Dynamic Light Scattering for the Measurement of Transport Properties of Fluids

Abstract

The present article summarizes experimental and theoretical considerations required for a proper use of dynamic light scattering (DLS) for the measurement of transport properties of fluids. It addresses not only recent advancements of the method, but also aims to provide recommendations to researchers who intend to apply the technique in the future. As outlined in this study, DLS is based on the analysis of scattered light governed by microscopic statistical or periodic fluctuations that originate from the thermal movement of molecules and/or particles at macroscopic thermodynamic equilibrium. The dynamics of these hydrodynamic fluctuations in the bulk of fluids or at their phase boundaries are related to the underlying diffusive processes and, thus, to the associated transport properties, and are reflected by the time-dependent correlation function of the scattered light intensity. The fundamentals of this type of detection, known as photon correlation spectroscopy (PCS), will be discussed in the present contribution in some more detail. It is emphasized that the experiments need to be designed carefully in accordance with theory in order to assign the measurement signals to the corresponding hydrodynamic fluctuations. If the necessary conditions are fulfilled, DLS allows the accurate determination of several transport properties including kinematic and dynamic viscosity, thermal diffusivity, mutual diffusivity, and sound attenuation, which may be accessed together with other thermophysical properties such as speed of sound and surface or interfacial tension. In some instances, also the simultaneous determination of several transport properties is possible. With the exception of the sound attenuation, expanded uncertainties for the mentioned transport properties down to 1 % can be achieved for various types of fluid systems over a wide range of thermodynamic states up to elevated temperatures and pressures as well as in the vicinity of critical points. This performance and versatility of the DLS technique is documented in the present study by highlighting measurement examples from recent thermophysical property research on different classes of working fluids relevant for process and energy technology.

1 Introduction

The knowledge of the thermophysical properties of fluids is of importance in nearly every area of scientific and technological interest. The most common thermophysical properties of fluids can be classified as thermodynamic or transport properties. The first ones, e.g., pressure–volume–temperature or density data and the numerous properties derived from these, determine the change in state of a fluid resulting from an external stimulus. Transport properties, e.g., thermal conductivity, thermal diffusivity, mass diffusivity, and viscosity, which are in the focus of this article, are concerned with the rate of change in the state of a fluid as a result of the variation of external conditions.

To satisfy the claim for reliable and accurate information about the transport properties of working fluids in various fields of process and energy engineering, measurements are most relevant and frequently the only available way. Modeling approaches for transport properties, e.g., estimations and predictions including molecular simulations, are very useful, but need to be tested against accurate experimental data of representative fluids. Thus, all models can only be as accurate as the experimental data used for the evaluation of their performance. A fundamental problem which arises with the experimental determination of transport properties by using conventional techniques is that a macroscopic gradient has to be applied which must be large enough to give rise to a measurable effect and small enough to cause only very little perturbation in the system under investigation [1]. This difficulty is strongly enhanced in the vicinity of critical points, where induced gradients may make any reasonable measurement impossible. However, information collected in these regions is of great importance both for a fundamental understanding of critical phenomena [2] and for setting up correlations for the individual properties of a substance [3].

A successful approach to overcome these limitations is the application of optical methods for the measurement of transport properties of fluids. An overview about optical methods commonly used in thermophysical property research can be found in Ref. [4]. In this context, light scattering techniques are of particular interest here which rely on the analysis of light scattering caused by equilibrium, non-equilibrium, or induced fluctuations. These microscopic fluctuations analyzed by optical methods are related to various diffusive processes and, thus, to the associated transport properties. Measurements may be performed in or close to thermodynamic equilibrium with a very small input of energy. Since the experiments are carried out in a non-contact mode, regions of thermodynamic state can be investigated that can hardly be probed by other techniques. Light scattering techniques allow for the determination of various transport properties and other thermophysical properties, sometimes simultaneously. Since these techniques often rely on strictly valid working equations, no calibration procedure is needed and the reliability of the measurements can be easily checked. However, light scattering techniques are usually restricted to the study of optically transparent fluids.

For the study of fluids that are not at equilibrium, non-equilibrium Rayleigh scattering in the fear field [5,6,7] or in the near field [8,9,10,11] and the forced Rayleigh scattering technique [12,13,14,15,16] can be used to determine transport properties. Here, the application of macroscopic or induced temperature gradients causes an intensification of the probed non-equilibrium fluctuations in comparison to the naturally occurring, typically weak equilibrium fluctuations. This allows not only access to thermal diffusivity, mutual diffusivity, and/or viscosity, but also to the Soret coefficient and thermodiffusion coefficient which cannot be determined under equilibrium conditions. Despite significant advancements in recent decades, light scattering techniques applied at non-equilibrium are not used for a routine determination of transport properties of fluids to date [4].

The most frequent application of light scattering techniques for the measurement of transport properties of fluids is given by DLS applied in macroscopic thermodynamic equilibrium [17,18,19]. The technique makes use of statistical or periodic fluctuations at equilibrium in the bulk of fluids [20, 21] and at their phase boundaries [22] originating from the thermal movement of molecules and/or particles. From the analysis of the Rayleigh and/or Brillouin components of the scattered light in the far field, DLS allows the determination and, in some cases, the simultaneous determination of several transport and other thermophysical properties in an absolute way over a broad range of thermodynamic state including regions close to critical points, see, e.g., Refs. [18, 19, 23,24,25,26,27,28]. To these properties belong, e.g., kinematic or dynamic viscosity, thermal diffusivity, mutual diffusivity, sound attenuation, speed of sound, and surface or interfacial tension. At present, DLS is mostly employed to study diffusion coefficients in dispersed systems for the characterization of macromolecular or particle size and size distribution. Only a few research groups use DLS to a greater or lesser extent for the determination of transport properties in molecular fluids. This is documented by scientific activities carried out mainly in the working groups of Fröba [18, 23, 29,30,31,32,33], Nagasaka [28, 34, 35], Wu [25, 36,37,38], and He [26, 39, 40]. As a result of a continuous development of the method detailed in Ref. [19], DLS is nowadays an established tool for thermophysical property research of fluids.

The present contribution aims to provide recommendations for a proper execution of DLS for the accurate and reliable determination of thermophysical properties including transport properties of fluids. It is not only addressed to researchers already familiar with DLS, but also to those who intend to apply the technique in the future. First, basic principles of DLS are provided in the second section, discussing light scattering from the bulk of fluids and at phase boundaries. In the third section, basic instructions for the design and operation of DLS setups and experiments are given. As a main part of this article, the fourth section shows representative measurement examples for technically relevant working fluids of chemical and energy engineering obtained at the Institute of Advanced Optical Technologies – Thermophysical Properties (AOT-TP) of the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) especially within the past 10 years. It will be demonstrated that the proper evaluation and interpretation of the measurement signals is essential to reliably access thermophysical properties, focusing on the transport properties thermal diffusivity, mutual diffusivity, sound attenuation, and viscosity. In addition to the benefits of the DLS technique and the achievable uncertainties, also limitations and open questions with respect to its application for thermophysical property research are addressed. The fifth section summarizes the main conclusions of this article.

2 Basic Principles

The theoretical principles of dynamic light scattering (DLS) from the bulk of fluids, often referred to as conventional DLS, and its application to phase boundaries, also called surface light scattering (SLS), are only briefly summarized in the following. A more comprehensive description of the theory for conventional DLS [17, 18, 41, 42] and SLS [23, 29, 43] is given in the cited literature.

The focus in this article is on hydrodynamic fluctuations in the bulk of fluids and at phase boundaries at macroscopic thermodynamic equilibrium. The dynamics of these equilibrium fluctuations are reflected by time-dependent correlation functions [42]. In the hydrodynamic regime, the mean free path r₀ of the molecules is much smaller than the reciprocal value of the modulus of the scattering or wave vector q probed by light scattering, i.e. r₀ ≪ q^–1. Thus, Onsager’s regression hypothesis holds [44, 45], which states that the time correlation functions of the fluctuations in the hydrodynamic variables of defined q, g_q(τ), also called hydrodynamic modes, can be derived from classical hydrodynamics. Here, the conservation of mass, energy, and momentum is satisfied for the fluids that are assumed to be Newtonian and incompressible. Since the fluctuations around the equilibrium values are very small, the set of the linearized equations of fluid mechanics can be used.

The key concept behind DLS is to access the dynamics of the hydrodynamic fluctuations via their interaction with light [17]. When a fluid sample in macroscopic thermodynamic equilibrium is irradiated by coherent laser light, light scattered from the bulk of the fluid and/or its phase boundary can be observed in all directions. By defining the detection direction of the scattered light, a particular scattering or wave vector of the probed fluctuations can be analyzed. The underlying quasi-elastic scattering process is governed by the associated macroscopic transport and equilibrium properties of the fluid. In most cases, the spectrum of the scattered light can only be analyzed in a post-detection filtering scheme, where the total intensity is first detected and the detector signal is later filtered and processed. In this detection type known as photon correlation spectroscopy (PCS), one measures the time-dependent normalized intensity correlation function (CF), g⁽²⁾(τ), at a point in the far field. This experimentally accessible CF is directly related to the corresponding g_q(τ) of the observed fluctuation. More details to the calculation of g⁽²⁾(τ) are given in Refs. [17, 18]. In the following, we focus mainly on the application of a heterodyne detection scheme which can be accomplished by deliberately adding reference light of much stronger intensity to the light scattered by the fluctuations. A homodyne detection scheme, which is due to the light scattered by the fluctuations alone, can hardly be realized, especially for comparatively weak signals, e.g., far away from the critical point.

2.1 Light Scattering from the Bulk of the Fluid—Conventional DLS

For the bulk of the fluid, the underlying scattering process is governed by microscopic fluctuations of temperature or entropy, of pressure, and of species concentration in mixtures. The relaxations of these statistical fluctuations follow the same laws which are valid for the relaxation of macroscopic systems. In the balance equations for energy and momentum, Fourier’s law of heat conduction and the definition of the stress tensor according to Newton are considered. For a binary fluid mixture, a mass balance for both species is additionally required and involves the application of Fick’s first law of diffusion. The result is a set of linearized hydrodynamic equations for the fluctuating independent variables, i.e. the tensorial quantity velocity v and the scalar quantities density ρ, temperature T, and concentration c [42]. From these equations, g_q(τ) of the observed fluctuations can be found [17].

A typical scattering geometry for light scattering from the bulk of the fluid is shown in Fig. 1. With the scattering angle Θ_S, i.e. the angle between the direction of observation and the incident light, the scattering vector is given by $\overrightarrow {q} = \overrightarrow {k}_{{\text{I}}} - \overrightarrow {k}_{{\text{S}}} .$ Here, the wave vectors of the incident and scattered light are represented by $\overrightarrow {k}_{{\text{I}}}$ and $\overrightarrow {k}_{{\text{S}}} .$ Assuming elastic scattering ($\left| \overrightarrow {k}_{{\text{I}}} \right| \cong \left| \overrightarrow {k}_{{\text{S}}} \right|$), the modulus of the scattering vector q depends on the fluid’s refractive index n, the laser wavelength in vacuo λ₀, and Θ_S via [18]

$$q = \left| {\overrightarrow {k}_{{\text{I}}} - \overrightarrow {k}_{{\text{S}}} } \right| \cong 2k_{{\text{I}}} \sin (\varTheta_{{\text{S}}} /2) = \frac{4\pi n}{{\lambda_{0} }}\sin (\varTheta_{{\text{S}}} /2).$$

(1)

Fig. 1

Scattering geometry for light scattering from the bulk of the fluid including the scenario of dispersed particles in the fluid

The different relaxation processes result in a characteristic spectrum of the scattered light, which is known as Rayleigh–Brillouin triplet for a molecular binary mixture [17]. The statistical non-periodic fluctuations in T and/or c contribute to the central or unshifted Rayleigh component of the spectrum of the scattered light. The periodic pressure fluctuations are responsible for the Brillouin lines shifted relative to the frequency of the incident light ω_I by ω_S. In the case of particles dispersed in a fluid, a further frequency-unshifted Rayleigh line appears. The widths of all these lines, exhibiting in good approximation a Lorentzian form, are characterized by a corresponding damping coefficient Γ, which represents the half width at half maximum in the power spectrum. The inverse of the damping τ_C = Γ^–1 is called the “relaxation time” or “correlation time”, which reflects the mean life-time of the fluctuation observed. This quantity yields information on the values of the associated transport coefficients, as shown in the following.

