R EPORTS This number is one-seventh of the value References and Notes 23. J. F. Fernandez, M. Puma, J. Low Temp. Phys. 17, 131 reported in (1) as the upper limit of viscos- 1. P. Kapitza, Nature 141, 74 (1938). (1974). 2. J. F. Allen, A. D. Misener, Nature 141, 75 (1938). 24. R. A. Guyer, Phys. Rev. Lett. 26, 174 (1970). ity of superfluid helium. For comparison, 3. P. Nozie`res, D. Pines, Theory of Quantum Liquids 25. W. M. Saslow, Phys. Rev. Lett. 36, 1151 (1976). the viscosity of normal fluid 4He is 2 (Addison-Wesley Reading, MA, 1989), vol. 2, chap. 6. 26. D. Ceperley, G. V. Chester, M. H. Kalos, Phys. Rev. B 10j6 PaIs. This calculation invalidates this 4. P. E. Sokol, in BoseMatter Physics Program. exceeding 40 bars. The phase diagram of 17. D. J. Bishop, J. D. Reppy, Phys. Rev. Lett. 40, 1727 Supporting Online Material 4 (1978). He, including both the superfluid and 18. J. M. Kosterlitz, D. J. Thouless, J. Phys. C 6, 1181 (1973). www.sciencemag.org/cgi/content/full/1101501/DC1 supersolid phases, is shown in Fig. 4. 19. V. Ambegaokar, B. I. Halperin, D. R. Nelson, E. D. Fig. S1 A number of experiments have searched Siggia, Phys. Rev. Lett. 40, 783 (1978). 20. A. L. Fetter, J. Low Temp. Phys. 16, 533 (1974). 15 June 2004; accepted 12 August 2004 for the supersolid phase in bulk solid helium 21. E. J. Mueller, personal communication. Published online 2 September 2004; without success (27). A recent experiment 22. R. P. Feynman, in Progress in Low Temperature 10.1126/science.1101501 found an unexpected interaction between Physics I (North-Holland, Amsterdam, 1955), chap. 2. Include this information when citing this paper. 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 Visualization of Dislocation at temperatures above 200 mK and therefore is not related to the findings reported here. A Dynamics in Colloidal Crystals prior torsional oscillator experiment (29) 1 1,2 1,2 1 reported the absence of any supersolid Peter Schall, * Itai Cohen, David A. Weitz, Frans Spaepen decoupling in a solid 4He sample that contains 411 parts per million of 3He—a The dominant mechanism for creating large irreversible strain in atomic crys- concentration, as we found in the Vycor tals is the motion of dislocations, a class of line defects in the crystalline lattice. experiment (15), sufficient to quench the Here we show that the motion of dislocations can also be observed in strained supersolid phase. The authors of this prior colloidal crystals, allowing detailed investigation of their topology and experiment also reported that they failed to propagation. We describe a laser diffraction microscopy setup used to study find evidence of the supersolid phase in the growth and structure of misfit dislocations in colloidal crystalline films. samples of high-purity 4He (29). Our results Complementary microscopic information at the single-particle level is obtained disagree with this conclusion. Other experi- with a laser scanning confocal microscope. The combination of these two ments involve the search of motion of a fall- techniques enables us to study dislocations over a range of length scales, ing dense object in solid helium or flow of allowing us to determine important parameters of misfit dislocations such as solid helium from one chamber of higher critical film thickness, dislocation density, Burgers vector, and lattice resistance pressure to one of lower pressure (30–34). A to dislocation motion. We identify the observed dislocations as Shockley par- likely explanation of why these experiments tials that bound stacking faults of vanishing energy. Remarkably, we find that found null results is that such pressure- even on the scale of a few lattice vectors, the dislocation behavior is well driven mass flow requires a difference in described by the continuum approach commonly used to describe dislocations the supersolid fraction between regions or in atomic crystals. chambers of different pressures. Our mea- surements showed that the supersolid frac- Dislocations in a crystalline lattice are cen- the crystal has been uniformly translated tion is insensitive to the pressure of the solid tral to our understanding of yield, work hard- with respect to the other (2–4). A complete 4He samples. ening, fracture, fatigue, and time-dependent understanding of dislocations and their We noted above that Bose-Einstein elasticity in atomic crystals (1). Such dis- dynamics requires an analysis that bridges a condensation is found together with super- locations are line defects that mark the range of length scales (5). On the atomic fluidity in bulk liquid helium and in alkali boundary of a surface at which one part of scale, the interatomic potential determines the gases. In contrast, superfluidity at T 9 0is structure of the dislocation core. On the

