J. Phys. Chem. B 2008, 112, 14409–14414
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Do Directional Primary and Secondary Crack Patterns in Thin Films of Maghemite Nanocrystals Follow a Universal Scaling Law?† Anh Tu Ngo, Johannes Richardi, and Marie Paule Pileni* Laboratoire des Mate´riaux Me´soscopiques et Nanome´triques (LM2N), UMR CNRS 7070, UniVersite´ Pierre et Marie Curie, baˆt F, BP 52, 4 Place Jussieu, 75252 Paris Cedex 05, France ReceiVed: March 26, 2008; ReVised Manuscript ReceiVed: May 7, 2008
Cracks due to a shrinking film restricted by adhesion to a surface are observed in nature at various length scales ranging from tiny crack segments in nanoparticle films to enormous domains observed in the earth’s crust. Here, we study the formation of cracks in magnetic films made of maghemite (γ-Fe2O3) nanocrystals. The cracks are oriented by an external magnetic field applied during the drying process which presents a new method to produce directional crack patterns. It is shown that directional and isotropic crack patterns follow the same universal scaling law with the film height varying from micrometer to centimeter scales. Former experimental studies of scaling laws were limited to small variations in height (1 order of magnitude). The large variation in height in our expriments becomes possible due to the combined use of nanocrystals and electron microscopy. A simple two-dimensional computer model for elastic fracture leads to structural and scaling behaviors, which match those observed in the experiments. Introduction Crack patterns in various materials have been studied over several decades. The similarities between the geometries of the different patterns are observed over a wide range of scales,1–5 from millimeters to kilometers. This suggests a universal mechanism. Different morphologies of crack patterns, such as isotropic, directional, radial, ring, spiral, and others, can be observed depending on the drying process of a colloidal suspension.6–10 Recent experiments1,10 have shown that the areas and distances between cracks increase with film height. However, different scaling laws were, surprisingly, observed for linear (1D) and isotropic (2D) cracks.1,10 Thus, Allain and Limat1 observed a deviation from linear scaling of the average crack distance with height, h, in their directional drying experiments, in good agreement with a theoretical model developed by Komatsu and Sasa11 which predicted that the distance between cracks varied as h2/3. This is different from the linear relationship between the square root of the area and the height found by Groisman and Kaplan10 and theoretically confirmed by Leung and Ne´da12 for the isotropic crack patterns. To solve this puzzle, the scaling behaviors of 1D and 2D crack patterns are here directly compared in the same maghemite nanoparticle films over 2 orders of magnitude. Moreover, a new method to align the cracks is employed by applying a magnetic field to the film during the drying process. To compare with simulation results, 1D and 2D crack formation is theoretically studied using a recently developped model.12 Experimental Methods Chemicals. Iron(II) chloride, Fe(Cl)2, and iron(III) chloride, Fe(Cl)3, were from Prolabo. Dodecanoic acid, CH3(CH2)10COOH, and ammonium hydroxide, NH4OH, were from Aldrich. Apparatus. A JEOL (100 kV) model JEM-1011 transmission electron microscope (TEM) was used to characterize the size † Part of the “Janos H. Fendler Memorial Issue”. * To whom correspondence should be addressed. E-mail: pileni@ sri.jussieu.fr.
