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Lecture 6 – Diffusion
Ch 24
pages 619-625
Summary of lecture 5
 We have introduced the general problem of random walk and
provided the remarkable simple result relating the mean square
displacement and therefore the root mean square displacements
with the number of steps N taken and the length of each step
 r  Nl
2
2
r 
2
1/ 2
N
1/ 2
l
 For the random diffusion of molecules in a gas, the mean square
displacement of each molecule can be expressed in terms of
number of collisions and mean free path:
 r 2  zl 2
Summary of lecture 5
 The probability of having a certain displacement x can then be
expressed in terms of the step length l:
1
W ( x) 
2Nl 2
e x
2
/ 2l 2 N
 The frequency (if the number of hops or steps per unit time is
N’, then the number of hops N=N’t
W
1
2N   t
2
e
 x2 /2l 2 N

1
2N   t
2
e
 x 2 / 2 l 2 N t
Microscopic Diffusion
We have introduced the concept of diffusion from a microscopic
perspective when discussing the motion of molecules in a gas
The diffusion coefficient has been defined by Einstein as a
measure of the distance traveled over time (on average) by a
particle undergoing diffusion
r 2  r 2  6Dt
An equivalent description can be provided in macroscopic terms
when we consider the concentration C(x,t) of a solute in a solvent
system (e.g. a protein in water)
Macroscopic Diffusion
When a solution is at equilibrium the concentration of solute is
uniform throughout. If the solute concentration is not uniform, a
solute concentration exits that must be reduced to zero if the
system is to attain equilibrium
Diffusion is the process whereby concentration gradients in a
solution are reduced spontaneously until a uniform homogeneous
solution is obtained
Diffusion occurs whenever there is a concentration difference
As a consequence of diffusion, an equilibrium state of uniform
concentration (or heat, if it heat diffusion) is reached
Macroscopic Diffusion
Let us think of molecules in solution as they move across a certain
surface as they ‘diffuse’. Flux is the amount of matter (heat,
charge, etc.) that crosses an area per unit time in a direction
perpendicular to the surface
The flux of the solute J2 is related to its concentration C2 (how
much solute there is) and its transport velocity v2 (how fast it
moves) by:
J2  v2 C2
units are mol/s cm2
Macroscopic Diffusion
If there is no difference in concentration (concentration gradient),
there will be no flux; if the concentration is higher on the right,
solute will go from right to left to equalize the concentration and
reduce the gradient
There is a net transport of material in the direction opposite the
concentration gradient; the steeper the concentration gradient, the
larger the flux
These considerations lead to Fick’s First Law of diffusion:
 dc 
J 2 ( x )   D 2 
 dx 
D is a phenomenological property called the diffusion coefficient;
the units of D are cm2/s
Macroscopic Diffusion
Fick’s First Law of diffusion:
 dc 
J 2 ( x )   D 2 
 dx 
This equation expresses the fact that, as diffusion occurs, the
gradient of concentration decreases, reducing flux until, at
equilibrium, the next flux ceases and diffusion stops
Heat Conduction
We shall now generalize the concept of diffusion
If a temperature gradient exists in a material, heat will be
conducted through the material from the region of higher
temperature to a region of lower temperature; this process is
called heat conduction
The thermal conductivity is the rate at which heat is transferred
through a material per unit temperature gradient
We can define an analogous of the diffusion coefficient (thermal
conductivity) as the constant of proportionality that relates the
heat flux h (Joules per m2 per second) to the thermal gradient
dT/dx (degrees K per meter)
Heat Conduction
The thermal conductivity is the constant of proportionality that
relates the heat flux h (Joules per m2 per second) to the thermal
gradient dT/dx (degrees K per meter)
We have a relationship analogous to Fick’s First Law of Diffusion
which relates the solute mass flux J2 (kg per m2 per second) to the
concentration gradient dC(x)/dx (kg/ m3 per m):
J 2   D2
dC2  x 
dx
 h  
dT  x 
dx
Heat Conduction
From Fick’s First Law, D2 must have units of m2/s
From the heat flux equation it is clear that  must have units of
J  K 1  m 1  s 1
J 2   D2
dC2  x 
dx
 h  
dT  x 
dx
Because of their units, fluxes like h and J2 are sometimes called
current densities, because they measure the amount of a quantity
that passes through a unit area per unit time
Chemical Potential
Generally, we think of a force acting on an object as inducing
movement; since we observe flow, we can think that there must be
a ‘force’ that induces the solute to ‘flow’
A force occurs when a potential difference exits
In the case of an electric charge, the potential difference is
electrostatic (measured as a voltage difference)
In the case of heat flow, a thermal gradient induces heat transfer
In the case of solute transport the potential difference results from
a concentration gradient
Chemical Potential
In analogy to the classical concept of force as of a potential
gradient, we can introduce a chemical potential to express
differences in free energy that induce flux
If the concentration of solute C2 is a function of x, the chemical
potential has the general form:
G 2 ( x )  G 20  RT ln C 2 ( x )
A difference in chemical potential exercises a force on the solute
molecules; the force that induces solute flow is related to the
chemical potential by the equation
Fext  
dG2 ( x )
d ln C2 ( x )
RT dC2 ( x )
  RT

