Demonstration of variable angle Super-Heterodyne Dynamic Light Scattering for measuring colloidal dynamics

We demonstrate a prototype light scattering instrument combining a frequency domain approach to the intermediate scattering function from Super-Heterodyning Doppler Velocimetry with the versatility of a standard homodyne Dynamic Light Scattering goniometer setup for investigations over a large range of scattering vectors. Comparing to reference experiments in correlation-time domain, we show that the novel approach can determine diffusion constants and hence hydrodynamic radii with high precision and accuracy. Possible future applications are discussed shortly.


Introduction
Structured fluids offer a wide range of interesting fundamental problems and technological applications [1]. Colloidal suspensions in particular consist of (spherical) solid particles suspended in a carrier liquid. Being able to form colloidal fluids, glasses and crystals, they have proven to be valuable mesoscopic models for a large variety of fundamental problems of Statistical Mechanics and Condensed Matter Physics. These range from the very existence of atoms [2], over studies on gas-liquid interfaces [3], to matter in external fields [4], and to active matter [5]. In many instances, such systems pose considerable experimental challenges like meta-stability, multiple relaxation times, non-ergodicity or a pronounced turbidity.
The latter issue creates particularly demanding difficulties for any experiment employing photon correlation spectroscopy (PCS) or dynamic light scattering (DLS), which is the standard technique to obtain colloidal dynamics [6,7,8]. More specifically, it concerns effects due to the difference in refractive index of particles and solvent. This is also known for W/O emulsions [9] or suspensions in organic media [10] but much more pronounced in aqueous suspensions. It leads to extinction as well as to multiple scattering and strongly distorts measured correlation functions [11,12]. In water, only for systems of low refractive index, index-matching is feasible, e.g. for perfluorinated polymer latex spheres [13]. However, full dynamic information can also be obtained for weakly or moderately multiple scattering systems from cross-correlation instruments [14,15,16,17], some of which are even available commercially [18]. A second issue is the non-egodicity of samples displaying slow structural relaxations like polycrystalline solids, glasses or gels. Here, time averages do not coincide with ensemble averages. This forbids a straightforward use of the Siegert relation ĝI(q,t) = 1 + [ĝE(q,t)] 2 to relate the measurable normalized intensity auto-correlation function ĝI(q,t)) to the desired normalized field auto-correlation function ĝE(q,t) [19]. ĝE(q,t), also known as intermediate scattering function or dynamic structure factor, is the Fourier-transform of the van Hove space-time correlation function describing the dynamics in real space. In DLS, typical diameters of the detection volume are restricted to some 50 -100µm to preserve coherent illumination. Thus, a DLS measurement records time averaged data from small ensembles. To obtain ĝE(q,t) in slow systems, additional measures must be taken to ensure correct ensemble averaging [20]. Several procedures have been reported for that step [21,22,23,24,25]. The most recent ones even simultaneously address multiple scattering and non-ergodicity, and they return ensemble averaged single-scattering dynamic structure factors [26]. However, all of these techniques still rely on the Siegert-relation and do not supply direct access to ĝE(q,t), In addition, they often demand some sophisticated optical and mechanical instrumentation. [3] We here report on a prototype light scattering instrument combining the versatility of a goniometer setup for investigations over a large range of scattering vectors with the frequency domain approach of Super Heterodyne Dynamic Light Scattering (SH-DLS) [27]. Analysis in frequency space was introduced already early for homodyne (self-beating) light scattering experiments to study diffusive properties of biological macro-molecules [28] but not much followed after the rapid development of time domain DLS. By contrast, the optical mixing technique of heterodyning (beating of scattered light with a local oscillator) works very well in frequency space [29] and became a standard technique in flow or Doppler velocimetry [30]. Super-heterodyning (SH), in addition, allows separating any homodyne contribution and low frequency noise by adding an additional frequency shift, wSH, between scattered light and local oscillator [31]. Integral measurements collect scattered light from a large observation volume or even the complete cross section of the sample cell and thus provide an excellent ensemble average [32]. The heterodyne part of the super-heterodyne spectra contains all the relevant information in terms of the Fourier transform of the intermediate scattering function [33]. A small angle configuration is optimally suited for Doppler velocimetric investigations of flow and selfdiffusion irrespective of sample structure [34,35]. Very recently, we also implemented a facile way of correcting for multiple scattering in small angle scattering [27]. This allowed detecting dynamics in turbid samples with a transmission as low as 20%, which is close to the detection limit of typical cross correlation experiments [15,26] or heterodyne near field scattering (HNFS [36]). Modulated 3D cross correlation was shown to push that limit even further to values around 1% [18].
