t o be symmetric according t o Onsager's r e c i p r o c a l r e l a t i o n s [ 6 ] . The thermal c o n d u c t i v i t y , k , i s equal t o L ~ ~ / T ~ .

There have been a number of measurements of isothermal multlcomponent (more t h a n two independent s p e c i e s ) d i f f u s i o n i n g l a s s forming systems [ 7 ] . While t h e c r o s s terms a r e o f t e n l a r g e t h e y a r e r e a d i l y understood c o n c e p t u a l l y .

On

t h e o t h e r hand, thermal d i f f u s i o n c r o s s c o e f f i c i e n t s L

BT

and LAT a r e l e s s well c h a r a c t e r i z e d

[8] b u t t h e i r v a l u e s a r e thought t o be low. Thus, although a non-isothermal g l a s s melt

w i l l

tend toward a non-uniform composition, t h e magnitude of t h i s e f f e c t i s probably small.

P o t e n t i a l dependence of c o n d u c t i v i t i e s provides a n o t h e r b a s i s f o r i n t e r a c t i o n between t r a n s p o r t p r o c e s s e s , e . g . s h e a r v i s c o s i t y , t h e c o n d u c t i v i t y f o r momentum, depends s t r o n g l y on temperature and composition. Species d i f f u s i v i t y and e l e c t r i c a l c o n d u c t i v i t y a r e s i m i l a r l y a f f e c t e d by temperature and by composition. Thls r e s u l t s , f o r example, i n a need t o know t h e temperature and composition d i s t r i b u t i o n b e f o r e t h e f l u i d flow problem can be completed.

Sources of momentum and h e a t a r e c o n t r i b u t e d by t r a n s p o r t . The product of t h e

--

f l u x d e n s i t y and t h e p o t e n t i a l g r a d i e n t of any t r a n s p o r t p r o c e s s d e f i n e s a l o c a l h e a t source. E l e c t r i c m e l t i n g [9] depends on t h i s f a c t . The f r i c t i o n caused by f l u i d flow i s another source of h e a t which can be important [ l o ] i n t h e formation of g l a s s f i b e r s . G r a v i t a t i o n a l a c c e l e r a t i o n , g, p r o v i d e s a source f o r momentum of s t r e n g t h amg where

pm

i s t h e mass d e n s i t y . Since

p m = P

( T ) , i f t h e system i s not

m

i s o t h e r m a l , t h e r e w i l l be d i f f e r i n g source s t r e n g t h s i n d i f f e r e n t l o c a t i o n s c a u s i n g t h e c i r c u l a t i n g flows, termed thermal convection, t h a t a r e s o important i n g l a s s m e l t i n g . Obviously t o c a l c u l a t e t h e s e flows one needs t o know t h e temperature

*Rank depends on t h e d i r e c t i o n a l c h a r a c t e r of t h e q u a n t i t y being t r a n s p o r t e d ,

h e a t , s p e c i e s , and e l e c t r i c charge a r e a l l s c a l a r q u a n t i t i e s a n d h e n c e o f t h e s a m e r a n k .

d i s t r i b u t i o n .

Convection, t h e motion of a l l s p e c i e s t o g e t h e r , i s o f t e n t h e dominant component of t h e f l u x d e n s i t y . To know t h e convective f l u x d e n s i t y f o r h e a t flow f o r example r e q u i r e s knowing t h e v e l o c i t y v e c t o r a s a f u n c t i o n of p o s i t i o n .

These i n t e r a c t i o n s cause each t r a n s p o r t p r o c e s s i n glassmaking t o depend on t h e o t h e r s and r e q u i r e c o n s i d e r a t i o n of simultaneous s o l u t i o n of s e v e r a l e q u a t i o n s of t h e form of equation

7 .

They a r e a l s o a reason why t h e use of p h y s i c a l models [ I l l which n a t u r a l l y but i m p e r f e c t l y [I21 couple e f f e c t s a r e s o p o p u l a r .

6 .

R e l a t i o n s h i p Between t h e Transport C o e f f i c i e n t s .

