Ballistic deposition on deterministic fractals: On the observation of discrete scale invariance

Ballistic deposition on deterministic fractals: On the observation of discrete scale invariance

Claudio M. Horowitz , Federico Romá and Ezequiel V. Albano . 1- Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, (INIFTA), UNLP, CCT La Plata-CONICET, Sucursal 4, Casilla de Correo 16, (1900) La Plata. Argentina
2- Centro Atómico Bariloche, 8400 S. C. de Bariloche, Río Negro, Argentina
September 27, 2019
Abstract

The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established Family-Vicsek dynamic scaling approach. Systematic deviations from that standard scaling law are observed, suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface. Subsequently, we study the internal structure of the growing aggregates that can be rationalized in terms of the scaling behavior of frozen trees, i.e., structures inhibited for further growth, lying below the growing interface. It is shown that the rms height () and width () of the trees of size obey power laws of the form and , respectively. Also, the tree-size distribution () behaves according to . Here, and are the correlation length exponents in the directions parallel and perpendicular to the interface, respectively. Also, is a critical exponent. However, due to the interplay between the discrete scale invariance of the underlying fractal substrates and the dynamics of the growing process, all these power laws are modulated by logarithmic periodic oscillations. The fundamental scaling ratios, characteristic of these oscillations, can be linked to the (spatial) fundamental scaling ratio of the underlying fractal by means of relationships involving critical exponents. We argue that the interplay between the spatial discrete scale invariance of the fractal substrate and the dynamics of the physical process occurring in those media is a quite general phenomenon that leads to the observation of logarithmic-periodic modulations of physical observables.

PACS numbers: 68.35.Ct, 05.45.Df, 02.50.-r, 81.15.Aa

I Introduction

The study, characterization, and understanding of growth processes that take place under far-from equilibrium conditions are topics that have attracted great attention due to their relevance in many fields of science and technology BS ; 2 ; 3 ; 4 . Growing aggregates of biological origin (bacteria, fungi, tumors, etc.), the deposition of thin films, the growth of magnetic materials, alloys and polymers, among others, have recently become the subject of extensive studies BS ; 61 ; 6 ; 8 ; 5 . Also, the dynamic evolution of interfaces is closely related to almost all growth processes BS . The characterization of the properties of interfaces has achieved considerable progress during the last two decades, mostly due to the success of the concepts of the dynamic scaling theory developed by Family and Vicsek 5b ; 5a , which allows a comprehensive description of interfaces in terms of universality classes that group systems described by the same set of physically meaningful exponents.

In order to provide a more complete description of growing aggregates, it is desirable not only to focus on the properties of the evolving interface, but also to attempt the simultaneous characterization of bulk properties. In fact, in some cases, growing processes lead to the formation of porous materials that inherently have very interesting physical and chemical properties with many potential practical applications.

Within this broad context, ballistic deposition (BD), which was originally proposed by Vold BD as a model for the description of sedimentary rock formation, has become an archetypical system for the study of growing aggregates BS . BD aggregates are characterized by a porous structure in the bulk and a rough evolving interface. The interface roughness of BD has extensively been studied by means of computer simulations and analytical approaches in connection to the Kardar-Parisi-Zhang (KPZ) theory BS ; KPZ .

In spite of the considerable effort devoted to understanding growing aggregates in Euclidean substrates BS ; 2 ; 3 ; 4 , to the best of our knowledge little attention has been drawn to the study of deposition models on fractal media. It should be expected that the interplay between the self-similarity of the substrates and the growing mechanisms would lead to the formation of interesting and complex porous (bulk) structures. Also, the fractality of the substrate would affect the self-affine nature of the growing interface. So, it would not be surprising to find nonvanishing scaling corrections to the well-established phenomenological dynamic approach of Family-Vicsek 5b ; 5a ; BS . In fact, in some pioneering works bessis ; brigita54 and in more recently ones we2 ; we ; lett it has shown that the power-law behavior of some observables becomes modulated by logarithmic oscillations due to the effect of the underlying lattice structure. Also, in the last years two interesting books have been published where this issue is addressed libro1 ; libro2 .

