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

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Diffusion coefficient Reasonable values of diffusion coefficient –in gas: ~ 10 -1 cm 2 /sec –in liquid: ~ 10 -5 cm 2 /sec –in solids: ~ 10 -10 cm 2 /sec (strong function of temperature) –in polymer/glasses: ~ 10 -8 cm 2 /sec (strong function of solute concentration)

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Diffusion coefficient in gases

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One atmosphere and near room temperature, values between 10 -1 ~ 10 0 cm 2 /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)

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Chapman-Enskog theory Theoretical estimation of gaseous diffusion:

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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: Average molecular velocity Mean free path of the molecules Concentration gradient Molecular mass Diameter of the spheres

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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!

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Diffusion coefficients in liquids

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Most values are close to 10 -5 cm 2 /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 cm 2 /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

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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)

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~ const. Stoke - Einstein equation

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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

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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:

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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 R 0 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.

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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.

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Stoke - Einstein equation (for dilute concentration) We found that D = f (solute concentration) Derive the Stoke - Einstein equation? Add hydrodynamic interaction among different spheres: Diffusion in concentrated solutions (Batchelor, 1972) The volume fraction of the solute Not very good for small solutes

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Empirical relations Activity coefficient Arithmetic mean (Darken, 1948; Hartley and Crank, 1949) Geometric mean (Vigness, 1966; Kosanovich and Cullinan, 1976) works better! (Table 5.2-3 page 117)

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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 cm 2 /sec; in pure water, it is 4.68 x 10 -5 cm 2 /sec. Geometric mean (Vigness, 1966; Kosanovich and Cullinan, 1976): Very close to the experimental measurement

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Diffusion coefficients in solids

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Most values are very small. The range is very wide ~ 10 10 (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.

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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

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Any theory? not very accurate (although theory for face-centered-cubic metals is available) (Franklin, 1975; Stark, 1976) The spacing between atoms (estimated from crystallographic data) The fraction of sites vacant in the crystal (estimated from the Gibbs free energy of mixing) The jump frequency (estimated by reaction-rate theories for the concentration of activated complexes, atoms midway between adjacent sites)

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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 j 1 = Number of atoms per unit area at z + z Number of atoms per unit area at z 4N4N The average number of vacant sites The rate of jumps The factor of 4 reflects the face that the FCC structure has 4 sites into which jumps can occur

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Diffusion in polymers

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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

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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)

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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 (R 2 ) 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.

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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.

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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

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

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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

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Diaphragm cell Can obtain ~ 99.8% accuracy Diffusion in gases or liquids or across membrane Two well-stirred (m.r. @ 60 rpm) compartments are separated by either a glass frit or by a porous membrane. Area available for diffusion Effective thickness of the diaphragm

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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.

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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 concentration at the end of the bar The average concentration in the bar

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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.

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

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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.

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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

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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%

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