2.3. Heterodyne Dynamic light scattering (DLS)
For molecular binary systems such as liquids containing a dissolved gas, DLS has proven to allow the accurate and simultaneous determination of mutual diffusivity D11 and thermal diffusivity a in an absolute way [18–21]. The same also holds true for the determination of the translational particle diffusion coefficient DT in free media and under confinement in particulate systems consisting of isotropic particles dispersed in liquids [22] as well as of DT and the rotational particle diffusion coefficient DR in dispersions containing anisotropic particles [23]. In contrast to conventional techniques, DLS does not require a calibration and is applied in macroscopic thermodynamic equilibrium. Thus, no macroscopic gradient, which may cause advection, has to be applied. The fundamentals and the application of the DLS method within thermophysical property research [24, 25] have been described in detail in the example literature sources cited above. In the following, only the essential information about the method and the experimental realization in the present study is given.
When a bulk fluid sample in macroscopic thermodynamic equilibrium is irradiated by coherent laser light, scattered light from the sample can be observed in all directions. The underlying scattering process is governed by statistical microscopic fluctuations in temperature or entropy, in pressure, and in species concentration in mixtures. By analyzing the temporal behavior of the scattered light intensity and, thus, the relaxation of these statistical fluctuations following the same laws as the equilibration of gradients in macroscopic systems [26], the DLS method allows the absolute determination of multiple thermophysical properties. In the case of a binary mixture and neglecting pressure fluctuations by focusing only on the Rayleigh components of the scattered light, information about the dissipation processes of the temperature and concentration fluctuations can be derived by the temporal analysis of the scattered light intensity by calculating the second-order correlation function (CF). In DLS experiments using a heterodyne detection scheme, the scattered light from the sample fluid is coherently superimposed with a reference light, also called local oscillator, of a much larger intensity. The reference light can be generated in a defined way or be present from further scattering processes in the optical path, e.g., from the windows of the sample container as well as from the inside of the sample. In the heterodyne case, the recorded normalized intensity CF takes the form
$${g^{(2)}}(\tau )={a_0}+{b_\text{t}}\exp \left( { - \frac{{\left| \tau \right|}}{{{\tau _{\text{C},\text{t}}}}}} \right)+{b_\text{c}}\exp \left( { - \frac{{\left| \tau \right|}}{{{\tau _{\text{C},\text{c}}}}}} \right)$$
1
,
where the experimental constants a0, bt, and bc are defined by the characteristics of the experimental setup, the statics of the fluctuations in temperature and concentration at a given thermodynamic state, and the strength of the local oscillator field. τC,t and τC,c are the mean lifetimes of fluctuations in temperature and concentration, and are linked to the thermal diffusivity a and the mutual diffusivity D11 by
$$a=\frac{1}{{{\tau _{\text{C},\text{t}}}{q^2}}}$$
2
and
$${D_{11}}=\frac{1}{{{\tau _{\text{C},\text{c}}}{q^2}}}$$
3
.
Here,
$$q=\frac{{4\pi {n_{\text{f}\text{l}\text{u}\text{i}\text{d}}}}}{{{\lambda _0}}}\sin \left( {\frac{{{\Theta _\text{S}}}}{2}} \right)$$
4
is the modulus of the scattering vector for a given fluid sample with the refractive index nfluid at the laser wavelength in vacuo λ0. The scattering angle ΘS inside the sample is determined via the Snell-Descartes law using nfluid and the incident angle Θi defined by the directions of the incident light outside the sample container and that of the scattered light. Due to the direct connection between the observed mean lifetimes of fluctuations and q, diffusive processes on the order of (10−13 to 10−10) m2⋅s−1 are analyzed at large q values with the optical configuration used in the present study. Therefore, a, which is typically on the order of 10−8 m2⋅s−1, is not accessible here. In this case, the normalized intensity CF reduces to
$${g^{(2)}}(\tau )={a_0}+{b_\text{c}}\exp \left( { - \frac{{\left| \tau \right|}}{{{\tau _{\text{C},\text{c}}}}}} \right)$$
5
.
While for small ΘS values, nfluid has a negligible influence on the obtained diffusivities, at large values of ΘS, nfluid becomes a significant quantity and was therefore also determined in the present work, cf. section 2.4.
