3D Pore scale network model for the transport of liquid water, water vapor and

oxygen in polymeric films.

Jose E. Paza and Luis A. Seguraa

a Food Engineering Department, Universidad del Bío-Bío, Chillán, Chile (lsegura@ubiobio.cl)

ABSTRACT

Polymeric materials are used in food packaging mainly due to their capability to control the exchange of low

molecular weight compounds between foodstuff and external environment. Indeed, the mass transport control

through the packaging film is extremely important to guarantee the proper shelf-life of the packaged

products. Most mathematical models developed to describe those phenomena are based on modifications of

Fick’s law in a continuous approach and the transport parameters such as diffusivity or permeability are the

result of the process history in these kinds of models. The objective of this work is to present a 3D pore scale

model for the transport of liquid water, water vapor and oxygen in polymeric films. The porous media is

represented by a three-dimensional cubic network with pore segments randomly assigned and the model is

solved using Monte Carlo method. As water vapor flows into the polymeric film, condensation of water

occurs at the pore walls of the network. Liquid in pore corners allows hydraulic connectivity throughout the

network at all time and capillary pressure is determined by augmented Young-Laplace equation. Here we

report pore-level distribution of liquid and vapor as transport phenomena advanced, effective water vapor,

liquid and oxygen diffusivity and absolute permeability are calculated. The vapor and oxygen diffusivity

diminish as vapor condensation occurs from 8.97x10-8 to 2.21x10-9 [cm2/s] and 7.83x10-8 to 1.89x10-9 [cm2/s]

respectively. The vapor and oxygen permeability diminish from 1.10x10-21

to 2.72x10-23

[m2] and 3.16x10

-21

to 7.67x10-23

[m2]. On the other hand, the hydraulic permeability increases from 4.49x10

-25 to 8.48x10

-21 [m

2].

The transport properties obtained by the model were compared with experimental results obtained by

specialized literature given a good agreement for the oxygen and water vapor.

Keywords: condensation; diffusivity; pore-level; oxygen; water vapor

INTRODUCTION

The deterioration of packaged foodstuffs largely depends on the transfers that may occur between the internal

environment of the packaged food and the external environment [1]. Polymers have been commonly used for

food packing due to their characteristics such as: low cost, easy manufacture, low weight, versatility in size

and shape, light weight among other properties. However, plastic materials due to their peculiar morphology

allow it mass transport of low molecular weight compounds such as permeant gases, water vapor, odors,

plastic residues, and additives within the environment/package/food system [2].

Many studies have been focused on the characterization and understanding of this process due to the

important effect of the mass transport in the quality of food.

The classical macroscopic mathematical models developed to describe the transport of water vapor and gases

through polymeric films have been described in terms of the solubilization-diffusion mechanism, governed

by both thermodynamics and kinetics factors [3-5]. Most mathematical models developed to describe those

phenomena are based on modifications of Fick’s law in a continuous approach and the transport parameters

such a diffusivity or permeability are the result of the process history. In this approach, as mass transfer

process advance it is impossible to obtain fluid distribution at pore level and a complete comprehension of

the involved transport mechanisms is still far from satisfactory.

There is another approach to solve the problem, the microscopic models. These kinds of models are based on

transport properties and physical characteristics at pore scale. Pore-level models have not yet used to model

the transport phenomena at nanopore-scale but they have been used to model the vapor water condensation

and transport phenomena at micropore-scale, mainly in oil reservoirs [6-7].

Pore network models are a suitable tool to understand the role of the pore structure on the transport

parameters. In-situ condensation is an important phenomenon that occurs at a pore-level when a gas phase is

present in the porous structure. During capillary condensation the pores are partially or completely blocked

by the condensed phase preventing the flow of non-condensed gases through the polymer. Therefore

condensable gases can be transported through porous media as a gas or liquid. Under certain conditions both

phases could be present, rendering the quantitative description of the transport a challenging problem. The

phenomena are not fully understood despite numerous studies published on gas transport by capillary

condensation, and there is a lack of agreement on how to predict the permeability if capillary condensation

occurs in small pores [8].

