Water-water correlations in electrolyte solutions probed by hyper-Rayleigh scattering

Second-harmonic or hyper-Rayleigh scattering (HRS) is a nonlinear light scattering technique widely used to measure the first hyperpolarizability β of molecules in solution,^1,2 and HRS has largely replaced the previous alternative electric-field-induced second harmonic generation (ESHG or EFISH) technique.³ It is usually assumed that only incoherent scattering from the individual molecules contributes to HRS. However, for electrolyte solutions, there is a coherent ESHG contribution to the HRS signal due to the orientation correlations induced by the electric field of the ions in solution.⁴ The ion-induced HRS intensity increases with ion concentration and saturates at high ionic strength due to the spatial correlations of the ions, as described by the Debye-Huckel theory.

In this model for ion-induced HRS,⁴ the radial electric field of an ion induces a radial orientation of the surrounding water molecular dipoles which decreases as r⁻² with distance r from the ion. This weak induced radial orientation of the molecules produces scattered field contributions that add coherently with a sum that increases with r until it is limited by phase differences when Kr > π, where $K$ is the magnitude of the scattering wave vector. Most of the ion-induced HRS signal comes from many weakly oriented molecules at large distances r > π/K.⁵ At low ion concentration, the ion positions are uncorrelated, so the scattered fields induced by each ion add incoherently and the ion-induced HRS intensity is proportional to ion concentration. At high ion concentration, the Debye-Huckel ion-ion spatial correlations⁶ cause destructive interference of the scattered fields induced by the individual ions and the ion-induced HRS intensity reaches a limiting value at high ion concentration. At small distances compared to the separation of the ions, the ion-field-induced orientation distribution of the water dipole vectors is radial, and at larger distances, it is a random vector field with only longitudinal spatial Fourier components.^4,7

Previous measurements of the intensity of the ion-induced HRS contribution for KCl–D₂O solutions did not quantitatively agree with the theoretical predictions of this model,⁴ and more recent measurements for several electrolyte solutions find even larger differences between theory and experiment for the ion-induced HRS intensity and its ion concentration dependence.⁸ These recent experiments also find dramatic differences between the results measured for electrolyte solutions with H₂O and D₂O.⁸ These discrepancies have motivated a number of subsequent investigations.^9,10 Quantitative understanding of ion-induced HRS is relevant to second harmonic generation (SHG) from aqueous interfaces since the bulk SHG can interfere with the surface contribution and because ion-induced HRS probes the electric field induced effects that are modeled in surface and interface studies.^11–13 

In this work, the ion-induced HRS contribution has been re-measured for solutions of KCl in H₂O and D₂O. The ionic strength dependence of ion-induced HRS is found to be nearly identical for H₂O and D₂O and is in quantitative agreement with the predictions of the Debye-Huckel theory. There is also quantitative agreement between theory and experiment for the intensity of ion-induced HRS, when the orientation correlations in pure water are taken into account. In the following, the experimental methods and results will be presented and analyzed, and an attempt will be made to reconcile the present results with the previous observations.

The HRS experimental apparatus and methods are similar to those previously employed^4,5,14–19 and are described and discussed in detail in Ref. 14. Linearly polarized pulses from an Nd:YAG (yttrium aluminium garnet) laser (operating at λ₀ = 1064 nm, 4.3 kHz repetition rate, 100 ns pulse duration) are focused to a 4.5 μm beam waist radius in the liquid sample in a standard square 10 mm fluorimeter cuvette. Scattered light at θ_s = 90° is collected and collimated by an aspheric lens (f = 13.8 mm), analyzed by a linear polarizer, focused into an optical fiber, and fiber-coupled to a 60 cm⁻¹ spectral filter [532 nm center, 2 nm transmission bandwidth, full width at half maximum (FWHM)] followed by the photon counting detector measuring the integrated intensity. The laser beam average power was P_av = 2.5 W for D₂O but was reduced to 0.75 W for H₂O due to absorption and thermal defocusing of the laser beam by the sample. The experiments were done with multi-longitudinal mode laser output (0.6 cm⁻¹ FWHM) to increase the HRS signal. Several measurements were also made using a wider spectral filter (536 nm center, 46 nm FWHM).

