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1、arXiv:cond-mat/9903094v1 cond-mat.stat-mech 5 Mar 1999 MICROSCOPIC EQUATION FOR GROWING INTERFACES IN QUENCHED DISORDERED MEDIA L. A. Braunstein, R. C. Buceta and A. D az-S anchez Departamento de F sica, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, (76
2、00) Mar del Plata, Argentina Departamento de F sica, Universidad de Murcia, E-30071 Murcia, Espa na Abstract We present the microscopic equation of growing interface with quenched noise for the Tang and Leschhorn model L. H. Tang and H. Leschhorn, Phys. Rev. A 45, R8309 (1992). The evolution equatio
3、n for the height, the mean height, and the roughness are reached in a simple way.An equation for the interface activity density (or free sites density) as function of time is obtained. The microscopic equation allows us to express these equations into two contributions: the diff usion and the substr
4、atum contributions. All these equations shows the strong interplay between the diff usion and the substratum contribution in the dynamics. PACS numbers: 47.55.Mh, 68.35.Fx Typeset using REVTEX 1 I. INTRODUCTION The investigation of rough surfaces and interfaces has attracted much attention, for deca
5、des, due to its importance in many fi elds, such as the motion of liquids in porous media, growth of bacterial colonies, crystal growth, etc. Much eff ort has been done in understanding the properties in these processes 1. When a fl uid wet a porous medium, a nonequilibrium self-affi ne rough interf
6、ace is generated. The interface has been characterized through scaling of the interfacial width w = hhi hhii2i1/2with time t and lateral size L. The result is the determination of two exponents, and called dynamical and roughness exponents respectively. The interfacial width w Lfor t L/and w tfor t
7、L/. The crossover time between this two regimes is of the order of L/. The formation of interfaces is determinated by several factors, it is very diffi cult to theoretically discriminate all of them. An understanding of the dynamical nonlinearities, the disorder of the media, and the theoretical mod
8、el representing experimental results is diffi cult to arrive at due the complex nature of the growth. The disorder aff ects the motion of the interface and leads to its roughness. Two main kinds of disorder have been proposed: the “annealed” noise that depends only of time and the “quenched” disorde
9、r due to the inhomogeneity of the media in which the moving phase is propagating. Some experiments such as the growth of bacterial colonies and the motion of liquids in porous media, where the disorder is quenched, are well described by the directed percolation depining model. This model was propose
10、d simultaneously by Tang and Leschhorn 2 and Buldyrev et al. 3. Braunstein and Buceta 4 showed that the power law scaling for the roughness only holds at criticality for t L. Also, starting from the macroscopic equation for the roughness the dynamical exponent has been theoretically calculated. They
11、 found = 0.629 for the critical value qc= 0.539. In this paper, we use the TL model in order to investigate the imbibition of a viscous fl uid in a porous media driven by capillary forces. We write a microscopic equation (ME), starting from the microscopic rules, for the evolution of the fl uid heig
12、ht as function of time. 2 The ME allows us t0 identify two contributions that dominates the dynamics of the system, the “diff usion” and the “substratum” contributions. In this context we study the mean height speed (MHS), the interface activity density (IAD), i.e the density of actives sites of the
13、 interface, and the roughness as function of time. We show that the diff usion contribution smooth out the surface for q well below the criticality but enhances the roughness near the critical value. To our knowledge, the separation into two contributions for all the quantities studied in this paper
14、 and the important role of the diff usion contribution to the critical power-law behaviour has never been studied before. The paper is organized as follows. In section II we derive the microscopic equation for the evolution of height for the TL model. In section III we separate two contributions of
15、the MHS: the diff usion and the substratum one. We fi nd a relation between these contributions that allows us to write an analytical equation for the IAD. In Section IV the temporal derivative of square interface width as function of time is derived from the ME and the two contributions are identif
16、i ed. These two contributions allow us to explain the mechanism of roughness. Finally, we conclude with a discussion in Section V. II. THE MICROSCOPIC MODEL In the model introduced by Tang and Leschhorn (TL) 2 the interface growth takes place in a square lattice of edge L with periodic boundary cond
