Configurational and energy landscape in one-dimensional Coulomb systems

We study a one-dimensional Coulomb system, where two charged colloids are neutralized by a collection of point counterions, with global neutrality. With temperature being given, two situations are addressed: Either the colloids are kept at fixed positions (canonical ensemble) or the force acting on the colloids is fixed (isobaric-isothermal ensemble). The corresponding partition functions are worked out exactly, in view of determining which arrangement of counterions is optimal. How many counterions should be in the confined segment between the colloids? For the remaining ions outside, is there a left-right symmetry breakdown? We evidence a cascade of transitions as system size is varied in the canonical treatment or as pressure is increased in the isobaric formulation.

Figure 1

Sketch of the model studied in [13].

Figure 2

Representation of the modified model studied here.

Figure 3

Fundamental configuration for $P=0$ and $N=2n=26$ (even) and $n=13$ (odd). In this case, the first threshold is precisely $P_0=0$. As soon as $P$ increases and $0<P\le P_1$, one particle from each interior side will go to the outside, leaving $(n-1)/2$ particles in each interior side of the colloid and $(n+1)/2$ on each outer side.

Figure 4

Fundamental configuration for $P=0$ and $N=2n=28$ (even) and $n=14$ (even).

Figure 5

Gibbs energies for $N=4$. The solid lines represent the configurations with minimal Gibbs energy for a fixed number of unbounded particles.

Figure 6

Configuration that minimizes the Gibbs energy for $N=4$. The figure shows the value of $m=M$, which is half the number of unbounded counterions.

Figure 7

Fundamental configuration for $P=-1/4$ (meaning that the intercolloid distance is large) and $N=2n+1=27$ (odd) and $n=13$ (odd).

Figure 8

Same as Fig. 7, with still an odd $N=2n+1$, but now $n$ even: $P=-1/4, N=29$, and $n=14$.

Figure 9

Gibbs energies $N=5$. The solid lines represent the configurations with minimal Gibbs energy for a fixed number of unbounded particles.

Figure 10

Configuration that minimizes the Gibbs energy $N=5$. The values of both $m$ and $M$ are reported; the total number of unbounded ions is $m+M$.

Figure 11

Fundamental isobaric length $L_P^F=\langle L\rangle$ for (a) $N=4$ and (b) $N=5$.

Figure 12

(a) Isobaric fundamental length $L_P^F\left(n\right)$ when $P=0$ as a function of $n$. Here $L_P^F\left(n\right)\in [8,10)$, approaching asymptotically 10. (b) Corresponding values of $m_n^c=M_n^c$ as a function of $n$. Both graphs emphasize the large-$n$ region, where a doublet structure is apparent.

Figure 13

Behavior of the Helmholtz free energy.

Figure 14

(a) Fundamental Helmholtz energy $A_F$. (b) Pressure $P_c^F$ for $N=4$.

Figure 15

(a) Fundamental Helmholtz energy $A_F$. (b) Pressure $P_c^F$ for $N=5$.

Figure 16

Density plot for $r=2$ and (a) $\ell =0$, (b) $\ell =1$, (c) $\ell =2$, and (d) $\ell =3$.