The water sample (H₂O, 18 MΩ cm, Millipore Milli-Q; D₂O, 99.9 at. % D, Sigma-Aldrich) was de-ionized by continuous flow in a closed loop containing a PTFE (polytetrafluoroethylene)-tube peristaltic pump, ion-exchange resin column (Dowex Monosphere MR-450 UPW), 0.2 μm PTFE filter, sample cell, conductivity cell, and reservoir, in that order. D₂O was exchanged for H₂O in the resin before loading the column used for the D₂O measurements. The column with 1 cm³ of DI resin can maintain ionic strength 0.1 μM continuously for days or weeks in the 20 cm³ circulating fluid volume. Samples with controlled larger ion concentration were obtained by adding concentrated KCl (99.98%, Baker) solution to the de-ionized circulating fluid, with the ion-exchange column by-passed. The temperature of the continuously circulating fluid was T = 25.0 °C, and the ion concentration was determined from the conductivity recorded before and after each HRS measurement, assuming that H⁺, OH⁻ (or D⁺, OD⁻), K⁺, and Cl⁻ are the only ions present.^20–22 The conductivity sensor was calibrated with 3 mM and 10 mM KCl–H₂O solutions.²² 

Contamination by ions desorbed from the walls of the loop is minimized by using the DI column to remove all ions but H⁺ and OH⁻ (or D⁺ and OD⁻), adding KCl, and completing the HRS measurement before significant contamination could occur. Ionic strength determined from the sample conductivity may have a systematic error if the desorbed ions have mobility or charge different from that of K⁺ and Cl⁻. Desorption rate about 2 μM/h was estimated from the conductivity increase of the circulating liquid after closing the DI column. Several H₂O data points in Fig. 1 at low KCl concentration have larger error bars as a result of low HRS signal for H₂O and short measurement time to avoid errors due to desorbed ions.

Scattering configurations with incident and scattered light polarized either perpendicular or parallel to the horizontal scattering plane are denoted VV, HV, VH, and HH, where V denotes vertical polarization, H denotes horizontal polarization, and the first and second letters refer to the incident and scattered light, respectively. Ratios of HRS intensities were measured (⁠$IVV/IHV$ and $IHV/IVH$⁠).¹⁴ The effect of intensity drift was cancelled by using several hundred alternate 10 s measurements of the two polarization configurations for each ratio. Rapid switching between polarization configurations was enabled using a liquid crystal variable wave plate (LCVWP) to control the laser polarization and a fast rotator¹⁵ to control the analyzing polarizer for the scattered light. The collection numerical aperture (NA = n sin θ) was controlled by a circular aperture following the collection lens, and the HRS intensity ratio at NA = 0 was obtained by extrapolating measurements in the range 0.07 < NA < 0.15 to zero collection aperture with a fit function a + b (NA)².¹⁴ HRS intensity I_VV corrected for thermal lensing²³ due to 7% (0.8%) absorption^24,25 of the input laser beam by the H₂O (D₂O) sample was obtained from the P = 0 limit of $SVV/P2$⁠, where S_VV is the HRS signal measured with laser power P. Thermal lensing has no effect on the HRS intensity ratio measurements.

