Spontaneous pattern growth on chocolate surface: simulation and experiments

The natural variation of temperature at ambient conditions produces spontaneous patterns on the surface of chocolate, which result from fat bloom. These patterns are peculiar because of their shape and cannot be obtained by controlled temperature conditions. The formation of these spontaneous grains on the surface of chocolate is studied on experimental and theoretical grounds.Three different kinds of experiments were conducted: observation of formed patterns in time, atomic force microscopy of the initial events on the grain formation and rheology of the melted chocolate. The patterns observed in our experiments follow the trends described by the Avrami model, which considers that is possible to define a characteristic time scale that governs the growth of grains starting from germ nuclei. Through computer simulations, in the NVT ensemble using a coarse-grained model of triacylglycerides molecules, we studied the process of nucleation that starts the pattern growth and that is consistent with the Avrami model.


I. INTRODUCTION
The study of chocolate is interesting for many reasons. Despite its evident importance in the food industry, this complex mixture is also a good example of some scientific curiosities [1]. For sure, some of these curiosities were found because of its technological importance and this is the case of our research: if a chocolate bar on a lab table had not become aged by chance, the observation of beautiful patterns on its surface had not inspired this ongoing study. Most of us have seen the appearance of patterns on a chocolate bar when left unperturbed for a long enough period of time, allowing it to age, some examples of such patterns are shown in Fig. 1. It is easy to see that the contours formed call for the use of fractals to explain them, but shape generalities might not be easy to describe in scientific terms. On a close inspection of patterns in Fig. 1, in the same sample, one finds a variety of shapes as different as "mostly circular", "snowflake-like", "almost circular" and "hexagonal-like".
The formation of chocolate phases and their properties can be revised extensively in the literature [2]. Fig. 2 summarizes the phases observed as function of temperature [3,4]. In the temperature range of 50 to 60 • , all the history of the sample can be erased. The volume change in the cooling-heating processes presents an important hysteresis. On one side, under cooling it is understood that the polymorphic forms β(V) and β(VI) form low temperature phases without a clear formation of the phases α and β . In contrast, under heating, volume changes are noticeable when different phases are formed. As depicted in Fig. 2, previous works suggest that volume changes are not monotonous [4]. Phase change temperatures are approximate and, in general, temperatures reported differ by several degrees. Moreover, glass transition temperatures are also reported in the same ranges as the ones for phase change temperatures [5]. As a consequence, it is easy to suggest metastable lines that practically span all over the volume temperature diagram due to the proximity between the phase change temperatures [6,7]. The story is a little different for the formation of patterns on the surface of chocolate, the phenomenon is not clearly understood, and few studies analyze the shape of these patterns [8]. In addition, from the collection of patterns previously reported, it is difficult to draw conclusions about the shapes formed, or about the reasons that gave rise to them.
However, it is a consensus to consider the onset of the whitish color on the chocolate surface a consequence of the "fat bloom" event: the physical separation between β(V) and β(VI) phases during the aging process of chocolate. In the literature, one can find some "recipes" to produce "fat bloom", most of them involve the use of temperature cycles [3,9,10].
However, in our initial attempts to create such patterns, we had difficulties obtaining them using periodic cycles or constant temperatures. As a consequence, the patterns observed in Fig. 1 were produced using a different procedure to what is usually reported in the literature.
In this paper, we reproduce the common patterns observed during the aging of chocolate and give a glimpse on the reasons for obtaining them, based on a theoretical description given by the Avrami model for nucleation [11,12]. For our study we use both experiments and simulations. Definitively, the complex mixture of at least triglycerides, polyphenols and carbohydrates of high molecular weight called "chocolate", represents a challenge for experimental reproducibility. Despite being difficult to conduct an experiment expecting the same pattern twice, it is possible to reproduce some trends of events necessary to obtain patterns that share some common characteristics. Here, it is crucial to identify, as is going to be discussed, that different events play a role in the final shape observed.