-

Thus f a r we have considered t h e phenomenology of t r a n s p o r t p r o c e s s e s : t h e d i f f e r e n t modes of t r a n s p o r t , t h e s i m i l a r i t y of t h e f l u x e q u a t i o n s and t h e c o n s i d e r a b l e i n t e r a c t i o n between t h e s e p r o c e s s e s i n g l a s s m e l t i n g and forming. W e t u r n now t o a d i f f e r e n t m a t t e r : t h e microscopic b a s i s f o r t h e t r a n s p o r t c o e f f i c i e n t s and t h e r e l a t i o n s between t h e s e c o e f f i c i e n t s .

7 .

S e l f D i f f u s i o n C o e f f i c i e n t s [13]

-

Dynamic v i s c o s i t y ,

Q ,

s p e c i e s c o n d u c t i v i t y , L , and e l e c t r i c a l c o n d u c t i v i t y ,

a ,

depend on t h e s e l f d i f f u s i o n c o e f f i c i e n t of t h e s p e c i e s , while t h e thermal c o n d u c t i v i t y k depends on t h e s e l f d i f f u s i o n c o e f f i c i e n t f o r phonons and photons.* S e l f d i f f u s i o n c o e f f i c i e n t s

D

f o r q type p a r t i c l e s a r e

4 always d e f i n e d by [14]

D

=

g < ~ 2 > / < ~

1

>

4

9

9

(91

where

A

i s t h e f r e e path and .r i s t h e i n t e r v a l between the completions of

successive

jumps and < > i n d i c a t e s averaging. For photons, phonons and molecules i n a gas t h e r e i s continuous motion of t h e p a r t i c l e s and t h e f r e e p a t h s a r e randomly d i s t r l - buted with t h e r e s u l t t h a t

< A ~ > =

2

<A>'.

This p e r m i t s m o d i f i c a t i o n of e q u a t i o n 9 t o

D =

? < v > < X q >

1

9 9 (101

w h e r e < v > i s t h e average speed of q p a r t i c l e s (photons, phonons o r g a s molecules).

9

The problem of o b t a i n i n g a microscopic d e s c r i p t i o n of t h e d i f f u s i o n c o e f f i c i e n t r e - duces t h e r e f o r e t o f i n d i n g t h e a p p r o p r i a t e expression f o r < A > s i n c e <v> i s u s u a l l y known.

P a r t i c l e m o b i l i t y , t h e p a r t i c l e ' s v e l o c i t y per u n i t f o r c e , was shown by Ein- s t e i n [15] t o be simply r e l a t e d t o

D

i . e .

B = D

/kT.

9' 9 9

8.

Thermal Conductivity.

-

The important c a r r i e r s of h e a t a r e molecules, e l e c t r o n s , h o l e s , phonons, and photons. The thermal c o n d u c t i v i t y , kT from n d i f f e r e n t t y p e c a r r i e r s i s given by:

kT

=

i g l ~ i ( a ~ i / a ~ ) n (11)

*The term " s e l f " i s n o t c o n v e n t i o n a l l y used when i d e n t i f y i n g t h e d i f f u s i o n

c o e f f i c i e n t s f o r phonons and photons. I t i s n e c e s s a r y , however, t o make a d i s t i n c -

t i o n between t h e d i f f u s i o n of i n d i v i d u a l s p e c i e s and t h e i r combined e f f e c t i n b i n a r y

o r multicomponent d i f f u s i o n . Although t h e d i s t i n c t i o n i s n o t necessary f o r photons

and phonons, we use

it

f o r c o n s i s t e n c y .

C9-374 JOURNAL

DE PHYSIQUE

where T d e n s i t y t h e

i

t h

i s t e m p e r a t u r e , t h e p o t e n t i a l f o r thermal conduction,and

p .

i s t h e energy f o r t h e ith c a r r i e r , i . e . api/aT i s t h e s p e c i f i c h e a t p e r l u n i t volume due t o p r o c e s s . For nonmetallic g l a s s e s o n l y phonons, a , and photons, b , a r e i m - p o r t a n t . Thus:

