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 M.K. thanks P. Mironescu for communicating his unpublished results and P. Mironescu and E.A. Carlen for interesting discussions.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. 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.

UR - http://www.scopus.com/inward/record.url?scp=77953244464&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953244464&partnerID=8YFLogxK

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 -