Visualization of Dislocation Dynamics in Colloidal Crystals - Itai Cohen

REPORTS This number is one-seventh of the value reported in (1) as the upper limit of viscosity of superfluid helium. For comparison, the viscosity of normal fluid 4He is 2
10j6 PaIs. This calculation invalidates this soft shear mode interpretation and supports the superflow interpretation. The smooth decay in Ds/D with increasing temperature near 250 mK makes it difficult to determine with precision the supersolid transition temperature Tc. We estimate the value of Tc by assuming a linear dependence of Ds/D on temperature near Tc and then use the data with Ds/D between 0.01 and 0.1 to fit for Tc. The resultant transition temperatures, with uncertainty of 20 mK, are likely to be lower than the Btrue[ values. The transition temperatures we have found show a weak pressure dependence decreasing from 315 mK at 26 bars down to 230 mK at pressures exceeding 40 bars. The phase diagram of 4 He, including both the superfluid and supersolid phases, is shown in Fig. 4. A number of experiments have searched for the supersolid phase in bulk solid helium without success (27). A recent experiment found an unexpected interaction between acoustic and heat pulses in solid 4He with several tens of parts per million of 3He (28). The result led to the interpretation of a Bose condensate of thermally activated vacancies at temperatures above 200 mK and therefore is not related to the findings reported here. A prior torsional oscillator experiment (29) reported the absence of any supersolid decoupling in a solid 4He sample that contains 411 parts per million of 3He—a concentration, as we found in the Vycor experiment (15), sufficient to quench the supersolid phase. The authors of this prior experiment also reported that they failed to find evidence of the supersolid phase in samples of high-purity 4He (29). Our results disagree with this conclusion. Other experiments involve the search of motion of a falling dense object in solid helium or flow of solid helium from one chamber of higher pressure to one of lower pressure (30–34). A likely explanation of why these experiments found null results is that such pressuredriven mass flow requires a difference in the supersolid fraction between regions or chambers of different pressures. Our measurements showed that the supersolid fraction is insensitive to the pressure of the solid 4He samples. We noted above that Bose-Einstein condensation is found together with superfluidity in bulk liquid helium and in alkali gases. In contrast, superfluidity at T 9 0 is found in two-dimensional liquid helium films without Bose-Einstein condensation (18). An intriguing question is whether the supersolid phase is associated with BoseEinstein condensation.

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References and Notes 1. P. Kapitza, Nature 141, 74 (1938). 2. J. F. Allen, A. D. Misener, Nature 141, 75 (1938). 3. P. Nozie`res, D. Pines, Theory of Quantum Liquids (Addison-Wesley Reading, MA, 1989), vol. 2, chap. 6. 4. P. E. Sokol, in BoseE(0(1 – 8) cos "^, where rc and R are the core and outer radii of the dislocation strain field, respectively (18), (0 is the misfit strain imposed by the template, and " is the angle between the Burgers vector and its projection onto the template. We take rc to be b/4

t =t 0

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(1, 19), R to be the film thickness (17), and (0 to be 0.015. Because E/6 0 2(1 þ 8) for an isotropic elastic medium (1) and the Poisson ratio can be set at 8 0 1/3 (7), we find hc 0 22 6m. Consistent with this prediction, at a thickness of 22 6m the colloidal crystalline film is still free of dislocations. However, as the film thickness is increased to 31 6m, the film develops a large number of dislocations (Fig. 1, D to G), again consistent with the model. The misfit strain associated with the dislocations is (p0ffiffiffi b cos "/3. Using b 0 0.94 6m, cos " 0 1= 3, and 3 0 53 6m, we find ( 0 0.010, which corresponds to two-thirds of the total misfit strain (0 0 0.015 (20). A further test of the continuum model is to calculate the height at which the dislocations rest above the template. A dislocation is driven toward the template by the force Fel 0 ½3E(02, which results from the elastic stress in the layers below the dislocation core, which are still strained (21). The boundary condition, set by the requirement that the particles in the first layer must adopt the template spacing, gives rise to an image force that repels the dislocation away from the template. The image force acting on a dislocation whose Burgers vector b¶ is parallel to the template and whose core rests at a distance z above the template is Fi 0 6b¶2/E4>(1 – 8)z^ (1). We estimate this force for our system by neglecting the vertical component of the Burgers vector and taking b¶ 0 b cos ". The image force balances the elastic force when the dislocation rests at a distance z0 0 6(b cos ")2/ Using E2>3E(1 – 8)(02^ from the template. pffiffiffi 3 0 53 6m, b 0 0.94, cos " 0 1= 3, and (0 0 0.015, we find z0 0 2.1 6m, in good agreement with our observation of z0 0 3 6m.