As one main application of DLS is particle sizing, a more informative quantity would be the limiting transmission for obtaining relaxation times with a certain precision (discussed e.g. in [37,38] and references therein). An important benchmark for SH-DLS is the limit at which the average diffusion coefficient (and thus particle size) can still be determined with a statistical uncertainty of s £ DD0/D0 » 0.02 corresponding to the accepted statistical uncertainty in homodyne time domain DLS particle sizing [39,40]. This was possible in small angle SH-DLS at transmissions of 40% and larger [27].
With this approach we studied diffusion in turbid systems including flowing suspensions, active matter, and systems undergoing phase transitions [27,41,42]. For the competing cross correlation techniques, similar values have been reported, with modulated 3D cross correlation showing somewhat better marks due to the modulation induced increase of the intercept [18]. The limits for determining both mean particle size and width of a size distribution in poly-disperse samples have not been explored in a systematic way in the literature. It has, however, been shown that cross correlation techniques are excellently suited to extract multiple scattering free form factors as well as static and dynamic structure factors from cross correlation DLS data [18]. Cross correlation may, in addition, also give some limited access to information on advective and turbulent dynamics [43,44]. These features [4] and in particular the applicability of frequency domain analysis to the study of the length scale dependent diffusive dynamics in ordered and/or non-ergodic samples remain to be demonstrated for

SH-DLS.
In what follows, we go a first important step beyond our previous work and introduce the first prototype instrument of a variable angle integral Super Heterodyne Dynamic Light Scattering (SH-DLS) instrument. This new instrument allows to cover the full range of scattering vectors known from goniometer-based DLS. We provide proof of principle for simple diffusion measurements on noninteracting, dilute suspensions at scattering angles between 15° and 135° from comparing to homodyne time domain DLS performed on the very same samples. We anticipate that SH-DLS may become a facile and versatile approach also to more complex situations including size dispersity analysis, non-ergodic materials and multiple scattering systems. wBR range between 1 and 8 kHz. The goniometer is a double-arm construction allowing to adjust the angle Q between spatially fixed illuminating beam (Iill) and reference beam (Iref) by rotating the ringshaped platform on which the reference beam sending and collecting optics are mounted. In this prototype, we still adjust Q manually with a nonius reading precision of 0.02°, but integration of a motorized and piezo-controlled drive at a later stage is anticipated. The mechanical design allows both positive and negative angles. For the illumination beam sending optics (SI), the lens (L) and a polarizer (P) are adjusted past the rear FC to obtain a parallel, vertically (V) polarized illuminating beam of 1-2 mm diameter within the sample cell (C) placed in the centre of the index match bath (IMB) filled with fused silica matching liquid (50350, Cargille Laboratories, Inc., France). Depending on beam diameter, the impinging integrated power of 5.1mW (determined at the outlet of SI and SR) [5] results in 0.4 to 1.6 mW/mm 2 within the sample. The same power is used for measurements at all angles. After passing the sample and the IMB the remaining light is stopped by a beam stop (B). The reference beam is treated similarly, albeit the sending optics (SR) are adjusted to result in a smaller beam diameter of 0.5 mm. The detection side is aligned co-linear with the reference beam. A polarizer assures V/V detection and a horizontal slit aperture (HSA) rejects light received from out of plane directions. A combination of a cylindrical lens (CL) and a spherical lens (L), focus the light in the plane of a precision vertical slit aperture (VSA, Newport) which defines the detected scattering vector q, by rejecting all light focused off axis. The distance between the fiber coupler grin lens and the VSA defines the maximal extension of the observation volume. We collect singly scattered light emerging off the region illuminated by the illumination beam as well as multiply scattered light from within