*2.1.1 Flow field considerations*

conditions and chemical reactions. SAXS is clearly a versatile technique that can provide useful information on systems that are well behaved, and can also be applied to samples that may not display ideal behaviour (for example aggregation prone nanoparticles, or time dependent mixtures of particles). However, the measurement does have some drawbacks. SAXS analyses are heavily reliant on complementary information. SAXS cannot provide information at an atomic resolution, so high resolution structural information is lacking, and needs to be obtained by alternative methods such as NMR, chemical crystallography, or electron microscopy. Further, SAXS does not provide information on changes in chemical environment so correlating the particle shape and size evolution with changes in the chemistry of a system requires the use of other techniques that are sensitive to the chemical environment. Additionally, for *in situ* experiments, SAXS on high intensity beamlines has the disadvantage that intense dose of radiation are required to obtain high quality data at short time frames. This can result in radiation damage in

In SAXS analyses, there are a range of disadvantages that the current sample environments struggle to address. First and foremost is that in most solution SAXS measurements there needs to be a high concentration of particles in the solution to achieve a scattering signal with high enough signal to noise to be of use in further analysis. For the most part, this is not a significant issue as most samples are generally amenable to reasonably high concentrations. However, in a number of cases, the amount of sample can prohibit the use of standard sample environments, and limits the use of SAXS to samples that are not in limited quantities, or expensive to produce. Further, for a continuous flow mixing device, where many exposures are required at each time point, the sample consumption can reach many millilitres; again this may be prohibitive for a majority of samples. Additionally it can be difficult to apply high throughput methodologies to systems where flow, volume and data quality constraints limit the number of

The limitations of the current sample environments can be significantly mitigated by the use of custom microfluidic devices. The very low internal volumes mean that sample consumption is reduced, and the time that a volume of sample can be measured over under flow is increased, leading to a general improvement in measurement statistics. The lower spatial footprint, and lower sample consumption rates, means that a large number of measurements can be conducted in a very short period of time in parallel; increasing throughput for screening measurements. The lower volumes, and thus much more efficient mixing allows for much lower deadtimes then would otherwise be possible, and with the increasing access to microbeam SAXS measurements, the time resolution of the mixing experiments are greatly improved over conventional approaches. Further, the ease of design and modification of devices means that bespoke devices for specific applications can be achieved rapidly. Given that microfluidic devices can address many of the limitations of conventional SAXS sample environments, we believe that there will be increasing uptake and incorporation of these devices into SAXS measurements.

To successfully investigate time-resolved reactions in microfluidic devices, the channel design has to be carefully adapted to the requirements of each application.

the sample that can significantly influence results.

measurements that can physically be conducted in a period of time.

**2. Microfluidics for time-resolved studies**

**2.1 Device design**

**16**

**1.3 Microfluidic devices and X-rays**

*Advances in Microfluidics and Nanofluids*

An understanding of flow fields at the microscale is required to understand the function of hydrodynamic focusing and device design considerations. No turbulent mixing occurs inside a microfluidic channel, as typically *Re* numbers below 100 are achieved, thus liquids can only mix by diffusion. This has the advantage of allowing predictions of the exact movement of particles by calculation, as no chaotic (turbulent) mixing needs to be considered.

For microfluidic channels, assuming no-slip conditions in combination with pressure driven flow, Poiseuille flow with a parabolic shaped flow profile arises. Here, the highest velocity is in the middle of the channel, which decreases parabolically towards the walls until it reaches zero. For cylindrical shaped channel geometries with coordinate length *x*, radius *r* and azimuthal angle Φ, the velocity field can be derived as:

$$\nu\_{\mathbf{x}}(r,\phi) = -\frac{\Delta p}{4\eta L} \left(a^2 - r^2\right) \tag{13}$$

With pressure *p* and viscosity η over the channel length L and channel radius a. The hydraulic resistance *R* results then as:

$$R = \frac{8\eta L}{\pi a^4} \tag{14}$$

For rectangular shaped channels with height *h*, width *w* and small aspect ratio (*w* > *h*) the velocity field over the coordinates *x, y, z* is:

$$\nu\_{\mathbf{x}} = \frac{4h^2 \Delta p}{\pi^3 \eta L} \sum\_{n,odd}^{\infty} \frac{1}{n^3} \left[ 1 - \frac{\cosh\left(n\pi \frac{y}{h}\right)}{\cosh\left(n\pi \frac{w}{2h}\right)} \right] \sin\left(n\pi \frac{z}{h}\right) \tag{15}$$

and the hydraulic resistance R is then [14]:

$$R = \frac{12\eta L}{wh^3} \left[ 1 - \frac{h}{w} \left( \frac{192}{\pi^5} \sum\_{n=1,3,5}^{\infty} \frac{1}{n^5} \tanh\left(\frac{n\pi w}{2h}\right) \right) \right]^{-1} \tag{16}$$

This understanding of this pressure-driven, steady-state flow in microfluidic channels is the basis of liquid handling in lab-on-chip systems. Especially in microfluidics, the channel cross-sections can be of various shapes, depending on the application and fabrication method. Eqs. (13) to (16) describe the velocity field and hydraulic resistance for spherical and rectangular cross-sections, which are the most common geometries used for the devices described in this chapter. The derivation of those values is exceedingly more complicated for arbitrary channel cross-section shapes.