The main results of this experiment are the values of $IHV/IVH$ measured for KCl–H₂O and KCl–D₂O solutions as a function of ionic strength $c=(1/2)∑iZi2ρi$⁠, where Z_i is the charge and ρ_i is the concentration of ion species i. The results of these measurements are shown in Table I and Fig. 1. The curves fit to the data points in Fig. 1 have the form

where $x=c/c0$⁠, $A=(IHV/IVH)c=0$ and $B=(IVHi)c→∞/(IVH)c=0$ = $IVH,∞i/IVH,0$⁠. The fit coefficients c₀, A, and B are given in Table II. Equation (1) is the ratio of HRS intensity I_HV, which is independent of ion concentration, and $IVH=IVH,0+IVHi$⁠, which has an ion-induced contribution $IVHi$ that saturates at high ion concentration. The reason for measuring the ratio $IHV/IVH$ is that I_HV serves as an ion-independent internal reference for the measurements of the ion concentration dependent intensity I_VH, producing more accurate results for the ion-induced HRS contribution $IVHi$⁠. Figure 1 shows that Eq. (1) is a good fit to the HRS data, and that the HRS results for H₂O and D₂O are nearly the same.

The HRS intensity I_VV and polarization ratio $IVV/IHV$ were also measured and are slightly different for H₂O and D₂O. Results for $IVV/IHV$⁠, and the result for $IVV(H2O)/IVV(D2O)$⁠, are given in Table III. The $IVV(H2O)/IVV(D2O)$ result accounts for both absorption and the thermal lensing effect shown in Fig. 2. These HRS measurements were made with a 2 nm spectral filter and de-ionized samples but are insensitive to dissolved ions. The intensities I_VV, I_HV, and I_VH for de-ionized H₂O (0.2 μM) and D₂O (0.1 μM) were also measured using a 46 nm spectral filter. The intensity measured with the wider filter is larger due to inclusion of the 660 cm⁻¹ (460 cm⁻¹) libration mode hyper-Raman band.^26–28 The measurements made using 2 nm and 46 nm filters were combined to also determine the ratios $IVV/IHV$ and $IHV/IVH$ for this hyper-Raman band, with the results given in Table III. The ratio $IHV/IVH$ < 1 indicates that this is a longitudinal collective mode.²⁹ 

Expressions for c₀ and B, quantifying the concentration dependence and intensity of ion-induced HRS, were previously derived.⁴ [In the notation of Ref. 4, $c0=Z2ρD$ and $B=(S/B)∞$⁠.] The expression for c₀ is

where K is the magnitude of the scattering wave vector, ε₀ is the vacuum permittivity, ε_s is the static relative dielectric constant,^30,k_B is the Boltzmann constant, and e is the electronic charge. The scattering wave number K for HRS at θ_s = 90° is¹⁹ 

where λ₀ is the vacuum wavelength of the incident light and n_i and n_s are refractive indices at the incident and scattered light frequencies.^25,31–33 Table IV gives data needed to evaluate Eqs. (2) and (3) and the resulting values for c₀. The difference between the experimental and theoretical results for c₀ in Tables II and IV is insignificant, −0.9 ± 1.7 μM for H₂O and 0.7 ± 0.9 μM for D₂O.

The expression for B from Ref. 4 is

where ρ_s is the number density of solvent molecules,^21,μ₀ is the permanent dipole moment,^34,β and γ are the molecular first and second hyperpolarizabilities,^35,$f(0)=εs(ε∞+2)/(ε∞+2εs)$ is the Onsager local field factor with ε_∞ being the high frequency dielectric constant,^36,$y=ρsμ02/(9ε0kBT)$ is the dimensionless dipole strength, and the isotropic average $βZXX2$ gives the HRS intensity for a pure liquid with uncorrelated randomly oriented molecules. Far from resonance or in the static limit, where the tensors $βαβγ$ and $γαβγδ$ are invariant under all permutations of the indices (Kleinman symmetry), one has^37–40 

For molecules that also have C_2v symmetry, one has

In the case that $γ⊥$ can be neglected, and $|β(3)|$= 0 so that $β⊥2/βZXX2=3$⁠, Eq. (4) becomes

which is the simplest approximate estimate for B. Evaluating Eq. (11) using the data in Table V gives B₀ which is about twice the experimentally measured value for B in Table II. Also in this approximation, $IVV/IHV=βZZZ/βXZZ$ = 9 and $IHV/IVH,0=βXZZ/βZXX$ = 1.

