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A three-dimensional phase diagram of growth-induced surface instabilities.

Wang Q, Zhao X - Sci Rep (2015)

Bottom Line: However, a general model that accounts for the formation and evolution of these various surface-instability patterns still does not exist.The predicted wavelengths and amplitudes of various instability patterns match well with our experimental data.It is expected that the unified phase diagram will not only advance the understanding of biological morphogenesis, but also significantly facilitate the design of new materials and structures by rationally harnessing surface instabilities.

View Article: PubMed Central - PubMed

Affiliation: 1] Soft Active Materials Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA [2] Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27708, USA.

ABSTRACT
A variety of fascinating morphological patterns arise on surfaces of growing, developing or aging tissues, organs and microorganism colonies. These patterns can be classified into creases, wrinkles, folds, period-doubles, ridges and delaminated-buckles according to their distinctive topographical characteristics. One universal mechanism for the pattern formation has been long believed to be the mismatch strains between biological layers with different expanding or shrinking rates, which induce mechanical instabilities. However, a general model that accounts for the formation and evolution of these various surface-instability patterns still does not exist. Here, we take biological structures at their current states as thermodynamic systems, treat each instability pattern as a thermodynamic phase, and construct a unified phase diagram that can quantitatively predict various types of growth-induced surface instabilities. We further validate the phase diagram with our experiments on surface instabilities induced by mismatch strains as well as the reported data on growth-induced instabilities in various biological systems. The predicted wavelengths and amplitudes of various instability patterns match well with our experimental data. It is expected that the unified phase diagram will not only advance the understanding of biological morphogenesis, but also significantly facilitate the design of new materials and structures by rationally harnessing surface instabilities.

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Illustrations of examples, schematics and potential energies of various growth-induced surface instabilities.(A) Examples of growth-induced surface instabilities on (i) the pumpkin skin, (ii) the cerebral cortex, (iii) the biofilm and (iv) the dog skin. (B) Schematics of growth-induced surface instabilities: (i) wrinkle, (ii) crease, (iii) delaminated-buckle, (iv) fold, (v) period-double and (vi) ridge. (C) One example pathway to induce the mismatch strain in the film-substrate structure: (i) The film and substrate is first assumed to be detached from each other to form a stress-free state; (ii) the detached stress-free substrate is then pre-stretched by a ratio of Lf/Ls and adhered to the film; (iii) relaxed to length L; and (iv) eventually relaxed to length Ls at the current state. Other pathways to induce mismatch strains are illustrated in Supplementary Figs S5 and S7. (D) Evolution of potential energy of the film-substrate structure with increasing mismatch strain following the pathway in (C). The red dash line denotes the surface patterns with the minimum potential energy. The potential energy of the film-substrate structure with mismatch strain εM is denoted by the black solid circle. Image (Ai) is reprinted with permission from Yin, et al., Proc. Natl. Acad. Sci. U.S.A., 105, 49 (2008). Copyright 2008, National Academy of Sciences, USA. Image (Aii) is reprinted from Bradbury, PLOS Biol., 3, 3 (2005) under Open-Access License. Image (Aiii) is reprinted with permission from Asally, et al., Proc. Natl. Acad. Sci. U.S.A., 109, 46 (2012). Copyright 2012, National Academy of Sciences, USA. Image (Aiv) is reprinted with permission from Alison Ruhe.
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f1: Illustrations of examples, schematics and potential energies of various growth-induced surface instabilities.(A) Examples of growth-induced surface instabilities on (i) the pumpkin skin, (ii) the cerebral cortex, (iii) the biofilm and (iv) the dog skin. (B) Schematics of growth-induced surface instabilities: (i) wrinkle, (ii) crease, (iii) delaminated-buckle, (iv) fold, (v) period-double and (vi) ridge. (C) One example pathway to induce the mismatch strain in the film-substrate structure: (i) The film and substrate is first assumed to be detached from each other to form a stress-free state; (ii) the detached stress-free substrate is then pre-stretched by a ratio of Lf/Ls and adhered to the film; (iii) relaxed to length L; and (iv) eventually relaxed to length Ls at the current state. Other pathways to induce mismatch strains are illustrated in Supplementary Figs S5 and S7. (D) Evolution of potential energy of the film-substrate structure with increasing mismatch strain following the pathway in (C). The red dash line denotes the surface patterns with the minimum potential energy. The potential energy of the film-substrate structure with mismatch strain εM is denoted by the black solid circle. Image (Ai) is reprinted with permission from Yin, et al., Proc. Natl. Acad. Sci. U.S.A., 105, 49 (2008). Copyright 2008, National Academy of Sciences, USA. Image (Aii) is reprinted from Bradbury, PLOS Biol., 3, 3 (2005) under Open-Access License. Image (Aiii) is reprinted with permission from Asally, et al., Proc. Natl. Acad. Sci. U.S.A., 109, 46 (2012). Copyright 2012, National Academy of Sciences, USA. Image (Aiv) is reprinted with permission from Alison Ruhe.

