Elastic interaction and the phase transition in coherent metal-hydrogen systems

Wagner, Hermann‐Josef; Horner, H.

doi:10.1080/00018737400101401

Cited by 293 publications

(102 citation statements)

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“…Although the expression in Eq. 3 is similar to the interaction potential for the physical system of force dipoles in an external strain field, it has the opposite sign, because in the physical system, one has to minimize the composite energy of defect and medium (19,20). The physical potential has been used before to model elastic interactions of cells without any regulatory response and has been shown to lead to aggregation behavior similar to that of electric quadrupoles (21), whereas the model introduced here leads to aggregation behavior similar to that of electric dipoles (31).…”

Section: Discussionmentioning

confidence: 99%

“…We account for this constraint by considering only pairs of opposing forces. In elasticity theory, such a pinching force pattern is known as an anisotropic force contraction dipole, that is the tensor P ij ϭ Pn i n j , where P is the dipole strength, the product of force magnitude and force separation, and n ជ is its orientation (19)(20)(21). Typical cellular dipoles have been measured to be of the order of P Ϸ Ϫ10 Ϫ11 J (this corresponds to two forces of 200 nN each, separated by a distance of 60 m) (22).…”

Section: Theorymentioning

confidence: 99%

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Cell organization in soft media due to active mechanosensing

Bischofs

Schwarz

2003

Proc. Natl. Acad. Sci. U.S.A.

291

263

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Adhering cells actively probe the mechanical properties of their environment and use the resulting information to position and orient themselves. We show that a large body of experimental observations can be consistently explained from one unifying principle, namely that cells strengthen contacts and cytoskeleton in the direction of large effective stiffness. Using linear elasticity theory to model the extracellular environment, we calculate optimal cell organization for several situations of interest and find excellent agreement with experiments for fibroblasts, both on elastic substrates and in collagen gels: cells orient in the direction of external tensile strain; they orient parallel and normal to free and clamped surfaces, respectively; and they interact elastically to form strings. Our method can be applied for rational design of tissue equivalents. Moreover, our results indicate that the concept of contact guidance has to be reevaluated. We also suggest that cell-matrix contacts are up-regulated by large effective stiffness in the environment because, in this way, build-up of force is more efficient.T he mechanical activity of adherent cells usually is attributed to their physiological function. For example, fibroblasts are believed to maintain the structural integrity of connective tissue and to participate in wound healing by actively pulling on their environment. During recent years, it has become clear that there is another important role for mechanical activity of adherent cells: by pulling on their environment, cells can actively sense its mechanical properties and react to it in a specific way (1-3). Harris et al. (4) observed surprisingly large tension fields for fibroblasts on elastic substrates, which induce mechanical activity of other cells, even when located at considerable distance. When plated on elastic substrates of increased rigidity, many cell types show increased spreading and better developed stress fibers and focal adhesions (5). Fibroblasts on elastic substrates orient in the direction of tensile strain (6) and locomote in favor of regions of larger rigidity or tensile strain (7). The same response has been reported for vascular smooth muscle cells on rigidity gradients (8). Similar observations have been reported numerous times also for tissue cells in hydrogels. For fibroblasts in collagen gels, Bell et al. (9) not only found that traction considerably contracts the gel, but also reported orientational effects: cells align along the direction of pull between fixed points and parallel to free surfaces. When a collagen gel is stretched uniaxially, cells orient in the direction of principal strain (10). Moreover, cells align in a nose-to-tail configuration, thus forming strings running in parallel to the direction of external strain. If a collagen gel is cut perpendicular to the direction of tensile strain and if cells are present in sufficient numbers, they round up and reorient parallel to the free surface introduced (11).The response of adherent animal cells to mechanical input ...

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Section: Discussionmentioning

confidence: 99%

Section: Theorymentioning

confidence: 99%

Cell organization in soft media due to active mechanosensing

Bischofs

Schwarz

2003

Proc. Natl. Acad. Sci. U.S.A.

291

263

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“…The essential difference between active and passive particles is that cells respond to strain in an opposite way as do defects. For defects the interaction with the environment leads to a linearized potential V = −P ij u ij [6]. For example, an anisotropic defect attracting the atoms of its host lattice turns away from tensile strain, in this way enhancing the displacement fields in the medium.…”

mentioning

confidence: 99%

“…hydrogen in metal [6,7]. For a local force distribution in the absence of external fields, the overall force (monopole) applied to the elastic medium vanishes due to Newton's third law [8].…”

mentioning

confidence: 99%

Effect of Poisson Ratio on Cellular Structure Formation

Bischofs

Schwarz

2005

Phys. Rev. Lett.

