Growth Kinetics in a Phase Field Model with Continuous Symmetry

We discuss the static and kinetic properties of a Ginzburg-Landau spherically symmetric $O(N)$ model recently introduced (Phys. Rev. Lett. {\bf 75}, 2176, (1995)) in order to generalize the so called Phase field model of Langer. The Hamiltonian contains two $O(N)$ invariant fields $\phi$ and $U$ bilinearly coupled. The order parameter field $\phi$ evolves according to a non conserved dynamics, whereas the diffusive field $U$ follows a conserved dynamics. In the limit $N \to \infty$ we obtain an exact solution, which displays an interesting kinetic behavior characterized by three different growth regimes. In the early regime the system displays normal scaling and the average domain size grows as $t^{1/2}$, in the intermediate regime one observes a finite wavevector instability, which is related to the Mullins-Sekerka instability; finally, in the late stage the structure function has a multiscaling behavior, while the domain size grows as $t^{1/4}$.Comment: 9 pages RevTeX, 9 figures included, files packed with uufiles to appear on Phy. Rev.

Area-preserving dynamics of a long slender finger by curvature: a test case for the globally conserved phase ordering

A long and slender finger can serve as a simple ``test bed'' for different phase ordering models. In this work, the globally-conserved, interface-controlled dynamics of a long finger is investigated, analytically and numerically, in two dimensions. An important limit is considered when the finger dynamics are reducible to the area-preserving motion by curvature. A free boundary problem for the finger shape is formulated. An asymptotic perturbation theory is developed that uses the finger aspect ratio as a small parameter. The leading-order approximation is a modification of ``the Mullins finger" (a well-known analytic solution) which width is allowed to slowly vary with time. This time dependence is described, in the leading order, by an exponential law with the characteristic time proportional to the (constant) finger area. The subleading terms of the asymptotic theory are also calculated. Finally, the finger dynamics is investigated numerically, employing the Ginzburg-Landau equation with a global conservation law. The theory is in a very good agreement with the numerical solution.Comment: 8 pages, 4 figures, Latex; corrected typo

Dynamics of an Unbounded Interface Between Ordered Phases

We investigate the evolution of a single unbounded interface between ordered phases in two-dimensional Ising ferromagnets that are endowed with single-spin-flip zero-temperature Glauber dynamics. We examine specifically the cases where the interface initially has either one or two corners. In both examples, the interface evolves to a limiting self-similar form. We apply the continuum time-dependent Ginzburg-Landau equation and a microscopic approach to calculate the interface shape. For the single corner system, we also discuss a correspondence between the interface and the Young tableau that represents the partition of the integers.Comment: 9 pages, 11 figures, 2-column revtex4 format. V2: references added and discussion section expanded slightly. Final version for PRE. V3: A few small additional editorial change

Coarsening of Surface Structures in Unstable Epitaxial Growth

We study unstable epitaxy on singular surfaces using continuum equations with a prescribed slope-dependent surface current. We derive scaling relations for the late stage of growth, where power law coarsening of the mound morphology is observed. For the lateral size of mounds we obtain $\xi \sim t^{1/z}$ with $z \geq 4$. An analytic treatment within a self-consistent mean-field approximation predicts multiscaling of the height-height correlation function, while the direct numerical solution of the continuum equation shows conventional scaling with z=4, independent of the shape of the surface current.Comment: 15 pages, Latex. Submitted to PR

A Soluble Phase Field Model

The kinetics of an initially undercooled solid-liquid melt is studied by means of a generalized Phase Field model, which describes the dynamics of an ordering non-conserved field phi (e.g. solid-liquid order parameter) coupled to a conserved field (e.g. thermal field). After obtaining the rules governing the evolution process, by means of analytical arguments, we present a discussion of the asymptotic time-dependent solutions. The full solutions of the exact self-consistent equations for the model are also obtained and compared with computer simulation results. In addition, in order to check the validity of the present model we confronted its predictions against those of the standard Phase field model and found reasonable agreement. Interestingly, we find that the system relaxes towards a mixed phase, depending on the average value of the conserved field, i.e. on the initial condition. Such a phase is characterized by large fluctuations of the phi field.Comment: 13 pages, 8 figures, RevTeX 3.1, submitted to Physical Review

Pattern formation in directional solidification under shear flow. I: Linear stability analysis and basic patterns

