Article pubs.acs.org/JPCC
Laser-Driven Accelerated Growth of Dendritic Patterns in Liquids Himanish Basu,† Kiran M. Kolwankar,‡ Aditya K. Dharmadhikari,† Jayashree A. Dharmadhikari,† Kapil Bambardekar,† Shobhona Sharma,† and Deepak Mathur†,* †
Tata Institute of Fundamental Research, Mumbai 400 005, India Department of Physics, Ramniranjan Jhunjhunwala College, Ghatkopar (West), Mumbai 400 086, India
‡
S Supporting Information *
ABSTRACT: We report a scheme for very significantly accelerated growth of dendritic patterns in organic, inorganic, polymeric, and biological liquids, using laser power as low as a few hundred microwatts in the presence of an efficient absorber such as carbon nanotubes (CNTs). The CNTs act as a heat source that drives dendritic growth; their anisotropy ensures a rich diversity of branched patterns. We have studied the time evolution of the accelerated growth patterns; growth patterns are seen on millisecond time scales. We rationalize such unprecedented speed of dendritic growth using a diffusion equation for the temperature field with an additional source term. The predictions of the well-established microscopic solvability theory (MST) are seen to hold on time scales in excess of 100 ms even with the introduction of an additional source term to account for our laser beam. On the other hand, on time scales shorter than 100 ms, MST is seen to break down. Our method opens new vistas for studies on the dynamics of dendritic patterns and crystal growth.
1. INTRODUCTION Humans have long contemplated patterns in nature, especially in the context of beauty and the diversity of form. Insights into how stable patterns are formed in systems that are out-ofequilibrium continue to be elusive,1−3 despite the significance of pattern formation in emerging areas such as the tailoring of new materials with selected properties and applications such as bacterial growth, electrodeposition, viscous-fingering, and solidification. Experiments on dendritic growth have hitherto relied on two archetypal scenarios: (i) dentritic solidification of a pure substance from its supercooled melt, where pattern formation is driven by supercooling and is controlled by diffusion of the latent heat that is generated away from the interface in the course of the transformation; this usually leads to dendritic patterns with branching instability; and (ii) the well-known Hele-Shaw cell, in which two immiscible liquids are constrained to move between narrowly separated plates2 such that viscous fingering patterns arise as high-viscosity fluid is displaced by low-viscosity fluid under pressure; such patterns usually show tip splitting instability. Newer experiments have involved organic molecules,4 Langmuir monolayers,5 and other materials:6,7 the parameters of importance in such work have been tip radius and velocity. In our experiments, we have formed branched dendritic structures in liquids of different types on time scales that are 2 orders of magnitude faster than hitherto encountered. Such accelerated growth, which is observed by us in diverse solutions (organic, inorganic, polymeric, and biological) is achieved in our laser-based experiments by relying on carbon nanotubes as inert mediators that act as efficient converters of optical energy into localized thermal energy.8 This amounts to an additional physical term © 2012 American Chemical Society
describing the heat source in the usual equation of interface growth. The success of our technique may have implications in the development of new strategies for accelerated crystal growth and in diverse chemical applications that demand the creation of temperature gradients. We probe dendritic pattern formation in various solutions such as poly-L-lysine, agarose, dimethyl amino stilbazolium tosylate (DAST), gelatin, and a protein solution, bovine serum albumin (BSA). Typically, a solution is irradiated by a tightly focused but low-powered (0.2−5 mW) infrared laser beam. Absorbers (carbon nanotubes) are suspended in the liquid medium. The heat that is generated within the laser focal volume (by absorption) is transferred to the liquid with an efficiency that depends on the absorber’s thermal conductivity. In other words, our experimental arrangement ensures that the laser-irradiated absorber acts as a localized heat source, which leads to dehydration of the proximate liquid, giving rise to an interface between a dried or semidried region and the liquid region. This interface, in turn, becomes unstable and results in a branched structure. The instability is a consequence of the outgrowth in the interface growing faster due to large temperature gradients9 and offers a unique opportunity to test the applicability of the well-established microscopic solvability theory.2 Pattern formation in our experiments is not permanent in that once the laser is switched off, the patterns dissolve unless the solution itself is on the verge of drying. Although short laser pulses of high peak power have Received: April 3, 2012 Revised: April 28, 2012 Published: May 8, 2012 11480
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Figure 1. Time-evolution of dendritic patterns growing in diverse materials irradiated by 5 mW of 1064 nm laser light.