Within this context, the aim of this paper is to study the scaling behavior of the BD model on deterministic fractal substrates. The study is based on Monte Carlo numerical simulations analyzed by means of the Family-Vicsek phenomenological scaling approach 5b ; 5a and on extensive numerical investigations of the scaling behavior of the internal structures of the growing system RV ; Meakin0 ; zinc ; Meakin ; trees ; arpascu . For this purpose the manuscript is organized as follows: firstly, in Section II we provide a brief theoretical background on the scaling behavior of growing aggregates. The fractal substrates used for the growth of BD aggregates and the concepts of space and time discrete scale invariance are described and discussed in Section III. Subsequently, in Section IV, we describe the BD model on fractal media, as well as the determination of the internal structure of the aggregates in terms of frozen and growing trees. The results obtained by applying the standard dynamic scaling approach to the data are presented and discussed in Section V, while the analysis of the data of the internal structure of the aggregates is performed in Section VI. Finally, our conclusions are stated in Section VII.

Ii Brief theoretical background

Models aimed to describe growing aggregates may be defined and studied by means of both continuous approaches, which involve the formulation of analytical equations, and discrete lattice models, which consider the deposition of individual particles. A discrete model is defined by means of a set of deposition rules that provides a detailed microscopic description of the evolution of the aggregate. In these discrete models, the growing interface of the aggregate is described by a discrete set , which represents the height of site at time . The interface then has sites, where is the linear size and is the dimensionality of the substrate (as usual, is assumed to be an integer).

The dynamic evolution of the aggregate interface is characterized through the scaling behavior of the interface width , given by

(1)

For this purpose, the Family-Vicsek phenomenological scaling approach 5b ; 5a , which has proved to be very successful, can be written as

(2)

where is a scaling function. In fact, it may be expected that will show the spatiotemporal scaling behavior given by 5b ; 5a : for and for , where is the crossover time between these two regimes. The scaling exponents , , and are called roughness, growth and dynamic exponents, respectively. Also, different models can be grouped into universality classes when they share the same scaling exponents.

An alternative method that can also be used for the characterization of interfaces is related to the description of the internal structure of the growing system. This approach is based on the fact that any growing system can effectively be rationalized on the basis of a treeing process, i.e., any growing structure can be thought as the superposition of individual trees RV ; Meakin0 ; zinc ; Meakin ; MM . Those trees that spread out incorporating additional growing centers, e.g., capturing particles, developing new branches, are said to be alive. In contrast, other trees that may stop growing due to shadowing by surrounding growing trees are termed dead trees. The structure of dead trees remains frozen because it cannot be modified by any further growth. It is well known that for growing aggregates on substrates having integer dimension, the rms height ( ) and the rms width ( ) of dead trees of size ( is the number of particles belonging to the tree) obey simple power laws given by

(3)

and

(4)

where and are the correlation length exponents parallel and perpendicular to the main growing direction of the aggregate RV ; Meakin0 ; zinc ; Meakin ; arpascu ; MM , respectively.

By assuming both the proportionality between the correlation length perpendicular to the main growing direction and , as well as , one has that the dynamic exponent is given by Krug

(5)

Furthermore, one also expects that during the competition among trees along the evolution of the aggregate, the existence of large neighboring trees may inhibit the growing of smaller ones. This competing process ultimately leads to the death of some trees that become frozen within the underlying aggregate. These prevailing large trees continue the competition within more distant trees in a dynamic process. Since this situation takes place on all scales, it is reasonable to expect that the tree size distribution () should also exhibit a power-law behavior, so that RV ; Meakin0 ; zinc ; Meakin ; MM ; Krug

(6)

where is an exponent.

Iii The fractal substrates and discrete scale invariance

In the present paper we used Sierpinski carpets (SC) as substrates for the growth of ballistic aggregates. In fact, SC’s provide generic models for the building of both deterministic and nondeterministic fractals. In order to generate a SC embedded in dimensions, one proceeds as follows: a square is divided into subsquares, and then subsquares are deleted from the initial square ( is the number of occupied subsquares). This process is iterated in the remaining subsquares times, where accounts for the number of different generations. If the deleted subsquares are chosen in the same way in all iterations, the resulting fractal is deterministic, but if the deleted subsquares are selected at random, one generates a nondeterministic fractal. The mathematical fractal, obtained in the limit , is generically called . Also, the fractal associated with a finite number of segmentation steps is denoted by . In both cases the index refers to the topological features of the generating cell (). The size of the lattice, where the finite fractal is embedded, then is .

