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Periodic ordering of clusters in a one-dimensional lattice model
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10.1063/1.4799264
/content/aip/journal/jcp/138/14/10.1063/1.4799264
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/14/10.1063/1.4799264
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Ground state of the considered model. The repulsion to attraction ratio J* and the chemical potential μ* are dimensionless (see (5) ). The coexistence lines are , , and . Schematic illustration of the three phases is shown in the insets inside the region of stability of each phase.

Image of FIG. 2.
FIG. 2.

given by (16) . J* = 0.05, 1/9, 0.5, 1, 1.5 from the bottom to the top line on the left.

Image of FIG. 3.
FIG. 3.

Lines of instability (solid) of the homogeneous phase in the (μ*, T*) variables for a range of J*. Similar behavior was obtained in Ref. 45 . The coexistence lines at T* = 0 are shown as dashed lines.

Image of FIG. 4.
FIG. 4.

Illustration of the method used for obtaining the phase coexistence. In the bottom panel the left, central, and right lines correspond to the gas, periodic, and disordered liquid phases, respectively.

Image of FIG. 5.
FIG. 5.

MF phase diagram for J* = 1/4 in variables (μ*, T*) (a) and (ρ, T*) (b). The symmetry axis in (a) is μ* = −3/4 and ρ = 1/2 in (b). Only half of the phase diagram is shown because of the symmetry. Dashed and solid lines represent continuous and first-order transitions. The dotted line is the λ-line. The coexisting phases in the two-phase regions in (b) are labeled by g for gas, d for dense fluid, and p 1, p 2 for the periodic phases with the smaller and the larger period, respectively. The density range of stability of the large-period phase is within the thickness of the line. The periodic phases in (a) are stable inside the lens (p 2) and inside the loop (p 1).

Image of FIG. 6.
FIG. 6.

MF phase diagram for J* = 1/3 in variables (μ*, T*) (a) and (ρ, T*) (b). The symmetry axis is μ* = −2/3 and ρ = 1/2 in (a) and (b), respectively. Only half of the phase diagram is shown because of the symmetry. Dashed and solid lines represent continuous and first-order transitions between the disordered fluid and the periodic phase. The dotted line is the λ-line. The periodic phase in (a) is stable inside the loop (thick line). The two-phase regions in (b) are shaded.

Image of FIG. 7.
FIG. 7.

MF phase diagram for J* = 3 in variables (μ*, T*) (a) and (ρ, T*) (b). The symmetry axis is μ* = 2 and ρ = 1/2 in (a) and (b), respectively. Only half of the phase diagram is shown because of the symmetry. Dashed and solid lines represent continuous and first-order transitions. The dotted line is the λ-line. The two-phase regions in (b) are shaded with different shades for different phase equilibria. The high-amplitude periodic phase coexists with gas (for μ* < 2 or ρ < 1/2) or liquid (for μ* > 2 or ρ > 1/2) for , and with the low-amplitude periodic phase for . The coexistence line between the two periodic phases (short solid line above the dashed line in (a)) begins at and terminates at the critical point with . Note that the point where the transition between the disordered and the periodic phases changes order is not the TCP. The disordered phase coexists with one periodic phase for , and undergoes a continuous transition to the other periodic phase for , whereas at the TCP the transition between the same phases changes order.

Image of FIG. 8.
FIG. 8.

Amplitudes of the density profiles in the two periodic phases for J* = 3. (a) As a function of temperature along the coexistence line (the lines meet at ); (b) as a function of μ* for .

Image of FIG. 9.
FIG. 9.

Density profiles (a) in the coexisting high- and low-amplitude phases for T* = 0.3. The lines are shifted horizontally for clarity. (b) In the low-amplitude phase close to the continuous transition to the fluid at T* = 0.3 and (c) for T* = 1.3 and μ* = 1. The quasi-periodic structure with a period incommensurate with the lattice is obtained from a density profile with a large-period when 2π/k b is noninteger. The lines connecting the results for integer x are to guide the eyes.

Image of FIG. 10.
FIG. 10.

p(μ*) obtained from Eq. (25) (a) J* = 3; top line: T* = 0.1, bottom line: T* = 1 and (b) J* = 1/4; top line: T* = 0.05, bottom line: T* = 0.5.