been used to study side-branching initiation in dyes,10 to the best of our knowledge, a low-power (microwatts) continuous wave (cw) laser has never been employed with absorbing mediators to drive growth and to achieve growth speeds of the type we encounter in our experiments. Recently, a high power (1W) cw laser, focused onto a supersaturated D2O solution of glycine, was shown to form glycine crystals.11,12
variation is ascribed to differences in absorber size as well as variations in the thermal properties of the different solutions. To help develop insights into the dynamics, we focus attention in the following on dendritic growth in one of the solutions that we have studied, BSA in phosphorus buffer saline (PBS) solution. BSA (1−2% w/v) was dissolved in 300 mOsm PBS; 0.5−1 mg of carbon nanotubes (CNTs) was added to the solution and dispersed by ultrasonication (for 15 s, five cycles). The CNTs acted as efficient absorbers of 1064 nm light. A small quantity of prepared solution was placed in a thin (0.1 mm) glass cell and was irradiated by a cw, 1064 nm, Nd:YVO4 laser. The initial laser beam diameter (2 mm) was passed through a beam expander so as to produce a parallel beam of 8 mm diameter, which was then transmitted through a 1:1 telescopic arrangement and steered through a 45° mirror onto a
2. EXPERIMENTAL SECTION With 3 orders of magnitude lower laser power, we observe dendritic patterns when diverse solutions are irradiated in the presence of an absorber (Figure 1) using only 5 mW laser power. Pattern formation commences on fast time scales, ranging from a few tens of milliseconds to a few seconds. This 11481
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large numerical aperture (NA = 1.3) 100× microscope objective, producing a focused spot of ∼1 μm diameter. This was the initial dry-region size that determined primal tip size. Patterns resulting from irradiating the samples at this spot were imaged onto a CCD camera (either JVC TKC-1480E operating at 25 Hz or PCO-1200HS operating at 1 kHz) coupled to a computer for real-time recording. Individual frames from movie clips were extracted for further analysis of growth dynamics, including quantification of growth velocity and dendritic tip radius. The imaging system was calibrated using standard 2 μm diameter polystyrene beads. Details of our experimental apparatus and methodology have been recently published.8,13−15 In order to experimentally confirm the crucial role played by CNTs in the dynamics that we explore in this work, we conducted experiments in which multiple laser beams were focused on the microscope slide containing the solution of interest. The experimental arrangement was such that only some of the focused laser spots coincided with the location of CNT bundles. The observation that dendritic patterns only occurred when there was a spatial coincident between CNT bundles and focused laser spots confirmed the role of the CNT absorber. Our multiple laser beams were created using a wiremesh technique developed in our laboratory.16 A Ni mesh with 175 μm spacing and 95 μm wire thickness (50% transmission) was placed in the beam path. Our wire mesh creates a diffraction pattern in the two-dimensional plane that lies transverse to the laser propagation direction. This pattern is reproduced at specific distances that are obtained from Fresnel’s diffraction law. Such a diffraction pattern, when focused using a microscope objective, has been shown to produce multiple arrays, each of which constitutes an optical trap.16 The multiple focused spots thus created can readily be translated and rotated by fixing the wire mesh on a translation stage whose motion can be precisely controlled, and the distance between adjacent focused spots can be easily altered. We used single-walled (SWNT) and multiwalled (MWNT) carbon nanotubes as well as graphite as absorbers. The absorption coefficients for 1064 nm light are very large in all three cases: SWNT (105 cm−1), MWNT (8 × 103 cm−1), graphite (5 × 105 cm−1).17 The thermal conductivity of SWNT (3500 W mK−1) is far in excess of that for MWNT (50−150 W mK−1) and graphite (80−240 W mK−1).17 On the basis of these numbers, it would be expected that SWNTs are the best of the three mediators; our experiments, indeed, confirmed that SWNT-mediated growth was most efficient in terms of accelerated growth. All samples used in the present experiments, including SWNT, MWNT, and graphite, were obtained from Sigma Aldrich. pH values for all solutions were maintained at 7.2.
