Kink-antikink asymmetry and impurity interactions in topological mechanical chains

We study the dynamical response of a diatomic periodic chain of rotors coupled by springs, whose unit cell breaks spatial inversion symmetry. In the continuum description, we derive a nonlinear field theory which admits topological kinks and antikinks as nonlinear excitations but where a topological boundary term breaks the symmetry between the two and energetically favors the kink configuration. Using a cobweb plot, we develop a fixed-point analysis for the kink motion and demonstrate that kinks propagate without the Peierls-Nabarro potential energy barrier typically associated with lattice models. Using continuum elasticity theory, we trace the absence of the Peierls-Nabarro barrier for the kink motion to the topological boundary term which ensures that only the kink configuration, and not the antikink, costs zero potential energy. Further, we study the eigenmodes around the kink and antikink configurations using a tangent stiffness matrix approach appropriate for prestressed structures to explicitly show how the usual energy degeneracy between the two no longer holds. We show how the kink-antikink asymmetry also manifests in the way these nonlinear excitations interact with impurities introduced in the chain as disorder in the spring stiffness. Finally, we discuss the effect of impurities in the (bond) spring length and build prototypes based on simple linkages that verify our predictions.

Figure 1

A kink (a) and an antikink (b) configuration in a topological chain (TC) model of rotors (blue) and springs (red dashed lines) in the presence of a single impurity (green solid lines) modeled as a spring with a different stiffness. For the kink profile, the springs in the chain are at their rest length, while for the antikink, they are stretched. A sketch of kink and antikink profiles in terms of the continuum field variable $u=sin\theta$ (where $\theta$ is the rotor angle) is shown below each configuration. (c) A two-rotor system. The masses are the blue dots, the rigid rotors are the black lines, the pivots are the crosses, and the spring is the dashed red line. Here $a$ is the lattice spacing, $r$ is the rotor length, $\overline{l}$ is the rest length of the springs, and ${\theta }_{1,2}$ are the rotor angles with respect to the vertical.

Figure 2

Illustrated are the possible scenarios for how the kink and antikink interact with a single impurity of spring stiffness. As indicated by the arrow, an initial kink or antikink approaches the impurity site (indicated by the green star) from the right. After scattering, the incident kink is either (I) perfectly transmitted or (II) splits into a reflected kink, a transmitted kink, and an antikink that gets trapped at the impurity site. The incident antikink isr (I) perfectly transmitted, (II) trapped at the impurity site, or (III) perfectly reflected.

Figure 3

The configuration (a) and the corresponding cobweb plot (b) for the kink in a topological rotor chain with $r/a=0.8,|\overline{\theta }|=0.58$. The springs are at their rest lengths. In (b), the black curve is the constraint equation which ensures that the springs are unstretched, the gray diagonal line satisfies ${\theta }_{i+1}={\theta }_i$, the blue point $({\theta }_i,{\theta }_i)$ represents rotor $i$, the red point $({\theta }_i,{\theta }_{i+1})$ represents the spring connecting rotors $i$ and $i+1$, and the red dashed lines with arrows indicates the iterative process that generates the kink profile. The iteration steps from ${\theta }_7$ to ${\theta }_{10}$ are shown.

Figure 4

The configuration (a) and the corresponding cobweb plot (b) for an antikink profile in the topological rotor chain with $r/a=0.8,|\overline{\theta }|=0.58$, where we see that the springs are stretched. In (b) the same graphic notation as in Fig. 3 is used except that we have not used an iterative process for constructing the antikink profile; rather, depicted is only a visualization of the configuration of the rotor chain. The red points are obtained by first reflecting the red points in Fig. 2 across the diagonal line and then relaxing the springs using dissipative Newtonian dynamics. Note that the two rotors at the edges need to be collinear with the springs to ensure force balance. This results in the angles overshooting at the fixed points.

Figure 5

(a) The $\theta$ profile (rotor angles) for the antikink profile in Fig. 4 and the corresponding continuum prediction from Eq. (13). Note that the two rotors at the edges need to be collinear with the springs to ensure force balance and this results in the rotor angles overshooting the equilibrium value $\overline{\theta }=\pm 0.58$. (b) The amount of spring stretching for the antikink profile.

Figure 6

The normalized potential energy plotted against the equilibrium angle $\overline{\theta }$, for a static antikink configuration in a topological rotor chain with with $r/a=0.8$. The discrete model has 60 rotors. Note that the wobbler transition [23] is around $\overline{\theta }={sin}^{-1}\left(\frac{a}{2r}\right)=0.67$, which is close to where the continuum theory starts to significantly deviate from the discrete model.

Figure 7

The configurations of (a) the kink translation mode, (b) the kink shape mode, (c) the antikink translation mode, and (d) the antikink shape mode. The green arrows depict the mode component of each rotor.

Figure 8

The frequencies $\omega$ of localized mode(s) for (a) the kink and (b) the antikink as a function of $\overline{\theta }$ for a rotor chain with $r/a=0.8$. The data points are numerically obtained from the tangent stiffness matrix approach, filled circles correspond to the shape mode $\left({\omega }_s\right)$, while open circles correspond to the translation mode $\left({\omega }_t\right)$. The curves are from the continuum theory. The frequencies for the kink translation mode for all $\overline{\theta }$ and the frequencies for the antikink translation mode for $\overline{\theta }<0.1$ are effectively zero at machine precision and, thus, not visible in the figure.

