POLYMER PROCESS ENGINEERING

Mass transfer phenomenon in polymers: Diffusivity and solubility of

gases in polymers

Prof. Shishir Sinha

Department of Chemical Engineering

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 Topic previously covered
• Mass transfer coefficient in laminar flow
• Mass transfer in falling film
• Laminar falling film in inclined surface
• Mass transfer coefficient in turbulent flow
• Boundary layer theory
• Film theory

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 Table of content

• Diffusion in polymers
• Gas diffusivities in molten polymers
• Gas solubility's in molten polymers
• Examples

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 Diffusion and Solution in polymers
• Fick's first law gives the simplest representation of the molecular
diffusion of a species in a system.
• This expression relates the molar flux JA of species A to a gradient
dCA/dy by means of a transport or diffusion coefficient DAB
dC A
J A = − DAB
dy
• DAB is a property of the system in much the same way
that thermal conductivity is for the transfer of heat.
• Diffusivity is the ratio of molar flux to concentration
gradient whereas thermal conductivity is the ratio of
heat flux to temperature gradient.
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 Diffusion and Solution in polymers
• For diffusion of molecules into a liquid, semisolid, or solid system, there is
an important limiting case, which is solution.
• The analogy to this situation is chemical reaction, which is a kinetic
process that is ultimately limited by chemical equilibrium.
• Both the diffusivities and solubilities of small molecules are important in
polymer processing operations.
• The rate of transport of small molecules within the polymer being
processed is related to the diffusivity.
• Solubility can fix the retention of the molecule in
the system for a given temperature and pressure.

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 Diffusion and Solution in polymers
• A third coefficient of some importance is the permeability, which is
defined as the amount of a diffusing molecule that passes through a
polymer film of unit thickness per second per unit area and a unit
difference of pressure.

• Diffusivity, solubility, and permeability are related by the following:

P= D x H
Where, D is diffusivity and H the Henry's law constant
obtained from the relationship
H Pi = Xi
Where, Pi is the partial pressure of the diffusing
component and X is the gas concentration in the polymer.
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 Diffusion and Solution in polymers
• Permeabilities apply mainly to molecular transport in solid polymers,
particularly polymer films.
• Diffusivities are usually obtained in conjunction with permeability
measurements.
• The principal role of processing with respect to permeabilities and
diffusivities in solid polymers is development of the material's structural
characteristics.
• Mass transport of gaseous molecules in molten or
thermally softened polymer systems is a particularly
pertinent area. This is so because polymer processing
operations generally involve gases diffusion systems.

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 Diffusion and Solution in polymers
• Measurement of solubilities and diffusivities have been accomplished by
various techniques as shown in table.
Method Used for Used for
diffusivity solubility
Movement of a Yes Yes
color boundary
Gas Yes No
chromatography
Piezoelectric Yes Yes
Special diffusion Yes Yes
cell
J. A. Wesselingh and R. Krishna, Mass Transfer in Multicomponen Mixtures, Delft Academic Press. Edition 1st, 2000.
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 Gas diffusivities in polymer systems
• A listing of experimentally determined diffusivities for a variety of gases
and polymers is given in Table for a temperature of 188°C.
Gases Diffusivities
Polyethylene Polypropylene Polystyrene
Helium 17.09 10.51 12.96
Argon 9.19 7.40 5.18
Methane 5.50 4.02 0.42
Nitrogen 6.04 3.51 0.348
Carbon 5.69 4.25 0.39
dioxide
J. A. Wesselingh and R. Krishna, Mass Transfer in Multicomponen Mixtures, Delft Academic Press. Edition 1st, 2000.
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 Gas diffusivities in polymer systems
• The diffusivities of gases in polymers are related to a variety of parameters:
temperature, pressure, the nature of the gas, and the nature of the
polymer.
• In order to deal properly with processing operations, it is necessary to be able
to take these parameters into account, which means that correlations are
needed.
• Diffusivities can be related to temperature by an exponential function given
by:
Where,
(
D = D0 exp − Ed / RT )
• Ed is the activation energy of diffusion (kcal/mole).
D0 is the pre-exponential function (an empirical
constant; its units are cm2/s)
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 Gas diffusivities in polymer systems
• Experimentally determined values of Ed and D0 are given in Table below:
Polymer Gas Ed D0x105 Temp
(kcal/mol) cm2/s (ᴼC)
Polyethylene N2 2.0 53.414 125-188
CO2 4.4 688.13 188-224
Polypropylene CO2 3.0 111.76 188-224
Polystyrene H2 10.1 218.06 120-188
N2 9.6 21.11 119-188
CH4 3.6 21.24 125-188

J. A. Wesselingh and R. Krishna, Mass Transfer in Multicomponen Mixtures, Delft Academic Press. Edition 1st, 2000.
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 Gas diffusivities in polymer systems
• Values of Ed and D0 can be estimated for other gases and polymers by using
Figs 1 and 2. 1)

2)
Fig 1: Logarithm of D0
vs. activation energy
divided by gas constant

Fig 2: Logarithm of D0 vs.

activation energy divided by
gas constant
J. A. Wesselingh and R. Krishna, Mass Transfer in Multicomponen Mixtures, Delft Academic Press. Edition 1st, 2000.
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 Gas diffusivities in polymer systems
• In Fig. 1 the logarithm of D0 is plotted against Ed/R.