An improved theoretical estimate for B makes use of the hyperpolarizability tensor components given in Table VI, calculated in Ref. 35 for molecules in liquid water. This is an ab initio calculation at the Moller-Plesset MP2 level of theory, for the static hyperpolarizability of the water molecule in a local environment with a strong axial field and symmetric (model II) or asymmetric (model III) field gradient. The effect of the local environment is to induce an increment Δβ large enough to reverse the sign of β (with respect to the dipole). The non-zero octupolar $|β(3)|2$ contribution reduces the value of $β⊥2/βZXX2$⁠, and non-zero $γ⊥$ increases the last factor of Eq. (4). The values for B₁ in Table VI evaluated using the calculated hyperpolarizabilities are 1.6–3.1× the experimentally measured B value for H₂O in Table II. The discrepancy between theoretical and experimental values for B is smaller for model II. This model has a larger octupolar $|β(3)|2$ contribution and is also a better fit to the HRS data from Ref. 18 that is analyzed in Ref. 41.

The orientation correlation of molecules in the pure liquid, which enhances HRS due to the vector $β(1)$ part of β, is neglected in Eq. (4). Including the transverse mode enhancement factor C_T, the HRS intensity I_VH,0 for the pure liquid is given by^19,41

and the revised theoretical expression for $B$ is

Table VII gives the values for I_VH,0 and B₂ calculated using Eqs. (14) and (15) with the hyperpolarizabilities from Ref. 35 and C_T from Ref. 41. The values for B₂ in Table VII are 1.2–2.0× the experimentally measured B value for H₂O, and the values calculated for B₂, $IVV/IHV$⁠, and $IHV/IVH,0$ increase as $|β(3)|/|β(1)|$ decreases going from model II to model III. The values for B₂, $IVV/IHV$⁠, and $IHV/IVH,0$ obtained using model II differ from the experimental values by 23%, 5%, and 2%, respectively.

The final issue is the contribution of background signals included in the measured HRS intensity but not described by the expressions given above. Short range interactions during molecular collisions induce rapid fluctuations in the molecular hyperpolarizabilities and produce additional HRS with a broad spectrum. This collision-induced HRS accounts for the HRS observed from centro-symmetric molecules and is also present in the HRS spectra previously measured for D₂O.⁴² The D₂O HRS spectrum at T = 25 °C is the sum of two narrow Lorentzian functions (1.2 cm⁻¹ FWHM), a broad Lorentzian (10.6 cm⁻¹ FWHM) and a broad exponential function (70 cm⁻¹), where the integrated intensity of the broad components is 35% of the total HV intensity. Identifying the Lorentzian components as orientational HRS and the exponential component as collision-induced HRS, the fit functions from Ref. 42 were integrated over a 60 cm⁻¹ band to determine the relative contributions of orientational and collision-induced HRS for the present HRS measurements. The collision-induced/orientational VH HRS intensity ratio obtained is 0.148. The effect of this additional spectral component is to reduce B₂ by the factor 0.871, giving the final estimates B₃ at the bottom of Table VII. The value of B₃ for model II differs from the experimental value by 7%.

Ion-induced HRS was previously measured for KCl–D₂O solutions using nearly the same apparatus,⁴ with the result c₀ = 40 ± 8 μM, in agreement with the present result, and B = 0.667 ± 0.008, significantly lower than the present result. In those experiments, the ion-induced contribution was determined from high resolution VH HRS spectra obtained using a Fabry-Perot interferometer. The ion-induced HRS peak spectral width increases with ion concentration and reaches 100 MHz at the highest ion concentration in those experiments, which is comparable to the 750 MHz free-spectral range of the interferometer. For solutions with the highest ion concentration and largest peak width, the background contribution estimated from the fit to the data becomes sensitive to the assumed shape for the wings of the spectral peak. This is due to overlap of successive orders of the Fabry-Perot transmission function, and the systematic error introduced by this effect could account for the 20% difference between the previous and present results for B.

