Softness dependence of the anomalies for the continuous shouldered well potential

By molecular dynamic simulations we study a system of particles interacting through a continuous isotropic pairwise core-softened potential consisting of a repulsive shoulder and an attractive well. The model displays a phase diagram with three fluid phases: a gas-liquid critical point, a liquid-liquid critical point, and anomalies in density, diffusion, and structure. The hierarchy of the anomalies is the same as for water. Here we study in a systematic way the effect on the anomalies of varying the softness of the potential. We find that, making the soft-core steeper and more penetrable, the regions of density and diffusion anomalies contract in the T-rho plane, while the region of structural anomaly is weakly affected. Therefore, a liquid can have anomalous structural behavior without having density or diffusion anomalies. We show that, by considering as effective distances those corresponding to the maxima of the first two peaks of the radial distribution function g(r) in the high-density liquid, we can generalize to continuous two-scale potentials a criterion for the occurrence of the anomalies of density and diffusion, originally proposed for discontinuous potentials. However, we observe that the knowledge of the structural behavior within the first two coordination shells of the liquid is not enough to establish, in general, the occurrence of the anomalies. By introducing the density derivative of the cumulative order integral of the excess entropy, measuring shell by shell the amount of order in the liquid, we show that the anomalous behavior is regulated by the structural order at distances as large as the fourth coordination shell. By comparing the results for different softness of the potential, we conclude that the disappearance of the density and diffusion anomalies for the steeper potentials is due to a more structured short-range order. All these results increase our understanding on how, knowing the interaction potential, we can evaluate the possible presence of anomalies for a liquid.