of γ-Fe2O3 nanocrystals. The directional cracks were produced with an electromagnet (Oxford Instruments N38). The crack patterns are visualized with a scanning electron microscope (SEM) JEOL 5510 LV and with a C-740 Olympus digital camera. Procedure. Synthesis of Solutions of Maghemite Nanocrystals. The γ-Fe2O3 nanocrystals are synthesized according to the well-known synthesis method described elsewhere13 with slight changes. An acidic aqueous mixture of iron(III) chloride, FeCl3 (80 mL, 1 M), and iron(II) chloride, FeCl2 (20 mL, 2 M, 2 M HCl), is added to an ammonium hydroxide solution, NH4OH (800 mL, 0.6 M). At this stage, uncoated nanocrystals are produced. The precipitate of uncoated nanocrystals is washed with a large excess of ethanol. Then, a solution of dodecanoic acid, CH3(CH2)10COOH (0.5 M), solubilized in ethanol, is added. The solution is sonicated for 2 h at 90 °C. The resulting precipitate is washed with a large excess of ethanol, and the powder is dried in air. The nanocrystals coated with dodecanoic acid are dispersed in chloroform, to give an oily ferrofluid. This leads to the synthesis of γ-Fe2O3 nanocrystals with an average diameter and polydispersity of 10 nm and 25%, respectively. Fabrication of Crack Patterns. A solution of 10-nm γ-Fe2O3 nanocrystals dispersed in chloroform is injected above a silicon wafer placed at the bottom of a glass beaker at room temperature, and the solution is subjected to an applied magnetic field (0.2 T) parallel to the substrate. Crack patterns are formed during the drying of the ferrofluid. After complete evaporation of the solvent, the crack patterns are visualized with a scanning electron microscope and with a digital camera. The area analysis of the individual fragments of primary and secondary 2D cracks, Ap and As, is made by using a public domain image processing and analysis program (NIH Image)14 on about 150 and 400 domains on average, respectively. The heights, h, are measured over the whole substrate on tilting by 45°. Simulation Method. Simulation of Crack Formation. The simulation method used here was first introduced by Leung and Néda.12 Simulations are carried out on a hexagonal array of 250 000 blocks connected by springs and separated by an initial
10.1021/jp802736g CCC: $40.75 2008 American Chemical Society Published on Web 07/30/2008
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Figure 1. Photographs (A-E) and SEM (F) images of directional (1D-2D) crack patterns made of 10-nm γ-Fe2O3 nanocrystals produced by applying a magnetic field (0.2 T) during the evaporation process with an average layer height of 415 µm. (A-E) Photographs of the formation of centimeter-size directional cracks. The solution volume and concentration are 5 mL and 28% by weight before drying. Formation of the first crack in the direction of the applied field, at the middle of the substrate (A). Successive domain divisions occur mainly parallel to the applied field, leading to equally spaced linear cracks (B), (C) obtained after 15 and 65 min of the formation of the first crack, respectively. Formation of perpendicular cracks (D) and final crack pattern (E) obtained after 105 and 1060 min of the formation of the first crack, respectively. (F) SEM image of primary 2D crack (solid lines) with secondary 2D cracks (dashed lines).
distance a g l, where l is the relaxed spring length. Initially, f the blocks are slightly randomly displaced by ∆r where f |∆r| < 0.001. The force at a block i is given by a harmonic approximation using Hooke’s law:12
Fi )
f
∑ j
f
hijFij
with
Fij ) k(rij - l)
f
f r ij
rij
j denotes the neighboring blocks at a distance rij. The relaxed spring length and constant are fixed at k ) 1 and l ) 0.9, respectively. The dimensionless number hij corresponds to the number of springs between the blocks i and j. Before the simulation starts, hij is set to the value of h. The harmonic approximation is used here, since in realistic situations the stress is primarily tensile and the strain is low. At the start of each simulation, a strain is applied to the relaxed array in only one or both directions by extension of the array by a factor of 1/l > 1 to produce 1D or 2D crack patterns, respectively. In the experiment, the stress is built up by the contraction of the layer due to the drying. The contraction is restricted by friction from the substrate. Therefore, in the simulation, a block is only moved to a new mechanical equilibrium position when the force Fi at the block is larger than a slipping threshold denoted by Fs. Whenever the force
Fij on a spring exceeds a cracking threshold denoted by Fc, a spring is broken (hij f hij - 1). During the simulation, the two thresholds are systematically decreased to induce slipping and cracking. The decrease in the thresholds is generally used to describe the gradual drying of the film.12 Since both thresholds decrease by the same physical means, the ratio κ ) Fc/Fs remains fixed. A simulation step consists of the following: (1) It starts with the calculation of the forces of all blocks in the array. (2) Both thresholds Fs and Fc are lowered keeping κ constant until either threshold is exeeded somewhere in the system. (3) If Fi > Fs, block i is slipped to a new position where Fi ) 0 using a Newton-Raphson method.15 (4) If Fij > Fc, a spring is broken (hij f hij - 1). (5) The forces of the neighboring blocks are corrected and blocks are moved or springs are broken until all forces in the system respect the conditions Fi < Fs and Fij < Fc. To avoid the instantaneous breaking of the whole bundle of springs before the neighboring stress has at all relaxed, only one spring of a bundle is broken during a simulation step (see discussions in ref 12). Each simulation takes 100 million steps (several hours on a modern work station). In 1D systems, the statistical accuracy has to be improved by using a very large array of 250 000 blocks in comparison to