An asymptotic interface equation for directional solidification near the absolute stabiliy limit is extended by a nonlocal term describing a shear flow parallel to the interface. In the long-wave limit considered, the flow acts destabilizing on a planar interface. Moreover, linear stability analysis suggests that the morphology diagram is modified by the flow near the onset of the Mullins-Sekerka instability. Via numerical analysis, the bifurcation structure of the system is shown to change. Besides the known hexagonal cells, structures consisting of stripes arise. Due to its symmetry-breaking properties, the flow term induces a lateral drift of the whole pattern, once the instability has become active. The drift velocity is measured numerically and described analytically in the framework of a linear analysis. At large flow strength, the linear description breaks down, which is accompanied by a transition to flow-dominated morphologies, described in a companion paper. Small and intermediate flows lead to increased order in the lattice structure of the pattern, facilitating the elimination of defects. Locally oscillating structures appear closer to the instability threshold with flow than without.Comment: 20 pages, Latex, accepted for Physical Review

Time evolution of in vivo articular cartilage repair induced by bone marrow stimulation and scaffold implantation in rabbits

Purpose: Tissue engineering techniques were used to study cartilage repair over a 12-month period in a rabbit model. Methods: A full-depth chondral defect along with subchondral bone injury were originated in the knee joint, where a biostable porous scaffold was implanted, synthesized of poly(ethyl acrylate-co-hydroxyethyl acrylate) copolymer. Morphological evolution of cartilage repair was studied 1 and 2 weeks, and 1, 3, and 12 months after implantation by histological techniques. The 3-month group was chosen to compare cartilage repair to an additional group where scaffolds were preseeded with allogeneic chondrocytes before implantation, and also to controls, who underwent the same surgery procedure, with no scaffold implantation. Results: Neotissue growth was first observed in the deepest scaffold pores 1 week after implantation, which spread thereafter; 3 months later scaffold pores were filled mostly with cartilaginous tissue in superficial and middle zones, and with bone tissue adjacent to subchondral bone. Simultaneously, native chondrocytes at the edges of the defect started to proliferate 1 week after implantation; within a month those edges had grown centripetally and seemed to embed the scaffold, and after 3 months, hyaline-like cartilage was observed on the condylar surface. Preseeded scaffolds slightly improved tissue growth, although the quality of repair tissue was similar to non-preseeded scaffolds. Controls showed that fibrous cartilage was mainly filling the repair area 3 months after surgery. In the 12-month group, articular cartilage resembled the untreated surface. Conclusions: Scaffolds guided cartilaginous tissue growth in vivo, suggesting their importance in stress transmission to the cells for cartilage repair.This study was supported by the Spanish Ministry of Science and Innovation through MAT2010-21611-C03-00 project (including the FEDER financial support), by Conselleria de Educacion (Generalitat Valenciana, Spain) PROMETEO/2011/084 grant, and by CIBER-BBN en Bioingenieria, Biomateriales y Nanomedicina. The work of JLGR was partially supported by funds from the Generalitat Valenciana, ACOMP/2012/075 project. CIBER-BBN is an initiative funded by the VI National R&D&i Plan 2008-2011, Iniciativa Ingenio 2010, Consolider Program, CIBER Actions and financed by the - Instituto de Salud Carlos III with assistance from the European Regional Development Fund.Sancho-Tello Valls, M.; Forriol, F.; Gastaldi, P.; Ruiz Sauri, A.; Martín De Llano, JJ.; Novella-Maestre, E.; Antolinos Turpín, CM.... (2015). Time evolution of in vivo articular cartilage repair induced by bone marrow stimulation and scaffold implantation in rabbits. International Journal of Artificial Organs. 