the [6] complete observation volume. Of both contributions, the photons propagating exactly in the direction of the reference beam are selectively fed into another fiber leading to the detector (Hamamatsu H5783, SN:831-5833). A crucial point in the alignment of the double illumination path is avoidance of parasitic reflections under any detection angle. The top inset of Fig. 1 demonstrates the absence of reflections over the complete range of angles (with the exception of angles close to 0° and 180° due to interference with the illumination optics mounted there). The slightly lower reading (dips), here obtained around angles of 100° and 340°, can be shifted to other angles at constant angular difference between the dips by changing the orientation of the sample vial. It is therefore attributed to slight deviations from perfect circularity of the vial. It does not interfere with dynamic measurements but has to be accounted for if static data are requested. On the detector, the reference beam (acting as the local oscillator) and the scattered light superimpose, which gives rise to beats in the intensity observed at the detector. These are analyzed by a four channel Fast Fourier Transform analyzer (Ono Sokki D-3200, Compumess, Germany) to yield the power spectrum as a function of frequency, f = w / 2p. A smartphone snap shot of our first spectrum obtained with the new prototype is shown in the lower inset of Fig. 1. Note the broad homodyne signal extending from the origin and the smaller, but clearly discriminated peaked SH-DLS signal at 4 kHz corresponding to the frequency difference wSH between reference and illuminating beam. To obtain a noise level below 0.1 typically some 200 subsequent spectra are averaged corresponding to some five minutes measurement time. The noise level can be lowered to 10 -3 by increasing the measurement duration accordingly. For particle characterization and as reference homodyne DLS-experiment we also employed a custom made multi-purpose instrument, described in detail elsewhere [45]. Evaluation of DLS data followed [38]. Initially, we assumed a constant ambient temperature to apply for all measurements. Since the instruments are located in differently climatised labs, we found it necessary to further note also the evolution of the ambient temperatures separately during each SH-DLS and DLS measurement series.

Samples and sample conditioning
Our sample for diffusion measurements in dilute aqueous suspension (lab code PnBAPS80) consisted of 35:65 W/W copolymer particles of Poly-n-Butylacrylamide (PnBA) and Polystyrene (PS), kindly provided by BASF, Ludwigshafen. Their nominal diameter and standard error based relative size dispersity index, PI, are given by the manufacturer as 2anom = 80.5 nm (DLS) and PI = 0.19 (Hydrodynamic Chromatography), respectively. Our own form factor measurements using SAXS yield 2aS-AXS = 86.9 nm and PI » 0.08. The effective charges for PnBAPS80 are Zeff,G = 365.1±2.3 (via TRS [45]) and Zeff,s = 513±3 (from conductivity [46]). Samples were prepared using batch conditioning procedures to first obtain a thoroughly deionized sample of number density n = 5.1 10 18 m -3 (from [7] Debye Scherrer type Static Light Scattering) [47]. This was then further diluted with Milli-Q water to adjust the number density to n = 1.2 10 17 m -3 (volume fraction F = 0.0018) and refilled into freshly cleaned and dried vials (Supelco, Bellefonte, PA, USA) without ion exchange resin added, capped to avoid contamination with dust, but not tightly sealed to allow equilibration with the CO2 of ambient air. This results in so-called realistically salt free conditions. At n = 1.2 10 17 m -3 in the presence of 5.7 10 -6 mol/L of carbonate ions, the system is non-interacting and takes an isotropic gas-like structure (as corroborated by Static Light Scattering showing a static structure factor of S(q) = 1 ± 0.02 for angles 15° ≤ Q ≤ 165°).

Data processing and evaluation.
Our theoretical frame for single scattering is based on earlier work on homo-and heterodyning techniques in dynamic light scattering [6,7,48]. A theory of conventional heterodyne LDV using an integral reference beam set-up was outlined in [29]. Super heterodyne theory for integral measurements at low angles has been detailed in [33] and extended in [27] to include moderate multiple scattering. We therefore here only recall some basics relevant for the present experiments. The .
