Langmuir 1994,10,4698-4702
4698
Random Walk in Cellular Media Andrey Milchev,*?+Victor Perera,$>$ and Victor Fleurov” Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1040 Sofia, Bulgaria, Institut fur Physik, Universitat Mainz, 0-55099Mainz, Federal Republic of Germany, and Beverly and Raymond Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978,Israel Received April 11, 1994. I n Final Form: September 19, 1994@ We introduce a model for diffusion of tracer particles in cellular media in which the walls of a cell are characterized by stronglyreduced permeability. Our analytical results, confirmed also by extensiveMonte Carlo simulations, reveal several distinct regimes of diffusion behavior in time whereby an initially normal diffusion at very short times turns into transient one at a characteristic crossover time ts and later, after , back to normal. At fured permeabilityp of the a period marked by another characteristic time, t ~returns cell walls we find that these crossover times scale as ts = L2 and t~ = L with the size of the cells L whereas for L = constant one has t~ = p - l . Our results for the frequency-dependentconductivity 40)show that at low frequency the real and imaginary parts of d w ) vary as w2 and w , respectively, while saturating at By measurement of the dc and ac conductivity of charge carriers, it appears constant values for w possible to determine both the size of the cells and the permeability of their walls.
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00.
One of the most interesting features of stochastic transport predicted and observed in nonhomogeneous media is the (anomalous) non-Gaussian character of diffusive The problem is also relevant for a number of applications, for instance, the ability ofbarrier materials to drastically resist the gas flux through them, the capacity of membranes for the separation of gases, etc. The transport of small molecules through cellular media could be ascribed to related problems of significant technological importance for a number of new materials, such as zeolites, soap froths, biological tissues, etc. Apart from peculiarities of periodicity structure and form of the cells, one can view such cellular media as been composed by a more or less regular array of cavities, separated by walls. One may consider these cells as filled by a homogeneous material so that a tracer molecule performs a conventional random walk before hitting the walls of the cage which are characterized by some (reduced) permeability. The random walks are thus constantly disrupted by interaction of the carrier with the cell walls Bulgarian Academv of Sciences.
* U n g e r s i t a t Mainz: +
5 Permanent address: INTEQUI-Universidad Nacional de S a n Luis-Chacabuco y Pedernera, 5700 San Luis, Argentina. Tel Aviv University. @Abstractpublished in Advance ACS Abstracts, November 1, 1994. (1)Haus, J. W.; Kehr, K. W. Phys.Rep. 1987,150,263. Kehr, K. W.; Richter, D.; Swendson, R. H. J . Phys. (Paris) 1978, 8, 433. ( 2 ) Havlin, S.; Ben-Avraham, D. Adu. Phys. 1987, 36, 695. ( 3 )Alexander, S.; Orbach, R. J . Phys. Lett. 1882, 43, L625. BenAvraham D.; Havlin, S. J.Phys. A 1982, 15, L691. (4) Bouchard, J. P.; Georges, A. Phys. Rep. 1990, 195, 127. (5) Limoge,Y.;Bocquet, J. L.Actu.MetuZZ.1988,36,1717. Kirchheim, R.; Stolz, U.ActuMetuZl. 1987,35,281. Limoge,Y.;Bocquet, J. L. Phys. Rev. Lett. 1990, 65, 60. (6) Avramov, I.; Milchev, A.; Argyrakis, P. Phys. Rev. E 1993, 47, 2303. (7) Sapag, K.; Pereyra, V.; Riccardo, J. L.; Zgrablich, G. Surf. Sci. 1993,295, 433. (8)Bulnes, F.; Sapag, K.; Riccardo, J. L.; Pereyra, V.; Zgrablich, G. J . Phys.: Condens. Mutter. 1993, 5, A223. (9)Arinshtein, A. E.; Moroz, A. P. Sou. Phys. JETP 1992, 75, 117. (10) Avramov, I.; Milchev, A. J.Non-Cryst. Solids 1988, 104, 253. (11)Guzev, A.;Arizzi, S.; Suter, U.; Moll, D. J.J.Chem. Phys. 1993, 99, 2221. Guzev, A.; Suter, U. J . Chem. Phys. 1993,99, 2228. (12) Muller-Plathe, F.; Rogers, S. C.; van Gunsteren, W. F. Chen. Phys. Lett. 1992, 199, 237. ~~
~
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Figure 1. Schematic representation of a cellular medium in 1D.