Figure 9

Time evolution of the kinetic energy for a kink [Fig. 7] and an antikink [Fig. 7] in a topological rotor chain with nondimensional parameters $M=1,k=10000,r/a=0.8,\mathrm{and}\overline{\theta }=0.58$. The magnitude of initial velocity in both cases is $v_0=2.4$. The units of energy and velocity are determined by the aforementioned physical parameters. The kink propagation only results in small oscillation of the KE, whereas we see significant fluctuations during the propagation of an antikink. These can be traced to the Peierls-Nabarro potential as shown in Fig. 10.

Figure 10

Two equilibrium configurations in the potential energy landscape of a static antikink: (a) a minimum and (b) a saddle point, respectively. The topological chain has the same configuration parameters as in Fig. 9.

Figure 11

The dependence of the normalized PN barrier $\left(V_{\mathrm{PNB}}/k{a}^2\right)$ on the normalized antikink width $\left(w/a\right)$ for both the discrete model (black circles) and the continuum theory (solid line). The slope of the dashed line (fit to simulation) is $-10.6$, in reasonable agreement with the predictions from the continuum theory in Eq. (18), which gives a slope $-{\pi }^2\approx -9.9$.

Figure 12

The finite-size effect on $V_{\mathrm{PNB}}$. $\mathrm{\Delta }{V}_{\mathrm{PNB}}$ is defined as $V_{\mathrm{PNB}}\left(L\right)-V_{\mathrm{PNB}}(L=60)$. The configuration parameters are $r/a=0.8$ and $\overline{\theta }=0.40$.

Figure 13

Perspective view of a moving antikink trapped in its Peierls-Nabarro barrier around $\mathrm{Time}=20$ near Rotor 35. The topological rotor chain has the same configuration parameters as in Fig. 9 and the initial antikink velocity is $v_0=1.1$ in nondimensional units.

Figure 14

A kink interacts with an impurity (different spring stiffness) and is either (a) transmitted, shown here for $v_0=4.0$ and $k_i/k=0.10$, or (b) splits into a transmitted kink, a reflected kink, and an antikink trapped at the impurity, shown here for $v_0=9.6$ and $k_i/k=0.01$. The nondimensional parameters are $M=1,k=10000,r/a=0.8,\mathrm{and}\overline{\theta }=0.28$.

Figure 15

An antikink interacts with an impurity and is either (a) transmitted, shown here for $v_0=4.0$ and $k_i/k=0.80$, (b) trapped, shown here for $v_0=4.0$ and $k_i/k=0.70$, or (c) reflected, shown here for $v_0=4.8$ and $k_i/k=0.20$. The system parameters are the same as in Fig. 14.

Figure 16

The phase diagram of the scattering behavior in the parameter space of normalized spring constant of impurity $k_i/k$ and kink initial velocity $v_0$ for (a) the kink and (b) the antikink. The system parameters are the same as Fig. 14. The lower limit of $v_0$ for the antikink is around 0.7, below which even the PN barrier in a perfect chain will capture the antikink.

Figure 17

The zero vibrational mode (a), the soft vibrational mode (b), and the soft tensional mode (c) of a topological chain with a longer spring in the middle as an impurity. The configuration parameters are $\overline{\theta }=0.58,r/a=0.8,\overline{l}/a=1.68,l_0/a=2.30$, and $l_{\mathrm{critical}}/a=2.31$. The soft mode frequency is $7.7\times {10}^{-9}$ in the unit of $(r/a)\sqrt{k/M}$, which means the mode is much “softer” than the kink shape mode whose frequency is of the order ${10}^{-2}$. In (a) and (b), the arrows indicate the mode amplitude of the displacement of each rotor. In (c), the thickness of the green bars indicates the tensional mode amplitude on each spring. All the springs, both normal ones and the impurity, have the same stiffness. Panel (d) shows a LEGO demonstration (see movie S2 in the Supplemental Material [41]).

Figure 18

(a) Illustration of the coordinate system of a topological rotor linkage chain with $\overline{\theta }=0.58,r/a=0.8,\overline{l}/a=1.68$, and $l_{\mathrm{critical}}/a=2.31$. The linkage bars are the solid lines and the impurity spring is the dashed line. In (b), the upper panels show the potential functions in 2D configuration space for various $l_0$. One corner of the function is trimmed for visualization. The red curve corresponds to the potential for Kink 1 in the one d.o.f. case where Kink 2 is fixed at $x_2=0$. The lower panels show the phase portraits of Kink 1.

Figure 19

The trajectories of the chain generated by simulations of Newtonian dynamics on the theoretical potential function in the configuration space at (a) $l_0<l_{\mathrm{critical}},E<E_c$; (b) $l_0<l_{\mathrm{critical}},E>E_c$; (c) $l_0=l_{\mathrm{critical}},E=E_c=0$; and (d) $l_0>l_{\mathrm{critical}}$. In the top figures of (a) and (b), the color scale of the trajectories indicates the potential energy of the chain in arbitrary units. The big red dots correspond to the configuration of the real-space chains shown in the bottom figures of each panel.

Figure 20

The parameter space of the total energy $E$ and the impurity spring length $l_0$. The critical energy $E_c$ as a function of $l_0$ forms a parabola. The chain shows different dynamical behaviors across the left branch of the parabola. The vertical dashed line of $l_0=l_{\mathrm{critical}}$ is the boundary line across which the shape of the potential function transitions qualitatively. The gray area below the right branch of the parabola is energetically forbidden.

Figure 21

Detailed configurations around a single spring $p$.