9.3 Electrolyte Solution Surfaces
229
Validity of χ (2) ≈ N · α (2) : In the above paragraph we argued that the positive
perturbation of Im[χ (2) ] is indicative of the surface preference of I − more than
Na + . When we evaluate the perturbation on the spectral amplitude quantitatively,
further insight is obtained into the mechanism of SFG spectra for electrolyte solution
surfaces. The amplitude of χ (2) is conventionally presented with Eq. (3.41),
χ
(2)
pqr ≈
N
l=1
α
(2)
l,pqr = N · α
(2)
pqr ,
(3.41)
on the basis of an assumption that the nonlinear susceptibility χ (2) is the sum of
hyperpolarizabilities α (2) of constituent molecules. Equation (3.41) implies that
χ (2) is governed by two factors, number density N and orientational order α (2) .
The electric double layer orients the water molecules, as illustrated in Fig. 9.5, and
thus enhances the orientational average α
(2)
pqr . This interpretation is qualitatively in
accord with the above picture of electric double layer.
However, the actual enhancement of the χ (2) amplitude is significantly smaller
than that predicted by Eq. (3.41) in the SSP polarization. χ (2) is expressed by
Eq. (5.27),
χ
(2),res
pqr ((, ω 1 , ω 2 ) =
iω 2
k B T
∞
0
dt
A eff, pq (t)M r (0)
exp(iω 2 t),
(5.27)
where
A eff,pq (t) =
N
l=1
α eff,pq (l, t) M r (0) =
N
m=1
μ eff,r (m, 0).
(5.28)
On the other hand, Eq. (3.41) represents the χ (2) as the sum of molecular hyperpolarizabilities α
(2)
l , and accordingly corresponds to
χ
(2),res
pqr ((, ω 1 , ω 2 ) ≈
N
l=1
α
(2)
l,pqr =
N
l=1
iω 2
k B T
∞
0
dt
α eff,pq (l, t)μ eff,r (l, 0)
exp(iω 2 t)
(5.27 )
in the form of time correlation function. We notice that Eq. (5.27 ) includes only the
self correlation (l = m) of Eq. (5.27). Equation (5.27 ) should be regarded as an
approximation of Eq. (5.27) by omitting the intermolecular correlation (l = m).
The calculated Im[χ (2) ] spectra of NaI solution by Eqs. (5.27) and (5.27 ) are
compared in panels (b) and (b ) of Fig. 9.4, respectively. The perturbation of NaI
(red lines) from pure water (black) exhibits noticeable differences in the two panels.
The self correlation of Im[χ (2) ] spectrum in panel (b ) exhibits a remarkable positive
amplitude (red line), because of the orientational order of water molecules induced
229
Validity of χ (2) ≈ N · α (2) : In the above paragraph we argued that the positive
perturbation of Im[χ (2) ] is indicative of the surface preference of I − more than
Na + . When we evaluate the perturbation on the spectral amplitude quantitatively,
further insight is obtained into the mechanism of SFG spectra for electrolyte solution
surfaces. The amplitude of χ (2) is conventionally presented with Eq. (3.41),
χ
(2)
pqr ≈
N
l=1
α
(2)
l,pqr = N · α
(2)
pqr ,
(3.41)
on the basis of an assumption that the nonlinear susceptibility χ (2) is the sum of
hyperpolarizabilities α (2) of constituent molecules. Equation (3.41) implies that
χ (2) is governed by two factors, number density N and orientational order α (2) .
The electric double layer orients the water molecules, as illustrated in Fig. 9.5, and
thus enhances the orientational average α
(2)
pqr . This interpretation is qualitatively in
accord with the above picture of electric double layer.
However, the actual enhancement of the χ (2) amplitude is significantly smaller
than that predicted by Eq. (3.41) in the SSP polarization. χ (2) is expressed by
Eq. (5.27),
χ
(2),res
pqr ((, ω 1 , ω 2 ) =
iω 2
k B T
∞
0
dt
A eff, pq (t)M r (0)
exp(iω 2 t),
(5.27)
where
A eff,pq (t) =
N
l=1
α eff,pq (l, t) M r (0) =
N
m=1
μ eff,r (m, 0).
(5.28)
On the other hand, Eq. (3.41) represents the χ (2) as the sum of molecular hyperpolarizabilities α
(2)
l , and accordingly corresponds to
χ
(2),res
pqr ((, ω 1 , ω 2 ) ≈
N
l=1
α
(2)
l,pqr =
N
l=1
iω 2
k B T
∞
0
dt
α eff,pq (l, t)μ eff,r (l, 0)
exp(iω 2 t)
(5.27 )
in the form of time correlation function. We notice that Eq. (5.27 ) includes only the
self correlation (l = m) of Eq. (5.27). Equation (5.27 ) should be regarded as an
approximation of Eq. (5.27) by omitting the intermolecular correlation (l = m).
The calculated Im[χ (2) ] spectra of NaI solution by Eqs. (5.27) and (5.27 ) are
compared in panels (b) and (b ) of Fig. 9.4, respectively. The perturbation of NaI
(red lines) from pure water (black) exhibits noticeable differences in the two panels.
The self correlation of Im[χ (2) ] spectrum in panel (b ) exhibits a remarkable positive
amplitude (red line), because of the orientational order of water molecules induced