7.3.8 Diffusion in fractal matrices
In section 5.3 we focussed the attention on some initial and boundary conditions (finite release environment volume, partitioning, initial solute distribution and presence of a stagnant layer) that are possible cause of macroscopic (or apparent) non fickian release. In all cases, however, Fick’s law (eq.(5.38)) holds at the microscopic level, i.e. solute flux is equal to the product of the diffusion coefficient and the concentration gradient. Inside an infinite delivery system, this is equivalent to say that the mean square displacement () went through by a diffusing solute molecule after avery long time (t → ∞) is proportional to t or, equivalently, [64]. Notably, thisis no longer true in the case of fractals, objects with a very peculiar topology characterised by a high disorder degree where closed loops and dead ends hindersolute diffusion [65]. Theword fractal means fraction form and derives from the Latin verb frangere (frango, frangis, fregi, fractum, frangere) meaning tofracture orto break.Mandelbrot observedthat the form of fractal objects does not depend on the scale (magnification) at which they are observed. Indeed, the shape of a fractal object is given by assembling smaller objects having the same shape of the fractal object. In turn, this smaller objects are constituted by much smaller objects sharing the same shape with the fractal. This is the reason why fractal objects are said autosimilar (from the Latin words auto = self + similes = same).The consequence of auto-similarity is that, roughly, regardlessof the magnification used, a microscopic analysis of a fractal object wouldalways reveal the same images. Although only theoretical fractals are trulyautosimilar (i.e. autosimilarity holds at every length scale), naturally occurring forms show autosimilarity only in a certainrange. One of the best example of natural fractal we have ever seen is the particular kind of cabbage, very common in Italy (broccolo romanesco – Brassica oleracea italica), depicted in Figure 5.7. It is evident that the conical structure of the cabbage is built up by smaller cones helicoidally arranged on cone surface. In turn, each cone is constituted by even smaller cones arranged in the same manner. Obviously, the autosimilar character of the cabbage is limited in the length rage spanning, approximately, from 1 mm to 10 cm. By means of the fractal geometry it is possible totheoretically recreate naturally occurring forms (coastlines, clouds, mountains,trees, and so on) with an incredible grade of similitude, and that iswhy fractal geometry is also called the geometry of nature. The essence of fractals topology lies inthe fractal dimension df that always differs from the dimension nof the Euclidean space (1, 2 or 3) into which the fractal is embedded. Since a rigorous mathematical definition of df for a set of points contained in one-, two- or three –dimensional Euclidean space is out of the scope of this chapter, we would like to focus the attention on the physical meaning of df recurring to a famous experiment by Richardson who was asked to measure the length of Great Britain coastline. Basically, he presumed that for a sufficiently small measure unit (Km, hm, dam, m, cm, mm, mm, ….) the coastline length L would have been independent on . On the contrary, he found that when , L was still increasing. In the bi-logarithmic plane Log(L) vs Log(), this means that the following relation holds:
m < 0(5.95)
If the coastline were a Euclidean object, m would be zero and the coastline length would be equal to . Thus, m must be connected to the fractal character of the coastline and the simplest relation connecting m to df and to the dimension n of the Euclidean space containing the fractal object is m = n- df (obviously, in the particular case of the coastline, a one dimensional object, we have m = 1 – df ). Finally, Eq.(5.95) can be re-written to give the correct mathematical definition of df:
(5.96)
where N() is the number of measure units (segments, squares,cubes, or iper-cubes in one-, two-, three-, or n-dimensional Euclidean space, respectively) necessary to give the measure L.
Interestingly, due to the high disorder degree, the diffusion of a solute molecule inside a fractal object does not follow Fick’s law in the sense that the mean displacement went through by the solute molecule after avery long time (t → ∞) is not proportional to t0.5.In particular, solute diffusion is simultaneously ruled bydfand by the fracton dimension ds, connected to the probability p(t) of a random walker returning to its original position after a very long (infinite) time t [64]:
(5.97)
If inside an Euclidean object ds = df = n, in fractal objects we have: dsdf n. Thus, inside a fractal object, the mean square displacement is ruled by the following law:
(5.98)
beingdwthe so called random walk dimensionand its value is usually > 2 [64]. It is evident that, for Euclidean objects, dw = 2.
