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Depinning dynamics of two-dimensional magnetized colloids on a substrate with periodic pinning centers
Physica A 391 (2012) 2940–2947
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Physica A journal homepage: www.elsevier.com/locate/physa
Depinning dynamics of two-dimensional magnetized colloids on a substrate with periodic pinning centers Y.G. Cao ∗ , Z.F. Zhang, M.H. Zhao, G.Y. Fu, D.X. Ouyang School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China
article
info
Article history: Received 15 November 2011 Received in revised form 5 January 2012 Available online 16 January 2012 Keywords: Depinning Magnetized colloid Plastic flow Elastic smectic flows Elastic crystal flow
abstract We study, by Langevin simulations, the depinning dynamics of two-dimensional magnetized colloids on a substrate with periodic pinning centers. When the number ratios of pinnings to colloids are 1:1 matching and at finite temperature, we find for the first time crossovers from plastic flow through elastic smectic flow to elastic crystal flow near the depinning with increasing the pinning strength. There exists a power-law scaling relationship between the average velocity of colloids and the external driving force for all the three types of flows. It is found that the critical driving force and the power-law scaling exponent as well as the average intensity of Bragg peaks are invariant basically in the region of elastic smectic flow. We also examine the temperature effect and it reveals that the above dynamic moving phases and their transitions could be attributed to the interplay between thermal fluctuation and pinning potential. At sufficiently low temperature, the thermal fluctuation could be neglected and the colloids near the depinning move in the elastic crystal flow no matter what the pinning strength. In addition, the number of pinning centers is changed and when it is close to the number of colloids, there appears a peak in the critical driving force and a dip in the power-law scaling exponent, respectively. The peak and dip are more pronounced for higher pinning strength. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Colloids could display intriguing phase transitions among gas, liquid, solid and liquid crystalline phases [1], which are highly relevant in everyday applications and in basic research [2]. The micron size of colloids allows to image them in videomicroscopy experiments [3–9]. Thus, colloids present an ideal model system for approaching various problems in physical chemistry [10], material science [11], and condensed matter physics [12]. Recently, the dynamics of colloids under external field has attracted much attention [5,7,13]. By investigating the response of colloids to an external field, one can get an insight into the dynamics of colloids, and through examining the dynamics, one can tailor and control the external field [2,14]. Sometimes, the external field may complicate the problem by causing a system to go into a nonequilibrium moving state. When a magnetic field is applied, a magnetic moment will be induced in superparamagnetic colloidal particles. The magnitude of the induced moment scales linearly with the magnetic field strength, implying that the colloid interaction strength could be controlled by the external magnetic field completely [15–17]. Consequently, a near ideal two-dimensional (2D) system could be experimentally realized in magnetized colloids. In reality, colloids interact with pinnings in the solid or liquid substrates [18]. The depinning dynamics of 2D charged colloids on disordered pinning substrates has been studied numerically [19–21]. With an increase in the pinning strength (or a decrease in the colloid interaction strength), a crossover from elastic crystal to
∗
Corresponding author. E-mail address: [email protected] (Y.G. Cao).
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.01.020
Y.G. Cao et al. / Physica A 391 (2012) 2940–2947
2941
plastic flows was observed above the depinning. The crossover was confirmed by recent experiments [7]. Also, the average velocity of colloids near the depinning was found to scale with the drive in a power-law scaling relationship for both elastic crystal and plastic flows. The power-law scaling exponent was observed to be about 2/3 and 2.0 for the elastic crystal and plastic flows, respectively [19,7]. Most recently, we have studied systematically the depinning dynamics of 2D magnetized colloids on a quenched substrate [22,23]. Decreasing the magnetic field strength, we found crossovers from the elastic crystal through the elastic smectic flows to the plastic flows above the depinning [23]. Although there have been many studies of the dynamics of driven colloids on disordered pinning substrates, there are far fewer studies of the dynamics of colloids driven over periodic pinning potentials [24,25]. In this paper, we investigate systematically the depinning dynamics of 2D magnetized colloids on a substrate with periodic pinning centers. When the number ratios of pinnings to colloids are 1:1 matching and at a finite temperature, we find that the colloids near the depinning move in a plastic flow for sufficiently weak pinning potential, as pointed out in phase-field crystal model [26]. But an elastic crystal flow is observed for strong enough pinning potential. These are contrary to the results on disordered pinning substrates [19,20,22]. Increasing the pinning strength, we find for the first time crossovers from the plastic flow through the elastic smectic flow to the elastic crystal flow near the depinning. It is shown that the interplay between thermal fluctuation and pinning potential could be responsible for the above dynamic moving phases and their transitions. The paper is organized as follows. In Section 2, we describe the model and the procedure we have used in the Langevin simulations. In Section 3, we give the numerical results and discussions. We summarize in Section 4. 2. Model The motion of magnetized colloids is described by the Langevin equation
η
dRi dt
=−
∇i Ucc (Ri − Rj ) −
∇i Ucp (Ri − rj′ ) + fLi + f,
(1)
j′
i̸=j
