## 1. Introduction

In nature, fractal structures emerge in a wide variety of systexms as a local optimization of diverse growth processes restricted to the entropic and energetic inputs from the environment. Even more, the fractality of these systems determines many of their physical, chemical, and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology [1–3].

One of the most successful approaches to this problem employs stochastic growth processes of particle aggregation. In general, aggregation phenomena are out-of-equilibrium processes of fractal pattern formation that are ubiquitous in nature [4]. As such, since the introduction of the diffusion-limited aggregation (DLA) and ballistic aggregation (BA) models, a plethora of studies has been developed trying to understand the ultimate aspects of the aggregation dynamics that give rise to *self-similar* or fractal clusters, the relationship of this *fractality* with their physical and chemical properties, and the most effective methods and techniques to control the fractal growth.

In particular, one striking feature of these systems is the morphological transition that they undergo as a result of the interplay of the entropic and energetic aspects of their growth dynamics that ultimately manifest themselves in the geometry of their structure [5]. It is here, where despite of their complexity, great insight can be obtained into the fundamental elements of their dynamics from the powerful concepts of fractal geometry [6, 7].

One example of this is the well-known dielectric breakdown model (DBM) or generalized Laplacian growth model, which has importantly contributed to our understanding of far-from-equilibrium growth phenomena, to such extent that seemingly unrelated patterns found in nature, as river networks or bacterial colonies, are understood in terms of a single framework of complex growth [8, 9]. However, we are still in need for a complete scaling theory of growth for systems far-from-equilibrium, as well as a comprehensive description of the fractality of systems that exhibits fractal to nonfractal morphological transitions [10].

In this chapter, starting from the mean-field result for the fractal dimension of Laplacian growth, we present a theoretical framework for the study of these transitions. Using a statistical approach to fundamental particle-cluster aggregation dynamics, under which it is possible to create four nontrivial fractal to nonfractal transitions that will capture all the main features of fractal growth, we show that, regardless of their space symmetry-breaking mechanism, they are well described by a universal dimensionality function, including the Laplacian one.

In order to show this, we consider the following: first, we introduce a general dimensionality function that is able to describe the measured fractal dimensions and scaling of clusters generated form particle aggregation. Second, we apply this equation to a set of fractal to nonfractal morphological transitions, created by identifying the fundamental dynamics that drive the fractal growth in particle aggregation and by combining three fundamental *off-lattice* particle-cluster aggregation models, the DLA, BA, and a recently introduced infinite-range mean-field (MF) attractive model [11, 12] under two different schemes. Afterwards, the scaling of the clusters along the transitions is measured for different values of their control parameters using two standard methods: the two-point density correlation function and the radius of gyration. Finally, we show how all measurements for the scaling of the DLA-MF, and BA-MF transitions collapse to a single universal curve valid for any embedding Euclidean space, under the appropriate variable transformations of the general dimensionality function.

## 2. Fundamental models

In the Laplacian theory of growth, the growth probability at a given point in space,
**Figure 1**). It has been found that the structure that emerges from this process exhibits self-similar properties described by a fractal dimension,

For

Here, for
**Figure 1**), whereas for

However, one of the most challenging aspect of the theory comes when the growth is not purely limited by diffusion, e.g., when it takes place under the presence of long-range attractive interactions, where strong screening and anisotropic effects must be considered [1, 5, 7]. In this case, a clever generalization to the Laplacian growth process was proposed within the context of the DBM, assuming

provides a good approximation to the dimensions of this transition but due to its mean-field limitations, it does not have a good correspondence with the numerical results [25, 26]. Nonetheless, as shown here, Eq. (3) is the starting point to clarify this aspect of the theory and, even, to establish a suitable and general framework to analyze more complex morphological transitions in stochastic growth processes.

This is done by considering that the fundamental dynamical elements of aggregation, which drive the fractal growth, are mainly two: a stochastic one, coming from the particles’ trajectories randomness, and an energetic one, coming from attractive interactions. With regard to the latter, there are two physical mechanisms related to these interactions and two models that are able to reproduce their effects. First, the model we will refer to as
**Figure 2a**–**c**. Second, the model referred here to as the
**Figure 2d**. Therefore, by controlling the interplay of any of these two mechanisms with a pure stochastic model (in this case the DLA or BA models), one is able to generate fractal to nonfractal morphological transitions.

