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(a)

High initial

concentration of
i

Low initial

concentration of
i

(b)

T
/ °C

t
0

1400

1350

1300

t
2

t
1

c
i

-22

t
3

Basalt

t
∞

-24

Andesite

-26

Flux of ions

-28

0.60

0.62

0.64

x

10
3

/ K
-1

Section across sample

T

Figure 3.6
(a) The diffusion of ions in response to a non-uniform concentration distribution. (b) Arrhenius plot of diffusion

coefficients for the diffusion of cobalt ions in silicate melts. The activation energies measured from the slopes of the two graphs are:

basalt: 220 kJ mol
−1

andesite: 280 kJ mol
−1

The units of
D
are m
2
s
−1
. Temperature is expressed in kelvins (K). (Source: Adapted from Lowry
et al
. (1982).

Reproduced with permission from Springer Science and Business Media.)

randomly even through a homogeneous substance,

but only where a concentration gradient exists is a
net

flow of chemical components observed.

Imagine a crystal with an abrupt internal discont-

inuity in the concentration of a component
i
, as shown

in Figure 3.6a. Suppose the crystal is maintained at a

constant temperature
T,
high enough for solid-state

diffusion to occur at a significant rate. If we were to

measure the distribution of component
i
on several

successive occasions
t
1
,
t
2
,
t
3
, we would see the devel-

opment of a progressively smoother concentration

profile, leading eventually to a uniform distribution of

i
(at
t
∞
in Figure 3.6a). These changes point to a net flux

of component
i
from left to right through the plane of

the original discontinuity. The magnitude of the flux in

moles per second will depend on the surface area of

this interface. One therefore expresses the flux
f
i
as the

amount of component
i
(in
moles
) that migrates

through a unit area of the plane per second, so the

units are 'moles per square metre per second'

(mol m
−2
s
−1
). Common sense suggests that the flux will

depend upon the steepness of the concentration grad-

ient
d

d

uniform and the gradient zero, whereas a high flux

will occur if the gradient is steep. The 19th century

German physicist Adolf Fick backed up this hunch

theoretically in 1855, showing that:

f D
c

x

d

d

i

=−

(3.11)

i

i

This equation is known as
Fick's First Law of Diffusion
.

The negative sign indicates that the direction of the net

flux
f
i
is
down
the concentration gradient (i.e. towards

the right in Figure 3.6a). The constant
D
i
is called the

diffusion coefficient
for the species
i
in the crystal con-

cerned (at a given temperature). The units of concen-

tration
c
i
are 'moles per cubic metre' (mol m
−3
), so the

concentration gradient d
c
i
/d
x
will have units of moles

per cubic metre
per metre
in the
x
direction ([mol m
−3
]

m
−1
= mol m
−4
) from which it is simple to show that the

units of
D
i
must be m
2
s
−1
(see Exercise 3.5).

Many experimental determinations of diffusion

coefficients have been made for various elements in a

range of silicate materials at various temperatures.

Figure 3.6b shows how the measured diffusion

c

x

i
; so no net flux will occur if the concentration is

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