Here, Ir is the reference beam intensity, and <I1(q)> is the time-averaged singly scattered intensity, and ĝE(q,t) = gE(q,t) / <I1(q)> is the normalized field autocorrelation function. For homogeneous suspensions of interaction monodisperse, optically homogeneous, mono-sized spherical particles, <I1(q)> factorizes as: Here, I0 is a constant comprising experimental boundary conditions like illuminating intensity, distance from the sample to the detector, and polarization details. b 2 (0) is the single particle forward [8] scattering cross section, n is the particle number density, P(q) = b(q) 2 / b(0) 2 denotes the particle form factor and S(q) the static structure factor [6,7,47]. For isotropic, fluid-like ordered samples, S(q) = S(q) and <I1(q)> = <I1(q)>. In the present case, the suspension is in addition adjusted to be noninteracting and thus S(q) = 1. We assume the particles to undergo only Brownian motion with a single effective diffusion coefficient Deff. For this simple case, the field correlation function is ĝE(q,t) = exp(-Deff q 2 ½t½). The power spectrum reads: For the equivalent expression in the presence of an additional drift velocity and of multiply scattered light, see [27]. Deff(q) depends on the probed length scale, in case collective diffusion is measured [49]. In the present case of unstructured suspensions, we expect collective and self-diffusion to coincide and hence to show no dependence on length scale. In fact, apart from hydrodynamic corrections [50], we expect to measure the free diffusion Stokes-Einstein-Sutherland Diffusion coefficient: Deff = D0 = kBT / 6phah. Here, kB is Boltzmann's constant, h(T) the temperature dependent viscosity of water [51], and ah is the hydrodynamic radius).
The spectrum in Eqn. (4) contains three contributions: a trivial constant term centered at zero frequency, two super-heterodyne Lorentzians of spectral width q 2 Deff shifted away from the origin by the Bragg frequency, and the homodyne Lorentzian of double width which is again centered at the origin. This description ignores electronic and other noise, which is assumed to be uncorrelated to the signal and which can, therefore, be simply subtracted after recording and averaging the spectra. The homodyne term is known to be seriously affected by multiple scattering but also by convection or shear [6,52]. In particular for larger scattering volumes as used here, both result in the loss of coherence of the scattered light, which renders the homodyne term ill-defined. From Eqn. (4), however, we note that the desired information about the diffusive particle motion is also fully contained in each of the super-heterodyne Lorentzians which are symmetric about the origin. In Fig.   2. and also the following, we therefore display the measured data only for positive frequencies centered about the positive Bragg shift frequency.
Eqn. (4) neglects any contribution from non-diffusive particle motions. For a further treatment of this term for flowing systems, see e.g. [42]. Active suspensions are addressed in [41]. Eqn. (4) further [9] ignores any contribution not stemming from single scattering events to the spectra. There are, however, several noise sources including detector shot noise (resulting in a frequency independent background), delta peak type electronic noise and a systematic contribution from the superposition of reference beam light with light scattered off the illuminating beam by parasitic reflection at the cell or IMB surface (resulting in a delta peak at exactly the Bragg frequency).

Results
Before evaluation, we first mask the data at wB and subtract the frequency independent noise background. We show a background corrected spectrum in Fig. 2a.  For comparison we show in Fig. 2c a representative autocorrelation function as obtained in a measurement using homodyne DLS at an angle of Q = 110° (q = 21.6 µm -1 ). For a total measurement duration of 120 min the signal to noise ratio of better than 10 3 was obtained between 2 and 2´ 10 4 ms. For comparison to the SH-DLS data, we first fitted a single exponential to obtain and an intercept of ĝE(q,0) = 0.972 and a decay constant corresponding to an effective diffusion coefficient of Deff = (4.46 ± 0.07) µm 2 s -1 . We further performed a 2 nd order cumulant fit to the same data (not shown), yielding Deff = (4.39 ± 0.04) µm 2 s -1 and PIDLS = 0.035±0.009, which is to be compared to the PI from SAXS of PISAXS = 0.08. This large discrepancy should be resolvable using non-linear cumulant analysis [38]. In all cases, the quoted uncertainties derive from the standard error for fits performed at a confidence level 0.95 and for an assumed temperature of T = (298.15±0.3) K. [11] In Fig. 2d,  Temperature readings can be further improved by directly measuring the IMB fluid temperature.
