Journal of the Korean Physical Society, Vol. 40, No. 6, June 2002, pp. 1000∼1004
Pattern Dynamics in a Thin Granular Layer under Vertical Vibration Kipom Kim and Hyuk Kyu Pak∗ Department of Physics, Pusan National University, Busan 609-735 (Received 10 October 2001) The process of pattern formation in granular layers was experimentally studied. Ten layers of granular materials were prepared inside a vacuum container under a vertical vibration of A sin 2πf t. The control parameters were the dimensionless acceleration Γ = A(2πf )2 /g and the vibration frequency f . When the system was quenched from a flat pattern state to a striped pattern state by increasing Γ, it took more than 104 periods before a full steady striped pattern appeared. This nonequilibrium and non-steady process showed a dynamic scaling behavior. The growth exponent of the characteristic length scale of the ordered domain was 0.25, which agreed with that of the SwiftHohenberg system. Furthermore, we show that the ordering process in the pattern is dominated by defect annihilation with the topological constraint of conserving the Burger’s vector. PACS numbers: 45.70.-n, 47.54.+r, 47.20.-k Keywords: Patter formation, Granular system
I. INTRODUCTION With an increase in the control parameters, certain dissipative systems show various instabilities from no pattern to various organized patterns [1]. In particular, shallow granular layers under vertical vibration show a rich variety of patterns including flat, stripe, square, hexagon, phase discontinuity, and solitary structures [2]. Each pattern in these driven systems corresponds to a non-equilibrium steady state. The coarsening process of patterns in driven systems has been studied much less than it has in thermodynamics systems. It is often useful to describe a nonlinear system near the instability threshold by using a phenomenological model equation that shows the same linear instability as the experimental system [1]. The coarsening process of striped patterns in some nonlinear dissipative systems with type I instability can be well described by the SwiftHohenberg model [3]: ∂u = (ε − (∇2 + kc2 )2 )u − u3 , ∂t
(1)
where u(r, t) is an order parameter with a typical wavevector kc , and ε is the distance from the threshold. Several numerical methods have been used to describe the pattern dynamics of this model [4–8]. During the formation of stripes in the Swift-Hohenberg model, the width of the structure factor, w(t), has been shown to decay in two distinct stages [4,5]. In the early-time region, due to the small amplitude of the order parameter, the linear term in Eq. (1) dominates the system, and ∗ E-mail:
w(t) decays rapidly as w(t) ∼ t−1/2 . In the late-time region, due to the large amplitude of the order parameter, nonlinear effects emerge, and w(t) decays slowly as w(t) ∼ t−1/5 in the case of no noise. Moreover, when the correlation function of the local orientation order parameter is computed in the late-time region in real space, the characteristic length grows algebraically as L(t) ∼ tz with z = 0.25 in the case of no noise [6,7]. In real space analysis, the density ρ(t) of topological defects decays algebraically as ρ(t) ∼ t−y with y = 0.25 in the case of no noise [8]. Since most topological defects can be found in borders between ordered domains, the defect density scales in the same way as the perimeter density of the domains. Therefore, the scaling exponents z and y have the same value. These numerical results agree with the two recent experiments on micro-phase separation in block copolymers [9] and electroconvection in liquid crystals [10]. The former is a thermodynamic system, and the latter is a driven system. For a driven system, initial and final states are not in thermodynamic equilibrium. Instead, the final states are in non-equilibrium steady states. The coarsening process in the electroconvection does not have the same symmetry as the SwiftHohenberg model [10]. In this paper, we present an experimental study on the coarsening dynamics of striped patterns in granular layers under the vertical vibration with a single frequency. In this system, the energy supplied by external forcing is in balance with the dissipation due to inelastic collisions and frictions. Immediately after a rapid change (quench) of a control parameter from a flat pattern to a striped pattern, rolls of a well-defined wavelength developed. Initially the rolls were randomly oriented. As
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Pattern Dynamics in a Thin Granular Layer under Vertical Vibration – Kipom Kim and Hyuk Kyu Pak
Fig. 1. Schematic diagram of the experimental setup.
time progressed, however, they locally aligned in parallel, thereby creating an increasingly ordered pattern. The characteristic length scale of the ordered rolls was measured and compared with that of the Swift-Hohenberg model.