2.1.1 Thermal Diffusivity

For the measurement of the thermal diffusivity a of pure fluids or fluid mixtures, the temporal behavior of statistical non-periodic fluctuations of temperature T or entropy is analyzed in a DLS experiment and is reflected by the heterodyne CF according to [18]

$${\text{g}}^{(2)} (\tau ) = a_{{\text{t}}} + b_{{\text{t}}} \cdot \exp ( - \left| \tau \right|/\tau_{{\text{C,t}}} ).$$

(2)

Here and in the following, the baseline a_t and the contrast b_t are experiment-specific parameters which mainly depend on the intensities of the scattered and the reference light as well as effects caused by an imperfect signal collection [18]. The index “t” indicates that these parameters are connected to the study of temperature fluctuations. The central quantity of interest τ_C,t reflects the mean lifetime or decay time of fluctuations in T and allows the determination of a via

$$a = \frac{1}{{\tau_{{\text{C,t}}} \,q^{2} }}.$$

(3)

2.1.2 Mutual Diffusivity

For fluid mixtures, information on the mutual diffusivity can be accessed by DLS. Here, besides fluctuations in temperature also fluctuations in concentration are present at the same time and are connected to the purely diffusive mass transport. In the following, only the molecular diffusion process in mixtures containing non-electrolytic components is addressed. For the special case of mixtures containing electrolytic components, the reader is referred to Ref. [17]. In the simplest case of a binary fluid mixture, i.e. N = 2 with the number of components N, the heterodyne CF related to c fluctuations decays analogously as given in Eq. 3 in the form of a single exponential by [18]

$${\text{g}}^{(2)} (\tau ) = a_{{\text{c}}} + b_{{\text{c}}} \cdot \exp ( - \left| \tau \right|/\tau_{{\text{C,c}}} ).$$

(4)

From the extracted decay time of the concentration fluctuations τ_C,c, the single mutual diffusivity or Fick diffusion coefficient D₁₁ is accessible by the relation

$$D_{11} = \frac{1}{{\tau_{{\text{C,c}}} \,q^{2} }}.$$

(5)

Following Fick’s generalized formalism for the description of diffusive mass transport, the index “11” connected to D₁₁ indicates the diffusive molar flux of component 1 (first index) due to a gradient in the concentration of component 1 (second index) as the driving force. It should be mentioned that one single D₁₁ value accessible via Eq. 5 describes the diffusive mass transport also in, e.g., binary mixtures consisting of one non-electrolytic component and one electrolytic component consisting of a cation and an anion as a result of the electroneutrality condition [46, 47].

For multi-component mixtures, the diffusive mass transport involves (N – 1) independent diffusion fluxes which can be related to the Fick diffusion matrix with (N – 1)² diffusion coefficients. For N ≥ 3, the numerical values of the Fick diffusion coefficients depend on the order of the components as well as on the reference frame. DLS experiments relying on the volume-averaged velocity reference frame allow access to (N – 1) exponentially decaying hydrodynamic modes associated with c fluctuations. The corresponding decay times of these modes are related to (N – 1) eigenvalues of the Fick diffusion matrix. More details on the application of DLS for the study of Fick diffusion coefficients in multi-component mixtures can be found in Refs. [17, 48, 49].

2.1.3 Sound Attenuation

For the determination of the sound attenuation D_S of pure fluids and fluid mixtures as well as their sound speed, the local periodic fluctuations of pressure p at constant entropy need to be analyzed. These fluctuations can be represented, to a good approximation, by propagating sound waves. To analyze the p fluctuations in a heterodyne detection scheme, the frequency of the local oscillator is shifted relative to the frequency ω_I of the laser light by ω_M applying a modulator system. The frequency shift ω_M is in the same order of magnitude of about (10 to 200) MHz as the frequency ω_S of the pressure fluctuations observed (ω_M ≈ ω_S). By a proper choice of the intensity of the local oscillator, the signal governed by the p fluctuations may be strongly enhanced relative to that related to T and/or c fluctuations. In this case, the normalized CF takes the form of a damped oscillation [18],

$${\text{g}}^{(2)} (\tau ) = a_{{\text{S}}} + b_{{\text{S}}} \cdot \cos (\Delta \omega_{{\text{S}}} \left| \tau \right|) \cdot \exp ( - \left| \tau \right|/\tau_{{\text{C,S}}} ),$$

(6)

where the characteristic decay time τ_C,S is connected to D_S via

$$D_{{\text{S}}} = \frac{1}{{\tau_{{\text{C,S}}} \,q^{2} }}.$$

(7)

The sound attenuation is related to the further transport properties dynamic shear viscosity η, bulk viscosity η_b, and thermal conductivity λ_c according to [17, 42]

$$D_{{\text{S}}} = \frac{1}{2\rho }\left[ {\frac{4}{3}\eta + \eta_{{\text{b}}} + \left( {\frac{1}{{c_{{\text{v}}} }} - \frac{1}{{c_{{\text{p}}} }}} \right)\lambda_{{\text{c}}} } \right].$$

(8)

In Eq. 8, ρ is the density and c_v and c_p are the specific heats at constant volume and pressure, respectively. The speed of sound c_S can be found from the knowledge of the adjusted modulator frequency ω_M and the residual detuning ∆ω_S =|ω_S − ω_M| of the CF according to c_S = ω_S/q.

2.1.4 Dynamic Viscosity

The determination of the dynamic viscosity η of fluids by DLS differs from the measurement of the other transport properties discussed so far, as seed particles need to be added to the fluid. The statistical Brownian motion of the particles results in a further hydrodynamic mode related to the decay of fluctuations in the particle number density or concentration. Since light scattering intensities from particles are often much larger than from molecules, a homodyne detection scheme can be easier realized. The corresponding homodyne CF describing the relaxation behavior of fluctuations in the number density of monodisperse spherical particles is given by an exponential via [18]

$${\text{g}}^{(2)} (\tau ) = a_{{\text{P}}} + b_{{\text{P}}} \cdot \exp ( - 2\left| \tau \right|/\tau_{{\text{C,P}}} ).$$

(9)

In Eq. 9, the factor 2 reflects the homodyne term in the CF, whereas it is replaced by a factor of 1 in the case of a pure heterodyne detection scheme. With the obtained decay time of the fluctuations in the particle number density τ_C,P, the translational particle diffusion coefficient D_P is accessible by

$$D_{{\text{P}}} = \frac{1}{{\tau_{{\text{C,P}}} \,q^{2} }}.$$

(10)

Based on D_p, which is the actual quantity measured by DLS, η can be determined from the application of the Stokes–Einstein equation according to [50]

$$\eta = \frac{{k_{\text{B}} T}}{{3\pi D_{{\text{P}}} \,d_{{\text{P}}} }}.$$

(11)

In Eq. 11, which is only valid at very small particle volume fractions below around 10^–5, k_B is the Boltzmann constant, T the fluid temperature, and d_P the hydrodynamic diameter of the assumed monodisperse spherical particles. Thus, the value for d_P has to be known precisely and can be determined by, e.g., performing DLS measurements of D_P in a fluid of well-known viscosity.

2.2 Light Scattering by Surface Waves—Surface Light Scattering (SLS)

For the measurement of the kinematic viscosity ν or dynamic viscosity η and, in some cases, also the surface or interfacial tension σ, the SLS method can be used. The technique makes use fluctuations at phase boundaries between two fluid phases. In macroscopic thermodynamic equilibrium, these surface waves are caused by the thermal motion of molecules and are quantized in so-called “ripplons” [51]. The rough surface structure caused by these ripplons can be represented by a superposition of surface waves with different amplitudes ξ_q and wave vectors $\overrightarrow {q} .$ The correlation function of the vertical displacement of a surface wave with a particular modulus of the wave vector q, g_q(τ), can be obtained from the solution of the linearized hydrodynamic equations expressing mass and momentum conservation; see details in Refs. [29, 43]. Each Fourier component of the rough surface behaves optically like a weak phase grating and scatters a small fraction of the incident light in different directions around both the reflected and refracted beams. Surface fluctuations observable in the SLS experiment cover wavelengths Λ = 2π/q from about (0.1 to 1000) μm, see, e.g., Refs. [43, 52, 53]. The total root-mean-square amplitude of the surface roughness integrated over all wavelengths is between about (1 and 100) nm [43]. Here, the amplitude of a given Fourier component is usually below 1 nm.

The scattering geometry typically used in SLS experiments is shown in Fig. 2. Here, scattered light can be observed in transmission direction near the refracted light or in reflection direction near the reflected light. In this section, only the scattering geometry in transmission direction is considered. By choice of the angle of incidence ε, which results in a specific angle δ of the refracted light, and the scattering angle Θ_S, the wave vector of the probed surface vibration mode, $\vec{q} = \vec{k^{\prime}}_{{\text{I}}} - \vec{k^{\prime}}_{{\text{S}}} ,$ is determined. Here, $\vec{k}^{\prime}_{{\text{I}}}$ and $\vec{k}^{\prime}_{{\text{S}}}$ denote the projections of the wave vectors of the reflected ($\vec{k}_{{\text{I}}}$) and scattered light ($\vec{k}_{{\text{S}}}$) to the surface plane. By observing scattered light within the irradiation plane and assuming elastic scattering (i.e. $k_{{\text{I}}} \cong k_{{\text{S}}}$), the modulus of the wave vector is given by [54]

$$q = \left| {\vec{k}^{\prime}_{{\text{I}}} - \vec{k}^{\prime}_{{\text{S}}} } \right| \cong 2k_{{\text{I}}} \,\sin (\varTheta_{{\text{S}}} /2)\,{\kern 1pt} \cos (\delta - \varTheta_{{\text{S}}} /2) = \frac{4\,\pi \,n}{{\lambda_{0} }}\sin (\varTheta_{{\text{S}}} /2)\,{\kern 1pt} \cos (\delta - \varTheta_{{\text{S}}} /2).$$

(12)

Fig. 2

Scattering geometry for light scattering by surface waves at the phase boundary between two fluids

The power spectrum of the scattered electric field at a point in the far field reflects ideally the power spectrum of a particular surface mode. An exponential decay of surface waves results solely in a broadening of the spectrum, whereas an oscillatory damping gives rise to a Brillouin doublet. In the following, a simplified view on capillary waves on a free liquid surface is given, which neglects the presence of a second fluid phase. Moreover, the theoretical approach discussed below neglects the time of propagation and dissipation of the rotational flow in the bulk, which is discussed later in Sect. 4.4.2. The relaxation behavior of surface fluctuations is also reflected in the normalized intensity CF, resulting in two cases depending on the reduced capillary number Y = (σ·ρ)/(4 η²·q).