The objective of this study was to develop a 3D pore scale model for the transport of liquid water, water

vapor and oxygen at pore-level in polymeric films. A network model was selected to represent the pore space

and the model was solved using a Monte Carlo method based on a previous model by Bustos & Toledo [7].

In polymeric films, the water-vapor condensation and the gases transport occurs at the nanopore scale rather

than at the micropore scale, as in the Bustos & Toledo model [7]. This distinction is very important because

at the nanopore scale it is possible to find some transport mechanisms and condensation effects that are not

significant at the micropore scale, such as Knudsen diffusion and disjoining pressure.

MATERIALS & METHODS

A three-dimensional cubic network of pore segments represents porous media. Nodes at which the pore

segments are connected act only as volumeless junctions with infinite conductance. Pore segments are

rectilinear with polygonal cross sections circumscribing circles with distributed radii. Figure 1 displays a

typical portion of the pore network and corresponding parameters. Pore segment radii rt are randomly

assigned according to Log- Normal distribution function for chitosan film [9]. According to Assis & Da Silva

[10], the film throats radii range from 5 to 30 nm. The mean pore radius is 6.425 nm, with the standard

deviation of 1.825 nm.

Figure 1. Pore network (cross section) and parameters: rt represent the radii of the pore throats; L is the pore-body-center

to pore-body-center distance, Lb is the pore-body-side half-length and Lt is the pore-throat length, 1 and 2 are pore center

(nodes).

The water vapor and oxygen transport process is considered under isothermal condition. The gravitational

effects are negligible. As vapor flows into the polymeric film, condensation of water vapor occurs at pore

walls of the network. Liquid in pore corners allows hydraulic connectivity throughout the network at all time

and capillary pressure (Pc) is determined by augmented Young-Laplace equation (Eq.1).

( ) γHhPc 2+Π≡ [1]

Where Π(h) is the disjoining pressure, H is the mean curvature and γ is the interfacial tension. Using Monte

Carlo simulation, we find pore-level distribution of liquid (condensate) and vapor as transport phenomena

occurs and effective water vapor and oxygen diffusivity and water vapor permeability are determined. A

modified form of Poiseuille’s law defines the gas conductance, g, for each pore segment, i.e., pgQvo ∆⋅= ,

where voQ is the volumetric flow of gas (water vapor or oxygen) through the pore space and p∆ is the

pressure drop across the pore; g depends on the configuration adopted by the fluid phase [9, 11-12]. For any

given gas-condensate capillary pressure each phase develops its own flow network to which conductance can

be assigned in much the same way as for single-phase flow. Several approaches are available for computing

the pressure fields in either phase once the pore-level saturations are established. Here we used the direct

solution of performing a nodal-material balances for each phase (see, for instance, [7, 9, 13]). A nodal-

material balance for each phase leads to a system of linear equations of the form G·p=B, where G is a matrix

of conductances, p is a vector containing the unknown pressures and B is a vector dependent on the pore

pressures at the upper and lower boundaries of the network and the conductances of the throats connected to

these boundaries [7]. To find the distribution of nodal pressures in each flow network once an external

pressure gradient was imposed, we used an iterative solution to the system of equations. The system was

Liquid

βL

L

2rt 1

L

βL

2

Solid

Gas

(1-β)L/2

2rt (1-β)L/2

(1-β)L/2

optimally stored and solved with a conjugate-gradient method with successive overrelaxation. This method is

part of the ITPACK routine libraries, which are publicly available at the web site

http://rene.ma.utexas.edu/CNA/ITPACK. The relaxation parameter was chosen as 1.84. With the nodal pressure of a given flow network in hand, the flow rate everywhere was calculated and the

network conductance computed for the water-vapor and oxygen diffusivities from (Eq. 2) [14]:

vo N

vo

vo

Q LRTD

A M p

ρ =

∆

[2]

where Dvo is the water-vapor or oxygen effective diffusivity (m2/s), Qvo is the water-vapor or oxygen flux