17、itions. We assign a random pinning force g(r) uniformly distributed in the interval 0,1 to every cell of the square lattice. For a given applied pressure p 0 , we can divide the cells into two groups: those with g(r) p (free or active cells), and those with g(r) p (blocked or inactive cells). Denoti
18、ng by q the density of inactive cells on the lattice, we have q = 1 p for 0 p 1 and q = 0 for p 1. The interface is specifi ed completely by a set of integer column heights hi(i = 1,.,L). At t = 0 all columns are assume to have the same height, equal to zero. During growth, a column is selected at r
19、andom, say column i, and compared its height with those of neighbor columns (i1) and (i+1). The growth event is defi ned as follow. If hiis greater than either 3 hi1or hi+1by two or more units, the height of the lower of the two columns (i1) and (i+1) is incremented in one (in case of the two being
20、equal, one of the two is chosen with equal probability). In the opposite case, hi min(hi1,hi+1) + 2, the column i advances by one unit provided that the cell to be occupied is an active cell. Otherwise no growth takes place. In this model, the time unit is defi ned as one growth attempt. In numerica
21、l simulations at each growth attempt the time t is increased by t, where t = 1/L. Thus, after L growth attempts the time is increased in one unit. In our simulations we used L = 8192 and a time interval much less than the crossover time to the static regime. We consider the evolution of the height o
22、f the i-th site for the process described above. We assume periodic boundary conditions in a one-dimensional lattice of L sites. At the time t a site j is chosen at random with probability 1/L. Let us denote by hi(t) the height of the i-th generic site at time t. The set of hi ,i = 1,.,L defi nes th
23、e interface between wet and dry cells. The time evolution for the interface in a time step t = 1/L is hi(t + t) = hi(t) + 1 LGi(hi1,hi,hi+1) , (1) where Gi= Wi+1+ Wi1+ Fi(hi)Wi,(2) with Wi1= (hi1 hi 2)1 (hi hi2) + hi,hi2/2 , Wi= 1 (hi min(hi1,hi+1) 2) . Here hi= hi + 1 and (x) is the unit step funct
24、ion defi ned as (x) = 1 for x 0 and equals to 0 otherwise. Fi(hi) equals to 1 if the cell at the height hiis active (i.e. the growth may occur at the next step) or 0 if the cell is inactive. Fiis the interface activity function. Gitakes into account all the possible ways the site i can grow. The hei
25、ght in the site i is increased by one with probability 1.1if j = i + 1 and hi+1 hi+ 2 and hi hi+2, 4 2.1/2if j = i + 1 and hi+1 hi+ 2 and hi= hi+2, 3.1if j = i 1 and hi1 hi+ 2 and hi hi2, 4.1/2if j = i 1 and hi1 hi+ 2 and hi= hi2, 5.1if j = i and hi 0, as we shall see bellow. Using the Eq. (4), the
26、IAD is f = ph1 Wii + hFiWii .(5) Figure 2 shows both sides of this equation as function of time showing that Eq. (4) holds. Notice the similarity between Eq. (3) and Eq. (5). Figure 1 shows the diff usion and the substratum contributions as a function of the time for various values of q. At the init
27、ial time dh/dt = f = p.In the early time regime the substratum contribution dominates the diff usion one, because 1 Wiis very small. The substratum contribution dominates the behavior of f and dh/dt in the early regime.As growth continues, the probability that growth will occur by diff usion becomes
28、 larger; the diff usion contribution increases and the substratum one decreases. This can be explained heuristically: inactive sites generate a diff erence of heights greater than two between any site and his neighbor, enhancing the growth by diff usion. As time goes on, long chains of pinned sites
29、are generated, slowing down the diff usion contribution and hence the substratum one. For q qcthese contributions, which in turn dominate, saturate to equilibrium in the asymptotic regime; while, for q qc, both contributions go to zero because the system becomes pinned. At the critical value both co
30、ntributions gives rise to a power law in the IAD and the MHS. Notice that only at the critical value does a power-law scaling holds for the MHS (see Figure 3), which contradicts 2. This was shown for the roughness by Braunstein and Buceta 4. 6 IV. ROUGHNESS From the Eq. (1), the temporal derivative
31、of the square interface width (DSIW) is: dw2 dt = 2h(hi hhii)Gii .(6) Replacing Gifrom Eq. (2), the DSIW can also be expressed by means of substratum and diff usion additive contributions. The diff usion contribution is 2h(1 Wi) min(hi1,hi+1)i h1 Wiihhii ,(7) and the substratum contribution is 2hhiF