where η is a damping coefficient and fixed to unity, Ri and rj′ denote the coordinates of the ith colloidal particle and the j′ th pinning center in the substrate, Ucc is the interaction potential between colloids, Ucp is the interaction potential between colloids and pinning in the substrate, respectively. f is the external driving force. The random fluctuating force exerted on the ith colloid fLi is described by the correlation functions [19,20,22–24,27] ⟨fiL ⟩ = 0 and ⟨fiαL (t )fjβL (t )⟩ = 2T δij δαβ δ(t − t ′ ), where subscripts α and β take values 1, 2 or 3, and T is the temperature of the system. The interaction between magnetized colloids has the same form as parallel dipoles and could be written as [22,23,28,29] Ucc (Ri − Rj ) =
µ0 M2 , 4π |Ri − Rj |3
(2)
with µ0 the magnetic permeability of free space. M = χ B is the magnetic moment. Here χ is the effective susceptibility µ M2
and B is the magnetic field strength. We keep 40 π = 0.05 in our numerical calculations. The interaction between magnetized colloids and pinnings in the substrate is chosen as conventional attracting Gaussian potential [20,22,23,27,30] −
Ucp (Ri − rj′ ) = −Ap e
|Ri −rj′ |2 rp2
,
(3)
where Ap is a constant, characterizing the strength of pinning potential. The size of the pinning center rp is chosen to be rp = 0.2a0 , with a0 the lattice constant of the perfect triangular lattice. In the above simulations, we use the substrate with Np point-like pinning centers distributed in a triangular lattice subjected to periodic boundary conditions. Nc (=400) magnetized colloids are initially placed at the pinning sites. All the lengths are scaled with respect to a0 . The external driving force f is increased from zero by small increments along the Nc ˆ horizontal symmetry axis (y axis) and the average velocity v = N1 i=1 v · y is measured at each increment. c
3. Results and discussions 3.1. Influence of the strength of pinning potential At first, we investigate the influence of the pinning strength on the depinning of 2D magnetized colloids. The number ratios of pinnings to colloids are 1:1 matching. The temperature T is fixed to be Tm0 of the ‘‘bare’’ Kosterlitz–Thouless melting temperature of 2D system [31], and the pure 2D magnetized colloids (i.e., no pinnings and external driving forces exist) are in a plastic phase at such a temperature. The initially pinned colloids are found to depin when the external driving force f is increased above the threshold (i.e., the critical driving force) fc , below which the advances of pinned colloids generated mainly by rearrangements of topological defects are so small that they can be neglected. For weak pinning potential with a small value of Ap , the v –f dependence is nonlinear and shows an evident concave upward curvature near fc , as shown in Fig. 1(a) where we give the v –f curve
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a
b
c
Fig. 1. Power-law scaling relationship between the average velocity v and the driving force f above the depinning for Ap = 0.005 (a), 0.05 (b) and 0.15 (c), respectively. The corresponding history dependencies are also shown.
for Ap = 0.005 and an evident history dependence is observed. The concave upward feature and history dependence are typical characteristics of plastic depinning where colloids depin individually. Detailed analysis shows that a power-law scaling relationship could be derived between v and f near fc , i.e., v ∼ (f − fc )ζ , where the scaling exponent ζ is found to be close to 2.0 (e.g., ζ = 1.94 for Ap = 0.005, as shown in Fig. 1(a)). This is coincident with recent experimental measurement on plastic flow of 2D charged colloids driven over a disordered pinning potential [7]. The plastic depinning in weak periodic pinning potential has been recently pointed out in phase-field crystal model [26]. To identify the flows near the depinning, we present the trajectories and structure factors defined as S (k) = ⟨| N1 Ni=c 1 exp[ik · Ri (t )]|2 ⟩ in Fig. 2. For the weak pinning potential, we find that colloids move in channels between different pinning centers, as shown in Fig. 2(a), and the channels are unstable and changeable with time. Correspondingly, no Bragg peaks are found in S (k) and only a peak is observed at the center (kx = ky = 0), as seen in Fig. 2(d). This is contrary to the results on disordered substrates, where an elastic crystal flow was found for the weak pinning potential [19,20,22]. The reason is, as discussed in the following, that the thermal fluctuation at Tm0 exceeds the pinning potential, resulting in the plastic flow above fc . For strong pinning potential with a large value of Ap , the v –f curve is found to show a distinct convex upward curvature near fc and the history dependence disappears, as shown in Fig. 1(c) where the v –f curve as well as the history dependence is presented for Ap = 0.15. The convex upward curvature indicates that colloids depin collectively. A power-law scaling relationship between v and f could be also obtained near fc , and the power-law scaling exponent ζ is to be about 2/3 (e.g., ζ = 0.76 for Ap = 0.15), consistent with previous simulations [19] and experiments [7] on the elastic crystal flow of colloids. To prove the elastic crystal flow, we give the trajectories and structure factors in Fig. 2(c) and (f) for the strong pinning potential with Ap = 0.15. We can find in Fig. 2(c) that colloids move in channels parallel to the drive. The channels are stable and do not change with time. The sixhold coordinated Bragg peaks could be observed distinctly in S (k), as shown in Fig. 2(f). This may be due to the fact that all the colloids are localized along the pinning arrays for sufficiently strong pinning potential and the thermal fluctuation could not compete with the pinning potential in this case, leading to the elastic crystal flow above fc . With a suitable value of Ap , we find a intermediate moving phase, i.e., elastic smectic flow, between the plastic and elastic crystal flows near fc . This can be seen in Fig. 2(b) and (e) where we show the trajectories and structure factors for Ap = 0.05. From Fig. 2(b), we can find that most colloids move in channels parallel to f but some move away from the direction of f . Correspondingly, there are only two Bragg peaks along the driving direction, as shown in Fig. 2(e). In this case, thermal fluctuation could compete with the pinning potential. The elastic smectic flow has been observed in our previous simulations of 2D magnetized colloids on a disordered pinning substrate [22,23]. Scaling analysis tells us that the v –f dependence also satisfies a power-law scaling relationship, and the exponent ζ is found to be slightly larger than 1.0, as seen in the Fig. 1(b) where the v –f curve is given for Ap = 0.05 and ζ = 1.02 is observed. The history dependence is not found in the v –f curve, as shown in the Fig. 1(b).
