1. Introduction

According to Einstein-Smoluchowski theory, scattering of light in a particle-free medium is due to fluctuations in the dielectric constant (ε), caused by the random motion of molecules. Following this theory, the volume scattering function at 90°, β(90), for a pure liquid is modeled as [1, 2]:

where $\overline{\Delta {\epsilon }^2}$ represents the fluctuations of dielectric constant within a volume of ΔV, which is small as compared to the wavelength λ and yet large enough for the laws of statistical thermodynamics to apply. The Cabannes factor $f=\frac{6+6\delta }{6-7\delta }$, where δ is the depolarization ratio due to the fluctuation in the orientation of anisotropic molecules, such as H₂O. For pure liquid, Δε (hence Δn, where n is the refractive index and ε = n ²) can be expanded as a function of two independent thermodynamic variables, density ρ and temperature T [3],

The second term in Eq. (2), the fluctuation of refractive index due to temperature, is typically < 1% of the first term, the fluctuation due to density [2, 3]. Omitting the second term in Eq. (2) and using the thermodynamic statistics, $\overline{\Delta {\rho }^2}=\frac{{\rho }^2\mathrm{kT}{\beta }_T}{\Delta V}$, Eqs. (1) and (2) lead to

where k is the Boltzmann constant, β_T is the isothermal compressibility, and T is absolute temperature.

Scattering by pure water has been measured experimentally by Morel [4, 5] at five wavelengths of 366, 405, 436, 546, and 578 nm with a relative experimental error of 2% (his results are listed in row 1, Table 1). Comparing to the earlier experimental determinations, the measurements by Morel gave the smallest values (Table 1 of [1]). For the theoretical calculation of molecular scattering by pure water, various representations of ρ∂n ²/∂ρ have been tried but none of them agreed with the experiment [1, 2]. Kerker [6] compared the scattering measurements for a variety of liquids with the corresponding estimates calculated using five different expressions of ρ∂n ²/∂ρ, either derived theoretically or empirically, and concluded that none of them gave satisfactory agreement for all liquids. As an example, the estimates of β(90) using the density derivative that are deduced from two classical equations are shown in Table 1 (row 2 for the Lorentz-Lorenz equation and row 3 for the Laplace equation), each with an average difference of 32% and -18%, respectively.

Table 1. β(90) measured by Morel (row 1) and its comparison with various theoretical calculations with δ = 0.039 at 20U°C. In the last column, the numbers in the parenthesis are the results if δ = 0.051 or 0.09 were used in the calculations.

View Table

Because of these discrepancy, the density derivative was replaced with the pressure derivative, i.e., ${\left(\rho \frac{\partial n^2}{\partial \rho }\right)}_T=\frac{2n}{{\beta }_T}{\left(\frac{\partial n}{\partial P}\right)}_T$, which can be measured relatively easier. Alternatively,

Equation (4) has been used by Morel [1], Shifrin [2], and Buiteveld et al. [7] to estimate scattering by pure water. Twardowski et al [8] reviewed these early studies and recommended the use of the estimate by Buiteveld et al [7], which was based on the most recent experimental results and showed a better agreement with the measurements by Morel [1]. Their estimate was also used by Morel et al. [9] in their recent study of Southern Pacific Ocean. The results by Buiteveld et al, estimated as having a relative error of 6%, are shown in row 4 of Table 1. The percentage difference between Morel’s measurement and their estimates is ~3.2%. Note, Buiteveld et al used a value of 0.051 for δ, which would give a percentage difference of ~5.9%, and in Table 1 we used a value of 0.039, which has led to a better agreement.

Typical values of (∂ n/∂P)_T for water are ~1.5×10^-10 Pa^-1 varying about ±10% with the temperature. Given that the typical experimental precision in measuring n is on the order of 10^-5, ΔP needs to span at least ~ 1 Atm, which we believe is beyond the microscopic fluctuation range for pressure. Among the few experiments determining this pressure derivative, ΔP ranged from 1.5 Atm [10, 11] to ~100 Atm [12]. Austin and Halikas [13] commented that among all the measurements, n(P) is of the least quality as compared to those of n(T), n(λ), and n(S) (S is salinity), and it is well known that a derivative such as ∂n/∂P is very sensitive to the errors in the function of n.

In the meantime, several theoretical relationships between refractive index and density for liquid water have been developed and verified with experimental measurements [14–16]. Given the inherent uncertainties associated with determining (∂n/∂P)_T experimentally, it is of great interest to re-evaluating the scattering of pure water directly from its underlying physics: density fluctuation of the refractive index.

