(392g) Modeling Mass Transport Properties in Binary Composite Systems

Modeling mass transport
properties in binary composite systems

Matteo
Minelli^1,2, Ferruccio Doghieri¹, John Petropoulos³

¹ Department of Chemical Engineering, Mining and Environmental Technology
(DICMA)

 Alma Mater Studiorum
- Università di Bologna

² Advanced
Applications in Mechanical Engineering and Materials Technology
Interdepartmental Center for Industrial Research, CIRI-MAM

³ National Center
for Scientific Research Demokritos, Institute of Physical Chemistry, Athens, Greece

The description of mass transport properties of low molecular weight
species in heterogeneous polymeric systems is of great importance. The
theoretical analysis of composite materials behaviors is indeed fundamental for
the proper material design of such systems for a variety of different
applications. Among the others, immiscible polymer blends as well as block
copolymers are often of interest for their ability to combine properties of two
or more materials. Semicrystalline polymers are also
systems of great industrial importance, which require a two phases modeling in
order to describe their behavior.

In recent years, the development of nanocomposite materials was employed for barrier applications by incorporating
impermeable platelets in the polymer phase. This technology is also suitable
for the design of membranes for gas separation, and to this aim, particles
selective to one or more gases, such as for instance zeolites
or metallic organic frameworks, are dispersed in the
polymer phase.

All these examples are constituted by two, or more, well
defined and well distinguished regions (of different length scale) characterized
by their value of gas permeability P_i. The resulting permeability of
the complete heterogeneous medium is then related to
these P_i values but also to the relative weight of the two phases as
well as the system morphology, i.e. the shape of the dispersed phase.

The aim of this work is to develop a valuable tool for the description
of transport properties in composite media, able to evaluate the permeability as
function of system morphology. Numerical calculations are thus employed to
model the two-phases system in a wide range of
relative fraction of the two components and varying their characteristics. Results
are then compared with existent model equations in
order to investigate their predictive ability and ranges of validity.

The investigation of overall permeability P of
a heterogeneous medium with a dispersed phase A, in a continuous phase B has
been investigated by many authors. The main parameters assumed for such modeling effort are the
permeability coefficients of the two moiety, P_A
and P_B, (assumed to be constant for a given permeating species,
independent of the concentration of the permeant),
the composition of the system, as volume fraction of A and B (v_A and v_B).
The shape of the dispersed particles is also relevant as well as their
arrangement in the composite.

One of the most widely used model for the description of mass transport
in heterogeneous systems is the relation derived by Maxwell for a diluted
dispersion of spheres [1,2]:

                                                                                     (1)

The model is based on the assumption of interparticle
distances sufficiently large to ensure that the behavior around any sphere is
practically unaffected by the presence of the others, and this is a clear
limitation on its applicability. Maxwell equation indeed has an upper v_A bound and analytical expressions are developed based on
regular lattices of congruent spheres, which can hardly extend beyond v_A≈ 0.5 [3]. On the other hand,
approximate expression based on a simple cubic (s.c.) lattice has
been shown to be useful at v_A → 1 [4,5]. Hence, it is of importance the investigation of the
behavior of a regular s.c.
lattice of congruent cubes over the full range v_A
= 0 ? 1, for a complete coverage of the relative concentrations of the two
components.

If non-isometric particles are then considered, the shape of the
dispersed domains becomes also relevant, as for the case of long fiber-like
objects or layered platelets, and the aspect ratio of such elements has to be
accounted in modeling the transport properties.

A general relation has been derived in this
respect, known as Wiener equation [5]:

                                                                                    (2)

where A → ∞ or A = 0
leads to arithmetic mean permeability (parallel resistances) and harmonic mean
permeability (series resistances) and A = 2 or 1 yields to the Maxwell equation
for spheres or long transverse cylinders, respectively. However, the complete
description of the present problem outside these known cases is still missing.

In this work, a continuous model is used to
describe the diffusion process in a heterogeneous system consisting of an
ordinate s.c. lattice of cubic (or
parallelepipeds) of B dispersed in a medium A. The numerical solution of the
problem was then approached by discretizing
the computational domain and solving Fick's law
equations with the appropriate boundary conditions with the control volume
technique [6]. This approach has been already applied to the case of ordered
and disordered dispersions of impermeable particles in a continuous media [7,8].

The modeling procedure was at first validated by comparing the
predictions given by Maxwell's equations to the results obtained from numerical
calculations on heterogeneous systems with a regular lattice of spherical
inclusion (limited to 0.50 of inclusion loading). This also allowed the
comparison of properties in heterogeous systems with particles of different
shapes.

Therefore, the idea of this works relies on
the possibility to extend Maxwell's equation to higher inclusion volume
fractions assuming a regular lattice of cubic inclusions.

To this aim, numerical calculations of mass
transport in heterogeous systems with s.c. lattice of cubes were
performed in a wide range of volumetric concentration of particles inclusions
between 0.10 and 0.90, also exploring a complete permeability ratio range (P_A/P_B
= 0, 0.01, 0.1, 10 and 100).

The analysis of the results shows that Maxwell
equation is able to describe the permeability behavior of the composite systems
in the whole range of concentration inspected and also
for all the P_A/P_B considered.

The same numerical calculations were also
performed for the case of infinitely long square rods, which reflects a 2-D geometry
with square inclusions. The obtained results were in good agreement with the
predictions given by the Wiener's equation.

On the basis of these analyses, other configurations were
explored, square rods of different (finite) lengths, and square plates of
various aspect ratio.

The results pointed out that Wiener's equation is suitable to the
description of more complicated geometries once a proper value of factor A is
given. The analysis of the investigated cases provides a simple relation between
this parameter A and particle inclusion aspect ratio. This can
also be supported by a set of numerical calculations purposely carried
out on systems at very low loadings of inclusions of different aspect ratios.

References:

[1]       R.M. Barrer,
in Diffusion in Polymers, J. Crank, G.S. Park, Eds., Academic, New      York, 1968, Chap. 6.

[2]       J.H.
Petropolulos, A comparative Study of Approaches
Applied to the Permeability of Binary Composite Polymeric Materials, J. Polym.
Sci. Polym. Phys Ed. 23 (1985) 1309.

[3]       Lord
Rayleigh Philos. Mag. 34 (1982) 481.

[4]       R.E.
Meredith, C.W. Tobias, J. Appl. Phys. 32 (1960) 1271.

[5]       D.A. de Vries, Bull. Inst. Int. Froid Annexe 115 (1952) 1.

[6]       H.K.
Hersteeg, W. Malalasekera,
An Introduction to Computational Fluid Dynamics: The Finite Volume Method, 2nd
edition, Harlow, Prentice Hall, 2007.

[7]       M.  Minelli, M. Giacinti  Baschetti, F. Doghieri, Analysis
of modeling results for barrier properties in ordered nanocomposite systems, J.
Membr. Sci. 327 (2009) 208.

[8]       M.  Minelli, M.
Giacinti  Baschetti, F. Doghieri, A comprehensive model 
for mass transport
properties in nanocompositesJ.
Membr. Sci. 381 (2011) 10.