Y.G. Cao et al. / Physica A 391 (2012) 2940–2947
a
d
b
e
c
f
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Fig. 2. Colloid trajectories ((a), (b) and (c)) and structure factors ((d), (e) and (f)) near the depinning. (a) and (d) are for Ap = 0.001, (b) and (e) are for Ap = 0.05, (c) and (f) are for Ap = 0.15, respectively.
Fig. 3. Critical driving force fc versus Ap .
Increasing the value of Ap , we find crossovers from the plastic flow through the elastic smectic flow to the elastic crystal flow. Comparing Fig. 2((a)–(c)) or ((d)–(f)), we can see this point. It should be pointed out that such crossovers also occur at another finite temperature. Accompanying with each crossover, fc and ζ as well as the average intensity of Bragg peaks S¯ (k) defined as the average height of sixhold coordinated Bragg peaks are found to change suddenly, as shown in Figs. 3–5 where we present the Ap dependencies of fc and ζ as well as S¯ (k), respectively. A remarkable feature is that fc , ζ and S¯ (k) remain invariant basically in the region of the elastic smectic flow, as also seen in the Figs. 3–5. 3.2. Influence of the temperature In order to explain why the above dynamic moving phases and their transitions occur, we change the temperature systematically. It is pointed out that thermal fluctuation might affect the depinning dynamics of colloids through phase
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Fig. 4. Power-law scaling exponent ζ versus Ap .
Fig. 5. Average intensity of Bragg peaks S¯ (k) versus Ap .
fluctuations [26]. At sufficiently low temperature, the thermal fluctuation is so small that it could be neglected and the pinning potential dominates and the colloids move in an elastic crystal no matter what the pinning strength. We can see this from comparing Fig. 6((b) and (d)) with 7((b) and (d)) where we present the colloid trajectories (6(b) and 7(b)) and structure factors (6(d) and 7(d)) above the depinning at T = 0.001Tm0 for Ap = 0.15 and Ap = 0.001, respectively. With an increase in the temperature, the thermal fluctuation increases. For the strong pinning potential (e.g., Ap = 0.15), this might bring the elastic crystal flow, as shown in Fig. 6(b) and (d), into the elastic smectic flow when the thermal fluctuation could compete with the pinning potential, as shown in Fig. 6(a) and (c) where we present the colloid trajectories and structure factors above the depinning for Ap = 0.15 and at T = 10Tm0 . But for the weak pinning potential (e.g., Ap = 0.001), this might bring the elastic crystal flow, as shown in Fig. 7(b) and (d), into the plastic flow when the thermal fluctuation exceeds the pinning potential, as seen in Fig. 7(a) and (c) where we give the colloid trajectories and structure factors above the depinning for Ap = 0.001 and at T = 10Tm0 . In a word, the interplay between thermal fluctuation and pinning potential results in the above dynamic moving phases and their transitions. When the thermal fluctuation is small enough, the pinning potential dominates and the colloids move along the pinning arrays and an elastic crystal flow occurs. The colloids move in an elastic smectic crystal when the thermal fluctuation could compete with the pinning potential. Finally, the plastic flow takes place in the moving colloids when the thermal fluctuation overcomes the pinning potential.
Y.G. Cao et al. / Physica A 391 (2012) 2940–2947
a
c
b
d
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Fig. 6. Colloid trajectories ((a), (b)) and structure factors ((c), (d)) near the depinning for Ap = 0.15. ((a), (c)) are at T = 10Tm0 ; ((b), (d)) are at T = 0.001Tm0 .
a
c
b
d
Fig. 7. Colloidal trajectories ((a), (b)) and structure factors ((c), (d)) near the depinning for Ap = 0.001. ((a), (c)) are at T = 10Tm0 ; ((b), (d)) are at T = 0.001Tm0 .
