**1.1 Microfluidic principles**

Fundamentally, the fluid dynamics in micro-dimensions are different from macroscopic systems. Fluid flows in these tiny systems are characterised by nonchaotic, smooth flow, where the fluid travels in parallel layers and the only interaction between those layers of flow is diffusion. By adapting reactions to microfluidic environments, the time axis of a reaction is converted into a distance axis along the outlet channel of the microfluidic device. This is key to enabling time-resolved studies *in situ* in a microfluidic channel.

The Navier–Stokes equation describes the motion of fluids mathematically, and is derived from Newton's second law of motion (F = ma), resulting in a set of two partial differential equations. For an incompressible Newtonian fluid, the Navier– Stokes equation is defined as:

$$
\rho \left[ \frac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \cdot \nabla) \boldsymbol{u} \right] = \eta \nabla^2 \boldsymbol{u} - \nabla p + \boldsymbol{F} \tag{1}
$$

where ρ is the density and η the viscosity of the fluid, *p* is the pressure, *u* the vector of the fluid flow, ∇ is the Nabla-Operator and *F* stands for any additional forces, that are directed at the fluid. The left side of the equation represents internal accelerations, and the right side represents the stress force per unit volume resulting from a pressure gradient and the viscosity of the fluid. In microfluidics, body forces are negligible, leading to a simplified, linear equation:

$$
\eta \nabla^2 u = \nabla p \tag{2}
$$

Microfluidic channel systems have generally a high surface-to-volume ratio, thus surface properties have significant effects on flow resistance and the velocity profile. To describe the interaction of a flowing liquid and a solid surface in microfluidic devices, Navier defined boundary conditions. The flow velocity tangential to the surface *υ<sup>x</sup>* is proportional to the shear stress at the surface and

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

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

*<sup>υ</sup><sup>x</sup>* <sup>¼</sup> *<sup>β</sup> <sup>d</sup>υ<sup>x</sup>*

β is the slip length, or Navier length, and is defined as the distance from a point inside the channel to the surface, where the velocity is zero. Where *β* ¼ 0 is a "no-

Every biological process or chemical reaction is limited by the converging and mixing of the reactants. Mixing in fluidic systems can generally occur *via* two methods – diffusion or advection. On the macroscopic scale, mixing is achieved by "chaotic advection" or turbulence, while on the micron-scale it is driven by diffusion. Diffusion specifies the migration of particles along a concentration gradient, and thereby always takes place from an area of high concentration to an area of lower concentration. This flux is in proportion to the diffusion coefficient, *D,* given by Fick's first law of diffusion. Solving Fick's diffusion law for adequate boundary conditions, the diffusion coefficient can be described for spherical particles with

> *<sup>D</sup>* <sup>¼</sup> *kBT* 6*πηr*

with *kB* as the Boltzmann constant,*T* as the temperature and η as the solvent viscosity. The relation between advection and diffusion for mass transport is

*diffusion* <sup>¼</sup> *<sup>υ</sup><sup>d</sup>*

For turbulent mixing, advection dominates the above equation, leading to high *Pe* numbers. In microfluidics, turbulent chaotic mixing is very difficult to achieve, because the Reynolds numbers are almost always very low. Thus, in microfluidic channels, advection is almost always very small, and diffusion dominates, resulting in *Pe* numbers that are low. As such mixer design in microfluidics devices seeks to optimise diffusion [10, 11]. Along microfluidic channels, diffusion becomes insignificant when compared to convection occurring far downstream at the outlet channel. Thus most mixing devices incorporate some method for laminating flows to reduce diffusion distances, and reduce mixing times. Most commonly, these mixers are simple Y- or T-shaped cross channels, and diffusive mixing in these types of mixers for kinetic experiments can be described by the following equations:

*Pe* � *advection*

*d*1 *d*2 ¼ *η*1 *η*2 *Q*1 *Q*2

<sup>¼</sup> *<sup>η</sup>MC ηSC*<sup>1</sup> þ *ηSC*<sup>2</sup>

where d is the thickness of the relevant layer, η the viscosity and *Q* the volume flow. The following assumptions must be fulfilled for these equations to be true:

*QMC QSC*<sup>1</sup> þ *QSC*<sup>2</sup>

*dMC dSC*<sup>1</sup> þ *dSC*<sup>2</sup>

**13**

slip" condition, describing the interaction between fluids and walls [9].

radius *r* in low *Re* numbers by the Stokes-Einstein relation:

described by the Péclet number (*Pe*) [8].

