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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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A calculated three-dimensional phase diagram of various surface instability patterns induced by mismatch strains.The instability pattern is determined by three non-dimensional parameters: mismatch strain εM, modulus ratio μf/μs and normalized adhesion energy Γ/(μsHf).
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f2: A calculated three-dimensional phase diagram of various surface instability patterns induced by mismatch strains.The instability pattern is determined by three non-dimensional parameters: mismatch strain εM, modulus ratio μf/μs and normalized adhesion energy Γ/(μsHf).

Mentions: Next, we discuss the process to quantitatively construct the phase diagram. A plane-strain finite element model is developed to calculate the formation of instability patterns (Methods and SI). To induce mismatch strains in the model, we assume the detached stress-free substrate in Fig. 1Ci is pre-stretched by a ratio of Lf/Ls, adhered to the film (Fig. 1Cii), and then relaxed to length L (Fig. 1Ciii), during which all deformation occurs in plane-strain condition. The overall compressive strain in the film is defined as ε = (Lf − L)/Lf (Fig. 1Ciii). As ε increases to critical values, patterns of surface instabilities can initiate and transit into others (Fig. 1D). Force perturbations and mesh defects have been introduced into the model as fluctuations to facilitate the system to seek minimum-potential energy states (Fig. 1D). When the substrate is fully relaxed (i.e., L = Ls and ε = εM, shown as the black solid circle on Fig. 1D), the resultant pattern is the instability pattern of the film-substrate system with mismatch strain εM, which represents a point of one phase in the phase diagram (Fig. 2). The boundaries between regions of different phases give the phase boundaries on the phase diagram. We can also determine the phase boundaries by comparing the potential energies of different patterns with the same set of μf/μs, Γ/(μsHf) and εM394344, i.e.,where Πi and Πj are the potential energies of two different patterns on film-substrate models with the same properties and dimensions (Fig. 1D). Following this method, we categorize all modes of surface instabilities patterns discussed above into a three-dimensional phase diagram with quantitatively determined phase boundaries (Fig. 2).


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

Wang Q, Zhao X - Sci Rep (2015)

A calculated three-dimensional phase diagram of various surface instability patterns induced by mismatch strains.The instability pattern is determined by three non-dimensional parameters: mismatch strain εM, modulus ratio μf/μs and normalized adhesion energy Γ/(μsHf).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4352857&req=5

f2: A calculated three-dimensional phase diagram of various surface instability patterns induced by mismatch strains.The instability pattern is determined by three non-dimensional parameters: mismatch strain εM, modulus ratio μf/μs and normalized adhesion energy Γ/(μsHf).
Mentions: Next, we discuss the process to quantitatively construct the phase diagram. A plane-strain finite element model is developed to calculate the formation of instability patterns (Methods and SI). To induce mismatch strains in the model, we assume the detached stress-free substrate in Fig. 1Ci is pre-stretched by a ratio of Lf/Ls, adhered to the film (Fig. 1Cii), and then relaxed to length L (Fig. 1Ciii), during which all deformation occurs in plane-strain condition. The overall compressive strain in the film is defined as ε = (Lf − L)/Lf (Fig. 1Ciii). As ε increases to critical values, patterns of surface instabilities can initiate and transit into others (Fig. 1D). Force perturbations and mesh defects have been introduced into the model as fluctuations to facilitate the system to seek minimum-potential energy states (Fig. 1D). When the substrate is fully relaxed (i.e., L = Ls and ε = εM, shown as the black solid circle on Fig. 1D), the resultant pattern is the instability pattern of the film-substrate system with mismatch strain εM, which represents a point of one phase in the phase diagram (Fig. 2). The boundaries between regions of different phases give the phase boundaries on the phase diagram. We can also determine the phase boundaries by comparing the potential energies of different patterns with the same set of μf/μs, Γ/(μsHf) and εM394344, i.e.,where Πi and Πj are the potential energies of two different patterns on film-substrate models with the same properties and dimensions (Fig. 1D). Following this method, we categorize all modes of surface instabilities patterns discussed above into a three-dimensional phase diagram with quantitatively determined phase boundaries (Fig. 2).

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