Multifractal Growth of Crystalline NaCl Aggregates in a Gelatin

Article pubs.acs.org/crystal

Multifractal Growth of Crystalline NaCl Aggregates in a Gelatin Medium Abhra Giri,†,‡ Moutushi Dutta Choudhury,‡ Tapati Dutta,† and Sujata Tarafdar*,‡ †

Physics Department, St. Xavier’s College, Kolkata 700016, India Condensed Matter Physics Research Centre, Physics Department, Jadavpur University, Kolkata 700032, India

‡

ABSTRACT: Sodium chloride, or common salt, crystallizes in a variety of interesting patterns during desiccation in different gel type solvents. In gelatin, it forms the usual cubic crystals of macroscopic dimensions, but in the regions in between these, it forms highly intricate ramified dendritic patterns with a 4-fold symmetry, visible under a microscope. We analyze these patterns and show that they are multifractal in nature. The multifractal character stems from the strongly inhomogeneous growth process at work during drying of the viscous gel.

1. INTRODUCTION

Interesting crack patterns are observed in addition to crystal aggregates in drying albumin-salt droplets.8−10 These highlight the relevance of the present topic in tackling problems of cracking and peeling in layers of complex fluids used as paint or coatings. Highly inhomogeneous systems that do not obey a selfsimilar scaling law with a single exponent, may actually consist of several intertwined fractal sets with a spectrum of fractal dimensions. Such systems are said to be multif ractal. The system is not simply a collection of superposed fractal sets; rather, the complex distribution arises from a peculiarity in the extremely inhomogeneous growth process itself.11,12 Usually, multifractality arises from a spatial distribution of points, each having a strength or weight associated with it. The weights also have a nontrivial distribution. A growing diffusionlimited aggregate (DLA) with weights proportional to the growth probability assigned to each site is an example of such a multifractal. However, a system with equal weights associated with each point may also have purely geometrical multifractality arising from the growth process.11 Our system belongs to this class of geometrical multifractals, when the photograph of the aggregate is digitized after grayscale conversion. The multifractal system has local fractal dimensions α(x), which may be determined by applying the “sandbox” method at different points x on the system. Different regions distributed over the whole system may have the same local dimensions α(x). If we collect the regions with the same local α(x) and see that their spatial distribution over the whole system follows fractal scaling law, we have identified one fractal subset of the multifractal. The scaling exponent for this subset has, say, a value of f(α). The plot of the fractal dimension f(α) against the

Sodium chloride is one of the most common and well-known crystalline materials. Its structure consists of a face-centered cubic (fcc) lattice of Na+ ions and another fcc of Cl− ions, together forming a simple cubic lattice with alternating Na+ and Cl− ions. A concentrated solution of NaCl dries to form rectangular parallelepiped crystals of salt. Their structure is very different from highly ramified diffusion-limited aggregate (DLA) patterns, which are the paradigm of fractal growth. However, if the crystal growth process is slowed down by allowing crystallization in a highly viscous medium, interesting crystalline aggregates are seen. We study such growth in a gelatin medium and find beautiful aggregates that we show to be multifractal in character. The subject has important practical applications, in addition to being of academic interest. Studies on desiccating colloid-salt systems have been pursued since the 1950s.1,2 In these early works, varied aggregation patterns involving different length scales were recorded and their use in medical diagnostics was discussed. Crystallization under different conditions and development of Mullins−Sekerka instabilities in various forms are also well-documented.3 Pathological examinations and pharmaceutical procedures are concerned with analyzing biological fluids and proteins in combination with biomedically important salts; the topic of the present study is relevant to such fields. Several recent works in this area may be mentioned, for example, designing of high-density stain libraries for highthroughput drug screening4 and application of crystallographic methods in differential analysis of biomaterials.5 A related problem is the development of protein solutions used as lubricating fluids in prosthetics; diffusion of macromolecules in living tissue is found to depend strongly on the presence of ionic salts.6 Different microstructural patterns observed in protein-salt thin films are expected to be suitable for development of patterned matrixes for sensor applications.7 © 2012 American Chemical Society

