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Latin American applied research

versión On-line ISSN 1851-8796

Lat. Am. appl. res. vol.42 no.2 Bahía Blanca abr. 2012

 

A simple 2d pore-scale network model for the transport of water vapor and oxygen in polymeric films

L. A. Segura, P. A. Mayorga, J. M. Gonzalez and J. E. Paz

Food Engineering Department, Universidad del Bío-Bío, PO Box 447, Chillán, Chile.
lsegura@ubiobio.cl

Abstract — A mathematical model was developed to describe mass transport in polymeric films based on modifications of Fick's law in a continuous macroscopic approach. The purpose of this work was specifically to model the transport of water vapor and oxygen in polymeric films at the pore level, based on physical aspects of the condensation process and the morphology and connectivity of the porous medium, with two-dimensional pore networks representing pore spaces. Using a Pore scale discrete model, we found that the pore-level distributions of liquid (condensate) and vapor as transport phenomena occurred. The vapor and oxygen diffusivities ranged from 4.22×10-6 to 3.16×10-7 cm2/s and from 3.66×10-6 to 2.86×10-7 cm2/s, respectively. Also, the vapor and oxygen permeabilities ranged from 2.61×10-3 to 3.83×10-4 nD (1nD=10-21m2) and from 6.77×10-3 to 1.35×10-3 nD, respectively. The transport properties obtained by the model were compared with the corresponding results for chitosan films obtained in previous experimental studies, showing a partial agreement.

Keywords — Condensation; Diffusivity; Porous Film; Oxygen; Water Vapor.

I. INTRODUCTION

Polymers have been commonly used for food packaging due to their characteristics such as versatility in the manufacturing of different packages, low cost, light weight and chemical inertness (Del-Valle et al., 2003). It is well known that polymeric films can control the transport of low-molecular-weight compounds through the packaging, thus preserving the packaged foods during storage (Del Nobile et al., 2003a, 2003b). The barrier properties of flexible films can be measured by the use of permeability and diffusivity coefficients (water vapor or gas). These are intensive macroscopic properties that represent the overall resistance of the films offer to the passage of gas fluid (water vapor or oxygen).

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 thermodynamic and kinetic factors (Hernandez, 1994; Del Nobile et al., 2003a, 2003b, 2004). There is another approach to solving this problem, the microscopic models. These kinds of models are based on the transport properties and physical characteristics of the film at the pore-level scale such as: wettability in function of the contact angle, pore size distribution (a measure of the porosity, pores size), pore geometry (a measure of the pore shape cross-section, circular, polygonal, etc.), and topology (connectivity between the pores). The traditional mathematical models used to explain the mass transfer in polymeric films do not make difference between the two main mechanisms of mass transfer (in gas phase and in liquid phase), i.e., the models are based on Fick's law modifications with and adjust parameter variable on time. On the other hand, those kinds of models do not able to show the liquid distribution as condensation occurs during the process. The percolation model has enough skills to model the condensation in-situ, because it is possible to simulate the liquid quantity that residing into the nanopores walls in function of the morphology and topology of the porous structure and also it is possible to simulate the process evolution on the time. Pore-level models have not yet been used to model the transport phenomena at the nanopore-scale but they have been used to model water-vapor condensation and transport phenomena at the micropore-scale, mainly in oil reservoirs (Mohammadi et al., 1990; Fang et al., 1996; Li and Firozabaadi, 2000; Bustos and Toledo, 2003a, 2003b).

An important phenomenon that occurs at the pore level when a gas phase is present is in-situ condensation. When the total pressure in a porous media goes down condensation occurs and this phenomenon is named retrograde condensation. At the beginning of the condensation the gas effective permeability or diffusivity remains high, but as total pressure decreases condensate tends to fill more space within pore structure diminishing the transport properties in the gas phase (gas permeability or gas diffusivity). Once the critical condensation is reached, the transport phenomena in the gas phase become zero (Bustos and Toledo, 2003a). How abruptly it decreases depends on the pore structure, fluid properties and operating conditions. Bustos and Toledo (2003a, 2003b) studied the effects of contact-angle hysteresis, pore-size distribution and pore shape on the relative permeability of gas and condensate in two- and three-dimensional pore networks at the micropore-scale.