and, as shown below, for certain values of the cell size L and the permeability of the walls p, the diffusive motion presents a crossover between two well-characterized regions. Thus during transient periods of time which may last many decades, a periodical system exhibits all characteristic features of transport in amorphous materials. The cellular medium in our model is characterized by a regular array of cells which form an infinite lattice. The linear size L of these cells is given by the number of sites (potential wells) a t which a tracer particle may reside before jumping to a nearest neighbor site, separated from this by an energy barrier. We consider two possible values for the energy barriers between sites (a dichotomicbarrier model13)so that the jump probability to overcome a barrier also has two values, r a n d r, (see Figure 1).We normalize for simplicity the values of the jump rates to that of the smaller barrier, then r = 112 and r, =pIZ, where Z is the number of nearest neighbors a n d p denotes the jump rate a t which the particles overcome the higher barrier, r,. The transition rate between sites within a cell is then given by r while the rate of crossing the cell borders by r,. Since we generally assume that the cell walls are much more difficult for the tracer to penetrate than the space within the cells, we study the case r, = 3 2 p + l
-[
1
2(1
- e-(1+2p)t/2
3pt (17) (1 2p) For larger system we obtain the time dependence of (x2(t)) numerically. It is seen from eqs 16 and 17 that the deviations from purely diffusive behavior, represented by the first terms in the right-hand side of eqs 16 and 17, vanish exponentially with time, provided p < 1. The relaxation times are indeed given by sl. Forp = 1one has r = r,, i.e. the cell borders are equally permeable as the inner space and diffusion is normal. In the coefficient a t the linear terms in the rhs of eqs 16 and 17 one easily recovers the general result (11). The diffusion coefficient D(w) a t frequency w is given by the following equation15 +
(15)
For the time dependence of the MSD there follows for L =2
+
D ( w ) = -cu2hw(x2(t))e-iWt dt
(18)
and related to the conductivity d w ) by the generalized Einstein relation
(19) where. I is the density of effective carriers of charge e. The conductivity d w ) depends only on equilibrium properties of the system in the absence of an applied electric field. If we insert eq 16 and eq 17 into eq 18,we obtain D(w) for L = 2, 3 etc., as for L = 2
+
(13)
(x&)) =
for L = 3
+ u2
w(p i 2[w2 (1+PI2] (20) and for L = 3
i
w(p - 112 (21) 3[w2 (1 2pl2/41
+ +
This frequency-dependence of D underlines the distinction of our model from the case of normal diffusion where D(w)= constant. Again the condition for this i s p < 1.In Figure 2 we show ReD(w) and ImD(0) for L = 2, 3, 4. Evidently, with growing w the real part of the diffusion coefficient initially increases quadratically with w and then a t high w saturates at a new value, different from the low-frequency limit. It is interesting to note that this frequency dependence, R e d w ) = w 2 , is similar to the dominant behavior obtained in the case of electrons hopping between localized eigenstates in a disordered system a t very low temperatures.16-18 The imaginary part changes linearly with w a t low frequencies and goes as a t high frequencies thus vanishing in the limit w 00. In between, ImD(w)has a maximum; cf. Figure 2, which gets sharper with growing L. It can be inferred from eqs 20-21 that the exact position of this peak is determined by w = SI. A general expression for the low- and high frequency D(w1for arbitraryL andp may be derived even more easily
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(15) Scher, H.; Lax, M. Phys. Rev. B 1973,7 , 4491. Heinrichs, J. Phys. Rev. B 1980,22,3093. (16) Odagaki, T.; Lax, M. Phys. Reu. B 1981,24,5284. (17)Thouless, D. J. Phys. Rep. C 1974,13, 93. (18)Kaner, E. A.; Chebotarev, L. V. Phys. Rep. 1987,150, 179.
Langmuir, Vol. 10,No. 12, 1994 4701
Random Walk in Cellular Media 0.8
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