found in two-dimensional liquid helium 1 medium-range scale, the strain field of the Division of Engineering and Applied Sciences, films without Bose-Einstein condensation 2Department of Physics, Harvard University, 9 Oxford dislocations determines their interactions. On (18). An intriguing question is whether the Street, Cambridge, MA 02138, USA. the macroscopic scale, the behavior of the supersolid phase is associated with Bose- *To whom correspondence should be addressed. dislocations determines the deformation of Einstein condensation. E-mail: [email protected] the crystal. It is difficult to observe dis-

1944 24 SEPTEMBER 2004 VOL 305 SCIENCE www.sciencemag.org R EPORTS d 0 6 locations simultaneously on these length lattice constant 0 1.63 mthatnearly investigate dislocations in atomic systems scales. Transmission electron microscopy matches the equilibrium interparticle separa- (13). We use a HeNe laser with a wavelength (TEM) is best suited for in situ observations tion for colloidal crystals that are about 30 6m of 632 nm, which scatters coherently from on the medium-range scale. Computer sim- thick. This preferred lattice spacing changes the colloidal crystal. When the incident beam ulations can bridge the length scales, but the slightly with film thickness as a result of the is perpendicular to the template, a symmetric size of their systems and length of the difference in the pressure head (12). fcc (100) diffraction pattern is observed. By evolution time remain limited. Thus, there We add 3.5 ml of the dilute suspension, slightly tilting the sample to change the remains a need for techniques that link which after sedimentation gives rise to about direction of the incident beam, we maximize investigations of dislocations on these differ- 11 crystalline layers corresponding to a film the intensity in the (220) diffracted spot. We ent length scales. 13 6m thick. After several days, we remove then use two lenses to project the light in the We show that colloidal crystals offer a two-thirds of the supernatant and replace it with diffracted beam onto a screen (Fig. 1A). The unique opportunity for just such a bridging another dose of the dilute silica suspension. image on the screen corresponds to a region of length scales. Because colloidal particles Using this procedure, we grow the crystal by in the crystal that is illuminated by the in- are several orders of magnitude larger than 9-6m increments until the film is 31 6m thick. cident beam. A perfect crystal shows a atoms, they can be studied in real time and We image dislocations in the crystalline uniformly bright image. When the crystal their positions in three dimensions can be films with a simple LDM setup that is contains dislocations, however, dark lines determined accurately by confocal micros- inspired by the TEM techniques used to appear in the image: The bending of lattice copy (6). Concentrated hard-sphere colloidal suspensions form crystals to increase their Fig. 1. Laser diffraction k0 Laser beam entropy, thereby lowering their free energy. microscopy (LDM) tech- nique and images. (A) A Such crystals have a finite stiffness (7), which Schematic of the LDM Colloidal is essential for the existence of dislocations. Screen instrument: A laser beam Projector Crystal To study these dislocations on the medium is sent through a colloidal lens Objective crystal. One of the dif- scale, we developed a laser diffraction micros- lens copy (LDM) technique that images the strain fracted beams is imaged field in a manner analogous to TEM in atomic on a screen by means of k an objective and a projec- systems. Confocal microscopy and LDM tor lens. (B and C)LDM make a powerful combination for studying images of the colloidal dislocation dynamics simultaneously over two crystal grown on the tem- Diffraction pattern qualitatively different length scales. plate with the ideal lattice 0 6 We focus on misfit dislocations formed constant d0 1.63 m. Arrows indicate disloca- 0 0 when a film is grown via particle sedimenta- k g tions. The upper left inset 0 k0 g = q tion on a substrate with a different lattice shows the diffraction pat- k q k parameter. This configuration allows us to in- tern from the crystalline troduce the dislocations in a controlled way film; 0 indicates the trans- s and to study nucleation and propagation of mitted beam, and an dislocations (8) during a process analogous to arrow indicates the dif- the industrially important epitaxial growth of fractedbeamusedfor imaging. The upper right thin atomic crystalline films. In thin epitaxial inset illustrates the wave films, the misfit strain ( is accommodated 0 vectors of the incident and BB CC purely elastically and results in a uniform diffracted beams k0 and k, strain ( . The total elastic energy increases the diffraction vector q 0 el 0 0 k – k0, and the cor- linearly with the film thickness. At some k g responding reciprocal lat- 0 k0 g critical thickness h , the crystal can lower its = q c tice vector g. In (B) the k q k energy by incorporating dislocations that re- diffraction vector q coin- lieve some of the elastic strain. As the film cides with the reciprocal s thickness increases, an increasing portion of lattice vector g and the the misfit strain is accommodated. diffracted beam intensity We determine the important parameters of is maximum. In (C), the the dislocation array: number per length, sample is tilted, so that q differs from g by the Burgers vector, position and range of the strain excitation error s 0 q – g, x field, and mobility. We find that many of these which gives rise to an DD y E features can be accounted for by the continuum inversion of the image theory used for epitaxial growth of atomic thin contrast. (D to G)LDM k k0 k 0 k films. Some features, however, are unique to images of a colloidal crys- talline film grown on a colloidal crystals, such as the negligibly small 0 0 q stretched template with stacking fault energy and the slight variation of 0 lattice constant d1 1.65 q = g g s the lattice parameter with height (due to the 6m. (D) and (E) show that pressure head of the upper layers). using the (220) diffraction We grow colloidal face-centered cubic vector, which lies along (fcc) single crystals by slowly sedimenting the y direction, gives colloidal particles onto a patterned template images of dislocations ori- ented in the x direction. (9). We use silica particles with a diameter of (F) and (G) show that FF GG 1.55 6m and a polydispersity of less than choosing the (220) diffrac- 100 µm 3.5% (10). We prepare a pattern (11)with tion vector, which lies along the x direction, images dislocations oriented in the y direction.

