Ind. Eng. Chem. Res. 2008, 47, 6459–6464
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Crack Formation in Drying Laponite Sujata Tarafdar* and Suparna Sinha Condensed Matter Physics Research Centre, Physics Department, JadaVpur UniVersity, Kolkata 700032, India
Natural clay and other granular materials form crack networks with typical patterns on desiccation. The present paper reviews a series of studies on crack patterns produced by drying laponite suspensions in different solvents, on different substrates. Laponite is a synthetic clay composed of nanosized platelets of uniform size and composition. The area covered by the cracks shows a scaling behavior with layer thickness. The salient features of the experimental observations can be reproduced by a computer simulation with a chain of springs modeling the clay. We review earlier work done on crack patterns in drying laponite, suspended in methanol as well as water. Experiments on a modification of the setup where the samples are allowed to dry in an electric field are also reviewed. We present new results done on a different geometrical arrangement of the electrodes. The crack patterns are again found to follow the symmetry of the field. Another new observation is that gelation starts earlier when suspensions with excess water are placed in an electric field. 1. Introduction
2. Experiments and Analysis
Granular materials form typical crack patterns when a suspension dries out.1–5 Pictures of fractured mud-flats in drought-stricken regions is a familiar sight, as are crack networks formed on cement floors, old painted surfaces, and glaze on pottery. The patterns are characteristic of the granular material, the solvent, the substrate, and thickness of the drying film. External conditions such as the temperature, humidity, and the desiccation rate also affect the patterns. There are a large number of experimental studies of laboratory produced crack patterns under carefully monitored or controlled conditions.4–7 Several analytical and computer simulation models have also been developed to explain the mechanism of formation of the patterns.8–12 The development of cracks is a problem of practical importance. One objective of such a study would be to prevent formation of cracks where a smooth texture is required. In other cases cracks are intentionally produced for artistic effect, e.g. in crackle glazed pottery.13 In the field of earth science, flow of fluids through fractures in the ground is a useful study. Formation of dislocations or the spectacular hexagonal “columnar joints” 14 and their small-scale laboratory replica15 are relevant examples.
2.1. Procedure. Laponite is a synthetic clay with the chemical formula
The clay films appear to retain a “memory” of the treatment prior to drying. Films vibrated or rotated in a particular manner exhibit crack patterns with the symmetry of the disturbance.16 In the next section, we discuss experimental studies on cracks produced in drying laponiteswhich is a synthetic clay. Laponite is used in many commercial products such as toothpaste, cosmetics, and personal care products, it also serves as a thickening agent in paints and ointments,17 so such studies on laponite may be useful to industry. We discuss in the following section, a simple spring network model,10 which reproduces the principal features of the experimental patterns. Finally, a new experimental setup is discussed where the film dries in an electrostatic field.18,19 It is seen that the cracks are affected by the electric field and the pattern follows the symmetry of the field lines. * To whom correspondence should be addressed. E-mail: sujata_tarafdar@ hotmail.com. Tel.: +913324138917. Fax: +913324146584.
Na0.7+[(Si8Mg5.5Li0.4)O20(OH)4]0.7It consists of monodisperse flat platelike particles about 25 nm in diameter and of 1 nm thickness. We have used two different solvents with the laponites(i) methanol, in which case a suspension is formed and the film dries rapidly, and (ii) water, in aqueous solution, laponite forms a gel on standing, for concentrations above a few percent. When the gel dries, cracks are formed. (i) The experimental procedure is as follows: Laponite (RD) is mixed with methanol and stirred for 1-2 h to make a suspension. The mixture is poured in a Petri-dish and allowed to dry. The crack pattern is photographed at intervals as it forms. The Petri-dish is placed on a digital balance during drying, so the weight loss can be recorded continuously. The same procedure is repeated on glass substrates. (ii) In water, the following method is adapted. A 2.5 g portion of laponite RD (Rockwood Additives) is mixed with 50 mL distilled water. The mixture is stirred for 15 min in a magnetic stirrer and deposited in circular Petri-dishes of 10 cm diameter and allowed to dry. The thick suspension just before formation of the gel is mildly alkaline with a pH of 9.5. The crack development starts early for group i and is complete within a few hours. For the aqueous suspension ii, a drying time of about 48 h or more is required. Figure 1 shows typical patterns of a methanol-laponite suspension drying on polypropylene (PP) and glass. 2.2. Analysis of the Crack Patterns. A. Fractal Dimension. Crack patterns are known to be fractal. In a coarse-grained view of the cracks where the resolution is lower, a reduced version of the whole pattern looks similar to the original. We measure the fractal dimension of the pattern formed on the PP substrate (with methanol solvent), since the self-similar character is more evident here. In the pattern with the glass substrate, our sample is not large enough to exhibit a wide range of crack widths. To measure the fractal dimension by box-counting,20 obviously the maximum box size should be smaller than the upper cutoff for self-similarity. In this case, the upper cutoff is the maximum crack width or the average size of the final peds
10.1021/ie071375x CCC: $40.75 2008 American Chemical Society Published on Web 05/07/2008
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Figure 1. Crack patterns developed in drying laponite-methanol suspension on (A) a polypropylene (PP) substrate and (B) a glass substrate, respectively.
