DIFFUSION IN POLYMERS

CHARLES M. HANSEN

OUTLINE

Laws of Diffusion Generalized Solutions to these Laws Concentration Dependent Coefficients Surface Condition can be significant Combine These - No Anomalies Predict Missing Data from Limited Results Control Solvent Retention

FICK’S FIRST AND SECOND LAWS

Law 1: F = - D0(c/x)

For Steady State Flux in the x Direction, and

Law 2: c/t = /x (D0c/x)

This is also called the Diffusion Equation

DIMENSIONLESS VARIABLES

Dimensionless time:

T = D0t/L2 (cm2/s)(s/cm2)

Dimensionless distance:

X = x/L

Dimensionless concentration:

C = (c – c0)/(c - c0)

FOR STEADY STATE PERMEATIONAt low concentrations (≤1%) D(c) = D0

F = - D0(c1 – c2)/L

For Concentration Dependent Diffusion -

D(c) increases by a factor of 10 for each

3%v increase in concentration (See Below)

MEASURING DIFFUSION COEFFICIENTS

Half-time (t½) equation for measuring D0

Corrections required for concentration

dependence (M) and surface resistance (B)

D0 = 0.049 L2/t½

½

2049.0)(

t

LFFcD BM

CORRECTIONS FOR CONCENTRATION DEPENDENCE

ALONE Note huge corrections for

desorption

Desorption Absorption Dmax (Fd)1/2 (Fd)1/4 (Fa)1/2

1 1.00 1.00 1.002 1.56 1.55 1.305 2.70 2.61 1.70101 4.00 3.84 2.01102 13.40 10.20 3.30103 43.30 23.10 4.85104 138.7 47.40 6.14105 443.0 89.0 7.63106 1,370.0 160.5 8.97107 4,300.0 290.0 10.60108 13,670.0 506.0 12.10

SURFACE CONDITION Fs = -DsCs/x = h(Ceq – Cs)

External Flux at surface, Fs, equals mass transfer coefficient (cm/s) times concentration difference, g/cc giving g/cm2s

In dimensionless terms the ratio of diffusion resistance to surface resistance is given by B

Corrections best by curve fitting (See Below).

B = Rd/Rs = (L/D0)/(1/h) = hL/D0

CORRECTIONS FOR SURFACE RESISTANCE FOR D0 = CONST.

B = hL/D = Rd/Rs

B 1/B FB

0 1.0

10 0.1 1.45

2 0.5 3.14

1 1 4.95

0.5 2 6.8

0.1 10 37.5

PERMEATION WITH SURFACE AND/OR EXTERNAL

RESISTANCESF = p/(L/Papp) = p/(L/P + R1 + R2 + R3 …)

L/Papp = L/P + R1 + R2 + R3 ….

1/Papp = 1/P + (R1 + R2 + R3 ….)/L

Use Plot of 1/P Versus 1/L

TRUE PERMEATION COEFFICIENT (P∞)

BY EXTRAPOLATION (ACRYLIC FILMS)

20

15

10

5

0 5 10 15 20 25

P

Papp

1 x 10-12

L1 x 10-3

DIFFUSION SIDE EFFECTS

Film: Thickness (L), length (l), width (w)

D0 = Dapp /(1 + L/l + L/w)2

Circular Film: Thickness (b), Radius (R)

D0 = Dapp/(1 + b/R)2

For L = 1mm and w = 10mm: Dapp/D0 = 1.21

Tensile bars (L = 2-4mm, w=10mm): Do not use!

UNIQUE DATA USED IN FOLLOWING The system chlorobenzene in poly(vinyl acetate)

has been studied extensively with all relevant data reported in my thesis and subsequent journal articles. See the next slides. Absorption data from one equilibrium to another, desorption data from different equilibria to vacuum, and film drying (years) all present a unified and coherent picture of solvent diffusion in polymers, if one accounts for concentration dependence and significant surface effects when present.

