Lecture 4 Diffusion coefficient

1
????? ??Lecture 4Diffusion coefficient

• ??? ????

2
Diffusion coefficient

• Reasonable values of diffusion coefficient
• in gas 10-1 cm2/sec
• in liquid 10-5 cm2/sec
• in solids 10-10 cm2/sec (strong function of
  temperature)
• in polymer/glasses 10-8 cm2/sec (strong
  function of solute concentration)

3
Diffusion coefficient in gases
4
Diffusion coefficient in gases

• One atmosphere and near room temperature, values
  between 10-1 100 cm2/sec (Reid, Sherwood, and
  Prausnitz, 1977)
• approximation
• inversely proportional to pressure
• 1.5 to 1.8 power of the temperature
• vary with molecular weight
• When , the diffusion process has
  proceed significantly (i.e., the diffusion has
  penetrated a distance z in time t)

5
Chapman-Enskog theory

• Theoretical estimation of gaseous diffusion

6
Theory? Kinetic theory - Molecular motion in
dilute gases

• Molecular interactions involve collisions between
  only two molecules at a time (cf lattice
  interaction in solids)
• Chunningham and Williams (1980)
• a gas of rigid spheres of very small molecular
  dimensions
• the diffusion flux

Concentration gradient
Mean free path of the molecules
Average molecular velocity
Diameter of the spheres
Molecular mass
7
Empirical relations
(Fuller, Schettler, and Giddings, 1966)
The above two methods allow prediction of
diffusion coefficient in dilute gases to within
the average of eight percent. Not very accurate
in high pressure system!
8
Diffusion coefficients in liquids
9
Diffusion coefficients in liquids

• Most values are close to 10-5 cm2/sec, including
  common organic solvents, mercury, and molten
  iron, etc.... (Cussler, 1976 Reid et al. 1977)
• High molecular-weight solutes (like albumin and
  polystyrene) can be must slower 10-7 cm2/sec
• The sloth characteristic liquid diffusion means
  that diffusion often limits the overall rate of
  process occurring in the liquid
• chemistry rate of acid - bas reaction
• physiology rate of digestion
• metallurgy rate of surface corrosion
• industry rate of liquid-liquid extractions

10
Assumption a single rigid solute sphere moving
slowly through a continuum of solvent (cf
molecular motion as in the kinetic theories used
for gases). The net velocity of this sphere is
proportional to the force acting on it
Friction coefficient
Stokes law (Stokes, 1850)
Thermodynamic virtual force The negative of the
chemical potential gradient (Einstein, 1905)
11
const.
Stoke - Einstein equation
12
Stoke - Einstein equation

• Most common basis for estimating diffusion
  coefficients in liquids (accurate 20, Reid et
  al., 1977)
• Derived by assuming a rigid solute sphere
  diffusion in a continuum of solvent (ratio of the
  size of solute to that of solvent gt 5)

13
Diffusion coefficient is inversely proportional
to the viscosity of solvent

• Limitations
• When the solute size is less than 5 times that of
  solvent, the Stoke-Einstein equation breaks!
  (Chen, Davis, and Evan, 1981)
• High-viscosity solvent (Hiss and
  Cussler, 1973)
• Extremely viscosity solvent

14

• For small solute, the factor is often replaced by
  a factor of 4? or of 2.
• Used to estimate the radius of macromolecules
  such as protein in dilute aqueous solution.
• The radius of the solute-solvent complex, not the
  solute itself if the solute is hydrated or
  solvated in some way.
• If the solute is not spherical, the radius R0
  will represent some average over this shape.