*dy* (6)

*<sup>D</sup>* (8)

(7)

(9)

(10)

therefore given by:

Treating the incompressible liquid as a continuum, the Navier–Stokes equation can be expressed as the continuity equation:

$$\nabla \cdot \boldsymbol{u} = \mathbf{0} \tag{3}$$

This means that the flux of liquid into a volume is the same as the flux out of a volume over a period of time. Additionally, the continuity equation is time-independent, restricting fluid flow in microfluidic channels to be symmetric in time [8].

To describe and compare phenomena on different scales, various dimensionless numbers for microfluidics were introduced. The most important is the Reynolds number (*Re*), showing the relation of inertial and viscous forces of a fluid. It is defined as:

$$Re \equiv \frac{inertial\ forces}{viscous\ forces} = \frac{|\rho \ (u \cdot \nabla) \ u|}{|\eta \nabla^2 u|} = \frac{\rho v}{\eta} d \tag{4}$$

where *υ* is the flow velocity and *d* the characteristic length of the system, which in microfluidics is the diameter of the channel. The Reynolds number decreases with decreasing size of the system, reflecting the increased importance of viscous forces. The transition from turbulent to laminar flow is represented by *Re* being below 2040 � 10.

The next most important dimensionless number is the Weber number (*We*), which describes the relation of the fluid surface tension to its internal forces, where γ is the surface tension of the fluid:

$$\text{We} \equiv \frac{\rho u^2 v}{\gamma} d \tag{5}$$

*Microfluidics for Time-Resolved Small-Angle X-Ray Scattering DOI: http://dx.doi.org/10.5772/intechopen.95059*

**1.1 Microfluidic principles**

*Advances in Microfluidics and Nanofluids*

Stokes equation is defined as:

studies *in situ* in a microfluidic channel.

*ρ ∂u ∂t*

are negligible, leading to a simplified, linear equation:

can be expressed as the continuity equation:

defined as:

below 2040 � 10.

**12**

γ is the surface tension of the fluid:

þ ð Þ *u* � ∇ *u* 

Fundamentally, the fluid dynamics in micro-dimensions are different from macroscopic systems. Fluid flows in these tiny systems are characterised by nonchaotic, smooth flow, where the fluid travels in parallel layers and the only interaction between those layers of flow is diffusion. By adapting reactions to microfluidic environments, the time axis of a reaction is converted into a distance axis along the outlet channel of the microfluidic device. This is key to enabling time-resolved

The Navier–Stokes equation describes the motion of fluids mathematically, and is derived from Newton's second law of motion (F = ma), resulting in a set of two partial differential equations. For an incompressible Newtonian fluid, the Navier–

<sup>¼</sup> *<sup>η</sup>*∇<sup>2</sup>

where ρ is the density and η the viscosity of the fluid, *p* is the pressure, *u* the vector of the fluid flow, ∇ is the Nabla-Operator and *F* stands for any additional forces, that are directed at the fluid. The left side of the equation represents internal accelerations, and the right side represents the stress force per unit volume resulting from a pressure gradient and the viscosity of the fluid. In microfluidics, body forces

*η*∇<sup>2</sup>

over a period of time. Additionally, the continuity equation is time-independent, restricting fluid flow in microfluidic channels to be symmetric in time [8].

*Re* � *inertial forces*

Treating the incompressible liquid as a continuum, the Navier–Stokes equation

This means that the flux of liquid into a volume is the same as the flux out of a volume

To describe and compare phenomena on different scales, various dimensionless numbers for microfluidics were introduced. The most important is the Reynolds number (*Re*), showing the relation of inertial and viscous forces of a fluid. It is

*viscous forces* <sup>¼</sup> j j *<sup>ρ</sup>* ð Þ *<sup>u</sup>* � <sup>∇</sup> *<sup>u</sup>*

where *υ* is the flow velocity and *d* the characteristic length of the system, which in microfluidics is the diameter of the channel. The Reynolds number decreases with decreasing size of the system, reflecting the increased importance of viscous forces. The transition from turbulent to laminar flow is represented by *Re* being

The next most important dimensionless number is the Weber number (*We*), which describes the relation of the fluid surface tension to its internal forces, where

> *We* � *<sup>ρ</sup>u*<sup>2</sup>*<sup>υ</sup> γ*

*η*∇<sup>2</sup> *u* 

 <sup>¼</sup> *ρυ η*

*d* (4)

*d* (5)

*u* � ∇*p* þ *F* (1)

*u* ¼ ∇*p* (2)

∇ � *u* ¼ 0 (3)

Microfluidic channel systems have generally a high surface-to-volume ratio, thus surface properties have significant effects on flow resistance and the velocity profile. To describe the interaction of a flowing liquid and a solid surface in microfluidic devices, Navier defined boundary conditions. The flow velocity tangential to the surface *υ<sup>x</sup>* is proportional to the shear stress at the surface and therefore given by:

$$
\rho\_{\mathbf{x}} = \beta \left. \frac{d\nu\_{\mathbf{x}}}{dy} \right. \tag{6}
$$

β is the slip length, or Navier length, and is defined as the distance from a point inside the channel to the surface, where the velocity is zero. Where *β* ¼ 0 is a "noslip" condition, describing the interaction between fluids and walls [9].

