Discrete kink dynamics in hydrogen-bonded chains: The two-component model

We study discrete topological solitary waves (kinks and antikinks) in two nonlinear diatomic chain models that describe the collective dynamics of proton transfers in one-dimensional hydrogen-bonded networks. The essential ingredients of the models are (i) a realistic (anharmonic) ion-proton interaction in the hydrogen bond, (ii) a harmonic coupling between the protons in adjacent hydrogen bonds, and (iii) a harmonic coupling between the nearest-neighbor heavy ions (an isolated diatomic chain with the lowest acoustic band) or instead a harmonic on-site potential for the heavy ions (a diatomic chain subject to a substrate with two optical bands), both providing a bistability of the hydrogen-bonded proton. Exact two-component (kink and antikink) discrete solutions for these models are found numerically. We compare the soliton solutions and their properties in both the one- (when the heavy ions are fixed) and two-component models. The effect of stability switchings, discovered previously for a class of one-component kink-bearing models, is shown to exist in these two-component models as well. However, the presence of the second component, i.e., the softness of the heavy-ion sublattice, brings principal differences, like a significant difference in the stability switchings behavior for the kinks and the antikinks. Water-filled carbon nanotubes are briefly discussed as possible realistic systems, where topological discrete (anti)kink states might exist.

Figure 1

Schematics of interactions in one unit cell of a 2C HB chain.

Figure 2

Schematics of (a) 2CO and (b) 2CA chain models.

Figure 3

Two bands of small-amplitude oscillations for $\beta =10$: (a) ${\kappa }_p=10$, ${\kappa }_i=100$, and ${\kappa }_o=10$; (b) ${\kappa }_p=10$, ${\kappa }_i=5000$, and ${\kappa }_o=10$; (c) ${\kappa }_p=50$, ${\kappa }_i=100$, and ${\kappa }_o=0$; (d) ${\kappa }_p=3$, ${\kappa }_i=150$, and ${\kappa }_o=0$.

Figure 4

Two-component profiles of monotonic symmetric (anti)kinks in the 2CO chain (${\kappa }_i=0$ and ${\kappa }_o=5$) with ${\kappa }_p=20$ and $\beta =5$: (a) ion-centered kink, (b) ion-centered antikink, (c) proton-centered kink, and (d) proton-centered antikink.

Figure 5

Two-component profiles of monotonic symmetric (anti)kinks in the 2CA chain (${\kappa }_i=5$ and ${\kappa }_o=0$) with ${\kappa }_p=20$ and $\beta =5$: (a) ion-centered kink, (b) ion-centered antikink, (c) proton-centered kink, and (d) proton-centered antikink.

Figure 6

Dependence of the energy of symmetric ion-centered antikink (curve 1) and symmetric proton-centered antikink (curves 2 and 4) on coupling constant ${\kappa }_p$ in the 2CO chain (${\kappa }_i=0$ and ${\kappa }_o=5$) with $\beta =5$. The upper inset shows more detailed behavior of the antikink energy $E$ in the vicinity of the stability switching, whereas curve 3 corresponds to the antikink with asymmetric profile. The lower four insets show the profiles of proton-centered antikinks at fixed ${\kappa }_p=5$. In all figures, solid lines correspond to stable and dashed lines to unstable states.

Figure 7

Dependence of the antikink energy $E$ on the coupling constant ${\kappa }_p$ in the 2CO chain (${\kappa }_i=0$ and ${\kappa }_o=5$) with (a) $\beta =10$ and (b) $\beta =20$. Solid lines show stable and dashed lines unstable states.

Figure 8

Dependence of the kink energy $E$ on the coupling constant ${\kappa }_p$ in the 2CO chain (${\kappa }_i=0$ and ${\kappa }_o=70$) with $\beta =10$. Solid lines show stable and dashed lines unstable states.

Figure 9

Dependence of the antikink energy $E$ on the coupling constant ${\kappa }_p$ in the 2CA chain (${\kappa }_i=5$ and ${\kappa }_o=0$) with (a) $\beta =5$, (b) $\beta =10$, and (c) $\beta =20$. Solid lines show stable and dashed lines unstable states.

Figure 10

Dependence of the kink energy $E$ on the coupling constant ${\kappa }_p$ in the 2CA chain (${\kappa }_i=60$ and ${\kappa }_0=0$) with $\beta =20$. Solid lines show stable and dashed lines unstable states.