Thus, surprisingly, the predictions of continuum theory, which are typically applied on length scales above 100 lattice constants, hold even for a very small number of layers.

We also used LDM to study the dislocation dynamics during the epitaxial growth of the film. We started with a crystal 22 6m thick, added a final dose of particles, waited 14 hours, and then tracked the evolution of dislocations over a period of 2.5 hours (movie S1). Three snapshots from the movie are shown in Fig. 3, A to C. The images show the spreading of existing dislocations and the nucleation and growth of new ones. Figure 4A shows the dislocation length as a function of time for the four dislocations indicated in Fig. 3C. Initially, the dislocations grow rapidly, at a rate of about 1 6m sj1. As the dislocation length increases, however, the growth rate decreases and eventually falls to zero. To quantitatively analyze this spreading behavior, it is necessary to account for the forces acting on a dislocation. A diagram depicting a typical dislocation line in our thin film is shown in Fig. 4B. The dislocation consists of an edge segment that runs parallel to the template and joins two screw segments that terminate at the crystal surface. The edge dislocation expands through the lateral motion of the screw segments (22). Figure 4B also depicts the forces acting on one of the screw segments. The expansion of a dislocation is driven by the Peach-Koehler force FPK due to the elastic stress (22) and is resisted by the dislocation line tension Fl (1) and a drag force Fd associated with moving the screw segment through the crystal. For colloidal crystals, Fd has been shown to be

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100 µm Fig. 3. (A to C) Snapshots from an LDM movie (movie S1) taken during the epitaxial growth of a colloidal crystal film grown on the stretched template. The time t0 corresponds to 14 hours after adding a final dose of particles. The dots in (C) mark four dislocations whose growth is tracked and displayed in Fig. 4.

L / L inf

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0.8 0.6 0.4 0.2 0.0

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Fig. 4. (A) Length versus time for the four dislocations marked in Fig. 3C. (B) Schematic diagram of a dislocation line that lies in a hcp plane shown in red. The dislocation edge segment runs parallel to the template and joins two screw segments that terminate at the crystal surface. The forces acting on the dislocation line are marked. (C) The symbols indicate the rescaled dislocation length L/Linf versus rescaled time t/I for the four growth curves marked in (A). The solid line is the theoretical prediction of the model and has the functional form [1 – exp(t/I)].