www.sciencemag.org SCIENCE VOL 305 24 SEPTEMBER 2004 1945 R EPORTS planes in the strain field of the dislocation number of dislocations nucleated and grew reconstruction of a 55 6mby556mby176m gives rise to a local change in the Bragg (Fig. 1, D and E). We determined the average section of the crystal is shown in Fig. 2A. The condition and results in a dark line in the dislocation line separation in the direction x, y,andz axes correspond to the (110), (110), image of the diffracted beam. A LDM image perpendicular to the dislocation lines, 3, and (001) directions of the fcc lattice, of a 0.3 mm by 0.3 mm section of the from the images. 3j1 is the number of respectively. Particles with three opposing colloidal crystal grown on the template with dislocations per unit length. Measuring 3 in nearest neighbor pairs are shown in red, and d 0 6 the ideal lattice constant 0 1.63 mis three different 0.3 mm by 0.3 mm regions, those with six opposing nearest neighbor pairs shown in Fig. 1B. Even these crystals, grown we obtained an average value of 53 (T10) 6m in blue. The red particles lie along intersecting on templates with an ideal lattice constant, for the 31-6m crystal. planes embedded in the fcc lattice. By display- contain some dislocations (indicated by Remarkably, although the template was ing only the red particles, we show that the arrows). Such dislocations are most fre- stretched in both spatial directions, dislocation planes are hcp (Fig. 2B). The red planes quently observed near the template border. lines were seen in one direction only (Fig. 1, D sandwich a stacking fault where the stacking We can further exploit the analogy with and E). The dislocation contrast is visible only order of the hcp planes changes from ABC- the TEM technique and use contrast inversion if the particle displacements in the dislocation ABCABC to ABCBCABCA. The stacking fault to verify that the dark lines in Fig. 1B indeed strain field have a component parallel to the lies along the (111) plane where the disloca- result from scattering due to the dislocations. diffraction vector used for imaging. The (220) tions move most easily as a result of the When the sample is tilted slightly further, the diffraction vector chosen for imaging in Fig. shallow potential wells. This defines the glide Bragg condition is no longer fulfilled in the 1, B to E, lies along the y direction; therefore, plane of the dislocations in the fcc structure. perfect lattice and is instead locally satisfied only lattice distortions with a component For a closer look at the strain field asso- in the region corresponding to the dislocation along the y direction showed up in the image. ciated with the stacking faults, we display a strain field. Thus, the image contrast on the When we instead chose the (220) diffraction typical y-z cut through a stacking fault. The screen inverts (Fig. 1C). The additional tilt vector, which lies along the x direction, we fault ends above the template and is termi- of the sample introduces an excitation error, observed a second set of dislocations (Fig. 1, nated by a dislocation (Fig. 2C). The first s 0 q – g (Fig. 1C, upper right inset), which F and G). Comparing the images in Fig. 1, D row of particles sits in the template holes. In causes the intensity in the diffracted beam to and F, we conclude that the strain field of the the second row of particles, we recognize the decrease (Fig. 1C, upper left inset). Close to dislocations is strictly perpendicular to the emergence of a strain field in the y-z plane the dislocation, however, the bending of imaged dislocation lines. associated with a dislocation line oriented lattice planes locally scatters light into the To elucidate the defect structure on the perpendicular to this plane. The dislocation direction of the diffracted beam, making the microscopic scale, we used confocal micros- core (±) lies about two lattice constants dark lines appear light. copy to image the individual particles and to above the template. The Burgers circuit To investigate the effect of a lattice mis- determine their positions (6). The 31-6m-thick illustrated by the red line, which would close match, we grew a crystal on a template with fcc colloidal crystal grown on the stretched in the perfect lattice, exhibits a closure fail- d 0 6 lattice constant 1 1.65 m, which is 1.5% template contains characteristic defects. At ure around the dislocation core. The Burgers d larger than 0. The crystal grown on the these defects, the nearest neighbor particle vector b, which connects the starting and stretched template exhibited a similarly low configuration changes so that particles have ending points of the Burgers circuit, is density of dislocations at a crystal thickness three opposing nearest neighbor pairs, as is 1/6(112). This type of dislocation is known of 22 6m. Strikingly, as the crystal was the case in the hexagonal close-packed (hcp) as a Shockley partial dislocation and is the grown to a thickness of 31 6m, a large lattice, rather than six as in the fcc lattice. A most prominent dislocation observed in fcc

