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Updated: Jul 3, 2026

Impacts of Free-falling Spheres on a Deep Liquid Pool with Altered Fluid and Impactor Surface Conditions
Published on: February 17, 2019
Penetrable square-well fluids: exact results in one dimension
Andrés Santos1, Riccardo Fantoni, Achille Giacometti
1Departamento de Física, Universidad de Extremadura, E-06071 Badajoz, Spain. andres@unex.es
We present a new model for attractive penetrable spheres, analyzing its properties in one dimension. This model provides accurate benchmarks for testing theories of star polymers in dilute solutions.
Area of Science:
- Statistical Mechanics
- Soft Matter Physics
Background:
- Attractive penetrable sphere models are crucial for understanding star polymers.
- Existing models require validation, especially under dilute conditions.
Purpose of the Study:
- Introduce and analyze a novel model of attractive penetrable spheres.
- Provide exact solutions for low-density expansions to benchmark approximate theories.
- Investigate thermodynamic stability and the conditions for a well-defined thermodynamic limit.
Main Methods:
- Developed a one-dimensional model of attractive penetrable spheres with a square-well potential.
- Derived exact coefficients for low-density expansions of the radial distribution function and virial expansion.
- Utilized the exact impenetrable counterpart as a reference point.
- Analyzed limiting cases, including the sticky penetrable-sphere model.
Main Results:
- Derived exact second-order coefficients for the radial distribution function and fourth-order for the virial expansion.
- Validated the accuracy of Percus-Yevick and hypernetted chain theories using derived benchmarks.
- Provided a complete solution for the sticky penetrable-sphere model in one dimension.
- Identified the region for a well-defined thermodynamic limit and analyzed Ruelle's stability criterion.
Conclusions:
- The proposed model offers precise predictions for star polymer behavior in dilute good and moderate solvents.
- Exact low-density results serve as reliable benchmarks for theoretical approximations.
- The study clarifies thermodynamic stability conditions for such systems in one dimension.
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