Every biological process or chemical reaction is limited by the converging and mixing of the reactants. Mixing in fluidic systems can generally occur *via* two methods – diffusion or advection. On the macroscopic scale, mixing is achieved by "chaotic advection" or turbulence, while on the micron-scale it is driven by diffusion. Diffusion specifies the migration of particles along a concentration gradient, and thereby always takes place from an area of high concentration to an area of lower concentration. This flux is in proportion to the diffusion coefficient, *D,* given by Fick's first law of diffusion. Solving Fick's diffusion law for adequate boundary conditions, the diffusion coefficient can be described for spherical particles with radius *r* in low *Re* numbers by the Stokes-Einstein relation:

$$D = \frac{k\_B T}{6\pi\eta r} \tag{7}$$

with *kB* as the Boltzmann constant,*T* as the temperature and η as the solvent viscosity. The relation between advection and diffusion for mass transport is described by the Péclet number (*Pe*) [8].

$$Pe \equiv \frac{advection}{d\text{diffusion}} = \frac{\nu d}{D} \tag{8}$$

For turbulent mixing, advection dominates the above equation, leading to high *Pe* numbers. In microfluidics, turbulent chaotic mixing is very difficult to achieve, because the Reynolds numbers are almost always very low. Thus, in microfluidic channels, advection is almost always very small, and diffusion dominates, resulting in *Pe* numbers that are low. As such mixer design in microfluidics devices seeks to optimise diffusion [10, 11]. Along microfluidic channels, diffusion becomes insignificant when compared to convection occurring far downstream at the outlet channel. Thus most mixing devices incorporate some method for laminating flows to reduce diffusion distances, and reduce mixing times. Most commonly, these mixers are simple Y- or T-shaped cross channels, and diffusive mixing in these types of mixers for kinetic experiments can be described by the following equations:

$$\frac{d\_1}{d\_2} = \frac{\eta\_1}{\eta\_2} \frac{Q\_1}{Q\_2} \tag{9}$$

$$\frac{d\_{\rm MC}}{d\_{\rm SC\_1} + d\_{\rm SC\_2}} = \frac{\eta\_{\rm MC}}{\eta\_{\rm SC\_1} + \eta\_{\rm SC\_2}} \frac{Q\_{\rm MC}}{Q\_{\rm SC\_1} + Q\_{\rm SC\_2}} \tag{10}$$

where d is the thickness of the relevant layer, η the viscosity and *Q* the volume flow. The following assumptions must be fulfilled for these equations to be true:


*<sup>q</sup>* <sup>¼</sup> <sup>4</sup>*<sup>π</sup> sin <sup>θ</sup>*

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

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

Where *θ* is the angle from the incident X-ray beam to the point on the detector where the intensity is measured, and *λ* is the wavelength of the incident X-rays (see **Figure 1A**). The derivation of the dependence of scattered intensity on the volume, concentration and electron density contrast of a particle described in Eq. 12 is given in detail in [13], which we highly recommend for further reading. The form factor *P (q)* is typically a defined function, and varies depending on the physical parameters of the particle; for example a sphere with homogenous electron density has a different form factor function to that of a hollow sphere of the same size.

The structure factor component (*S(q)*) of Eq. (11) is a further analytical function that describes how the particles are arranged in the solution, e.g. forming large ordered structures with defined correlation lengths. Largely, samples are measured in a dilute condition, where the concentration of the particle is kept low enough to avoid these secondary interference effects, and thus *S(q)* can be ignored. Where this effect cannot be avoided by reducing concentration, the use of hard sphere packing models or ionic charge–charge interaction models defining the effect as a function of *q* may be used to account for this effect, and provide information on

Thus, for a sufficiently monodisperse sample, or a defined mixture of particles, it is possible to define an analytical model that provides volume, size and shape information. In polymer and colloid science, SAXS is used for many applications, including analysing the hierarchical nature of polymers in solution to assess clumping, local structure, overall morphology, and subunit arrangement, assessing the shape, size and dispersity of nanoparticles in solution, and investigating the dynamics, and evolution of particle size and shape under varying solution

*(A) Schematic illustration of the Bragg equation with incident and reflected X-rays on two scattering planes, showing the lattice distance* d*, the half scattering angle θ, the wavelength λ and the path difference defined by Bragg's law. (B) Geometric construction of the scattering vector* q *from the incident wave vector* k0 *and the scattered wave*

*vector* k *with the half scattering angle θ. (C) Schematic setup of a small-angle X-ray scattering setup.*

changes in long range order in a sample.