## 3. Methods

In the following and as explained below, all data for

### 3.1. Aggregation dynamics

In all of the numerical calculations, we choose as a unit of distance, the particles’ diameter here is set to one. For generating aggregates based on BA or MF (**Figure 1a** and **1c**), a standard procedure was used in which particles are launched at random, with equal probability in position and direction of motion, from a circumference of radius
**Figure 1c**). Finally, in the case of aggregates generated using DLA dynamics (**Figure 1b**), particles were launched from a circumference of radius

Regarding the

### 3.2. Fractal and scaling analysis

In all measurements, we performed an ensemble average over

where the double bracket indicates an average over all possible origins

where
*i*th-particle in the cluster and,

In particular, for the
**Figure 6**) over the interval
^{4}, and

## 4. Fractality prescriptions

Despite the complexity leading to morphological transitions, simple models can be established to describe their fractality or scaling as a function of the control parameter, in our case, the branching parameter

with

In this form, by setting

where

with

Furthermore, let us introduce the reduced parameter,

where

is an effective parameter associated to a generalized screening/anisotropy-driven force. Its first-order approximation is then,

where the effective parameter is now given as,

With this prescription, the dynamical change in growth regime is now located at

## 5. Morphological transitions

### 5.1. The
λ
-model: screening-driven transition

In the first approach to morphological transitions, we will consider the case when long-range attractive interactions are introduced in the growth dynamics. In this case, the way to obtain self-similar clusters, that is, clusters with a single fractal dimension, is to maintain a proper balance between the energetic and entropic contributions to the growth process. This can be done by considering an aggregation radius,

For example, for
*direct-contact* interaction, the usual DLA or BA models are recovered (see **Figure 3a** and **4a**, respectively). When
*multiscaling* branching growth and a crossover in fractality, from
**Figures 3** and **4**.

It can be appreciated that this growth presents three well-defined stages as illustrated in **Figures 3d** and **4d**. In the first one, the growth is limited by the interactions and is characterized by
*interaction-limited* to *trajectory-limited*. In the third stage, when the distance among the tips of the main branches becomes much larger than

However, taking into account that the spatial size of the clusters is proportional to the radius of gyration
*branching* parameter. Given a fixed value of
**Figure 5**). Here, one can clearly appreciate the transition in growth regimes from entropic, when

Additionally, this model allows one to estimate

### 5.2. The
p
-model: anisotropy-driven transition

In the second approach, a general stochastic aggregation process can be model under a Monte Carlo scheme involving three fundamental and simple *off-lattice* models of particle-cluster aggregation. On one hand, the well-known BA and DLA models provide disordered/fractal structures through their stochastic (entropic) dynamics (**Figure 1a** and **1b**). On the other, we introduce a mean-field (MF) model of long-range interactive particle-cluster aggregation [11, 12] that provides the most energetic (and noiseless) aggregation dynamics that, simultaneously, acts as the main source of anisotropy. We must remark that this anisotropy is purely generated by the growth dynamics and not from lattice effects [28] (see **Figure 1c**). Then, the statistical combination of these models results in an off-lattice DLA-MF and BA-MF dynamics, whose morphological transitions can be controlled by the *mixing* parameter
**Figure 6**).

## 6. Universal description

It is necessary to remark that the DLA-MF and BA-MF transitions in the
**Figure 6c** and **d**, in contrast with the ones present in BA-DLA [15, 21] and the DBM [23, 25] characterized by monofractals. These multiscaling features reveal a crossover behavior that can be properly quantified by measuring a local or effective,
**Figure 7a** (details for the values of the parameter used to produce **Figure 7** are presented in **Table 1**). Analytically, all measurements can be described by Eqs. (8) and (9), using
**Figure 7d**.

In the first block, we present the parameter values used to describe
*gnuplot* embedded algorithms.

The labels

By observing the description of the transitions based on the function
**Figure 7**, one can clearly appreciate their continuous nature for both the
**Table 2**). Even more, by plotting all data as a function of
**Figure 7c** and the bottom pane in **Figure 7d**.

A final important implication of the previous findings is that the DBM and BA-MF transitions (for both
*universality* of these morphological transitions must be understood in the sense that they are described by the same scaling in their fractal dimension. In fact, by defining the reduced codimension,

where the effective parameter
**Figure 8**. This finding makes it clear that it is possible to define *universal* transitional point,
**Figure 8**). Then, the universal transitional points,
**Table 2**).

## 7. Conclusion

It has been stated above that the entropic and energetic elements are the two aspects of the complex aggregation dynamics which in nature are strongly correlated. Nonetheless, this reductionist approach that essentially encapsulates the information of all the finer details of the dynamics into an effective interaction (in the

Additionally, the descriptive framework for the scaling of fractal to nonfractal morphological transitions in stochastic growth processes, which includes the concept of an effective screening/anisotropy force and reduced codimensionality transformations, has revealed that the DLA-MF, BA-MF, and DBM transitions exhibit a well-defined universal scaling

The results and models discussed in this chapter represent an important unifying step toward a complete scaling theory of fractal growth and far-from-equilibrium pattern formation. Additionally, the possibility of applying the dimensionality function to discuss complex structures in other research areas, ranging from biology [4, 1], intelligent materials engineering [31, 32] to medicine [33], seems to be in some cases straightforward.