Lecture 18

Lecture 18
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
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 -
• At lower temperature,
where permanent
vacancies dominate,
we might see different
behavior because:
 = me-EB /RT + ne-(EH +EB )/RT
Determining Diffusion
• 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 ¶V 2v
¶V ¶n r
Surface Free Energy
• We now define the Surface Free Energy as:
æ ¶G ö
sf = ç ÷
è ¶A ø T ,P,n
• The surface free energy represents those energetic
effects that arise because of the difference in
atomic environment on the surface of a phase.
• Surface free energy is closely related to surface
• The total surface free energy of a phase is
minimized by minimizing the phase’s surface area.
o Thus a water-drop in the absence of other forces will tend to form a
sphere, the shape that minimizes surface area.
Incorporating Surface Free
• When surface effects must be considered, we can
revise equation 3.14 to read:
dG = VdP - SdT + å µ dn + ås dA
• where the second sum is taken over all interfaces
and surfaces of the phase.
o In general, the surface free energy depends on the nature of the two
phases in contact. It will be different for water in contact with a mineral
than for water in contact with air.
o For an isotropic phase immersed in a homogeneous medium, we need
concern ourselves with only one interface free energy.
o For crystals, different faces can have different interfacial free energies,
even if in contact with the same phase (but we won’t concern ourselves
with this complexity).
o In a rock, for example, a given crystal might be in contact with several
different minerals - each would have a different interfacial free energy.
The Kelvin Effect
When the size of phases involved is sufficiently small, surface free energy
can have the effect of displacing equilibrium. For an equilibrium system
at constant temperature and pressure, eqn. 5.109 becomes:
0 = ån i µio + RT ån i lnai + ås k dAk
The first term on the right is ∆G˚, which is equal to –RT ln K. This is the
“normal” equilibrium constant, so we’ll call it K˚. We’ll call the summation
in the second term Ks, the equilibrium influenced by surface free energy.
Making these substitutions and rearranging, we have:
s k dAk
ln K s = ln K˚ - k
Thus we predict that equilibrium can be shifted due to surface free
energy, and the shift will depend on the surface or interfacial area. This is
known as the Kelvin effect.
There are a number of examples of this effect. For example, fine, and therefore high
surface area, particles are more soluble than coarser particles of the same composition.
Water has a surface free energy of about 70 mJ/m2. So, for example, humidity in clouds
and fogs can reach 110% when droplet size is small.
Surface Free Energy &
• One usual effect of
metamorphism is an
increase in grain size.
This occurs even in
monomineralic rocks
like limestone and
• The free energy of the
system is reduced by
reducing grain-to-grain
interfaces, which is
lower in coarser
grained rocks.
Liquids can become significantly supersaturated but
crystallization will often begin as soon as seed crystals are added.
This suggests that nucleation is an important barrier to crystallization.
This barrier arises because the formation of a crystal requires a local increase in free energy due to
the surface free energy at the solid–liquid interface.
Let’s explore this a bit further. For a crystal growing in a liquid, the
complete free energy change is:
dG = dσ+dGxtl
For a spherical crystal of phase ϕ growing from a liquid solution of
component ϕ. The free energy change over some finite growth
interval is:
4 3 ∆ Gxt
∆ Gtot = 4p r s + p r
r is radius. We divide by the molar volume to convert J/mol to
J/m3. (In fact, per volume units, rather than per mass or per mole,
turn out to generally be more convenient in kinetics).
The first term on the right is always positive, so exactly at
saturation, ∆Gtot is positive and there will be no crystallization.
Nucleation & Growth
∆ Gxt
∆ Gtot = 4p r 2s + p r 3
• For small r, first term
increases more rapidly.
• ‘Turnaround’ occurs at
∂∆G/∂r = 0
¶∆ Gtot
∆ Gxt
= 8p rs + 4p r 2
• Setting ∂∆G/∂r = 0, we
find the critical radius:
∆ Gxtl / V
Total free energy as a function of r for various
amounts of undercooling. We approximate
the ∆G term as ∆G ≈ -∆T∆Sxtl, where ∆T is the
difference between actual temperature and
the saturation temperature.
Surface Free Energy & Viscosity
• The surface free energy term correlates with
viscosity. Thus nucleation should require less
supersaturation for aqueous solutions than silicate
o Among silicate melts, nucleation should occur more readily in basaltic
ones, which have low viscosities, than in rhyolitic ones, which have high
viscosities. This is what one observes.
• Also, we might expect rapid cooling to lead to
greater supersaturation than slow cooling. This is
because there is an element of chance involved in
formation of a crystal nucleus (the chance of
bringing enough of the necessary components
together in the liquid so that r exceeds rcrit).