compression, or focusing, of the central flow, and decreases the mixing times significantly by reducing the diffusion length. There are two main categories for microfluidic devices with hydrodynamic focusing: coaxial tube and planar on-chip devices. The design can be defined depending on the use, with adapted geometries for fast mixing (**Figure 2A**), gradients (**Figure 2B**), specific nanoparticle growth

On-chip hydrodynamic focusing devices can be differentiated in twodimensional (2D) and three-dimensional (3D) devices. In two-dimensional hydrodynamic focusing devices, the central flow is focused in the horizontal plane. The simplest geometry for a 2D device is a Y- or T-shaped mixer (**Figure 2E**), where the cross-sectional diffusion is broadened at the channel walls in comparison to the centre. However, this design is highly limited regarding flow stability and focusing and susceptible to variation in these parameters in operation. To avoid this, cross-

*Left: Scheme of microfluidic features for kinetic investigations in flow in a cross shaped mixer. (A)*

*Hydrodynamically focused Centre stream for fast mixing experiments. (B) Mapping of concentration gradient across and along the channel through interdiffusion of different liquids from main and side channels. (C) Nucleation and growth of nanoparticles or self-assembly processes of nanomaterials as a function of time along the outlet channel. Schematic comparison of the provided time scales in continuous and stopped flow microfluidic devices. Right: Schematic illustration of microfluidic devices with various channel cross designs with the corresponding cross-sections through the outlet channel. (D) Coaxial tube reactor with two concentric channels/capillaries. (E) Y-shaped design, where mixing is solely based on diffusion. (F) Cross-shaped geometry at the inlets for hydrodynamic focusing. (G) Two-cross-section geometry, also known as double-focus device, where three different solutions can be introduced into the channels. Solutions introduced into the first sidechannel (SC1) act as an inert buffer between reactants in main channel (MC) and second side-channel (SC2). (H) and (J) multilayer designs of the geometries from (F) and (G), respectively, avoiding contact between the central stream and channel walls. (K) Hybrid device consisting of multilayer focusing device (J) and an inserted*

The simplest type of coaxial tube reactors is a device consisting of two concentric capillaries (**Figure 2D**), which are connected to a channel where a central flow is injected through the inner capillary, with sheath flow injected from the outer layer. Coaxial tube microreactors find various applications, but are typically used as the interface for droplet-based reactors, as the transition from flow to droplet generation, dripping and jetting is defined by the flowrate of the outer sheath flow. However, this approach is of limited utility in time resolved mixing applications, as it only offers limited mixing geometries. Further limitations abound in the fabrication process for coaxial designs, which requires multiple steps and precise align-

reactions or self-assembly processes (**Figure 2C**) [15].

*Microfluidics for Time-Resolved Small-Angle X-Ray Scattering*

*DOI: http://dx.doi.org/10.5772/intechopen.95059*

ment and assembly of the parts [15].

**Figure 2.**

**19**

*glass capillary as outlet channel.*

#### *2.1.2 Continuous flow vs. stopped flow*

The first design consideration is how the device will enable time-resolved measurement of phenomenon. This can be achieved in two different ways. The first, and conceptually simplest method, is a static experiment, where a sample is firstly mixed and then introduced into a monitoring chamber and measured repeatedly at defined time periods. The most common apparatus for this style of measurement this is a *stopped flow* device, where mixing is achieved rapidly, and then flow is stopped as soon as the homogenously mixed sample fills the monitoring chamber. The measurement is triggered as soon as the flow is stopped, and generally continues as rapidly as possible until the reaction reaches completion. The second method is to use a continuous flow system, where the mixed sample is introduced into a flow-through system, and temporal measurements are achieved by varying the distance between the mixing point and the sampling point.

Both styles of devices have advantages and disadvantages, and the choice depends strongly on several experimental considerations, including the time domain of the reaction, mixing efficiency, sample volume constraints, and sample chemistry constraints (e.g. resistance to photobleaching, or radiation damage). Stopped flow measurements are favoured when there is a small volume of sample that is resistant to measurement induced damage (for example a flurophore that is resistant to photobleaching), where the reaction is not extremely fast, and where the experimental measurements are not slow. In stopped flow measurements the initial point in the measurement is always some degree of time post the start of the reaction (given the time it takes to fill the sample cell, stop the flow and take the first measurement), and the temporal resolution of the measurement is given by the speed at which the measurement can be taken. However, agglomeration of the reacting sample on the channel walls can influence the quality of measurements and, due to the ongoing reaction, leads to only a small window that can be detected before the experiment needs to be repeated. Alternatively, continuous flow measurements favour samples that are sensitive to the measurement, are very rapid, and require temporal resolution finer then the measurement speed of the instrument. Continuous flow measurements allow for measurement very close to the point of mixing, temporal resolution is given by the spatial resolution of the measurement, and the time taken to travel to the point of measurement. Further, the deadtime and temporal resolution is heavily influenced by flowrate, allowing for fine control across many temporal regions. As a result, the observation of the reaction can be precisely controlled. It needs to be considered that continuous flow measurements need more sample volume in comparison to stopped flow methods, to provide a constant flow profile.

#### *2.1.3 Hydrodynamic focusing*

The basis of hydrodynamic focusing lies in a central solution that flows with a lower flowrate within an outer sheath fluid with a higher flowrate. This enables the