Figure 2. Cumulative area covered by cracks with widths more than wmin for layers of different thickness. Samples are dried laponite-mathanol suspensions on PP substrate.
(these two are comparable and of the order ∼2 mm.). The lower cutoff for self-similarity of this pattern is the minimum crack width visible in the magnified pattern; this is about 0.2 mm. For two patterns with layer thicknesses of 0.21 and 0.25 mm, respectively, the fractal dimensions come out to be 1.65 and 1.66.21 This agrees well with earlier work.22 This fractal dimension refers to the part of the pattern covered by clay. B. Area Covered by Cracks. Now, we look at the gaps left by the shrinking clay, i.e. the cracks. The final crack pattern, when observed under slowly increasing resolution shows a typical shape, which is highly reproducible. We measure the cumulative area Acum covered by cracks as a function of the minimum observable crack width wmin. Acum is the total area covered by cracks of width wmin or more. We show in Figure 2 the curves for a range of film thicknesses t. The thicker samples have wider cracks and larger peds (i.e. the solid fragments separated by cracks) than the thinner samples. This agrees with previous observations. The same quantity for the pattern with the glass substrate has a gentler gradient compared with the PP substrate case; this is discussed later and compared with simulation results. 2.3. Scaling of the Crack Area with Layer Thickness. We suggest two different crack formation mechanisms to be operative in the low and high resolution sides. The region to the right of the knee of the area curve is the desiccation regime, and the region on the left is the relaxation regime. In the
desiccation regime, which occurs earlier in time, there is a high rate of loss of solvent and cracks are wide. In the relaxation regime, cracks are very narrow and do not contribute significantly to the cumulative area. The dominating process here is rearrangement of the particles to energetically favorable configurations. The rate of mass reduction is much higher in the earlier desiccation regime, compared to the relaxation regime.21 In Figure 2, the high resolution (low wmin) side of the curves for the crack area all show the relaxation regime, where crack width is negligible and Acum changes very slowly. If the curves of Figure 2 are represented on a log-log scale with the fractional area (relative to the maximum area) on the y-axis and the x-axis (i.e., the minimum observable crack width) is scaled by the thickness of the film t in each case, we find that the curves all collapse onto a single curve shown in Figure 3.23 So, the pattern scales with the film thickness; this has been found in other work as well.2 3. Simple Model for Crack Formation A quasi-one-dimensional spring chain model reproduces the area vs wmin curves. It also exhibits the difference in the PP and glass substrate cases, through a variation in the adhesion between the laponite and substrate. The model is described in brief below, details are to be found in ref 10.
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Figure 3. Curves from Figure 2 collapsed onto a single curve, when the area relative to the maximum area is plotted on the y-axis and the crack widths are scaled by the thickness “t” on the x-axis on a log-log plot.
Figure 4. (upper diagram) Unstretched spring chain. The nodes a, b, c, and so on are connected by the horizontal springs to each other and by vertical springs to the nodes A, B, C, and so on on the substrate shown as a thick line. (lower diagram) Strained chain on desiccation. Strain is maximum at the center for the horizontal springs, but maximum at the ends for the vertical springs.
