Now consider the regime of s
Ã
[ 0:5. At any given value of s
à , if a 1 ′ decreases
towards zero, a 2 ′ will increase towards a finite value. If a 2 ′ increases indefinitely
while a 1 ′ decreases towards zero, the value of s
à will approach 0.5, again for a
point-plane geometry. But if a 2 ′ remains unchanged while a 1 ′ decreases, the value
of s
à will increase towards a limit which is less than unity, because in this case the
parameter s
à describes only a particular range of point-sphere geometries but not the
geometric limit of a pair of point particles.
5 Conclusions
A dimensionless, scaled surface-to-surface separation distance s
Ã
2 [0, 1] has been
derived from the bispherical coordinate system to describe geometries of spheresphere interactions. It serves as a measure of how close a system of interacting
spheres is to the geometric limit of two point particles or two planes. A value close
to unity indicates that the system is close to the limit of two point particles, and a
value close to zero indicates that the system is close to that of two planes. This
approach applies to all possible combinations of sphere size and separation distance, including ambiguous cases where a description of the interacting bodies as
spheres becomes questionable.
Acknowledgments EB gratefully acknowledges an ERC Consolidator Grant for financial support. EBL is supported by a PhD scholarship from the Brazilian Government’s Science Without
Borders programme (CAPES: 0702/13-7).
References
1. Bichoutskaia E, Boatwright AL, Khachatourian A, Stace AJ (2010) J Chem Phys 133:024105
2. Khachatourian A, Chan HK, Stace AJ, Bichoutskaia E (2014) J Chem Phys 140:074107
3. Ohshima H (1998) J Colloid Interface Sci 198:42
4. Matsuyama T, Yamamoto H, Washizu M (1995) J Electrost 36:195
5. Mahanty J, Michalewicz MT (1987) Aust J Phys 40:413
6. Raggi G, Stace AJ, Bichoutskaia E (2013) Phys Chem Chem Phys 15:20115
7. Messina R (2002) J Chem Phys 117:11062
8. Arfken G (1970) Mathematical Methods for Physicists, 2nd edn. Academic Press, Orlando,
pp 115–117
9. Morse PM, Feshbach H (1953) Methods of theoretical physics, part II. McGraw-Hill, New
York, p 1298
10. Munirov VR, Filippov AV (2013) J Exp Theor Phys 117:809
36
H.-K. Chan et al.
Ã
[ 0:5. At any given value of s
à , if a 1 ′ decreases
towards zero, a 2 ′ will increase towards a finite value. If a 2 ′ increases indefinitely
while a 1 ′ decreases towards zero, the value of s
à will approach 0.5, again for a
point-plane geometry. But if a 2 ′ remains unchanged while a 1 ′ decreases, the value
of s
à will increase towards a limit which is less than unity, because in this case the
parameter s
à describes only a particular range of point-sphere geometries but not the
geometric limit of a pair of point particles.
5 Conclusions
A dimensionless, scaled surface-to-surface separation distance s
Ã
2 [0, 1] has been
derived from the bispherical coordinate system to describe geometries of spheresphere interactions. It serves as a measure of how close a system of interacting
spheres is to the geometric limit of two point particles or two planes. A value close
to unity indicates that the system is close to the limit of two point particles, and a
value close to zero indicates that the system is close to that of two planes. This
approach applies to all possible combinations of sphere size and separation distance, including ambiguous cases where a description of the interacting bodies as
spheres becomes questionable.
Acknowledgments EB gratefully acknowledges an ERC Consolidator Grant for financial support. EBL is supported by a PhD scholarship from the Brazilian Government’s Science Without
Borders programme (CAPES: 0702/13-7).
References
1. Bichoutskaia E, Boatwright AL, Khachatourian A, Stace AJ (2010) J Chem Phys 133:024105
2. Khachatourian A, Chan HK, Stace AJ, Bichoutskaia E (2014) J Chem Phys 140:074107
3. Ohshima H (1998) J Colloid Interface Sci 198:42
4. Matsuyama T, Yamamoto H, Washizu M (1995) J Electrost 36:195
5. Mahanty J, Michalewicz MT (1987) Aust J Phys 40:413
6. Raggi G, Stace AJ, Bichoutskaia E (2013) Phys Chem Chem Phys 15:20115
7. Messina R (2002) J Chem Phys 117:11062
8. Arfken G (1970) Mathematical Methods for Physicists, 2nd edn. Academic Press, Orlando,
pp 115–117
9. Morse PM, Feshbach H (1953) Methods of theoretical physics, part II. McGraw-Hill, New
York, p 1298
10. Munirov VR, Filippov AV (2013) J Exp Theor Phys 117:809
36
H.-K. Chan et al.