Empirical relations for liquid diffusion
coefficients
Several correlations have been developed (Table
5.2-3, page 117). They seem all have very similar
form as the Stoke - Einstein equation.
15
Estimate the diffusion at 25ºC for oxygen
dissolved in water using the Stoke-Einstein model.
Estimate the radius of the oxygen molecule? We
assume that his is half the collision diameter in
the gas
About 30 lower than the experimental measurement.
16
Diffusion in concentrated solutions

• Stoke - Einstein equation (for dilute
  concentration)
• We found that D f (solute concentration)
• Derive the Stoke - Einstein equation? Add
  hydrodynamic interaction among different spheres

(Batchelor, 1972)
The volume fraction of the solute
Not very good for small solutes
17
Empirical relations
(Table 5.2-3 page 117)
Activity coefficient
Arithmetic mean (Darken, 1948 Hartley and Crank,
1949)
Geometric mean (Vigness, 1966 Kosanovich and
Cullinan, 1976) works better!
18
Diffusion in an acetone-water mixture
Estimate the diffusion coefficient in a 50-mole
mixture of acetone (1) and water (2). This
solution is highly non-ideal, so that
. In pure acetone, the diffusion
coefficient is 1.26 x 10-5 cm2/sec in pure
water, it is 4.68 x 10-5 cm2/sec.
Geometric mean (Vigness, 1966 Kosanovich and
Cullinan, 1976)
Very close to the experimental measurement
19
Diffusion coefficients in solids
20
Diffusion coefficients in solids

• Most values are very small. The range is very
  wide 1010 (Barrer, 1941 Cussler, 1976)
• very sensitive to the temperature and the
  dependence is nonlinear
• A very wide range of materials metals, ionic and
  molecular solids, and non-crystalline materials.
• The penetration distance of hydrogen in iron
• after 1 second, hydrogen penetrates about 1
  micron
• after 1 minutes, hydrogen penetrates about 6
  micron
• after 1 hour, hydrogen penetrates about 50 micron
• Hydrogen diffuses much more rapidly than almost
  any other solute.

21
Diffusion mechanisms in solids

• Isotropic diffusion through the interstitial
  spaces in the crystal - lattice theory
• diffusion depends on vacancies between the
  missing atoms or ions in the crystal - vacancy
  diffusion
• Anisotropic crystal lattice leads to anisotropic
  diffusion
• Noncrystal diffusion
• Compare the driving forces
• Liquid/Gas concentration gradient/pressure
  driven flows
• Solids concentration gradient/stress that
  locally increases atomic energy

22
Any theory? not very accurate (although theory
for face-centered-cubic metals is available)
(Franklin, 1975 Stark, 1976)
The jump frequency (estimated by reaction-rate
theories for the concentration of activated
complexes, atoms midway between adjacent sites)
The fraction of sites vacant in the crystal
(estimated from the Gibbs free energy of mixing)
The spacing between atoms (estimated from
crystallographic data)
23
Lattice Theory We consider a face-centered-cubic
crystal in which diffusion occurs by means of the
interstitial mechanism (Stark, 1976). The net
diffusion flux is the flux of atoms from z to (z
?z) minus the flux from (z ?z) to z
The rate of jumps
The average number of vacant sites
The factor of 4 reflects the face that the FCC
structure has 4 sites into which jumps can occur
24
Diffusion in polymers
25
Diffusion in polymers

• Its value lies between the coefficients of
  liquids and those of solids
• Diffusion coefficient is a strong function of
  concentration.
• Dilute concentration
• a polymer molecule is easily imagined as a solute
  sphere moving through a continuum of solvent
• Highly concentrated solution
• small solvent molecules like benzene can be
  imagined to squeeze through a polymer matrix
• Mixture of two polymers

26
Polymer solutes in dilute solution

• Imagined as a necklace consisting of spherical
  beads connected by string that does not have any
  resistance to flow. The necklace is floating in a
  neutrally buoyant solvent continuum (Vrentas and
  Duda, 1980)

Polymer in poor solvent
Polymer in good solvent
(Ferry, 1980)
27
Between the two extremes, the segment of the
polymer necklace is randomly distributed. (i.e.,
the ideal polymer solution). A solvent showing
these characteristics is called a ? solvent.
Stoke-Einstein equation may be used
Equivalent radius of polymer 0.676 (R2)1/2
Root-mean-square radius of gyration
In good solvents, the diffusion coefficient can
increase sharply with polymer concentration
(i.e., the viscosity). This is apparently the
result of a highly nonideal solution.
28
Highly concentrated solution