The results of the other recent ion-induced HRS measurements⁸ differ much more from the present work. The results from Ref. 8 for H₂O are c₀ = 55 ± 5 μM and B = 0.30 ± 0.03, and for D₂O they are c₀ = 310 ± 71 μM and B = 0.09 ± 0.03 [however $IVV(H2O)/IVV(D2O)$ = 0.8 in Fig. S4 of Ref. 8 agrees with the result 0.82 ± 0.02 in Table III]. Those experiments measured the ratio of HRS intensities for electrolyte solutions and pure solvent samples, for solutions of H₂O with 21 different electrolytes and solutions of D₂O with NaCl. A 50 nm wide spectral filter was used, and the filter pass band (500-550 nm, −540 < $Δν$ < 1270 cm⁻¹ down-shift Δν from the hyper-Rayleigh peak at 514 nm) includes both the hyper-Rayleigh band and the intermolecular libration hyper-Raman band (at 660 cm⁻¹ in H₂O and 460 cm⁻¹ in D₂O).^26–28 Including the hyper-Raman band increases I_VH,0 and reduces the measured value for B. The effect of the wide filter can be estimated from the results of the present experiment using the 46 nm wide filter (pass band −690 < $Δν$ < 920 cm⁻¹) which also includes the hyper-Raman band. The factor by which $IVH,0$ is increased and B is decreased is given by $IVH/IVH(2 nm)$ in Table III. Dividing B in Table II by this factor gives the value for B in Table III.

Accounting for the wider pass band, the results for H₂O from Ref. 8 are in fair agreement with c₀ in Table II and B in Table III, with differences 10 ± 5 μM and 0.03 ± 0.03, respectively. The results for D₂O still do not agree with differences for c₀ and B that are 4× and 9× the stated uncertainties. Excess ions due to contamination of the nominally pure D₂O solvent could explain the results for D₂O in Ref. 8. If the sample and reference are both contaminated, then a larger additional ion concentration is required for a significant change in the sample/reference intensity ratio, and the maximum change is reduced. The predicted effect of 100 μM ionic contamination would be to reduce the maximum change to 0.09, with half-maximum change at 145 μM so that the differences between predicted and observed c₀ and B would be 165 ± 71 μM and 0.00 ± 0.03, respectively. Since the ion concentration was not measured in situ, such contamination is not ruled out.

In summary, the theoretical predictions for c₀ are in very good agreement with the present experimental results for H₂O and D₂O, with no adjustable or uncertain parameters entering the theoretical calculation. The ionic strength dependence for ion-induced HRS is nearly the same for H₂O and D₂O, and the calculated results using the Debye-Huckel theory are quantitatively accurate. The agreement between experiment and theory is not as good for the ion-induced HRS intensity B, but in this case there is more uncertainty in the theoretical calculation. This calculation requires the first and second molecular hyperpolarizability tensor components and orientation correlation functions for the molecules in the liquid, and information about any additional spectral components in the region of interest. Not all the required data are available, definitive, and accurate. However, the difference between experiment and the final theoretical result B₃ with model II is small. This indicates that the present experimental results for H₂O, and also the observed differences between H₂O and D₂O, could be fit by the theory with small adjustments that are within the uncertainty of the model parameters. Not all the previous experimental results agree with the present results to within the stated uncertainties. However, the sources of the largest discrepancies have been identified and accounted for, and there are plausible explanations for the remaining apparent differences. All considered, the evidence indicates that the simple theoretical model for ion-induced HRS is correct and accurate.

This work was supported by the National Science Foundation (NSF) through Grant No. CHE-1212114.