For fluids of small viscosity and/or large surface tension (Y > 0.145), surface fluctuations propagate. Here, the heterodyne CF is given by a damped oscillation in the form of [23]

$${\text{g}}^{(2)} (\tau ) = a_{{\text{R}}} + b_{{\text{R}}} \cos (\omega_{{\text{R}}} |\tau | - \phi ) \cdot \exp ( - |\tau |/\tau_{{\text{C,R}}} ).$$

(13)

In Eq. 13, the phase term ϕ largely accounts for deviations of the spectrum from the Lorentzian form. The correlation time τ_C,R and the frequency ω_R are identical to the mean life-time of the studied ripplons or surface waves and their frequency of propagation. In a first-order approximation, τ_C,R is related to the kinematic viscosity ν by

$$v \approx \frac{1}{{2\,\tau_{{\text{C,R}}} \,q^{2} }},$$

(14)

while ω_R gives access to the surface tension through

$$\sigma \approx \frac{{\omega_{{\text{R}}}^{2} \rho }}{{q^{3} }}.$$

(15)

Thus, the oscillatory case allows for a simultaneous determination of ν or η and σ.

For fluids of large viscosity and/or small surface tension (Y < 0.145), surface fluctuations are over-damped and do not propagate (ω_R = 0). The associated heterodyne CF according to [23]

$${\text{g}}^{(2)} (\tau ) = a_{{\text{R}}} + b_{{{\text{R1}}}} \exp ( - \left| \tau \right|/\tau_{{\text{C,R1}}} ) - b_{{{\text{R2}}}} \exp ( - \left| \tau \right|/\tau_{{\text{C, R2}}} )$$

(16)

contains two exponentially decaying modes with the amplitudes b_R1 and b_R2 as well as the respective decay times τ_C,R1 and τ_C,R2. In first-order approximation, the slower decaying mode with τ_C,R1 gives access to the ratio of η to σ by

$$\frac{\eta }{\sigma } \approx \frac{{\tau_{{\text{C,R1}}} \,q}}{{2}}.$$

(17)

The faster decaying mode with τ_C,R2 can be related to the kinematic viscosity via

$$\nu \approx \frac{1}{{\tau_{{\text{C,R2}}} \,q^{{2}} }}.$$

(18)

Only near the critical damping (Y = 0.145), both modes in Eq. 16 can be resolved, which enables in principle a simultaneous determination of η and σ, yet only with relatively large uncertainties. If Y ≪ 0.145, Eq. 16 reduces to a single exponential associated with τ_C,R1. Therefore, in order to determine η by SLS in the over-damped case, knowledge about σ is required.

It is important to stress out that the first-order approximations [54] provided here and often used in literature cannot be used for a reliable determination of η and σ, especially for Y ~ 1. Instead, an exact description of the dynamics of surface fluctuations, as given by classical hydrodynamic theory in form of their dispersion relation D(τ_CR, ω_R, q, ρ, η, σ) or D(τ_C,R1, τ_C,R2, q, ρ, η, σ), needs to be considered, which is detailed in, e.g., Refs. [23, 29, 43, 54]. Within thermophysical property research, an exact description of the dispersion relation considering also a second fluid phase has always to be used to obtain reliable data for viscosity and surface tension.

3 Dynamic Light Scattering Experiment

In this section, fundamentals, recommendations, and practical hints for the design of DLS setups used for the measurement of thermophysical properties, in particular transport properties, are given. Section 3.1 discusses a setup that is specifically designed for the measurement of a, D₁₁, D_S, and c_S by light scattering from the bulk of the fluid on a molecular level. In principle, such a setup can also be used for light scattering by dispersed particles, allowing the determination of η via the measurement of D_P. For the determination of η and/or σ via light scattering by surface waves, a slight modification of the setup is required, which is explained in Sect. 3.2. For each light scattering experiment, the main components are quite similar, consisting of a sample cell, laser, illumination and detection optics, detectors, and a correlator system. The exact choice and arrangement of the individual components strongly depends on the exact goal of the experiment, i.e. which property is to be measured and at which state. In either case, it needs to be ensured that all optical components such as mirrors and sample cell windows show a surface quality or planarity better than λ₀/10. This prevents that roughness of optical surfaces cause disturbing signals that are primarily due to instabilities of the beam direction of the laser system. Since all mechanical vibrations of the experimental setup caused by, e.g., floor or air vibrations and thermal changes, can significantly impair the quality of the experiment, it is recommended to place the DLS setup on a vibration-damped optical table. For the selection of the main components of DLS experiments, more detailed information can be found in Refs. [24, 29, 41] and in the literature cited in the following subsections.

Since DLS measurements are performed at or close to thermodynamic equilibrium, the proper adjustment of the thermodynamic state is an essential requirement. For this, the accurate measurement of T and p by corresponding probes with low uncertainties is needed. The measurements of temperature and pressure have to be carried out in close vicinity of the sample of interest. Furthermore, a precise adjustment of T and p by control systems is important to maintain temperature and pressures stabilities of around 1 mK and 1 mbar, respectively, and below during the experiments. If these stabilities become too large, disturbing signal contributions caused by, e.g., temperature gradients resulting in convection inside the sample may be present. A more detailed description of the temperature and/or pressure control in DLS setups is given in the literature, see, e.g., Refs. [55, 56].

3.1 Light Scattering from the Bulk of the Fluid

Figure 3 shows the schematics for the optical and electronic arrangement of an experimental setup for analyzing the Rayleigh and Brillouin scattering processes on a molecular level at small scattering angles (Θ_S ~ 2.5°–6.0°) [55, 57, 58]. As a coherent light source, a solid-state laser such as a frequency-doubled Nd:YVO₄-laser (λ₀ = 532 nm) or gas lasers such as a He–Ne laser (λ₀ = 632.8 nm) or an Ar-ion laser (λ₀ = 488.0 nm or 514.5 nm) are often used. The main portion of the laser light is irradiated into a thermostatted sample cell whose T should be kept as close as possible to thermodynamic equilibrium. To analyze the Rayleigh line for fluids showing small scattering intensities, it turned out that by using small Θ_S, enough light is scattered by the cell windows to ensure a sufficiently high degree of heterodyning. In this case, to analyze the Brillouin lines, an additional frequency-shifted reference beam or local oscillator is added. For this, a part of the incident beam is reflected by a beam splitter, shifted in frequency by ω_M using a modulator system such as an acousto-optical modulator, and superimposed on the scattered light behind the sample cell. If large scattering intensities are available in experiments focusing on the analysis of the Rayleigh line, heterodyne conditions must be ensured by an additional frequency-unshifted local oscillator. In case the distance between the laser and the sample cell is more than ca. 1 m, a confocal lens of sufficiently large focal length may optionally be placed behind the laser to focus the main beam inside the sample volume.

Fig. 3

Possible setup for a DLS experiment for light scattering from the bulk of the fluids using relatively small scattering angles

The laser power required is mainly determined by the scattering cross-section of the fluctuations to be investigated. Typical laser powers impinging on the sample are up to 300 mW far from the critical point and only a few mW in the critical region. As the reference beam is to be added coherently on the detector within a heterodyne detection scheme, the coherence length of the laser must be large enough to ensure that the experiment is not affected by possibly different path lengths of the main and the reference part of the light. Thus, the laser has to be operated in a single longitudinal mode. This operation mode is also an essential requirement to perform heterodyne measurements that rely on the interference of the scattered light and the reference light. The combination of lambda-half (λ/2) wave plates and polarization beam splitters enables not only the adjustment of the intensity of both the incident beam and reference beam, but also ensures that the vector of the electric field is perpendicular to the scattering plane spanned by the incident beam and the axis of observation. The latter is given by two circular apertures with diameters of about 1 mm and separated by a distance of about 4 m in order to define q precisely. The system is aligned so that the irradiating beam and the axis of observation intersect within the sample. A part of the scattered light is imaged by means of two circular apertures onto the detector system which consists of single-photon counting units. These detector units typically consist of two photomultiplier tubes (PMTs), as exemplarily shown in Fig. 3, or Avalanche photodiodes (APDs). The two PMTs or APDs receiving the scattered light are images of each other with respect to a semi-transparent mirror or beam splitter. Here, attention needs to be paid to the maximum count rate of about 5 MHz and 20 MHz for PMTs and APDs. To further reduce the intensity of the local oscillator guided directly on the detectors, an interference filter can be integrated in the beam path. The signals are amplified, discriminated, and fed into a correlator which is a special purpose computer for the computation of the pseudo-cross CF. A signal processing by a cross-correlation scheme avoids that after-pulses and dead-times of the detectors may distort the CF at short τ if only one detector is used which gives only access to the autocorrelation function [59].

For the calculation of g⁽²⁾(τ), digital correlators with either a multiple-tau or a single-tau structure can be optionally used. An important characteristic of both types of correlators is the sample time, which is the time interval used to collect the detector signals and which determines the resolution of the CF. A multi-tau structure means that the sample time of the channels increases with increasing delay time. A possible design of such a structure is that after a block of eight or sixteen correlator channels with a constant sample time, the latter is doubled after each subsequent block. With this quasi-logarithmic channel spacing, sample times span over a range from a few ns up to many seconds, which allows to probe processes on completely different time scales simultaneously. In this way, thermal and mutual diffusivity may be probed at the same time. Multiple-tau correlation has proven to obtain optimum statistics for exponentially decaying CFs. Yet, light scattering experiments may show also oscillating structures in the CF, e.g., when the periodic pressure fluctuations are analyzed. Since multiple-tau correlators, which integrate over these fluctuations quite quickly, may no longer provide optimum statistics in these cases, a single-tau correlator should be used instead. Its structure is composed of equally spaced correlator channels, whereas the sample time is chosen manually, typically between a few ns and up to several ms. In general, this type of correlator is of advantage to analyze very fast and/or periodic signals, but it can also be used to obtain maximum information on a desired time scale by a manual input of the sample time. In this case, the simultaneous use of a single-tau and a multiple-tau structure is advisable. Here, the multiple-tau correlator serves to inspect the CF for possible further signals, which may need to be considered when evaluating data from the CF recorded by the single-tau correlator.

For small scattering angles, Θ_S can be deduced from the angle of incidence ε. The latter is defined by the angle between the optical axis of the incident laser beam outside the sample cell and the detection direction, see Fig. 3. Θ_S and ε are related by Snell’s refraction law n·sin(Θ_S) = n_air·sin(ε), where the refractive index of air is assumed to be n_air = 1. In this case, knowledge on the refractive index of the fluid n with an uncertainty of about 10 % is sufficient to determine q from the combination of Eq. 1 and Snell’s law with an uncertainty of less than 0.2 % [60]. For larger scattering angles, where Θ_S = ε = 90° is often used especially for molecular systems with very slow relaxation processes or particulate systems, n needs to be known more precisely since its uncertainty directly affects the uncertainty in q. For the measurement of ε, the laser beam is first adjusted through the detection system, before then the laser beam is set to the desired angle. The actual angle of incidence is measured by back reflection either from a mirror mounted to a precision rotation table placed on top of the sample cell for small ε or from the cell window itself for ε = 90°. The error in the angle measurement can be adjusted to be approximately ± 0.01°, which results in an uncertainty of less than 1 % for the studied transport properties.

