4 Diffusion coefficient in gases One atmosphere and near room temperature, values between 10-1 ~ 100 cm2/sec (Reid, Sherwood, and Prausnitz, 1977)approximationinversely proportional to pressure1.5 to 1.8 power of the temperaturevary with molecular weightWhen , 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 dimensionsthe diffusion flux:Concentration gradientMean free path of the moleculesAverage molecular velocityMolecular massDiameter 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!
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/secThe sloth characteristic liquid diffusion means that diffusion often limits the overall rate of process occurring in the liquidchemistry: rate of acid - bas reactionphysiology: rate of digestionmetallurgy: rate of surface corrosionindustry: 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 coefficientStokes’ 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 soluteBoltzmann’s constantSolvent viscositySolute 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 coefficientsSeveral 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 soluteNot very good for small solutes
17 Empirical relations Activity coefficient (Table page 117)Activity coefficientArithmetic 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
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 nonlinearA 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 micronafter 1 minutes, hydrogen penetrates about 6 micronafter 1 hour, hydrogen penetrates about 50 micronHydrogen diffuses much more rapidly than almost any other solute.
21 Diffusion mechanisms in solids Isotropic diffusion through the interstitial spaces in the crystal - lattice theorydiffusion depends on vacancies between the missing atoms or ions in the crystal - vacancy diffusionAnisotropic crystal lattice leads to anisotropic diffusionNoncrystal diffusionCompare the driving forcesLiquid/Gas: concentration gradient/pressure driven flowsSolids: 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 TheoryWe 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 fluxj1=Number of atoms per unit area at z + zNumber of atoms per unit area at z4NThe rate of jumpsThe average number of vacant sitesThe factor of 4 reflects the face that the FCC structure has 4 sites into which jumps can occur
25 Diffusion in polymersIts value lies between the coefficients of liquids and those of solidsDiffusion coefficient is a strong function of concentration.Dilute concentration:a polymer molecule is easily imagined as a solute sphere moving through a continuum of solventHighly concentrated solution:small solvent molecules like benzene can be imagined to squeeze through a polymer matrixMixture 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” solventPolymer 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)1/2Root-mean-square radius of gyrationIn 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 valuein devolatilization (i.e., the removal of solvent and unreact monomer from commercial polymers)in drying many solvent based coatingsSometimes, 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 soluteVolume 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 energy2. Effected by any space or “free volume” between the polymer chains
31 Polymer solute in Polymer solvent Practical importance:adhesion, material failure, polymer fabricationNo accurate model availablethe 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 surroundingsDegree of polymerizationOK 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.131Three methods give accuracies sufficient for most practical purposedDiaphragm cellInfinite coupleTaylor dispersion
33 Diaphragm cell Can obtain ~ 99.8% accuracy Diffusion in gases or liquids or across membraneTwo well-stirred 60 rpm) compartments are separated by either a glass frit or by a porous membrane.Effective thickness of the diaphragmArea available for diffusion
34 Issues for diaphragm cell For accurate work, the diaphragm should be a glass frit and the experiments may take several daysFor routine laboratory work, the diaphragm can be a piece of filter paper and the experiments may take a few hoursFor 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 barThe 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 indexA 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 systemDynamic light scatteringdose not requires initial concentration difference, suitable for highly viscous solutions of polymersIf 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 timeMach-Zehnder and Rayleigh interferometerssolid 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%