Crack Patterns in Thin Films of γ-Fe2O3 Nanocrystals 10 000 blocks for 2D crack patterns used in ref 12. Free boundary conditions are applied. To ensure that our results do not depend on the κ value chosen, simulations are carried out for κ ranging from 0.25 to 3.0. For all κ values, the same scaling laws were obtained within the statistical accuracy of the method. The experimental slope of the crack distance as a function of height gives a ratio of cracking to slipping forces of 3.4 which applies for the experiments. The κ value can be estimated from a simple model. Employing typical interaction parameters for van der Waals surfaces as used from the cobblestone model,16 the κ value varies from 4 to 20. This range of values compares well with that found experimentally. All simulations were carried out using the homemade C package CrackT. To study the evolution of the crack patterns over a large variation with height, a multiscale approach is developped. Within this approach, several blocks are reassembled in one “super” block. Therefore, the distance and spring length between super blocks in comparison to the original blocks are increased by a factor denoted by R > 1. To obtain the same average crack patterns at given values of h and κ for the super blocks as for the original block array, κ must be multiplied by the factor R. This is explained as follows. To apply the same physical conditions during the simulation and, thus, obtain the same average crack properties as with the original block array, the ratio between slipping and cracking forces, κ, must be changed for the superblock array. The slipping force is proportional to the surface occupied by the super block which increases by a factor of R2 compared to the original array (Fs ∼ R2). The cracking force is proportional to the block distance which is multiplied by R (Fc ∼ R). When both results are taken together, κ ) Fc/Fs must be decreased by a factor of R to obtain the same average crack properties during the simulation for the super block array as with the original array. The simulation for the super block array is carried out in the same way as for the original array. Only the distance, spring contant, and the value of κ change. Results and Discussion Fabrication of Directional Crack Patterns and Its Characterization. The crack patterns are obtained by injecting 10nm γ-Fe2O3 nanocrystals dispersed in chloroform above a silicon wafer placed at the bottom of a glass beaker. The beaker is covered with a glass plate to reduce the evaporation rate. The evaporation of the solvent occurs at room temperature. During this process, the solution is subjected to an applied magnetic field (0.2 T) parallel to the substrate. As the chloroform evaporates, the concentration of the solution increases with formation of a gel that adheres to the silicon wafer. Further evaporation induces shrinkage of the thin film layer due to large stresses, but the adhesion to the silicon substrate limits the contraction of the film. This results in large stresses which are at the origin of the crack formation. The crack patterns spanning the whole substrate appear after drying. They are visualized with a scanning electron microscope and a digital camera. The layer height of the crack patterns is controlled by solutions containing 0.35-28% by weight of nanocrystals. Samples with average layer heights, h, varying from 7 ( 0.2 to 107 ( 3.5 µm are obtained using a 6 × 6 mm2 silicon wafer. For thicker samples (229 ( 2.2 and 415 ( 5 µm), we used a larger silicon wafer (25 × 25 mm2). We first follow the formation of cracks for the thickest sample, h ) 415 µm, by taking photographs at different times using a digital camera (see Figure 1A-C). Figure 1A shows the formation of the first crack in the direction of the applied
J. Phys. Chem. B, Vol. 112, No. 46, 2008 14411 field, at the middle of the substrate where the stress in the fragment is the largest.17 Successive domain divisions occur mainly parallel to the applied field leading to equally spaced linear cracks (see Figure 1B and C). One hour after the formation of the first crack, cracks perpendicular to the directional ones appear and invade the sample as the evaporation continues (Figure 1D). The formation of directional cracks produced by applying a magnetic field parallel to the substrate is explained as follows. In the liquid state, the nanocrystals are free to rotate and their magnetic moments are aligned along the applied magnetic field direction.18 This favors the attractions between nanocrystals in the direction parallel to the applied field, whereas repulsions take place in the perpendicular direction, leading to an anisotropic stress within the film. The stress σyy perpendicular to the applied field is larger than that (σxx) parallel to the field according to the following equation:19