Mentions: Numerous intriguing morphologies and phenomena on surfaces of growing animals, plants and microorganism colonies have fascinated artists and scientists for decades12. Abundant examples (Fig. 1A) can be found in various types of living creatures across multiple size scales, such as wrinkles on skins of mammalians, plants and fruits345678, undulations in developing biofilms91011, grooves on the cerebral cortex12131415, mucosal villi and folds of airways, esophagi and guts16171819202122, buckled tumor surfaces2324, epithelial cell delamination due to tissue crowding2526, and crumpled membranes of blood cells27. Although these biological patterns may be results of complex genetic, biological and biochemical processes, recent studies have suggested that growth-induced mechanical forces regulate the formation and evolution of biological patterns21618282930. Biological structures usually consist of multiple layers with strikingly different biochemical compositions and mechanical properties; for example, epidermis on the dermis or hypodermis of mammalian skins345, the epidermis on the ground tissue of plant skins678, biofilms on culture gels91011, the grey matter on the white matter of cerebral cortexes1213, the mucosa on the muscle layer of airways, esophagi and guts16171819202122, outer proliferative cells on the inner necrotic core of a tumor23, epithelial cell monolayer on the underlying tissue2526, membranes on the cytoskeleton of blood cells27. During growth, development or aging, different layers of biological structures usually have different expanding or shrinking rates, thus resulting in mismatch strains between the biological layers. The surface topographical patterns have long been believed to be results of mismatch-induced compressive strains in the skin layers which have higher growth rates or lower shrinkage rates than the underlying biological layers1415161831. Once the mismatch compressive strain rises to critical values, the initially flat surface of the film becomes unstable and bifurcate into different types of corrugated patterns (Fig. 1B), including (i) wrinkle — the film undulates sinusoidally but remains locally smooth (e.g., the pumpkin skin in Figs 1Ai and 1Bi)6, (ii) crease — the surface of the film folds into dispersed regions of self-contacts with sharp tips (e.g., the cerebral cortex in Figs 1Aii and 1Bii)1232, and (iii) delaminated-buckle — the film delaminates from the substrate to form buckled regions (e.g., the biofilm in Figs 1Aiii and 1Biii)10. As the mismatch strain further increases, the wrinkles may further bifurcate into more complicated surface patterns, including (iv) fold – some valleys of the wrinkle fold into self-contacts with sharp tips (e.g., the dog skin in Figs 1Aiv and 1Biv)33, (v) period-double — the sinusoidal wrinkle transits into a pattern with twice of the wavelength (Fig. 1Bv), and (vi) ridge — the wrinkle drastically increases its amplitude but decreases its wavelength, forming a high-aspect-ratio pattern that ceases to follow sinusoidal shape (Fig. 1Bvi). These instability patterns with distinctive topographical characteristics have been only studied and identified separately in different biological systems under varied physical and biological conditions610161825. However, a general model that can quantitatively predict the formation and evolution of various types of surface-instability patterns still does not exist; primarily because existing theories such as linear stability analysis cannot systematically analyze all modes of instabilities12, and existing experiments did not systematically vary the mechanical properties of film-substrate systems.


A three-dimensional phase diagram of growth-induced surface instabilities.