II. EXPERIMENTAL SETUP Figure 1 is a schematic diagram of the experimental setup in this study. Ten layers of phos-bronze spherical particles with diameters of d = 0.2 mm filled a cylindrical plexiglas cell of 110 mm in diameter. The cell was mounted on an electrical vibration exciter (Ling Dynamics, V408) with its principal axis parallel to the gravitational direction. The exciter was driven by an AC power source/analyzer (HP 6813A) which was controlled by a computer. The cell vibrated vertically with the vertical position of z(t) = A sin 2πf t. Therefore, the maximum value of the dimensionless acceleration was Γ = A(2πf )2 /g, where g is the gravitational acceleration. The dimensionless acceleration amplitude Γ and the vibration frequency f were monitored by using a PCB piezoelectric accelerometer. The combination of the cell and the exciter was mounted inside a pressure-controlled container which was evacuated down to P = 0.1 Torr, where the volumetric effects of the gas were negligible and heaping was not observed [11]. In order to see the image of the pattern more clearly, we illuminated the free surface of the granular system at low angle with a halogen ring light. A high-speed camera (Kodak SR Ultra-C) was positioned above the container to record the image of the free surface. Bright parts in the recorded images corresponded to the high area of the free surface, and dark parts corresponded to the low area of the free surface. In this experiment, the vibration frequency was fixed at f = 50 Hz, where the transition from a flat state to a
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striped pattern state occurred at a critical dimensionless acceleration of Γc = 2.6. The system was quenched from Γi = 2.4 (flat state) to Γf = 2.8 (striped pattern state) by instantly increasing the vibration amplitude A with a rise time of 50µsec, which was much shorter than the vibration period T = 1/f (=0.02 sec). Typically, it took more than 104 periods before a full steady striped pattern appeared. Since the striped pattern state is sub-harmonic waves, in order to compare the striped patterns at the same phase, we captured images at every two vibration periods by triggering the high-speed camera externally. Each image was taken at the moment just before the granular layer and the bottom plate collided with each other; that is where the cleanest image of pattern appeared. In this letter, the unit of time is the vibration period T , and the origin of time is the moment of the first layer-plate collision after the quench.
III. RESULTS AND DISCUSSION Figures 2(a)-(d) show the real images of the coarsening process for Γf = 2.8. The bright parts correspond to the crests of the free surface, and the dark parts correspond to the troughs of the free surface. Immediately after the quench, rolls of well-defined wavelength developed with random orientation. Figure 2(a) shows the pattern at two periods after the quench. As time progressed, the rolls locally aligned in parallel, thereby creating an increasingly ordered pattern [Fig. 2(b), (c)]. After a long time, a fully ordered striped pattern finally appeared [Fig. 2(d)]. Figures 2(e)-(h) show the results for the Swift-Hohenberg equation for ε = 0.2 in time, starting from a featureless noisy state. Here, the coarsening dynamics of the striped pattern shows very similar spatio-temporal morphology in both the experiment and the numerical calculation. The Swift-Hohenberg model is based on three phenomenological rules. First, the system undergoes a type-I instability which generates a pattern with a given wavelength around a critical value. Second, the equation is invariant under a u → −u transformation. In this experiment, that corresponded to an exchange of the crest and the trough parts in the striped pattern. Third, due to the nonlinear interaction term, u3 , near the threshold, the system has a supercritical bifurcation, and a striped pattern is selected as the state with the lowest Lyapunov energy functional [12]. The experimental transition from a flat state to a striped pattern state did not show any hysteresis, confirming a supercritical bifurcation. Therefore, the coarsening process of striped patterns in this experiment satisfied all three properties. Thus, we conjecture that the coarsening dynamics of the striped patterns in granular layers under a vertical vibration and that of the Swift-Hohenberg model are in the same universality class near the critical control parameter. The average distance between neighbor-
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Journal of the Korean Physical Society, Vol. 40, No. 6, June 2002
Fig. 2. Evolution of patterns in time. The real images of the free surface of granular layers for Γf = 2.8 at (a) t = 2, (b) t = 10, (c) t = 200, and (d) t = 1000. The bright parts correspond to the crests of the free surface, and the dark parts correspond to the troughs of the free surface. (e)-(h): The images of the numerical results for the 2D Swift-Hohenberg equation with ε = 0.2 in time.