The objective of this work was to present a nanopore-level model of the transport of water vapor and oxygen applicable to polymeric films. We consider that the polymer is theoretically a system compose of impermeable matter through which co-exist interconnected pores (nanopores) where the diffusion mechanisms could occur. A two-dimensional 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 and Toledo (2003a, 2003b). In polymeric films, the water-vapor condensation and the water-vapor and other gas transport occurs at the nanopore scale rather than at the micropore scale, as in the Bustos model. This distinction is very important because at the nanopore scale there are transport mechanisms and condensation effects present that are not significant at the micropore scale, such as Knudsen diffusion and disjoining pressure.

II. METHODS

A. Network Model

In this first approach nanopore structure of the polymers were represented by two-dimensional square networks of prismatic pore bodies connected by narrow pore throats of rectangular cross-section circumscribing circles of given radii and depth. Usually we can describe a pore network using pore throats (narrow pores that control the transport properties) and pore bodies (large pores that control the fluid inventory of the porous media). Figure 1 displays a typical portion of the pore network and the corresponding parameters. Assis and Da Silva (2003) show experimental evidence of the nanoporous structure features of the Chitosan films. Those researches prepared chitosan films at different concentrations (3, 10, 15, 20 and 50 g/L) and then observed by Atomic Force Microscopy (AFM) fine cross-section zones of the film in order to study its porous structure. They conclude that the chitosan films obtained have a nano porous structure with an irregular topography. Also they determined that the nanoporous structure obey to a log-normal pore size distribution function.


Fig. 1: Pore network and parameters: rt values 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 (βL), 1 and 2 are pore center.

The radii of the pore throats, rt, or equivalently the throat section short-side half-lengths, are randomly assigned according to a probability-density function, f(rt). The pore-throat depth z is a constant, and the pore-body-center to pore-body-center distance, L, is a chosen constant. Pore-throat length, Lt, is constant and equal to a fraction, β, of L. Pore-body-side half-length, Lb, is constant and equal to ((1-β)L/2); body depth is also constant and equal to the throat depth. Once f(rt ) and L are defined, the porosity of the network is altered by changing the factor β (Segura and Toledo, 2005a; 2005b; 2005c; Segura, 2007; San Martín et al., 2011).

To validate the model we used the experimental results for the pore-size distribution in chitosan films given by Assis and Da Silva, 2003. The film-throat radii ranged from 2.5 to 15 nm. The mean pore radius was 6.425 nm, with a standard deviation of 1.825 nm. The pore-throat radius was randomly assigned according to a log-normal distribution function for chitosan film (Eq. 1) (Assis and Da Silva, 2003; Bustos and Toledo, 2003b):

(1)

B. Physics of Pore-level Condensation

The water-vapor- and oxygen-transport processes were considered under isothermal conditions. As vapor flows into the polymeric film, condensation of water vapor occurs at the pore walls of the network (Fig. 2).


Fig. 2: Condensate phase in a pore corner.

The accumulation of the condensate in pore corners allows for condensate connectivity throughout the network. The volume of condensate remaining in the corners of a pore segment of arbitrary polygonal cross-section is given by the general formula (Eq. 2) below (Bustos and Toledo, 2003b; Segura and Toledo, 2005a):

(2)

where Vw is the volume of condensate at corners of a pore, n is the number of sides of the polygonal cross section of the pore and rw is the radius of the curvature of the longitudinal meniscus, defined as follows:

(3)

where γ is the interfacial tension, Pcap is the capillary pressure, defined as Pg - Pc, where Pg is the pressure in the gas phase and Pc in the condensate phase (liquid), α is the corner half-angle and θ is the contact angle.

The pore volume of the network is V = (Vw +Vg), where Vg is the gas-phase volume. Liquid saturation is calculated as Sw = (Vw/V) × 100.

Derjaguin (1957) modified the traditional Young-Laplace equation, which relates interfacial tension, , mean curvature, H, and the pressure difference across an interface at equilibrium, by assuming that the disjoining pressure and capillary pressure are additive (Eq. 4):

(4)

The difference between the chemical potentials in the film and bulk states is called the disjoining potential, and the derived force per unit area is the disjoining pressure (∏ ). The disjoining-pressure contribution is significant only for thin films less than 1 μm thick (Novy et al., 1989). Disjoining thin-film forces repel the meniscus at small distances from the solid wall and prevent the meniscus from touching the pore walls. The meniscus then lies parallel to the wall with a thin liquid film in between. This would explain the high values of pressure in the system to low levels of liquid saturation, when water is found present at mono-cape level.

