Subcriticality of the zigzag transition: A nonlinear bifurcation analysis

When repelling particles are confined by a transverse potential in quasi-one-dimensional geometry, the straight line equilibrium configuration becomes unstable at small confinement, in favor of a staggered row that may be inhomogeneous or homogeneous. This conformational phase transition is a pitchfork bifurcation called the zigzag transition. We study the zigzag transition in infinite and periodic finite systems with short-range interactions. We provide numerical evidence that in this case the bifurcation is subcritical since it exhibits phase coexistence and hysteretic behavior. The physical mechanism responsible for the change in the bifurcation character is the nonlinear coupling between the transverse soft mode at the transition and the longitudinal Goldstone mode linked to the translational or rotational invariance of the zigzag pattern. An asymptotic analysis, near the bifurcation threshold and assuming an infinite system, gives an explicit expression for the normal form of the bifurcation. We establish the subcriticality, and we describe with excellent precision the inhomogeneous zigzag patterns observed in the simulations. A direct test of the physical mechanism responsible for the bifurcation character evidences a quantitative agreement.

Figure 1

A schematic illustration of the zigzag pattern. The solid lines indicate the interactions that are taken into account in the theoretical description.

Figure 2

Schematic plot of the energy $E\left(h\right)$ as a function of the zigzag height $h$, for $\beta <{\beta }_{ZZ}$, with a sketch of the relevant patterns [see Eq. (1)].

Figure 3

(a) Zigzag height $h$ (mm) as a function of $\epsilon$ for 8 (thick red curve), 16 (thick green curve), and 32 (thick blue curve) particles. The fine solid black line indicates the expected height for a zigzag pattern $h^{*}$. For inhomogeneous zigzag, $h$ is the maximum height of the pattern. In the insets, we show two typical inhomogeneous patterns. The dots are simulations data, while the lines are only guide to the eyes. (b) Onset of the zigzag transition zoomed near the bifurcation threshold for 32 particles. The solid line is the theoretical height $h^{*}$ of a zigzag pattern, while the dots are simulations data.

Figure 4

Evolution of the zigzag height $h$ (mm) for increasing (orange points, see arrows) and decreasing (blue points, see arrows) $\beta /{\beta }_{ZZ}$. The relevant particle number $N$ is indicated in the upper right corner of each plot. The mean interparticle distance $d$ is the same in all plots.

Figure 5

Reduced stiffness $\beta /{\beta }_{ZZ}$ at which a transition between aligned particles and localized zigzag happens as a function of the particle number $N$, at constant $d$. Blue dots are the measured values ${\beta }_{\mathrm{down}}$ for decreasing $\beta$. Orange dots and stars are the measured values ${\beta }_{\mathrm{up}}$ for increasing $\beta$. The thermodynamic temperature in the simulations (see Appendix pp1) is such that $U\left(d\right)/k_B{T}_b={10}^{-2}$ (dots) and $U\left(d\right)/k_B{T}_b={10}^{-4}$ (stars). The dashed line is the maximum height of the hysteresis calculated from the normal form [see Eq. (11)]. The inset is a log-log plot of the deviation to this maximum height as a function of $N$. The dotted line is of slope $-1$.

Figure 6

(a) In red (dark gray), we show the bifurcation diagram corresponding to (13). Solid (dotted) lines represent stable (unstable) solutions. In cyan (light gray), we sketch an hysteresis cycle. (b) : Homogeneous zigzag height $h$ (for $h>0$ only) as a function of the distance to threshold $\epsilon$. Red solid line, stable solutions $h=0$ and $h=h_{+}$. Red dotted line, unstable solution $h=h_{-}$ [see (13)]. Dashed black line, maximum height of bubbles solutions $h=h_{\mathrm{bubble}}$ [see (18)]. The intersection of the lines $h_{\mathrm{bubble}}$ and $h_{+}$ corresponds to a wall solution and is emphasized by the large black dot.

Figure 7

Plot of the dimensionless potential $W\left(h\right)$, Eq. (14), as a function of the dimensionless zigzag height $h$. We indicate the location $h_{-}$ of the local minimum and the location $h_{+}$ of the local maximum. The plot is done for the particular value $\epsilon ={\epsilon }_{\mathrm{wall}}$, such that $W\left(h_{+}\right)=W\left(0\right)=0$.

Figure 8

Plot of the dimensionless potential $W\left(h\right)$ for ${\epsilon }_{\mathrm{wall}}<\epsilon <0$ [see Eq. (14)], as a function of the dimensionless zigzag height $h$. The line $W\left(h\right)=0$ defines a bubble of height $h_{\mathrm{bubble}}$.

Figure 9

Plot of the dimensionless potential $W\left(h\right)$ for ${\epsilon }_{\mathrm{wall}}<\epsilon \le 0$, Eq. (14), as a function of the dimensionless zigzag height $h$. The straight line of ordinate $W_0$ defines a nonlinear modulation between the minimum height $h_1$ and the maximum height $h_2$.

Figure 10

Comparison between simulations (blue points) and theoretical predictions (thick solid lines) for localized zigzag equilibrium patterns (bubbles). From top to bottom and left to right, $N=32$ and ${\epsilon }_{\mathrm{num}}=0.07,N=32$ and ${\epsilon }_{\mathrm{num}}=0.48$ [see the lower inset in Fig. 3 (a)], $N=64$ and ${\epsilon }_{\mathrm{num}}=0.0264$, and $N=64$ and ${\epsilon }_{\mathrm{num}}=0.261$. For clarity, the patterns maxima are placed at the center of the frame even if they can be anywhere in the periodic cell.

Figure 11

Modulated zigzag patterns. From top to bottom and left to right: $N=32$ and ${\epsilon }_{\text{num}}=0.05$, and then $N=64$ and ${\epsilon }_{\text{num}}=0.0173,0.0185$, and $0.0207$. Blue points denote simulation data, while the thick red solid line represents the theoretical prediction, Eq. (22).

Figure 12

Modulated zigzag patterns for $N=16$. From top to bottom and left to right: ${\epsilon }_{\text{num}}=0.132,0.150,0.187$, and $0.206$. Blue points denote simulation data, while the thick red solid line represents the theoretical prediction, Eq. (22). The thick green solid line is the theoretical prediction (20), because this pattern cannot be described by a nonlinear modulation.

Figure 13

Plot of $\alpha \left(u\right)$ (linear scale) as a function of $u$ (log scale) for the modified Bessel function potential (blue curve) and the Yukawa potential (red curve).