38(4):210-223. https://doi.org/10.5301/ijao.5000404S210223384Becerra, J., Andrades, J. A., Guerado, E., Zamora-Navas, P., López-Puertas, J. M., & Reddi, A. H. (2010). Articular Cartilage: Structure and Regeneration. Tissue Engineering Part B: Reviews, 16(6), 617-627. doi:10.1089/ten.teb.2010.0191Nelson, L., Fairclough, J., & Archer, C. (2009). Use of stem cells in the biological repair of articular cartilage. Expert Opinion on Biological Therapy, 10(1), 43-55. doi:10.1517/14712590903321470MAINIL-VARLET, P., AIGNER, T., BRITTBERG, M., BULLOUGH, P., HOLLANDER, A., HUNZIKER, E., … STAUFFER, E. (2003). HISTOLOGICAL ASSESSMENT OF CARTILAGE REPAIR. The Journal of Bone and Joint Surgery-American Volume, 85, 45-57. doi:10.2106/00004623-200300002-00007Hunziker, E. B., Kapfinger, E., & Geiss, J. (2007). The structural architecture of adult mammalian articular cartilage evolves by a synchronized process of tissue resorption and neoformation during postnatal development. Osteoarthritis and Cartilage, 15(4), 403-413. doi:10.1016/j.joca.2006.09.010Onyekwelu, I., Goldring, M. B., & Hidaka, C. (2009). Chondrogenesis, joint formation, and articular cartilage regeneration. Journal of Cellular Biochemistry, 107(3), 383-392. doi:10.1002/jcb.22149Ahmed, T. A. E., & Hincke, M. T. (2010). Strategies for Articular Cartilage Lesion Repair and Functional Restoration. Tissue Engineering Part B: Reviews, 16(3), 305-329. doi:10.1089/ten.teb.2009.0590Hangody, L., Kish, G., Kárpáti, Z., Udvarhelyi, I., Szigeti, I., & Bély, M. (1998). Mosaicplasty for the Treatment of Articular Cartilage Defects: Application in Clinical Practice. Orthopedics, 21(7), 751-756. doi:10.3928/0147-7447-19980701-04Steinwachs, M. R., Guggi, T., & Kreuz, P. C. (2008). Marrow stimulation techniques. Injury, 39(1), 26-31. doi:10.1016/j.injury.2008.01.042Brittberg, M., Lindahl, A., Nilsson, A., Ohlsson, C., Isaksson, O., & Peterson, L. (1994). Treatment of Deep Cartilage Defects in the Knee with Autologous Chondrocyte Transplantation. New England Journal of Medicine, 331(14), 889-895. doi:10.1056/nejm199410063311401Richter, W. (2009). Mesenchymal stem cells and cartilagein situregeneration. Journal of Internal Medicine, 266(4), 390-405. doi:10.1111/j.1365-2796.2009.02153.xBartlett, W., Skinner, J. A., Gooding, C. R., Carrington, R. W. J., Flanagan, A. M., Briggs, T. W. R., & Bentley, G. (2005). Autologous chondrocyte implantationversusmatrix-induced autologous chondrocyte implantation for osteochondral defects of the knee. The Journal of Bone and Joint Surgery. British volume, 87-B(5), 640-645. doi:10.1302/0301-620x.87b5.15905Little, C. J., Bawolin, N. K., & Chen, X. (2011). Mechanical Properties of Natural Cartilage and Tissue-Engineered Constructs. Tissue Engineering Part B: Reviews, 17(4), 213-227. doi:10.1089/ten.teb.2010.0572Vikingsson, L., Gallego Ferrer, G., Gómez-Tejedor, J. A., & Gómez Ribelles, J. L. (2014). An «in vitro» experimental model to predict the mechanical behavior of macroporous scaffolds implanted in articular cartilage. Journal of the Mechanical Behavior of Biomedical Materials, 32, 125-131. doi:10.1016/j.jmbbm.2013.12.024Weber, J. F., & Waldman, S. D. (2014). Calcium signaling as a novel method to optimize the biosynthetic response of chondrocytes to dynamic mechanical loading. Biomechanics and Modeling in Mechanobiology, 13(6), 1387-1397. doi:10.1007/s10237-014-0580-xMauck, R. L., Soltz, M. A., Wang, C. C. B., Wong, D. D., Chao, P.