The sketch of the optical and electronic arrangement of the DLS setup used in the present work is shown in Fig. 1. Single-mode continuous wave fiber lasers with λ0 = (488 or 532) nm were used as light sources. The laser light is directed and focused via a set of mirrors (M) and a lens (L) with a focal length of 2 m into the sample cell represented by the cuvette holder (CH).
The incident laser beam is split by a beam splitter (BS) into a main beam and a reference beam represented by the solid and the dashed lines. With the reference beam, the alignment of the detection direction and the heterodyne detection scheme are realized. To achieve the latter, the reference beam with much larger intensity is coherently superimposed to the scattered light. The intensity and polarization of both the main beam and the reference beam are controlled using combinations of a half-wave plate (λ/2) and a polarization beam splitter (PBS) being aligned so that only vertically polarized light is transmitted. For the reference light, a neutral density filter (NDF) was additionally inserted into the beam path to further reduce its intensity. The detection direction of the scattered light is fixed by the combination of an aperture (A) and the collimator (C) of a single-mode fiber (SMF) mounted on a 3-axis translational stage. In the first step, the optical setup is adjusted by aligning the main and reference beams so that their reflected beams from the cuvette are in their directions of incidence. This yields an incident angle of Θi = 90°. The collimator is adjusted until the maximum intensity is observed from the fiber output. In a second step, with help of a rotational stage carrying mirror M7, the desired incident angle is adjusted relative to the reference value. The uncertainty in Θi is estimated by the double maximum resolution of the rotational stage being 0.04°. The scattered light is collected by the collimator and fed into the SMF connected to a fiber splitter which equally splits the scattered light to two avalanche photodiodes (APDs) operated in a pseudo cross-correlation scheme which limits afterpulsing and dead-time effects. To ensure that only scattered and reference light as well as vertically polarized light are detected by the APDs, a band-pass filter (BPF) centered at the employed λ0 and a further PBS are placed in front of the collimator. The dynamics of the scattered light intensity is temporally resolved by the two APDs. Their pulses are discriminated, amplified, and fed into two different digital correlators. The linear-tau correlator (LTC) features 2048 equally spaced correlation channels with adjustable interval time or sample time Δt. The multiple-tau correlator (MTC) has 263 channels with a fixed but broader quasi-logarithmic time structure, i.e., Δt is increased with increasing lag time. While the MTC can provide a scan of all signals present at very short up to very large time scales, the LTC gives the highest temporal resolution for the time range of interest by selecting a suitable Δt. To avoid disturbances rising from laser heating, the incident laser power measured at the incident beam inlet to the cuvette holder is limited to 47 mW. The power of the reference beam is only a few µW and below. Preliminary studies have shown no influence of the adjusted laser power in the applied range on the obtained D11.
The aluminum core of the temperature-controlled cuvette holder features four optical accesses covering both sides of the cuvette. This design combined with the optical arrangement allows the analysis of the scattered light for Θi = (70 to 110)°, where Θi = (70 to 90)° were adjusted in the present study due to temporal resolution limitations of the used detectors at larger angles combined with the properties of the probed samples. A quartz glass cuvette with a cross section of 10 mm2 and a total volume of 3.5 ml is positioned and sealed in the center of the core part of the cuvette holder by two flanges with protrusions and embedded O-rings. The entire core is wrapped with resistance heating wires (RW) and is further enclosed by a stainless-steel casing with inner liquid channels. The temperatures of the aluminum core, the fluid in the cuvette, and the casing are measured by two calibrated Pt100 resistance probes (Pt1 and Pt2) with an expanded absolute uncertainty (k = 2) of 20 mK and one uncalibrated Pt100 (Pt3), respectively, using a millikelvin thermometer. The temperature measured with Pt1 is used in a PID algorithm to regulate the temperature of the core in combination of a digital-to-analog converter, a self-constructed heating amplifier, and a RW. In addition, a laboratory thermostat connected to the liquid circuits in the outer casing is used to control the ambient T. The average stability of the sample temperature measured with Pt2 inside the cuvette is 0.7 mK, which is quantified by the double standard deviation of the temperatures measured within one measurement series typically lasting 1.5 h. The reported T in the figures and tables for the DLS experiments represent the mean values of the data recorded by Pt2 during one complete measurement series.