D(c) FOR CHLOROBENZENE IN PVAc FOR ALL CONCENTRATIONS

(HANSEN, 1967)

- L

OG

D,

cm²/

sec

0.2

Desorption

Absorption

Absorption

0.03 Vf1 decade

~

0.2 Vf 1 decade~

DAPP

DC

D1 (dry film)

Isotope technique

Self-diffusion

0 0.4 0.6 0.8 1.0Vf

14

12

10

8

6

4

DROP IN CURVE ABOVE 0.2 Vf When apparent diffusion coefficients are

measured by absorption above a break point, the surface condition becomes progressively more important and the apparent diffusion coefficients become lower and lower. Proper interpretation allows these to be corrected to values expected from other measurements. Initial S-curvature indicates surface resistance is important. The consequences are shown in the following slides.

DESORPTION AND ABSORPTION GIVE SAME D(c) WITH CORRECTION

(HANSEN 1967, 2004)

14

12

10

8

6

- L

OG

dif

fusi

on c

oeff

icie

nt a

t 20

°C

, cm

²/se

c

0.1 0.2 0.3 0.4 0.5 0.6

Desorption(to vacuum)

Absorption

Isotope

F = 1.8a

F = 40d

F = 144d

F = F x F= 1.3 x 1.25= 1.63

a B F = F x F= 1.2 x 250= 300

a B

Vf

ABSORPTION WITH CORRECTIONS (Fa) REQUIRED FOR D(c) AND FB FOR Rs

1

Chlorobenzene / polyvinyl acetate

2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1.0

M

/ Mt

min ½t

L = 118 µm

C = 0.22 V0

C = 0.27 V

F = 1.3a

B

½

F = 1.25

F x F = 1.63a½ B

B ~ 15D = 1.8(10)-8 cm²sec

,

f

f

ADDITIONAL EXAMPLES OF SURFACE RESISTANCE – COC POLYMER (NIELSEN, HANSEN

2005)Absorption of selected solvents in a COC polymer

0

200

400

600

800

1000

1200

0 20 40 60 80 100 120 140 160

Sqrt time in min

We

igh

t c

ha

ng

e i

n m

g/g

Hexane

THF

Diethylether

1,2-Dichloroethylene

0

100

200

300

0 5 10 15 20

S-SHAPED CURVES CAUSED BY SURFACE RESISTANCE (NIELSEN,

HANSEN 2005)Absorption of selected solvents in a COC polymer

0

10

20

30

40

50

60

0 50 100 150 200 250 300 350 400

Sqrt time in min

Wei

gth

ch

ang

e in

mg

/g

Butylacetate

Ethylacetate

ABSORPTION – CASE II AND SUPER CASE II CAUSED BY COMBINED

( Hansen, 1980)

Rd and Rs for D = D0ekc

0.0

0.2

0.4

0.6

0.8

1.0

Mt

T x 10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.46

106

107108

109

/ M

B:

CONCENTRATION GRADIENTS COMBINED Rd AND Rs FOR D = D0e

kc

( Hansen, 1980)

0.0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.00.0

0.2

0.4

0.6

0.8

1.0

ØR

X

0.5620.4670.3860.3190.2160.1460.0980.037

0.869

DRYING OF A LACQUER FILM (Hansen, 1967, 1968)

10 -7 10 -6 10 -5 10 -4 10 -3 10 -210 -2

10 -1

10

10 1

B=106

B=107

CA CA

Exptl.165 microns

Exptl.22 microns

B=105

~ MO

C S = O

For B=107 C S = O

For B=106

C S = O

For B=105

Experimental

Calculated

One day L=30 microns

Effect of water - a steeper slope

DO t

(L) 2T, Dimensionsless

Vol

ume

Solv

ent /

Vol

ume

Poly

mer

V2 = 10 6

Vt = 10 10

CA = 0·2

B as indicated

RELATIVE SOLVENT RETENTION (HANSEN, 1967)