We represent the thin layer of clay suspension by a chain of nodes connected by horizontal springs with spring constant Sh to each other, and by vertical springs with spring constant Sv, to the substrate. The chain is shown in Figure 4. Initially, all springs have a natural length d0 ) 1 and the nodes and their points of attachment to the substrate lie on a unit lattice. The drying process is represented by a reduction in d0 of all the horizontal springs. The nodes a, b, c, and so on are allowed to move horizontally to relieve stress. As the stress builds up, either the horizontal spring breaks when the strain exceeds a threshold Hth or the vertical spring allows a “slip” in the lower node on exceeding a threshold Vth. To bring about the effect of desiccation in our simulation, the length of each link in the upper chain is reduced according to the rule dn+1 ) dn(1 - b/rn)
(1)
where dn+1 and dn represent the natural length between two nodes in the (n + 1)th and nth time step, b is a constant, and r
is a parameter which controls the rate of decrease in the natural length from one time step to the next higher time step. The value of b is kept constant at 0.1 and initial time corresponds to n ) 0. After one desiccation step, the energy of the system is minimized by the conjugate gradient method and all the nodes are allowed to relax to their minimum energy position. After each desiccation step, it is checked whether the strain in any of the springs exceeds the breaking or slipping limit. If so, the necessary break and/or slip is implemented, and the energy minimization procedure is rerun, until an equilibrium is reached. If one or more cracks appear, their widths and position are noted. The rate of desiccation decreases with time, so eventually the system stabilizes; we now consider the system to be completely dry. The results show that Vth is the parameter which determines the nature of the area vs wmin curve. Glass being smoother is represented by a lower Vth compared to the rougher PP surface. Comparison of simulation results with experiments are shown
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Figure 5. Simulation results for different Vth values compared with the experimental results for the cumulative area for the crack patterns shown in Figure 1. The symbols represent experimental results, and the lines represent the simulation for the smooth (rough) substratesglass (PP) with a low (high) threshold.
in Figure 5. The simulation results have been multiplied by a constant factor, so that their magnitude becomes comparable to the experimental results. The agreement is pretty good and the qualitative difference in the nature of the curve on the different substrates is reproducd. The parameters in the model are b and r controlling the drying, and the thresholds Hth and Vth control the breaking of springs and slipping on the substrate. The parameter b and the vertical and horizontal spring constants have not been varied in this model, Sh ) Sv ) 1. So physically, we may say r determines the rate of decrease of the natural length of the springs, i.e. the rate of drying. The power law ensures that the rate of drying decreases with time. The value of r has been kept in the range 1.2-1.3. This ensures that after full desiccation when the drying rate has become negligibly small, the total sample size has reduced to about 60-70% of its initial size (the sample should not vanish all together on drying). We have shown that Vth corresponds to the slipping on the substrate; this is connected to the adhesion between the clay and substrate, or roughness of the substrate and controls the nature of the crack area vs wmin curve. The parameter Hth, which represents the cohesion between clay particles, simply controls the maximum crack width. Variation in the spring constants and establishing a more quantitative equivalence between the material properties and the model parameters may be a subject of further work. Our approach is somewhat similar to the model developed by Kitsunezaki;9 they have not, however, studied the scaling of the area covered by cracks of different width. Another approach worked on earlier11,12 was to replace the springs modeling the links by electrical conductors so that the network becomes a system of fuses. So here, a blown fuse corresponds to a broken spring. The electric analogue is a scalar problem and hence simpler to deal with. This model was used to study the roughness of the crack paths, which look like a random walk. 4. Crack Pattern in an Electrostatic Field Clay platelets in general have a surface charge in aqueous suspension, and it is natural to expect that an electrostatic field may have some effect on orientation and aggregation of particles during drying. We have carried out another set of experiments to record and analyze this novel effect.16,17 In this experiment, an aqueous suspension is prepared, which finally forms a gel. A 2.5 g portion of laponite RD (Rockwood
Figure 6. Crack pattern formed in a radial electric field of 200 V. The central electrode is positive and the cylidrical negative electrode is along the periphery of the petridish.