• Figure 2 gives a plot of Ed vs. a function of the polymer's glass

temperature and ε/k (a gas molecule parameter).

• In order to use the data of Fig. 1 and 2, first establish the Ed value from
Fig. 2 for a given polymer by using the polymer's glass temperature and
ε/k for the gas.

• Next, obtain the D0 value from Fig. 1.

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 Gas diffusivities in polymer systems
• The polymer's glass temperature and ε/k for the gas is used from the table
below
Gas Collision diameter, σ (Aᴼ) ϵ/k, K
H2 2.915 380
He 2.576 10.2
N2 3.681 91.5
CO 3.590 110
CO2 3.996 190
CH4 3.822 137
C2H4 4.232 205
R. Taylor and R. Krishna, Multicomponent Mass Transfer, John Wiley & Sons Inc. Edition 1st, 1993
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 Gas diffusivities in polymer systems
• The smaller the molecule, the larger the
diffusivity. This becomes apparent when a
parameter (the logarithm of the diffusivity
divided by the square of the diameter of the
gas molecule).
• The diffusivity decreases with increasing
polymer weight, which is evident from a
correlation and plot
developed by Lundberg
et al., (1969)
Fig 3: Diffusivities of gases
versus polymer mer weight
R. Taylor and R. Krishna, Multicomponent Mass Transfer, John Wiley & Sons Inc. Edition 1st, 1993
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 Gas diffusivities in polymer systems
The explanation for the behavior of Fig-3 are:

• Chain flexibility is not an appropriate explanation as the glass temperature

of polyisobutylene lies between that for polyethylene and polypropylene
whereas the correlation shows the polyisobutylene diffusivity data to be
lower than either of the other two polymers.

• Diffusivity to be related directly to the mobility of the

diffusing species. The materials could be arranged (in
order of increasing consistency or decreasing
diffusing species mobility) as polyethylene,
polypropylene, polyisobutylene, and polystyrene.
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 Gas diffusivities in polymer systems
• The relation indicated in Fig. 3 between diffusivity and mer weight is
actually between the diffusivity and mobility of the diffusing species
since these polymers differ structurally in the pendant groups attached
to the basic polyethylene chain.

• There are actually two relations between the diffusion coefficient and
the reciprocal of the absolute temperature. One of
the relations holds up to 150°C, and the second from
150 to 170°C.

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 Gas diffusivities in polymer systems
• The resultant correlation involves two complex equations with diffusivity
equation. In order to use the correlation, it is necessary to know:

1. Polymer density data as a function of temperature

2. Density data for the pure solvent as a function of temperature
3. Several values of the polymer-solvent system diffusivity for at least two
temperatures
4. Sorption equilibrium data for the polymer solvent
system
5. Rheological flow data for the polymer
6. Solvent viscosity data

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 Gas diffusivities in polymer systems
• Lists 1-6 are then used to calculate a number of quantities such as a
solvent free-volume parameter, a polymer free-volume parameter, and
molar volumes at 0 K for the polymer and solvent.

• These quantities are then used with the three continuity equation sets to
calculate diffusivity.

• This method has been applied not to semi-crystalline

molten polymers (such as polyethylene) but rather to
amorphous polymers (such as polystyrene and
polymethyl acrylate) that thermally soften.

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 Gas diffusivities in polymer systems
• It appears that the simpler generalized techniques are adequate for the
higher temperature ranges for both polystyrene and polyvinyl acetate
(above 420 K for polystyrene and 358 K for polyvinyl acetate).

• These temperatures are, respectively, 1.1 and 1.2 times the polymers' glass
temperature (in degrees Kelvin).

• It is, therefore, suggested that the free-volume model

be used in the region from the glass temperature up
to 1.1 to 1.2 times its value.

