Conceptos básicos

I will now consider the behavior of counterions near the critical point and begin with deter- mining the precise location of the critical Manning parameter,ξc.

To this end, I shall employ a procedure similar to the method of locating the transition temperature in bulk critical phenomena [101, 117, 121]. Namely, one expects that the transi- tion point is reflected by a singular behavior in thermodynamic quantities such as energy or heat capacity as already indicated by the mean-field results obtained in Section 3.3.3. The mean (internal) energy,EN, and the excess heat capacity, CN, may be obtained directly from

the simulations and in rescaled units as

˜ E = EN N kBT = _H N N kBT , (3.77) ˜ C= CN N kB =N δHN N kBT 2 , (3.78)

where the configurational Hamiltonian HN is defined through Eq. (3.10) and δHN =HN −

hHNi.

Simulation results for the rescaled energy, ˜E, and the rescaled excess heat capacity, ˜C, in Figure 3.8 (symbols) show a non-monotonic behavior as a function ofξ. The energy develops a

0 0.25 0.5 0.75 1 0 0.5 1 _ξ 1.5 2

E

∆

∼ 0.9 1 1.1 0.85 0.9 0.95 1 0 0.5 1 _ξ 1.5 2

a)

0 0.5 1 1.5 2

C

∆

∼

b)

Ξ=102, ∆=200 Ξ=0.1, ∆=300 Ξ=0.1, ∆=200 Ξ=0.1, ∆=100

Figure 3.8: a) The rescaled internal energy, ˜E = EN/(N kBT) and b) the rescaled excess heat capacity, ˜C =CN/(N kB), of the counterion-cylinder system as a function of Manning parameter, ξ. Open symbols show the data for small coupling parameter Ξ = 0.1 and for lateral extension parameters ∆ = 100,200 and 300 as indicated on the graph. Filled symbols are the data for large coupling Ξ = 102 and ∆ = 200. Here number of counterions isN= 100. Solid curves show the asymptotic PB prediction for ∆_{→ ∞}, Eqs. (3.60) and (3.61). Dashed curves are the full PB result for ∆ = 100 and 300 (from bottom to top), which are calculated numerically using Eqs. (3.58) and (3.80). The inset shows a closer view of the energy peak.

pronounced peak and the heat capacity exhibits a jump at intermediate Manning parameters, which become singular for ∆ increasing to infinity. The general behavior of energy and heat capacity can be understood using simple arguments as follows.

For sufficiently small ξ, counterions are all unbound and the electrostatic potential in space is roughly given by the bare potential of the charged cylinder, i.e. ψ(˜r) _≃2ξln(˜r/R˜). This yields the rescaled internal energy, ˜E, (via integrating over the square electric field, see Eq. (3.58)) as ˜ E = 1 4ξ Z D˜ ˜ R r˜d˜r _dψ d˜r 2 ≃ξ∆ (3.79)

for ∆ = ln(D/R) ≫ 1. Intuitively, this result may be obtained also by assuming that counterions experience the potential of the cylinder at the outer boundary; thus one simply has ˜E_≃ψ( ˜D)/2_≃ξ∆, which explains the linear increase of the left tail of the energy curve with both ξ and ∆ (Figure 3.8a). Now using the following thermodynamic relation

ξ∂E˜

∂ξ = ˜E−C,˜ (3.80)

the excess heat capacity is obtained to vanish in the de-condensation regime, i.e. C˜ ≃ 0 (Figure 3.8b). Hence, the heat capacity reduces to that of an ideal gas of particles.

For largeξ, the electrostatic potential of the cylinder is screened due to counterionic bind- ing. If one estimates the screened electrostatic potential of the cylinder as ψ(˜r)≃2 ln(˜r/R˜), which can be verified systematically within the PB theory [38, 105], one obtains the energy and the heat capacity as

˜

3.6 Simulation results in 3D 43

These results may also be obtained by noting that only the fraction 1/ξof de-condensed coun- terions (Section 3.6.1) contributes to the energy on the leading order; thus ˜E _≃ψ( ˜D)/(2ξ)_≃ ∆/ξ. The above asymptotic estimates in fact coincide with the asymptotic (∆ → ∞) PB results (3.60) and (3.61), which are shown by solid curves in Figure 3.8.