32、iWii hhiihFiWii ,(8) where the relation (xx)+(xx)(xx ) = 1 has been used to derive the diff usion contribution. In Figure 4 we plot both contributions as a function of time for various values of q. At short times, the diff usion process is unimportant because h is mostly less than two. As t increase
33、s, the behaviour of this contribution depends on q. Notice, from Eq. (7), that the diff usion contribution may be either negative or positive. The negative contribution tends to smooth out the surface. Figure 4 shows that this case dominates for small q. The positive diff usion contribution enhances
34、 the roughness. This last eff ect is very important at the critical value. At this value, the substratum contribution is practically constant, but the diff usion contribution is very strong, enhancing the roughness. This last contribution has important duties on the power-law behaviour. We think tha
35、t it is amazing how the diff usion plays a dominant role in roughening the surface. To our knowledge the strong eff ect on the roughness, at the criticallity, of the diff usion contribution has never been proven before. Generally speaking, the substratum roughens the interface while the diff usion f
36、l attens it for small q, but the diff usion also roughens the interface when q increases. The diff usion is enhanced by substratum growth. The growth by diff usion may also increase the probability of substratum growth. This crossing interaction mechanism makes the growth by diff usion dominant near
37、 the criticality. 7 V. CONCLUSIONS We wrote the ME for the evolution of the height in the TL model. The ME allows us to separate the substratum and the diff usion contributions and to explain the great interplay between them.We found that both contributions to the MHS are related in simple way.We fo
38、und an amazing numerical result that allows to derive the IAD in a simple way. The analytical prove of this numerical result is still open. All the quantities studied shows, the strong interplay of the diff usion and the substratum contribution in the dynamics. The substratum growth enhances the dif
39、f usion; increasing the growth by diff usion may increase the probability of substratum growth, and vice versa. This crossing interaction mechanism makes both dominant at the criticality. The diff usion contribution of the DSIW shows diff erent behaviour depending of q. In the intermediate regime, w
40、hen q is small, this contribution is negative, smoothing out the surface. It is astonishing that as q increases the contribution became positive, roughing the surface. Finally, we are sure that other DPD growth models would permit separation in two contributions with the same features of the TL mode
41、l. ACKNOWLEDGMENTS A. D az-S anchez acknowledges fi nancial support from the INTERCAMPUS E.AL.96 Program to attend to UNMdP. 8 REFERENCES E-mail address: lbraunsmdp.edu.ar 1 Family F 1986 J. Phys. A: Math. Gen. 19 L441; Kardar M, Parisi G and Zhang Y C 1986 Phys. Rev. Lett. 56 889; Horvath V and Sta
42、nley H E 1995 Phys. Rev. E 52 5166; Nielaba P and Privman V 1995 ibid. 51 2022. 2 Tang L H and Leschhorn H 1992, Phys. Rev. A 45 R8309. 3 Buldyrev S V, Barab asi A -L, Caserta F, Havlin S, Stanley H E and Viscek T 1992 Phys. Rev. A 45 R8313. 4 Braunstein L A and Buceta R C 1998 Phys. Rev. Lett. 81 6
43、30. 5 Braunstein L A and Buceta R C 1996a Phys Rev E 53 3414; Braunstein L A and Buceta R C 1996b ibid. 54 6125. 6 Yang J and Hu G 1997 Phys. Rev E 55 1525. 7 To compute the Eq (3) or any equation derived from the ME we froze the simulation at a time t. For this confi guration we compute for the i-t
44、h site all the contributions to the growth of this site in the next time t + t without changing the confi guration. Then, we average over the lattice and over realizations. This technique has been employed in other systems, see 5 9 FIGURES FIG. 1. ln-ln plot of hFiWii (2) and h1 Wii (?) versus t. Th
45、e parameter q is (A) 0.51 (B) 0.539 (C) 0.6. FIG. 2. ln-ln plot of f/p versus t. The symbols show the right-side of Eq. (4) (in units of p) compute as is explained in reference 5. The full curve shows the left-hand side of the same equation (in units of p) where f = hFii. The parameter q is 0.51 (2)
46、, 0.539 (?) and 0.6 (). The critical case shows that the IAD goes as twith 0.40. FIG. 3. ln-ln plot of p1dh/dt versus t. The parameter q is 0.51 (), 0.539 (?) and 0.6 (). All cases shows the same behavior in the early time regime. The subcritical case shows that the MHS asymptotically goes to certain constant. The critical case shows that the mean height goes as t. The supercritical case shows that the mean height is asymptotically constant. FIG. 4. DSIW (full curve), and its diff usion (?) and substratum (2) contributions versus lnt; for q equal to 0.3 (A), 0.539 (B) and 0.6 (C). 10