Diffusion
(continued)
Lecture 17
Fick’s First Law
Last time we derived Fick’s First Law
¶c
J = -D
¶x
Continuity Equation
Imagine a volume and a flux of a
species into that volume, Jx, at X
and a flux out, Jx+dx, at x+dx.
• Suppose nx atom/sec at x and nx+dx
atoms/sec pass through the plane
at x+dx.
• The fluxes at the two planes are
Jx = nx/dx2-sec
and
Jx+dx = nx+dx/dx2-sec.
• Conservation of mass dictates that
the increase in the number of
atoms, dn, within the volume is just
what goes in minus what comes
out.
• Over an increment of time dt this
will be dn = (Jx - Jx+dx)dt. The
concentration change is
•
dc =
(J x - J x+dx )dt
dx
or:
dc -(J x+dx - J x )
=
dt
dx
At fixed t and x
æ ¶c ö
æ ¶J ö
=
çè ÷ø
çè ÷ø
¶t x
¶x t
Deriving Fick’s Second
Law
æ ¶c ö
æ ¶J ö
=
çè ÷ø
çè ÷ø
¶t x
¶x t
• Continuity equation
relates any change in
flux along the gradient
to change in
concentration with
time.
• Since: æ ¶c ö æ ¶(-D ¶c / ¶x) ö
çè ÷ø = - çè
¶t x
¶x
• we have:
æ ¶2 c ö
æ ¶c ö
çè ÷ø = D ç 2 ÷
¶t x
è ¶x ø t
÷ø
t
Fick’s Second Law
æ ¶2 c ö
æ ¶c ö
çè ÷ø = D ç 2 ÷
¶t x
è ¶x ø t
• The second law says that
the change in
concentration with time at
some point on a conc.
gradient depends on the
second derivative of the
concentration gradient.
• The straight line (∂2c/∂x2) is
the steady state case. It is
also the situation towards
which a system will evolve
over time.
Solutions to the Second
Law
• There is no ‘solution’ to
the second law, rather
there are a variety of
solutions that depend on
the boundary conditions
of a particular situation.
• As a first example,
consider a thin layer (e.g.,
Ir at K-T boundary) of
material at x = 0 at t = 0.
These, together with mass
conservation, are our
boundary conditions.
• Mass conservation
is
¥
M=
ò c dx
-¥
Thin Flim Solution
• Boundary conditions
were x = 0 at t = 0 and
¥
M=
ò c dx
-¥
• Solution, from Crank
(1975), fitting these
conditions is:
c(x,t) =
M
- x 2 /Dt
e
2(p Dt)1/2
• If diffusion can occur
only in one direction:
c(x,t) =
M
- x 2 /Dt
e
(p Dt)1/2
Adjacent Layers
• Now suppose with have
one layer with
concentration co
abutting a second with
c=0.
• Crank’s solution is to
superimpose the solutions
for an infinite number of
thin flims, dξ, and to sum
their contributions to the
total concentration at Xp.
• Solution is:
¥
C0
- x 2 /4 Dt
c(x,t) =
e
dx
2(p Dt)1/2 òx
Adjacent Layers
• The solution to:
¥
• is
C0
- x 2 /4 Dt
c(x,t) =
e
dx
1/2 ò
2(p Dt) x
c(x,t) =
C0 ì
æ x öü
1erf
í
çè
÷ý
2 î
2 Dt ø þ
• where erf is the error
function and has the
approximate form:
erf (x) = 1- exp(-4x 2 / p )
• also available as the ERF
function in Excel.
Other examples
• Book has several other
examples such as
diffusion of Ni in a
spherical, zoned olivine
crystal with an initially
homogeneous conc. of
2000 ppm and a conc.
fixed at the rim by
equilibrium with a
magma at 500 ppm.
Multicomponents
Systems
• So far, we have considered only what is called
tracer or self diffusion - diffusion of a species who
concentration is so small it has not affect on the
system.
• By definition, diffusion is a process in which there is
no net transport across the boundary of interest
(otherwise it is advection). Mass & charge balance
require diffusion in one direction to be balanced by
diffusion in another.