The second term i n t h i s e x p r e s s i o n was f i r s t c o r r e c t l y e v a l u a t e d i n g l a s s melt- ing by Czerny and Genzel [16]. Photons move a t t h e v e l o c i t y of l i g h t i e - l o 5 f a s t e r t h a n t h e v e l o c i t y of phonons. Also t h e mean f r e e p a t h f o r photons -0.07m i s - l o 8

l a r g e r t h a n t h e mean f r e e p a t h of phonons i n a g l a s s . However, t h e h e a t capacity 1 6 0 1 1 ~ ~ ~

a s s o c i a t e d with photons, ---

Vb (where n i s t h e index of r e f r a c t i o n ) , i s s o much l e s s t h a n t h e o r d i n a r y h e a t c a p a c i t y from phonons ( l a t t i c e v i b r a t i o n s ) t h a t

it

1s only a t high temperatures t h a t photons dominate thermal c o n d u c t i v i t y . The v a l u e of

A

v a r i e s from g l a s s t o g l a s s , and dark g l a s s e s have mean f r e e paths much l e s s than 0.07m. Often f o r r e l a t i v e l y c l e a r g l a s s t h e c h a r a c t e r i s t i c s i z e of a g l a s s a r t i c l e i s much l e s s than t h e mean f r e e p a t h significantly reducing t h e photon c o n t r i b u t i o n t o k.

A particularly

s t r i k i n g c a s e occurs i n t h e drawing of f i b e r s where r a d i a l d i s t a n c e s a r e l e s s t h a n of t h e photon mean f r e e path, b u t along t h e a x i s of t h e f i b e r t h e d i s t a n c e can be

substantially

g r e a t e r t h a n t h e f r e e p a t h .

Species s e l f d i f f u s i o n I n g l a s s e s and g l a s s forming l i q u i d s i s q u i t e d i f f e r e n t . There i s no c h a r a c t e r i s t i c v e l o c i t y and atoms spend t h e g r e a t e s t f r a c t i o n of time i n o s c i l l a t o r y non-neighbor exchanging motions ( t h e concept of neighbor i n n o n c r y s t a l -

l i n e m a t e r i a l i s made p r e c i s e by t h e use of Voronoi p o l y h e d r a ) . The time i n t e r v a l ,

T,

i s p r i m a r i l y composed of t h e pauses between neighbor exchanging motions (jumps).

The mean f r e e p a t h f o r jumps i s u s u a l l y assumed t o b e of t h e o r d e r o f t h e atomic spacing. The frequency,

( T - 1 1 ,

of jumping i s t h e product of t h e atomic v i b r a t i o n a l frequency, t h e p r o b a b i l i t y of t h e atom having s u f f i c i e n t f r e e energy t o break enough bonds t o permit an exchange of neighbors and t h e p r o b a b i l i t y of t h e e x i s t e n c e of s u f f i c i e n t space t o allow such an exchange.

Atoms of a s i n g l e s p e c i e s may d i f f u s e independently from t h e jumping of o t h e r s p e c i e s . In t h i s c a s e t h e r e i s analogy t o vacancy, i n t e r s t i t i a l , and i n t e r s t i t i a l l y d i f f u s i o n i n c r y s t a l s except t h a t t h e d e f e c t s a r e d e f i n e d by t h e topology r a t h e r t h a n by t h e geometry. I t i s a l s o p o s s i b l e t h a t s e l f d i f f u s i o n of d i f f e r e n t s p e c i e s a r e c o r r e l a t e d . For example d i f f u s i o n of oxygen and t h e network forming c a t i o n s may occur by a p r o c e s s which i n v o l v e s t h e c o o p e r a t i v e motion of both i o n s . I t i s i n t e r - e s t i n g t h a t t h i s d i f f u s i o n p r o c e s s a l s o r e s u l t s i n a change i n t h e r i n g s i z e d i s t r i - b u t i o n .

9. E l e c t r i c a l Conductivity.

-

Ions a r e t h e c a r r i e r s f o r e l e c t r i c a l charge i n oxide

*In f a c t photons and phonons have a continuous range of f r e q u e n c i e s , and p a r t i c u l a r l y f o r photons t h e mean f r e e p a t h depends d r a s t i c a l l y on frequency.

Thus t h e v a l u e s i n e q u a t i o n 12 r e p r e s e n t a p p r o p r i a t e l y averaged v a l u e s of Xa and

lh

-

g l a s s e s . Most mobile i o n s a r e a l k a l i i o n s , and sodium (Na), i s t h e most mobile of t h e s e . Hence, we make t h e s i m p l i f y i n g assumption t h a t sodium i o n s a r e t h e only c a r r i e r .