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REPORTS the dominant force associated with the motion of such screw dislocations (19). At the onset of dislocation growth, we estimated the magnitude of each of the forces to be on the order of 100 f N. The length of a dislocation as a function of time can be calculated from the force balance, FPK 0 Fl þ Fd (23). As the dislocation grows and accommodates an increasing portion of the misfit strain, the elastic force driving the expansion decreases. Thus, the force balance predicts that L 0 LinfE1 – exp(–t/I)^, where Linf is the final dislocation length and I is a constant, proportional to the ratio of the viscosity of the solvent to the elastic modulus. To test this prediction, we scaled the entire data set and plotted L/Linf versus t/I (Fig. 4C). The data are in excellent agreement with the theoretical prediction. As a final check, we used the average value of I, which we determined to be 130 (T40) s, to estimate the elastic modulus of the colloidal crystal (23). This estimate yields a value of 0.3 Pa, in reasonable agreement with theoretical estimates that predict a value on the order of 1 Pa (7). This is one of the only techniques available for directly determining the elastic modulus of thin colloidal crystal films. The combination of imaging techniques presented and the close similarity of dislocations in colloidal and atomic crystals lays the groundwork for investigating further important phenomena that cannot be directly studied on the atomic scale, such as the nucleation and interaction of dislocations in very constrained systems. The remarkable and unexpected correspondence between continuum model predictions and the phenomena we observe on the scale of just a few lattice constants suggests that continuum models may also be applied to describe dislocation behavior even in highly constrained structures, such as those being made as nanoscale science pushes to ever smaller devices. Finally, the effects of the vanishing stacking fault energy and the pressure head on dislocations in colloidal crystals highlight some of the unique features of this class of condensed matter. References and Notes 1. J. P. Hirth, J. Lothe, Theory of Dislocations (Wiley, New York, ed. 2, 1982). 2. G. I. Taylor, Proc. R. Soc. A 145, 362 (1934). 3. E. Orowan, Z. Phys. 89, 605 (1934). 4. M. Polanyi, Z. Phys. 89, 660 (1934). 5. J. Lepinoux, D. Maziere, V. Pontikis, G. Saada, Eds., NATO Science Series E, Multiscale Phenomena in Plasticity: From Experiments to Phenomenology, Modeling and Materials Engineering (Kluwer Academic, Dordrecht, Netherlands, 2000), vol. 367. 6. A. D. Dinsmore, E. R. Weeks, V. Prasad, A. C. Levitt, D. A. Weitz, Appl. Opt. 40, 4152 (2001). 7. D. Frenkel, A. J. C. Ladd, Phys. Rev. Lett. 59, 1169 (1987). 8. R. Hull, J. C. Bean, D. J. Werder, R. E. Leibenguth, Appl. Phys. Lett. 52, 1605 (1988). 9. A. van Blaaderen, R. Ruel, P. Wiltzius, Nature 385, 321 (1997). 10. The silica particles (Micromod, Sicastar, 1.5 6m) are suspended in an index-matching mixture of water and dimethyl sulfoxide at an initial volume fraction

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12. 13. 14. 15. 16. 17. 18.

of 0.03%. We add a small amount of a fluoresceinNaOH mixture to the solvent, allowing us to image the particles with fluorescence under the confocal microscope; the particles appear as dark spheres in a light background. We use standard photolithographic processes to prepare a template with a square array of 3000 by 3000 holes in a polymethyl methacrylate layer 500 nm thick over an area of about 5 mm by 5 mm. For example, we estimate that increasing the film thickness from 20 6m to 30 6m produces a 0.4% decrease in the preferred lattice spacing. D. B. Williams, C. B. Carter, Transmission Electron Microscopy: A Textbook for Materials Science (Plenum, New York, 1996). D. Hull, D. J. Bacon, Introduction to Dislocations (Pergamon, New York, 1984). S. Pronk, D. Frenkel, J. Chem. Phys. 110, 4589 (1999). F. C. Frank, J. H. van der Merwe, Proc. R. Soc. London Ser. A 198, 216 (1949). J. W. Matthews, A. E. Blakeslee, J. Cryst. Growth 27, 118 (1974). In accordance with the one-dimensional dislocation model, we assume that the x and y directions are uncoupled and study the accommodation of the misfit strain in one spatial direction. The elastic energy stored in a strained film of thickness h is Uel 0 ½(el2Eh per unit area, where the elastic strain is (el 0

(0 – (, and ( is the strain relieved by the dislocations. The energy cost per unit area associated with the misfit dislocations is Ul 0 (3j1){6b2 ln(R/rc)/[4>(1 – 8)]}, where b is the magnitude of the Burgers vector of the dislocations, and rc and R are the core and outer radii of the dislocation strain field, respectively (1). Minimizing the total energy Uel þ Ul with respect to the number of dislocations per unit length 3j1, we find that, for film thicknesses greater than hc, the equilibrium dislocation density is 3j1 0 [(0/(b cos ")] – {6 ln(R/rc)/[4>(1 – 8)E cos2 " h]}. As the film thickness increases, the dislocation density approaches the limit [(0/(b cos ")], where the entire misfit strain is accommodated by dislocations. The critical film thickness, where 3j1 0 0, is hc 0 6b ln(R/rc)/[4>E(0(1 – 8)cos "]; this yields a critical thickness hc 0 22 6m.