MOLECULAR SIZE AND SHAPE

Cl

O

CH3

O

CH3

OH

CH3CH3

O

CH3

CH3

CH3

CH3

O

CH3

CH3

CH3

O

CH3

CH3

CH3

O

N+

O O

CH3CH3

Cl

CH3

O

O

O

O

CH3 O CH3

O

OOH

CH3

N+

O O

CH3

OOH

CH3

CH3 O

O

CH3

CH3

N+

O O

OOH CH3

CH3

OH

Effect of Molecular Properties on D0

Compare Methanol with Iodine

GENERAL ARTICLE APPEARS EXPLAINING “ANOMALIES” USING

DIFFUSION EQUATION Much of the above has been presented in

Chapter 16 of Second Edition of Hansen Solubility Parameters: A User’s Handbook, CRC Press, 2007. The following article: Hansen CM. The significance of the surface condition in solutions to the diffusion equation: explaining "anomalous" sigmoidal, Case II, and Super Case II absorption behavior. Eur Polym J 2010;46;651-662 contains the next slides.

SIGNIFICANT SURFACE CONDITION FOR ABSORPTION OF WATER INTO PVALC FROM BONE DRY TO 0.748 VOLUME FRACTION

CASE II ABSORPTION WITH LINEAR UPTAKE WITH LINEAR TIME. THE

SURFACE CONCENTRATION INCREASES SLOWLY

SUPER CASE II WITH SLOWLY INCREASING RATE OF ABSORPTION

WITH TIME. CONCENTRATION GRADIENTS SHOW A FRONT.

HANSEN IS “EXTRANEOUS”:

PETROPOULOS et.al Petropoulos JH Sanopoulou M Papadokostaki KG. Physically insightful modeling of non-Fickian kinetic energy regimes encountered in fundamental studies of isothermal sorption of swelling agents in polymeric media. Eur Polym J 2011;47:2053-2062.

Hansen extraneous, challenges included

Hansen cannot explain these data!

Next two slides do explain these data

CALCULATED ABSORPTION CURVE AND GRADIENTS MATCH EXPERIMENTAL DATA FOR

ABSORPTION PERPENDICULAR TO STRETCH DIRECTION: METHYLENE CHLORIDE IN

CELLULOSE ACETATE.

CALCULATED ABSORPTION CURVE IS PERFECT, FRONT NOT A SHARP STEP, BUT CLOSE TO

EXPERIMENTAL. METHYLENE CHLORIDE IN STRETCHED CELLULOSE ACETATE STRETCH

DIRECTION. ARE INITIAL CONDITIONS MAINTAINED?

Thomas and Windle Case II ExampleMethanol/PMMA with Iodine Tracer

Straight line absorption

with linear time cited as

excellent example of

Case II behavior.

This result is duplicated:

Diffusion equation with

significant surface effect

and exponential D(c)

Thomas and Windle Case II ExampleWindle, “Case II Sorption” in Comyn, Polymer Permeability (1985) Iodine tracer lags methanol

in PMMA at 30°C showing

apparent step-like gradient.

Methanol does not have this

“advancing sharp front”.

Iodine tracer far too slow

as shown in the next slide.

Methanol gradients become

flat at longer time.

Methanol/PMMA Absorption at 30ºC

Calculated Concentration Gradients Flat at 13 hours

Super Case II: n-Hexane/Polystyrene

Hopfenberg and Coworkers

Hopfenberg and Coworkers Super Case II

Correctly Modeled Absorption, D0, and h.

CONCLUSION: STRESS RELAXATION NEED NOT BE

INVOKED. Stress relaxation phenomena need not be

invoked to explain the cases examined including Thomas and Windle Case II, Super Case II, and Sigmoidal examples or the studies of Petropoulos and coworkers.

The diffusion equation seems to fully describe all of these studies when the a significant surface condition is included and exponential diffusion coefficients are used.

DIFFUSION IN POLYMERS SUMMARY

Laws of Diffusion Generalized Solutions to these Laws Concentration Dependent Coefficients Surface Condition involved with ”Anomalies” Combine These - No Anomalies Predict Missing Data from Limited Results Estimate Behavior at Different Conditions Improved understanding

Thank you for your attention!

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