Additives) is mixed with 50 mL distilled water. The mixture is stirred for 15 min in a magnetic stirrer and deposited in circular Petri-dishes of 10 cm in diameter and allowed to dry. The thick suspension just before formation of the gel is mildly alkaline with a pH of 9.5. Two electrodes constructed from aluminum foil are fitted to the Petri-dishes. One is in the form of a thin rod placed at the center of the dish, and the counterelectrode consists of an aluminum strip placed at the edge of the Petridish in the form of a short cylinder. A static field is applied from a constant voltage power supply, between the two electrodes. For comparison, we dry a set of samples in an identical arrangement but without the applied voltage. The observations are as follows: (i) We find that radial crack patterns start developing earlier in the samples with applied field than in the sample without field. With the positive terminal in the center, the cracks appear at the central electrode and move out radially in a very straight and symmetric array. The radial cracks always start from the positive terminal. If the outer cylindrical electrode is connected to the positive terminal of the power supply, cracks start at the periphery. Presumably because the field at the periphery is weaker, the gel dries before these cracks reach the center. A pattern in a radial field is shown in Figure 6. The cracks start from the central positive electrode and follow the “lines of force” of the electrostatic field. Another pattern in a different geometry is shown in Figure 7. Here, the two elecrodes are in the form of two semicircles, as indicated in the figure. Again, the cracks emanate from the positive electrode along the lines of force. The clay seems to be repelled from the negative electrode, probably because it carries negative charge; this creates an air gap between the clay and the electrode. Very few or no cracks are observed at the negative electrode. This is true for all the different geometrical arrangements we studied. It appears further that the field gradient has a role to play. A uniform electric field setup between planar electrodes on opposite sides of a rectangular trough failed to affect the formation of desiccation cracks.16 The laponite platelets carry a nonzero quadrupole24 moment, which interacts with the field gradient in an inhomogenous field; this may affect the crack formation. We conjecture that the disk shaped laponite particles crowd near the positive electrode due to their net negative charge
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Figure 8. Closeup of the cracks proceeding from the positive terminal (indicated by + in the figure). The cracks clearly originate at the surface, and the tip has a wedge shape. Eventually, the cracks reach the bottom surface. Figure 7. Electric field of 200 V applied between two semicircular electrodes marked in the figure. The cracks start from the positive terminal and follow the lines of force.
and the field gradient tends to align them. Adjacent particles repel their neighbors in the crowded regions due to their similar surface charge, thus forming the cracks early. However, more detailed analysis and observation of the microstructure is needed to understand the processes fully. (ii) Another interesting observation is that if the sample is left to dry in the electric field with excess water, the solid gel starts to form within about 15 min near the positive electrode. In a sample without a field, this takes several hours or even days. Detailed observation of this phenomenon is in progress. Since the mechanism of gelation in laponite is still not fully understood and conflicting theories exist,25 these observations may be of importance. The texture of this field-induced gel appears different from the homogeneous gel formed when a more concentrated sample near gel point is allowed to set. Detailed investigation of the microstructure is in progress. (iii) The drying of the gel and its fracture take place slowly compared to the samples with methanol solvent. This allows us to observe and photograph the cracks during formation. There is some debate over whether the cracks form first at the upper surface of the sample or start from the substrate.26 It is possible that either may happen, depending on the conditions. But, in this set of experiments, the cracks clearly form at the upper surface, the wedge-shaped tip cuts into the sample like a knife, and finally reach the lower surface. Figure 8 shows the cracks from an oblique angle. In absence of the field, similar cracks form at the surface, but these advance symmetrically, in both directions. If pinning at the substrate, impeding slipping, is primarily responsible for crack formation, it is likely to start at the substrate, i.e. at the bottom of the sample. On the other hand, if cohesion between clay particles is weaker than the adhesion to the surface, failure should start at the surface. Another determining factor is the rate of drying at the surface, compared to the rate of downward penetration of the drying front. For a thick layer (like the swollen aqueous gels), evaporation at the surface is faster than the speed of escape of moisture from the bottom, so cracks are more likely to form at the surface. For the methanol suspensions, crack initiation at the bottom surface
cannot be ruled out, but in our experiments, the cracks form too rapidly to monitor their formation kinetics. 5. Conclusion Different aspects of desiccation crack patterns in clay were discussed. The pattern formation process is fascinating as an academic problem, especially the effect of an electric field with a specific symmetry has been observed only recently18,19 and is not yet fully understood. It needs to be explained through theory or simulations. The scaling behavior of the area covered by cracks is yet another interesting aspect. Since the study involved clay, which in its natural or synthetic form is a very familiar and ubiquitous material, the study should be of practical importance as well. Laponite17 is used in many commercial products, so both control and stimulation of cracking or peeling may have important applications in industry and the physics and chemistry behind the processes must be well understood. Acknowledgment The authors sincerely thank D. Mal, T. Dutta, and Kaushik Das who have helped with the experiments and participated actively in this study. Thanks are due to T. R. Middya for encouragement and valuable suggestions. S.T. is indebted to Unilever and INSA for providing assistance in attending the first International Conference organized by Unilever in Beijing where this work was presented as a poster. Finally, Rockwood Additives Ltd. is gratefully acknowledged for gifting the laponite sample. Literature Cited (1) Mal, D.; Sinha, S.; Mitra, S.; Tarafdar, S. Field induced radial crack patterns in drying laponite gel. Physica A 2005, 346, 110–115. (2) Groisman, A.; Kaplan, E. An Experimental Study of Cracking Induced by Desiccation. Europhys. Lett. 1994, 25, 415. (3) Bohn, S.; Platkiewicz, J.; Andreotti, B.; Adda-Bedia, M.; Couder, Y. Hierarchical crack pattern as formed by successive domain divisions. II. From disordered to deterministic behavior. Phys. ReV. E 2005, 71, 046215. (4) Allain, C.; Limat, L. Regular Patterns of Cracks Formed by Directional Drying of a Collodial Suspension. Phys. ReV. Lett. 1995, 74, 2981.