(m3/s), (∆p/LN)vo is the pressure gradient across the network on the gas phase (Pa), A is the total network

cross-section (m2), R is the ideal gas constant (J/mol K), T is the system temperature (K), ρ is the gas density

(kg/m3) and M is the molecular weight of the gas (kg/mol). On the other hand, the liquid diffusivity (DL) is

obtained from the equation 3 [15]:

−=

l

cH

LdS

dPkD

µ

[3]

Where kH is the liquid permeability, µ is the liquid viscosity, and c ldP dS is the change of capillary pressure

with the network liquid saturation (Sl). The liquid permeability is computed from the Darcy’s law (Eq. 4):

pA

QLk N

H∆

=µ

[4]

where Q is the liquid flux (m3/s). For flow of gases, since the volumetric flow rate varies with pressure, it is

necessary to use either an integrated form of the equation or alternatively an average value of the flow rate. If

an average pressure is used, the volume at mean pressure has to be converted to a volume measured at one

atmosphere, so that Darcy's law may be expressed as [16]:

( )22

2

oi

ogvo

voPPA

PLQk

−=

µ

[5]

where kvo is the water-vapor or oxygen effective permeability (m2), Pi is the water-vapor or oxygen inlet

pressure of the network (Pa), Po is the water-vapor or oxygen outlet pressure of the network (Pa) and µg is the

gas viscosity (cP)

The mathematical model was developed using Fortran (Compaq Visual Fortran 6.6). All the

graphics results were obtained using SigmaPlot 9.0. The reported simulation results correspond to 95%-

confidence intervals around the mean of ten repetitions of each pore-size distribution. The 3D graphs were

made using the software Noesys 1.3 and T3D.

RESULTS & DISCUSSION

The mechanistic pore-level model of the transport of water vapor and oxygen is used here to find pore-level

distributions of the gas and liquid (condensate) and water vapor and oxygen permeability and diffusivity. Our

simulations results are compared with experimental results for water vapor and oxygen diffusivity given by

Del Nobile et al. [3-5], Buonocore et al. [17] and Van Krevelen [18].

Figure 2, shows images of liquid-gas patterns as the condensation occurs. At each condensation step, we

considered that a fixed volume of 100 (nm3) per each duct was condensed (as a condensation criterion), then

as the condensate volume increased some pores were sealed against water-vapor and oxygen transport; when

the percolation threshold was reached (the fraction of connected bonds was 0.248), the network was

disconnected from the water-vapor and oxygen transport.

p=1, GC=0, Sl (%)= 0 p=0.988, GC=1151, Sl (%)=2.77 p=0.941, GC=5842, Sl (%)= 3.00

p=0.858, GC= 14037,Sl (%) =3.27 p=0.648, GC= 34843, Sl (%) =3.67 p=0.248, GC= 74520, Sl (%) = 4.51

Figure 2 Condensation sequence for a three-dimensional pore network (40×20×40) with 32,000 pores and 99,200 of pore

throats ; p is the fraction of connected bonds; GC is the number of totally condensed pore throats and Sl is the network

liquid saturation (%).

Figure 3 shows liquid water, water-vapor and oxygen diffusivity curves from the simulation in the three-

dimensional network. As condensation advanced, some pores were sealed and both water-vapor and oxygen

diffusivities were diminished. On the other hand, as condensation occurs the liquid content in the pore

structure increases and the liquid permeability increases too. Here the effect of water re-evaporation and

oxygen transport in the liquid phase were not considered.