The f l u x d e n s i t y , j f o r e l e c t r i c c h a r g e , q , i s given by t h e product of t h e c a r r i e r m o b i l i t y DNa/HRkT, t h e d e n s i t y o f c a r r i e r s pNaZNae, and t h e f o r c e -ZNaeW,ie. 9'

j

= -

{ D ~ ~ P ~ ~ z ~ ~ ~ ~ / H ~ ~ T ~ v Y (13)

The q u a n t i t y i n

{

1 i s t h e e l e c t r i c a l c o n d u c t i v i t y , ~ . The r a t h e r s l i g h t d i f f e r e n c e s between D and DNa due t o d i f f e r e n c e s I n jump c o r r e l a t i o n s a r e accounted f o r by what

9

i s termed t h e Haven R a t i o , "H " [17]. Estimates of e l e c t r i c a l c o n d u c t i v i t y of com- m e r c i a l g l a s s e s a r e r e a d i l y obtained from s e l f d i f f u s i o n c o e f f i c i e n t measurements of Na. Conversely, measurements of

DC

e l e c t r i c a l c o n d u c t i v i t y a r e o f t e n used t o i n f e r t h e sodium s e l f d i f f u s i o n c o e f f i c i e n t .

10. I n t e r d i f f u s i o n .

-

Processes important t o glassmaking l i k e r e f r a c t o r y d i s s o l u t i o n , e v a p o r a t i o n , homogenization, b a t c h m e l t i n g , s p i n o d a l decomposition and d e v i t r i f i c a - t i o n a r e c o n t r o l l e d by i n t e r d i f f u s i o n , t h e t r a n s p o r t of atoms i n a c o n c e n t r a t i o n g r a d i e n t i n which some i o n s a r e moving i n one d i r e c t i o n and o t h e r s i n t h e o p p o s i t e d i r e c t i o n . In a b i n a r y system where t h e r e i s o n l y a s i n g l e independent s p e c i e s , i n t e r d i f f u s i o n i s d e s c r i b e d by a s i n g l e number, t h e b i n a r y d i f f u s i o n c o e f f i c i e n t . In a multicomponent system of m independent s p e c i e s i n t e r d i f f u s i o n i s d e s c r i b e d by a

mxm

m a t r i x [ I & ] .

For d i f f u s i o n i n a b i n a r y (A,B) system, F i c k s Law can be w r i t t e n j A

= -DABVCA = - L G V C

AB AB A

(14)

a (uA

- UB)

where

CA

i s t h e molar c o n c e n t r a t i o n of

A,

and

G ⁼

AB

(u i s chemical poten-

~ C A

t i a l ) , and LAB i s a c o e f f i c i e n t depending only on t h e i n d i v i d u a l s p e c i e s ' m o b i l i t y . The v a l u e s of LAB a r e deduced e i t h e r f r o m t h e Darken Model [19] which assumes t h e e x i s t e n c e of a r e l a x a t i o n v e l o c i t y t o compensate f o r t h e d i f f e r e n c e i n atomic mobil- i t i e s of

A

and

B

atoms o r from t h e Nernst Planck Model [20] which p e r m i t s t h e devel- opment of a l o c a l e l e c t r i c f i e l d t o compensate f o r d i f f e r e n t i o n i c m o b i l i t i e s .

For multicomponent d i f f u s i o n i n a system of

m

independent s p e c i e s t h e following g e n e r a l i z a t i o n t o equation 14 a p p l i e s

-

j

=

- [ D ] v ~

= -

[L] [ G ] v ~ (15)

The f l u x d e n s i t y v e c t o r , 7, i s j u s t t h e f l u x d e n s i t y of each independent, i . e . s p e c i e -

j = j l ,

j

2,.

. . , j m . Likewise, t h e molar composition ?

= C

c2, c3, . . ., cm.

The

G

m a t r i x with elements i s determined from thermodynamics while t h e

L

matrix (symmetric according t o & s a g e r 1 s r e c i p r o c a l r e l a t i o n ) has elements d e t e r - mined i n a manner analogous t o t h e b i n a r y case by a g e n e r a l i z a t i o n of t h e Darken o r t h e Nernst Planck model [21].

11.

V i s c o s i t y .

