pubs.acs.org/Langmuir © 2009 American Chemical Society
Determination of Elastic Constants of Two-Dimensional Close-Packed Colloidal Crystals Ke-Qin Zhang*,†,‡ and Xiang Y. Liu‡ †
National Engineering Laboratory for Modern Silk, Soochow University, Suzhou, 215006 Jiangsu, People’s Republic of China, and ‡Department of Physics, National University of Singapore, Singapore 117542 Received December 30, 2008. Revised Manuscript Received April 13, 2009
We present a noncontact, accurate, and efficient methodology for determination of elastic constants in twodimensional colloidal crystals via the calculation of the local strain fluctuation of particles. The hexagonally closepacked colloidal crystals form from microsized particles subjected to an alternating electric field. The elastic constants in the thermodynamic limit are obtained by the extrapolation of finite-size scaling of the elastic moduli as the functions of the frequency and field strength. It is found that the elastic constants in our system are larger than those in nonclose-packed colloidal crystals reported before. This technique could be a rational method to study the elasticity of soft solids.
Introduction During the last two decades, interest in colloidal systems has grown substantially. This is due to their widespread technological applications and the availability of precisely calibrated particles as model systems for studying phenomena in classical condensed matter physics.1-3 In contrast to atomic crystals, it is possible to control characteristics of the colloidal crystals by adjusting parameters such as the size, surface charge, and concentration of the colloidal particles and concentration of added salts and by applying the external fields. On the other hand, as colloidal spheres have a dimension of about 103-104 times larger than the atomic length scale, the mechanical properties of colloidal crystals are different from those of the conventional solids in elasticity.4 Despite this, the knowledge on how a soft material deforms under external mechanical stresses is essential in material science and of fundamental interest. Currently, the study of elasticity receives a lot of attention in soft solids because of their great potential for the development of novel materials, such as photonic and phononic materials.5-8 We noticed that the conventional indentation methods have been intensively applied to measure the mechanical properties of hard and soft materials. However, the conventional methods, such as nanoindentation or atomic force probes, are not applicable to a large variety of soft solids under certain circumstances. First, some of the soft solids normally consist of polymeric solids and aqueous phases, leading to the overdamping of the long*Corresponding author. E-mail:
[email protected]. (1) Pusey, P. N. In Liquids, Freezing and the Glass Transition; Hansen, J. P., Levesque, D., Zinn-Justin, J., Eds; North-Holland: Amsterdam, 1991. (2) Schall, P.; Weitz, D. A.; Spaepen, F. Science 2007, 318, 1895. (3) Lu, P. J.; Zaccarelli, E.; Ciulla, F.; Schofield, A. B.; Sciortino, F.; Weitz, D. A. Nature (London) 2008, 453, 499. (4) Shinohara, T.; Yoshiyama, T.; Sogami, I. S.; Konishi, T.; Ise, N. Langmuir 2001, 17, 8010. (5) Yethiraj, A.; van Blaaderen, A. Nature (London) 2003, 421, 513. (6) Lawrence, J.; Ying, Y.; Jiang, P.; Foulger, S. H. Adv. Mater. 2006, 18, 300. (7) Cheng, W.; Wang, J.; Jonas, U.; Fytas, G.; Stefanou, N. Nat. Mater. 2006, 5, 830. (8) Baumgartl, J.; Zvyagolskaya, M.; Bechinger, C. Phys. Rev. Lett. 2007, 99, 205503.
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itudinal sound modes and small elastic constants.4,9 Consequently their mechanical response to external perturbations is inappreciable and far beyond the sensitivity of conventional methodologies/techniques for measuring microelastic and microrheological properties. Second, most of the soft solids are difficult to grow to the bulk material with a macroscale size. This is especially challenging when trying to investigate the elastic behavior by applying the external stresses on them. Lastly, the two-dimensional (2D) or three-dimensional (3D) soft solids are often confined by the hard walls. These causes the samples to be inaccessible when measuring the elastic properties by the conventional means. Recently, methods on measuring the elastic moduli of soft solids from fluctuations of particle coordinates were developed.9-12 These methods focus on the 2D crystals of superparamagnetic particles with the assumption that the 2D colloidal lattice has isotropic elastic behavior, which means the crystal undergoes the homogeneous expansion (or compression) and rotation. The general expression of the stress tensor in terms of the strain tensor for an isotropic body is determined by the elastic moduli and the components of the strain and stress tensors. Under the homogeneous deformation, the nondiagonal components of the strain tensor are zero, which leads to the large simplification of the strain-stress relation. Therefore, the elasticity of 2D crystals can be completely described by two independent elastic constants, the bulk modulus K and the shear modulus μ according to the elasticity theory.13 In this Letter, we report the determination of the elastic constants of 2D close-packed colloidal crystals from a sequence of particle positions based on the application of an alternating electric field on a cell. The time-dependent strain distribution is (9) Zahn, K.; Wille, A.; Maret, G.; Sengupta, S.; Nielaba, P. Phys. Rev. Lett. 2003, 90, 155506. (10) Sengupta, S.; Nielaba, P.; Rao, M.; Binder, K. Phys. Rev. E 2000, 61, 1072. (11) Keim, P.; Maret, G.; Herz, U.; von Grunberg, H. Phys. Rev. Lett. 2004, 93, 215504. (12) Franzrahe, K.; Keim, P.; Maret, G.; Nielaba, P.; Sengupta, S. Phys. Rev. E 2008, 78, 026106. (13) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity, 3rd ed.; Pergamon: Oxford, 1986.