A. Sample preparation
Two types of commercial chocolate were used in this work: a commercial 100% cacao product from the brand Mayordomo without added sugar or soy lecithin (https://chocolatemayordomo.com.mx) and a chocolate from the brand New Art Xocolatl with 85% cacao (https://www.newartxocolatl.com). The latter has added sugar and soy lechitin. These two extra components are usually the main difference between pure cacao and the mixture called "chocolate". In these mixtures, soy lechitin allows emulsification of the product with water and milk while sugar provides sweetness, which is completely absent in pure cacao. Despite the differences in composition between the two chocolates used, the experimental results were similar when changing the type of chocolate used.
Three different kinds of experiments were conducted: observation of formed patterns in time on chocolate surfaces at different temperatures, atomic force microscopy of the initial events on the pattern formation and rheology of melted chocolate. Patterns on chocolate surfaces were obtained by heating up samples of chocolate up to 60 • C and pouring the viscous liquid in pre-heated petri dishes, which were then subjected to different temperature conditions. Time and temperature were recorded during pattern formation. The images analyzed in this paper were captured from these petri dishes during the evolution of the patterns.
The topography Atomic Force Microscopy (AFM) images were also obtained from the petri dishes samples using a Witec alpha-300 microscope in a tapping mode (100 µm × 100 µm and 150 × 150 lines). Rheology was performed in a HR-3 Discovery Hybrid Rheometer (TA Instruments) using a cone-plate geometry of 40 mm diameter and 0.5 deg. The temperature was controlled in the rheometer using a Peltier plate.

B. Computer simulations
We have developed a two-dimensional (2D) model for the experimental system which allow us to have a glance at the self assembly of the β(V) molecules. Our system consist of rigid bodies, each made with spheres of diameter σ, that have a shape close to one of a β(V) molecule [13][14][15], see Fig. 3. We have chosen a Mie-type potential to model the short-range attractive behaviour between fat molecules. This type of short-range interactions have been previously used to model units of fat crystals and its assembly into bigger systems [16]. The spherical units in each molecule interact only with the spheres of neighboring molecules via the pair potential, Where r is the center-of-mass distance between two spheres, ij indicates interaction between molecules i and j, σ is the approximated diameter of the repulsive core and is the strength accordingly such that u ij (r cut ) = 0. For simplicity, we have set = 1, σ = 1 and the unit time τ sim = mσ 2 /k B .

All simulations have been performed with the open-source MD simulation package
LAMMPS [17], which has a dynamical integrator for rigid bodies [18,19]. In our 2D simulations, the spherical units that form our molecules are restricted to a plane and keep the same initial z position. Each molecule segment of length 1σ is composed of two spheres, with a total of 43 spheres per molecule and 1020 molecules in our 2D simulation box. To explore the assembly of our molecules, we use the packing fractions φ = 0.22, φ = 0.36 and φ = 0.48. For each case, we start with a 2D box with an initial isotropic configuration, which was created with purely repulsive molecules (r cut = σ). The interaction between the molecules is turned on in a constant number, volume and temperature (NVT) ensemble using a Nose-Hoover thermostat [20]. We have set the temperature of our systems to k B T / = 1.
The simulation time step was taken as δt = 0.005τ sim and the total simulation times used were of 2 × 10 5 τ sim to 5 × 10 5 τ sim .

III. RESULTS
A. Pattern formation formed by controlled cycles, or just by maintaining a constant temperature, do form different types of patterns, but they are qualitatively different from those in Fig. 1. Perhaps the main phenomenological differences between patterns in both figures are the occurrence of clear circular centers and more symmetric shapes, see the two patterns presented in Fig. 1. What we observe in Fig. 1 is the most commonly observed phenomenology and it can only be produced when temperature fluctuates in the range between 20 to 30 • C, as observed in the sequence of pictures presented in Fig. 5 with the correspondent registry of temperature.
These fluctuations are, apparently, necessary to obtain the characteristic morphology of these patterns. The initial steps to obtain this morphology are shown in Fig. 5. At the very beginning, a circular center acts as a germ nucleus for a new phase, and in this way the polymorphic forms evolve from β to β, as reported in the literature [4]. Some of the