Network liquid saturation, %Sw

2 3 4 5 6 7

Effe

ctive

Diffu

siv

ity (

cm

2/s

)

0.0

2.0e-8

4.0e-8

6.0e-8

8.0e-8

1.0e-7

1.2e-7

1.4e-7

Oxygen

Water vapor

Water liquid

Figure 3. Effective diffusivity curves for three-dimensional pore network (40×20×40) obtained in this work (mean value

of ten repetitions).

Del Nobile et al. [3] present experimental values of water vapor diffusivity for three kinds of polyamides,

those values ranges between 0.5x10-9

and 5.5x10-9

(cm2/s). Specifically for quitosan films, Del Nobile et al.

[5] present two continuum mathematical models to estimate the water vapor diffusivity in function of the

water activity (aw). Their results ranges between 2x10-10 and 2.5x10-8 (cm2/s) for an ideal Fick’s model,

whereas using another modified Fick’s model their results are nearly to 2x10-10

(cm2/s). Buonocore at al. [17]

shows experimental water vapor diffusion coefficients for quitosan and their results ranges between 1x10-8

and 8x10-8

(cm2/s) for a water activity ranges between 0.3 and 0.7. On the other hand, Van Krevelen [18]

shows that the effective oxygen diffusivity for syntetic polymers ranges between 1x10-9

and 1x10-6

(cm2/s).

From Figure 3, we observe that from our model the effective water vapor diffusivity ranges between 6.4x10-9

and 4.1x10-8 (cm2/s) and the effective oxygen diffusivity obtained through the present model gives values

between 7.2x10-9

and 3.2x10-8

(cm2/s), which is agreeable with the experimental results presented by Del

Nobile et al. [5], Buonocore et al. [17] and Van Krevelen [18]. Then we can infer that the model proposed at

present work has a good predictive capability. It is important to note that the present model is based only in

morphological information of the porous medium, physical properties of the vapor water and oxygen and in

the general transport laws.

From figures 2 and 3 also it is possible to see the strong dependency of the mass transport mechanisms with

the liquid network saturation. As condensation process advances and pore throats are sealed for the transport

of water vapor and oxygen, the Knudsen transport mechanism becomes less important and other transport

mechanism like liquid transport becomes important.

Network liquid saturation, %Sw

2 3 4 5 6 7 8

Eff

ective

Pe

rme

ab

ility

(m

2)

0

2e-21

4e-21

6e-21

8e-21

1e-20

Oxygen

Water vapour

Water liquid

Figure 4. Effective permeability curves for three-dimensional pore network (40×20×40) obtained in this work (mean

value of ten repetitions).

In figure 4 we present simulation results of water-vapor, liquid water and oxygen effective permeabilities in

units of square meter (m2). A comparison between values of permeabilities obtained at present work with the

ones present in bibliography was not possible. The values reported in bibliography for the gas permeability of

films are measured gravimetrically according to the ASTM E96-80 or ASTM E96-95 standard and adapted to

edible films. These experimental results depend of the way of the experiments were carried out and also the

particular experimental conditions, presenting varied units of measurement, e.g. cm3/mm

2•s•atm. On the

other hand, the results of permeability obtained in this work (in m2) are based in a macroscopic law of

transportation (Darcy law). Moreover, this work is pioneer because is the first pore network model applied to

a packaging food material, for which, until now scientific comparable studies do not exist.

CONCLUSION

We have developed a simple pore-scale network model in order to study the water vapor, liquid water and

oxygen transport in polymeric films used in food packing. The model is based on microscopic properties.

The macroscopic transport properties obtained by the model were compared with experimental results

obtained by specialized literature given a good agreement between both. Then we can infer that the model

proposed at present work has a good predictive capability. This simple model can reproduce the transport

parameters values without any adjust parameters and only considering physical aspects of condensation

process and morphology and connectivity of the porous medium.

ACKNOWLEDGMENTS

The financial support from CONICYT-Chile through project FONDECYT N° 11060081 is gratefully

acknowledged.

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