2. Density fluctuation of the refractive index

From the spherical cavity model, Proutiere et al. [16] derived,

where ε₀ is the dielectric constant for vacuum, N the number of molecules per unit volume, α the molecular polarizability, ν the volume of the molecular cavity, and the bar indicates a mean value within ΔV. The authors referred Eq. (5) as the generalized formulation for the Lorentz-Lorenz equation. Assuming that N̄ ≅ 1/ν̄ and n ≅ n̄, the classic Lorentz-Lorenz equation,

can be obtained, where ρ = NM/N_A, M is the molar mass and N_A the Avogadro number. Proutiere et al. [16] pointed out that since the classic equations (such as Lorentz-Lorenz or Laplace) are only approximations, they may fail to give accurate values for the derivatives, such as ρ∂n ²/∂ρ. By imposing the same approximations but only after differentiating n ² in Eq. (5), they derived,

The comparison with the measurements of 75 different liquids including water showed that Eq. (7) has an average error of 4.1% [16]. Note, Proutiere et al. [16] only derived the isobaric term, ρ(∂n ²/∂ρ)_P, but from their theory, it can be shown that the isothermal term is equal to the isobaric term, i.e., ρ(∂n ²/∂ρ)_T = ρ(∂n ²/∂ρ)_P [17]. We will refer Eq. (7) as PMH model.

Based on Van der Waals’s equation of state and assuming a liquid be represented by uniformly distributed, statistically permanent pairs of spherical molecules, Niedrich [14] deduced the general formulae for both ρ(∂n ²/∂ρ)_T and ρ(∂n ²/∂ρ)_P. Experimental verification with 15 polar and nonpolar liquids showed that his models have an average error of 3%. Note, in PMH model, the isothermal density derivative ρ(∂n ²/∂ρ)_T is equal to the isobaric density derivative ρ(∂n ²/∂ρ)_P, but in his study, Niedrich showed that ρ(∂n ²/∂ρ)_T > ρ(∂n ²/∂ρ)_P. However they differ with each other at most by 2%. For water, the isothermal density derivative is,

We will refer Eq. (8) as Niedrich model.

By examining the data of Tilton and Taylor of the refractive of index of pure water [18], which, with its 7 significant decimal digits, remains by far the most accurate measurement, Eisenberg [15] found that the right hand side of Eq. (6), which was supposed to be independent of T and P, actually decreases continuously with the increase in temperature of water. This finding further confirmed that the Lorentz-Lorenz equation cannot be used directly in calculating water scattering. He proposed an improved Lorentz-Lorenz equation,

where A, B, and C are coefficients dependent on the wavelength only. Equation (9) is valid for water between 0° and 60°C at any wavelength in the visible and agrees with the Tilton and Taylor’s measurements [18] to the 7^th decimal digit. Both coefficients B and C, though derived empirically, have thermodynamic meaning and are related to the thermal expansion coefficient at constant pressure. From Eq. (9), we have,

Except for a correction parameter B, Eq. (10) is the same as that derived directly from the Lorentz-Lorenz equation (row 2, Table 1). From the same relationship as Eq. (9), Jonasz and Fournier [Eq. 2.68 in 19], however, derived a different representation of ρ∂n ²/∂ρ = B(n ² -1), which is in the form similar to the Laplace equation (row 3, Table 1). Equation (9) is based on the measurements taken under one atmospheric pressure, but the test with the measurements by Waxler et al [12] showed that it applies to the pressure up to 1100 bar. We will refer to Eq. (10) as Eisenberg model.

3. Results and discussion

In their respective theoretical estimates of β(90), Morel [1] and Shifrin [2] use a value of 0.09 for the depolarization ratio, δ, and Buiteveld et al. [7] adopted a value of 0.051. Recently, Jonasz and Fournier [19] recommended a value of 0.039 determined by Farinato and Rowell [20] and showed that a better agreement with the measurements by Morel [4] could be achieved if the lower value of δ was used in Buiteveld et al. (0.039 vs. 0.051). Farinato and Rowell [20] attributed the lowered value of δ to the correction of the detector’s finite acceptance angle effect and to the reduction of stray light contamination. By progressively reducing the stray light from using no filter, to a medium band filter (bandwidth = 22.5 nm) and finally to a narrow band filter (bandwidth = 0.46 nm), their experiment led to a series of determinations of δ with decreasing values of 0.051, 0.045, and 0.039, respectively. They also reported that similar trends of improvement had been observed for other molecules, such as gaseous methane and liquid benzene.

We would assume that 0.039 is the most accurate for δ by far because the narrow band filter would remove most (if not completely) of the stray light. However, there is still uncertainty in this parameter; and we will use the other two values for δ to show its effect. Everything else the same, using 0.039 for δ instead of 0.051 or 0.09 would translate into a reduction in the values of β(90) by a factor of 0.97 or 0.89, respectively.