3.3. Influence of the number of pinning centers Here we keep the number of colloids Nc fixed and change the number of pinning centers Np . The temperature T is fixed to be Tm0 . Fig. 8 presents the variations of the critical driving force fc with the ratio of Np /Nc for Ap = 0.05 and 0.15. One can find that fc increases with Np when Np < Nc . A peak is found in fc when Np ∼ Nc . Then fc is found to decrease with a further increase in Np because more colloids are pinned between different pinning centers at higher Np and thereby a lower fc would drive the pinned colloids to depin. Comparing the left and right insets of Fig. 8, we can see this point clearly. Further analysis on power-law scaling exponent shows that there exists a dip in the dependence of ζ on Np /Nc , as shown in Fig. 9 where we give the curves of ζ versus Np /Nc for Ap = 0.05 and 0.15. Also, more pronounced peak and dip effects are found for larger values of Ap , as seen in Figs. 8 and 9. In fact, it has been pointed out in Type-II superconductors [24] and charged colloids [32] that the formation of elastic crystal flow for the strong pinning potential is associated with the peak effects of the critical driving force. Our results here indicate that there will appear a peak in the critical driving force and a dip in the power-law scaling exponent when the number of pinning centers is close to the number of colloids.
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Fig. 8. Critical driving force fc versus number ratio of Np /Nc . The left and right insets give the snapshots of colloid locations (hollow circle dots) below fc (f /fc = 0.9) for Np = 400 and 3600, respectively. The pinning centers are shown with crosses.
Fig. 9. Power-law scaling exponent ζ versus number ratio of Np /Nc .
4. Summary We have studied numerically the depinning dynamics of 2D magnetized colloids on a substrate with periodic pinning centers. Increasing the pinning strength, we found crossovers from plastic flow through elastic smectic flow to elastic crystal flows above the depinning at finite temperature when the number ratios of pinning to colloids are 1:1 matching. This is quite different from the case of the disordered pinning substrate where crossovers from elastic crystal flow through elastic smectic flow to plastic flow were observed with an increase in the pinning strength. A power-law scaling relationship was obtained between the average velocity of magnetized colloids and the driving force above the depinning for all the three types of flows, and we found that the power-law scaling exponent and the critical driving force as well as the average intensity of Bragg peaks remain invariant basically in the region of elastic smectic flows. The interplay between thermal fluctuation and pinning potential was shown to be responsible for the abnormal dynamic phases and their transitions. The thermal fluctuation increases with the temperature, turning the elastic crystal flow into elastic smectic flow when the thermal fluctuation could compete with the pinning potential or the plastic flow when the pinning potential is exceeded. Finally, we found a peak in the critical driving force and a dip in the power-law scaling exponent, which are associated with the
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formation of the commensurate states between colloids and pinnings where the colloids move in an elastic crystal flow at sufficiently low temperature. Acknowledgments The work was supported by the Natural Science Research Foundation of Henan Provincial Department of Science and Technology under Grant No. 112300410151 and the Scientific Research Foundation of Graduate School of Zhengzhou University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
V.J. Anderson, H.N.W. Lekkerkerker, Nature (London) 416 (2002) 811. G. Foffi, G. Kahl, J. Phys.: Condens. Matter 22 (2010) 100301. A. Pertsinidis, X.S. Ling, New J. Phys. 7 (2005) 33. C. Eisenmann, U. Gasser, P. Keim, G. Maret, H.H. von Grünberg, Phys. Rev. Lett. 95 (2005) 185502. M. Köppl, P. Henseler, A. Erbe, P. Nielaba, P. Leiderer, Phys. Rev. Lett. 97 (2006) 208302. D. Reinke, H. Stark, H.H. von Grünberg, A.B. Schofield, G. Maret, I. Gasser, Phys. Rev. Lett. 98 (2007) 038301. A. Pertsinidis, X.S. Ling, Phys. Rev. Lett. 100 (2008) 028303. F. Ebert, G. Maret, P. Keim, Eur. Phys. J. E 29 (2009) 311. T.O.E. Skinner, D.G.A.L. Aarts, R.P.A. Dullens, Phys. Rev. Lett. 105 (2010) 168301. J.H. Holtz, S.A. Asher, Nature (London) 389 (1997) 829. J.E.G.J. Wijnhoven, W.L. Vos, Science 281 (1998) 802. A.P. Gast, W.B. Russel, Phys. Today 51 (12) (1998) 24. C. Reichhardt, C.J. Olson Reichhardt, Phys. Rev. Lett. 106 (2011) 060603. H. Löwen, J. Phys.: Condens. Matter 13 (2001) R415. C. Bechinger, M. Brunner, P. Leiderer, Phys. Rev. Lett. 86 (2001) 930. K. Franzrahe, P. Nielaba, A. Ricci, K. Binder, S. Sengupta, P. Keim, G. Maret, J. Phys.: Condens. Matter 20 (2008) 404218. F. Ebert, P. Dillmann, G. Maret, P. Keim, Rev. Sci. Instrum. 