高等輸送二 — 質傳

Lecture 4Diffusion coefficient

郭修伯 助理教授

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)

Diffusion coefficient in gases

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)

1~4

2

Dt

z

Chapman-Enskog theory

• Theoretical estimation of gaseous diffusion:

212

21

21

233

~1

~1

1086.1

pMM

T

D

1~)(:

)(2

1:

:~

:::

2112

2

speciestwothebetweenactioninterf

Adiametercollision

weightmolecularMatmpKTsec

cmD i

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: 01

11 3

1vc

dz

dclvn

Average molecular velocity

Mean free path of the molecules

Concentration gradient

NM

Tkv B

~~

8

Molecular mass

2112p

Tkl

B

Diameter of the spheres

2

21

23

212

3 ~1~

3

2

3

1

pM

TN

klvD B

),

~,,( MpTf

Empirical relations

2

31

23

1

1

21

21

75.1

3

~1~1

10

i ii i VVp

MMT

D

(Fuller, Schettler, and Giddings, 1966)

jmoleculetheofpartsofvolumesV

atmpKTsec

cmD

ij :

:::2

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!

Diffusion coefficients in liquids

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

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:

1vfforce

Friction coefficient 06 Rf

Stokes’ law (Stokes, 1850)

Thermodynamic “virtual force”The negative of the chemical potential gradient (Einstein, 1905)

1force

101 6 vR

101 6 vR

2101

21

1011

011 lnlnlnln ccTk

cc

cTkxTk BBB

~ const. 101

01 6ln vRcTkB

1011

6 vRcc

TkB

1111106

cDccR

TkB nv 06 R

TkD B

Stoke - Einstein equation

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)

06 R

Tk

f

TkD BB

Friction coefficient of the solute

Boltzmann’s constant

Solvent viscositySolute radius

06 R

Tk

f

TkD BB

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:

32

D

)(fD

06 R

Tk

f

TkD BB

• 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.

Estimate the diffusion at 25ºC for oxygen dissolved in water using the Stoke-Einstein model.

06 R

TkD B

cmR 810 1073.1

2

1 Estimate the radius of the oxygen molecule?We assume that his is half the collision diameter in the gas:

sec/103.11073.1)

sec01.0(6

)298)(sec

1038.1(

625

8

2

216

0

cmcm

cmg

KK

cmg

R

TkD B

About 30% lower than the experimental measurement.

• Stoke - Einstein equation (for dilute concentration)

• We found that D = f (solute concentration)• Derive the Stoke - Einstein equation? Add

hydrodynamic interaction among different spheres:

06 R

Tk

f

TkD BB

Diffusion in concentrated solutions

...)5.61(6 10 Rf (Batchelor, 1972)

The volume fraction of the solute

...)5.11(6 1

0

RTk

D BNot very good for small solutes

Empirical relations

11 vf 11

1v

f 110 v

RT

D

11011 ln cTkB

11

1011 ln

ln1 c

cDc

v Activity coefficient

1

10 ln

ln1

cDD

)1()1( 2021010 xDxxDxD

21 )1()1( 20100xx xDxDD

Arithmetic mean (Darken, 1948; Hartley and Crank, 1949)

Geometric mean (Vigness, 1966; Kosanovich and Cullinan, 1976) works better!

(Table 5.2-3 page 117)

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.

69.0ln

ln

1

1

x

sec/1043.2

sec)/1068.4(sec)/1026.1(

)1()1(

25

5.0255.025

2010021

cm

cmcm

xDxDD xx

Geometric mean (Vigness, 1966; Kosanovich and Cullinan, 1976):

sec/1075.0

69.011043.2

ln

ln1

25

5

1

10

cm

cDD

Very close to the experimental measurement

Diffusion coefficients in solids

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.

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

Any theory? not very accurate (although theory for face-centered-cubic metals is available)

NaD 20 (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)

RT

H

eDD

0

melt

melt

RT

H

veaD

200

frequencyvparameterempirical

etemperaturmeltingTmeltingofenthalpytheH meltmelt

:4.0~:

::

12

20

~2

aM

Hv meltT

Kmolg

calH

36

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 + z

Number of atoms per unit area at z

4N

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

z

caNj

1

2

01 2

4

Diffusion in polymers

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

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)

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:

eB

R

TkD

6

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.

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.

For binary diffusion coefficient:

1

10 ln

ln1

DD 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 energy2. Effected by any space or “free volume” between the polymer chains

2211

202101

00KK

VV

RT

E

eeDD

parametersvolumefreeadditionaltheTTK

volumesfreecriticalspecifictheV

fractionmassthe

gi

i

i

:),(

:

:

0

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:

NTk

RouseD B)(

Degree of polymerization

Friction coefficient characteristic of the interaction of a bead with its surroundings

OK for low molecular weight

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

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.

tupperlower

initialupperlower

cc

cc

tD

,1,1

,1,11

lowerupper VVl

A 11

Area available for diffusion

Effective thickness of the diaphragm

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.

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.

Dt

z

cc

cc

4erf

11

11

The concentration at the end of the bar

The average concentration in the bar

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.

The concentration profile found is that for the decay of a pulse:

Et

z

eEt

RMc 4

20

1

2

4

D

RvtcoefficiendispersionE

solventflowingtheofvelocityaveragev

radiustubetheR

injectedsolutetotalM

48:

:

:

:

2

00

0

0

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

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.

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

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%