For isothermal compressibility, β_T, we used the equation reported in Kell [21],

where β_T in bar^-1, T_c in Degree Celsius, and the values for a _0-5 and b _0-1 are 50.88630, 0.7171582, 0.7819867×10^-3, 31.62214×10^-6, -0.1323594×10^-6, 0.6345750×10^-9, 1.0, 21.65928×10^-3, respectively. Equation (11) is valid for T_c from 0 to 110°C at one atmospheric pressure with a standard error of 0.002×10^-6 bar^-1. Note, Eq. (11) is effectively the same as the one used in Buiteveld et al. [7], who generated a quadratic fit to the data produced by Eq. (11).

For n, we used the empirical equation by Quan and Fry [22], who fitted the experimental data selected by Austin and Halikas [13] with an average error of 1.5×10^-5. Their equation for pure water is:

The coefficients have the following values: n ₀ = 1.31405, n ₄ = -2.02×10^-6, n ₅ = 15.868, n ₇ = -0.00423, n ₈ = -4382, n ₉ = 1.1455×10⁶. Their model, originally developed from the visible region, fits the available data well over an extended range covering the UV to the near-IR (200 – 1100 nm) [23]. The index of refraction of water in Austin and Halikas [13] was defined relative to the air, and so is Eq. (12). The n in Eqs. (7), (8) and (10) is the index in vacuum, which can be derived by multiplying Eq. (12) with the refractive index of air [24],

where ν̄ is the wavenumber (reciprocal of the vacuum wavelength) in μm^-1, and the other coefficients have the following values: k₀ = 238.0185, k₁ = 5792105, k₂ = 57.362, and k₃ = 167917 μm^-2. Equation (13) is for standard air at 15°C, 1 Atm, and 0% humidity. The refractive index of air does change with the temperature, pressure and water content. The variations, however, would affect β(90) only on the 8^th decimal digit, and therefore can be safely ignored for the purpose of this study.

We calculated β(90) using Eq. (3) with ρ(∂n ²/∂ρ)_T estimated using the three models, Eqs. (7), (8), and (10), respectively and with δ = 0.039. The results are shown in Table 1 (rows 5, 6, and 7 respectively). Apparently, in terms of comparison with Morel’s measurements, the latest development in characterizing density fluctuation of the refractive index has led to an improved estimate of scattering by water molecules. The average differences with the measurements (-0.67%, 0.72%, and 1.89%, for Eqs. (7), (8), and (10), respectively) are all within the relative experimental error (2%) and much smaller than those for the earlier estimates using the same equation (Eq. 3). They also perform better than the model by Buiteveld et al., whose results were based on a different equation (Eq. 4). In particular, estimates based on PMH model are about 0.5–1.5% lower than Morel’s experimental values, which might be expected if there were extremely low levels of unavoidable contamination in Morel’s sample, which in turn would cause additional scattering.

We also calculated β(90) with δ = 0.051 and 0.09 and their comparisons with the measurements are shown in the last column (inside the parentheses) of Table 1. Except for the estimates based on the Laplace equation (row 3), δ = 0.09 leads to significantly larger differences between various models and the measurements, which might suggest that this value be too big for the depolarization ratio of water molecules. With δ = 0.051, although all the three new models performed better than the earlier models, only the estimates based on PMH model agree with Morel’s measurements (1.97%) within the experimental error (2%). Based on Table 1 alone, however, we still cannot completely reject 0.051 as a possible value of δ. Regardless of the values used for δ, the two theoretical models (PMH and Niedrich) performed better than the empirical one (Eisenberg).

Spectral β(90) from 350 to 700 nm calculated by Buiteveld et al. [7], Morel [Table 4 of Ref. 1], Niedrich model, and Eisenberg model relative to that by PMH model are plotted in Fig. 1 (a) for δ = 0.039 and (b) for δ = 0.051. Also shown in Fig. 1 are the measured β(90) by Morel [4, 5]. It can be seen from Fig. 1, estimates based on the three models agree to each other within 3.5%, with better agreement in the shorter wavelengths. It can also be seen from Fig. 1(a), where δ = 0.039 was used, that the values of the measurements by Morel fall between the estimates based on the two theoretical models, PMH and Niedrich.