80 (2009) 083902. M. Brunner, C. Bechinger, Phys. Rev. Lett. 88 (2002) 248302. C. Reichhardt, C.J. Olson, Phys. Rev. Lett. 89 (2002) 078301. Y.G. Cao, J.X. Chen, Y.M. Wang, Z.K. Jiao, W.K. Chow, Phys. Rev. B 68 (2003) 134209. J.X. Chen, Y.G. Cao, Z.K. Jiao, Phys. Rev. E 69 (2004) 041403. Y.G. Cao, Q.X. Li, Physica A 387 (2008) 4755. Y.G. Cao, Q.X. Li, G.Y. Fu, J. Liu, H.Z. Guo, X. Hu, X.J. Li, J. Phys.: Condens. Matter 22 (2010) 155101. C. Reichhardt, C.J. Olson, Phys. Rev. Lett. 88 (2002) 248301. K. Xiao, D.G. Grier, Phys. Rev. Lett. 104 (2010) 028302. C.V. Achim, J.A.P. Ramos, M. Karttunen, K.R. Elder, E. Granato, T. Ala-Nissila, S.C. Ying, Phys. Rev. E 79 (2009) 011606. Y.G. Cao, Z.K. Jiao, H.P. Ying, Phys. Rev. B 62 (2000) 4163. L. Assoud, F. Ebert, P. Keim, R. Messina, G. Maret, H. Löwen, J. Phys.: Condens. Matter 21 (2009) 46114. F. Ebert, P. Keim, G. Maret, Eur. Phys. J. E 26 (2008) 161. E.H. Brandt, Phys. Rev. Lett. 50 (1983) 1599. D.S. Fisher, Phys. Rev. B 22 (1980) 1190. A. Sarlah, E. Frey, T. Franosch, Phys. Rev. E 75 (2007) 021402.
Contents lists available at SciVerse ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Depinning dynamics of two-dimensional magnetized colloids on a substrate with periodic pinning centers Y.G. Cao ∗ , Z.F. Zhang, M.H. Zhao, G.Y. Fu, D.X. Ouyang School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China
article
info
Article history: Received 15 November 2011 Received in revised form 5 January 2012 Available online 16 January 2012 Keywords: Depinning Magnetized colloid Plastic flow Elastic smectic flows Elastic crystal flow
abstract We study, by Langevin simulations, the depinning dynamics of two-dimensional magnetized colloids on a substrate with periodic pinning centers. When the number ratios of pinnings to colloids are 1:1 matching and at finite temperature, we find for the first time crossovers from plastic flow through elastic smectic flow to elastic crystal flow near the depinning with increasing the pinning strength. There exists a power-law scaling relationship between the average velocity of colloids and the external driving force for all the three types of flows. It is found that the critical driving force and the power-law scaling exponent as well as the average intensity of Bragg peaks are invariant basically in the region of elastic smectic flow. We also examine the temperature effect and it reveals that the above dynamic moving phases and their transitions could be attributed to the interplay between thermal fluctuation and pinning potential. At sufficiently low temperature, the thermal fluctuation could be neglected and the colloids near the depinning move in the elastic crystal flow no matter what the pinning strength. In addition, the number of pinning centers is changed and when it is close to the number of colloids, there appears a peak in the critical driving force and a dip in the power-law scaling exponent, respectively. The peak and dip are more pronounced for higher pinning strength. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Colloids could display intriguing phase transitions among gas, liquid, solid and liquid crystalline phases [1], which are highly relevant in everyday applications and in basic research [2]. The micron size of colloids allows to image them in videomicroscopy experiments [3–9]. Thus, colloids present an ideal model system for approaching various problems in physical chemistry [10], material science [11], and condensed matter physics [12]. Recently, the dynamics of colloids under external field has attracted much attention [5,7,13]. By investigating the response of colloids to an external field, one can get an insight into the dynamics of colloids, and through examining the dynamics, one can tailor and control the external field [2,14]. Sometimes, the external field may complicate the problem by causing a system to go into a nonequilibrium moving state. When a magnetic field is applied, a magnetic moment will be induced in superparamagnetic colloidal particles. The magnitude of the induced moment scales linearly with the magnetic field strength, implying that the colloid interaction strength could be controlled by the external magnetic field completely [15–17]. Consequently, a near ideal two-dimensional (2D) system could be experimentally realized in magnetized colloids. In reality, colloids interact with pinnings in the solid or liquid substrates [18]. The depinning dynamics of 2D charged colloids on disordered pinning substrates has been studied numerically [19–21]. With an increase in the pinning strength (or a decrease in the colloid interaction strength), a crossover from elastic crystal to
∗
Corresponding author. E-mail address: [email protected] (Y.G. Cao).