Very similar to the setup shown in Fig. 3, an experimental setup for light scattering by dispersed particles can be found in, e.g., Refs. [61,62,63,64]. In the following, only experimental issues specifically relevant for the study of dispersions by DLS are addressed. Due to the large scattering cross-section of particles, a power of the incident beam of about 10 mW and below is sufficient. Scattered light is usually analyzed at relatively large scattering angles (Θ_S ~ 30° – 170°). Here, the detection of the scattered light in reflection direction at Θ_S > 90° is required for opaque or non-transparent dispersions. In this case, the scattering volume can be positioned at the boundary between the sample and the cell container, which is also beneficial for ensuring heterodyne conditions. Despite the relatively strong scattering signals, a pure homodyne detection scheme is still difficult to realize, yet is often taken for granted in literature, see, e.g., the discussion in Refs. [65, 66]. The determination of η of liquids requires the measurement of D_P of the added small portion of particles. A key to a successful determination of η is the choice of suitable particles in a size range of about (20 to 500) nm. These are to be spherical and monodisperse, or at least are to exhibit a narrow size distribution. This ensures an experimental CF that matches the model of a single exponential based on Eq. 9, which is essential for a reliable data evaluation. Moreover, the particles must be chemically stable and form stable dispersions. For these reasons, it is essential to use low particle volume fractions of typically 10^–5 or below. Excessively large particle concentrations may not only cause particle aggregation, but also multiple scattering effects giving rise to additional signal contributions that can hardly be modeled.

3.2 Light Scattering by Surface Waves

Although it has been shown that light scattering by surface waves may be performed in complete analogy to light scattering measurements from the bulk of the fluid using an identical setup [67,68,69], some differing design features have to be employed for an accurate determination of η and σ. One key difference is that SLS normally investigates interfaces in horizontal orientation within a sample cell with light impinging from above. Commonly, scattered light is observed near the reflected beam (see Fig. 2), which eases the optical access and is essential for non-transparent fluids. A corresponding setup for SLS measurements in reflection direction can be found in Ref. [70]. Alternatively, for the measurement of η and σ of transparent fluids from light scattering by surface waves, a possible setup is shown in Fig. 4, where scattered light is observed in the forward direction near refraction [29, 32, 36, 54, 71, 72]. This arrangement is chosen due to signal and stability considerations.

Fig. 4

Possible setup for a DLS experiment for light scattering by surface waves in transmission geometry

To perform SLS experiments on the phase boundary between two fluid phases, the thermostatted sample cell is mounted in a vertical position. This arrangement is highlighted by the dashed line around the setup part involving the sample cell. The laser beam is focused with a lens of sufficiently large focal length of about 2 m onto the phase boundary of the fluid inside the sample cell. For the observation of light scattered by surface waves, the optical path has to be aligned in such a way that the laser beam and the direction of detection intersect on the phase boundary. This can be achieved by moving the mirror in front of the sample cell to a position, where a SLS signal with a maximum signal strength is registered. Care has to be taken that the inner surface of the optical access on the top of the sample cell is free of liquid droplets or films. In the latter case, for example, the dynamics of the surface waves on the liquid film layer, which is influenced by the film thickness, may also be registered in the SLS signal [73]. To ensure a condensation-free surface of the window, a further heating element may be integrated next to the cell window. A special feature of the sample cell used for SLS experiments is that it needs to provide a sufficiently large area for the phase boundary of the fluid with a diameter of about 7 cm [32, 54]. By this, disturbing capillary effects of the liquid at the edges of the cell on the probed region are avoided. The detection system, the adjustment of heterodyne conditions by the addition of reference light, and the signal processing are all realized in a similar way as for the experimental setup shown in Fig. 3 for light scattering from the bulk of the fluid. The laser power is set typically between a few mW and 300 mW depending on the properties of the fluid, especially its surface or interfacial tension. For the study of fluids with relatively small values for σ, as it is given, e.g., close to critical points, low incident laser powers are required due to the larger vertical displacements of the probed surface fluctuations, yielding strong light scattering signals.

The main feature of the optical arrangement is based on the analysis of scattered light at variable and relatively high wave vectors of capillary waves in the order of about 10⁶ m^–1. By this, instrumental line-broadening effects caused by the relative increase in the spread of q for small wave vectors detected are negligible [29, 70]. Large wave vectors can be achieved more easily using the transmission geometry due to intensity considerations. Light scattered by the liquid surface is detected perpendicularly to the surface plane, i.e. Θ_S = δ, see Fig. 2. For this arrangement, with the help of Snell’s refraction law and simple trigonometric identities applied to Eq. 12, the modulus of the wave vector q can be deduced exactly as a function of the easily accessible angle of incidence ε, i.e.

$$q = \frac{2\,\pi }{{\lambda_{0} }}\sin (\varepsilon ),$$

(19)

without requiring knowledge of n of the sample. In experiments, ε is typically set between 3° and 6°, and is measured with a high-precision rotation table on which the tilted mirror in front of the sample cell is placed. Here, the error in the angle measurement is approximately ± 0.005°, which results in an uncertainty of less than 1 % for the desired thermophysical properties.

4 Measurement Examples and Data Evaluation

In the following subsections, a brief overview about the application of DLS in the field of thermophysical property research is given first. Here, the present paper is not intended to give a historical review of the development of DLS for its application in the characterization of fluids. Instead, selected and mostly recent publications from research groups are referenced which are currently predominantly active in thermophysical property research by DLS. Thereafter, the focus is on the presentation and evaluation of representative recent measurement examples obtained at AOT-TP. This should document latest advancements in thermophysical property research by DLS, but also provide evaluation strategies and hints how to deal with challenging experimental situations.

Throughout this section, the evaluation of the CFs for the central quantities decay times and, in some cases, additionally the frequency is discussed for different experimental conditions. For this, the theoretical model is fitted to the experimental CF, which may be done using a non-linear algorithm according to Marquard and Levenberg [74]. It often turns out that the CF cannot be described only by a single exponential or a damped oscillation, as discussed in Sect. 2, because different scattering signals may be present simultaneously. In order to extract decay times and/or frequencies with low uncertainty, it is desirable to ensure experimental conditions for which the CF takes the simplest possible form. Moreover, it is also of great importance to make sure that the CF actually matches the theoretical model by checking the residuals between the experimental and theoretical data points.

4.1 Rayleigh Scattering—Thermal Diffusivity and Mutual Diffusivity

Since 1975, the DLS method applied to the bulk of fluids was originally used extensively in three main directions. Besides the investigation of the decay behavior of critical fluctuations in pure fluids and fluid mixtures [75, 76], the focus was on the measurement of the Fick diffusion coefficient as a function of composition in several binary liquid mixtures at a constant temperature [77, 78] and the thermal diffusivity of pure fluids and selected binary mixtures as a function of temperature in the vicinity of the critical point [79, 80]. Here, the thermal diffusivity a is the transport property for which the development of the DLS technique is most advanced. Measurements for a in the order from (10^–6 to 10^–9) m²·s^–1 can be carried out with expanded uncertainties down to about 2 % or even below over a wide range of T and p for the liquid phase and, due to the lower signal levels, in an extended vicinity of the critical point for the vapor phase, see, e.g., studies from AOT-TP [19, 24, 57, 81,82,83] and the research groups of Wu [27, 37] and He [26, 39]. In this connection, DLS has especially contributed to an improvement of the data situation for refrigerants [37, 57, 83,84,85,86]. The use of a heterodyne detection scheme was particularly beneficial to study fluids far away from the critical point. This holds also for the determination of the mutual diffusivity D₁₁ that could often be obtained simultaneously with a in binary mixtures [27, 47, 56, 60, 87]. Mainly investigations at AOT-TP, but also those from the research group of Wu on model or reference systems and, in particular, systems relevant for process and energy engineering could demonstrate the reliable access to D₁₁ with typical expanded uncertainties of 5 % and below over a wide range of D₁₁ with about six orders of magnitude from around (10^–6 to 10^–12) m²·s^–1. For binary mixtures, objects of investigations comprise gaseous and supercritical systems far or near to the critical region given by, e.g., mixtures of n-alkanes [55, 88] or mixtures of an n-alkane with carbon dioxide (CO₂) [88] or hydrogen (H₂) [30] and liquid systems at saturation conditions or in the compressed liquid phase in the form of, e.g., fuel-related mixtures [56], mixtures of liquid organic hydrogen carriers (LOHCs) [89], mixtures of an ionic liquid (IL) with a molecular solvent [47, 90], and electrolyte systems [91, 92]. Moreover, various types of binary mixtures of liquids with dissolved gases could be investigated at saturation conditions up to 20 MPa and 573 K. Here, besides selected model systems involving organic liquids or ILs [25, 27, 33, 60, 93,94,95,96,97], also technically relevant systems given by, e.g., an LOHC with H₂ [69], aqueous solutions with H₂ [98], polymer melts with nitrogen (N₂) as blowing agent [99], and a long-chain n-alkane with H₂, carbon monoxide (CO), or water (H₂O) [87, 100] can be mentioned. Based on the latter systems, fundamental questions regarding the diffusivities accessible in ternary mixtures by DLS were also addressed [48].

While the measurement of a in pure fluids is a straightforward task, this is clearly more difficult in fluid mixtures. Here, fluctuations in T and c are present at the same time. Even in the simplest case of a binary fluid mixture and neglecting the Brillouin component, the heterodyne CF related to the Rayleigh line includes two exponentials according to Eqs. 2 and 4 and is expressed by [24]

$${\text{g}}^{(2)} (\tau ) = a + b_{{\text{t}}} \exp ( - |\tau |/\tau_{{\text{C,t}}} ) + b_{{\text{c}}} \exp ( - |\tau |/\tau_{{\text{C,c}}} ).$$

(20)

Whether it is possible to determine signals from T and/or c fluctuations is discussed in more detail in Sects. 4.1.1 and 4.1.2. It is evident that even for the simplified function equation 20, the determination of the decay times τ_C,t and τ_C,c and, thus, of a and D₁₁ is more complicated and is associated with a higher degree of uncertainty than in the case of a pure fluid, because a larger number of parameters are to be fitted. To determine, e.g., a, the situation becomes easier if the refractive indices of the two components nearly match. In this case, the signal from the fluctuations in c is then relatively weak and may be either treated as a low-amplitude perturbation or completely neglected [19]. Furthermore, to ensure true heterodyne conditions, the signal contrast of the exponential or hydrodynamic mode, which is governed by the ratio of the intensity of the scattered light to that of the local oscillator, needs to be kept below about 0.01. This condition is not only valid for b_c and b_t in the present case given by Eq. 20, but also for all other amplitudes associated with hydrodynamic modes.

In the following, three experimentally very challenging scenarios complicating the analysis of the Rayleigh signal in DLS measurements on binary fluid mixtures are highlighted. While the first two scenarios address the coupling and separation of the two hydrodynamic modes related to temperature and concentration fluctuations, the third one deals with the consideration of signal contributions related to disturbances that may appear especially in the long-time range of DLS signals.