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 Problem - 1
Question: What are the diffusivities at 220°C for the following systems:
Methane-polyethylene
Nitrogen-polypropylene
Krypton-polyisobutylene
Given: D0 = 2.2 x 10-5 cm2/s ; -Ed (polyethylene) = 3 kcal/mol ; -Ed
(polypropylene) = 2.78 kcal/mol ; -Ed (polyisobutylene) = 5.4 kcal/mol

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 Problem - 2

Questions: Estimate a value of diffusivity for the system carbon dioxide-

polyvinyl chloride at 200ᴼC (473.16 K) ?
Given –Ed (polyvinyl chloride) = 2.84 kcal/mol and D0 (461.16 K) = 2.2 x 10-
5 cm2/s

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 Gas solubilities in polymer systems
• The limiting effect of the diffusion of a gas into a molten or thermally
softened polymer is its solution.

• Such behavior can be expressed in the form of Henry's law.

Experimentally determined values of H, the Henry's law constant are
given in Table below:

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 Gas solubilities in polymer systems
Table shows henry's law constants for various gas-polymer systems.
Henry's Law Constant: cm3/g atm
Polymer Nitrogen Carbon Argon Helium
dioxide
Polyethylene 0.111 0.275 0.113 0.038
Polypropylene 0.133 0.228 0.176 0.086
Polyisobutylene 0.057 0.210 0.102 0.043
Polystyrene 0.049 0.220 0.093 0.029
Polymethyl 0.045 0.260 0.105 0.066
methacrylate
McCabe W.L., Smith J.C. and Harriott P., “Unit Operations of Chemical Engineering”, 6th Ed., 2001, McGraw Hill
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 Gas solubilities in polymer systems
• As with diffusion, solution is a function of a number of parameters,
including temperature, pressure, and the nature of the gases and
polymers involved.

• The relation of the Henry's law constant to temperature is an exponential

one, represented by:

 − ES 
H = H 0 exp 
 RT 
where ES is the heat of solution in kcal/ mole.
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 Gas solubilities in polymer systems
• Experimental values of ES are given in table below:
Polymer Gases
H2 CO2 N2 CH4
Polyethylene - 0.80 0.95 -
Polypropylene - 1.7 - -
polystyrene 1.9 - 1.7 1.05

McCabe W.L., Smith J.C. and Harriott P., “Unit Operations of Chemical Engineering”, 6th Ed., 2001, McGraw Hill
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 Gas solubilities in polymer systems
• As evident from table, the heat of solution changes with gas structure for a
given polymer.

• This phenomenon was also observed in solid amorphous polyethylene and

natural rubber, where the heats of solution moved from endothermic to
exothermic as the gas collision diameter increased.

• Figure 4 compares data for molten polyethylene to

data for solid amorphous polyethylene

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 Gas solubilities in polymer systems
• The dotted line of Fig. 4 can be used to
estimate heats of solution for gases
other than nitrogen or carbon dioxide in
polyethylene.
• References in the literature indicate that
the Henry's law constant is independent
at pressures up to at least 1.01 x 107 Pa
and possibly up to
Fig 4: Heats of solution for gases in 3.03 x 107 Pa
solid amorphous (solid line, circles)
polyethylene and for molten
polyethylene (dotted line, triangles)
Basmadjian D., “Mass Transfer and Separation Processes: Principles and Applications”, 2007, CRC Press
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 Problem -3
Question: What are the Henry's law constants for nitrogen, helium, argon,
carbon dioxide in solid amorphous polyethylene at 461.16 K?
Given
H0 (N2) = 0.0351 cm3/g. atm; -ES (N2) = 950 kcal/ mol
H0 (He) = 0.0102 cm3/g. atm; -ES (He) = 2300 kcal/ mol
H0 (A) = 0.0878 cm3/g. atm; -ES (A) = 500 kcal/mol
H0 (CO2) = 0.384 cm3/g. atm; -ES (CO2) = 800 kcal/ mol

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 References
• Fundamental of Heat and Mass Transfer, Incropera and Dewitt, 5th Edn.,John Wiley & Sons.
• Basmadjian D., “Mass Transfer and Separation Processes: Principles and Applications”,
2007, CRC Press
• Treybal R.E., “Mass Transfer Operation”, 3rd Ed., 1980, McGraw Hill.
• McCabe W.L., Smith J.C. and Harriott P., “Unit Operations of Chemical Engineering”, 6th
Ed., 2001, McGraw Hill
• Foust A. S., Wenzel L. A., Clump C. W., Maus L. and Andersen L.B., “Principles of Unit
Operations”, 2nd Ed., 2008,Wiley-India.
• Brown G. G. and Associates, “Unit Operations”,1995, CBS Publishers.
• Wankat P. C., “Separation Process Engineering”, 2nd Ed., 2006,
Prentice Hall.
• R. Taylor and R. Krishna, Multicomponent Mass Transfer, John
Wiley & Sons Inc. Edition 1st, 1993
• J. A. Wesselingh and R. Krishna, Mass Transfer in Multicomponent
Mixtures, Delft Academic Press. Edition 1st, 2000.
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Thank You

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