σyy - σxx ) µ0(H2 + MH) where µ0, M, and H are the permeability of the vacuum and the magnetization of the film and the field, respectively. The higher stress σyy is relieved first, leading to the parallel cracks shown in Figure 1A-C. Later on, the stress parallel to the direction of the field will also be relieved, inducing the formation of perpendicular cracks (Figure 1D). Primary 1D and 2D Cracks and Secondary 2D Cracks. To analyze the experimental data quantitatively, we need to define primary 1D and 2D cracks and secondary 2D cracks. The straight cracks parallel to the direction of the applied field correspond to the first generation of cracks and are called primary 1D cracks. The fragments bounded by these primary 1D cracks and the perpendicular ones (second generation of cracks) are called primary 2D cracks. A magnification of a region in Figure 1E by using SEM shows that a typical primary 2D crack fragment delimited by full lines (Figure 1F) is foursided with 90° angles. The mean crack aperture of parallel cracks (first generation of primary 1D cracks) is about twice as large as that of the perpendicular ones (second generation of cracks) and is clearly seen in Figure 1F. At the end of the drying process, the crack patterns (Figure 1E) show that the primary 2D crack fragments are broken into smaller ones (from two to eight domains) corresponding to the third and later generations of cracks, as indicated by the dashed lines in Figure 1F. These new generations of cracks are called secondary 2D cracks. Note that the new generation of cracks tends to appear at the middle of the sides of existing fragments bounded by primary cracks (see arrows in Figure 1F), which is consistent with the theoretical model of Hornig17 and the simulations. It has been checked that all primary and secondary cracks go straight from the top to the bottom of the film. Figure 2 shows how a typical secondary 2D crack goes all the way through the film. Scaling Behavior of 1D and 2D Crack Patterns. We now study the variation of crack patterns with height. The crack distances between regular linear cracks, D, are measured over the whole substrate and their heights, h, on tilting the sample by 45°. The area analysis of the individual fragments of primary and secondary 2D cracks, Ap and As, is made by using a public domain image processing and analysis program (NIH Image)14 over the whole sample. The size of the crack patterns depends on the initial concentration of nanocrystals deposited with formation of directional cracks at the micrometer (Figure 3A) to millimeter (Figure 3B) and centimeter scales (Figure 3C). Note that samples having 7 µm (Figure 3A) and 14 µm as average layer heights are characterized by only parallel cracks (primary 1D cracks), whereas, at 24 µm and above, perpen-
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Figure 2. SEM images at zero angle (A) and by 45° tilt (B, C) of directional (1D-2D) crack patterns made of 10-nm γ-Fe2O3 nanocrystals produced by applying a magnetic field (0.2 T) during the evaporation process. The average height is 107 ( 4 µm. The solution volume and concentration are 0.2 mL and 5% by weight before drying.
dicular and secondary cracks are observed within the bands formed by the primary linear cracks. On tilting the samples by 45° tilt, the insets (A, B, C) in Figure 3 show typical cracks which go from the top to the bottom of the film. Let us first consider the primary linear cracks to determine the average crack distance (D) as a function of layer height (h). A linear variation, on a logarithmic scale, of the average crack distance (D) between primary (1D) cracks with the correspond-
Ngo et al.
Figure 3. SEM (A, B) and photograph (C) images of directional (1D-2D) crack patterns made of 10-nm γ-Fe2O3 nanocrystals produced by applying a magnetic field (0.2 T) during the evaporation process. Inset (A, B, C): SEM images at 45° tilt. (A) Micrometer-size directional crack patterns obtained at the end of the drying process. The average layer height is 7 ( 0.2 µm. The solution volume and concentration are 0.2 mL and 0.35% by weight before drying. (B) Millimeter-size directional crack patterns with an average height of 107 ( 4 µm. The solution volume and concentration are 0.2 mL and 5% by weight before drying. (C) Centimeter-size directional crack patterns with an average height of 415 ( 5 µm. The solution volume and concentration are 5 mL and 28% by weight before drying.
ing height (h) is observed (squares in Figure 4). Using the average experimental value of the dimensionless ratio K ) D/h (K ) 5.31 ( 0.25), the linear fit gives a slope of 1.00 in good
Crack Patterns in Thin Films of γ-Fe2O3 Nanocrystals
Figure 4. Dependence of the average crack distance (D), the average square root areas of fragments of primary and secondary cracks (Ap, As) on the layer height. The average crack distance, D, on the layer height, h, designated by squares, fits a linear curve on the logarithmic scale with log D ) 0.72 + 1.00 log h (full lines). Full triangles correspond to the average square root area of primary 2D cracks, Ap. The average square root area of secondary 2D cracks, As, on the layer height, h, designated by triangles, fits a linear curve on the logarithmic scale with log As ) 0.42 + 1.00 log h (dashed lines).