Wang Q, Zhao X - Sci Rep (2015)

Illustrations of examples, schematics and potential energies of various growth-induced surface instabilities.(A) Examples of growth-induced surface instabilities on (i) the pumpkin skin, (ii) the cerebral cortex, (iii) the biofilm and (iv) the dog skin. (B) Schematics of growth-induced surface instabilities: (i) wrinkle, (ii) crease, (iii) delaminated-buckle, (iv) fold, (v) period-double and (vi) ridge. (C) One example pathway to induce the mismatch strain in the film-substrate structure: (i) The film and substrate is first assumed to be detached from each other to form a stress-free state; (ii) the detached stress-free substrate is then pre-stretched by a ratio of Lf/Ls and adhered to the film; (iii) relaxed to length L; and (iv) eventually relaxed to length Ls at the current state. Other pathways to induce mismatch strains are illustrated in Supplementary Figs S5 and S7. (D) Evolution of potential energy of the film-substrate structure with increasing mismatch strain following the pathway in (C). The red dash line denotes the surface patterns with the minimum potential energy. The potential energy of the film-substrate structure with mismatch strain εM is denoted by the black solid circle. Image (Ai) is reprinted with permission from Yin, et al., Proc. Natl. Acad. Sci. U.S.A., 105, 49 (2008). Copyright 2008, National Academy of Sciences, USA. Image (Aii) is reprinted from Bradbury, PLOS Biol., 3, 3 (2005) under Open-Access License. Image (Aiii) is reprinted with permission from Asally, et al., Proc. Natl. Acad. Sci. U.S.A., 109, 46 (2012). Copyright 2012, National Academy of Sciences, USA. Image (Aiv) is reprinted with permission from Alison Ruhe.
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f1: Illustrations of examples, schematics and potential energies of various growth-induced surface instabilities.(A) Examples of growth-induced surface instabilities on (i) the pumpkin skin, (ii) the cerebral cortex, (iii) the biofilm and (iv) the dog skin. (B) Schematics of growth-induced surface instabilities: (i) wrinkle, (ii) crease, (iii) delaminated-buckle, (iv) fold, (v) period-double and (vi) ridge. (C) One example pathway to induce the mismatch strain in the film-substrate structure: (i) The film and substrate is first assumed to be detached from each other to form a stress-free state; (ii) the detached stress-free substrate is then pre-stretched by a ratio of Lf/Ls and adhered to the film; (iii) relaxed to length L; and (iv) eventually relaxed to length Ls at the current state. Other pathways to induce mismatch strains are illustrated in Supplementary Figs S5 and S7. (D) Evolution of potential energy of the film-substrate structure with increasing mismatch strain following the pathway in (C). The red dash line denotes the surface patterns with the minimum potential energy. The potential energy of the film-substrate structure with mismatch strain εM is denoted by the black solid circle. Image (Ai) is reprinted with permission from Yin, et al., Proc. Natl. Acad. Sci. U.S.A., 105, 49 (2008). Copyright 2008, National Academy of Sciences, USA. Image (Aii) is reprinted from Bradbury, PLOS Biol., 3, 3 (2005) under Open-Access License. Image (Aiii) is reprinted with permission from Asally, et al., Proc. Natl. Acad. Sci. U.S.A., 109, 46 (2012). Copyright 2012, National Academy of Sciences, USA. Image (Aiv) is reprinted with permission from Alison Ruhe.
Mentions: Numerous intriguing morphologies and phenomena on surfaces of growing animals, plants and microorganism colonies have fascinated artists and scientists for decades12. Abundant examples (Fig. 1A) can be found in various types of living creatures across multiple size scales, such as wrinkles on skins of mammalians, plants and fruits345678, undulations in developing biofilms91011, grooves on the cerebral cortex12131415, mucosal villi and folds of airways, esophagi and guts16171819202122, buckled tumor surfaces2324, epithelial cell delamination due to tissue crowding2526, and crumpled membranes of blood cells27. Although these biological patterns may be results of complex genetic, biological and biochemical processes, recent studies have suggested that growth-induced mechanical forces regulate the formation and evolution of biological patterns21618282930. Biological structures usually consist of multiple layers with strikingly different biochemical compositions and mechanical properties; for example, epidermis on the dermis or hypodermis of mammalian skins345, the epidermis on the ground tissue of plant skins678, biofilms on culture gels91011, the grey matter on the white matter of cerebral cortexes1213, the mucosa on the muscle layer of airways, esophagi and guts16171819202122, outer proliferative cells on the inner necrotic core of a tumor23, epithelial cell monolayer on the underlying tissue2526, membranes on the cytoskeleton of blood cells27. During growth, development or aging, different layers of biological structures usually have different expanding or shrinking rates, thus resulting in mismatch strains between the biological layers. The surface topographical patterns have long been believed to be results of mismatch-induced compressive strains in the skin layers which have higher growth rates or lower shrinkage rates than the underlying biological layers1415161831. Once the mismatch compressive strain rises to critical values, the initially flat surface of the film becomes unstable and bifurcate into different types of corrugated patterns (Fig. 1B), including (i) wrinkle — the film undulates sinusoidally but remains locally smooth (e.g., the pumpkin skin in Figs 1Ai and 1Bi)6, (ii) crease — the surface of the film folds into dispersed regions of self-contacts with sharp tips (e.g., the cerebral cortex in Figs 1Aii and 1Bii)1232, and (iii) delaminated-buckle — the film delaminates from the substrate to form buckled regions (e.g., the biofilm in Figs 1Aiii and 1Biii)10. As the mismatch strain further increases, the wrinkles may further bifurcate into more complicated surface patterns, including (iv) fold – some valleys of the wrinkle fold into self-contacts with sharp tips (e.g., the dog skin in Figs 1Aiv and 1Biv)33, (v) period-double — the sinusoidal wrinkle transits into a pattern with twice of the wavelength (Fig. 1Bv), and (vi) ridge — the wrinkle drastically increases its amplitude but decreases its wavelength, forming a high-aspect-ratio pattern that ceases to follow sinusoidal shape (Fig. 1Bvi). These instability patterns with distinctive topographical characteristics have been only studied and identified separately in different biological systems under varied physical and biological conditions610161825. However, a general model that can quantitatively predict the formation and evolution of various types of surface-instability patterns still does not exist; primarily because existing theories such as linear stability analysis cannot systematically analyze all modes of instabilities12, and existing experiments did not systematically vary the mechanical properties of film-substrate systems.

