Dynamics and rheology under continuous shear flow studied by X-ray photon correlation spectroscopy

X-ray Photon Correlation Spectroscopy (XPCS) has emerged as a unique technique allowing the measurement of dynamics in materials on mesoscopic lengthscales. In particular, applications in soft matter physics cover a broad range of topics which include, but are not limited to, nanostructured materials such as colloidal suspensions or polymers, dynamics at liquid surfaces, membranes and interfaces, and the glass or gel transition. One of the most common problems associated with the use of bright X-ray beams with soft materials is beam induced radiation damage, and this is likely to become an even more limiting factor at future synchrotron and free electron laser sources. Flowing the sample during data acquisition is one of the simplest method allowing to limit the radiation damage. In addition to distributing the dose over many different scatterers, the method also enables new functionalities such as time-resolved studies in mixing cells. Here, we further develop an experimental technique that was recently proposed combining XPCS and continuously flowing samples. More specifically, we use a model system to show how the macroscopic advective response to flow and the microscopic dissipative dynamics (diffusion) can be quantified from the X-ray data. The method has many potential applications, e.g. dynamics of glasses and gels under continuous shear/flow, protein aggregations processes, the interplay between dynamics and rheology in complex fluids.


Introduction
Over the past several years, X-ray Photon Correlation Spectroscopy (XPCS) has become a well established experimental technique which allows the direct measurement of dynamics in materials (see e.g. Refs [1,2,3,4]). XPCS provides a tool complementary to Dynamic Light Scattering(DLS) [5] by giving access to dynamics of fluctuations on length-scales ranging from nanometer to micron.
In an important number of systems -especially soft or biological materialsradiation damage can seriously limit the applications of XPCS. This will become an even more important issue at new (or upgraded) high brilliance third generation synchrotron sources, and at fourth-generation light sources -free electron lasers and energy-recovery linacs -with their unprecedented brilliance several orders of magnitude stronger than available today [6].
Performing XPCS under continuous flow is a method that can limit the beam damage effects in a number X-ray sensitive systems. An additional benefit of such an experimental strategy is the possibility to time-resolve processes taking place in mixing flowcells [7,8], or the possibility to study the response of a system to applied shear.
In the experiments reported here, we further develop the method presented in [9,10] demonstrating that under specific conditions, it is possible to extract the diffusive component of the dynamics of nanoparticles suspended in a fluid undergoing shear flow. More specifically, we use the XPCS data to quantitatively measure both the diffusive and the advective, flow-induced, motion of the particles. This is achieved by taking advantage of the anisotropy of the measured correlation functions. Unlike the case of a non-flowing sample where (for isotropic systems) the characteristic times depend only on the magnitude of the scattering vector q = |q|, the correlation times measured in a flowing system depend on the relative orientation between the scattering vector and the flow direction. In a transverse flow scattering geometry (scattering wavevector q ⊥ flow direction) the correlation functions are mostly sensitive to the diffusive motion of the scatterers. On the contrary, in a longitudinal flow geometry (q flow) the correlation functions are strongly affected by the rheological properties of the flow (shear profile). The method has the unique ability of being able to simultaneously study the interplay between the advective motion of the scatterers in response to applied shear and the dissipative microscopic dynamics due to thermal diffusion.

