16
2 A New View on Mechanism of Functional Expression …
another large sphere or a planar wall, fortuitous cancellation of errors occurs in the
calculation of Φ Wall (0) using the AO theory, leading to the almost exact value of
Φ Wall (0) [11]. (Such cancellation does not occur for the contact of solutes with more
complex geometric properties [11]).
Here, we discuss the physical origin of the oscillatory behavior of the entropic
potential Φ Wall (h), the solid line shown in Fig. 2.6b [20]. When h is close to nd S (n
= 1, 2, …), Φ Wall (h) takes a negative, local-minimum value. We previously showed
that as d L increases, the local-minimum positions become closer to nd S . On the other
hand, the surface separation h which is not close to nd S (n = 1, 2, …) is entropically
unfavorable with the result of positive Φ Wall (h). These results can be rationalized
as follows [20]. Importantly, the presence of a small sphere also generates an EV
for the other small spheres. In this sense, all the small spheres in the system are
entropically correlated and this entropic correlation is referred to as the “crowding
of small spheres”. The entropy of small spheres decreases as the crowding becomes
more significant. The small spheres are driven to be packed within the space confined
between the large sphere and the wall for increasing the overlap of EVs generated by
the large sphere, wall, and small spheres, increasing the total volume available for the
translational displacement of small spheres in the system, and reducing the crowding
of small spheres. The small spheres within the confined domain are entropically
unfavorable, but the effect of the reduced crowding dominates. Close packing of
the small spheres within the confined domain is achieved for h ~ nd S . Hence, the
entropy of small spheres for h ~ nd S becomes higher than that for h → ∞, leading
to a negative, local-minimum value of Φ Wall (h). When h ~ nd S does not hold, it is
clear from simple geometric consideration that a significant amount of void space is
inevitably formed within the confined domain. This formation causes a decrease in
the total volume available for the translational displacement of small spheres in the
system. As a consequence, the entropy of small spheres becomes lower than that for
h → ∞, giving rise to positive Φ Wall (h).
When the small neutral hard spheres are replaced by water molecules, the energetic
factor as well as the entropic one comes into play. For the energetic factor, the
induced interaction is considerably more influenced by the properties of large-sphere
and wall surfaces. The mean force and the PMF then exhibit downward shifts with
smaller amplitudes of the oscillatory curves [21]. This can readily be understood
because the large-sphere and wall surfaces, which cannot form hydrogen bonds with
water molecules, become highly unfavorable, driving their contact more strongly for
reducing the area of the surfaces exposed to water. Suppose that the large neutral hard
sphere and the neutral hard wall are replaced by a large sphere and a wall possessing
the surfaces comprising atoms with positive and negative partial charges, respectively.
Due to the surface-water electrostatic attractive interactions, the large-sphere and
wall surfaces become energetically more affinitive for water. This factor hinders
their contact because a larger area of the surfaces exposed to water is more favored.
The mean force and the PMF then exhibit upward shifts with larger amplitudes of
the oscillatory curves [21]. Consequently, the resultant mean force and PMF look
2 A New View on Mechanism of Functional Expression …
another large sphere or a planar wall, fortuitous cancellation of errors occurs in the
calculation of Φ Wall (0) using the AO theory, leading to the almost exact value of
Φ Wall (0) [11]. (Such cancellation does not occur for the contact of solutes with more
complex geometric properties [11]).
Here, we discuss the physical origin of the oscillatory behavior of the entropic
potential Φ Wall (h), the solid line shown in Fig. 2.6b [20]. When h is close to nd S (n
= 1, 2, …), Φ Wall (h) takes a negative, local-minimum value. We previously showed
that as d L increases, the local-minimum positions become closer to nd S . On the other
hand, the surface separation h which is not close to nd S (n = 1, 2, …) is entropically
unfavorable with the result of positive Φ Wall (h). These results can be rationalized
as follows [20]. Importantly, the presence of a small sphere also generates an EV
for the other small spheres. In this sense, all the small spheres in the system are
entropically correlated and this entropic correlation is referred to as the “crowding
of small spheres”. The entropy of small spheres decreases as the crowding becomes
more significant. The small spheres are driven to be packed within the space confined
between the large sphere and the wall for increasing the overlap of EVs generated by
the large sphere, wall, and small spheres, increasing the total volume available for the
translational displacement of small spheres in the system, and reducing the crowding
of small spheres. The small spheres within the confined domain are entropically
unfavorable, but the effect of the reduced crowding dominates. Close packing of
the small spheres within the confined domain is achieved for h ~ nd S . Hence, the
entropy of small spheres for h ~ nd S becomes higher than that for h → ∞, leading
to a negative, local-minimum value of Φ Wall (h). When h ~ nd S does not hold, it is
clear from simple geometric consideration that a significant amount of void space is
inevitably formed within the confined domain. This formation causes a decrease in
the total volume available for the translational displacement of small spheres in the
system. As a consequence, the entropy of small spheres becomes lower than that for
h → ∞, giving rise to positive Φ Wall (h).
When the small neutral hard spheres are replaced by water molecules, the energetic
factor as well as the entropic one comes into play. For the energetic factor, the
induced interaction is considerably more influenced by the properties of large-sphere
and wall surfaces. The mean force and the PMF then exhibit downward shifts with
smaller amplitudes of the oscillatory curves [21]. This can readily be understood
because the large-sphere and wall surfaces, which cannot form hydrogen bonds with
water molecules, become highly unfavorable, driving their contact more strongly for
reducing the area of the surfaces exposed to water. Suppose that the large neutral hard
sphere and the neutral hard wall are replaced by a large sphere and a wall possessing
the surfaces comprising atoms with positive and negative partial charges, respectively.
Due to the surface-water electrostatic attractive interactions, the large-sphere and
wall surfaces become energetically more affinitive for water. This factor hinders
their contact because a larger area of the surfaces exposed to water is more favored.
The mean force and the PMF then exhibit upward shifts with larger amplitudes of
the oscillatory curves [21]. Consequently, the resultant mean force and PMF look