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Mechanically active cells in soft media act as force dipoles. The resulting elastic interactions are long-ranged and favor the formation of strings. We show analytically that due to screening, the effective interaction between strings decays exponentially, with a decay length determined only by geometry. Both for disordered and ordered arrangements of cells, we predict novel phase transitions from paraelastic to ferroelastic and anti-ferroelastic phases as a function of Poisson ratio.Predicting structure formation and phase behaviour from the microscopic interaction laws is a formidable task in statistical mechanics, especially if the interaction laws are long-ranged or anisotropic. In biological systems, the situation is further complicated because interacting components are active in the sense that informed by internal instructions (e.g. genetic programmes for cells) and fueled by energy reservoirs (e.g. ATP), they react to input signals in a complicated way, which usually does not follow from an energy functional. Therefore these systems are often described by stochastic equations [1,2]. One drawback of this approach is that typically the stochastic equations have to be analyzed by numerical rather than analytical methods. However, for specific systems structure formation of active particles can be predicted from extremum principles. In this case, analytical progress might become feasible again, in particular if analogies exist to classical systems of passive particles. One example of this kind might be hydrodynamic interactions of active particles like swimming bacteria [3]. Here this is demonstrated for another example, namely mechanically active cells interacting through their elastic environment [4,5].Our starting point is the observation that generation and propagation of elastic fields for active particles proceed in a similar way as they do for passive particles like defects in a host crystal, e.g. hydrogen in metal [6,7]. For a local force distribution in the absence of external fields, the overall force (monopole) applied to the elastic medium vanishes due to Newton's third law [8]. Therefore each particle is characterized in leading order of a multipolar expansion by a force dipole tensor P ij . For many situations of interest, including cells in soft media, this force dipole will be anisotropic and can be written as P ij = P n i n j , where n is the unit vector describing particle orientation and P is the force dipolar moment. The perturbation of the surrounding medium resulting from a force dipole P ′ kl positioned at r ′ is described by the strain tensor u ij ( r) = ∂ j ∂ ′ l G ik ( r, r ′ )P ′ kl , where summation over repeated indices is implied and G ij is the Green function for the given geometry, boundary conditions and material properties of the sample. The strain u ij generated by one particle causes a reaction of another particle leading to elastic interactions. The essential difference between active and passive particles is that cells respond to strain in an opposite way as do defects. ...

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“…Treating V in a perturbation expansion and performing the trace over the variables { ui, pi ), one is left with a renormalized XI which now describes a compressible generalized Ising model [7]. Such (2) (ii) MONTE CARLO COMPUTER SIMULATIONS [Is].…”

mentioning

confidence: 99%

Theories of Static and Dynamic Critical Phenomena at Order-Disorder and Unmixing Transitions

Binder

1977

J. Phys. Colloques

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RbumC. -Une b r k e revue de quelques concepts rkcents sur les transformations de phase dans les solides est donnke. Tout d'abord, des modeles de rkseaux microscopiques simples sont introduits, des modkles d'lsing qui decrivent soit la dkmixtion (si le parametre d'ordre de la transition est conservk) soit l'ordre (si le parametre d'ordre n'est pas conserve). On montre que les approximations de Bragg-Williams ou de champ moyen dkcrivent de f a~o n imprCcise la transition parce qu'elles negligent les fluctuations, sauf dans le cas des forces longue distance (hydroghe dans les mktaux etc.). Tandis que les calculs de simulation Monte Carlo donnent une description assez precise dans tout le domaine de tempbrature, des informations tres prkcises sur les exposants critiques sont obtenues par la mkthode groupe de renormalisation bask sur la notion d'kchelle. Par l'approche de cluster dynamics ces idkes sont Ctendues aux phknomknes cinktiques loin de l'kquilibre.Abstract. -Some recent concepts on phase transformations in solids are briefly reviewed. First simpIe microscopic lattice models are introduced, kinetic Ising models which describe either unmixing (if the order parameter of the transition is conserved) or ordering (if the order parameter is not conserved). Bragg-Williams or related meanfield-approximations are shown to describe the transition inaccurately due to the neglect of fluctuations, except for the case of long-range forces (hydrogen in metals etc.). While Monte Carlo computer simulations give a fairly accura5e description over the whole temperature range, most accurate information on critical exponents is due to the renormalization group approach based on scaling ideas. By the cluster dynamics approach these ideas are extended also to kinetic phenomena far from equilibrium.

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Elastic interaction and the phase transition in coherent metal-hydrogen systems

Cited by 293 publications

References 16 publications

Cell organization in soft media due to active mechanosensing

Cell organization in soft media due to active mechanosensing

Effect of Poisson Ratio on Cellular Structure Formation

Theories of Static and Dynamic Critical Phenomena at Order-Disorder and Unmixing Transitions

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