Published on Web 4/17/2009
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measured and visualized by calculating the symmetric part of the best affine deformation tensor with a different algorithm.12,14 In the meantime, the system size-dependent fluctuation of the strain and rotation is related to the elastic constants using the equipartition theorem. Finally, the elastic constants at the thermodynamic limit (infinite system) are determined by the finite scaling analysis as the functions of field strength and frequency of external electric field.
Experimental Section Our experimental setup15,16 is composed of polystyrene spherical colloids of diameter d = 1.8 μm with a polydispersity of 600 Hz).19 Furthermore, our observation shows that the repulsive dipolar interaction between the in-plane polarized particles is coupled with the electrohydrodynamic attraction and proportional to the local electric field. As a consequence, the changes of the field strength and frequency inherently affect the interparticle interaction. From the thermodynamic point of (17) Ristenpart, W. D.; Aksay, I. A.; Saville, D. A. Phys. Rev. E 2004, 69, 021405. (18) Ristenpart, W. D.; Aksay, I. A.; Saville, D. A. J. Fluid Mech. 2007, 575, 83. (19) Zhang, K.-Q.; Liu, X. Y. J. Chem. Phys., accepted.
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Figure 4. Dependence of the measured elastic constants (in units of kBT/a2, where a is the lattice constant) on the frequency (the most left frequency is close to the melting point fc = 3.7 kHz) in (a) and the applied voltage (the most right voltage is close to the melting point Vc = 1.25 V which corresponds to a critical strength σc = 0.9 104 V/m) in (b). The dashed lines are the best linear fits to guide the eye.
view, the field strength and frequency play the role of the pseudotemperature,20 which governs the elastic properties of the 2D colloidal crystals. The bulk (K) and shear (μ) moduli are plotted as the functions of the field strength and frequency for our colloidal system in Figure 4. The elastic moduli decrease with increasing frequency and decreasing field strength. By approaching the melting points ( fc = 3.7 kHz and σc = 0.9 104 V/m),16 the bulk modulus dramatically drops to about 500kBT/a2 and the shear modulus is also close to a finite value ∼50kBT/a2. The distances between the particles are increased due to thermal expansion, hence the restoring forces between the particles are reduced, and therefore the shear and bulk moduli decrease with rising pseudotemperature (increasing frequency or decreasing field strength). The softening of the shear modulus leads to a mechanical instability of the solid structure and finally to a collapse of the crystal lattice at some temperature. Interestingly, this scenario is consistent with the Born criterion for the stability of a crystal lattice.21 Moreover, the measured bulk modulus of 2D close-packed crystals lies in the range 500-5000kBT/a2 and the shear modulus is approximately 50-500kBT/a2, both of which are 10 times larger than those measured in the 2D superparamagnetic crystals.9 This may be (20) Yethiraj, A.; van Blaaderen, A. Nature (London) 2003, 421, 513. (21) Max, B. J. Chem. Phys. 1939, 7, 591–603.
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attributed to the close-packed configuration in our system. Apart from this, it follows that that the ratio of K/μ ∼ 10 when the frequency or field strength is approaching the melting point of the phase transition.16,22 This relationship is predicted in the deformation of a perfect static triangular solid of particles interacting with a r-3 potential.9 This agreement may provide us a clue to understand the interparticle interactions of the colloidal system driven by an alternating electric field in a quantitative way.
Conclusions In this work, we clearly demonstrate the determination of the elastic moduli of a 2D close-packed colloidal crystal governed by an alternating electric field. The local strain fluctuation is visualized by calculating the stain tensors of the particles under the affine deformation, which spatiotemporally evolutes as a function of the field strength and frequency. The equiparti:: (22) von Grunberg, H. H.; Keim, P.; Zahn, K.; Maret, G. Phys. Rev. Lett. 2004, 93, 255703.
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tion theorem relates the thermal average of strain fluctuation to the elastic constants. As a consequence, the elastic constants are obtained by the extrapolation of the finite-size scaling at the thermodynamic limit. Our results show that the elastic moduli are larger than that of the non-close-packed colloidal crystals. Also, our results imply that the interparticle interaction between colloids could be an inverse cubic potential, which is necessary to be clarified in a further investigation. An interesting feature in our system is that the elastic properties can be controlled by two independent parameters. This could provide plenty of room to study the elastic behavior in the presence of dislocations and grain boundaries for further research. Finally, we remark that this method is a noninvasive, accurate technique to study the micro- and macroscopic elastic properties for soft solids. Acknowledgment. This research has been supported by Academic Research Fund (ARF) of the Ministry of Education of Singapore (Grant No.: R-144-000-160-112).
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