agreement with a universal scaling law, D ) Kh. The same linear variation behavior Ap ) Kh is observed for the average square root area fragments of the primary 2D crack domains defined above (see Figure 1F) and denoted by Ap with the layer height (see full triangles in Figure 4). From this, similar dimensionless K values are obtained for 1D and 2D primary cracks. To our knowledge, this is the first time that the same scaling law is found experimentally for 1D and 2D in the same cracking system. Surprisingly, the variation of the average square root area fragments of the secondary 2D crack domains denoted by As as a function of the layer height (triangles in Figure 4) follows also a linear scaling as the primary cracks (dashed lines in Figure 4). However note that the origin of the plot differs for the primary cracks and the secondary ones. The dimensionless ratio K differs by a factor of 2 between the primary cracks (K ) 5.31 ( 0.25) and the secondary ones (K ) 2.66 ( 0.21). This leads to a linear relationship As ≈ 0.5 D. Hence, the length scale of the pattern for secondary cracks is divided by a factor of 2 compared to the primary cracks; this can be explained by the theoretical model of Hornig17 which predicted that cracks tend to appear in the middle of existing fragments (see arrows in Figure 1F), so the length scale of the pattern decreases by a factor of 2 with each generation of cracks. To check if the magnetic field influences the mean size of the crack fragments, a sample was prepared in the absence of an applied magnetic field during the evaporation process with the same volume and nanocrystal concentration as that shown in Figure 1, and characterized with the same thickness layer (h ) 415 ( 5 µm). The sample shows 2D-plane isotropic crack domains with almost the same average area fragments for the secondary cracks (Ais ) 1.09 ( 0.06 mm2) as that observed in directional cracks (As ) 1.08 ( 0.07 mm2). Hence, the magnetic field only induces the alignment of the cracks without any effect on the mean size of the crack fragments. Simulation Studies of the Scaling of Crack Patterns. The scaling of crack patterns with height is now theoretically studied using a recently introduced bundle-spring block model.12 Simple spring-block models have been widely used to successfully explain experimentally observed crack patterns.8,12,20,21 Within
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Figure 5. Simulation of linear (1D) and isotropic (2D) cracks. Scaling of mean distance D (squares) and square root area A (triangles) of cracks with height for κ ) 1 for the 1D and 2D cracks. κ is the ratio between the two force thresholds which induce breaking of the springs or slipping of the blocks. A linear increase D ∼ h and A ∼ h well describes the data within the statistical accuracy of simulation. The insets show typical crack patterns obtained by simulations for 1D (left) and 2D (right) cracking systems (κ ) 0.25 and h ) 50). The values for heights greater than 20 and 200 were obtained using a super block array with distances between the superblocks a ) 10 and 100, respectively.
this approach, the layer is represented by a large twodimensional hexagonal array of blocks where each neighbor is connected by coil springs. To study the variation of crack patterns with height, the single springs between blocks were replaced by a bundle of h springs with the bundle-spring model.12 h is a dimensionless number which plays the role of the height. In ref 12, it was shown that for 2D crack patterns the bundlespring model predicts a linear increase in the square root of the average fragment area bounded by cracks with height, in good agreement with our experiments (full triangles in Figure 4). Here, the bundle-spring model is further developed to answer two questions posed by the experiments: (1) How does the average distance D between cracks in a 1D system, such as that in Figures 1C and 3A, vary with height? A recent theory11 predicts that D increases as h2/3, which is in contradiction to the linear scaling observed in our experiments (squares in Figure 4). (2) Are the scaling laws still valid over a variation of height covering 2 orders of magnitude, the variation accessible by our experiments? Simulations are usually restricted to a height variation of 10. This is due to statistical inaccuracy which appears at larger heights, where the average fragment area bounded by the cracks increases and, consequently, the number of areas within the block array decreases. To answer the first question, at the start of each simulation, a strain is applied to the relaxed array in only one direction by a unilateral extension of the array by 10%. A detailed explanation of the crack simulation is given in the methods section. In the simulation, a pattern of regularly spaced linear cracks (left inset in Figure 5) is observed in good agreement with our experiments (Figures 1C and 3A). The log-log plot of D as a function of h in Figure 5 (squares) is well fitted by a straight line with a slope of 1. This shows a linear scaling of D with the height as observed in the experiment (squares in Figure 4). The nonlinear scaling found in ref 11 may be explained by the use of a slipping force which depends on the height in their theory in contrast to our model.