**Figure 1.**

**15**

*<sup>λ</sup>* (12)

4.The channel geometry is rectangular and all channel parts have the same height.

Eq. (9) applies to Y-shaped channel geometries where layer 1 and 2 are the spaces of two introduced liquid streams in the inlet channels, which merge in the outlet channel. For T-shaped channels where two side channels (SC1 and SC2) hydrodynamically focus a main channel (MC) stream, Eq. (10) applies [12].

#### **1.2 Principles of small-angle X-ray scattering**

Small angle X-ray Scattering (SAXS) is an extremely versatile technique used for investigating particle size, shape and dynamics that can be applied to a wide range of scientific problems. It is amenable to a wide range of particles, from the very small, of around a few nanometres, to very large sized structures in the order of a micron. It can be used to study mixtures, and the evolution of shape in reaction mixtures, and is widely used in biophysics and structural biology to confirm structure, and investigate structures that are not amenable to other structure investigations. SAXS can be used across all states of matter, including solids, liquids, gases, semisolid sample such as gels, and plasma. We will focus here on solution scattering, as this is the most applicable for microfluidic applications.

We aim to provide a brief overview of SAXS for solution scattering and time resolved measurements, but highly recommend Feigin and Svergun, 1987 [13] for a more comprehensive in depth review of SAXS measurements. In general, a solution SAXS experiment is relatively simple (which is one of the great attractions for the technique). A sample, in an appropriate sample cell, is exposed to a focussed, collimated monochromatic X-ray beam, and at a distance away from the sample the intensity of scattered X-rays is recorded using a 2D X-ray detector (**Figure 1B**). The resulting image is termed a scattering pattern. Similarly, the scattering from a matched pure background solvent is collected, and then subtracted from the sample scattering pattern to provide a scattering pattern that arises purely from the sample particles. The variation of the scattered intensity with angle, where the measured angles are very small, is related to differences in electron density between the sample and solvent, and the interatomic distances inside the sample particle, and thus contains information on the size and shape of the particle.

Scattering in solution is generally considered isotropic, as most particle systems adopt random orientations in solution. This allows for analytical mathematical descriptions of the scattering profile on the basis of particle shape. Scattered intensity (I) is described as a function of momentum transfer, *q*, and in a simplified form can be given as:

$$I(q) = \frac{N}{V} \mathbf{V}^2 (\rho \mathbf{1} - \rho \mathbf{2})^2 P(q) \mathbf{S}(q) \tag{11}$$

Where N is the concentration of the particle in the solution, V is the volume of the particle, *ρ*1 � *ρ*2 is the contrast in electron density between the solvent and the particle, and *q* is defined as:

*Microfluidics for Time-Resolved Small-Angle X-Ray Scattering DOI: http://dx.doi.org/10.5772/intechopen.95059*

1.The microchannel inhibits steady and laminar flow.

3.Density and viscosity of all fluids is the same in all channels and do not change

4.The channel geometry is rectangular and all channel parts have the same

Eq. (9) applies to Y-shaped channel geometries where layer 1 and 2 are the spaces of two introduced liquid streams in the inlet channels, which merge in the outlet channel. For T-shaped channels where two side channels (SC1 and SC2) hydrodynamically focus a main channel (MC) stream, Eq. (10) applies [12].

Small angle X-ray Scattering (SAXS) is an extremely versatile technique used for investigating particle size, shape and dynamics that can be applied to a wide range of scientific problems. It is amenable to a wide range of particles, from the very small, of around a few nanometres, to very large sized structures in the order of a micron. It can be used to study mixtures, and the evolution of shape in reaction mixtures, and is widely used in biophysics and structural biology to confirm structure, and investigate structures that are not amenable to other structure investigations. SAXS can be used across all states of matter, including solids, liquids, gases, semisolid sample such as gels, and plasma. We will focus here on solution scatter-