From a simulation viewpoint, water condensate in horizontal pore segments occurs in two steps: condensation until the capillary bridge is formed and further condensation until the condensate saturates the pore completely. The continuous process is discretized in steps, an idea first introduced by Fang et al. (1996). A small amount of condensate is added to each pore segment per step.

Using a Monte Carlo simulation, we found the pore-level distribution of liquid (condensate) and water vapor as transport phenomena occurred and the effective water-vapor and oxygen diffusivities and water-vapor permeability were determined.

As the condensation process advance into the pore structure the paths available for the gas phase transport diminish until the percolation threshold is reached, at this moment the network gas connectivity become zero, and the gas effective permeability becomes zero too.

C. Knudsen Diffusion

When a gas flows through porous media containing very small pores (size of about several nanometers), Knudsen diffusion is usually the dominant mechanism of the transport. This diffusion's type is produced because the radii of the pores are smaller than the molecular mean free path λ of the diffusant gas and a particular molecule will more often collide with the pore wall rather than with another molecule (Del-Valle et al., 2003).

From the kinetic theory of the gases, for a gaseous substance A, the Knudsen diffusion coefficient DKA in m2/s is a function of pore diameter d and the mean velocity of gas A , defined as (Youngquist, 1970):

(5)

where is the mean molecular velocity in m/s given by Youngquist (1970):

(6)

where R is the gas constant, T is the temperature in Kelvin degrees and MA is molar mass of gas A (kgmol-1).

D. Pore-level gas Conductance and Gas-Pressure Fields

A modified form of Poiseuille's law defines the gas conductance, g, for each pore segment, i.e., Qvo=gΔp, where Qvo is the volumetric flow of gas (water vapor or oxygen) through the pore space and Δp is the pressure drop across the pore; g depends on the configuration adopted by the fluid phase (Segura and Toledo, 2005a, 2005b, 2005c; Segura, 2007; San Martin et al., 2011).

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, Mogensen et al., 1999; Bustos and Toledo, 2003b, Segura and Toledo 2005a). 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 (Bustos and Toledo, 2003b).

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 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.

Boundary conditions for the partial-pressure field in the gas phase were the saturation pressure at the temperature of the system above each meniscus and the partial pressure of the evaporating species in the ambient air.

E. Effective Gas Diffusivity and Effective Gas Permeability

According to section D, 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 (Nowicki et al., 1992):

(7)

where Dvo is the water-vapor or oxygen effective diffusivity (m2/s), Qvo is the water-vapor or oxygen flux (m3/s), (Δp/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).

In the next paragraphs we will explain the contribution of the Knudsen diffusion to the overall flow of gas or vapor (Qvo). Firstly, through nodal balances applied to the network the partial pressure system both gas phase and liquid phase were determined. The diffusion coefficient through the Knudsen equation (Eq. 5 and 6) for the gas phase for each nodal balance were determined, i.e., the Knudsen coefficient diffusion was computed for each pore of the network through the Fortran code developed in this work, and obviously those values depended of the mean free path and molecule velocity.

It is important to establish that previously to compute the Knudsen coefficient diffusion the Knudsen number was determined by,

(8)

where kB is the Boltzman constant, di is the pore diameter; λ the mean free path, σii is the collision diameter and p the system total pressure (Cussler, 1984). As a numerical example, for water vapor and for a pore diameter of 10 (nm), the Knudsen coefficient is 64.4, such as the mean free path is greater than the pore diameter, we must use the Knudsen equation in our work. Finally the values of Qvo were determined as a sum of volumetric flow in a cross-section area of the network. For details see Segura and Toledo (2005a, 2005b, 200c), Bustos and Toledo (2003a, 2003b) San Martín et al. (2011).

The permeability of a porous medium was evaluated from the definition of Darcy's law. For flow of gases, since the volumetric flow rate varies with pressure, it was 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 (Scheidegger, 1974):

(9)

where kvo is the water-vapor or oxygen effective permeability (nD), 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).