Fig. 2. (A) Reconstruction of a 55 6mby556mby176m section of A B the colloidal crystal grown on the stretched template. The red par- z (µm) z (µm) ticles delineate stacking faults em- (001) (001) bedded in an otherwise perfect fcc lattice. (B) Crystal reconstruction showing only the particles adja- cent to the stacking faults. (C)A reconstructed y-z section through a stacking fault. The stacking fault y y is terminated by a Shockley partial (µ (µ dislocation whose core position is m m ) ) indicated by ±. The red loop in- (1 (1 dicates a Burgers circuit. The upper 10 10 10) ) (110) ) µ ) (1 right inset is a three-dimensional x (µm) x ( m illustration of the fcc unit cell. The y-z plane is gray; the hcp plane parallel to the stacking fault is red. C

z (µm)

y (µm)

1946 24 SEPTEMBER 2004 VOL 305 SCIENCE www.sciencemag.org R EPORTS 14 1 19 R 17 ( metals ( ). In metals, Shockley partial dis- ( , ), to be the film thickness ( ), and 0 Thus, surprisingly, the predictions of con- locations usually appear in pairs, held to- to be 0.015. Because E/6 0 2(1 þ 8)foran tinuum theory, which are typically applied gether by a stacking fault of nonzero energy. isotropic elastic medium (1) and the Poisson on length scales above 100 lattice constants, 8 0 7 h 0 In contrast, in the hard-sphere colloidal crys- ratio can be set at 1/3 ( ), we find c hold even for a very small number of layers. tal, the stacking faults extend to the crystal 22 6m. Consistent with this prediction, at a We also used LDM to study the dis- surface. This difference results from the thickness of 22 6m the colloidal crystalline location dynamics during the epitaxial vanishingly low energy cost associated with film is still free of dislocations. However, as growth of the film. We started with a crystal stacking faults in hard-sphere crystals (15), the film thickness is increased to 31 6m, the 22 6m thick, added a final dose of particles, where second nearest neighbor interactions film develops a large number of dislocations waited 14 hours, and then tracked the are almost negligible. (Fig. 1, D to G), again consistent with the evolution of dislocations over a period of h To calculate the critical film thickness c, model. The misfit strain associated with the 2.5 hours (movie S1). Three snapshots from where the crystal starts to nucleate disloca- dislocations is (p0ffiffiffib cos "/3.Usingb 0 0.94 the movie are shown in Fig. 3, A to C. The tions in order to relieve some of the elastic 6m, cos " 0 1= 3,and3 0 53 6m, we find images show the spreading of existing strain, we consider the crystalline film to be ( 0 0.010, which corresponds to two-thirds dislocations and the nucleation and growth ( 0 20 an isotropic, linear elastic medium with Young of the total misfit strain 0 0.015 ( ). of new ones. Figure 4A shows the disloca- modulus E, shear modulus 6, and Poisson A further test of the continuum model is to tion length as a function of time for the four ratio 8. The widely used one-dimensional calculate the height at which the dislocations dislocations indicated in Fig. 3C. Initially, dislocation models for misfit dislocations in rest above the template. A dislocation is driven the dislocations grow rapidly, at a rate of 16 17 h 0 6b R r F 0 3E( 2 6 j1 atomic thin films ( , )give c