^{1,a)}, Giancarlo Franzese

^{2,b)}, Paulo A. Netz

^{3,c)}and Marcia C. Barbosa

^{4,d)}

### Abstract

We investigate by molecular dynamics simulations a continuous isotropic core-softened potential with attractive well in three dimensions, introduced by Franzese [J. Mol. Liq.136, 267 (2007)], that displays liquid-liquid coexistence with a critical point and waterlike density anomaly. Besides the thermodynamic anomalies, here we find diffusion and structural anomalies. The anomalies, not observed in the discrete version of this model, occur with the same hierarchy that characterizes water. We discuss the differences in the anomalous behavior of the continuous and the discrete model in the framework of the excess entropy, calculated within the pair correlation approximation.

We thank the Brazilian Science Agencies, CNPq, CAPES, and FINEP and the Spanish Ministerio de Educación y Ciencia for the International Cooperation for financial support under IRCR Grant No. PHB2004-0057-PC. G.F. acknowledges financial support from the Spanish Ministerio de Educación y Ciencia within the Programa Ramón y Cajal and Grant No. FIS2004-03454.

I. INTRODUCTION

II. THE MODEL

III. DETAILS OF SIMULATIONS

IV. RESULTS AND DISCUSSION

A. Density anomaly

B. Diffusion anomaly

C. Structural anomaly

D. The hierarchy of anomalies and order map

E. The Widom line and the minima in

F. Excess entropy and anomalies

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Entropy
- 29.0
- Diffusion
- 26.0
- Critical point phenomena
- 12.0
- Silica
- 11.0
- Phase transitions
- 8.0

## Figures

Interaction potentials studied in this work. Dashed line represents the discontinuous shouldered well (DSW) potential (Refs. 27–30). Continuous line represents the continuous shouldered well (CSW) potential introduced in Ref. 25. The parameters are explained in the text.

Interaction potentials studied in this work. Dashed line represents the discontinuous shouldered well (DSW) potential (Refs. 27–30). Continuous line represents the continuous shouldered well (CSW) potential introduced in Ref. 25. The parameters are explained in the text.

(a) Pressure-temperature diagram for the CSW model. Lines correspond to isochores with, from bottom to top, , 0.18, 0.185, 0.19, 0.2, 0.21, 0.215, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.29, 0.3, 0.31, and 0.32. Triangles represent spinodal lines for the LDL-HDL phase transition, with LDL at low and HDL at high . We estimate the LDL-HDL critical point where the spinodal lines converge (large filled circle). Line with crosses represents our estimate of the liquid-liquid coexistence line. The TMD (bold continuous) line bends toward the LDL-HDL critical point at high . Dashed lines bound the region where the diffusion anomaly occurs (see Sec. IV B). (b) Experimental data for water anomalies adapted from Angell *et al.* (Ref. 1). Circles denote the line of temperatures of maximum density (TMD) at constant . Squares mark where the diffusion has a maximum value with increasing at constant , . (c) Simulation data for SPC/E water adapted from Netz *et al.* ^{13} Squares mark where the diffusion has a maximum value at constant , and diamonds mark for the local minima . Circles locate the TMD line. (d) Zoomed region from panel (a), showing good qualitative agreement between our simulations and the experiments.

(a) Pressure-temperature diagram for the CSW model. Lines correspond to isochores with, from bottom to top, , 0.18, 0.185, 0.19, 0.2, 0.21, 0.215, 0.22, 0.23, 0.24, 0.25, 0.26, 0.27, 0.28, 0.29, 0.3, 0.31, and 0.32. Triangles represent spinodal lines for the LDL-HDL phase transition, with LDL at low and HDL at high . We estimate the LDL-HDL critical point where the spinodal lines converge (large filled circle). Line with crosses represents our estimate of the liquid-liquid coexistence line. The TMD (bold continuous) line bends toward the LDL-HDL critical point at high . Dashed lines bound the region where the diffusion anomaly occurs (see Sec. IV B). (b) Experimental data for water anomalies adapted from Angell *et al.* (Ref. 1). Circles denote the line of temperatures of maximum density (TMD) at constant . Squares mark where the diffusion has a maximum value with increasing at constant , . (c) Simulation data for SPC/E water adapted from Netz *et al.* ^{13} Squares mark where the diffusion has a maximum value at constant , and diamonds mark for the local minima . Circles locate the TMD line. (d) Zoomed region from panel (a), showing good qualitative agreement between our simulations and the experiments.

The diffusion coefficient against the density for several isotherms. For the range of densities bracketed within the dashed lines, the particles move faster under compression for temperatures lower than 0.64. This is the opposite behavior which one expects for normal fluids. Dashed lines are guides for the eyes connecting and .