4.1.1 Coupling of Hydrodynamic Modes Related to Temperature and Concentration Fluctuations

As discussed in Sect. 2.1.1, the mean lifetimes of fluctuations in T or c accessed by DLS are directly connected to the corresponding transport properties a and D₁₁ via Eqs. 3 and 5. These relations are, however, only true if there is no distinct coupling between the hydrodynamic modes related to T and c fluctuations. According to the hydrodynamic theory [5, 42], the two accessible mean lifetimes τ_C,{1,2} are given by

$$\tau_{{{\text{C,}}\left\{ {1,2} \right\}}} = \frac{1}{{D_{{1,2}} \cdot q^{2} }},$$

(21)

where the two effective diffusivities are defined as

$$D_{{1,2}} = \frac{1}{2}\left\{ {\left[ {a + D_{{{11}}} (1 + \varepsilon_{{\text{c}}} )} \right] \mp \sqrt {\left[ {a + D_{{{11}}} (1 + \varepsilon_{{\text{c}}} )} \right]^{2} - 4aD_{{{11}}} } } \right\}.$$

(22)

In Eq. 22, the coupling parameter ε_c is calculated by

$$\varepsilon_{{\text{c}}} = c_{{\text{H}}}^{2} (1 - c_{{\text{H}}} )^{2} T\frac{{S_{{\text{T}}}^{2} }}{{c_{{\text{p}}} }}\left( {\frac{{\partial c_{{\text{H}}} }}{\partial \mu }} \right)^{ - 1}$$

(23)

and describes the coupling between heat and mass transfer, i.e. between the temperature and concentration fluctuations. Here, c_H is the mass fraction of the heavier component, S_T the Soret coefficient, c_p the isobaric heat capacity, and (∂c_H/∂μ) is the osmotic compressibility, where μ is the difference between the chemical potentials of the pure components. Far away from the critical plait point of a binary mixture in both the liquid and the gaseous state, ε_c tends to zero. In this case, the two decay times τ_C,{1,2} obtained via fit of the experimental CF to Eq. 20 reflect τ_C,c and τ_C,t which can be assigned to D₁ = D₁₁ and D₂ = a, where D₁₁ is typically smaller than a. Here, the two hydrodynamic modes are fully decoupled. A coupling of the modes with non-vanishing values for ε_c is usually only found around the critical point of the mixture. As can be seen from Eq. 22, neither of the two modes can be associated with a pure diffusion process related to mass or energy. In the following, experimental situations and findings are given, where a mode coupling may be predominant. A detailed discussion on this issue can be found in Refs. [5, 55, 101, 102].

In the vicinity of the plait critical point, D₁₁ vanishes, a and c_p remain nearly constant, while S_T and (∂c_H/∂μ) diverge. In the near-critical region along the critical isochore in the supercritical state, there is a temperature T > T_crit, with T_crit being the critical temperature, for which the Lewis number Le = a/D₁₁ is unity. The influence of the mode coupling parameter ε_c is stronger, the closer the temperature is to T_crit and the larger the concentration of the solute is, i.e. at conditions far away from infinite dilution [5, 102]. As a criterion, mode coupling is non-negligible until Le is much larger than ε_c. In general, the divergence of ε_c approaching T_crit is weaker than the divergence of Le [102]. Since knowledge of S_T and (∂c_H/∂μ) is often lacking, the calculation of ε_c is often precluded. Nevertheless, a possible influence of a coupling between heat and mass transfer can be evaluated by analyzing the crossover between the two diffusivities, i.e. by determining T where Le is unity, as elaborated in Ref. [101]. In DLS experiments along the critical isochore of methane-ethane or methane-propane mixtures, Ackerson and Hanley [103], Fröba et al. [55], and Piszko et al. [101] could resolve two hydrodynamic modes only at a distinct distance away from T_crit. Close to the critical point, only a single hydrodynamic mode associated with c fluctuations could be detected. Here, their amplitude becomes very large, which can be explained in connection with the Rayleigh ratio discussed in the next section.

4.1.2 Separation and Assignment of Hydrodynamic Modes

If mode coupling can be safely excluded, a further challenge could lie in the resolution of the two hydrodynamic modes and their assignment to the corresponding transport property. Whether both modes can be resolved experimentally depends on the magnitude of the Lewis number Le = a/D₁₁, as described in more detail in Ref. [30]. First, Le needs to be larger than 1.5 or smaller than 0.5 to allow a sufficient temporal separation of the two modes. Moreover, the signal amplitudes b_t and b_c should not differ too much from each other and for amplitude ratios b_c/b_t close to unity, Le needs to deviate relatively strongly from 1 in order to enable a separation of both contributions. If Le is very close to 1, only one mixed diffusivity D_m related to both transport properties, i.e. a and D₁₁, is accessible. Furthermore, even if Le deviates distinctly from unity, only a single mode may be accessible when either b_t or b_c are very small and tend to zero.

The individual signal amplitudes b_c and b_t cannot be determined experimentally in an absolute way due to unknown experimental constants. At least, the Rayleigh ratio representing the ratio b_c/b_t is accessible experimentally via Eq. 20. The Rayleigh ratio ℜ is defined by [5, 42]

$$\Re = \frac{{b_{{\text{c}}} }}{{b_{{\text{t}}} }} = \frac{{c_{{\text{p}}} }}{T}\left( {\frac{{\partial c_{{\text{H}}} }}{\partial \mu }} \right)\left( {\frac{\partial n}{{\partial c_{{\text{H}}} }}} \right)^{2} \left( {\frac{\partial n}{{\partial T}}} \right)^{ - 2} .$$

(24)

In Eq. 24, the derivatives of the refractive index with respect to composition and temperature, i.e. (∂n/∂c_H) and (∂n/∂T), are optical contrast factors. Information about ℜ gives hints about the type of fluctuation that mainly governs the experimental CFs for situations where only one hydrodynamic mode can be resolved in the DLS experiment. In the following, a strategy used to identify and interpret DLS signals is presented which is detailed in Refs. [30, 101]. For this, two heterodyne measurement examples free of mode-coupling effects are shown in Fig. 5 for an equimolar methane-propane mixture obtained with a linear-tau correlator at ε = 4.00° and p = 12.0 MPa for (a) T = 283.17 K in the compressed liquid state and (b) T = 362.61 K in the supercritical state, i.e. sufficiently far away from the critical point [101].

Fig. 5

Fit to CFs (upper parts) and residuals of CFs from fits (lower parts) recorded for an equimolar methane-propane mixture at ε = 4.00° and p = 12.0 MPa for (a) T = 283.17 K or (b) T = 362.61 K [101] (Color figure online)

For relatively large values for (∂n/∂c_H), which is usually accompanied by large differences in the refractive indices of the two pure components, ℜ is by trend larger than one. In this case, the signal in the CF with the larger amplitude is associated with fluctuations in c, while that with the smaller amplitude is associated with fluctuations in T. This is the case in Fig. 5a, where two hydrodynamic modes can be resolved and the fit model (red solid line) according to Eq. 20 describes the experimental CF well. Here, the slow mode (green dashed line) and fast mode (blue dotted line) are associated with τ_C,c and τ_C,t with expanded (coverage factor k = 2) uncertainties of (6.5 and 0.8) %, respectively. The identification of the two diffusivities is supported by analyzing their behavior as a function of T and p. While D₁₁ increases with increasing T and decreases with increasing p in the liquid state, a is typically only weakly affected by T and increases with increasing p. When ℜ ≪ 1 or ℜ ≫ 1, Eq. 20 reduces to a single exponential. This can be seen for the CF shown in Fig. 5b, where only a single mode with an uncertainty for τ_C of 0.8 % could be resolved, as confirmed by the residual plot in the lower part of the figure that is free of any systematics. For this supercritical state, the single diffusivity can be assigned to D₁₁ due to its slowing-down approaching T_crit and its weak variation as a function of T and ρ far away from the critical point [101].

Another example, where a and D₁₁ have very similar values and show a crossing, is given for the saturated liquid phase of binary mixtures of n-alkanes with dissolved H₂, CO, or H₂O [87, 100]. This can be observed in Fig. 6 from a series of CFs recorded at small incident angles ε around 5° for mixtures of n-dodecane and H₂ from (a) 398.22 K over (b) 473.45 K to (c) 523.57 K at varying p between (1.1 and 2.9) MPa, associated to H₂ mole fraction in the liquid phase between about 0.02 and 0.07 [100]. For such small mole fractions close to infinite dilution, a mode coupling can be excluded.

Fig. 6

Fit based on Eq. 20 to CFs recorded for the saturated liquid phase of a binary mixture of n-dodecane with dissolved H₂ at given ε and p for (a) T = 398.22 K, (b) T = 473.45 K, and (c) T = 523.57 K [100] (Color figure online)

At 398.22 K, τ_C,c related to the concentration mode (green dashed line) is about two times larger than τ_C,t related to the thermal mode (blue dash-dotted line), which corresponds to a two times smaller value for D₁₁ than for a. After increasing T to 473.45 K, the experimental CF shown in Fig. 6b only consists of one exponential. Here, it can be assumed that a and D₁₁ are identical and represent again a mixed diffusivity D_m. It should be noted that the uncertainty reported for D_m only quantifies how accurately the diffusivity could be determined by DLS, but not how well it matches a and/or D₁₁. At 523.57 K given in Fig. 6c, the c fluctuations relax faster than the T fluctuations, i.e. D₁₁ > a. This mode identification can again be related to the T-dependent behaviors for a and D₁₁. The systematic-free residuals of the CFs, which are not shown here, indicate the reliability of the theoretical working equations. As can be seen from the values in Fig. 6, a simultaneous determination of a and D₁₁ with typical uncertainties (k = 2) of 10 % or less can be achieved [100].

To prove that the aforementioned signals are related to hydrodynamic modes, the inverse of the mean lifetime of the fluctuations studied must be proportional to the squared modulus of the scattering vector; cf. Eqs. 3 and 5. This can be verified by performing experiments at varying q over a sufficiently large range. The measured results for 1/τ_C,t and 1/τ_C,c as a function of q² are to be given by a linear fit whose slope reflects the diffusivities a and D₁₁, as can be seen in, e.g., Refs. [60, 99]. If these values agree with the mean values obtained from the average of the individual results related to each q, the studied DLS signals can be clearly assigned to hydrodynamic modes.

4.1.3 Consideration of Signal Contributions in the Long-Time Domain

Among the relevant transport properties, the mutual or Fick diffusivity shows often the smallest values in liquid systems. This can be particularly pronounced for systems with a relatively large viscosity such as electrolytes [47, 60, 92] or polymers [99]. As the relaxation of the concentration fluctuations is then very slow, a relatively large time range needs to be investigated in the recorded CFs. Under such conditions, signals contributions in the long-time range are more likely to become dominant and are superimposed on the relevant molecular signals associated with the fluctuations in T and c. Such disturbances may originate from external influences that are not related to the fluid system itself, such as advection, incoherent external stray light, mechanical vibrations, and/or an imperfect signal acquisition. Moreover, also DLS measurements in the near-critical region are often subjected to signal disturbances in the long-time domain, although their physical origin could not be identified [101]. The task of the experimentalist is to reduce these signal contributions as much as possible or, if still present, to account for them in the working equations for the CFs accurately. Only if this is given, the determined thermophysical properties can be considered to be reliable.

One measurement example highlighting the aforementioned aspect is related to a binary mixture of n-octacosane with dissolved CO at T = 498.15 K and p = 3.85 MPa, for which a CF recorded in the liquid phase and focusing on the simultaneous analysis of fluctuations in T and c is shown in the upper part of Fig. 7 [87]. In the nonlinear regression to the measured CF, the theoretical model has to consider the disturbance which is visible in the long-time range. In general, the “disturbance term” can be expressed by a polynomial with respect to τ up to the third order. The larger the polynomial expansion is, the larger is the degree of freedom for the fitting procedure, which results in larger uncertainties for τ_C,t and τ_C,c. In the current case, a quadratic term given by c·τ² was sufficient to represent the disturbing signal well, resulting in a modified form of Eq. 20 according to

$${\text{g}}^{(2)} (\tau ) = a + b_{{\text{t}}} \exp ( - \left| \tau \right|/\tau_{{\text{C,t}}} ) + b_{{\text{c}}} \exp ( - \left| \tau \right|/\tau_{{\text{C,c}}} ) + c\,\tau^{2} .$$

(25)

Fig. 7

Fit based on Eq. 25 to CF (upper part) and residuals of CF from fit (lower part) recorded for the saturated liquid phase of a binary mixture of n-octacosane with dissolved CO at given ε, p, and T in the presence of a disturbing signal in the long-time range [87]

The residual plot in the lower part of Fig. 7 illustrating the deviation of the correlator data from the fit based on Eq. 25 shows no systematic behavior.