-H. G., Valhmu, W. B., … Ateshian, G. A. (2000). Functional Tissue Engineering of Articular Cartilage Through Dynamic Loading of Chondrocyte-Seeded Agarose Gels. Journal of Biomechanical Engineering, 122(3), 252-260. doi:10.1115/1.429656Palmoski, M. J., & Brandt, K. D. (1984). Effects of static and cyclic compressive loading on articular cartilage plugs in vitro. Arthritis & Rheumatism, 27(6), 675-681. doi:10.1002/art.1780270611Khoshgoftar, M., Ito, K., & van Donkelaar, C. C. (2014). The Influence of Cell-Matrix Attachment and Matrix Development on the Micromechanical Environment of the Chondrocyte in Tissue-Engineered Cartilage. Tissue Engineering Part A, 20(23-24), 3112-3121. doi:10.1089/ten.tea.2013.0676Agrawal, C. M., & Ray, R. B. (2001). Biodegradable polymeric scaffolds for musculoskeletal tissue engineering. Journal of Biomedical Materials Research, 55(2), 141-150. doi:10.1002/1097-4636(200105)55:23.0.co;2-jPérez Olmedilla, M., Garcia-Giralt, N., Pradas, M. M., Ruiz, P. B., Gómez Ribelles, J. L., Palou, E. C., & García, J. C. M. (2006). Response of human chondrocytes to a non-uniform distribution of hydrophilic domains on poly (ethyl acrylate-co-hydroxyethyl methacrylate) copolymers. Biomaterials, 27(7), 1003-1012. doi:10.1016/j.biomaterials.2005.07.030Horbett, T. A., & Schway, M. B. (1988). Correlations between mouse 3T3 cell spreading and serum fibronectin adsorption on glass and hydroxyethylmethacrylate-ethylmethacrylate copolymers. Journal of Biomedical Materials Research, 22(9), 763-793. doi:10.1002/jbm.820220903Kiremitçi, M., Peşmen, A., Pulat, M., & Gürhan, I. (1993). Relationship of Surface Characteristics to Cellular Attachment in PU and PHEMA. Journal of Biomaterials Applications, 7(3), 250-264. doi:10.1177/088532829300700304Lydon, M. ., Minett, T. ., & Tighe, B. . (1985). Cellular interactions with synthetic polymer surfaces in culture. Biomaterials, 6(6), 396-402. doi:10.1016/0142-9612(85)90100-0Campillo-Fernandez, A. J., Pastor, S., Abad-Collado, M., Bataille, L., Gomez-Ribelles, J. L., Meseguer-Dueñas, J. M., … Ruiz-Moreno, J. M. (2007). Future Design of a New Keratoprosthesis. Physical and Biological Analysis of Polymeric Substrates for Epithelial Cell Growth. Biomacromolecules, 8(8), 2429-2436. doi:10.1021/bm0703012Funayama, A., Niki, Y., Matsumoto, H., Maeno, S., Yatabe, T., Morioka, H., … Toyama, Y. (2008). Repair of full-thickness articular cartilage defects using injectable type II collagen gel embedded with cultured chondrocytes in a rabbit model. Journal of Orthopaedic Science, 13(3), 225-232. doi:10.1007/s00776-008-1220-zKitahara, S., Nakagawa, K., Sah, R. L., Wada, Y., Ogawa, T., Moriya, H., & Masuda, K. (2008). In Vivo Maturation of Scaffold-free Engineered Articular Cartilage on Hydroxyapatite. Tissue Engineering Part A, 14(11), 1905-1913. doi:10.1089/ten.tea.2006.0419Martinez-Diaz, S., Garcia-Giralt, N., Lebourg, M., Gómez-Tejedor, J.-A., Vila, G., Caceres, E., … Monllau, J. C. (2010). In Vivo Evaluation of 3-Dimensional Polycaprolactone Scaffolds for Cartilage Repair in Rabbits. The American Journal of Sports Medicine, 38(3), 509-519. doi:10.1177/0363546509352448Wang, Y., Bian, Y.-Z., Wu, Q., & Chen, G.-Q. (2008). Evaluation of three-dimensional scaffolds prepared from poly(3-hydroxybutyrate-co-3-hydroxyhexanoate) for growth of allogeneic chondrocytes for cartilage repair in rabbits. Biomaterials, 29(19), 2858-2868. doi:10.1016/j.biomaterials.2008.03.021Alió del Barrio, J. L., Chiesa, M., Gallego Ferrer, G., Garagorri, N., Briz, N., Fernandez-Delgado, J., … De Miguel, M. P. (2014). Biointegration of corneal macroporous membranes based on poly(ethyl acrylate) copolymers in an experimental animal model. Journal of Biomedical Materials Research Part A, 103(3), 1106-1118. doi:10.1002/jbm.a.35249Diego, R. B., Olmedilla, M. P., Aroca, A. S., Ribelles, J. L. G., Pradas, M. M., Ferrer, G. G., & Sánchez, M. S. (2005). Acrylic