• Small dilute solute diffuses in a concentrated
  polymer solvent.
• Considerable practical value
• in devolatilization (i.e., the removal of solvent
  and unreact monomer from commercial polymers)
• in drying many solvent based coatings

Sometimes, the dissolution of high polymers by a
good solvent has non-Fickian diffusion or type
II transport the speed with which the solvent
penetrates into a thick polymer slab may not be
proportional to the square root of time. This is
because the overall dissolution is controlled by
the relaxation kinetics (i.e., the polymer
molecules relax from hindered configuration into
a more randomly coiled shape), not by Fick law.
29
For binary diffusion coefficient
The activity coefficient of the small solute
Volume fraction, the appropriate concentration
variable to describe concentrations in a polymer
solution.
The correct coefficient (Zielinski and Duda,
1992) 1. function of solutes activation
energy 2. Effected by any space or free volume
between the polymer chains
30
(No Transcript)
31
Polymer solute in Polymer solvent

• Practical importance
• adhesion, material failure, polymer fabrication
• No accurate model available
• the simplest model by Rouse, who represents the
  polymer chain as a linear series of beads
  connected by springs , a linear harmonic
  oscillator

Friction coefficient characteristic of the
interaction of a bead with its surroundings
Degree of polymerization
OK for low molecular weight
32
Diffusion coefficient measurement

• It is reputed to be very difficult.
• Some methods are listed in Table 5.5-1, p.131
• Three methods give accuracies sufficient for most
  practical purposed
• Diaphragm cell
• Infinite couple
• Taylor dispersion

33
Diaphragm cell

• Can obtain 99.8 accuracy
• Diffusion in gases or liquids or across membrane
• Two well-stirred (m.r. _at_ 60 rpm) compartments are
  separated by either a glass frit or by a porous
  membrane.

Effective thickness of the diaphragm
Area available for diffusion
34
Issues for diaphragm cell

• For accurate work, the diaphragm should be a
  glass frit and the experiments may take several
  days
• For routine laboratory work, the diaphragm can be
  a piece of filter paper and the experiments may
  take a few hours
• For studies in gases, the entire diaphragm can be
  replaced by a long, thin capillary.

35
Infinite couple

• Limited to solids
• two bars are joined together and quickly raised
  to the temperature at which the experiment is to
  be made.
• After a known time, the bars are quenched, and
  the composition is measured as a function of
  position.
• For such a slow process, the compositions at the
  ends of the solid bars away from the interface do
  not change with time.

The average concentration in the bar
The concentration at the end of the bar
36
Taylor dispersion

• Valuable for both gases and liquids
• 99 accuracy
• employs a long tube filled with solvent that
  slowly moves in laminar flow.
• A sharp pulse of solute is injected near one end
  of the tube.
• When this pulse comes out the other end, its
  shape is measured with a differential
  refractometer.

37
The concentration profile found is that for the
decay of a pulse
Measured by the refractive index
A widely spread pulse means a large E and a small
D. A very sharp pulse indicates small dispersion
and hence fast diffusion.
38
Other methods

• Spin echo nuclear magnetic resonance
• 95
• dose not requires initial concentration
  difference, suitable for highly viscous system
• Dynamic light scattering
• dose not requires initial concentration
  difference, suitable for highly viscous solutions
  of polymers
• If high accuracy is required, interferometers
  should be used.

39
Interferometers

• Gouy interferometer
• measures the refractive-index gradient between
  two solutions that are diffusing into each other.
• the amount of this deflection is proportional to
  the refractive-index gradient, a function of cell
  position and time
• Mach-Zehnder and Rayleigh interferometers
• solid alternatives to the Gouy interferometer

40
Summary

• A great summary table at Table 5.6-1 p. 139
• In general diffusion coefficient in gases and in
  liquids can often be accurately estimated, but
  coefficients in solids and in polymers cannot.
• Prediction
• Chapman-Enskog kinetic theory for gases 8
• Stoke-Einstein equation or its empirical
  parallels for liquids with experimental data 20