Image of FIG. 11.
FIG. 11.

ρ(μ*) obtained from Eq. (26) for J* = 3 and T* = 0.1, 0.4, 0.7, 1 (top to bottom line on the right) (a) and J* = 1/4 and T* = 0.005, 0.05, 0.1, 0.15 (top to bottom line on the right) (b).

Image of FIG. 12.
FIG. 12.

EOS ρ(p*) isotherms obtained from Eqs. (25) and (26) for T* = 0.1, 0.2, 0.3, 0.4, 0.5, and 1 (top to bottom line on the left) for J* = 3 (a) and J* = 1/4 (b).

Image of FIG. 13.
FIG. 13.

EOS ρ(p*) isotherms obtained from Eqs. (25) and (26) for T* = 0.1. From the left to the right line J* = 0.1, 0.25, 1/3, 0.5, 0.75, …, 2.75, 3.

Image of FIG. 14.
FIG. 14.

The correlation function G(x) for x = 3k + i with i = 0, 1, 2 (Eq. (34) ) for J* = 3, μ* = 0, and T* = 0.1 (inside the MF stability region of the periodic phase). Solid line and the circles (black), dashed line and the asterisks (red), and dotted line and the squares (blue) correspond to i = 0, 1, 2, respectively. The bottom panel shows a small portion of the upper panel.

Image of FIG. 15.
FIG. 15.

The correlation function G(x) for x = 3k + i with i = 0, 1, 2 (Eq. (34) ) for J* = 3, μ* = −0.7, and T* = 0.1 (outside the MF stability region of the periodic phase). Black (circle), red (asterisk), and blue (square) symbols correspond to i = 0, 1, 2, respectively.

Image of FIG. 16.
FIG. 16.

The correlation length ξ (Eq. (32) ) for J* = 3 as a function of T* (a) outside the MF stability region of the periodic phase. From the top to the bottom line μ* = −2/3, −0.7, −0.8, −0.9, −1, and (b) inside the MF stability region of the periodic phase. From the bottom to the top line μ* = −0.65, −0.6, −0.55, −0.5, −0.45, and 2.

Image of FIG. 17.
FIG. 17.

The amplitude of the correlation function (see Eq. (34) and below) as a function of μ*. Dashed, solid, and dotted lines correspond to T* = 0.1, 0.2, 0.3, respectively. J* = 3 (a) and J* = 1/4 (b).

Image of FIG. 18.
FIG. 18.

(a) The wavenumber λ of the correlation function (Eq. (34) ). J* = 3 and μ* = −2/3, −0.7, −0.8, −0.9, −1 from the top to the bottom line (b) the period w (Eqs. (35)–(37) ) of the amplitude modulations. J* = 3 and μ* = −0.65, −0.6, −0.55, −0.5, −0.45 from the bottom to the top line.

Image of FIG. 19.
FIG. 19.

The wavenumber λ (a) and the correlation length ξ (Eq. (32) ) (b) of the correlation function (Eq. (34) ) as a function of T* for J* = 1/4. From the bottom to the top line in (a) and from the top to the bottom line in (b) μ* = −0.75, −0.75 ± 0.01, −0.75 ± 0.02, …, −0.75 ± 0.07.

Image of FIG. 20.
FIG. 20.

The correlation function G(x) for x = 3k + i with i = 0, 1, 2 (Eq. (34) ) for J* = 1/4 and μ* = −3/4 for T* = 0.05 (a) and T* = 0.2 (b). Black (circle), red (asterisk), and blue (square) symbols correspond to i = 0, 1, 2, respectively.

Image of FIG. 21.
FIG. 21.

The specific heat per particle (in k B units) as a function of density (dimensionless) for J* = 3. From the top to the bottom line on the left T* = 0.25, 0.5, 0.75, 1 with L = 1200.

Image of FIG. 22.
FIG. 22.

The specific heat (in k B units) as a function of density (dimensionless) for J* = 1/4. From the top to the bottom line on the left T* = 0.1, 0.25, 0.5, 0.75, 1 with L = 840.

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/content/aip/journal/jcp/138/14/10.1063/1.4799264
2013-04-11
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Periodic ordering of clusters in a one-dimensional lattice model
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/14/10.1063/1.4799264
10.1063/1.4799264
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