Thermal Fluctuations of Singular Bar-Joint Mechanisms

A bar-joint mechanism is a deformable assembly of freely rotating joints connected by stiff bars. Here we develop a formalism to study the equilibration of common bar-joint mechanisms with a thermal bath. When the constraints in a mechanism cease to be linearly independent, singularities can appear in its shape space, which is the part of its configuration space after discarding rigid motions. We show that the free-energy landscape of a mechanism at low temperatures is dominated by the neighborhoods of points that correspond to these singularities. We consider two example mechanisms with shape-space singularities and find that they are more likely to be found in configurations near the singularities than others. These findings are expected to help improve the design of nanomechanisms for various applications.

Figure 1

Shape space of the planar four-bar linkage visualized as two intersecting curves on a torus, each curve representing a “branch” of the shape space. The poloidal and toroidal angles along the branches correspond to the angles ${\theta }_1$ and ${\theta }_2$ of the linkage, which has two modes of deformation with ${\theta }_1={\theta }_2$ (blue curve) and ${\theta }_1\ne {\theta }_2$ (red curve).

Figure 2

Free-energy difference $\mathrm{\Delta }{\mathcal{A}}_{{\widehat{\theta }}_1}({\theta }_1)$ of a four-bar linkage with parameters $a=1$ and $\lambda =2$ in units of ${\beta }^{-1}$ at $\beta ={10}^4$. The inset shows the absolute errors between the numerical and asymptotic results using the harmonic approximation [Eq. (6), blue curve] and quartic approximation [Eq. (7), red curves].

Figure 3

(a) A triangulated origami modeled as a bar-joint mechanism and (b) its shape space visualized in the space of fold angles ${\rho }_1$, ${\rho }_2$, and ${\rho }_3$. (c) Free-energy difference $\mathrm{\Delta }{\mathcal{A}}_{{\widehat{\rho }}_1}({\rho }_1)$ in units of ${\beta }^{-1}$ at $\beta ={10}^4$. The inset shows the absolute errors between the numerical and asymptotic results using the harmonic and quartic approximations (blue and red curves, respectively).