C. Pattern growth
The nucleation process observed in experiments and simulations can be theoretically described using the Avrami model [11,12], which is based on a statistical description of the formation of grains starting from initial or germ nuclei. The basic assumption behind this model is that the factors that govern the tendency of the growth nuclei are similar to those which govern further growth, i.e, what Avrami described as an isokinetic process. If p(t) describes the probability of formation of growth nuclei per germ nucleus per unit time at temperature T , then where Q is the energy of activation per gram molecule, A is the work per gram molecule required for a germ nucleus to become a growth nucleus, R is the gas constant and K is a normalization constant. The probability p(t) is used to introduce a characteristic time τ of the growth process, defined as dτ dt = p(t).
In a similar way, a spatial scale r(t) that defines the size of the grain is given in terms of a rate of growth, G(t), according to the expression Both equations allow us to define a volume V and a surface area S of a grain, for the three-dimensional case (3D). For a strictly two-dimensional (2D) system we have where z is an initial value of the characteristic time, σ 3D and σ 2D are 3D and 2D shape factors that in the case of spherical or disk-like grains take the values σ 2D = π and σ 3D = 4π/3, respectively. The isokinetical regime consists in assuming that the rate of growth is proportional to the probability of formation, i.e., G(t) = αp(t), where α is a constant over concentration and temperatures. This assumption simplifies the evaluation of V (t) and S(t), resulting in the expressions and S(τ, z) = σ 2D (r − z) 2 (10) These expressions correspond to isolated growth nuclei. By considering two associating mechanisms for the growth of a grain, via the nucleation around a germ nucleus or the nucleation around an already growth nucleus, and using the expressions obtained for the V and S, as initial values for the grain sizes, Avrami [11] derives rate equations for the number density of nuclei germs N (t) and for the total extended volume of the grain. By coupling these equations and solving them in the general case of a random distribution of initial germ nuclei, a general equation for the actual value of V (t) is obtained, known in the literature as the Avrami equation, given by where k is a coefficient that depends on G(t) and p(t) and d is the Avrami exponent. Both quantities depend also on the geometric dimension of the grain. Typical theoretical values 13 obtained for d vary within the range 1 ≤ d ≤ 4, depending on the value of the probability of formation of growth nuclei p(t), which it is determined by the energy of activation Q and the work required to form a growth grain A [12]. In the case of quasi two-dimensional, plate-like, is small (i.e., a high total energy E = Q + A per mole required for a germ nucleus to become a growth one) whereas for high values of p(t) (i.e, small values of E), d ≈ 2. Linear-like grains will have 1 ≤ k ≤ 2, although it is important to bear in mind that the key factor that determines k is a complex one since not only depends on E but on G. For fluctuations in temperature, we could expect that Q becomes a determinant factor in the way that the grain growth formation is induced.
The growing area of the two pattern series observed in Fig. 5 was analyzed using the Avrami equation. Each pattern was drawn on a mask and this area in pixels was plotted vs. time. Results of the pattern growing for the two series, observed in the Fig. 9, show a remarkable similarity. The fitting of the data to the Avrami equation gives an exponent d ≈ 1. Furthermore, we analyzed the fat bloom homogeneous phenomena by performing a binarization that identifies regions of higher fat content, which presumably corresponds to β(V)-β(VI) rich regions, and has a lighter typical color. We focused on sections of the images where no patterns grow and fitted these data to obtain an Avrami exponent d ≈ 3.

IV. DISCUSSION
We have presented the different aspects required to obtain the most common patterns observed during the aging on the surface of the chocolate. One basic aspect is the necessity of having temperature fluctuations in the range of 20 to 30 • C, as observed in Fig. 5. Without them, it is practically impossible to obtain patterns with circular centers and high radial and angular symmetry. It is important to point out that the different phases observed with the variation of temperature have differences in volume and crystallinity (orientation).
In the literature there are descriptions of typical hysteresis in volume between the cooling and the heating histories, as depicted in Fig. 2, which is similar to what it is suggested from hysteresis in the normal stress coefficient versus temperature (Fig. 7). Following Fig. 2, we can also suggest that the proximity among the different stable and metastable chocolate phases produces easily different shapes or patterns on its surface, that depend on temperature fluctuations and on the previous history of the sample. Clearly, more than two phases could be involved to create a pattern, but interestingly the pattern growing can be captured using the Avrami model (Fig. 9). An Avrami exponent d in the range of one is obtained and, as previously mentioned, it can be linked, via the probability of formation of growth nuclei p(t), with a low value of energy E of the fat bloom event triggered in a pattern. By contrast, the homogeneous fat bloom formation also analyzed according to the Avrami model (Fig. 9, inset), present an exponent near 3, that corresponds to an energy higher than the one necessary to form a pattern.