0378-4371/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2012.01.020
Y.G. Cao et al. / Physica A 391 (2012) 2940–2947
2941
plastic flows was observed above the depinning. The crossover was confirmed by recent experiments [7]. Also, the average velocity of colloids near the depinning was found to scale with the drive in a power-law scaling relationship for both elastic crystal and plastic flows. The power-law scaling exponent was observed to be about 2/3 and 2.0 for the elastic crystal and plastic flows, respectively [19,7]. Most recently, we have studied systematically the depinning dynamics of 2D magnetized colloids on a quenched substrate [22,23]. Decreasing the magnetic field strength, we found crossovers from the elastic crystal through the elastic smectic flows to the plastic flows above the depinning [23]. Although there have been many studies of the dynamics of driven colloids on disordered pinning substrates, there are far fewer studies of the dynamics of colloids driven over periodic pinning potentials [24,25]. In this paper, we investigate systematically the depinning dynamics of 2D magnetized colloids on a substrate with periodic pinning centers. When the number ratios of pinnings to colloids are 1:1 matching and at a finite temperature, we find that the colloids near the depinning move in a plastic flow for sufficiently weak pinning potential, as pointed out in phase-field crystal model [26]. But an elastic crystal flow is observed for strong enough pinning potential. These are contrary to the results on disordered pinning substrates [19,20,22]. Increasing the pinning strength, we find for the first time crossovers from the plastic flow through the elastic smectic flow to the elastic crystal flow near the depinning. It is shown that the interplay between thermal fluctuation and pinning potential could be responsible for the above dynamic moving phases and their transitions. The paper is organized as follows. In Section 2, we describe the model and the procedure we have used in the Langevin simulations. In Section 3, we give the numerical results and discussions. We summarize in Section 4. 2. Model The motion of magnetized colloids is described by the Langevin equation
η
dRi dt
=−
∇i Ucc (Ri − Rj ) −
∇i Ucp (Ri − rj′ ) + fLi + f,
(1)
j′
i̸=j
where η is a damping coefficient and fixed to unity, Ri and rj′ denote the coordinates of the ith colloidal particle and the j′ th pinning center in the substrate, Ucc is the interaction potential between colloids, Ucp is the interaction potential between colloids and pinning in the substrate, respectively. f is the external driving force. The random fluctuating force exerted on the ith colloid fLi is described by the correlation functions [19,20,22–24,27] ⟨fiL ⟩ = 0 and ⟨fiαL (t )fjβL (t )⟩ = 2T δij δαβ δ(t − t ′ ), where subscripts α and β take values 1, 2 or 3, and T is the temperature of the system. The interaction between magnetized colloids has the same form as parallel dipoles and could be written as [22,23,28,29] Ucc (Ri − Rj ) =
µ0 M2 , 4π |Ri − Rj |3
(2)
with µ0 the magnetic permeability of free space. M = χ B is the magnetic moment. Here χ is the effective susceptibility µ M2
and B is the magnetic field strength. We keep 40 π = 0.05 in our numerical calculations. The interaction between magnetized colloids and pinnings in the substrate is chosen as conventional attracting Gaussian potential [20,22,23,27,30] −
Ucp (Ri − rj′ ) = −Ap e
|Ri −rj′ |2 rp2
,
(3)
where Ap is a constant, characterizing the strength of pinning potential. The size of the pinning center rp is chosen to be rp = 0.2a0 , with a0 the lattice constant of the perfect triangular lattice. In the above simulations, we use the substrate with Np point-like pinning centers distributed in a triangular lattice subjected to periodic boundary conditions. Nc (=400) magnetized colloids are initially placed at the pinning sites. All the lengths are scaled with respect to a0 . The external driving force f is increased from zero by small increments along the Nc ˆ horizontal symmetry axis (y axis) and the average velocity v = N1 i=1 v · y is measured at each increment. c
3. Results and discussions 3.1. Influence of the strength of pinning potential At first, we investigate the influence of the pinning strength on the depinning of 2D magnetized colloids. The number ratios of pinnings to colloids are 1:1 matching. The temperature T is fixed to be Tm0 of the ‘‘bare’’ Kosterlitz–Thouless melting temperature of 2D system [31], and the pure 2D magnetized colloids (i.e., no pinnings and external driving forces exist) are in a plastic phase at such a temperature. The initially pinned colloids are found to depin when the external driving force f is increased above the threshold (i.e., the critical driving force) fc , below which the advances of pinned colloids generated mainly by rearrangements of topological defects are so small that they can be neglected. For weak pinning potential with a small value of Ap , the v –f dependence is nonlinear and shows an evident concave upward curvature near fc , as shown in Fig. 1(a) where we give the v –f curve
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Fig. 1. Power-law scaling relationship between the average velocity v and the driving force f above the depinning for Ap = 0.005 (a), 0.05 (b) and 0.15 (c), respectively. The corresponding history dependencies are also shown.