-

V i s c o s i t y from 10 t o 1015 Pas i s perhaps t h e t r a n s p o r t c o e f f i c i e n t

most d e c i s i v e f o r t h e p r o c e s s i n g of g l a s s . A c t u a l l y t h e r e a r e two v i s c o s i t i e s , n ,

C9-376 JOURNAL DE PHYSIQUE

t h e s h e a r v i s c o s i t y , which measures t h e r e s i s t a n c e of a m a t e r i a l t o i r r e v e r s i b l e shape charge, and t h e volume v i s c o s i t y ,

X ,

a c o e f f i c i e n t r e l a t i n g t h e r a t e of f r a c - t i o n a l change o f molar volume with r e s p e c t t o time t o t h e d i f f e r e n c e between t h e a c t u a l p r e s s u r e and t h e p r e s s u r e which would be i n e q u i l i b r i u m with t h e c u r r e n t molar volume, i . e . i t measures t h e r e s i s t a n c e of t h e m a t e r i a l t o i r r e v e r s i b l e volume change. While we w i l l f o c u s on s h e a r v i s c o s i t y , we f i r s t n o t e t h a t volume v i s c o s i t y of g l a s s forming substances appears

[22]

t o b e of t h e same order a s s h e a r v i s c o s i t y

(perhaps a f a c t o r of t h r e e g r e a t e r with about t h e same temperature dependence).

For a monoatomic gas, molecules a r e t h e c a r r i e r s both f o r s p e c i e s , and f o r momentum. The f o l l o w i n g simple r e l a t i o n s between

D

and

n

a p p l i e s

D

= q / p m E

v

where pm i s t h e mass d e n s i t y of t h e g a s . Thus t h e d i f f u s i o n c o e f f i c i e n t f o r s p e c i e s

i s

i d e n t i c a l t o t h e d i f f u s i o n c o e f f i c i e n t f o r momentum, t h e kinematic v i s c o s i t y , v.

In l i q u i d s , however, t h e r e l a t i o n s a r e q u i t e d i f f e r e n t . Molecules o r atoms a r e s t i l l t h e i r own c a r r i e r s , b u t unbroken bonds a r e t h e c a r r i e r s f o r momentum. When t h e r e a r e no broken bonds a m a t e r i a l i s e l a s t i c ; momentum

i s

c a r r i e d without l o s s through a specimen; t h e v i s c o s i t y i s i n f i n i t e . S i n c e broken bonds a r e n e c e s s a r y f o r s p e c i e s d i f f u s i o n , s e l f d i f f u s i o n i s zero i n t h e absence of broken bonds.

Hence, i n l i q u i d s t h e v i s c o s i t y must be r e c i p r o c a l l y r e l a t e d t o t h e s p e c i e s d i f - f u s i o n c o e f f i c i e n t . This a p p l i e s slmply

i n

t h e c a s e of monoatomic l i q u i d s . I n mul- ticomponent l i q u i d s t h e v i s c o s i t y must depend on t h e r e c i p r o c a l of some combination of t h e i n d i v i d u a l s e l f d i f f u s i o n c o e f f i c i e n t s . In a d d i t i o n t o t h e broken bonds which permit neighbor exchange of atoms, i r r e v e r s i b l e shape deformation r e q u i r e s t h e d e s t r u c t i o n and c r e a t i o n of atomic s i t e s .

There a r e s e v e r a l d i f f e r e n t models l i n k i n g v i s c o s i t y t o s e l f d i f f u s i o n c o e f f i c i e n t : Combining S t o k e ' s s o l u t i o n f o r flow about a sphere with E i n s t e i n ' s r e l a t i o n between m o b i l i t y and s e l f d i f f u s i o n y i e l d s t h e well known S t o k e ' s E i n s t e i n equation [ 2 3 ]

where

r

i s t h e r a d i u s of t h e s p e c i e s whose s e l f d i f f u s i o n c o e f f i c i e n t

i s

given by D . This r e l a t i o n s h i p i s t y p i c a l l y used t o o b t a i n

D

knowing

n.

I t i s i n r e a s o n a b l e agreement with experimental r e s u l t s f o r aqueous s o l u t i o n s , molten s a l t s and molten m e t a l s . I t i s based on t h e concept t h a t t h e motion of a p a r t i c l e through a f l u i d depends on neighbor exchanges w i t h i n t h e f l u i d , i . e . on c o o p e r a t i v e motion.