Figure 2. (a) Typical tip splitting pattern obtained with two focused laser spots (located by the arrows). The black dots near the arrowheads denote the location of SWNT bundles which are coincident with the laser spots. A movie clip (M1) in the Supporting Information shows the dendritic growth. (b) Dimensional analysis indicates a fractal pattern with box dimension D = 1.72 ± 0.01.
fractal in nature. Such patterns, grown from more than a single seed, each separated by a small distance, have continued to arouse interest18−20 due to applications21 in diverse Laplacian processes, such as bacterial growth, electrodeposition, viscousfingering, and solidification. Materials such as NaCl are found to exhibit dramatically fast crystallization, as is depicted in Figure 3. Our experiments reveal what we believe to be the first visual evidence of crystal growth on time scales as fast as a few tens of millisconds. This is a manifestation of dramatic, locally induced concentration changes brought about by the localized heating of the CNT absorber by the tightly focused laser beam used in our experiments. We have also studied the time evolution of the accelerated growth patterns using two spatially separated laser beams (Figure 4a). Growth patterns are clearly established within only a few hundred milliseconds. In the case of natural drying (Figure 4b), the pattern formation is purely diffusion limited; it occurs on distinctly longer time scales, of the order of seconds. Of course, for natural drying, our experiments do not reveal the initial growth point, precluding quantitative comparison with laser-induced growth. However, earlier work has shown that for side-branching, the growth is faster at the branch than at the stem.22 In all cases, we found the speed at which the tips grow upon absorber-mediated laser irradiation to be several orders of magnitude faster than the growth rate of similar patterns upon natural drying. Growth velocities measured at different values of incident laser power are shown in Figure 5. Our data indicate that the accelerated growth is ascribable to (i) the coupling of energy from the laser (absorption) and (ii) its efficient utilization (conduction) by the absorber. The growth speed eventually decreases and reaches an asymptotic value that is the same as that obtained in natural drying (Figure 5). Figure 5 also highlights the very significant difference in the temporal evolution of v, the tip velocity, obtained for patterns formed upon natural drying and upon exposure to different values of incident laser power. Note that even with only 2 mW of laser power, the initial growth velocity increases by an order of magnitude. Measurements with different absorbers of various sizes establish that growth velocities obtained with SWNTs bundles of a given surface area are significantly higher than those obtained with graphite of approximately the same dimensions. Such difference is a consequence of the widely different thermal conductivity of SWNTs on one hand and graphite on the other, even though the respective absorption coefficients are very similar.
3. RESULTS AND DISCUSSION Figure 2 shows a typical example of pattern formation with multiple laser beams. Here it has been arranged that locations of two of these beams coincided with the locations of SWNT absorbers; only in these two locations were dendritic patterns seen to evolve, on time scales of 100−500 ms. This establishes the requirement of an absorber for laser-accelerated dendritic pattern formation. The absorber needs to efficiently convert incident optical energy into thermal energy, thereby creating a sharp temperature gradient (see later). A dimensional analysis of the resulting patterns yields a box dimension, D, of 1.72 ± 0.01, indicating that the accelerated growth pattern is, indeed, 11482
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Figure 3. Time-evolution of a sodium chloride crystal upon irradiation of a NaCl solution mixed with CNTs by 5 mW of laser power focused using a 10× objective. A real-time movie clip (M2) showing the accelerated growth of the NaCl crystal is available as Supporting Information.