The preceding considerations demonstrate that the non-monotonic behavior of the energy and excess heat capacity results directly from the screening effect due to the condensation of counterions as ξ increases. Hence, the singular peaks emerging in both quantities reflect the onset of the counterion-condensation transition, ξc, which occurs in the thermodynamic

infinite-system-size limit N → ∞ and ∆ → ∞. Within the PB theory (solid and dashed curves in Figure 3.8), the location of the peak of energy, ξ∗E,PB(N,∆), tends to the mean-field thresholdξPB

c = 1 from belowas ∆ increases obeying the finite-size-scaling relation

ξ_cPB−ξ_∗E,PB(∆)≃ 1

∆, (3.82)

which is obtained using the full PB energy (3.58). On the other hand, the location of the peak of the PB heat capacity approaches ξ_cPB fromabove.

I locate the critical point from the asymptotic behavior of the energy peak,ξE

∗, as N and ∆ increase. (The heat capacity peak is found to be located further away from the critical point than the energy peak, resembling the well-known behavior of the heat capacity peak in finite Magnetic systems [117], which makes it inconvenient for determining the critical point). In Figure 3.9, I show the simulation results for ξE

∗ (symbols) as a function of ∆−1 for Ξ = 0.1 and for various number of particles (main set). These data are obtained using the thermodynamic relation (3.80), which allows us to calculate the first derivative of the energy,

∂E/∂ξ˜ , including its error-bars, directly from the energy and the heat capacity data without referring to numerical differentiation methods which typically generate large errors near the peak. As seen, for increasing N, the data converge to and closely follow the mean-field prediction (solid curve) within the estimated error-bars; for N > 100, ξE

∗ lies within about 1% of the PB threshold ξ_cPB= 1. Since in the simulations I have used ∆≤300, the behavior of ξE_∗ for very small ∆−1 _→0 is not obtained, nevertheless, the excellent convergence of the data for Ξ = 0.1 to the PB prediction gives an accurate estimate for the critical Manning parameter as

ξc = 1.00±0.002. (3.83)

The results for larger values of the coupling parameter, Ξ, in the inset of Figure 3.9 show that the location of the energy peak does not vary with the coupling parameter. Therefore, the critical Manning parameter is found to be universal and given by the mean-field value

ξc = 1.0. Recall that the same threshold is obtained within the Onsager instability and the

strong-coupling analysis (Sections 3.2.2 and 3.4).

Another important result is that the CCT is not associated with a diverging singularity, in contrast to the Onsager instability prediction [39]. But, the energy at any finite value of

ξ, and also the heat capacity for ξ > 1, tend to infinity (as ∼ ∆) when the lateral exten- sion parameter, ∆, increases to infinity, which, as illustrated before, reflects the logarithmic divergency of the effective electrostatic potential in a charged cylindrical system. The CCT, however, exhibits auniversal discontinuous jump for the excess heat capacity atξc, and thus

0 0.002 0.004 0.006 0.008 0.01 0.012

1/∆

0.96 0.97 0.98 0.99 1

ξ

E

_∗

10-1

Ξ

0.993 0.995 0.997 10 103 105 ξE_∗ mean field

Figure 3.9: Main set: Location of the peak of energy,ξE

∗, as a function of the inverse lateral extension parameter, 1/∆ = 1/ln(D/R). Open symbols are simulation results for small coupling parameter Ξ = 0.1 and for various number of counterions (from bottom): N = 25 (triangle-downs), 50 (squares), 100 (triangle-ups), 200 (circles) and 300 (diamonds). Solid curve shows the mean-field PB prediction for the peak location as obtained by numerical evaluation of the full PB energy, Eq. (3.58). Inset: Location of the energy peak as a function of the coupling parameter, Ξ, forN = 200 and ∆ = 300.

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