^{1}, A. Ciach

^{1}and N. G. Almarza

^{2}

### Abstract

A generic lattice model for systems containing particles interacting with short-range attraction long-range repulsion (SALR) potential that can be solved exactly in one dimension is introduced. We assume attraction *J* _{1} between the first neighbors and repulsion *J* _{2} between the third neighbors. The ground state of the model shows existence of two homogeneous phases (gas and liquid) for *J* _{2}/*J* _{1} <1/3. In addition to the homogeneous phases, the third phase with periodically distributed clusters appears for *J* _{2}/*J* _{1} > 1/3. Phase diagrams obtained in the self-consistent mean-field approximation for a range of values of *J* _{2}/*J* _{1} show very rich behavior, including reentrant melting, and coexistence of two periodic phases (one with strong and the other one with weak order) terminated at a critical point. We present exact solutions for the equation of state as well as for the correlation function for characteristic values of *J* _{2}/*J* _{1}. Based on the exact results, for *J* _{2}/*J* _{1} > 1/3 we predict pseudo-phase transitions to the ordered cluster phase indicated by a rapid change of density for a very narrow range of pressure, and by a very large correlation length for thermodynamic states where the periodic phase is stable in mean field. For 1/9 < *J* _{2}/*J* _{1} < 1/3 the correlation function decays monotonically below certain temperature, whereas above this temperature exponentially damped oscillatory behavior is obtained. Thus, even though macroscopic phase separation is energetically favored and appears for weak repulsion at *T* = 0, local spatial inhomogeneities appear for finite *T*. Monte Carlo simulations in canonical ensemble show that specific heat has a maximum for low density ρ that we associate with formation of living clusters, and if the repulsion is strong, another maximum for ρ = 1/2.

We thank E. Lomba and W. T. Góźdź for discussions. A part of this work was realized within the International Ph.D. Projects Programme of the Foundation for Polish Science, cofinanced from European Regional Development Fund within Innovative Economy Operational Programme “Grants for innovation.” Partial support by the NCN Grant No. 2012/05/B/ST3/03302 is also acknowledged. N.G.A. gratefully acknowledges financial support from the Dirección General de Investigación Científica y Técnica under Grant No. FIS2010-15502, from the Dirección General de Universidades e Investigación de la Comunidad de Madrid under Grant No. S2009/ESP-1691 and Program MODELICO-CM.

I. INTRODUCTION

II. THE MODEL AND ITS GROUND STATE

A. The model

B. The ground state

III. MF APPROXIMATION

A. Short background

B. Stability analysis

C. MF phase diagrams

IV. EXACT SOLUTIONS

A. Transfer matrix and exact expressions

B. Results

1. Thermodynamic properties(equation of state)

2. Structure (correlation function)

V. MC SIMULATIONS IN CANONICAL ENSEMBLE

VI. SUMMARY AND DISCUSSION

### Key Topics

- Correlation functions
- 28.0
- Phase diagrams
- 20.0
- Phase transitions
- 16.0
- Equations of state
- 15.0
- Thermodynamic properties
- 13.0

## Figures

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.

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.

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

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.

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.

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.

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.

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}).

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}).

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.

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.

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.

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.

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 .

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 .

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.

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.

*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.

*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.

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).

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).

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).

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).

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.

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.

The correlation function *G*(*x*) for *x* = 3*k* + *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.

The correlation function *G*(*x*) for *x* = 3*k* + *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.

The correlation function *G*(*x*) for *x* = 3*k* + *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.

The correlation function *G*(*x*) for *x* = 3*k* + *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.

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.

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.

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).

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).

(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.

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.

The correlation function *G*(*x*) for *x* = 3*k* + *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.

The correlation function *G*(*x*) for *x* = 3*k* + *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.

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.

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.

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.

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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