Figure 5. (a) Time evolution of tip velocity for dendritic patterns generated upon irradiating a PBS/BSA mixture containing CNT bundles with 0.2 to 2 mW of 1064 nm light. The slow rate of growth upon natural drying (0 μW) is also shown (in black). (b) Comparative time evolution of laser-assisted growth and that obtained upon natural drying. (c) Variation of initial growth velocity with incident laser power.
individual molecules (such as BSA) that are dissolved in the host solution (such as PBS). We rationalize the growth patterns observed in our experiments by considering the heat flow in the classic two-phase problem:23 ciρi
∂ui = K i∇2 ui + Q i ∂t
(1)
where i = 1 refers to the solid phase, and i = 2 refers to the liquid phase. ui denotes temperatures in the two phases and ci, ρi, and Ki denote specific heat, density, and heat conductivity, respectively, which one can assume to be the same in both phases. Qi is the heat generation term. In our experiments, Q1 is a function peaked at the origin, which is also anisotropic owing to the anisotropy of absorbers, and Q2 = 0. The heat term makes the equation inhomogeneous. At the interface, we invoke Stefan’s boundary condition, which expresses the heat balance on the interface as
Figure 4. (a) Time evolution of a typical tip splitting pattern obtained using only 0.2 mW laser power with two focused laser spots (positions marked by arrows indicate the locations of the SWNT bundle coincident with laser spots). The solute was BSA in PBS (see text). (b) Longer-time formation of patterns upon natural drying (no laser applied).
Hitherto, theoretical rationalization of pattern formation has depended on Laplace’s equation and the diffusion equation along with relevant boundary conditions to gain insights into viscous fingering and dendritic solidification, respectively. This system of equations may also be replaced by equivalent integral equations (see, for instance, ref 3 for a review). Several analytical and numerical methods have been developed to study these systems. Earlier attempts used a perturbative approach, but it is only relatively recently that the singular nature of surface tension has become clear, necessitating the development of so-called microscopic solvability theory and recognition of the crucial importance of anisotropy. The anisotropy we refer to is a consequence of the random orientation of both the absorbing materials (such as CNT bundles) as well as the
−Lρνn = K 2n·̑ ∇u 2 − K1n·̑ ∇u1
(2)
where L is the latent heat, n̑ is the outward normal to the moving boundary, and vn is the velocity of this boundary along the normal. The heat source term Q1 in eq 1 and, consequently, the second term in eq 2 describe physics that does not exist in usual dendritic solidification or viscous fingering. It will be interesting to study the effect of Q on the stability of the interface or its implications in microscopic solvability. Equation 2 immediately reveals some insight into our experimental observations. The functional dependence of velocity on time 11483
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growth, a natural question arises as to whether the wellestablished microscopic solvability theory (MST) is still satisfied. MST seeks a mathematical solution to the steadystate growth equations that determine the time evolution, at constant velocity, of the solid−liquid interface while preserving its parabolic shape.24 Stability criteria in MST, as well as in earlier models, predict the mathematical form vr2 to be constant. This parameter is, consequently, useful to experimentally validate theoretical models. In our work, the parameter vr2 is experimentally accessible: the velocity was determined in our experiments by analyzing individual movie frames, each of which is temporally separated by an amount that is controlled by the camera frame rate at which specific measurements are made. In our experiments, frame rates in the range of 25−1250 frames per second were used, enabling dendritic growth patterns to be explored on time scales in the range 40−0.8 ms, respectively. The radius of the tip was also monitored in individual frames from movie clips; as noted above, a parabola was fitted to the tip observed in each frame, and values for the radii were extracted. Langer2 has conjectured the breakdown of MST in systems that are away from the steady state and possess a large anisotropy. As is well-known, in the case of dendritic solidification, supercooling drives growth, and one of the key signatures of MST is the invariance of vr2 over a wide range of supercooling. In our system, the drive depends on the laser power and also on the distance of the tip from the laser focus, which increases with time. In the context of MST, we note that the constant value of vr2 after about a hundred milliseconds is an experimental indicator of the robustness of the theory. However, in the first few tens of milliseconds, at high values of irradiating laser power and in close proximity to the heat source), the value of vr2 becomes up to an order of magnitude higher (Figure 6). This departure from a nonconstant value signals the breakdown of MST in the sense that the stability criterion vr2 = constant is violated because of the additional source term that is introduced by our laser beam. To the best of our knowledge, such a clear experimentally induced violation of MST on very short time scales has not been reported before. On the other hand, the predictions of MST are not violated at large time scales (>100 ms) even with the additional source term that our laser beam introduces. Is the violation of MST obvious since the system we have examined may be considered to be different in the sense that it requires an additional heat generation term? As is well-known for Laplacian growth phenomena, it is the existence of finite surface tension that introduces singular terms that lead to the breakdown of the original continuum of solutions, leading to the emergence of a discrete set of solutions, each with unique velocity. The MST has been seen to apply to various systems, such as Saffman−Taylor fingers and simple models of interfacial evolution. However, our results appear to indicate that the MST is certainly valid for low laser power and away from the heat source; it only breaks down for high laser powers and near the heat source.