When a s p e c i e s d i f f u s e s by an i n t e r s t i t i a l mechanism ( a l k a l i i o n s i n g l a s s ) e q u a t i o n 16 g r o s s l y u n d e r e s t i m a t e s t h e v i s c o s i t y . Equation 16 i s o f t e n used t o pro- v j d e an e s t i m a t e f o r t h e s e l f d i f f u s i o n c o e f f i c i e n t f o r d e v i t r i f i c a t i o n .

By u s e of a l a t t i c e model Eyring o b t a i n e d e x p r e s s i o n s f o r D and f o r

n [24]

i n a f l u i d . Comparing t h e r e s u l t s he o b t a i n e d

where 2r i s t h e f r e e path f o r jumping a s well a s t h e atomic diameter. For a given D E y r i n g ' s model p r e d i c t s a v i s c o s i t y about 10 times g r e a t e r t h a n S t o k e s - E i n s t e i n presumably because it r e q u i r e s v i s c o u s flow t o occur from t h e i n d i v i d u a l jumps of atoms and n o t from c o o p e r a t i v e motion. While e q u a t i o n 17 f i t s molten metal d a t a l e s s well t h a n t h e S t o k e s - E i n s t e i n e x p r e s s i o n , t h e v i s c o s i t y of Na20 CaO S i 0 2 a g r e e s well with t h e p r e d i c t i o n of e q u a t i o n 17 when t h e s e l f d i f f u s i o n c o e f f i c i e n t of oxy- gen i s used f o r D [25] and s i m i l a r behavior occurs f o r K20 SrO S i 0 2

[26!.

Nabarro and H e r r i n g ' s [27] e x p r e s s i o n f o r t h e v i s c o u s c r e e p of a p o l y c r y s t a l - l i n e s o l i d can be w r i t t e n

where a i s atomic r a d i u s , and w i s t h e r a d i u s of t h e g r a i n . Since w i s many o r d e r s of magnitude l a r g e r t h a n a , t h e v i s c o s i t y of c r y s t a l s i s f a r g r e a t e r t h a n t h a t of l i q u i d s w i t h t h e same v a l u e of

D .

This a r i s e s because l a t t i c e s i t e s i n a c r y s t a l can o n l y be c r e a t e d o r d e s t r o y e d a t g r a i n boundaries o r o t h e r d e f e c t s whereas i n a l i q u i d because of i t s d i s o r d e r , s i t e s can be c r e a t e d and d e s - t r o y e d by c o o p e r a t i v e motion n e a r l y everywhere. Considering a l i q u i d t o be a poly- c r y s t a l with each atom an i n d i v i d u a l g r a i n g i v e s r e a s o n a b l e agreement between Nabarro Herring (equation 18) (with w/d

=

1 ) and S t o k e s - E i n s t e i n equation 16.

12. Temperature Dependence of V i s c o s i t y . [ 2 8 ]

-

I n c r y s t a l l i n e m a t e r i a l s e l f d i f f u - s i o n by a vacancy mechanism r e s u l t s i n t h e following r e l a t i o n f o r

D

A% AH

D

= D

exp

- -

exp

- f

kT kT

where t h e f i r s t e x p o n e n t i a l g i v e s t h e p r o b a b i l i t y of an exchange of an atom with a neighboring vacancy and t h e second exponential f a c t o r g i v e s t h e p r o b a b i l i t y of a vacancy e x i s t i n g on a given s i t e .

Presumably d i f f u s i o n i n l i q u i d s h a s a s i m i l a r form t o e q u a t i o n l g e x c e p t t h a t t h e p r o b a b i l i t y of a vacancy i s r e p l a c e d by t h e p r o b a b i l i t y of f i n d i n g a ''hole" i n t h e l i q u i d s t r u c t u r e of s u f f i c i e n t s i z e ,

Vh,

t o p r o v i d e a s i t e f o r a neighbor ex- changing jump. Assumlng t h a t f r e e volume,

V f ,

p e r atom i s Poisson d i s t r i b u t e d and t h a t

Vf/Vo =

Aa(T- T

)

where Aa i s t h e d i f f e r e n c e between l i q u i d - l i k e and g l a s s - l i k e thermal expansion, T i s a temperature a t which t h e f r e e volume v a n i s h e s and t h e o v e r a l l molar volume, i s Vo, then t h e s e l f d i f f u s i o n c o e f f i c i e n t

A

B

D

= D

exp

- ( - )

exp

T (T

-

To) 1

where

A =

AHm/k and B

=

Vh/AaVo.