• More generally, we can consider
o Chemical diffusion
o Multicomponent diffusion
o Multicomponent chemical diffusion
Chemical Diffusion
• Chemical diffusion refers to non-ideal situations
where for one reason or another (e.g., medium is
inhomogeneous, or diffusing species affects
behavior of medium) we replace concentration
with the chemical potential gradient.
¶µ
J = -L
¶x
• where L is the chemical or phenomenological
coefficient
Multicomponent
Diffusion
• This is where the concentration of an species is great
enough that its diffusion depends on the diffusion of
another species, e.g., Fe and Mg in ferromagnesian
silicates (these elements tend to occupy the same site,
so Mg can move in only if Fe moves out, etc).
n
¶c
Ji = - å Di,k i
¶x
k=1
• Di,k is called the interdiffusion coefficient and computed
as:
ni Di + ni Di
Di,k =
o where n’s are the mole fractions.
ni + nk
• The total solution is J = -DC
(minus sign missing in book)
• where J and C are (vertical) vectors of species and D is
a matrix.
Multicomponent
Chemical
• Multicomponent
chemical diffusion is a
combination of the latter
two, where we use
interdiffusion coefficients
and chemical potential.
• Diffusion matrix is:
J = -LC
• Watson’s experiments
dissolving quartz in basalt
show an example where
chemical diffusion must
be used - note Na and K
diffusing up a
concentration gradient!
Mechanisms of Diffusion
in Solids
• Exchange: low probability
• Interstitial: pretty much
limited to He, etc.
• Interstitiality: also low
probability
• Vacancy: most likely
• Diffusion will then depend
on both the probability of
a jump and the
probability of a vacancy.
o Vacancies will increase with T:
N vac = N perm + ke- EH /RT
o where k is a rate constant and EH
and activation energy for
vacancy creation.
Diffusion Rates
• Probability of atom making a jump to vacancy is
P =Àe- EB /RT
• where  is number of attempts and EB is the activation or
barrier energy.
• Combining, total diffusion rate will be product of probability.
of a vacancy times probability of a jump:
 = me-EB /RT + ne-(EH +EB )/RT
o
where m and n are constants
• At higher T, thermal vacancies dominate and
 @ ne-(EH +EB )/RT
• Bottom line: like other things in kinetics, we expect diffusion
rates to increase exponentially with temperature.
Diffusion Rates
• In general, the
temperature of the
diffusion coefficient is
written as:
D = Doe- EB /RT
• In log form:
ln D = ln Do -
EA
RT
• At lower temperature,
where permanent
vacancies dominate,
we might see different
behavior because:
 = me-EB /RT + ne-(EH +EB )/RT
Determining Diffusion
Coefficients
• In practice, one would
do a experiments at a
series of temperatures
with a tracer, determine
the profile, solve Fick’s
second law for each T.
• Plotting ln D vs 1/T yields
Do and EB.
• Diffusion coefficient is
different for each species
in each material.
• Larger, more highly
charged ions diffuse
more slowly.
Reactions at Surfaces
Homogeneous reactions are those occurring within a single phase (e.g.,
an aqueous solution). Heterogeneous reactions are those occurring
between phases, e.g., two solids or a solid and a liquid. Heterogeneous
reactions necessarily occur across an interface.
Interfaces, Surfaces, and
partial molar area
• By definition, an interface is boundary between two
condensed phases (solids and liquids).
• A surface is the boundary between a condensed phase
and a gas (or vacuum).
o In practice, surface is often used in place of interface.
• We previously defined partial molar parameters as the
change in the parameter for an infinitesimal addition of
a component, e.g., vi = (∂V/∂n)T,P,nj. We define the partial
molar area of phase ϕ as:
æ ¶A ö
aif = ç
è ¶ni ÷ø T ,P,n j¹i
• where n is moles of substance.
• Unlike other molar quantities, partial molar area is not an
intrinsic property of the phase, but depends on shape,
size, roughness, etc. For a perfect sphere:
a=
¶A ¶V 2v
=
¶V ¶n r