for Ap = 0.005 and an evident history dependence is observed. The concave upward feature and history dependence are typical characteristics of plastic depinning where colloids depin individually. Detailed analysis shows that a power-law scaling relationship could be derived between v and f near fc , i.e., v ∼ (f − fc )ζ , where the scaling exponent ζ is found to be close to 2.0 (e.g., ζ = 1.94 for Ap = 0.005, as shown in Fig. 1(a)). This is coincident with recent experimental measurement on plastic flow of 2D charged colloids driven over a disordered pinning potential [7]. The plastic depinning in weak periodic pinning potential has been recently pointed out in phase-field crystal model [26]. To identify the flows near the depinning, we present the trajectories and structure factors defined as S (k) = ⟨| N1 Ni=c 1 exp[ik · Ri (t )]|2 ⟩ in Fig. 2. For the weak pinning potential, we find that colloids move in channels between different pinning centers, as shown in Fig. 2(a), and the channels are unstable and changeable with time. Correspondingly, no Bragg peaks are found in S (k) and only a peak is observed at the center (kx = ky = 0), as seen in Fig. 2(d). This is contrary to the results on disordered substrates, where an elastic crystal flow was found for the weak pinning potential [19,20,22]. The reason is, as discussed in the following, that the thermal fluctuation at Tm0 exceeds the pinning potential, resulting in the plastic flow above fc . For strong pinning potential with a large value of Ap , the v –f curve is found to show a distinct convex upward curvature near fc and the history dependence disappears, as shown in Fig. 1(c) where the v –f curve as well as the history dependence is presented for Ap = 0.15. The convex upward curvature indicates that colloids depin collectively. A power-law scaling relationship between v and f could be also obtained near fc , and the power-law scaling exponent ζ is to be about 2/3 (e.g., ζ = 0.76 for Ap = 0.15), consistent with previous simulations [19] and experiments [7] on the elastic crystal flow of colloids. To prove the elastic crystal flow, we give the trajectories and structure factors in Fig. 2(c) and (f) for the strong pinning potential with Ap = 0.15. We can find in Fig. 2(c) that colloids move in channels parallel to the drive. The channels are stable and do not change with time. The sixhold coordinated Bragg peaks could be observed distinctly in S (k), as shown in Fig. 2(f). This may be due to the fact that all the colloids are localized along the pinning arrays for sufficiently strong pinning potential and the thermal fluctuation could not compete with the pinning potential in this case, leading to the elastic crystal flow above fc . With a suitable value of Ap , we find a intermediate moving phase, i.e., elastic smectic flow, between the plastic and elastic crystal flows near fc . This can be seen in Fig. 2(b) and (e) where we show the trajectories and structure factors for Ap = 0.05. From Fig. 2(b), we can find that most colloids move in channels parallel to f but some move away from the direction of f . Correspondingly, there are only two Bragg peaks along the driving direction, as shown in Fig. 2(e). In this case, thermal fluctuation could compete with the pinning potential. The elastic smectic flow has been observed in our previous simulations of 2D magnetized colloids on a disordered pinning substrate [22,23]. Scaling analysis tells us that the v –f dependence also satisfies a power-law scaling relationship, and the exponent ζ is found to be slightly larger than 1.0, as seen in the Fig. 1(b) where the v –f curve is given for Ap = 0.05 and ζ = 1.02 is observed. The history dependence is not found in the v –f curve, as shown in the Fig. 1(b).
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Fig. 2. Colloid trajectories ((a), (b) and (c)) and structure factors ((d), (e) and (f)) near the depinning. (a) and (d) are for Ap = 0.001, (b) and (e) are for Ap = 0.05, (c) and (f) are for Ap = 0.15, respectively.
Fig. 3. Critical driving force fc versus Ap .
Increasing the value of Ap , we find crossovers from the plastic flow through the elastic smectic flow to the elastic crystal flow. Comparing Fig. 2((a)–(c)) or ((d)–(f)), we can see this point. It should be pointed out that such crossovers also occur at another finite temperature. Accompanying with each crossover, fc and ζ as well as the average intensity of Bragg peaks S¯ (k) defined as the average height of sixhold coordinated Bragg peaks are found to change suddenly, as shown in Figs. 3–5 where we present the Ap dependencies of fc and ζ as well as S¯ (k), respectively. A remarkable feature is that fc , ζ and S¯ (k) remain invariant basically in the region of the elastic smectic flow, as also seen in the Figs. 3–5. 3.2. Influence of the temperature In order to explain why the above dynamic moving phases and their transitions occur, we change the temperature systematically. It is pointed out that thermal fluctuation might affect the depinning dynamics of colloids through phase
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Fig. 4. Power-law scaling exponent ζ versus Ap .