Received: October 13, 2012 Revised: November 13, 2012 Published: November 29, 2012 341

dx.doi.org/10.1021/cg301499b | Cryst. Growth Des. 2013, 13, 341−345

Crystal Growth & Design

Article

corresponding local α then gives the typical multifractal spectrum. If the system were monofractal, α would be the same everywhere and f(α) versus α would reduce to a single point. Practically, however, it is not convenient to determine local fractal dimensions α(x) and, hence, f(α) as implied in the preceding paragraphs. The widely used technique actually followed is to determine the qth moments of the distribution of points on the system, with q varying from −∞ to +∞. It can be shown that αq determined for each q and the exponent fq corresponding to their distribution over the whole system have a one-to-one correspondence with the α versus f(α) picture described above. The exact procedure we follow is given in the following sections. For a monofractal of course, all moments with different q follow the same scaling exponent fq.

Figure 2. Large crystals in Figure 1 as seen under a microscope.

2. MATERIALS AND METHODS Chemicals used are from Lobachemie, Mumbai. The experiments are done with gelatin in water as the colloidal medium. NaCl (0.03 mol) is dissolved in 50 mL of water; then 0.5 g of gelatin is mixed with the solution and stirred for 1 h at 70 °C until a homogeneous gel forms. Drops of 0.5 mL in volume are allowed to dry on a glass plate at 30 °C; the humidity in the room is 35%.

3. RESULTS Cubic crystals of salt are formed more or less uniformly throughout the film. A photograph of the film taken with a Nikon CoolPix L120 is shown in Figure 1.

Figure 3. Region between the large crystals in Figure 1 when viewed under a microscope shows intricate dendritic patterns.

Figure 1. Photograph of dried gelatin droplet, showing large NaCl crystals formed all over the film. Magnified micrographs of the large crystals and the region in between large crystals are shown in Figures 2 and 3. The diameter of the dried film is roughly 1 cm.

3.1. Microscopic Images with Leica DM750. Under a microscope, we can see interesting details of the large crystals in Figure 2 and the region between the large crystals in Figure 3. Focusing on the large crystals shows a regular steplike structure of the crystal. The region in between the large crystals now shows beautiful patterns of aggregation of the salt, as shown in Figure 3. We analyze this pattern below and show that it is multifractal.

Figure 4. Black and white version of Figure 3. The regions enclosed by frames (red) marked (a)−(c) and the yellow shaded regions (d) and (e), enclosed by blue frames, correspond, respectively, to regions with lx = 936, 1236, 1536, 600tl, and 600bl, for which multifractal characterization has been done.

sections of the pattern of different sizes and positions, to check the reproducibility of our results. The patterns studied are (1) the whole system with a size of 1536 pixels, (2) a subsystem with a size of 1236 centered on the largest seed crystal, (3) a subsystem with a size of 936 centered on the largest seed crystal, (4) a subsystem with a size of 600 from the bottom left,

4. MULTIFRACTAL ANALYSIS The photo is grayscaled and digitized, as shown in Figure 4. We follow the procedure for multifractal analysis as elaborated in standard texts on fractals.11,12 We have analyzed different 342



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The multifractal spectrum f(α(q)) as a function of α(q) or equivalently f(α) versus α, thus obtained is shown in Figure 7.

and (5) a subsystem with a size of 600 from the top left region of the whole figure. These are labeled as (C), (B), (A), (E), (D), and their positions are shown in Figure 4. The whole system is covered with a d-dimensional hypercubic lattice with lattice constant δ. The measure of the qth moment of the mass distribution Md(q,δ) is defined as N

Md(q , δ) =

∑ μiq δ d i=1

(1)

Here, μi is the “mass” or probability assigned to the ith box and

μi = Ni /N

(2)

Ni is the number of mass points in the ith box, and N is the total number of points in the system. If ∑Ni = 1μqi in the limit δ → 0 crosses over from 0 to ∞ as d changes from a value less than τ(q) to a value greater than τ(q), then the measure has a mass exponent12

d = τ(q) The dimension of the qth exponent of the mass distribution is obtained by counting the number of boxes N(q,δ) of size δ needed to cover the system. N(q,δ) scales on varying δ as N (q , δ) = δ −τ(q)

Figure 6. d(q) versus q for regions with lx = 936, 1236, 1536, 600tl, and 600bl, corresponding to (a)−(e) in Figure 4.