Description of the XPCS-microfluidic experiment
The sample was prepared from a suspension of sterically stabilized colloidal silica spheres (purchased from Duke Scientific, radius 250 nm) by replacing the initial solvent (water) with Propylene Glycol (PG). The concentration of the suspension was low (<2%) in order to reduce inter-particle correlations.
The flowcell was made out of a Kapton tube with an inner radius of R = 0.66 mm and ≈100 µm thick walls. A syringe pump purchased from Harvard Apparatus Inc. g (2) (t) t (s) g (2) (t) t (s) g (2) (t) t (s) g (2) (t) b) and c)  was pushing the solution through PEEK TM polymer tubes from Upchurch Scientific Inc. to the Kapton flowcell. Leak-tight fittings and adapters were also purchased from Upchurch Scientific. The flow rates applied in these experiments were between Q =0 and 80 µl/hr. Translated into an average flow velocity via v 0 = Q/πR 2 , this gives v 0 ≈ 0-16 µm/s. The corresponding Reynolds number, defined as Re = v 0 2Rρ/η, where ρ is the density and η the dynamic viscosity of PG, is Re < 10 −6 . In these conditions the flow in the tube is laminar and can be well described by a parabolic Poiseuille profile for a Newtonian fluid, The XPCS experiments were performed in a small-angle x-ray scattering geometry ( Fig.1a) using partially coherent X-rays at the ID10A beamline (Troïka) of the European Synchrotron Radiation Facility (ESRF). A single bounce Si(111) crystal monochromator was used to select 8 keV X-rays, having a relative bandwidth of ∆λ/λ ≈ 10 −4 . A Si mirror downstream of the monochromator was employed to suppress radiation originating from higher order reflections. A transversely partially coherent beam was defined by using a set of high heat-load secondary slits placed at 33 m from the undulator source, a beryllium compound refractive lens (CRL) unit placed at 34 m from the source thereby focusing the beam near the sample location, at 46 m, and by a set of high precision slits with highly polished cylindrical edges, placed just upstream of the sample, at 45.5 m. The final beam size selected by the beam-defining slits was of 10 x 10 µm 2 . The parasitic scattering from the slits was limited by a guard slit placed a few cm upstream of the sample. Under these conditions, the partial coherent flux on the sample was ∼ 10 10 ph/s.
The scattering was recorded with a fast 2D pixelated sensor -the Maxipix detector -located 2.23 m downstream of the sample with a pixel size of 55 µm. Under these experimental conditions the measured speckle contrast was ∼20%, in good agreement with calculated values [11].
The intensity autocorrelation functions, were calculated with a standard multi-tau correlator [12] package developed in Python at ESRF [13]. Here . . . denotes that the correlation functions are both ensemble averaged, and averaged over time t 0 . The regions of ensemble averaging are schematically shown in Fig. 1a by dashed contour lines corresponding to the two scattering geometries considered here -longitudinal and transverse flow. The raw correlation functions can be seen in Fig. 1 (b-e). The transverse flow correlations (Fig. 1b) measured at one location in the flowcell (at the center) are, in the low-shear limit, basically insensitive to increasing flow rates. The longitudinal flow correlation times (Fig. 1c) become, however, increasingly faster at higher flow rates, as shown before [9,10]. Characteristic, self-beating oscillations due to the shear profile can also be clearly seen in the data.
The flow velocity profile can be probed by recording the scattering at different locations across the flow tube. This effect can be seen qualitatively in Figs. 1d & e. As expected, the transverse flow scans (Fig. 1d) are relatively insensitive to the position across the tube. In contrast, the longitudinal flow correlations have a strong dependence on the position across the flow, measuring the local velocity profile integrated along the direction parallel to the beam (Fig. 1e). The maximum shear-induced effect on the correlation functions is obtained near the center of the tube where the flow velocity reaches its maximum value. Near the edges of the wall, the velocity approaches zero, with a more flat profile along the tube, and hence the correlation functions shift towards their nominal zero-flow values.
The theory explaining XPCS measurements is briefly presented in the following section and will be followed by a detailed quantitative analysis of the effects described above.

XPCS in a laminar flow
In a homogeneous liquid, like the one used here, the temporal intensity fluctuations at a fixed q, are well described by a normal distribution. As a consequence, the normalized Intermediate Scattering Function (ISF), g (1) (q, t) = S(q, t)/S(q, 0), with the ISF (or is related to the the intensity autocorrelation functions g (2) by the Siegert relationship, Here β is the speckle contrast, in this setup around 20%. Due to a coupling between the diffusive motion of the particles and their flowinduced advective motion, the q-dependent ISFs describing the colloidal suspension in a cylindrical Poiseuille flow (Eq. 1) has been shown [10] to be well described, in the q-range probed by X-rays, by The expression above depends on three different relaxation rates. The Brownian diffusion is described by Γ D = D 0 q 2 . A shear-induced (oscillatory) decay of the measured correlation function is accounted for by Γ S ∝ q·v, with v representing the flow velocity at the maximum of the parabolic profile probed in the direction of the beam. The third factor, given by a relaxation rate ν tr , measures the rate at which the correlation functions decay because particles transit through a Gaussian-shaped beam. This effect is irrelevant here. Even at the highest flow rates, the transit time is on the order of 10 µm/16 µm·s −1 (size of the beam divided by the maximal velocities introduced previously), which is much larger than the other relaxation times measured here and hence this decay cannot be observed.