19. M. Jorand, F. Rothen, P. Pieranski, J. Phys. (Paris) 46, 245 (1985). 20. We note that this value for the strain ( is larger than that predicted for this 31-6m-thick film grown on a template where (0 0 0.015. However, the model prediction depends sensitively on the value of the misfit strain (0. For example, a 0.005

increase in the misfit strain can account for the discrepancy. Such an increase can result from the change in pressure head when the film thickness changes from 20 6m to 30 6m. 21. The elastic energy stored in the thin crystal layer below the dislocations is Uel 0 ½(02 Ez per area and decreases with decreasing distance, z, of the dislocations to the template. Hence, a single dislocation experiences the elastic force Fel 0 ½3E(02.

22. J. W. Matthews, S. Mader, T. B. Light, J. Appl. Phys. 41, 3800 (1970). 23. The forces acting on the screw dislocation are given by FPK 0 26(1 þ 8)hb cos " (el/(1 – 8), Fl 0 6b2 ln(R/rc)/ [4>(1 – 8)], and Fd 0 (h/sin "),v, where , 0 (b2/4>rc2, v is the velocity of the screw dislocations, and ( is the solvent viscosity. As the dislocation expands, it accommodates an increasing portion of the misfit strain. The rate at which the strain is accommodated is d(/dt 0 Dscrew b cos " v, where Dscrew is the number of mobile screw dislocations per unit area of the film. From the force balance FPK 0 Fl þ Fd, we find that ((t) 0 (inf[1 – exp(–t/I)], where the constant (inf 0 (0 – {b ln(R/rc)/[8>(1 þ 8)h cos "]} corresponds to the final misfit strain accommodated by the dislocation, and the time constant I 0 (1 – 8),/(Eb2 cos2 " sin " Dscrew). Because ( is proportional to the dislocation length L, this equation predicts the time dependence of L. To calculate the elastic modulus from the measured time constant I, we determine the density of mobile screw dislocations by counting the number of dislocations that are expanding, multiplying by 2, and dividing by the area corresponding to the field of view. We obtain Dscrew 0 3
10j4 6mj2. Taking ( 0 3.0
10j3 PaIs (24), we obtain a value of 0.3 Pa for the elastic modulus. 24. Y. Higashigaki, D. H. Christensen, C. H. Wang, J. Phys. Chem. 85, 2531 (1981). 25. We thank R. Christianson and D. Blair for their help with the image analysis, and E. Chen and Y. Lu for their help in manufacturing the templates. Supported by a Lynen Fellowship from the Alexander von Humboldt Foundation (P.S.), by NSF grant DMR0243715, and by Harvard MRSEC grant DMR0213805. Supporting Online Material www.sciencemag.org/cgi/content/full/305/5692/1944/ DC1 Movie S1 30 June 2004; accepted 20 August 2004

Ice Flow Direction Change in Interior West Antarctica Martin J. Siegert,1* Brian Welch,2 David Morse,3 Andreas Vieli,1 Donald D. Blankenship,3 Ian Joughin,4 Edward C. King,5 Gwendolyn J.-M. C. Leysinger Vieli,1 Antony J. Payne,1 Robert Jacobel2 Upstream of Byrd Station (West Antarctica), ice-penetrating radar data reveal a distinctive fold structure within the ice, in which isochronous layers are unusually deep. The fold has an axis more than 50 kilometers long, which is aligned up to 45- to the ice flow direction. Although explanations for the fold’s formation under the present flow are problematic, it can be explained if flow was parallel to the fold axis È1500 years ago. This flow change may be associated with ice stream alterations nearer the margin. If this is true, central West Antarctica may respond to future alterations more than previously thought. Ice-penetrating radar provides information on ice thickness, subglacial morphology, and internal layering, caused by electromagneticwave reflections from dielectric contrasts

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(such as high-acidity horizons formed from the aerosol product of volcanic events contained within ancient snow). Internal layers are believed to be isochronous, as

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