(3)

so the mass exponents τ(q) of the measure for different q are determined. For the present system, we have calculated the moments for q = −9 to q = +9. The zeroth-order moment τ(0) is, of course, equal to the fractal dimension of the support. The graph of τ(q) versus q is shown in Figure 5. It is clear that the

Figure 7. Multifractal spectrum for regions with lx = 936, 1236, 1536, 600tl, and 600bl, corresponding to (a)−(e) in Figure 4. The regions all show similar multifractal characteristics.

The curves are parabolic concave downward, which is typical of multifractals. All the curves have identical f(α(0)) values almost equal to 2.0, the dimension of the support. The width of the multifractal spectrum is defined as the difference between αmax corresponding to minimum q and αmin corresponding to maximum q. A greater width of the dimension spectrum indicates greater inhomogeneity in the mass distribution. From Figure 7, we observe that all the curves are asymmetric about the maximum f(α(0)). The width α(0) − αmin on the left side is smaller than the width αmax − α(0) on the right side. This indicates that there is greater heterogeneity of mass distribution of the aggregate in less-dense regions than the denser regions. One may expect that the graphs corresponding to the different regions should coincide as these regions belong to the same aggregate. However, though the graphs almost coincide for the left branch of the spectra, they are different for the right branches, that is, regions of low mass density. Examination of Figure 7 reveals that the graphs corresponding to the regions (b−e) are more bunched together on the right branch of the spectra, but that corresponding to region (a) has a greater

Figure 5. τ(q) versus q for regions with lx = 936, 1236, 1536, 600tl, and 600bl, corresponding to (a)−(e) in Figure 4.

slope for positive q values and negative q values are different. This indicates that the aggregate is a multifractal.13−17 The analysis has been done on different regions, as shown in Figure 4. Though the τ(q) values for positive q values almost coincide, there is some dispersion for negative values of q. This indicates greater heterogeneity in regions of lesser mass of the aggregate. The Lipschitz−Hölder exponent or “crowding index” α(q) corresponding to q is determined from Figure 5 as

α(q) = −

dτ(q) dq

(4) 343



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dispersion. If a region size less than 600 × 600 is chosen for analysis, the multifractal signature of the aggregate disappears. The aggregate is a multifractal only for a certain range of length scales. We conclude that, allowing for some scatter due to stochasticity, the five different patterns give similar singlehumped curves typical of multifractal systems. Another equivalent description18 of the multifractal system is obtained from the Dq versus q plot, where Dq is the fractal dimension of the qth moment of the mass distribution. Dq is defined by the relation Dq =

ln N (q , δ) 1 lim q − 1 δ→ 0 ln(δ)

(5)

Dq =

τ(q) (1 − q)

(6)

present experiment, the solvent, which is a gel, is viscous enough to prevent the convection currents responsible for the coffee-stain effect. The large crystals presumably grow in the beginning of the drying process, when there is enough water in the solvent to ensure high mobility of Na+ and Cl− ions. Therefore, they continue to aggregate according to their minimum energy cubic rock-salt structure, producing large macroscopic crystals. However, as desiccation continues, the gel thickens and becomes more viscous and ions diffuse very slowly. Moreover, the concentration of NaCl is now much diminished, since most of it has gone into making the large crystals. In between the depletion regions surrounding the large crystals, there is still enough NaCl to form new nucleation centers, but the growth process is now different. Growth around the cubic seed crystal is now according to a Laplacian process, which naturally favors growth at the tips, where the “field gradient” concentrates (in this case, the protruding “tips” are the corners of the rectangular seed crystals), dendritic growth results, with large gaps where the solute concentration is almost absent. Figure 3 clearly shows a highly ramified structure with branches growing preferably from the corners. The patterns are somewhat similar to noise reduced DLA growth.12 Normal DLA patterns branch out at all angles; they are generated by allowing a new particle to be added to the existing structure as soon as it makes contact with the growing aggregate. Noise reduced DLA is obtained if a new particle is added only after it makes contact to a site a certain number of times. This creates aggregates with reduced symmetry, depending on the underlying lattice. For a square lattice, it resembles the aggregates in the present experiment, with 4-fold symmetry predominating. On close examination, it is seen that, in the aggregate, most of the branches, particularly the longer branches, form at right angles to the parent branch, but many of the shorter branches are at an acute angle to the direction of growth of the branch. There appears to be a characteristic length above which noise is reduced. We cannot, however, explain the significance of this length at present.