For deposition models grown in integer-dimensional substrates one always has that all fragments or parts of the aggregate are connected to each other through paths of nearest-neighbor occupied sites. So, it is no longer possible to have isolated fragments of the aggregate on the substrate. On the other hand, for some fractal substrates with noninteger fractal dimension, it could be possible to observe the formation of isolated fragments of the aggregate, and consequently in order to avoid this shortcoming, one has to carefully select suitable substrates. So, this point is essential for the choice of the fractals that can be used in order to study ballistic deposition on this kind of substrate. Accordingly, in this paper we use deterministic fractal substrates generated by taking and , and in order to prevent the fragmentation of the aggregates, as mentioned above, and to account for the usual requirement of periodic boundary conditions, one ends up with only three generating cells, as shown in figure 1.

Figure 1: Sketches of the three different generating cells used to grow ballistic aggregates by keeping and .

Let us now discuss on the concepts of fractal dimensionality and discrete scale invariance associated with the SC’s used. The fractal dimension characterizes the dependence of the mass (or equivalently, the number of occupied sites of the fractal) as a function of the linear size of the system, so that if we now consider an amplification of the system of size , one has

(7)

which yields the following solution

(8)

where is a constant and is a noninteger dimension known, after Mandelbrot, as the fractal dimension. By applying equation (7) to the carpets one gets , which leads to for the carpets shown in figure 1 with and .

In general, the factor in equation (7) could be an arbitrary real number, leading to continuous scale invariance. However, deterministic fractals exhibit discrete scale invariance (DSI) refsornosa , which is a weak kind of scale invariance such that is no longer an arbitrary real number, but it can only take specific discrete values of the form , where is a fundamental scaling ratio. Then, for the case of DSI, the solution of equation (7) yields

(9)

where is a periodic function of period one. Notice that for the one has that . The measurement of soft oscillations in spatial domain refsornosa is a signature of spatial DSI.

Very recently one of us found evidence of discrete scale invariance in the time domain by measuring the relaxation of the magnetization in the Ising model on Sierpinski carpets we2 ; we . Subsequently, it has been conjectured that physical processes characterized by an observable , occurring in fractal media with DSI, and that develop a monotonically increasing time-dependent characteristic length , may also exhibit time DSI lett . In fact, by assuming

(10)

where is a dynamic exponent, it can be shown that has to obey time DSI according to lett

(11)

where and are constants, and is the relevant exponent in the expected power-law behavior of the observable . So, the conjecture given by equation (11) implies the existence of a logarithmic periodic modulation of time observables characterized by a time-scaling ratio given by

(12)

see also equation (9). It is worth mentioning that in the case of growth models, equation (10) can be identified, e.g., with the time development of the correlation length along the direction parallel to the interface given by .

As follows from figure 1, in this work only fractals with finite ramification order frac are used. A finite ramification implies that the fractal structure has weak points, where only a finite number of links connect two parts of arbitrary size together.

Iv The BD model on fractal substrates and definition of its internal structure

The lattice version of BD is simple to describe: particles fall vertically onto the substrate from a random position above the surface. When a particle reaches the surface, it sticks on the first site encountered that is a nearest-neighbor of an already deposited particle. Due to this constraint the growth of an interface essentially parallel to the substrate is observed. For substrates of integer dimension, the BD model can be described by the Kardar-Parisi-Zhang (KPZ) equation BS ; KPZ , namely,

(13)

In this equation the nonlinear term represents the lateral growth or the appearance of a driven force, accounts for the effective surface tension, is the flux of incoming particles, and is a Gaussian noise with zero configurational average. In dimension, equation (13) can be solved exactly and the resulting exponents are , and . For dimensions, one still lacks an analytical solution, but according to numerical simulations one has estimations of the exponents given by and cas .