Already now, however, we could demonstrate the excellent performance of our instrument. The statistical uncertainty (standard error at confidence level 0.95) of an individual measurement is on the order of 1.5%. The performance is therefore fully comparable to that of homodyne time domain DLS.
Note, however, that both approaches fall way short in comparison to other, albeit much more time consuming optical approaches, e.g optical tracking. There, for instance, Garbow et al. could resolve [12] all six species of a mixture with sizes ranging between 300 and 450 nm to obtain the average diameters with a combined statistical and systematic relative uncertainty of 0.7% in a single tracking experiment of 21h duration [53]. One further observes that both DLS and SH-DLS radii are consistently larger than the radius derived from static scattering. This, however, is a well-known effect that has been discussed extensively in the literature [8,54,55].
As compared to DLS, SH-DLS affords some additional instrumentation. It requires a set of acoustooptical modulators to realize the frequency difference between Iref and Iill and a double-arm goniometer. It is, however, much easier to align: feeding a laser through the detector-sided fibre, the obtained observation beam simply has to be made co-linear with the reference beam by coupling it into the reference beam sending optics fibre coupling, and its lateral extension has to be adjusted to encompass a major central portion of the illumination beam traversing the sample cell by adjusting the distance between the q-selecting vertical slit aperture and the detection side fibre coupler grin lens.
The present demonstration is certainly preliminary in the sense that several standard evaluation pro- diffusion is directly accessible without using the Siegert relation [33]. This could e.g. be used to obtain quantitative access to the coupling between structure and diffusivity in shear flow [56].
Even more interesting are measurements in turbid samples. Here, standard DLS is at loss even for weak multiple scattering contributions. In the limit of very strong multiple scattering Diffusive Wave Scattering can be applied, but for all intermediate cases only cross correlation schemes give access to the desired singly scattered light. Here we expect the present goniometer-based SH-DLS to show similar performance as the small angle instrument, where diffusion and directed motion have been studied with a precision of 2% at transmissions as low as 40% [27], while the limiting transmission for signal detection was 20%. Cross correlation experiments clearly performed better in the latter respect, since they detect singly scattered light only and can omit post recording discrimination. In fact, depending on the employed scheme, the detection threshold there is on the order of 1-5% transmission [15,16,17,18,25]. More important seems to be the precision with which the average relaxation time can be determined. We here obtained <Deff> = (4.19 ± 0.06) 10 -12 m 2 s -1 from the average of 10 measurements at different angles, each of about five min duration. This is on the order of the precision and duration for the single angle DLS reference measurement, and it can surely be improved further. The initial step will be the inclusion of size dispersity analysis. Here, several approaches are [13] conceivably. First, one may try a Fourier-back transform of the isolated SH contribution of the spectra followed by time domain employing known algorithms. Second, since ĝE(q,t) is the Laplace transform of the normalized distribution of decay rates, one may attempt a deconvolution directly in frequency space. Once that is achieved, one can test the precision obtainable in turbid samples against the current benchmarks set by cross correlation techniques. Moreover then, also the ability to measure q-dependent statics and dynamics in ordered samples can be tested.
Finally, in small angle SH-DLS, we also exploited the excellent ensemble average stemming from the large observation volume to study diffusion in non-ergodic (polycrystalline) materials. This should also be possible with the present instrument, however, now also covering the interesting q- Concluding, we have taken first steps towards extended diffusion measurements in frequency space reviving early approaches in a goniometer based integral super-heterodyning version. We have demonstrated the excellent performance of the new instrument with simple diffusion measurements and compared to results from conventional homodyne DLS instrumentation. We have discussed re- maining weaknesses and open questions as well as outlined the potential scope of our approach. Much work remains to be done. However, we anticipate that after solving the remaining challenges, SH-DLS may become a viable and versatile alternative to access turbid and non-ergodic systems.