so

Dq is defined so that τ(1) = 0 is satisfied automatically. The Dq curve as a function of q is shown in Figure 6. Because of the singularity in Dq at q = 1, D1 has to be evaluated as12

D1 = lim

δ→ 0

Σiμi ln μi ln δ

(7)

The first three generalized dimensions, D0, D1, and D2, are called the capacity dimension, the information (Shannon) dimension, and the correlation dimension, respectively. These three are usually used to parametrize a multifractal. The capacity dimension, D0, provides information about how abundantly the measure, defined by eq 1, is distributed over the scales of interest. We find here that the aggregate has a capacity dimension of 2.0. This indicates that the normal fractal dimension of the whole aggregate is 2; that is, the twodimensional Euclidean space of the support is covered in a nonfractal manner. The multifractality arises from the nontrivial distribution of mass on the dendrites. The entropy or information dimension D1 gives information on the concentration of the distribution in a certain region. A lower D1 indicates greater concentration of mass over a small size domain, that is, greater clustering. When D1 is close to 0, it will be be reflected as a sharp peak on a mass distribution curve. The information dimension for the aggregate ranges between 1.80 and 1.86 for the different regions of the NaCl aggregate. The correlation function D2 describes the uniformity of the measure (here mass) in different intervals. Smaller D2 values indicate long-range dependence, whereas higher values indicate domination of short-range dependence. The D2 value for the aggregate ranges between 1.81 and 1.84 for the different regions. In fact, Figure 6 shows that, for the first three generalized dimensions, the different regions of the aggregate yield almost identical values. Also, D0 > D1 > D2 further confirms the multifractal nature of the aggregate.

6. CONCLUSION To conclude, we have shown that a very simple method of drying salt in a colloidal gel produces extremely interesting multifractal patterns. Pattern formation in drying drops is a topic of considerable interest because, in addition to being a challenge to understand fundamentals of growth processes, it has several applications. The most important is in medical diagnosis;22 distribution and segregation of solutes during drying is also important in the technology of paints and coatings. Multifractal analysis of structures generated by inhomogeneous growth, such as sedimentary rocks and compacted soil,15 provides detailed statistics of the morphology, such as distribution of voids and solid particles in a soil matrix, range of variation in the morphological features, and so on. Multifractal analysis of growth probabilities of tips of DLA trees describes the inhomogeneity of the growth process.23,24 We hope that the observation and analysis of multifractal crystal growth presented here will be helpful in understanding and designing crystal growth.

5. DISCUSSION In the experiment described here, we find large crystals of NaCl distributed more or less uniformly over the whole film. This is quite contradictory to the well-known coffee-stain effect,19,20 which drives the solute toward the boundary of a drying drop. We have observed that, when a small droplet of a solution of NaCl in water is allowed to dry, salt crystallizes along the periphery of the dried droplet, leaving the interior nearly void, in a manner analogous to the coffee-stain effect.21 In the

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 344



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Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.G. and M.D.C. are grateful to IFCPAR (Project no. 4409-1) and the Alumni Association (JU) respectively, for the award of research fellowships.

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REFERENCES

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Multifractal Growth of Crystalline NaCl Aggregates in a Gelatin

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