In order to describe the internal structure of BD aggregates, one considers that trees are formed by assuming that any newly deposited particle belongs to the same tree as that of the nearest neighbor particle where it is attached trees . Also, if the deposited particle has more than one nearest neighbor belonging to different trees, one of them is selected at random and the particle is incorporated into that tree. Relevant exponents (see equations (3), (4), and (6)) have already been determined yielding , , and in dimension, and , , and in dimensions (see, e.g., trees and references therein).

On the other hand, for BD growth on fractal substrates, particles are deposited randomly on occupied sites of the fractal only by following the rules described above for the case of integer-dimensional substrates. Also the building procedure employed to determine the trees used for the description of the internal structure of the whole aggregate is independent of the substrate.

For the purpose of a numerical simulation, the Monte Carlo time step (mcs) involves the deposition of particles.

V Application of the Family-Vicsek phenomenological dynamic scaling approach

Figure 2: Log-log plots of the interface width () versus time for the BD model on with . The inset shows a log-log plot of versus . The dotted line with slope corresponds to the best fit of the data.

Figure 2 shows log-log plots of versus obtained for the BD model on fractal substrates obtained by using the generating cell . For short times, say mcs, the random growth of the interface is observed because the random deposition (RD) process dominates. At this stage, correlations have not been developed yet and one has that, according to equation (2), , with , holds. During an intermediate time regime, say mcs , correlations develop since the BD process now prevails, leading to the typical growth regime , see also equation (2). At a later stage, for , correlations can no longer develop due to the geometrical constraint of the lattice size, and the saturation regime () is observed, as expected from equation (2). Here, is the crossover time between the growing and the saturation regimes of the interface width.

Figure 3: Log-log plots of the interface width () versus time for the BD model on with . The inset shows a log-log plot of versus . The dashed line with slope 0.44 corresponds to the best fit of the data.
Figure 4: Log-log plots of the interface width () versus time for the BD model on with . The inset shows a log-log plot of versus . The dashed line with slope 0.48 have been drawn for the sake of comparison.

In order to obtain the roughness, growth, and dynamic exponents, we followed the standard procedure BS , e.g., the inset in figure 2 shows a log-log plot of versus . Here, a slight but noticeable systematic upward deviation of the data is observed. However, an effective exponent () can be determined and the best fit of the data yields . Also, we determined and for the BD model on the fractal substrate obtained by using the generating cell.

Figures 3 and 4 show log-log plots of versus obtained for the BD model on fractal substrates built up by using the generating cells and , respectively. Again, the standard procedure was attempted in order to determine the roughness, growth, and dynamic exponents. Accordingly, the insets in figures 3 and 4 show log-log plots of versus . For the case of the , again, a noticeable systematic upward deviation of the data is observed, but an effective exponent () can be determined, yielding . Also, we determined and . On the other hand, for the we observed strong systematic deviations of the data that prevent the evaluation of the exponents. This results strongly suggest that in the case of aggregates grown on fractals generated by either the finite-size effect are very important and one has to introduce important scaling corrections to the standard Family-Vicsek approach, or (instead of we are not be able to identify two competing processes leading to the formation of the aggregate) eventually the onset of a crossover effect is present. The understanding/clarification of which one of the two possibilities may be true is a very difficult task, which deserves further work, and is beyond the scope of the present paper.

Anyway, the obtained effective exponents are almost the same (within error bars) for both the and the , namely , , and . It is worth mentioning that these exponents nicely interpolates between the exact values corresponding to and the best estimates reported for , namely , , and .

Vi Characterization of the internal structure of the aggregates

The dependence of the rms width and rms height of the trees forming the aggregates on the tree size , for the cases of the substrates (), is shown in log-log plots in figures 5 and 6, respectively. For the growth of the BD model on nonfractal substrates it is known that the power laws given by equations (3) and (4) hold trees , but in the case of fractal media, one also clearly observes soft oscillations with a logarithmic period, which modulate the power laws. Figures 5 and 6 also show a direct relationship between the number of observed oscillations and the generation () of the Sierpinski carpets used as substrates.

Figure 5: Log-log plots of rms width of the trees as a function of the tree size for the BD model on substrates ().
Figure 6: Log-log plots of rms height of the trees as a function of the tree size for the BD model on substrates ().