6464 Ind. Eng. Chem. Res., Vol. 47, No. 17, 2008 (5) Okubo, T.; Nozawa, M.; Tsuchida, A. Kinetic aspects in the drying dissipative crack patterns of colloidal crystals. Colloid Polym. Sci. 2007, 285, 827. (6) Lecocq, N.; Vandewalle, N. Dynamics of crack opening in a onedimensional desiccation experiment. Physica 2003, 321, 431. (7) Shorlin, K. A.; de Bruyn, J. R.; Graham, M.; Morris, S. W. Development and geometry of isotropic and directional shrinkage-crack patterns. Phys. ReV. E 2000, 61, 6950–6957. (8) Komatsu, T. S.; Sasa, S. Pattern Selection of cracks in directionally drying fracture. Jpn. J. Appl. Phys. 1997, 36, 391. (9) Kitsunezaki, S. Fracture patterns induced by desiccation in a thin layer. Phys. ReV. E 1999, 60, 6449. (10) Sadhukhan, S.; Roy Majumder, S.; Mal, D.; Dutta, T.; Tarafdar, S. Desiccation cracks on different substrates: simulation by a spring networkmodel. J. Phys.: Condens. Matter 2007, 19, 356206. (11) Colina, H.; Arcangelis, L. De.; Roux, S. Model for Surface cracking. Phys. ReV. B, 1993, 48, 3666. (12) Colina, H.; Roux, S. Experimental model of cracking induced by drying shrinkage. Eur. Phys. J. Eng. 2000, 1, 189. (13) Iben, H. N.; O’brien, J. F. Generating surface crack patterns. Eurographics/ACM SIGGRAPH 2006, 177. (14) Spry, A. The origin of columnar jointing, particularly in basalt flows. J. Aus. Geol. Soc. 1962, 8, 192. (15) Goehring, L.; Morris, S. W.; Lin, Z. Experimental investigation of scaling of columnar joints. Phys. ReV. E 2006, 74, 036115. (16) Nakahara, A.; Matsuo, Y. Imprinting Memory into Paste and Its Visualization as Crack Patterns in Drying Process. J. Phys. Soc. Jpn. 2005, 74, 1362.
(17) Laponite technical information; Rockwood Additives limited. (18) Mal, D.; Sinha, S.; Middya, T. R.; Tarafdar, S. Field induced radial crack patterns in drying laponite gel. Physica A 2007, 384, 182. (19) Mal, D.; Sinha, S.; Middya, T. R.; Tarafdar, S. Desiccation Crack Patterns in Drying Laponite Gel Formed in an Electrostatic Field. Appl. Clay Sci. 2008, 39, 106. (20) Vicsek, T. Fractal Growth Phenomena; World Scientific: Singapore, 1992. (21) Mal, D.; Sinha, S.; Mitra, S.; Tarafdar, S. Fractal crack patterns in laponite films and their scaling behaviour. Fractals 2006, 14, 283–288. (22) Meakin, P. A simple model for elastic fracture in thin films. Thin Solid Films 1987, 151, 165. (23) Mal, D.; Sinha, S.; Dutta, T.; Mitra, S.; Tarafdar, S. Formation of crack patterns in clay films: desiccation and relaxation. J. Phys. Soc. Jpn. 2007, 76, 014801. (24) Dijkstra, M.; Hansen, J. P.; Madden, P. A. Gelation of a clay colloid suspension. Phys. ReV. Lett. 1995, 75, 2236. (25) Li, L.; Harnau, L.; Rosenfeldt, S.; Ballauff, M. Effective interaction of charged particles in aqueous solution: Investigations of colloid laponite suspensions by light scattering and small angle x-ray scattering. Phys. ReV. E 2005, 72, 051504. (26) Weinberger, R. Initiation and growth of cracks during desiccation of stratified muddy sediments. J. Struct. Geol. 1999, 21, 379.
ReceiVed for reView October 14, 2007 ReVised manuscript receiVed February 28, 2008 Accepted March 3, 2008 IE071375X