(Figure 5b) can be understood from Stefan’s boundary condition in eq 2, which decides the interface velocity. When the interface is in the proximity of an irradiated absorber, the temperature gradient in the liquid is very high compared to that pertaining to the case of natural drying. This higher temperature gradient results in a larger interface velocity. As the interface moves away from the laser focal volume, this temperature gradient decreases, and, consequently, the overall velocity decreases and approaches a steady state. This steady state velocity (Figure 5b) becomes smaller than that in the case of a normal drying pattern after a long time (>2 s). This is due to the temperature gradient in the solid phase, which is negligibly small for normal drying but not in the case of laser irradiation. Another feature of Figure 5a is that as the laser power increases, the tip velocity drops more rapidly. This is again a consequence of the second term in Stefan’s boundary condition. The initial large speed of the interface, for larger laser power, takes it quickly away from the heat source where the difference in the temperature gradients (which decides the velocity of the interface) is small. The general behavior of our measured data appears consistent with the expectations of eqs 1 and 2 even though quantitative comparison is precluded because of the paucity of information on the microscopic parameters in the equations. To explore the spatial features of the pattern generation dynamics in our experiments, it is instructive to consider the time evolution of the parameter vr2, where r is the tip radius (which we determine by fitting a parabola to the tip). vr2 enables us to account for the role of tip radius in determining growth rates. We found that for natural drying, a time-invariant vr2 relationship was obtained, as expected for steady-state conditions. For laser-induced growth, we obtained a very different temporal profile, with an extremely sharp falloff within the initial 50 ms (Figure 6). This reflects the accelerated growth of the dendritic pattern when spatial effects (the r2 term) are accounted for.
Figure 6. Time evolution of the parameter vr2, where v is the tip velocity and r is the tip radius for dendritic patterns, at different values of irradiating laser power. Note that at the lowest laser power, vr2 remains essentially constant over time, as is the case for normal drying. At laser powers ≥1 mW, vr2 values deviate from their constant value at times shorter than 100 ms.
4. SUMMARY In summary, we have shown that the growth of dendritic patterns in diverse liquids can be very significantly accelerated upon irradiation by even a few hundred microwatts of laser power in the presence of an absorber. Bundles of carbon nanotubes fulfill the role of absorber very efficiently. Our approach opens new vistas for studies of pattern formation,
We have considered the possibility that laser-induced increase in local concentration may also enhance the growth rate if r and the diffusion coefficient are invariable in the Péclet number, PeL. In our experiments, PeL ∼ 0 and, hence, convective dynamics dominate. For laser powers greater than 10 mW, growth was too fast for us to reliably plot the temporal evolution of vr2. In our studies, wherein a heat source drives the 11484
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(22) Honda, T.; Honjo, H.; Katsuragi, H. J. Cryst. Growth 2005, 275, e225−e228. (23) Crank, J. Free and Moving Boundary Problems; Oxford University Press: Oxford, U.K., 1984. (24) Kessler, D.; Koplik, J.; Levine, H. Adv. Phys. 1988, 37, 255−339.
theoretical as well as experimental, and should stimulate analytical investigations of the effect of the source term on the stability and microscopic solvability. Accelerated laser-induced patterns may have broader implications in future studies that involve temperature gradients in natural and man-made systems and phenomena.
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ASSOCIATED CONTENT
S Supporting Information *
Two movie clips, M1 and M2, are available as Supporting Information. This information is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Dr. Alpana Mishra is thanked for the DAST powder. Useful discussions with Deepak Dhar are acknowledged. K.M.K. thanks the University Grants Commission of India for financial support. J.A.D. thanks the Department of Science and Technology for assistance under the Women Scientists Scheme.
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REFERENCES
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