Figure 7 shows log-log plots of the tree-size distribution functions corresponding to the BD model grown on substrates (). Again, for the BD model on nonfractal substrates the power law given by equation (6) holds, but when a fractal carpet is used as substrate, a soft oscillation with a logarithmic period modulates the power-law behavior. Also, there is a one to one relationship between the number of observed oscillations and the number of generations of the carpet.

Figure 7: Log-log plots of the tree size distribution corresponding to the BD model on substrates ()
Figure 8: Log-log plots of rms width of the trees as a function of the tree size for the BD model on substrates ().
Figure 9: Log-log plots of rms height of the trees as a function of the tree size for the BD model on substrates ().
Figure 10: Log-log plots of the tree size distribution corresponding to the BD model on substrates ().
Figure 11: Schematic view of the dependence of the rms width of the trees () versus the tree size (), showing both the logarithmic period of the size of the trees (), and the logarithmic period of the rms width ().

After a systematic study, we found that the rms width, the rms height, and the tree-size distribution for BD aggregates grown on different deterministic fractal substrates, generated by and , exhibit a quite similar behavior, namely, soft oscillations with a logarithmic period, which modulate the expected power laws. Figures 8, 9, and 10 show log-log plots of the rms width, the rms height, and the tree-size distribution, obtained for these fractal substrates, respectively. Table I summarizes the exponents obtained by fitting the modulated power laws describing the properties of the internal structure of the aggregates. Also, data corresponding to the exponent obtained by means of the Family-Vicsek dynamic scaling approach have been included for the sake of comparison. The values of the exponents , , and , have been obtained by fitting the data with the function

(14)

where , , and , respectively andrade . Here, and are constants. When the fit is performed by taking , the presence of higher harmonics to the fundamental frequency is considered. So, the values of the exponents obtained by using could be improved andrade . However, when we considered higher harmonics, inspite of the fact that typically we have , the values of the exponents obtained by fitting the data with , and are almost indistinguishable (within errors bars).

Generating cell
Table 1: List of exponents describing the properties of the internal structure of ballistic aggregates on different fractals. The exponent ( column) is obtained by using equation (5) and the values of and are listed in the and columns, respectively. Also, the exponent ( column) is evaluated by using data obtained by applying the Family-Vicsek dynamic scaling (equation (2)).

In order to rationalize our findings, let us discuss the soft oscillations observed in the rms width mentioned above. Here, we can identify two logarithmic periods, one for the size of the trees () and the other for the rms width (), as is schematically shown in figure 11. These periods become evident by calculating the derivative of with respect to , and the derivative of with respect to , as shown in figure 12 for the case of the substrate. Also, it is worth mentioning that by following the same procedure, we can identify one logarithmic period for the rms height () and another one for the size distribution (). Furthermore, as mentioned above, there is a one to one relationship between the number of observed oscillations and the generations of the carpets. So, as all measurements share the same range of values of , the period of the oscillation observed for the size of the trees (), when tends towards infinity, has to be the same for all observables, namely, the rms width, the rms height, and the tree-size distribution.

Now, based on both the already discussed results and equation (4), we conjecture that for the case of a fractal substrate and in the limit , the derivative of with respect to is a periodic function (with a logarithmic period) that modulates a constant value. So, the conjecture can be written as

(15)

where is a periodic function. By integration of equation (15) it follows that

(16)

where is a constant and is a periodic function such that .

Let us now take two tree sizes, and , separated by a period (on a logarithmic scale), as shown in the horizontal axis of figure 11. Also, their corresponding rms widths, called and , respectively, are separated on a logarithmic scale by a period (see figure 11). Then, by assuming that equation (16) holds, it can be found that

(17)
Figure 12: Log-log plots of the rms width of the trees as a function of the tree size as obtained for ballistic aggregates on substrates. The derivative of with respect to is shown in the inset placed at the lower-right hand side, while the derivative of with respect to is shown in the inset placed on the upper-left hand side. The dashed line in the inset placed at the lower-right hand side correspond to the best fit of the data using Eq. 14.