 

Fig 1. Comparison of pure water scattering β(90) from the theories and the experiment. The percentage difference relative to the values of calculation using Eq. 3 and PMH model (Eq. 7) are plotted. The value used for the depolarization ratio δ was 0.039 in (a) and 0.051 in (b). Values for Eisenberg and Niedrich are calculated using Eq. 3, and the Eisenberg (Eq. 10) and Niedrich (Eq. 8) models, respectively. Other values are from Buiteveld et al. [7], Morel [4, 5] for experiment, and Morel [table 4 of 1] for his theoretical calculation. Note Morel and Buiteveld et al. used δ = 0.09 and 0.051 in their respective calculations, here we scaled their values correspondingly.

Download Full Size | PDF

As we have mentioned above, PMH model and the Lorentz-Lorenz equation share the same theoretical basis with difference in the stages where the approximations are applied. Eisenberg model is basically the same as the Lorentz-Lorenz equation but with an empirical adjustment based on the measurements. Niedrich model was derived from the Van de Waals equation of state. Yet, Table 1 and Fig. 1 show that they all have converged on estimating the molecular scattering by water, indicating a closure is achieved among different models in characterizing the density fluctuation of water, which in turn leads to a better closure between the theory and the observation.

Theoretically, Eqs. (3) and (4) are the same; therefore the uncertainty associated with the density fluctuation term determines which one to use in practice. The density derivative (ρ(∂n ²/∂ρ)_T) used in Eq. (3) can be expressed as a simple function of n, whose values for water have been measured with a relatively high precision (typically 10^-5 but up to 10^-7 as in [18]). Based on Eqs. (7), (8), and (10), the relative errors in ρ(∂n ²/∂ρ)_T due to the uncertainty in n are at most 0.001%. Therefore the total error is dominated by the modeling uncertainty of the density derivative. The convergence of the estimates of water scattering based on the three models as shown in Table 1 and Fig. 1 suggested that the uncertainty associated with the modeling of the density derivative has been significantly reduced.

On the other hand, the accuracy of (∂n/∂P)_T is limited by the experiment. With no analytical form existing, (∂n/∂P)_T can only be approximated as Δn/ΔP. Among the few experiments, ΔP were set at least ~ 1–2 Atm in order for Δn to be detectable (typical value for (∂n/∂P)_T is only ~ 1.5×10^-10 Pa^-1). Even though linearity has been assumed, the function n(P) does behave in a nonlinear manner over a large pressure range [13]. In principal, we still do not know (∂n/∂P)_T at the microscopic scales, under which the Einstein-Smoluchowski theory applies.

Among the three models of the density derivative, we recommend the PMH model because 1) the prediction based on it agrees closest to the measurement within the experimental error, even considering uncertainty in δ, ranging from -0.67% for δ = 0.039 to 1.97% for δ = 0.051; and 2) its underlying theoretical basis can also lead to the classic Lorentz-Lorenz equation. The MATLAB code to compute these coefficients will be provided upon request or can be downloaded from ftp://ftp.umac.org/zhang/betaw_ZH2009.m

The molecular scattering of water predicted with PMH model together with Eq. (3) show that the spectral dependence of scattering as in(λ/λ ₀)^s has a slope s of value -4.28 when the anchor wavelength λ₀ is at 450 nm. Fitting to Morel’s experimental data (row 1 of Table 1) rendered an s of -4.32 with λ₀ at 436 nm, the same value was also reported by Morel [1] based on his theoretical calculation. Fitting to Buiteveld et al. [7] values gave an s of -4.14, the same as that estimated by Twardowski et al. [8]. Shifrin [2] reported a value of -4.17. Based on our results, both estimates by Buiteveld et al. and Shifrin seemed a little bit low (in absolute values).

The minimum scattering occurs at ~26°C, which is close to the minimum of 22°C determined both theoretically and experimentally by Cohen and Eisenberg [25]. Note, Buiteveld et al. reported a maximum at 15°C, which is probably due to the relatively large uncertainty in the pressure derivative of the refractive index they used. The overall temperature variation is relatively small, about 3.7% for T from 0 to 26°C. Scattering decreases slowly as the pressure increases, by ~1.3% for an increase in P of 100 bar.

4. Conclusions

With the recent theoretical development in characterizing density derivative of the refractive index for liquid, we re-estimated the scattering by pure water using three different models (PMH, Niedrich, and Eisenberg). These theoretical estimates agree to each other within 3.5% and with the experiment [5] within 2% (δ = 0.039). Use of the PMH model would predict values about 0.5–1.5% lower than the measurements, which is expected because even extremely low level of unavoidable dust contamination in the sample would have caused additional scattering. Taking into account of uncertainty associated with the depolarization ratio, the prediction based on the PMH model still agrees with the measurement within 2%. Because of extremely low sensitivity in Δn/ΔP as an approximation of the pressure derivative of the refractive index, the use of density derivative offers numerical advantage.