##
278949 Modeling Mass Transport Properties in Binary Composite Systems

Modeling mass transport properties in binary composite systems

Matteo
Minelli^{1,2}, Ferruccio Doghieri^{1}, John Petropoulos^{3}

^{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]:

_{
}

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]:

_{
}

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.

**Extended Abstract:**File Uploaded

See more of this Group/Topical: Separations Division