The existence of a logarithmic period in the rms width of the trees () can be understood due to the fact that each tree can only spread over the fractal and, consequently, the rms width of the trees is constrained by the geometrical features of the underlying structure. In the case of a deterministic Siernspisky carpet, that structure is constructed by the iteration of a generating cell, so that the topological details of the generating cell are present in all the sample. In our simulations, a generating cell of side leads to the occurrence of discrete scale invariance, such that the underlying structures are self-similar only at scales of size , where is an integer. Based on theses concepts, it is expected that . In order to check this statement, in the inset shown on the left-hand side of figure 12 we have drawn a line of size , which nicely fits the logarithmic period of the oscillation of the rms width. This result has also been carefully checked for all the studied cases by measuring the peak-to-peak distance of the modulating oscillations. This finding leads us to conjecture that, for aggregate grown on deterministic fractal substrates generated by a generating cell , the logarithmic period in the rms width of the trees should obey the following relationship

(18)

Equation (18) reflects the fact that the size of the trees in the direction parallel to the substrate exhibits spatial discrete scale invariance with the same fundamental scaling ratio as that of the underlying fractal, namely, (see also equation (9) ).

In order to check this conjecture, figure 13 shows a log-log plot of the rms width as a function of the tree size for the case of an substrate. In this case, the size of the generating cell is , so according to the conjecture we expect . The inset placed on the left of figure 13 shows the derivative of with respect to , and a line of size , which nicely fits the value expected for , has been drawn.

Figure 13: Log-log plots of the rms width as a function of the tree size as obtained for the substrate. The derivative of with respect to is shown in the inset placed on the left.

In order to understand the oscillatory modulation of the power laws observed in the rms height of the trees, we followed the same method already discussed for the case of the rms width. Figure 14 shows the rms height as a function of the tree size , as obtained for the substrate. Also, both the derivative of with respect to and the derivative of with respect to are shown in the insets placed on the lower right-hand and the upper left-hand sides of figure 14, respectively

Figure 14: Log-log plots of the rms height as a function of the tree size as obtained for the substrate. The derivative of with respect to is shown in the inset placed on the right, and the derivative of the with respect to is shown in the inset placed on the left.

So, based on equation (3) and the obtained results, we now conjecture that

(19)

where is a periodic function. Then, after integration one gets

(20)

where is a constant and is a periodic function such that .

By using equation (20) and following the method already applied to the case of the rms width, it can also be found that

(21)

Then, by using equations (17), (18), and (21) one obtains

(22)

where equation (5) has also been used. Let us now recall that, according to our conjecture given by equation (11), it should be expected that the rms high of the trees, which is an observable that evolves along the growing (time) direction, will be coupled to the space discrete scale invariance of the underlying fractal, then exhibiting time discrete scale invariance. In fact, the coupling is nicely reflected by equation (22), which is the realization of equation (12), with being the fundamental time scaling ratio.

Finally, the same procedure can be applied to understand the oscillation observed around the power laws of the size distribution. In fact, figure 15 shows the tree size distribution function for the substrate. The derivative of with respect to is shown in the inset placed on the left-hand side, while the derivative of with respect to is shown in the inset placed on the right-hand side.

Figure 15: Log-log plot of the tree size distribution as obtained for the substrate. The derivative of with respect to is shown in the inset placed on the right, and the derivative of with respect to is shown in the inset placed on the left.

Based on the obtained results, now we conjecture that in the case of a fractal substrate where tends towards infinity, the logarithmic derivative of the size distribution with respect to should be a periodic function around a constant value. So, in this case one has

(23)

where is a periodic function. Then, after integration one gets

(24)

where is a constant and is a periodic function such that . Now, by using equation (24) and following the same method as in the case of the rms width, it is easy to find that

(25)

Table II summarizes the values of the different logarithmic periods obtained for the BD model on substrates (). The results were obtained by fitting the data shown in the insets of figures 12, 14 and 15, with the aid of equation 14 for . In the case of the BD model on fractals, the data exhibit some fluctuations and it is no longer possible to determine a reliable value of . The same shortcoming is found when attempting the evaluation of for the case of the substrate. In order to overcome this shortcoming, it should be necessary to perform the simulations on fractals with , but this task is very CPU-time demanding, so it is beyond our computational capabilities.

