CHARLES M. HANSEN

 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

Law 1: F = - D
0
(
 c/
 x)
For Steady State Flux in the x Direction, and
Law 2:
 c/
 t =

/
 x (D
0
 c/
 x)
This is also called the Diffusion Equation

Dimensionless time:
T = D
0 t/L
2
(cm
2
/s)(s/cm
2
)
Dimensionless distance:
X = x/L
Dimensionless concentration:
C = (c – c
0
)/(c

- c
0
)

At low concentrations ( ≤1%) D(c) = D
0
F = - D
0
(c
1
– c
2
)/L
For Concentration Dependent Diffusion -
D(c) increases by a factor of 10 for each
3%v increase in concentration (See Below)

Half-time (t
½
) equation for measuring D
0
D
0
= 0.049 L
2
/t
½
Corrections required for concentration dependence (M) and surface resistance (B)
D ( c )

F
M

F
B
0 .
049

L
2 t
½

CORRECTIONS FOR CONCENTRATION
DEPENDENCE ALONE
Note huge corrections for desorption
D max
1
2
5
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
Desorption Absorption
(F d
)
1/2
1.00
(F d
)
1/4
1.00
(F a
)
1/2
1.00
1.56
2.70
1.55
2.61
1.30
1.70
4.00
3.84
13.40
10.20
43.30
23.10
2.01
3.30
4.85
138.7
47.40
443.0
89.0
6.14
7.63
1,370.0 160.5
8.97
4,300.0 290.0
10.60
13,670.0 506.0
12.10

F s
SURFACE CONDITION
= -D s
 C s
/  x = h(C eq – C s
)
External Flux at surface, F s
, equals mass transfer coefficient (cm/s) times concentration
2 difference, g/cc giving g/cm s
In dimensionless terms the ratio of diffusion resistance to surface resistance is given by B
Corrections best by curve fitting (See Below).
B = R d
/R s
= (L/D
0
)/(1/h) = hL/D
0

CORRECTIONS FOR SURFACE RESISTANCE
FOR D
0
= CONST.
B = hL/D = R d
/R s
B

10
2
1
0.5
0.1
1/B
0
0.1
0.5
1
2
10
F
B
1.0
1.45
3.14
4.95
6.8
37.5

F =
 p/(L/P app
) =
 p/(L/P

+ R
1
+ R
2
+ R
3
…)
L/P app
= L/P

+ R
1
+ R
2
+ R
3
….
1/P app
= 1/P

+ (R
1
+ R
2
+ R
3
….)/L
Use Plot of 1/P

Versus 1/L

TRUE PERMEATION COEFFICIENT (P
∞
BY EXTRAPOLATION (ACRYLIC FILMS)
)
1
P app x 10
-12
20
15
10
5
P 
0 5 10 15 20 25
1 x 10
L
-3

 Film: Thickness (L), length (l), width (w)
D
0
= D app
/(1 + L/l + L/w)
2
 Circular Film: Thickness (b), Radius (R)
D
0
= D app
/(1 + b/R)
2
 For L = 1mm and w = 10mm: D app
/D
0
= 1.21
 Tensile bars (L = 2-4mm, w=10mm): Do not use!

 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)
4
6
0.2 Vf ~ 1 decade
Absorption
8
10
0.03 Vf
1 decade
~
Desorption
Absorption
D
APP
12
14
0 0.2
0.4
Vf
Isotope technique
D
C
D
1
(dry film)
0.6
0.8
Selfdiffusion
1.0

f
 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)
6
Isotope
8
Absorption F = F x F
B
= 1.3 x 1.25
= 1.63
F = F x F
B
= 1.2 x 250
= 300
10
Desorption
(to vacuum)
12
14
0.1
0.2
0.3
Vf
0.4
0.5
0.6

(F a
ABSORPTION WITH CORRECTIONS
) REQUIRED FOR D(c) AND F
B
FOR R s
Chlorobenzene / polyvinyl acetate
1.0
0.8
L = 118 µm f f
0.6
D = 1.8(10)
-8 cm² sec
B ~ 15
0.4
0.2
0
1 2 3 4 5 t , min ½
6 7 8

ADDITIONAL EXAMPLES OF SURFACE
RESISTANCE – COC POLYMER
(NIELSEN, HANSEN 2005)
Absorption of selected solvents in a COC polymer
1200
300
1000
200
800
100
600
400
0
0 5 10 15 20
Hexane
THF
Diethylether
1,2-Dichloroethylene
200
0
0 20 40 60 80
Sqrt time in min
100 120 140 160

S-SHAPED CURVES CAUSED BY SURFACE
RESISTANCE (NIELSEN, HANSEN 2005)
Absorption of selected solvents in a COC polymer
60
50
40
30
20
10
0
0 50 100 150 200 250 300 350 400
Sqrt time in min
Butylacetate
Ethylacetate

ABSORPTION – CASE II AND SUPER CASE II
CAUSED BY COMBINED ( Hansen, 1980)
R d and R s for D = D
0 e kc
1.0
0.8
0.6
B:
10
9
10
8
10
7
0.4
0.2
0.0
0.0
0.2
0.4
0.6
T x 10
6
0.8
1.0
1.2
10
6
1.4

CONCENTRATION GRADIENTS COMBINED
R d
AND R s
FOR D = D
0 e kc
( Hansen, 1980)
0.4
0.2
0.0
0.0
1.0
0.8
0.6
0.869
0.037
0.098
0.146
0.216
0.125
0.250
0.375
X
0.500
0.319
0.625
0.386
0.467
0.750
0.875
0.562
1.0

(Hansen, 1967, 1968)
10
1
V
2 = 10
6
V t = 10
10
C
A = 0·2
B as indicated
10
Exptl.
165 microns
10
-1
B=10
6
B=10
7
C
A
Exptl.
22 microns
C
S
= O
For B=10
7
~ M
O
B=10 5
C
S
= O
For B=10
6
C
A
C
S
= O
For B=10
5
Calculated
Experimental
Effect of water - a steeper slope
One day L=30 microns
10
-2
10
-7
10
-6
10
-5
10
-4
T,
D
O t
(L)
2
Dimensionsless
10
-3
10
-2

H
3
H
C
3
C
CH
3
OH
O
O
H
3
C
O
CH
3
CH
3
CH
3
CH
3
CH
3
H
3
C
CH
3
O
CH
3
RELATIVE SOLVENT RETENTION
(HANSEN, 1967)
MOLECULAR SIZE AND SHAPE
Cl
O
H
3
C CH
3
O
N
+
O
Cl
CH
3
O
H
3
C O
H O
O
CH
3
CH
3
O
H
3
C O CH
3
O
CH
3
H
3
C
H
3
C
O
N
+
O
O
O
O
O
H O
CH
3
O
N
+
O
O CH
3
H O
O
CH
3
H
3
C
OH

Compare Methanol with Iodine
D
0

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.


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

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?

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)

Windle, “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.

Calculated Concentration Gradients Flat at 13 hours

Hopfenberg and Coworkers Super Case II
Correctly Modeled Absorption, D
0
, 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.

 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!
For further contact please visit: www.hansen-solubility.com