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Fig. 3. Successive series of pinned interfaces of model A, showing the boundaries of avalanches, produced by removing a randomly chosen blocked cell from the previously pinned interface. L=400, p=0.5>p,. Correlation lengths €, and €, are the typical sizes of the avalanches.

Figure 3 Successive series of pinned interfaces of model A, showing the boundaries of avalanches, produced by removing a randomly chosen blocked cell from the previously pinned interface. L=400, p=0.5>p,. Correlation lengths €, and €, are the typical sizes of the avalanches.

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Critical exponents and percolation thresholds p*, p®, p°, for models A, B and C.
Critical exponents and percolation thresholds p*, p®, p°, for models A, B and C.
Fig. 1. Explanation of the surface growth models A, B and C in d=1+1. Cells are randomly blocked with probability p (indicated by shaded area) or unblocked with probability 1—p (indicated by a white area). Wet cells are marked by numbers indicating time step at which they have become wet. The line connecting the centers of the dry blocked cells indicates the spanning path of directed percolation that has stopped the growth. (a) Model A, proposed by Buldyrev et al. [13,14] (p, = 0.4698); (b) model B proposed by Tang and Leschhorn [21] (p, = 0.5390); (c) model C proposed by Dhar et al. [22] (p, = 0.6447). The configuration of blocked cells is the same for all three models, thus the largest cluster of model C corresponds to the largest p,. In model C diodes correspond to blocked cells resistors to the unblocked cells. The order of wetting differs from the original definition of ref. [22] in order to prevent overhangs in the moving parts of the interface. These changes, however, do not affect the shape of the pinned interface, which in both cases is the path of the bond directed percolation of diodes on the dual lattice. S.V. Buldyrev et al. | Anisotropic percolation and surface roughening
Fig. 1. Explanation of the surface growth models A, B and C in d=1+1. Cells are randomly blocked with probability p (indicated by shaded area) or unblocked with probability 1—p (indicated by a white area). Wet cells are marked by numbers indicating time step at which they have become wet. The line connecting the centers of the dry blocked cells indicates the spanning path of directed percolation that has stopped the growth. (a) Model A, proposed by Buldyrev et al. [13,14] (p, = 0.4698); (b) model B proposed by Tang and Leschhorn [21] (p, = 0.5390); (c) model C proposed by Dhar et al. [22] (p, = 0.6447). The configuration of blocked cells is the same for all three models, thus the largest cluster of model C corresponds to the largest p,. In model C diodes correspond to blocked cells resistors to the unblocked cells. The order of wetting differs from the original definition of ref. [22] in order to prevent overhangs in the moving parts of the interface. These changes, however, do not affect the shape of the pinned interface, which in both cases is the path of the bond directed percolation of diodes on the dual lattice. S.V. Buldyrev et al. | Anisotropic percolation and surface roughening
‘Thus the spanning path of directed percolation describes the final state of the yaper wetting experiment [13,14], where the wetting front is completely pinnec xy inhomogeneities in the paper. The theoretical value of a (eq. (3c)) is ir *xcellent agreement with both our simulations and our experiments [13-15]. A natural question is ““What is the dynamics of wetting?” To answer this, we tudy the dynamical behavior of the models below and above p,. Fig. 2 shows < napshot of the wetting front as it continues to propagate in the (1+ 1). limensional media when p<p,. Large sections of the interface are already sinned and the growth is occurring only in columns that contain unblockec ells on the wet boundary (shown as dark vertical lines). The average orizontal size of pinned sections is €,, while the average vertical size of pinnec ections is €,. The moving parts have constant steep slope (in model B it is ‘xactly equal to 2), and their vertical and horizontal dimensions are propor- ional to €,. As was shown in [25], the height—height correlation function (€,t), calculated for such an interface for €~€, is proportional tc 2Ca~ti2a — £97! The effective roughness exponent of this moving interfacc alculated directly from our numerical data near criticality in d=1+1 is ‘ayn = 9.70 £0.05, which is in good agreement with the above geometrical rguments. The growth is now mostly a fast erosion of steep slopes, propagat- Fig. 2. Two successive snapshots of the interface of model A still evolving near its pinning threshold (p = 0.4698). System size is 2°. (But the calculation is made for L <2'7.) Upper snapshot (a) taken at ¢ = 18000, lower (b) at ¢ = 19500. Light color indicates wet area, dark color indicates dry area, dark strips indicates “live” columns, i.e. columns that contain cells which become wet at the current time step. Arrows show the direction of the propagation of steep eroding slopes, which has approximately equal horizontal and vertical dimensions both of the order of €,. The horizontal dimension of the long blocked region is of the order of > while its vertical dimension is of the order of é,.
‘Thus the spanning path of directed percolation describes the final state of the yaper wetting experiment [13,14], where the wetting front is completely pinnec xy inhomogeneities in the paper. The theoretical value of a (eq. (3c)) is ir *xcellent agreement with both our simulations and our experiments [13-15]. A natural question is ““What is the dynamics of wetting?” To answer this, we tudy the dynamical behavior of the models below and above p,. Fig. 2 shows < napshot of the wetting front as it continues to propagate in the (1+ 1). limensional media when p<p,. Large sections of the interface are already sinned and the growth is occurring only in columns that contain unblockec ells on the wet boundary (shown as dark vertical lines). The average orizontal size of pinned sections is €,, while the average vertical size of pinnec ections is €,. The moving parts have constant steep slope (in model B it is ‘xactly equal to 2), and their vertical and horizontal dimensions are propor- ional to €,. As was shown in [25], the height—height correlation function (€,t), calculated for such an interface for €~€, is proportional tc 2Ca~ti2a — £97! The effective roughness exponent of this moving interfacc alculated directly from our numerical data near criticality in d=1+1 is ‘ayn = 9.70 £0.05, which is in good agreement with the above geometrical rguments. The growth is now mostly a fast erosion of steep slopes, propagat- Fig. 2. Two successive snapshots of the interface of model A still evolving near its pinning threshold (p = 0.4698). System size is 2°. (But the calculation is made for L <2'7.) Upper snapshot (a) taken at ¢ = 18000, lower (b) at ¢ = 19500. Light color indicates wet area, dark color indicates dry area, dark strips indicates “live” columns, i.e. columns that contain cells which become wet at the current time step. Arrows show the direction of the propagation of steep eroding slopes, which has approximately equal horizontal and vertical dimensions both of the order of €,. The horizontal dimension of the long blocked region is of the order of > while its vertical dimension is of the order of é,.
Fig. 4. Horizontal projection of the cluster of model B (p = 0.8009 = p,) which was started at the center of the screen 2"° time steps ago. The current diameter of the cluster is about 2°. The blue area shows the flat interface that is left dry since the beginning of the process. Darkest shades of gray correspond to the largest heights of the interface. Red dots forming “fractal dust’”’ indicate cells that become wet at the current time step.
Fig. 4. Horizontal projection of the cluster of model B (p = 0.8009 = p,) which was started at the center of the screen 2"° time steps ago. The current diameter of the cluster is about 2°. The blue area shows the flat interface that is left dry since the beginning of the process. Darkest shades of gray correspond to the largest heights of the interface. Red dots forming “fractal dust’”’ indicate cells that become wet at the current time step.
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