The diffusion coefficient against the density for several isotherms. For the range of densities bracketed within the dashed lines, the particles move faster under compression for temperatures lower than 0.64. This is the opposite behavior which one expects for normal fluids. Dashed lines are guides for the eyes connecting and .

The translational order parameter as a function of density. While for normal fluids compression leads to increase the translational order parameter, for the model of Eq. (1) this is the case only for high temperatures . Dashed lines bound the region where behaves anomalously.

The translational order parameter as a function of density. While for normal fluids compression leads to increase the translational order parameter, for the model of Eq. (1) this is the case only for high temperatures . Dashed lines bound the region where behaves anomalously.

The orientational order parameter against density. We observe that has a maximum at , meaning that decreases under compression for some range of densities. The maxima lie between the extrema points of the translational order parameter . Dashed line marks the location of maximum .

The orientational order parameter against density. We observe that has a maximum at , meaning that decreases under compression for some range of densities. The maxima lie between the extrema points of the translational order parameter . Dashed line marks the location of maximum .

Temperature-density plane containing all the anomalies found for the CSW potential. The TMD line bounds the innermost region with the density anomaly behavior. This region is surrounded by the extrema lines that encompass the region with diffusion anomaly. The outmost anomalous region, including the first two, is between curves B and A, where the system exhibits an anomalous behavior in structure as shown by the order parameters and . The curve C marks the maxima in occurring where has a normal behavior.

Temperature-density plane containing all the anomalies found for the CSW potential. The TMD line bounds the innermost region with the density anomaly behavior. This region is surrounded by the extrema lines that encompass the region with diffusion anomaly. The outmost anomalous region, including the first two, is between curves B and A, where the system exhibits an anomalous behavior in structure as shown by the order parameters and . The curve C marks the maxima in occurring where has a normal behavior.

The plane or order map. The arrows indicate the direction of increasing density. Each line correspond to an isotherm and from top to bottom they are , 0.55, 0.60, 0.62, 0.65, 0.70, 0.75, 0.80, 0.90, 1.0, 1.3, 1.5, 1.7, and 1.8. By increasing the density, at low both order parameters increase (normal fluid behavior), then at intermediate they both decrease (structural anomaly region), then at higher the orientational decreases, while the translational increases. As in the case of SPC/E water, silica, and other two-scale potentials the region with high and low is inaccessible. The inaccessible region is limited by a straight line with and .

The plane or order map. The arrows indicate the direction of increasing density. Each line correspond to an isotherm and from top to bottom they are , 0.55, 0.60, 0.62, 0.65, 0.70, 0.75, 0.80, 0.90, 1.0, 1.3, 1.5, 1.7, and 1.8. By increasing the density, at low both order parameters increase (normal fluid behavior), then at intermediate they both decrease (structural anomaly region), then at higher the orientational decreases, while the translational increases. As in the case of SPC/E water, silica, and other two-scale potentials the region with high and low is inaccessible. The inaccessible region is limited by a straight line with and .

Pressure-temperature phase diagram merging all the results found for the CSW potential. The meaning of the lines is described in the legend, where DE stands for diffusivity extrema, LL for liquid-liquid, and LG for liquid-gas. See text for more details.

Pressure-temperature phase diagram merging all the results found for the CSW potential. The meaning of the lines is described in the legend, where DE stands for diffusivity extrema, LL for liquid-liquid, and LG for liquid-gas. See text for more details.

(a) Pair contribution of excess entropy for the DSW potential (dashed line in Fig. 1) against density at constant . Circles are simulated data and lines are fifth order polynomial fit from data. (b) is shown for DSW potential. Panels (c) and (d) show the results for the CSW model. Horizontal lines mark the threshold value for anomaly in density , diffusion , and structure, as explained in the text. Solid, dotted, dashed, and dotted-dashed lines correspond to temperatures , 0.60, 0.75, and 0.90 in all panels.

(a) Pair contribution of excess entropy for the DSW potential (dashed line in Fig. 1) against density at constant . Circles are simulated data and lines are fifth order polynomial fit from data. (b) is shown for DSW potential. Panels (c) and (d) show the results for the CSW model. Horizontal lines mark the threshold value for anomaly in density , diffusion , and structure, as explained in the text. Solid, dotted, dashed, and dotted-dashed lines correspond to temperatures , 0.60, 0.75, and 0.90 in all panels.

## Tables

Critical temperatures and , pressures and , and densities and , for the gas-liquid critical point and the liquid-liquid critical point , for the CSW potential. The quantities are expressed in dimensionless units.

Critical temperatures and , pressures and , and densities and , for the gas-liquid critical point and the liquid-liquid critical point , for the CSW potential. The quantities are expressed in dimensionless units.

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