1. 高等輸送二 — 質傳 Lecture 4 Diffusion coefficient
郭修伯 助理教授

2. Diffusion coefficient
Reasonable values of diffusion coefficient
in gas: ~ 10-1 cm2/sec
in liquid: ~ 10-5 cm2/sec
in solids: ~ 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
Molecular mass
Diameter of the spheres

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. Stoke - Einstein equation
~ 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 > 5)
Friction coefficient of the solute
Boltzmann’s constant
Solvent viscosity
Solute radius

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. Empirical relations for liquid diffusion coefficients
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 Activity coefficient
(Table 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:
Net flux j1 =
Number of atoms per unit area at z + z
Number of atoms per unit area at z
4N
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 “good” solvent
Polymer in “poor” solvent
(Ferry, 1980)

27. Stoke-Einstein equation may be used:
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 ~ (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 solute’s activation energy
2. Effected by any space or “free volume” between the polymer chains

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 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%