TY - JOUR

T1 - Hard-sphere fluids with chemical self-potentials

AU - Kiessling, M. K.H.

AU - Percus, J. K.

N1 - Funding Information:
This work was initially supported through Grant No. NAG3-1414 to J.K.P. from NASA, and subsequently by NSF Grant No. DMS 96-23220 to M.K.; it was finalized while M.K. was supported by the NSF through Grant No. DMS 08-07705. M.K. thanks P. Mironescu for communicating his unpublished results and P. Mironescu and E.A. Carlen for interesting discussions. The authors thank the anonymous referee for pointing out Refs. 1 and 2 and for comments which prompted us to improve our introduction.

PY - 2010/1

Y1 - 2010/1

N2 - The existence, uniqueness, and stability of solutions are studied for a set of nonlinear fixed point equations which define self-consistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container Λ- R3 and in contact with either a heat and a matter reservoir, or just a heat reservoir. The local thermodynamics is furnished by the statistical mechanics of a system of hard balls, in the approximation of Carnahan-Starling. The fluid's local chemical potential per particle at r{small element of}Λ is the sum of the matter reservoir's contribution and a self-contribution -(V*ρ)(r), where ρ is the fluid density function and V a non-negative linear combination of the Newton kernel VN({pipe}r{pipe})=- {pipe}r{pipe}-1, the Yukawa kernel VY({pipe}r{pipe})=- {pipe}r{pipe}-1e-κ{p ipe}r{pipe}, and a van der Waals kernel VW({pipe}r{pipe})=- (1+κ{script}2{p ipe}r{pipe}2)-3. The fixed point equations involving the Yukawa and Newton kernels are equivalent to semilinear elliptic partial differential equations (PDEs) of second order with a nonlinear, nonlocal boundary condition. We prove the existence of a grand canonical phase transition and of a petit canonical phase transition which is embedded in the former. The proofs suggest that, except for boundary layers, the grand canonical transition is of the type "all gas↔all liquid" while the petit canonical one is of the type "all vapor↔liquid drop with vapor atmosphere." The latter proof, in particular, suggests the existence of solutions with interface structure which compromise between the all-liquid and all-gas density solutions.

AB - The existence, uniqueness, and stability of solutions are studied for a set of nonlinear fixed point equations which define self-consistent hydrostatic equilibria of a classical continuum fluid that is confined inside a container Λ- R3 and in contact with either a heat and a matter reservoir, or just a heat reservoir. The local thermodynamics is furnished by the statistical mechanics of a system of hard balls, in the approximation of Carnahan-Starling. The fluid's local chemical potential per particle at r{small element of}Λ is the sum of the matter reservoir's contribution and a self-contribution -(V*ρ)(r), where ρ is the fluid density function and V a non-negative linear combination of the Newton kernel VN({pipe}r{pipe})=- {pipe}r{pipe}-1, the Yukawa kernel VY({pipe}r{pipe})=- {pipe}r{pipe}-1e-κ{p ipe}r{pipe}, and a van der Waals kernel VW({pipe}r{pipe})=- (1+κ{script}2{p ipe}r{pipe}2)-3. The fixed point equations involving the Yukawa and Newton kernels are equivalent to semilinear elliptic partial differential equations (PDEs) of second order with a nonlinear, nonlocal boundary condition. We prove the existence of a grand canonical phase transition and of a petit canonical phase transition which is embedded in the former. The proofs suggest that, except for boundary layers, the grand canonical transition is of the type "all gas↔all liquid" while the petit canonical one is of the type "all vapor↔liquid drop with vapor atmosphere." The latter proof, in particular, suggests the existence of solutions with interface structure which compromise between the all-liquid and all-gas density solutions.

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U2 - 10.1063/1.3279598

DO - 10.1063/1.3279598

M3 - Article

AN - SCOPUS:77953244464

VL - 51

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 1

M1 - 013091JMP

ER -