ln( / c)/ toward the template by the force el ½ 0 , about 1 ms . As the dislocation length E >E( 8 "^ r R 4 0(1 – ) cos , where c and are the which results from the elastic stress in the increases, however, the growth rate de- core and outer radii of the dislocation strain layers below the dislocation core, which are creases and eventually falls to zero. 18 ( 21 field, respectively ( ), 0 is the misfit strain still strained ( ). The boundary condition, set To quantitatively analyze this spreading imposed by the template, and " is the angle by the requirement that the particles in the behavior, it is necessary to account for the between the Burgers vector and its projec- first layer must adopt the template spacing, forces acting on a dislocation. A diagram de- r b tion onto the template. We take c to be /4 gives rise to an image force that repels the picting a typical dislocation line in our thin dislocation away from the template. The film is shown in Fig. 4B. The dislocation image force acting on a dislocation whose consists of an edge segment that runs parallel Burgers vector b¶ is parallel to the template to the template and joins two screw segments t =t A 0 and whose core rests at a distance z above the that terminate at the crystal surface. The edge F 0 6b¶2 E > 8 z^ 1 template is i / 4 (1 – ) ( ). We dislocation expands through the lateral estimate this force for our system by neglect- motion of the screw segments (22). Figure ing the vertical component of the Burgers 4B also depicts the forces acting on one of vector and taking b¶ 0 b cos ". The image the screw segments. The expansion of a force balances the elastic force when the dislocation is driven by the Peach-Koehler z 0 6 b " 2 F 22 dislocation rests at a distance 0 ( cos ) / force PK due to the elastic stress ( ) and is E >3E 8 ( 2^ F 1 2 (1 – ) 0 from the template.pffiffiffi Using resisted by the dislocation line tension l ( ) 3 0 6 b 0 " 0 = ( 0 F 53 m, 0.94, cos 1 3,and 0 and a drag force d associated with moving z 0 6 0.015, we find 0 2.1 m, in good agree- the screw segment through the crystal. For ment with our observation of z 0 3 6m. colloidal crystals, F has been shown to be B t= t0+16 min 0 d

A B 80 1 2 F screw PK F 60 3 4 d F l edge

(µm) 40 L

20 C t =t0+260 min Fig. 4. (A) Length versus time for the four 0 dislocations marked in Fig. 3C. (B) Sche- 11 44 0 20 40 60 80 100 120 140 160 180 200 matic diagram of a dislocation line that lies t (min) in a hcp plane shown in red. The disloca- 1.2 tion edge segment runs parallel to the 22 C template and joins two screw segments 1.0 that terminate at the crystal surface. The 33 forces acting on the dislocation line are 0.8 marked. (C) The symbols indicate the inf rescaled dislocation length L/L versus

100 µm L inf 0.6 rescaled time t/I for the four growth /

L curves marked in (A). The solid line is the Fig. 3. (A to C) Snapshots from an LDM movie 0.4 (movie S1) taken during the epitaxial growth of theoretical prediction of the model and has the functional form [1 – exp(t/I)]. a colloidal crystal film grown on the stretched 0.2 template. The time t0 corresponds to 14 hours after adding a final dose of particles. The dots 0.0 in (C) mark four dislocations whose growth is 0 5 10 15 20 tracked and displayed in Fig. 4. t / τ