Nucleation Rate
• The first step in is the formation of small clusters of atoms
of the right composition. These so-called heterophase
fluctuations arise because of statistical fluctuations in the
distribution of atoms in the liquid. These fluctuations
cause local variations in free energy, and their
distribution can be described by the Boltzmann
distribution law. The number of clusters of critical size is:
N crit = NV e-∆ Gcrit /kT
• where Ncrit is the number of clusters of critical size per unit
volume containing i atoms, Nv is the number of atoms
per unit volume of the cluster, and ∆Gcrit is the total free
energy (∆Gtot) of clusters with critical radius (i.e., previous
equation with r = rcrit.
• For a spherical cluster, this is:
∆ Gcrit
16p s 3V
3 ∆ Gxt2
Nucleation Rate
• If EA is the energy necessary to attach an additional
atom to the cluster, then the probability of this is:
P = e- EA /kT
• According to transition state theory, the frequency of
attempts, ν, to overcome this energy is simply the
fundamental frequency, ν = kT/h. The attachment
frequency is then the number of atoms adjacent to the
cluster, N*, times the number of attempts, times the
probability of success:
N *n P = N *
kT - EA /kT
• Nucleation rate should be this times number of clusters of
critical radius:
N *n P = N *
kT - EA /kT -∆ Gcrit /kT
Diopside Nucleation
• Combining preexponential terms into
a frequency factor, A,
and using ∆G ≈ ∆S∆T
where ∆T is the offset
from the crystallization T
(the temperature
overstep), for a
spherical nucleus we
• or
I = Ae
-16 ps 3 V /(3∆ S∆ T )2 kT
I µe
-1/∆ T 2
The nucleation rate passes through a maximum.
This reflects the 1/T dependence of both
exponential terms; the formation and growth of
heterophase fluctuations falls with temperature.
Heterogeneous Nucleation
• Heterogeneous
nucleation refers to the
nucleation of a phase on
a pre-existing one. This
occurs when the surface
free energy between the
nucleating phase and
the pre-existing surface is
lower than between the
nucleating phase and
the phase from which it is
o Examples: dew, pyx-plag
Heterogeneous Nucleation
Consider, for example, a dew drop on a
The balance of surface forces at the threephase contact is:
s a s = s b s + s ab cosq
If the interfacial energy between the
nucleating phase, β, (the dew) and the
surface (σβs) (the leaf) is smaller than that
between phase α (air) and the surface
(σαs), then the angle of intersection, θ, will
be small so as to maximize the interfacial
surface area between β and s for a given
volume of β. In the limit where σβs ≪ σαs
then θ will approach 0 and β will form a film
coating the surface.
As σβs approaches σαs the nucleating
phase will form more spherical droplets. If
σβs ≥ σαs then θ will be 90° or greater, and
heterogeneous nucleation will not occur.
cosq =
s a s - s bs
s ab
Heterogeneous Nucleation
In metamorphic reactions,
nucleation will necessarily always
be heterogeneous. Provided the
necessary components of the
nucleating phase are available
and delivered rapidly enough by
fluid transport and diffusion,
interfacial energy will dictate
where new phases will nucleate,
nucleation being favored on
phases where the interfacial
energy is lowest.
(Where transport of components
limit growth, however, this may
not be the case, as phases will
nucleate where the components
necessary for growth are
Diffusion and heat-flow limited
Crystals can grow only as rapidly as the
necessary chemical components are
delivered to their surfaces and heat
removed (or added). Where diffusion is not
rapid enough to supply these components,
diffusion will limit growth.
Slow diffusion can change the apparent
distribution coefficient, because the crystal
“sees” the concentrations in the adjacent
boundary layer rather than the average
concentrations in the liquid. Thus the crystal
may become less depleted in elements
excluded from the crystal, and less
enriched in elements preferentially
incorporated in it, than equilibrium
thermodynamics would predict.
When crystals grow from a liquid there will
be a local increase in temperature due to
release of latent heat of fusion, ∆Hm, which
will retard crystal growth. In most cases, this
is at best a minor effect. The effect is
probably more important in prograde
metamorphic reactions (e.g., dehydration
reactions), which are usually endothermic
and hence require a continuous supply of
energy to maintain crystal growth.
• Diffusion limits the
ability of the interior of
a crystal to maintain
equilibrium with the
magma from which it is
• This leads to zoning in
crystals, apparent
under the microscope.
• Zoning can record
magma history.
Zoned pyx and plag in lava.

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