F. Mass transfer coefficients in the polymer-air interfaces.

In order to determine Qvo also it was necessary to compute the mass transfer coefficients in the polymer-air interfaces. Kmsup and Kminf correspond to the mass transfer coefficient to the external layer of the polymeric film (internal and external respectively). It is important to clarify that the molecular diffusion coefficients were only used to compute the mass transfer coefficients (water vapor and oxygen) in the external diffusion boundary layer of the film (to see Fig. 3). From those mass transfer coefficients we compute the input and output flow on the external boundary layer of the film (internal and external). Thereafter, these parameters were used in the follow equation: Qvo=gDp, with g=Km in order to determine the water-vapor or oxygen flow.


Fig. 3: Diagram that represents the interface polymer-air.

III. RESULT AND DISCUSSION

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 five repetitions of each pore-size distribution. Also we used the software Transform 3.3, included in the Noesys 2.4 to obtain the condensation sequences images (Fig. 4).


Fig. 4: Condensation sequence for a two-dimensional pore network (150×150) with 22,500 pores and 45,300 of pore throats ; gas phase (white), liquid phase (black); p is the fraction of connected bonds; CT is the number of totally condensed pore throats and Sw is the network liquid saturation (%).

A mechanistic pore-level model of the transport of water vapor and oxygen was used here to find the pore-level distributions of the gas and liquid (condensate) and water-vapor and oxygen permeabilities and diffusivities. The networks were arranged horizontally, thus gravity was not a factor. Parameters used in the simulation are given in Table 1. The physical properties of water and oxygen are given in Table 2 and Table 3, respectively.

Table 1: Pore-network parameters.

Table 2: Water properties at 293.15 K and atmospheric pressure (1.013×105 Pa) and boundary conditions.

Table 3: Oxygen properties at 293.15 K and atmospheric pressure (1.013×105 Pa) and boundary conditions.

Topological characteristics of Chistosan films have not been reported in the specialized literature. On the papers reviewed we only have found information about the pore size distribution obtained from image analysis that show the chitosan films such as nanoporous material with irregular topography (Assis and Da Silva, 2003). By the moment, we have not found information about coordination number and interconnectivity. Nevertheless we consider that this simple 2D network model with coordination number equal 4 is the first approach to reach the right comprehension of the problem, obviously the model must be improved and incorporate many other variables like 3D, swelling effects, etc. According to Table N°1, the network has big cubic pores with a constant diameter (195 nm). These pores are connected each other through pore throats. The radii of the pore throats are randomly assigned according to a probability-density function with values ranged between 2.5 to 15 nm. The pore throats length is a constant value of 805 nm. With those parameters, for a 100x100 network, we have 10,000 pore bodies and 20,200 pore throats. The total pore bodies and the total pore throats of the network correspond to a 96.24% and 3.76 % of the total network pore space volume respectively. From these saturation values it is clear that the pore bodies (big ones) control the porosity of the porous materials and the pore throats (less radii and longest length) control the flow properties of the system. Therefore a small inventory of liquid condensate in pore throats would produce greater changes on the transport properties of the system.

Figure 4 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 0.5), the network was disconnected from the water-vapor and oxygen transport.

In Fig. 5 we present simulation results of water-vapor and oxygen effective permeabilities in units of nanoDarcy (nD). A comparison between values of permeabilities obtained at present work with the ones present in bibliography was not possible. The values re ported 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/mm2•s•atm. On the other hand, the results of permeability obtained in this work (in nD) are based in a macroscopic law of transportation (Darcy law).


Fig. 5: Effective permeability curves for two-dimensional pore network (150×150) obtained in this work (mean value of five repetitions) for (a) oxygen and (b). water-vapor.

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.

Figure 6 shows water-vapor and oxygen diffusivity curves from the simulation in the two-dimensional network. As condensation advanced, some pores were sealed and both water-vapor and oxygen diffusivities were diminished. Here the effect of water re-evaporation and oxygen transport in the liquid phase were not considered.


Fig. 6: Effective diffusivity curves for two-dimensional pore network (150×150) obtained in this work (mean value of five repetitions) for (a) water-vapor and (b) oxygen.