dx
dx
C2 ( x ) dx
Friction
Consider a molecule in solution. If an external force F is applied to
the particle, the particle obviously accelerates according to F=ma
The particle will not accelerate for long. After a brief period, the
velocity becomes constant as a result of resistance from the
surrounding fluid. This velocity is called the steady state velocity
vT and fulfills the condition:
fvT  F
Friction
fvT  F
fv is the frictional force exerted by the surrounding fluid on the
particle and f is the frictional coefficient of the particle
The fictional coefficient depends on the size and shape of the
particle but not on its mass. For a spherical particle with radius R
f  6R
Where  is the viscosity of the fluid (Stoke’s Law)
Friction
The force acting on each single solute particle is:
Fext
k T dC2 ( x)
RT dC2 ( x)
 v2 f  
 B
N0
N 0 C2 ( x) dx
C2 ( x) dx
J 2  v 2 C2 ( x )  
k B T dC2 ( x )
f
dx
The diffusion coefficient is related to the temperature and to the
frictional coefficient f that depends on the solvent property
(viscosity) and on the molecular property of the solute (size, shape
and hydration)
The Diffusion Equation
If the solute flux J2(x) into a volume V=Ax is not equal to the flux
out of the volume J2(x+x), then the solute concentration in the
volume must change by the same amount (matter is conserved).
This means the change in solute concentration per unit time
equals the flux gradient (see diagram below)
C2
J 2 ( x)  J 2 ( x  x)

t
x
The Diffusion Equation
The relationship between the concentration change and the flux
gradient is combined with Fick’s First Law to Produce the
Diffusion Equation, which is also called Fick’s Second Law and
describes how the concentration gradient changes with time:
C2 J 2 ( x)  J 2 ( x  x)
dC2
dJ 2 ( x ) k B T d 2 C2




t
x
dt
dx
f dx 2
To be formally correct, it should be
expressed in terms of partial derivatives
since the concentration depends both on
time and space:
C 2 k b T  2 C 2
 2 C2

 D2
2
t
f x
x 2
The Diffusion Equation
C 2 k b T  2 C 2
 2 C2

 D2
2
t
f x
x 2
D is again the diffusion coefficient
The Diffusion Equation has the following general solution:
C 2 ( x, t ) 
C0
4D2 t
e
 x 2 / 4 D2 t
The Diffusion Equation
The Diffusion Equation has the following general solution:
C 2 ( x, t ) 
C0
4D2 t
e
 x 2 / 4 D2 t
0.3
0.25
0.2
0.15
0.1
0.05
0
-20
-10
Dt=1
Dt=4
Dt=16
0
x
10
20
The Diffusion Equation
C 2 ( x, t ) 
C0
4D2 t
e x
2
/ 4 D2 t
The function C2(x,t) has the form of a bell-shaped (i.e.
Gaussian) curve; it can be used to calculate the average
displacement of a solute particle:

x
 xC( x, t )dx 




xC0
4D2 t
e x
2
/ 4 D2 t
0
The mean squared displacement and root mean square
displacement are not 0:

x 
2
x

2
C ( x, t )dx  2 D2 t
x rms  x 2  2 D2 t
The Diffusion Equation

x 
2
2
x
 C ( x, t )dx  2D2 t

x rms  x 2  2 D2 t
Notice the similarity with the random walk problem
r 2  r 2  6Dt
The diffusion coefficient is half of the mean square displacement
per unit time. Because the solution to the diffusion equation has
gaussian shape, the diffusion coefficient is related to the width of
the gaussian at half height
The Diffusion Equation
The diffusion coefficient is half of the mean square displacement
per unit time. Because the solution to the diffusion equation has
gaussian shape, the diffusion coefficient is related to the width of
the gaussian at half height
0.3
0.25
0.2
0.15
0.1
0.05
0
-20
-10
Dt=1
Dt=4
Dt=16
0
x
10
20
The Diffusion Equation
The concentration c is obviously maximum at t=0; as time
increases, it spreads, while the concentration at half maximum is
 2 Dt ln 2
The width at half maximum is:
4 Dt ln 2
0.3
0.25
0.2
0.15
0.1
0.05
0
-20
-10
Dt=1
Dt=4
Dt=16
0
x
10
20
The Diffusion Equation
4 Dt ln 2
One can measure the diffusion coefficient by monitoring how a
concentration gradient disappears (non equilibrium experiment)
Because the diffusion coefficient is also related to the random
motion of molecules, it can be measured under equilibrium
conditions by measuring the random motion of molecules
directly
One such method is laser light scattering
0.3
0.25
0.2
0.15
0.1
0.05
0
-20
-10
Dt=1
Dt=4
Dt=16
0
x
10
20
The Diffusion Equation
Laser light scattering: The method monitors the scattering of
highly monochromatic light as it travels through the solution
Because each particle moves at a slightly different speed,
scattering will induce a distribution of Doppler shifts that will be
reflected in a broadening of the monochromatic light beam
The width at half height of the signal is proportional to D, when
the molecules are much smaller than the wavelength of the laser
beam (you need a laser because it is highly monochromatic)
0.3
0.25
0.2
0.15
0.1
0.05
0
-20
-10
Dt=1
Dt=4
Dt=16
0
x
10
20