Fractal
Table 2: List of the logarithmic periods obtained by fitting the log-periodic modulations of the power laws describing the internal structure of ballistic aggregates on different fractals. The data are also compared to scaling relationships that are explained in the text.

It is worth mentioning that all determined logarithmic periods can be compared to scaling relationships also involving exponents determined independently. In fact, the values of ( column) are compared to ( column), as it follows from equation (18), obtaining an excellent agreement. Also, according to equation (17), the period ( column) can be compared with the quotient given by ( column), where the exponent is taken from Table I, leading to an excellent agreement. On the other hand, the period ( column) can be compared with the relationships given by equations (22) ( column) and (21) ( column), where the exponents and are taken from Table I, obtaining agreement within error bars. Finally, the relationship given by equation (25), can be verified by comparing columns and , also observing agreement within error bars.

Finally, we would like to comment that for growing aggregates in dimensional homogeneous media, one expects that the average volume of the frozen trees of size (), will scale according to trees

(26)

so that one can define , where is an exponent. In previous work we showed that for ballistic aggregates in one has that , indicating that the trees are compact structures trees . By using this procedure, we determined , , and for the fractals of type a, b, and c (see figure 1), respectively. Since we evaluate the statistical error only, these exponents could be considered as preliminary evidence that frozen trees are also compact for ballistic aggregates in fractal media.

Vii Conclusions

We studied the ballistic deposition growth model in deterministic fractal substrates, embedded in dimensions, by means of numerical simulations, analyzing the data according to well established scaling theories.

While the standard Family-Vicsek scaling approach seems to hold at least qualitatively, i.e., the interface width exhibits an initial power-law increase followed by saturation, systematic deviations from the expectation are observed for large values. In fact, the measured values of the interface width are larger than the scaling predictions, suggesting that the standard scaling approach has to be modified in order to properly describe the behavior of the interface upon growth on fractal media. Anyway, the obtained (effective) scaling exponents , and , interpolate between the already known values corresponding to and Euclidean dimensions. Furthermore, the relationship holds, within error bars, and the determined dynamic exponent is in reasonable agreement with subsequent independent measurements, performed by studying the internal structure of the aggregates (see, e.g., Table I).

On the other hand, the study of the properties of the internal structure of the aggregates, in terms of scaling laws describing the properties of frozen trees, i.e., branched structures that are inhibited for further growth due to shadowing effects and lie below the interface, provides an interesting and rich physical behavior. In fact, the standard scaling theory applied to the deposition on substrates of integer dimension predicts the power-law behavior of relevant physical observables as a function of the size of the trees, such as the rms width and height of the trees, as well as for the tree size distribution. However, for fractal media with and , we show that these power laws become modulated by soft log-periodic oscillations. These oscillations reflect the interplay between the spatial discrete scale invariance (DSI) of the underlying fractal substrates and the growing process. While it is well known that dynamic observables become coupled to the spatial DSI given time DSI we2 ; we ; lett , growing aggregates in fractal media provide a richer scenario. On the one hand, the rms width of the frozen trees, i.e., an intrinsically spatial observable measured along the direction parallel to the substrate, exhibits spatial DSI with the same fundamental scaling ratio ( ) as that of the fractal. On the other hand, the rms height of the frozen trees, which is measured along the growing (time) direction perpendicular to the substrate, exhibits time DSI and its fundamental scaling ratio ( ) is coupled to through the dynamic exponent (see equation (22)), which is the realization of the more general relationship, also valid for any dynamic observable, given by equation (10). Of course, the size distribution of the frozen trees also exhibits DSI, which in a subtle way involves both spatial and time DSI, since it is an observable that depends on the structure of the trees developed along the directions parallel and perpendicular to the substrate.

We hope that this work will contribute to the understanding of growing aggregates in fractal media, also stimulating theoretical work aimed to describe the interplay between the underlying symmetries of the substrates and the behavior of physical observables characteristic of the processes occurring in those media.

ACKNOWLEDGEMENTS. This work was financially supported by CONICET, UNLP and ANPCyT (Argentina).

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