To compare our simulations results with both experimental and simulated results showed in the specialized literature, we make the following analysis. Cussler (1984) reported experimental diffusion coefficients for gases in polymers and crystals on the order of 10-8 cm2/s. Karbowiak et al. (2009) presented a diffusion coefficient for water vapor in a carrageenan film of 3.2×10-7 cm2/s. Del Nobile et al. (2003a) reported experimental values of water-vapor diffusivity for three kinds of polyamides, with values ranging between 0.5×10-9 and 5.5×10-9 cm2/s. Specifically, for chitosan films, Del Nobile et al. (2004) presented two continuum mathematical models to estimate the water-vapor diffusivity as a function of the water activity (aw). Their results ranged between 2×10-10 and 2.5×10-8 cm2/s for an ideal Fickian model, whereas using another modified Fickian model their results were closer to 2×10-10 cm2/s. Buonocore et al. (2005) reported experimental water-vapor diffusion coefficients in chitosan with results ranging between 1×10-8 and 8×10-8 cm2/s for water-activity ranges between 0.3 and 0.7. It is important to say that the last two works mentioned in the previous paragraph present results of diffusivities values obtained at the same ranges of water activity that those obtained in our mathematical model results. Additionally, Van Krevelen (1990) measured effective oxygen diffusivities for synthetic polymers ranging between 1×10-9 and 1×10-6 cm2/s. From Figure 6, we observe that from our model yielded an effective water-vapor diffusivity ranging between 3.16×10-7 and 4.22×10-6 cm2/s and the effective oxygen diffusivity was between 2.86×10-7 and 3.66×10-6 cm2/s. Comparing these with experimental values, the water-vapor diffusivity values are close to range of diffusivity reported by Karbowiak et al. (2009) for carrageenan films; however, they are around of one and two orders of magnitude greater than the values reported for chitosan films by Del Nobile et al. (2003a) and Buonocore et al. (2005).

With respect to the effective diffusivity of oxygen, the model results were in good agreement with the experimental results presented by Van Krevelen (1990).

Therefore, we can infer that the model proposed in the present work has a partial predictive capability, attributed to many factors such as: the two-dimensional character of the model, the swelling, gravity effects, liquid transport mechanisms, oxygen transport in liquid phase, etc. We do not know actually how these variables could affect the predictive capability of the model. Many authors like Hernandez (1994) and Del Nobile et al. (2003a, 2003b, 2004) mentioned that some of those variables affect the transport properties in the polymeric film, but actually we do not know how those variables could affect the transport properties, i.e., we do not know the influence on each variable on the effective transport properties. Here, it is important to note that the model is based only on the morphological information of the porous medium, the physical properties of water vapor and oxygen and general transport laws.

IV. CONCLUSIONS

A pore-level model of water-vapor and oxygen transport in polymeric films intended for food-packaging applications was used to determine water-vapor and oxygen diffusivity curves and water-vapor and oxygen permeability curves as functions of liquid content. Using a simple pore-scale network model, we simulated water-vapor condensation in a polymeric film, where a fraction of the pores became sealed as water vapor condensed in the pores walls of the polymeric material.

The transport properties obtained by the model were compared with the corresponding results obtained in previous experimental studies, giving a partial agreement, i.e., a good agreement for the oxygen-diffusivity and permeability values and between one or two orders of magnitude of difference for the water-vapor values. Thus, we can infer that the model proposed in the present work has a partial predictive capability, thereafter their results not correspond to a real proof for the proposed model. On the other hand, this simple model can produce useful values of transport-property coefficients without any adjustable parameters and by considering only the physical aspects of the condensation process and the morphology and connectivity of the porous medium. The main value of this paper is to explore a new kind of model in order to try to understand the transport mechanisms at nanopore-scale during the transport of gas and water through polymeric films

It remains for a future work to explore the effects of a three-dimensional network and those of re-evaporation, liquid transport and oxygen transport through the liquid phase on the transport-property coefficients of polymeric films. Indeed, if transport mechanisms in the liquid phase are considered in the model in order to compute the transport parameters, will be possible to compare the experimental values reported in the specialized literature with the simulation results of the present model.

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

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Received: June 11, 2010.
Accepted: August 5, 2011.
Recommended by Subject Editor Walter Ambrosini.