To further check the quality of the fit model applied to the CF in the presence of a disturbance, a multifit procedure (MFP) can be carried out, which is detailed in Ref. [87]. For each MFP run, where the considered time range in the measured CF is varied with respect to the first and last part of the related hydrodynamic mode, a so-called “start-of-fit” and “end-of-fit” variation is carried out. Here, correlator data within the two selected time windows are stepwise omitted and fits based on the same CF model as used in the initial fit are performed. If the variations of the individually obtained decay times for τ_C,t or τ_C,c from their mean value are within the calculated expanded uncertainties of the τ_C data, the experimental CF is described well by the theoretical model including the selected disturbance term. In this case, the decay times obtained from the initial regression are considered to be reliable and can be used for the determination of the diffusivities. It should be mentioned that in the CFs shown in Figs. 5 and 6, the present disturbances in the long-time range were already subtracted from the originally measured CFs for signal representation purposes.

Recently, it has been observed that disturbances in the long-time domain of DLS signals can be caused not only by external effects, but also by intrinsic effects originating from the sample itself. The latter phenomenon is demonstrated for mixtures of H₂O with sodium chloride (NaCl) at a salt concentration of 0.25 kg NaCl per kg of H₂O at saturation pressure [98]. For such binary mixtures consisting in total of three species, the diffusive mass transport is described by one single mutual diffusivity D₁₁ due to the electroneutrality of the system. For the present system, where D₁₁ is also often named as salt diffusivity D_salt, two exemplary CFs recorded by a multi-tau correlator at Θ_S = 90° are shown in Fig. 8 for T = 293 K in the upper part and T = 333 K in the lower part. At low T, the measured CF includes the signal related to the salt diffusivity D_salt in the short-time range with τ_C,c = 1.364 µs. In the long-time range up to 10 ms, a flat CF can be observed. At T above about 303 K, a strong disturbance in the form of a damped oscillation appeared in the experimental CF, see lower part of Fig. 8. Such type of oscillation could also be observed in other studies on molecular systems including electrolytes [92], and seems to be related to a quasi-static, slowly moving mesoscopic structuring in the liquid. For the present aqueous salt solution, this structuring was also observable for the bare eye by looking from the side at the incident laser beam guided through the sample. This structuring is reversible since reducing T to 293 K led to vanishing of the oscillation in the experimental CF. Although this disturbing signal is present in the lower part of Fig. 8, the relatively large amplitude and short decay time of the signal related to salt diffusion with τ_C = 0.620 µs still allowed to determine D_salt by adjusting the temporal resolution of the correlators to the short-time range of the CF [98].

Fig. 8

Measured CFs for mixtures of water with NaCl at salt concentration of 0.25 kg NaCl per kg of H₂O at Θ_S = 90° and saturation pressure for T = 293 K (upper part) and T = 333 K (lower part) [98]

4.2 Light Scattering by Dispersed Particles—Particle Diffusion Coefficients and Viscosity

To date, the most frequent application of conventional DLS refers to light scattering by macromolecular solutions or dispersed particles. Here, the study of particle diffusion coefficients is typically used for the characterization of size, size distribution, and shape of macromolecules or particles in dispersions [17, 104]. In this regard, the application of DLS can be found nowadays for, e.g., aerosols, colloidal dispersions, polymer solutions, and dispersions of biological macromolecules, see, e.g., Refs. [105,106,107]. Recent studies at AOT-TP contributed to the investigation of diffusion coefficients in various types of dispersions, such as aqueous solutions of polyethylene glycol [64] and nanofluids containing spherical [66, 108] or non-spherical [65] nanoparticles. Furthermore, particle diffusion coefficients were accessed for the characterization of technically relevant systems in form of, e.g., evaporating single droplets consisting of a particle dispersion [109], particle dispersions under the influence of the confinement of porous materials [110], and microemulsions containing micelles swollen with H₂O [111] or CO₂ [62]. In the field of thermophysical property research, the analysis of the translational particle diffusion coefficient via the Stokes–Einstein equation [50] can give access to the dynamic viscosity of the fluid for typical η values between mPa·s and Pa·s. This has been shown for, e.g., simple liquids [112, 113] and ILs and their mixtures with CO₂ [63].

Within this article, two selected topics in connections with the application of light scattering by dispersed particles for thermophysical property research are discussed representatively. The first topic focuses on the determination of viscosity from the analysis of the translational particle diffusion coefficient of particles dispersed in liquids. In the second subsection, it will be demonstrated that it may be possible under certain conditions to simultaneously resolve the hydrodynamic modes associated with concentration fluctuations of both molecular and particulate species, which enables to obtain both the molecular and particle diffusion coefficients from a single experiment.

4.2.1 Determination of Viscosity via Translational Particle Diffusion Coefficient

As discussed in Sect. 3.1, the determination of the viscosity of liquids can be performed in an indirect way by investigating the translational particle diffusion coefficient of the dispersed particles. This requires a calibration of the hydrodynamic diameter of ideally spherical particles in a reference liquid. For the latter, the viscosity needs to be known accurately and the properties should be relatively similar to those of the liquid under investigation. As an example, the study of the viscosity of ILs by analyzing D_P of dispersed spherical silica (SiO₂) or melamine resin particles is shown by Bi et al. [63]. Figure 9a illustrates a measurement example for a CF recorded for the IL 1-butyl-3-methylimidazolium tricyanomethanide ([BMIM][C(CN)₃]) with dispersed SiO₂ particles of a calibrated size of 221 nm at a particle volume fraction of 10^–4 and T = 343 K. In this case, it was possible to achieve a high degree of homodyning for the CF, which is reflected by the signal contrast b_P of more than 0.9. In the ideal case of monodisperse particles, the normalized homodyne CF is given by Eq. 9. The validity of the homodyne detection scheme is confirmed by the residual plot in the lower part of Fig. 9a that is free of systematics. Once τ_C is evaluated properly for stable and sufficiently diluted dispersions, reliable data for η with uncertainties (k = 2) of about (3 to 5) % can be achieved [63, 112, 113]. Here, the viscosity results have shown to agree with data obtained by other viscometers including SLS.

Fig. 9

Fit based on Eq. 9 to CFs (upper parts) and residuals of CFs from fits (lower parts) recorded for dispersions with 0.01 vol % SiO₂ particles in (a) [BMIM][C(CN)₃] and (b) [HMIM][B(CN)₄] at 343 K [63]

It should be emphasized that accurate determinations of η are not possible for situations where particle agglomeration, particle swelling due to penetrating liquid, and/or other particle-liquid or particle–particle interactions occur. Among these effects, particle agglomeration is often the most likely one, especially at large particle concentrations. Even at dilute conditions, the influence of aggregation on the recorded CF may be present and can be seen in Fig. 9b, where a dispersion of SiO₂ particles in the IL 1-hexyl-3-methylimidazolium tetracyanoborate ([HMIM][B(CN)₄]) was studied at the same thermodynamic state as in Fig. 9a. Although the signal in the upper part of Fig. 9b appears to be an exponential, the systematic deviations in the residuals in the lower part reveal distinct contributions from further terms in the long-time range [63]. These are mostly related to the presence of particle agglomeration, which leads to a distribution of decay times that can be evaluated via, e.g., CONTIN [114] or cumulant [115] analysis. Since the evaluated decay times and particle diffusion coefficients cannot be connected to a defined particle size, a determination of viscosity is impossible.

Although a homodyne detection scheme can be realized, it is advisable to apply a heterodyne detection scheme also for the study of diffusion processes of particles dispersed in liquids. The latter scheme enables not only an improved signal quality, but also allows for an unambiguous data evaluation [65, 66]. This is especially relevant for multimodal dispersions containing different particle fractions with various mean particle sizes. Here, additional homodyne cross-terms may appear in the CFs, which can be entirely suppressed using a heterodyne detection scheme [66]. For a unimodal dispersion with one defined particle fraction, the heterodyne CF is a single exponential like in Eq. 9, with the exception that the factor 2 is substituted by 1, resulting in the same form as in Eq. 4.

4.2.2 Simultaneous Analysis of Molecular Diffusion and Particle Diffusion

As the value of the molecular diffusion coefficient is usually between those of the thermal diffusivity and the translational particle diffusion coefficient, there are prospects of a simultaneous measurement of D₁₁ and D_P or η. This is only possible under special conditions and requires a simultaneous analysis of molecular and particulate diffusion processes which may cover a broad time range in respective DLS signals. An example is given in Fig. 10a which shows the CF recorded with a multi-tau correlator at ε = 168° without addition of reference light for a microemulsion consisting of a polyol mixture, a binary surfactant mixture, and CO₂ at T = 333 K and p = 13 MPa [62]. At this state, thermodynamically stable CO₂-swollen micelles of spherical shape are present in the liquid phase.

Fig. 10

(a) Fit to CF (upper part) and residuals of CF from fit (lower part) obtained for a polyol-based microemulsion with CO₂-swollen micelles at T = 333 K and p = 13 MPa for ε = 168° without defined addition of reference light. (b) A magnification of the short-time range for the same signal as shown in (a) [62] (Color figure online)

To obtain a systematic-free residual plot, the measured CF needs to be represented by a fit (solid black line) that contains the sum of three exponential functions. The latter were found to be associated with three hydrodynamic modes, which reflect three diffusion processes. Again, this requires an identification of the modes, named as modes 1, 2, and 3 in the following, which is often only possible by studying the corresponding subsystems. The most dominant and slowest mode (red dash-dotted line), referred to as mode 1 with a decay time τ_C,c,1 = 456.8 µs, is related to D_P of the CO₂-swollen micelles. This is not only due to the strong signal amplitude, but also due to the fact that this mode attributable to a homodyne signal was only visible in the presence of the surfactant required for micelle formation. By combining D_P with viscosity data for the continuous liquid phase, realistic micelle sizes of about 8 nm could be determined [62]. The fastest mode named mode 3 (blue dotted line) is more clearly visible in Fig. 10b by focusing on the short-time range of the CF. This weak signal with τ_C,c,3 = 2.996 µs can only be associated with fluctuations in c on a molecular level and was found to be related to the presence of CO₂ dissolved in the continuous liquid phase of the microemulsion, D_CO2. The remaining contribution in the CF, which is given by mode 2 (green dashed line) and has a correlation time τ_C,c,2 = 108.3 µs between those of the two other modes, could be assigned to a further molecular relaxation process that is connected to the diffusion of components of the polyol mixture, D_polyol. In connection with the heterodyne modes 2 and 3, it could be proven that the stronger light scattered from the micelles acts as reference light for the molecular diffusion signals and enables a heterodyning without the need for adding a further reference light [62]. Since microemulsions are multicomponent mixtures, the three evaluated diffusion coefficients can only be treated as “effective” or “apparent” diffusion coefficients.