For each sample and state point, at least 3 individual measurements were performed with different Θi, resulting in at least 6 final CFs from the LTC and the MTC altogether. Here, each final CF is the result of averaging multiple CFs with adjusted short integration time recorded within one measurement. For an individual measurement in this work, the number of CFs is (60 or 600) with typical integration time of (10 or 1) s. Thus, the total integration time or total experimental run time is constantly 10 min. For samples with low wPEG, e.g., of 0.01 and 0.001, the scattering originating from the fluctuations in concentration is extremely weak. Therefore, larger integration times were used to achieve sufficient signal-to-noise ratios (SNR) in the CFs. The described “multi-run” scheme is especially beneficial for DLS experiments encountering irregular fluctuations in the count rate, which result in pronounced disturbances in the long-term range of the recorded CFs [27]. Such irregularities may be caused by impurities, e.g., in form of particle contaminations crossing the scattering volume. By recording multiple CFs with short integration time, individual CFs which are affected by irregular count rate fluctuations can easily be discarded in the data evaluation procedure.
From the many CFs obtained in one measurement with the same incident angle, correlator, and thermodynamic state, one averaged CF is calculated using a post-processing algorithm. The applied data evaluation procedure is similar to the one detailed in our previous studies [28, 29]. The averaged experimental CFs are fitted by non-linear regression to find the experimental constants and the mean lifetimes of fluctuations in concentration τC,c with the appropriate theoretical model based on Eq. 5. Two typical experimental averaged CFs are shown in Fig. 2.
For the probed aqueous solutions of pure PEG 943 homolog with wPEG = 0.01 and technical PEG 4000 with wPEG = 0.05, the experimental CFs obtained under a heterodyne detection scheme can be appropriately described by a single exponential characterizing the dynamics of fluctuations in concentration and an additional first- or second polynomial term. In some cases, however, a second exponentials appeared in the experimental CFs and its contribution is more pronounced at low wPEG and normally has a decay time of around 1500 µs, which is at least 150 times larger than the one of the exponential associated with the mass diffusion of PEG molecules. The origin of the second exponential is expected to be related to particle or dust contaminations present in the solid PEG samples. This has been confirmed by the observation of a second exponential in the CFs for two solutions of PEG 4000 prepared with unfiltered and filtered deionized water. Filtration of the mixtures was avoided to prevent possible changes of the PEG concentration. Compared to the weak scattering at fluctuations on a molecular level, large contaminating particles in the solution scatter much more strongly, where the scattering intensity scales to the power of six of the particle diameter. Since the decay times of these two exponentials are temporally separated and the sample time of the LTC is always adjusted in a way that maximum temporal resolution of the first exponential is achieved, the second exponential can be well described by either a first- or a second-order polynomial in addition to the mathematical model shown in Eq. 5. In this case, any influences on the fitted decay time τC,c, and, thus, on D11 can be neglected. For clarity purposes, Fig. 2 shows the CFs after subtracting the long-term decay and, thus, the disturbances from the originally measured CFs. In all measurements of the present study, the fit was considered to be appropriate only if the residuals, i.e., the differences between the experimental data and the corresponding fit, were free of any systematic behaviors as given for the examples in Fig. 2. As expected, the SNR of the CF generally increases with increasing wPEG and allows to achieve lower fitting uncertainties even if the total integration time is clearly increased for small wPEG.
According to Eqs. 3 and 4, D11 is obtained from the fitted mean lifetimes of the fluctuations in concentration, the details on the optical arrangement, and the refractive index of the fluid at the laser wavelength applied in the DLS experiment. The reported D11 data for a defined mixture and thermodynamic state were determined by averaging the individual results for D11 from all averaged CFs obtained with the LTC and the MTC in one measurement series, where a weighting scheme based on the inverse relative uncertainties of the individual D11 data was applied. The latter uncertainties are given on a level of confidence of 0.95 (k = 2) and were determined by uncertainty propagation calculations accounting for the statistical uncertainty in the fitting of the CF as well as in the experimental uncertainties in nfluid and Θi.