scaffolds with interconnected spherical pores and controlled hydrophilicity for tissue engineering. Journal of Materials Science: Materials in Medicine, 16(8), 693-698. doi:10.1007/s10856-005-2604-7Serrano Aroca, A., Campillo Fernández, A. J., Gómez Ribelles, J. L., Monleón Pradas, M., Gallego Ferrer, G., & Pissis, P. (2004). Porous poly(2-hydroxyethyl acrylate) hydrogels prepared by radical polymerisation with methanol as diluent. Polymer, 45(26), 8949-8955. doi:10.1016/j.polymer.2004.10.033Diani, J., Fayolle, B., & Gilormini, P. (2009). A review on the Mullins effect. European Polymer Journal, 45(3), 601-612. doi:10.1016/j.eurpolymj.2008.11.017Mullins, L. (1969). Softening of Rubber by Deformation. Rubber Chemistry and Technology, 42(1), 339-362. doi:10.5254/1.3539210Jurvelin, J. S., Buschmann, M. D., & Hunziker, E. B. (2003). Mechanical anisotropy of the human knee articular cartilage in compression. Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 217(3), 215-219. doi:10.1243/095441103765212712Shapiro, F., Koide, S., & Glimcher, M. J. (1993). Cell origin and differentiation in the repair of full-thickness defects of articular cartilage. The Journal of Bone & Joint Surgery, 75(4), 532-553. doi:10.2106/00004623-199304000-00009SELLERS, R. S., ZHANG, R., GLASSON, S. S., KIM, H. D., PELUSO, D., D’AUGUSTA, D. A., … MORRIS, E. A. (2000). Repair of Articular Cartilage Defects One Year After Treatment with Recombinant Human Bone Morphogenetic Protein-2 (rhBMP-2)*. The Journal of Bone and Joint Surgery-American Volume, 82(2), 151-160. doi:10.2106/00004623-200002000-00001Hunziker, E. B., Michel, M., & Studer, D. (1997). Ultrastructure of adult human articular cartilage matrix after cryotechnical processing. Microscopy Research and Technique, 37(4), 271-284. doi:10.1002/(sici)1097-0029(19970515)37:43.0.co;2-oAppelman, T. P., Mizrahi, J., Elisseeff, J. H., & Seliktar, D. (2009). The differential effect of scaffold composition and architecture on chondrocyte response to mechanical stimulation. Biomaterials, 30(4), 518-525. doi:10.1016/j.biomaterials.2008.09.063Chung, C., & Burdick, J. A. (2008). Engineering cartilage tissue. Advanced Drug Delivery Reviews, 60(2), 243-262. doi:10.1016/j.addr.2007.08.027HUNZIKER, E. B., & ROSENBERG, L. C. (1996). Repair of Partial-Thickness Defects in Articular Cartilage. The Journal of Bone & Joint Surgery, 78(5), 721-33. doi:10.2106/00004623-199605000-00012Schulze-Tanzil, G. (2009). Activation and dedifferentiation of chondrocytes: Implications in cartilage injury and repair. Annals of Anatomy - Anatomischer Anzeiger, 191(4), 325-338. doi:10.1016/j.aanat.2009.05.003Umlauf, D., Frank, S., Pap, T., & Bertrand, J. (2010). Cartilage biology, pathology, and repair. Cellular and Molecular Life Sciences, 67(24), 4197-4211. doi:10.1007/s00018-010-0498-0Karystinou, A., Dell’Accio, F., Kurth, T. B. A., Wackerhage, H., Khan, I. M., Archer, C. W., … De Bari, C. (2009). Distinct mesenchymal progenitor cell subsets in the adult human synovium. Rheumatology, 48(9), 1057-1064. doi:10.1093/rheumatology/kep192Sakaguchi, Y., Sekiya, I., Yagishita, K., & Muneta, T. (2005). Comparison of human stem cells derived from various mesenchymal tissues: Superiority of synovium as a cell source. Arthritis & Rheumatism, 52(8), 2521-2529. doi:10.1002/art.21212Schaefer, D., Martin, I., Jundt, G., Seidel, J., Heberer, M., Grodzinsky, A., … Freed, L. E. (2002). Tissue-engineered composites for the repair of large osteochondral defects. Arthritis & Rheumatism, 46(9), 2524-2534. doi:10.1002/art.1049

Very Singular Diffusion Equations-Second and Fourth Order Problems

This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an H−1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example