4.3 Brillouin Scattering—Sound Attenuation and Speed of Sound

The measurement of sound attenuation D_S and/or speed of sound c_S requires the analysis of the Brillouin components of the spectrum. Since the Brillouin lines are shifted in frequency, a heterodyne detection scheme realized with the help of a local oscillator, which is also shifted in frequency by means of an acousto-optic modulator, is typically applied [116]. As a result, a CF in the form of a damped oscillation as described by Eq. 6 is obtained, where D_S is determined from τ_C and c_S is accessible from the known frequency shift by the modulator ω_M and the residual mistuning Δω of the correlogram. While τ_C can be evaluated as usual through a fit, the most straightforward way for determining Δω is to apply a Fourier transform to the experimental CF. Here, Δω is found from the maximum of the resulting spectrum [117]. Most DLS studies related to Brillouin scattering focus on the measurement of the speed of sound for which a small expanded (k = 2) uncertainty of about 0.5 % can be achieved. This has often been demonstrated for various types of refrigerants with respect to their liquid phase and, in parts, also their vapor phase, see, e.g., Refs. [57, 83,84,85,86, 118].

As an example, Fig. 11 shows the experimental CF G⁽²⁾(τ) and its Fourier transformation obtained at ε = 4.571° for the refrigerant mixture R507 in the liquid phase at saturation conditions at T = 333.11 K using an Ar ion laser (λ₀ = 488 nm) [84]. Here, the modulator frequency ν_M, the representation of the Fourier transformation, and the residual mistuning Δν are related to angular frequencies by ω = 2πν. The speed of sound can be obtained by representing the frequency shift of the Brillouin lines ω_S = c_S·q by a combination with ω_M and Δω, resulting in

$$c_{{\text{S}}}^{ \pm } = \frac{{\omega_{{\text{M}}} \pm \Delta \omega }}{q}.$$

(26)

Fig. 11

Measurement example for a CF and its Fourier transformation obtained for the refrigerant mixture R507 in the liquid phase at saturation conditions at T = 333 K [84]

In Eq. 26, two values c_S⁺ and c_S⁻ can in principle be calculated, depending on if ω_S is larger or smaller than the adjusted modulation frequency ω_M. In order to determine whether c_S⁺ and c_S⁻ is the correct value for the speed of sound, one can perform a small variation in q for a defined ω_M, which will cause a small variation in ω_S and, thus, also in Δω. From the analysis of the latter, the position of the Brillouin line relative to ω_M is clearly defined, which allows to assign the speed of sound value to either c_S⁺ or c_S⁻. The uncertainty for c_S depends on the uncertainty of the frequency determination as well as on that for the determination of q. While the latter is the major source of error for large c_S values, the need for an accurate frequency determination is enhanced at low values for the speed of sound. By performing the Fourier transformation, the frequency of the damped oscillation Δω can be calculated with uncertainties in the order of 10 kHz. This value can be considered to be small enough to guarantee speed of sound data with an expanded uncertainty below 1 %.

For the determination of D_S, two challenges need to be faced. First, due to the large width of the Brillouin lines, the decay time of CFs is typically only a few hundreds of nanoseconds, even if small q values are employed. Thus, it is required to use fast correlator systems. The second issue is of a more fundamental nature and is caused by a lack of definition of q [58, 116]. Due to the finite widths of the apertures in the detection optics, there is always some angular spread, which is a combination of a simple geometrical and a diffraction effect. This uncertainty in q does not have a significant effect on the frequency-unshifted Rayleigh line but causes a spread in the position of the frequency-shifted Brillouin lines, i.e. an instrumental line-broadening effect, as it is also possible in SLS signals. This means that the linewidth probed experimentally is a convolution of the distribution of positions and the actual linewidth. For measurements in the time domain, the presence of this effect leads to a decay time that is shorter than that associated with D_S. To suppress this effect, the angular uncertainty is to be kept as small as possible by, e.g., increasing the distance between the two apertures in the detection optics up to about (4 to 6) m. Nevertheless, the uncertainty in the determination of D_S by DLS is limited to about 10 % [58], which is often still better than what can be achieved by direct spectroscopic methods. It also needs to be considered that the speed of sound and attenuation show a frequency dependence. This dispersion effect has turned out to be negligible for c_S of many fluids at the frequencies probed by DLS, but is prominent for D_S, in particular for refrigerants [58].

4.4 Light Scattering by Surface Waves—Viscosity and Surface or Interfacial Tension

Light scattering by surface waves allows for the simultaneous determination of viscosity η and surface or interfacial tension σ of fluids under saturation conditions. Until the end of the 1990s, SLS had not been adequately explored for thermophysical property research and was restricted to investigations especially suited for the method, see, e.g., Ref. [119]. Nowadays, SLS is an established method for the accurate measurement of η and/or σ of two-phase fluid systems over a broad range of thermodynamic states up to elevated T and p, covering η values in the order of μPa·s up to several Pa·s [23]. For this purpose, it is necessary to apply the exact solution of the linearized hydrodynamic theory for the dynamics of surface fluctuations [43, 120]. Since SLS is based on a strict theory and represents an absolute method, it is accepted as a quasi-primary method for the measurement of viscosity [121, 122]. During the past about two decades, significant progress has been made in the application of SLS for thermophysical property research of different types of vapor–liquid systems, where a strong contribution is given by studies carried out at AOT-TP and by the research groups of Nagasaka and Wu. To name only a few of these studies, the systems investigated include low-viscosity [28, 54, 71] and high-viscosity [123, 124] standards, refrigerants [84, 86, 125, 126], high-temperature liquid melts [34], hydrocarbon-based solvents [25, 28, 32, 36, 127,128,129], ILs [130,131,132,133], and LOHCs [31, 72, 134], where the latter three fluid classes were also studied in the presence of pressurized gases such as H₂ or CO₂. In most cases, an experimental uncertainty (k = 2) for η and σ of 2 % and below could be achieved. By a proper execution of SLS, no measurable differences from the values obtained by conventional methods is found [23, 29, 54]. In certain cases, besides surface fluctuations also fluctuations in the bulk of the fluid can be analyzed at the same time, which enables a simultaneous determination of multiple transport properties [67]. It could also be shown that SLS can be applied to multiphase systems containing at least three phases. This allows the measurement of the viscosities of the individual phases and the interfacial tensions between them [135].

In this chapter, three aspects or challenges related to the application of SLS for thermophysical property research are emphasized, which also reflect new advancements in the field originating from the last five years. With respect to metrological requirements, the first topic concerns the chosen detection scheme in SLS experiments, while the second one is connected to the evaluation of SLS signals close to the critical damping of surface fluctuations. The third topic is of a system-specific nature and deals with the influence of monolayers at fluid interfaces that may induce a viscoelastic behavior, which impedes the accurate measurement of η and σ.

4.4.1 Detection in Transmission and Reflection Direction

In principle, two detection geometries can be applied in SLS experiments which mainly depends on the optical properties of the studied samples. For transparent fluids, a detection of the scattered light in transmission direction around the refracted light, as shown in Fig. 4, is the preferable choice and has been mainly used for thermophysical property research [23]. Such a detection scheme, which is ideally also combined with a detection perpendicularly to the fluid interface, is of advantage with respect to the stability of the scattered light and the increased light scattering intensity compared to a detection in reflection direction. In the latter case, the scattered light is registered close to the reflected beam, which is required for the study of non-transparent fluids. Here, the inherently weaker light scattering intensity is further reduced by the application of small laser powers needed to suppress possible laser heating effects on non-transparent fluids. To compensate for the weak signal strengths, small wave vectors q of surface fluctuations associated with larger scattering intensities need to be probed. However, line-broadening effects due to the uncertainty Δq may occur particularly for the frequency-shifted Brillouin lines, which would result in overestimations of η and underestimations of σ, see, e.g., Refs. [29, 70, 136]. Despite ongoing advancements in the representation of line-broadening effects, as recently shown by Knorr et al. [136], their presence does still not allow for an accurate determination of η and σ. This goal is only achievable by applying sufficiently large q in the order of 6 × 10⁵ m^–1 and above, especially for fluids of low η and large σ values, where line-broadening effects can be suppressed to a negligible level.

To demonstrate the applicability of SLS in reflection direction, two example CFs recorded over a period of 10 min are shown in Fig. 12 for the opaque IL 1-methyl-3-octylimidazolium hexafluorophosphate ([OMIM][PF₆]) at T = 353.15 K and p = 0.1 MPa argon (Ar) [133]. Following a concept introduced in Ref. [137] and further elaborated in Ref. [70], it is possible to realize heterodyne experiments under two different detection geometries, i.e. a perpendicular (P, Fig. 12a) or a non-perpendicular (NP, Fig. 12b) detection of the scattered light relative to the fluid interface. Based on the characteristic angles listed in Fig. 12 and visualized in Fig. 2, an about four times larger q of 5.8 × 10⁵ m⁻¹ was adjusted in the P geometry compared to q = 1.3 × 10⁵ m⁻¹ in the NP geometry [133]. For both geometries, the overdamped behavior of surface fluctuations (Y < 0.145) could be represented well by a single exponential with the decay time τ_C,R1 based on Eq. 16. Here, the presence of the faster mode with decay time τ_C,R2 was disregarded adequately by omitting the first few measured channels of the CF, which is detailed in Sect. 4.4.2. Despite the lower incident laser power needed to apply in the NP geometry (12 mW) close to the Brewster angle than in the P geometry (45 mW), a smaller uncertainty in τ_C,R1 below 1 % could be achieved in the former case due to the distinctly lower q probed. As the P geometry is beneficial with respect to an easier experimental realization, no clear preference towards either detection scheme can be given. By employing σ data for the same sample as input to solve the dispersion relation, matching results for η with typical expanded uncertainties of (2 to 3) % are found for both geometries and agree with data from conventional viscometry [133]. This also implies that line-broadening effects on the frequency-unshifted Rayleigh line are negligible.

Fig. 12

Fit to CFs (upper parts) and residuals of CFs from fits (lower part) obtained from SLS experiments for the IL [OMIM][PF₆] close to 0.1 MPa at 353.15 K in reflection direction using a perpendicular (a) or non-perpendicular (b) detection geometry [133]

4.4.2 Signal Analysis Close to the Critical Damping of Surface Fluctuations

Under certain experimental conditions, it occurs that the surface fluctuations probed by SLS are close to their critical damping, i.e. at Y ≈ 0.145. For q values in the order of 10⁶ m^–1, such situation is likely possible for fluids with σ values around 20 mN·m^–1 and moderate viscosities in the η range from about (0.5 to 5) mPa·s. By increasing, e.g., the temperature, it is typically the case for vapor–liquid systems that η of the liquid phase decreases much stronger than σ, which results in an increase in Y. Thus, it can occur that the surface fluctuations show a transition from the overdamped behavior at lower T to an oscillatory behavior at higher T after crossing the critical damping. This behavior is exemplarily shown in Fig. 13, where CFs are illustrated for the high-viscosity standard tris(2-ethylhexyl) trimellitate (TOTM) at four different temperatures between (273.15 and 523.15) K [124].