Fig. 5. Average intensity of Bragg peaks S¯ (k) versus Ap .
fluctuations [26]. At sufficiently low temperature, the thermal fluctuation is so small that it could be neglected and the pinning potential dominates and the colloids move in an elastic crystal no matter what the pinning strength. We can see this from comparing Fig. 6((b) and (d)) with 7((b) and (d)) where we present the colloid trajectories (6(b) and 7(b)) and structure factors (6(d) and 7(d)) above the depinning at T = 0.001Tm0 for Ap = 0.15 and Ap = 0.001, respectively. With an increase in the temperature, the thermal fluctuation increases. For the strong pinning potential (e.g., Ap = 0.15), this might bring the elastic crystal flow, as shown in Fig. 6(b) and (d), into the elastic smectic flow when the thermal fluctuation could compete with the pinning potential, as shown in Fig. 6(a) and (c) where we present the colloid trajectories and structure factors above the depinning for Ap = 0.15 and at T = 10Tm0 . But for the weak pinning potential (e.g., Ap = 0.001), this might bring the elastic crystal flow, as shown in Fig. 7(b) and (d), into the plastic flow when the thermal fluctuation exceeds the pinning potential, as seen in Fig. 7(a) and (c) where we give the colloid trajectories and structure factors above the depinning for Ap = 0.001 and at T = 10Tm0 . In a word, the interplay between thermal fluctuation and pinning potential results in the above dynamic moving phases and their transitions. When the thermal fluctuation is small enough, the pinning potential dominates and the colloids move along the pinning arrays and an elastic crystal flow occurs. The colloids move in an elastic smectic crystal when the thermal fluctuation could compete with the pinning potential. Finally, the plastic flow takes place in the moving colloids when the thermal fluctuation overcomes the pinning potential.
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Fig. 6. Colloid trajectories ((a), (b)) and structure factors ((c), (d)) near the depinning for Ap = 0.15. ((a), (c)) are at T = 10Tm0 ; ((b), (d)) are at T = 0.001Tm0 .
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Fig. 7. Colloidal trajectories ((a), (b)) and structure factors ((c), (d)) near the depinning for Ap = 0.001. ((a), (c)) are at T = 10Tm0 ; ((b), (d)) are at T = 0.001Tm0 .
3.3. Influence of the number of pinning centers Here we keep the number of colloids Nc fixed and change the number of pinning centers Np . The temperature T is fixed to be Tm0 . Fig. 8 presents the variations of the critical driving force fc with the ratio of Np /Nc for Ap = 0.05 and 0.15. One can find that fc increases with Np when Np < Nc . A peak is found in fc when Np ∼ Nc . Then fc is found to decrease with a further increase in Np because more colloids are pinned between different pinning centers at higher Np and thereby a lower fc would drive the pinned colloids to depin. Comparing the left and right insets of Fig. 8, we can see this point clearly. Further analysis on power-law scaling exponent shows that there exists a dip in the dependence of ζ on Np /Nc , as shown in Fig. 9 where we give the curves of ζ versus Np /Nc for Ap = 0.05 and 0.15. Also, more pronounced peak and dip effects are found for larger values of Ap , as seen in Figs. 8 and 9. In fact, it has been pointed out in Type-II superconductors [24] and charged colloids [32] that the formation of elastic crystal flow for the strong pinning potential is associated with the peak effects of the critical driving force. Our results here indicate that there will appear a peak in the critical driving force and a dip in the power-law scaling exponent when the number of pinning centers is close to the number of colloids.
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Fig. 8. Critical driving force fc versus number ratio of Np /Nc . The left and right insets give the snapshots of colloid locations (hollow circle dots) below fc (f /fc = 0.9) for Np = 400 and 3600, respectively. The pinning centers are shown with crosses.
Fig. 9. Power-law scaling exponent ζ versus number ratio of Np /Nc .
4. Summary We have studied numerically the depinning dynamics of 2D magnetized colloids on a substrate with periodic pinning centers. Increasing the pinning strength, we found crossovers from plastic flow through elastic smectic flow to elastic crystal flows above the depinning at finite temperature when the number ratios of pinning to colloids are 1:1 matching. This is quite different from the case of the disordered pinning substrate where crossovers from elastic crystal flow through elastic smectic flow to plastic flow were observed with an increase in the pinning strength. A power-law scaling relationship was obtained between the average velocity of magnetized colloids and the driving force above the depinning for all the three types of flows, and we found that the power-law scaling exponent and the critical driving force as well as the average intensity of Bragg peaks remain invariant basically in the region of elastic smectic flows. The interplay between thermal fluctuation and pinning potential was shown to be responsible for the abnormal dynamic phases and their transitions. The thermal fluctuation increases with the temperature, turning the elastic crystal flow into elastic smectic flow when the thermal fluctuation could compete with the pinning potential or the plastic flow when the pinning potential is exceeded. Finally, we found a peak in the critical driving force and a dip in the power-law scaling exponent, which are associated with the
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formation of the commensurate states between colloids and pinnings where the colloids move in an elastic crystal flow at sufficiently low temperature. Acknowledgments The work was supported by the Natural Science Research Foundation of Henan Provincial Department of Science and Technology under Grant No. 112300410151 and the Scientific Research Foundation of Graduate School of Zhengzhou University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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