Noting t h a t v i s c o s i t y i s i n v e r s e l y r e l a t e d t o d i f f u s i v i t y g i v e s

n

=

no e x p ( + + + ~ . (21)

Equation 21 i s s i m i l a r t o one proposed by Dienes [29] and Macedo and

L i t o v i t z [30]. When (B/(T- To))

<<

A/T o r when T

>>

To equation 21 reduces t o t h e

form proposed by Andrade [ 3 1 ] , Eyring

[24]

and d e Guzman [32]

C9-378

JOURNAL DE PHYSIQUE

When A/T

<<

B(T- T

)

t h e f a m i l i a r Fulcher-Vogel-Tamann equation r e s u l t s .

n

=

no exp

(

- _{T - To)} B ₍₂₃₎

Equation 23 r e q u i r e s one more a d j u s t a b l e parameter t h a n equation 22. I t g i v e s a f i t f a r s u p e r i o r [28] t o t h a t o b t a i n e d from equation 22 when v i s c o s i t y from 10 - 10' Pas i s considered.

The following q u i t e d i f f e r e n t e x p r e s s i o n was suggested by Adams and Gibbs [33]

based on t h e v a n i s h i n g of c o n f i g u r a t i o n a l e n t r o p y a t T E

=

exp ~ _ l n

T / T ~

(24)

In o x i d e m e l t s e q u a t i o n 24 f i t s v i s c o s i t y d a t a l e s s w e l l t h a n does t h e Fulcher- Vogel-Tamann e q u a t i o n .

Equations

20

and 21 and many o t h e r s l i k e them reviewed by Sturm d e s c r i b e t h e v i s c o s i t y of a g l a s s whose s t r u c t u r e i s i n e q u i l i b r i u m with temperature.

A t

temper-

a t u r e s w i t h i n t h e g l a s s t r a n s i t i o n range however where v i s c o u s flow i s important f o r s t r e s s r e l a x a t i o n i n a n n e a l i n g , tempering, and i n g l a s s t o metal s e a l s

it

i s neces-

,

s a r y t o consider g l a s s whose s t r u c t u r e i s n o t a t e q u i l i b r i u m . The use o f a s i n g l e parameter t h e f i c t i v e temperature, Tf, t h e temperature with which t h e g l a s s s t r u c - t u r e would be i n e q u i l i b r i u m , has proven t o be a u s e f u l approximation. Nayaranaswami

[34] used t h i s concept t o suggest an e x p r e s s i o n of t h e following form t o d e s c r i b e v i s c o u s flow i n t h e g l a s s t r a n s i t i o n r e g i o n

F G

q =

n e x p ( -

+ -)

.

Tf (25)

A t

"equilibrium" ( T = Tf) and t h i s e q u a t i o n becomes e q u i v a l e n t t o E y r i n g ' s e x p r e s s i o n equation 22 which i s notably unsuccessful i n d e s c r i b i n g t h e temperature dependence of v i s c o s i t y of most oxide g l a s s e s over a wide temperature i n t e r v a l . I t seems more a p p r o p r i a t e t o modify equation 21 a s suggested by Scherer and Rekhson [35] by n o t i n g t h a t f r e e volume should be a f u n c t i o n of Tf r a t h e r t h a n T t o o b t a i n

Modifying Adams and Gibbs e x p r e s s i o n , equation 24, i n an e q u i v a l e n t way by l e t t i n g c o n f i g u r a t i o n a l entropy depend on Tf r a t h e r t h a n T g i v e s

1 d n

, p r e d i c t e d by t h e s e e x p r e s s i o n s a r e The e f f e c t i v e a c t i v a t i o n energy,k- -

r e s p e c t i v e l y ;

kd In n / d ~ -

l = A +

B [T'/ ( T ~

- T ~ )

'1 d ~ ~ / d ~ (28) k d In q/dT-l

=

E ( l n ( T ~ / T ~ ) y2 [ l n ( T ~ / T ~ )