www.sciencemag.org SCIENCE VOL 305 24 SEPTEMBER 2004 1947 R EPORTS the dominant force associated with the mo- of 0.03%. We add a small amount of a fluorescein- increaseinthemisfitstraincanaccountforthe tion of such screw dislocations (19). At the NaOH mixture to the solvent, allowing us to image discrepancy. Such an increase can result from the the particles with fluorescence under the confocal change in pressure head when the film thickness onset of dislocation growth, we estimated the microscope; the particles appear as dark spheres in a changes from 20 6mto306m. magnitude of each of the forces to be on the light background. 21. The elastic energy stored in the thin crystal layer 11. We use standard photolithographic processes to below the dislocations is U 0 ½( 2 Ez per area and order of 100 f N. The length of a dislocation el 0 prepare a template with a square array of 3000 by decreases with decreasing distance, z,ofthedisloca- as a function of time can be calculated from 3000 holes in a polymethyl methacrylate layer 500 tions to the template. Hence, a single dislocation F 0 F þ F 23 0 3 ( 2 the force balance, PK l d ( ). As the nm thick over an area of about 5 mm by 5 mm. experiences the elastic force Fel ½ E 0 . dislocation grows and accommodates an 12. For example, we estimate that increasing the film 22. J. W. Matthews, S. Mader, T. B. Light, J. Appl. Phys. thickness from 20 6mto306m produces a 0.4% 41, 3800 (1970). increasing portion of the misfit strain, the decrease in the preferred lattice spacing. 23. The forces acting on the screw dislocation are given by 0 6 þ 8 " ( 8 0 6 2 elastic force driving the expansion decreases. 13. D. B. Williams, C. B. Carter, Transmission Electron FPK 2 (1 )hb cos el/(1 – ), Fl b ln(R/rc)/ > 8 0 " , , 0 ( 2 > 2 Thus, the force balance predicts that L 0 Microscopy: A Textbook for Materials Science (Plenum, [4 (1 – )], and Fd (h/sin ) v, where b /4 rc , L E t I ^ L New York, 1996). v is the velocity of the screw dislocations, and ( is inf 1 – exp(– / ) , where inf is the final 14. D. Hull, D. J. Bacon, Introduction to Dislocations the solvent viscosity. As the dislocation expands, it dislocation length and I is a constant, pro- (Pergamon, New York, 1984). accommodates an increasing portion of the misfit portional to the ratio of the viscosity of the 15. S. Pronk, D. Frenkel, J. Chem. Phys. 110, 4589 (1999). strain. The rate at which the strain is accommodated ( 0 D " D 16. F. C. Frank, J. H. van der Merwe, Proc. R. Soc. London is d /dt screw b cos v, where screw is the number solvent to the elastic modulus. Ser. A 198, 216 (1949). of mobile screw dislocations per unit area of the film. To test this prediction, we scaled the 0 þ 17. J. W. Matthews, A. E. Blakeslee, J. Cryst. Growth 27, From the force balance FPK Fl Fd, we find that L/L t/I 118 (1974). ((t) 0 ( [1 – exp(–t/I)], where the constant ( 0 entire data set and plotted inf versus inf inf 18. In accordance with the one-dimensional dislocation ( –{b ln(R/r )/[8>(1 þ 8)h cos "]} corresponds to the (Fig. 4C). The data are in excellent agree- 0 c model, we assume that the x and y directions are final misfit strain accommodated by the dislocation, ment with the theoretical prediction. As a final uncoupled and study the accommodation of the and the time constant I 0 (1 – 8),/(Eb2 cos2 " sin " I D ( check, we used the average value of , which misfit strain in one spatial direction. The elastic screw). Because is proportional to the dislocation 0 T energy stored in a strained film of thickness h is Uel length L, this equation predicts the time dependence we determined to be 130 ( 40) s, to estimate ( 2 ( 0 ½ el Eh per unit area, where the elastic strain is el of L. To calculate the elastic modulus from the the elastic modulus of the colloidal crystal ( ( ( I 0 – ,and is the strain relieved by the dislocations. measured time constant , we determine the density (23). This estimate yields a value of 0.3 Pa, The energy cost per unit area associated with the mis- of mobile screw dislocations by counting the number fit dislocations is U 0 (3j1){6b2 ln(R/r )/[4>(1 – 8)]}, in reasonable agreement with theoretical l c of dislocations that are expanding, multiplying by 2, where b is the magnitude of the Burgers vector of the and dividing by the area corresponding to the field of estimates that predict a value on the order D 0 j4 6 j2 ( 0 dislocations, and rc and R are the core and outer radii view. We obtain screw 3 10 m .Taking of 1 Pa (7). This is one of the only techniques of the dislocation strain field, respectively (1). Minimiz- 3.0 10j3 PaIs(24), we obtain a value of 0.3 Pa for the þ available for directly determining the elastic ing the total energy Uel Ul with respect to the elastic modulus. 