Fig. 13

Fits to CFs (upper parts) and residuals of CFs from fits (lower parts) obtained by SLS for TOTM close to 0.1 MPa at different ε for (a) 273.15 K, (b) 373.15 K, (c) 398.15 K, and (d) 523.15 K [124]

In the overdamped case, the CF given by Eq. 16 represents a sum of two exponentially decaying modes connected to the decay times τ_C,R1 and τ_C,R2. Whether both modes can be resolved depends mainly on the ratio of their signal amplitudes b_R1/b_R2, which is related to the ratio τ_C,R1/τ_C,R2 [23]. According to Fig. 13a for T = 273.15 K, where Y = 3.47 × 10^–6, only the slow mode associated with τ_C,R1 and the amplitude b_R1 is visible, because the second fast mode with a very short decay time τ_C,R2 and very small amplitude b_R2 cannot be resolved. Under such conditions, Eq. 16 reduces to a single exponential. By approaching the critical damping from the overdamped region, the decay times and amplitudes of the two modes increasingly converge, which is why both modes with τ_C,R1 and τ_C,R2 can be resolved, as illustrated in Fig. 13b at T = 373.15 K corresponding to Y = 0.117. In this case, it is in principle possible to simultaneously determine η and σ [23, 43]. However, a fit according to Eq. 16 with two exponentials of similar amplitudes and decay times leads to relatively large uncertainties in the decay times of the two modes. Instead, it is often recommended to evaluate the CF only with respect to τ_C,R1 which represents the parameter of interest for the determination of η. To reduce the uncertainty in τ_C,R1, a single exponential based on Eq. 16 is used for fitting, where the short-time range of the measured CF within the delay time of 3·τ_C,R2 is omitted. Here, the faster mode has an approximated decay time τ_C,R2 ≈ 0.9ρ/(η·q²) for the case of a free liquid surface [29].

In the oscillatory case, the measured CFs can be represented well by a damped oscillation as given in Eq. 13, which can be seen in Fig. 13c and d at T = 398.15 K and T = 523.15 K, respectively. Here, the dynamics of the surface fluctuations are described by their decay time τ_C,R and their frequency ω_R. Due to the generally only weak oscillation of CFs at Y ≈ 1, which is the case in Fig. 13c (Y = 0.456), the determination of η and σ can only be realized with larger uncertainties in the order of (5 to 10) %. In all CFs shown in Fig. 13, the residual plots are free of any systematics, confirming that the experiment matches the theoretical model. It should be mentioned that for an accurate access of τ_C,R and ω_R in Fig. 13d (Y = 5.74), it was necessary to omit the first few channels in the short-time range of the experimental CF. This is because SLS signals recorded near the critical damping, in particular for oscillatory signals at Y > 0.145, are affected by the rotational flow in the bulk of the fluid, which mainly originates from the liquid phase for vapor–liquid systems. In this case, the motion of the phase boundary is coupled to that of the liquid phase underneath that gives a delayed response to the dynamics of the surface fluctuations. To date, theoretical and experimental studies in the literature [138,139,140,141] focus on the physics behind this so-called “bulk shear mode.” Aiming at an accurate thermophysical property research, a new evaluation strategy for SLS signals recorded near the critical damping could be developed [142] which is briefly outlined in the following.

The presence of the bulk shear mode in the short-time range of the CF is evident in the case of an oscillatory behavior of surface fluctuations, especially for Y between about 0.4 and 15. This is exemplarily illustrated in Fig. 14a by a SLS signal recorded for pure n-hexadecane at T = 348.15 K, p = 0.1 MPa, and q = 6.30 × 10⁵ m^–1, where Y is about 4 [142]. To represent the signal by fit via Eq. 13, the first 15 data channels up to 0.75 μs indicated by the shaded gray area were excluded. While outside this area, the systematic-free residual plot in the lower part of Fig. 14a confirms the validity of Eq. 13, the distinct negative deviations of the measured data from the fit in the short-time range reflect the influence of the bulk shear mode. The latter has the shape of an exponentially decaying function and is present in a time range in the order of the characteristic viscous time τ₀ [139], which is given by τ₀ = ρ/(2η·q²) for a free liquid surface. By omitting the initial about three up to twenty data channels of the CF and a subsequent fit of the remaining signal by Eq. 13, reasonable values for τ_C,R and ω_R can be obtained, as it was also the case in Fig. 13d. Nevertheless, this procedure is not only somewhat arbitrary regarding how many channels need to be omitted, but also causes a partial loss of information on the signal of interest related to the surface fluctuations.

Fig. 14

(a) CF including fit by Eq. 13 and residuals obtained by SLS at ε = 3.1° for n-hexadecane at 348.15 K and 0.1 MPa excluding the shaded short-time range. (b) The same signal as given in (a) represented over the entire τ range by fit via Eq. 27 accounting for the rotational flow in the bulk of the fluid [142]

For the contribution of the bulk shear mode to the CF, an analytical solution was derived by Ohmasa et al. [141]. Since this function involves many parameters and can hardly be used for fitting to measurement signals, it was an objective to develop a simple and adequate working equation considering the presence of the bulk shear mode. The suggested normalized CF given by [142]

$${\text{g}}^{(2)} (\tau ) = a_{{\text{R}}} + b_{{\text{R}}} \cos (\omega_{{\text{R}}} \left| \tau \right| - \phi ) \cdot \exp ( - \left| \tau \right|/\tau_{{\text{C,R}}} ) - b_{{{\text{shear}}}} \exp ( - |\tau |/\tau_{{0}} )$$

(27)

reflects the superposition of the damped oscillation mode according to Eq. 13 and the bulk shear mode. The latter is modeled by an exponential with its amplitude b_shear and its effective mean response time which is modeled by τ₀. For the latter, it was found that it is sufficient to estimate η as mainly governing quantity with an uncertainty of about 10 %, which can still be achieved by using Eq. 13.

By applying Eq. 27 to the same CF as shown in Fig. 14a, a very good representation of the measurement signal with a systematic-free residual plot over the entire τ range is found in Fig. 14b. Although the further fit parameter b_shear is required in Eq. 27, the obtained values for τ_C,R and ω_R show typically smaller uncertainties than those using Eq. 13. The application of Eq. 27 to SLS signals recorded for several vapor–liquid systems near the critical damping for Y between 0.4 and 15 demonstrates the success of this evaluation strategy for an accurate determination of η and σ [142]. For both properties, expanded uncertainties in the order of 5 % at Y ~ 1 down to 2 % for Y up to 15 could be obtained.

4.4.3 Viscoelastic Effects on Dynamics of Surface Fluctuations Caused by Monolayers

The accurate determination of η and σ by SLS is only possible for fluids showing no viscoelastic behavior. In this case, the dispersion relation for hydrodynamic capillary waves at the phase boundary between two fluid phases [43, 120] can be applied. The presence of monolayers at interfaces formed by surface-active compounds introduces a viscoelastic behavior [120, 143, 144]. This leads to a coupling between the transverse capillary waves, which mainly contribute to the scattered light intensity, and the longitudinal dilatational waves originating from the monolayer. The general form of the dispersion relation for surface fluctuations considering a viscoelastic behavior is given in Ref. [120]. As mostly often shown for water surfaces covered with a monolayer of a highly amphiphilic surfactant around room T, the addition of small amounts of the surfactant strongly affects the dynamics of the surface fluctuations at vapor–liquid interfaces, see, e.g., Refs. [43, 145,146,147]. If the measured values for τ_C,R and ω_R are applied in the hydrodynamic theory neglecting a viscoelastic behavior, the values obtained by SLS are too small for σ and distinctly too large for η, in some cases up to 50 % and more, compared to results from conventional techniques [43, 146].

While the above studies allowed to derive the viscoelastic properties of the monolayer of “classical” surfactants, it is unclear if monolayer-like effects can already be introduced by structurally much less asymmetric molecules. Corresponding indications arose from thermophysical property research of the LOHC system based on diphenylmethane (H0-DPM) up to 573 K [134, 148,149,150]. To test the hypothesis that the reaction intermediate cyclohexylphenylmethane (H6-DPM) can exhibit a surfactant-like behavior, binary mixtures of H0-DPM or its hydrogenated analog dicyclohexylmethane (H12-DPM) with small amounts of H6-DPM of (0.1 or 1) mol % were studied by SLS from (303 to 473) K at 0.1 MPa Ar in combination with conventional viscometry and tensiometry [151]. The SLS signals showed a negligible influence of H6-DPM on the dynamics of the surface fluctuations for the mixtures with H12-DPM, where H6-DPM shows also no surface enrichment, as evidenced by molecular dynamics simulations. Here, the SLS results for η and σ of the mixtures agree with the data from capillary viscometry (CV) and the pendant-drop (PD) method.

For mixtures of H0-DPM with H6-DPM, the presence of the latter component affects the dynamics of surface fluctuations significantly. This can be visualized by comparing the two heterodyne CFs in Fig. 15 that are normalized to g⁽²⁾(τ = 0) and obtained for (a) H0-DPM and (b) H0-DPM with 1 mol % H6-DPM at q = 6.18 × 10⁵ m^–1 and T = 423 K. Compared to the values for pure H0-DPM, ω_R reduces by 8 % and the damping Γ_R (= 1/τ_C,R) increases by 28 % when 1 mol % H6-DPM is added to H0-DPM. In particular, the sharp increase in Γ_R hints at a viscoelastic behavior originating from surface-enriched H6-DPM molecules, as evidenced by molecular dynamics simulations [151]. This is corroborated in Fig. 15b by the clear systematics in the residuals of the measured data from the theoretical representation via Eq. 13 which is only valid if no monolayer is present at the phase boundary. By inserting the determined data for Γ_R and ω_q into the dispersion relation neglecting viscoelastic effects [43, 120], the apparent values for η_L and σ of the H0-DPM/H6-DPM mixtures are distinctly larger and slightly smaller than the data from CV and the PD method. In the case of η, the deviations vary with T and reach maximum values of about + 25 % at 373 K for 0.1 mol % H6-DPM and + 57 % at 473 K for 1 mol % H6-DPM [151].

Fig. 15

Fit based on Eq. 13 to CFs (upper part) and residuals of CFs from fits (lower parts) obtained by SLS at ε = 3.0° for (a) H0-DPM and (b) H0-DPM with 1 mol % H6-DPM at 423 K and 0.1 MPa [151]

The results enabled to develop a conception of the molecular surface orientation of surfactants, which is detailed in Ref. [151]. In analogy to the concentration-dependent change in the surfactant orientation of highly amphiphilic surfactants on the surface of water at ambient T [152, 153], the orientation of H6-DPM molecules with respect to the surface in the mixtures with H0-DPM appears to change from a preferentially perpendicular to a parallel alignment with increasing T. These findings should sensitize researchers using SLS for the determination of η and σ with respect to the high sensitivity of the method in resolving viscoelastic effects, including surface orientation effects, caused by a monolayer at the phase boundary. Here, the combination of SLS and conventional measurement techniques allows to deduce the viscoelastic properties of surfactant monolayers [43, 146].

5 Conclusions

Over the past four decades, DLS has been widely used to determine the thermophysical properties of fluids. Here, a particular focus has been on transport properties, including thermal diffusivity, mutual diffusivity, and viscosity. In some cases, these transport properties and other equilibrium properties are accessible simultaneously with DLS. The main advantage of the technique is the possibility of performing measurements in macroscopic thermodynamic equilibrium in a contactless way without the need for any calibration procedure. Research activities at AOT-TP during the past decade, which are documented and highlighted in the present review article, have contributed to the further development of the DLS technique and its application to working fluids of process and energy engineering. Besides offering a significant contribution to a reliable database for transport properties of different types of fluids over a broad range of thermodynamic state, light scattering from the bulk of fluids and from surface waves could also provide answers to fundamental questions regarding, e.g., the critical behavior of fluids, structure–property relationships, and the physics of interfaces. As has been emphasized in the present contribution, the proper application of DLS requires the consideration of key experimental and theoretical aspects. Only if this is given, reliable and accurate results can be obtained for various transport properties with expanded uncertainties down to 2 % and below. In summary, this article may serve as a practical guide for DLS experiments and encourage other researchers to use this technique for the accurate determination of thermophysical properties and especially transport properties.