+

T/TfdTf /dT] (29) While

~t

i s premature t o compare t h e i r v a l i d i t y , equation 27 is not i n c o n s i s t e n t with t h e o b s e r v a t i o n t h a t i s o s t r u c t u r a l (dT /dT

= 0)

v i s c o s i t y measurements of

f

m e t a l l i c g l a s s e s i n t h e t r a n s f o r m a t i o n show a c t i v a t i o n e n e r g i e s many times t h a t of

l i q u i d m e t a l s . Equation 28 on t h e o t h e r hand r e q u i r e s i s o s t r u c t u r a l v i s c o s i t y have

i s a lower a c t i v a t i o n energy than t h e high temperature v i s c o s i t y .

13. Composition- P r e s s u r e - and S t r a i n Rate Dependence.

-

V i s c o s i t y of l i q u i d s p a r t i - c u l a r l y g l a s s forming l i q u i d s depend on v a r i o u s o t h e r e x t e r n a l parameters l i k e com- p o s i t i o n , p r e s s u r e , and s h e a r s t r e s s

For example, f o r molten s i l i c a t e s B o t t i n g a and Weil [36] found than I n n a t c o n s t a n t T depends l i n e a r l y on oxide c o n c e n t r a t i o n s i f t h e A1203 content i s t r e a t e d a s aluminates. They provided c o e f f i c i e n t s t h a t allow s i l i c a t e v i s c o s i t y from 1200°C t o 1800°C over a wide composition range t o be a c c u r a t e l y e s t i m a t e d .

I t i s conventional wisdom t h a t because i n c r e a s e d p r e s s u r e reduces t h e f r e e v o l - ume of l i q u i d s t h e v i s c o s i t y of l i q u i d s

w i l l

i n c r e a s e with p r e s s u r e . However,

i t

i s w e l l e s t a b l i s h e d from compaction experiments [37] t h a t i r r e v e r s i b l e compaction only begins a f t e r a "threshold" p r e s s u r e i s reached. Also, r e c e n t high temperature meas- urements have shown t h a t v i s c o s i t y i n alumino s i l i c a t e m e l t s [38] and p u r e Ge02melts

[39] d e c r e a s e w i t h p r e s s u r e . That t h e v i s c o s i t y of d i f f e r e n t m e l t s behave d i f f e r - e n t l y with p r e s s u r e i s expected from equation 21 s i n c e

A

should d e c r e a s e and B i n - c r e a s e with p r e s s u r e .

Linear f l u x e q u a t i o n s imply t h a t t h e t r a n s p o r t c o e f f i c i e n t does not depend on t h e f l u x d e n s i t y . S i n c e t h e b a s i s f o r l i n e a r e q u a t i o n s

i s

t h e assumption

o f

small d e p a r t u r e s from e q u i l i b r i u m

it

i s t o be expected t h a t a t some p o i n t t h e l i n e a r i t y

w i l l

break down. In f a c t , a more g e n e r a l form of E y r i n g ' s e x p r e s s i o n p r e d i c t s t h e non l i n e a r , i . e . non Newtonian, behavior of v i s c o s i t y . While f o r most p r a c t i c a l purposes g l a s s forming systems behave i n a Newtonian manner, L i and Uhlman [40]

found t h a t a rubidium s i l i c a t e e x h i b i t e d non Newtonian behavior a t high v i s c o s i t i e s (1013 - 1016pas) when t h e s t r e s s exceeded about ~ o ' M P ~ , i . e . about 1/10 of t h e u s u a l f r a c t u r e s t r e n g t h of undamaged g l a s s . Rekhson e t a l . [41] o b t a i n e d about t h e same r e s u l t from c a l c u l a t i o n s u s i n g a Lennord-Jones p o t e n t i a l .

14. Summary.

-

Transport phenomena i s one of t h e many p o i n t s of connection between engineering s c i e n c e and g l a s s technology. I t i s hoped t h a t t h i s b r i e f review h a s demonstrated a few such j u n c t i o n s and r e v e a l e d a mutual b e n e f i t t o be gained from a d d i t i o n a l coupling.

15. Acknowledgement. - F i n a n c i a l support was provided by t h e U.S. Department of Energy under c o n t r a c t #DEA C02 79 EROY075. Conversations with S. Rekhson and

C .

Moynihan were b e n e f i c i a l .

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