ing stripe rolls was nearly time-independent in the time range of t > 15. This property agrees with the typical behavior of a type-I instability [1]. Experimentally, the free surface was illuminated at a low angle, and the scattering intensity was a nonlinear function of the height of the layer in the late-time region. Therefore, we could not accurately analyze the pattern dynamics by using the structure factor in kspace. Instead, we used the local properties of the striped pattern in real space [13]. We first calculated the local stripe orientations θ(r, t), which were defined as the local angles of the stripes [13]. Using the local stripe orientations, we could define the local order parameter ψ(r, t) = exp [2iθ(r)], where a factor of 2 was included because the orientation of the stripe is 2-fold degenerate. The angle-averaged correlation function C(r, t) was calculated from the local order parameters: C(r, t) = hRe[ψ(0)∗ ψ(r)]i = h[cos(2δθ(r))]i,
(2)
where δθ(r) is the difference in stripe orientation between the rolls at a distance of r. The expected scaling form for the correlation function is C(r, t) = g (r/L(t)) = g (r/tz ) ,
(3)
where L(t) ∼ tz is the characteristic length scale of the ordered domain. Figure 3(a) shows the time evolutions of L(t) in the experiment. In the late-time region of t = 20 − 200, the log-log plot of L(t) is a straight line, and the exponent z is nearly 0.25 for Γf = 2.8, 3.0, and 3.2 [Fig. 4(b)]. The value 0.25 agrees with the numerical and experimental results [6,7,9,10]. We could also observe the coarsening behavior when quenched from a flat pattern state to a square pattern state. Recently, Swinney’s group studied this feature using the disorder function [14]. Since the transition from a flat state to a square pattern state is subcritical, they had to add new terms to Eq. (1) in order to explain the coarsening process [15].
Fig. 3. Time evolution of L(t) shows that the characteristic length scale of the ordered domain grows with a power law of L(t) ∼ tz . (a) The log-log plot of L(t) is a straight line at different Γf . (b) The growth exponent, z, is 0.25 within experimental accuracy.
In order to understand the mechanism of this coarsening process, we analyzed the role of topological defects dislocations. Figures 4(a) and 4(b) are snapshots of the pattern for Γf = 2.8 at t = 60 and t = 220, where the dynamic scaling worked well. Soon after the quench, there were many defects in the system [Fig. 4(a)]. During the coarsening, the number of defects decreased, and the domain area having the same stripe orientation expanded [Fig. 4(b)]. From the viewpoint of a Lyapunov energy functional, excess energy is stored in each defect. Therefore, if the Lyapunov energy functional of the system is to be decreased, the number of defects should be decreased. Figures 4(c)-(e) show the coarsening processes of the A, B, and C parts in Fig. 4(a) in time, respectively. These time series of images show vividly how defects interacted with each other during the coarsening process. In Figs. 4(c) and 4(d), there are two nearby defects initially with Burger’s vectors of the same magnitude, but opposite sign. As time passed, the two nearby defects met and eventually disappeared. In Fig. 4(e), there are two nearby defects with Burger’s vectors of different magnitude and opposite sign. The two defects met
Pattern Dynamics in a Thin Granular Layer under Vertical Vibration – Kipom Kim and Hyuk Kyu Pak
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IV. CONCLUSION The dynamics of the coarsening process in granular layers under vertical vibration was experimentally studied. The growth exponent of the characteristic length scale of the ordered domain agreeed with that of the Swift-Hohenberg system. The coarsening mechanism of the striped pattern and the defect dynamics were also studied. The ordering process in the pattern was dominated by defect annihilation with the topological constraint of conserving the Burger’s vector. This work is the first experiment about the coarsening dynamics of a driven system that has the same symmetry as the SwiftHohenberg model.
ACKNOWLEDGMENTS
Fig. 4. Ordering process in patterns at (a) t=60 and (b) t=220 for Γf = 2.8. (c), (d), and (e) show the interaction of the defects of the A, B, and C parts in (a), respectively, in time.
We would like to thank I. Chang, J. Lee, and B. Kim for valuable discussions. This work was supported by grant No. 1999 − 2 − 114 − 007 − 3 from the interdisciplinary research program of the Korea Science and Engineering Foundation.
REFERENCES
and formed a new defect whose Burger’s vector was the sum of initial two values. In these three cases [Figs. 4(c)(e)], defects always interacted with each other, satisfying topological constraints for conserving the Burger’s vector except for the boundary of the system where defects can move out of system. We could see the same behavior in the numerical calculation with the Swift-Hohenberg model. Therefore, we believe that the ordering process in the pattern formation of granular layers is dominated by defect annihilation with the topological constraint of conserving the Burger’s vector. When the system was quenched to near the critical instability threshold in this study, the initial defects disappeared through annihilation processes, and a fully ordered steady pattern eventually appeared. However, when the system was quenched far above the critical instability threshold, due to the secondary nonlinear instabilities, defects, such as dislocations and spirals, were continuously generated, and we did not observe the fully ordered steady state even long after the quench. In this case, the disordering process of the stripe pattern was similar to the transition to spatio-temporal chaos in the Rayleigh-B´ enard convection system [16]. We believe that this disordering process is the main origin of Fig. 3, where as Γf is increased, the characteristic length scale of the ordered domain L(t) grows slowly in time and scaling exponent z decreases gradually from z = 0.25 .
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