Page 366 - Biosystems Engineering

Food Package Engineering 343

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where J = flux, mol/cm s
D = diffusion coefficient, cm /s
2
c = concentration of water vapor or gas, mol/cm 3
x = film thickness, cm
It is convenient to measure vapor pressure (p) than measure the actual
concentration of gas or vapor. Under steady-state conditions, Eq. (11.1)
can be integrated across the total thickness of the polymer film and
vapor pressure across the film [Eq. (11.2)]:
Δ
Q PA p
= (11.2)
t x
where Q = amount of gas or water vapor, cm 3
t = time, s
2
3
P = permeability cm cm/cm s atm
Over 30 different units for P are reported in the literature. Some of
2
the common units of permeability are cm cm/cm s Pa, mL mm/m 2
3
day kPa, and mL cm/m day atm. Equation (11.2) assumes (1)
2
steady-state diffusion, (2) a linear concentration gradient across the
film, (3) unidirectional diffusion, and (4) permeability independent
of concentration.
Unsteady-state transport of water vapor or gas through poly-
meric film is described using Fick’s second law [Eq. (11.3)]:
∂ 2 c dc
D = (11.3)
dx 2 dt
where t is time (s).
The permeability of a polymer is affected by the nature of the
polymer, the nature of the gas or water vapor, and ambient condi-
tions. The barrier properties of a polymer’s gas and vapor is increased
with the increasing polarity of certain types of polymer, regularity of
molecular structure, and close chain-to-chain packing in the polymer
matrix. Crystalline polymers have lower gas permeability than amor-
phous polymers. Permeability (diffusivity × solubility) also depends
8
on the molecular size of the gas and its chemical affinity to the poly-
mer. Larger gas molecules generally have lower diffusivity and higher
solubility compared to smaller molecules. Temperature and relative
humidity also affect a gas’s permeability.
Many methods are used for measurement of water vapor and gas
permeability of polymer-based packaging. One method is based on
1,2
the permeation rate equation [see Eq. (11.2)]. Quantity of water vapor
(Q) transferred through a film is measured using a precision balance,
whereas a sensitive detector is used in a permeation cell to quantify
the amount of gas transfer. The exposed surface area (A) of the poly-
mer film is accurately defined by the opening in the permeation
cell. The pressure difference (Δp) across the film is controlled by the