3j1 modulus of thin colloidal crystal films. number of dislocations per unit length ,wefind 24. Y. Higashigaki, D. H. Christensen, C. H. Wang, J. Phys. that, for film thicknesses greater than hc,theequi- Chem. 85, 2531 (1981). 3j1 0 ( " The combination of imaging techniques librium dislocation density is [ 0/(b cos )] – 25. We thank R. Christianson and D. Blair for their help 6 > 8 2 " presented and the close similarity of disloca- { ln(R/rc)/[4 (1 – )E cos h]}. As the film thickness with the image analysis, and E. Chen and Y. Lu for increases, the dislocation density approaches the limit their help in manufacturing the templates. Supported tions in colloidal and atomic crystals lays the ( " [ 0/(b cos )], where the entire misfit strain is accom- by a Lynen Fellowship from the Alexander von groundwork for investigating further impor- modated by dislocations. The critical film thickness, Humboldt Foundation (P.S.), by NSF grant DMR- 3j1 0 0 6 > ( 8 " tant phenomena that cannot be directly studied where 0, is hc b ln(R/rc)/[4 E 0(1 – )cos ]; 0243715, and by Harvard MRSEC grant DMR- this yields a critical thickness h 0 22 6m. 0213805. on the atomic scale, such as the nucleation and c 19. M. Jorand, F. Rothen, P. Pieranski, J. Phys. (Paris) 46, interaction of dislocations in very constrained 245 (1985). Supporting Online Material systems. The remarkable and unexpected 20. We note that this value for the strain ( is larger www.sciencemag.org/cgi/content/full/305/5692/1944/ than that predicted for this 31-6m-thick film DC1 correspondence between continuum model ( 0 Movie S1 grown on a template where 0 0.015. However, predictions and the phenomena we observe the model prediction depends sensitively on the ( on the scale of just a few lattice constants value of the misfit strain 0. For example, a 0.005 30 June 2004; accepted 20 August 2004 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 Ice Flow Direction Change in ever smaller devices. Finally, the effects of the vanishing stacking fault energy and the Interior West Antarctica pressure head on dislocations in colloidal 1 2 3 1 crystals highlight some of the unique features Martin J. Siegert, * Brian Welch, David Morse, Andreas Vieli, of this class of condensed matter. Donald D. Blankenship,3 Ian Joughin,4 Edward C. King,5 Gwendolyn J.-M. C. Leysinger Vieli,1 Antony J. Payne,1 References and Notes 2 1. J. P. Hirth, J. Lothe, Theory of Dislocations (Wiley, Robert Jacobel New York, ed. 2, 1982). 2. G. I. Taylor, Proc. R. Soc. A 145, 362 (1934). Upstream of Byrd Station (West Antarctica), ice-penetrating radar data reveal 3. E. Orowan, Z. Phys. 89, 605 (1934). a distinctive fold structure within the ice, in which isochronous layers are 4. M. Polanyi, Z. Phys. 89, 660 (1934). 5. J. Lepinoux, D. Maziere, V. Pontikis, G. Saada, Eds., unusually deep. The fold has an axis more than 50 kilometers long, which is NATO Science Series E, Multiscale Phenomena in aligned up to 45- to the ice flow direction. Although explanations for the Plasticity: From Experiments to Phenomenology, fold’s formation under the present flow are problematic, it can be explained if Modeling and Materials Engineering (Kluwer Aca- È demic, Dordrecht, Netherlands, 2000), vol. 367. flow was parallel to the fold axis 1500 years ago. This flow change may be 6. A. D. Dinsmore, E. R. Weeks, V. Prasad, A. C. Levitt, associated with ice stream alterations nearer the margin. If this is true, central D. A. Weitz, Appl. Opt. 40, 4152 (2001). West Antarctica may respond to future alterations more than previously 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, thought. Appl. Phys. Lett. 52, 1605 (1988). 9. A. van Blaaderen, R. Ruel, P. Wiltzius, Nature 385, Ice-penetrating radar provides information (such as high-acidity horizons formed from 321 (1997). on ice thickness, subglacial morphology, and the aerosol product of volcanic events 10. The silica particles (Micromod, Sicastar, 1.5 6m) are suspended in an index-matching mixture of water internal layering, caused by electromagnetic- contained within ancient snow). Internal and dimethyl sulfoxide at an initial volume fraction wave reflections from dielectric contrasts layers are believed to be isochronous, as

1948 24 SEPTEMBER 2004 VOL 305 SCIENCE www.sciencemag.org