Bottom Line: However, a general model that accounts for the formation and evolution of these various surface-instability patterns still does not exist.The predicted wavelengths and amplitudes of various instability patterns match well with our experimental data.It is expected that the unified phase diagram will not only advance the understanding of biological morphogenesis, but also significantly facilitate the design of new materials and structures by rationally harnessing surface instabilities.

View Article: PubMed Central - PubMed

Affiliation: 1] Soft Active Materials Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA [2] Department of Mechanical Engineering and Materials Science, Duke University, Durham, North Carolina 27708, USA.

ABSTRACT
A variety of fascinating morphological patterns arise on surfaces of growing, developing or aging tissues, organs and microorganism colonies. These patterns can be classified into creases, wrinkles, folds, period-doubles, ridges and delaminated-buckles according to their distinctive topographical characteristics. One universal mechanism for the pattern formation has been long believed to be the mismatch strains between biological layers with different expanding or shrinking rates, which induce mechanical instabilities. However, a general model that accounts for the formation and evolution of these various surface-instability patterns still does not exist. Here, we take biological structures at their current states as thermodynamic systems, treat each instability pattern as a thermodynamic phase, and construct a unified phase diagram that can quantitatively predict various types of growth-induced surface instabilities. We further validate the phase diagram with our experiments on surface instabilities induced by mismatch strains as well as the reported data on growth-induced instabilities in various biological systems. The predicted wavelengths and amplitudes of various instability patterns match well with our experimental data. It is expected that the unified phase diagram will not only advance the understanding of biological morphogenesis, but also significantly facilitate the design of new materials and structures by rationally harnessing surface instabilities.

Show MeSH
Related in: MedlinePlus