All these considerations make clear that the description of diffusion in fractal media can be undertaken provided that medium topology is properly accounted for. Of course, different possibilities exist. One of them consists in adopting the classical differential mass balance (see paragraph 4.3.1) assuming that both diffusion coefficient D and fractal object differential volume dV depend on d. In particular, remembering that, by definition, , it follows and, in the light of eq.(7.224), we finally have:
(7.225)
Eq.(7.225), showing that D decreases with r, reflects eq.(7.223) content concerning the reduction of fractal media density with r. In addition, remembering that, for Euclidean (i.e. not fractal) one (n = 1), two (n = 2) or three (n = 3) dimensions objects the length, surface or volume differential is proportional to , for fractal statistical radially symmetric objects we can assume [254]:
(7.226)
where l is a prefactor related to Mandelbrodt’s definition of lacunarity [252]. Eq(7.226) is nothing more that the Euclidean approximation for the volume scaling of a fractal structure and, thus, V(r) has the dimension [length]d. Relying on eqs.(7.225) and (7.226), O’Shaugnessy and Procaccia [255] get a macroscopic model for diffusion in fractal structures:
(7.227)
Assuming an infinite medium and an initial concentration distribution showing a -Dirac form localized in r = 0, eq.(7.227) solution reads:
(7.228)
where C0 is a normalisation constant depending on initial conditions [105].
Another possibility for the solution of diffusion in fractal media is to consider the traditional Euclidean mass balance (paragraph 4.3.1) and imposing that diffusion can take place only in some space zones (those belonging to the fractal structure) while others are forbidden. At this purpose, the use of percolation lattices is very useful. A percolation lattice is a spatially uncorrelated random distribution of empty and occupied sites characterized by a percolation probability p representing the frequency of empty sites (sites were diffusion can occur). Percolation theory [103] affirms that there exists a critical percolation probability pc above which transport is allowed. In other words, for p≥ pc, there exists a connected cluster of empty sites (the so called infinite cluster) so that a diffusing molecule can cross the entire lattice (it can be verified that, for two dimensional square lattices, pc ≈ 0.593 [103]). Interestingly, just above pc, a highly heterogeneous complex structure showing fractal properties for what concerns diffusion takes place. Accordingly, once the infinite cluster has been built up by means of a proper algorithm [256], in the case of two dimensional lattices, mass balance equation:
(7.229)
can be numerically solved only on empty sites. This implies eq.(7.229) discretisation on the lattice (now thought as a grid of mesh size L and composed, respectively, by h and k sites (squares) in the X and Y direction) according, for example, to the following explicit algorithm [257]:
(7.230)
where t is time, t is time step, ’ = Dt/L, (i, j) are generic lattice site coordinates, I(i,j) is the set of the first four nearest neighbours of (i,j), and are, respectively, diffusing molecules concentration in (i, j) at time (t + t) and t, while is the matter flux at time t:
(7.231)
Indicating with Mthe infinite cluster, its presence on the lattice can be accounted for by the following characteristic function (i, j):
(7.232)
Empty sites (where diffusion can take place) are characterised by = 1, while forbidden sites correspond to = 0. Remembering that eq.(7.230) is defined only on empty sites and that is zero in forbidden sites, the combination of eq.(7.230) and (7.232) leads to the model final form:
(7.233)
Notably, eq.(7.233) satisfies the conservation principle, i.e
(7.234)
which is a fundamental requisite for the applicability of finite difference scheme deriving from mass conservation. The stability criteria applied to eq.(7.233) require that < 0.5 [76]. Interestingly, this approach can be easily modified to perform more sophisticated numerical techniques (e. g. implicit algorithms) and to account for more complex situations such as the presence of two phases inside M.. Indeed, in this case, the = 1 set would be composed by two sub-sets 1 and 2 characterised, respectively, by to two different values, D1 and D2, of the diffusion coefficient.
Assuming sink conditions in the release environment and a uniform initial distribution of diffusing molecules concentration, eq.(7.233) can be numerically solved to yield the results shown in figure 7.33. It is evident that for 0.7 < p < 1 a typical fickian release occurs as the dimensionless amount of drug released is proportional to the square root of dimensionless time t+. On the contrary, when p approaches pc, depends on time according to a lower (< 0.5) exponent. In particular, for p = 0.65, we have t0.38. These results evidence how anomalous diffusion in fractal, disordered, media is associated with the reduction of the exponent ruling the dependence on time.
REFERENCES
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105. Giona, M., Statistical analysis of anomalous transport phenomena in complex media. AIChe Journal, 37 , 1249, 1991.
252. Mandelbrot, B. B., The fractal geometry of nature, Freeman, San Francisco, 1982.
253. Orbach, R., Dynamics of fractal structures. J. Stat. Phys., 36, 735, 1984.
254. Giona, M., First order reaction-diffusion kinetics in complex fractal media. Chem. Eng. Sci., 47, 1503, 1992.
255. O'Shaughnessy, B. and Procaccia, I., Analytical solutions for diffusion on fractal objects. Phys. Rev. Lett., 54, 455, 1985.
256. Vicksek, T., Fractal growth phenomena, World Scientific, Singapore, 1989.
257. Adrover, A., Giona, M., and Grassi, M., Analysis of controlled release in disordered structures: a percolation model. J. Membr. Sci., 113, 21, 1996.
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