corresponds to the interaction between two planes. As illustrated in Fig. 3, for any
possible combination of sphere size and separation distance, the value of s
à lies
within the range [0, 1]. A value of s
à close to unity indicates that the system is close
to the geometric limit of two point particles, and a value of s
à close to zero indicates
that it is close to the limit of two planes.
4 Graphical Representation of the Scaled
Surface-to-Surface Separation
Figure 4 illustrates how the scaled surface-to-surface separation s
à depends on a 1 ′
and a 2 ′. As shown in Fig. 4a, each value of s
à for 0\s
Ã
\1 does not correspond to a
unique geometry, but rather to a range of possible combinations of a 1 ′ and a 2 ′. If,
for example, a 2 ′ increases while a 1 ′ remains unchanged, the value of s
à would
decrease. To return to the original value of s
à , one can move in the direction of
decreasing a 1 ′ until the contour line of the original value of s
à is reached. Figure 4b
is a ln-ln plot of the same contour map, which illustrates a difference in the
dependence of s
à on a 1 ′ and a 2 ′ between cases of s
Ã
\0:5 and cases of s
Ã
[ 0:5.
Consider the regime of s
Ã
\0:5: At any given value of s
à , if a 1 ′ increases indefinitely, a 2 ′ will decrease towards a finite value. If a 2 ′ decreases towards zero while
a 1 ′ increases indefinitely, the value of s
à will instead approach 0.5 which describes,
among many others, a point-plane geometry. But if a 2 ′ remains unchanged for
increasing a 1 ′, the value of s
à will decrease towards a finite value, which describes a
particular range of sphere-plane geometries.
Fig. 4 Contour maps of the scaled surface-to-surface separation s
à s=2a, showing its
dependence on a 1 ′ and a 2 ′ in (a), and on ln(a 1 ′) and ln(a 2 ′) in (b). The dimensionless parameter
s
à approaches unity if the system is close to the geometric limit of two point particles at
a 1 ′ = a 2 ′ = 0. The values of s
à ranging between 0 and 1 correspond to particular combinations of
a 1 ′ and a 2 ′. The ln-ln plot in (b) illustrates a difference in the behaviour of s
à between cases of
s
à \0:5 and cases of s
à [ 0:5
A General Geometric Representation of Sphere-Sphere Interactions
35
possible combination of sphere size and separation distance, the value of s
à lies
within the range [0, 1]. A value of s
à close to unity indicates that the system is close
to the geometric limit of two point particles, and a value of s
à close to zero indicates
that it is close to the limit of two planes.
4 Graphical Representation of the Scaled
Surface-to-Surface Separation
Figure 4 illustrates how the scaled surface-to-surface separation s
à depends on a 1 ′
and a 2 ′. As shown in Fig. 4a, each value of s
à for 0\s
Ã
\1 does not correspond to a
unique geometry, but rather to a range of possible combinations of a 1 ′ and a 2 ′. If,
for example, a 2 ′ increases while a 1 ′ remains unchanged, the value of s
à would
decrease. To return to the original value of s
à , one can move in the direction of
decreasing a 1 ′ until the contour line of the original value of s
à is reached. Figure 4b
is a ln-ln plot of the same contour map, which illustrates a difference in the
dependence of s
à on a 1 ′ and a 2 ′ between cases of s
Ã
\0:5 and cases of s
Ã
[ 0:5.
Consider the regime of s
Ã
\0:5: At any given value of s
à , if a 1 ′ increases indefinitely, a 2 ′ will decrease towards a finite value. If a 2 ′ decreases towards zero while
a 1 ′ increases indefinitely, the value of s
à will instead approach 0.5 which describes,
among many others, a point-plane geometry. But if a 2 ′ remains unchanged for
increasing a 1 ′, the value of s
à will decrease towards a finite value, which describes a
particular range of sphere-plane geometries.
Fig. 4 Contour maps of the scaled surface-to-surface separation s
à s=2a, showing its
dependence on a 1 ′ and a 2 ′ in (a), and on ln(a 1 ′) and ln(a 2 ′) in (b). The dimensionless parameter
s
à approaches unity if the system is close to the geometric limit of two point particles at
a 1 ′ = a 2 ′ = 0. The values of s
à ranging between 0 and 1 correspond to particular combinations of
a 1 ′ and a 2 ′. The ln-ln plot in (b) illustrates a difference in the behaviour of s
à between cases of
s
à \0:5 and cases of s
à [ 0:5
A General Geometric Representation of Sphere-Sphere Interactions
35