Experimental Results
The free diffusion coefficient D 0 can be determined by fitting the transverse flow correlation functions to simple exponentials. The results are shown in Fig. 2. In the lowflow limit D 0 is, as expected, independent on both q and the flow rate. At higher flow rates (Q > 40 µl/hr) the correlation functions (shown in the inset) start deviating from single-exponential (straight lines in the lin-log representation) and the fitted correlation times shift towards faster values due to increasing longitudinal components of the flow velocity. These components are present because of the finite widths of the detector regions over which g (2) (q, t) are ensemble averaged. The value obtained from the simple exponential fits for D 0 is 2.25×10 6Å2 /s. The viscosity can be estimated using the Einstein-Stokes relationship, yielding η=0.038 Pa·s at room temperature. This is in fair agreement with tabulated values for PG, η=0.056 Pa·s (at 20 • C), the difference probably being due to a small amount of residual water present in the solvent. The speckle contrast β and baseline value g ∞ of the measured correlation functions were first obtained from fits with stretched exponentials following the Kohlrausch-Williams-Watts form β exp [−(Γt) γ ] + g ∞ . Subsequently, the normalized correlation functions g (2) norm = (g (2) − g ∞ )/β were fitted using Eq. (3), with the value measured for D 0 kept fixed. Examples of fits for normalized correlation functions measured in transverse flow (blue circles) and longitudinal flow (green triangles) geometries, for a single value of the scattering wavevector q = 2.8×10 −3Å−1 and a single position in the flowcell (at the center, x/R = 0) at different flow rates Q = 20 − 80 µl/hr, are shown in Fig.3. The position-dependence of the correlation functions for the same value of q, a single flow rate Q = 40 µl/hr and the same scattering geometries, is displayed in Fig.4.
The fits show that the very simple model used here assuming a parabolic-shaped Poiseuille flow and hence an exact form for g (2) (q, t) describes the experimental data remarkably well. The most significant fitting parameter here is the shear-induced relaxation rate, Γ S =q·v where v is the maximum flow velocity along the direction of the beam (y). Since this is obtained at different positions across the flow tube (x), the fitted values also depend on x: Γ S (x) = q · v(x). Here we omit the vectorial notation and the dot product because only values measured from transverse flow fits are used. The  x-dependence of the maximum flow velocity along the y direction in a Poiseuille-flow geometry can be easily obtained from (1), and from here, we obtain Fits with Eq. (5) for the shear-induced relaxation rates obtained from the fits at different locations in the flow tube are shown in Fig. 5a. The data shown here were measured at four different flow rates and two different values of q. From (5) one can also see that a scaled relaxation rate should follow a single, flow-and q-independent parabolic profile across the tube (Fig. 5b).

Conclusions
The results presented here show, for the first time, that XPCS can be used to measure both the advective response to applied shear and the diffusive dynamics of a colloidal suspension under continuous flow. The data shows very good quantitative agreement with a simple Poiseuille-flow hydrodynamical model.
Possible future applications of this method include the study of response to shear and the interplay between dissipative effects (diffusion) and advective motion in colloidal gels, glasses, and other non-Newtonian fluids (see for e.g. [14,15]). The XPCS-flow method provides also a very useful way of avoiding radiation damage. For many applications, using high aspect ratio flow tubes where the velocity profile is more constant (approaching a plug-flow shape [16]) is advantageous.

Acknowledgments
We wish to acknowledge Yuriy Chushkin for his help at the beamline and for useful discussions.