The distribution, g KO (r), of water around K+ in the presence (aq) and absence (g) of the external medium for the cluster. The radius of the coordination sphere λ = 3.7 Å. The pair correlation in the presence of the medium shows enhanced structure relative to that for the cluster in the absence of the bulk. This suggests that the external medium promotes a better packing of solvent around K+. (Inset) The distribution, , of the farthest water molecule from K+. Observe that in the absence of the bulk medium, this water molecule is closer to the boundary of the cluster.
The average solvent-solvent interaction energy, ⟨ε w − w ⟩, of ion-water clusters for different coordination states. For curves denoted by “aq” (filled symbols), ⟨ε w − w ⟩ is obtained by extracting the cluster from simulations in the presence of the bulk medium. Curves denoted by “g” (open symbols) correspond to clusters simulated in the absence of a bulk. The coordination radius λ = 3.7 Å. For higher coordination states, the average excess internal energy is lower in the case of a cluster without a bulk medium outside.
The local contribution, ξ(n), to the free energy of forming an ion plus n-water molecule cluster. The filled symbols, ξaq(n), are results in the presence of the bulk (aq) and the unfilled symbols, ξg(n), in its absence (g). The radius of the coordination sphere is 3.7 Å. For n = 1, 2, 3, ξaq(n) is insensitive to the presence of the bulk medium outside the coordination shell, whereas for n = 4, 5, 6, ξaq(n) is more favorable with the medium than without.
(Left) The local contribution, ξaq(n), of forming Na+ plus n-water clusters. (Right) The long-range contribution, , to the free energy of forming Na+ plus n-water clusters. Note that increasing the radius λ of the coordination sphere does not greatly affect ξaq(n) for n ⩽ 3 but increases it for n = 4, 5, 6. On the same scale, the response of the medium is more pronounced for the same change in λ and it tends to become insensitive to the presence of the coordinating solvents for large λ.
A maximum term approximation to the local chemical contribution (Eq. (3)), ln x 0 ≈ βW n − ln p n /p 0. For K+, neglecting the bulk medium in the local ion-water clustering reaction leads to the identification of the n = 3 coordination state as the optimal (or most probable) coordination state. Account for the medium indicates that n = 7 is the most probable coordination state (cf., Fig. 7 in Ref. 42). For Na+, in the presence (absence) of the medium n = 7 (n = 4) states are indicated to be optimal (cf., Fig. 3 in Ref. 42).
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