14414 J. Phys. Chem. B, Vol. 112, No. 46, 2008 Let us now turn to the second question, whether the scaling law is still valid over several orders of magnitude. To answer this question, the multiscale approach was developed, where several blocks are reassembled in one “super” block. The implementation of the super blocks in the simulation is explained in detail in the methods section. In Figure 5, the values for heights larger than 20 were obtained using the super block array. The super block approach was first applied for 2D crack patterns obtained by an isotropic stretching applied to the block array (right inset in Figure 5). This shows that the linear scaling observed in ref 12 for heights varying by a factor of 20 is still valid for a height variation over 3 orders of magnitude (triangles in Figure 5). Also, for 1D cracks (squares in Figure 5), the super block approach indicates a linear increase in the crack distance D on varying the height from 1 to 1000. Both results are in excellent agreement with the experimentally observed scaling laws. Conclusion In summary, it is shown that directional crack patterns in a magnetic film, on a large scale, are obtained by slow evaporation of a ferrofluid made of 10-nm γ-Fe2O3 nanocrystals in a magnetic field. We demonstrated that, with the same sample, 1D and 2D crack patterns are produced and follow the same scaling law over more than 2 orders of magnitude. The scaling law observed here indicates the universality of crack patterns independent of their dimensions. Further evidence is needed to confirm that the scaling law seen here is universal. To observe these scaling laws, it is very important to distinguish between different crack generations such as primary and secondary cracks. A simple spring-block model reproduces the same scaling in 1D and 2D systems over several orders of magnitude. Our results show the validity and limits of laboratory model systems to understand cracking at micrometer and centimeter scales and their extensions to geological scales.
Ngo et al. Acknowledgment. We thank Dr. D. Ingert, of our laboratory, for very fruitful discussions on the division in 1D crack patterns. References and Notes (1) Allain, C.; Limat, L. Phys. ReV. Lett. 1995, 74, 2981–2984. (2) Shorlin, K. A.; de Bruyn, J. R.; Graham, M.; Morris, S. W. Phys. ReV. E 2000, 61, 6950–6957. (3) Dufresne, E. R; et al. Phys. ReV. Lett. 2003, 91, 224501–224504. (4) Pileni, M. P.; Lalatonne, Y.; Ingert, D.; Lisiecki, I.; Courty, A. Faraday Discuss. 2004, 125, 251–264. (5) Belousov, V. V.; Gzovsky, M. V. Phys. Chem. Earth 1965, 6, 409– 472. (6) Colina, H.; Roux, S. Eur. Phys. J. E 2000, 1, 189–194. (7) Xia, Z. C.; Hutchinson, J. W. J. Mech. Phys. Solids 2000, 48, 1107– 1131. (8) Leung, K. T.; Jozsa, L.; Ravasz, M.; Néda, Z. Nature 2001, 410, 166. (9) Nakahara, A.; Matsuo, Y. Phys. ReV. E 2006, 74, 045102-1–045102-4. (10) Groisman, A.; Kaplan, E. Europhys. Lett. 1994, 25, 415–420. (11) Komatsu, T. S.; Sasa, S. Jpn. J. Appl. Phys. 1997, 36, 391–395. (12) Leung, K. T.; Ne´da, Z. Phys. ReV. Lett. 2000, 85, 662–665. (13) Massart, R. IEEE Trans. Magn. MAG-17 1981, 1247–1248. (14) NIH Image is a public domain image processing and analysis program for the Macintosh: http://rsb.info.nih.gov/nih-image/about.html. (15) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes: The Art of Scientific Computing; Cambridge University Press: Cambridge, U.K., 1986. (16) Israelachvili, J. Surface Forces and Nanorheology of Molecular Thin Films. In Springer Handbook of Nanotechnology, 2nd ed.; Bhushan, B., Ed.; Springer: Heidelberg, Germany, 2007; pp 1-66. (17) Hornig, T.; Sokolov, I.; Blumen, A. Phys. ReV. E 1996, 54, 4293– 4298. (18) Hendriksen, P. V.; Bodker, F.; Linderoth, S.; Wells, S.; Morup, S. J. Phys.: Condens. Matter 1994, 6, 3081–3090. (19) Rosensweig, R. E. Ferrohydrodynamics; Dover publications, Inc.: Mineola, NY, 1997; pp 103-108. (20) Skjeltorp, A. T.; Meakin, P. Nature 1998, 335, 424–426. (21) Walmann, T.; Malthe-Sørenssen, A.; Feder, J.; Jøssang, T.; Meakin, P.; Hardy, H. H. Phys. ReV. Lett. 1996, 77, 5393–5396.
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