We aim to provide a brief overview of SAXS for solution scattering and time resolved measurements, but highly recommend Feigin and Svergun, 1987 [13] for a more comprehensive in depth review of SAXS measurements. In general, a solution SAXS experiment is relatively simple (which is one of the great attractions for the technique). A sample, in an appropriate sample cell, is exposed to a focussed, collimated monochromatic X-ray beam, and at a distance away from the sample the intensity of scattered X-rays is recorded using a 2D X-ray detector (**Figure 1B**). The resulting image is termed a scattering pattern. Similarly, the scattering from a matched pure background solvent is collected, and then subtracted from the sample scattering pattern to provide a scattering pattern that arises purely from the sample particles. The variation of the scattered intensity with angle, where the measured angles are very small, is related to differences in electron density between the sample and solvent, and the interatomic distances inside the sample particle, and

Scattering in solution is generally considered isotropic, as most particle systems

ð Þ *<sup>ρ</sup>*<sup>1</sup> � *<sup>ρ</sup>*<sup>2</sup> <sup>2</sup>

Where N is the concentration of the particle in the solution, V is the volume of the particle, *ρ*1 � *ρ*2 is the contrast in electron density between the solvent and the

*P q*ð Þ*S q*ð Þ (11)

adopt random orientations in solution. This allows for analytical mathematical descriptions of the scattering profile on the basis of particle shape. Scattered intensity (I) is described as a function of momentum transfer, *q*, and in a simplified form

2.The fluids are all Newtonian.

*Advances in Microfluidics and Nanofluids*

**1.2 Principles of small-angle X-ray scattering**

ing, as this is the most applicable for microfluidic applications.

thus contains information on the size and shape of the particle.

*I q*ð Þ¼ *<sup>N</sup>*

*<sup>V</sup> <sup>V</sup>*<sup>2</sup>

during the experiment.

height.

can be given as:

**14**

particle, and *q* is defined as:

$$q = 4\pi \frac{\sin \theta}{\lambda} \tag{12}$$

Where *θ* is the angle from the incident X-ray beam to the point on the detector where the intensity is measured, and *λ* is the wavelength of the incident X-rays (see **Figure 1A**). The derivation of the dependence of scattered intensity on the volume, concentration and electron density contrast of a particle described in Eq. 12 is given in detail in [13], which we highly recommend for further reading. The form factor *P (q)* is typically a defined function, and varies depending on the physical parameters of the particle; for example a sphere with homogenous electron density has a different form factor function to that of a hollow sphere of the same size.

The structure factor component (*S(q)*) of Eq. (11) is a further analytical function that describes how the particles are arranged in the solution, e.g. forming large ordered structures with defined correlation lengths. Largely, samples are measured in a dilute condition, where the concentration of the particle is kept low enough to avoid these secondary interference effects, and thus *S(q)* can be ignored. Where this effect cannot be avoided by reducing concentration, the use of hard sphere packing models or ionic charge–charge interaction models defining the effect as a function of *q* may be used to account for this effect, and provide information on changes in long range order in a sample.

Thus, for a sufficiently monodisperse sample, or a defined mixture of particles, it is possible to define an analytical model that provides volume, size and shape information. In polymer and colloid science, SAXS is used for many applications, including analysing the hierarchical nature of polymers in solution to assess clumping, local structure, overall morphology, and subunit arrangement, assessing the shape, size and dispersity of nanoparticles in solution, and investigating the dynamics, and evolution of particle size and shape under varying solution

#### **Figure 1.**

*(A) Schematic illustration of the Bragg equation with incident and reflected X-rays on two scattering planes, showing the lattice distance* d*, the half scattering angle θ, the wavelength λ and the path difference defined by Bragg's law. (B) Geometric construction of the scattering vector* q *from the incident wave vector* k0 *and the scattered wave vector* k *with the half scattering angle θ. (C) Schematic setup of a small-angle X-ray scattering setup.*

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 the sample that can significantly influence results.

All experiments require planning and consideration of the simultaneous use of analysis, mixing and cleaning equipment due to the generally small dimensions of microfluidic devices. In the past decades, a very diverse range of microfluidic reactor devices have been designed for time-resolved studies of reactions. Designs such as continuous-flow, stopped-flow, droplet-based and digital microfluidics have been developed and applied to produce materials with sizes ranging from nanometres to almost millimetres. In this chapter, we are focusing on continuous and stopped flow devices, in particular on hydrodynamic focusing techniques. In comparison to droplet-based techniques, hydrodynamic focusing is a straightforward approach to implement, due to its pure hydrodynamic principles. It only includes surface tension effects at the liquid–liquid interface in the outlet channel of the microfluidic device without the need of consideration of surface tension effects at liquid–gas interfaces. These devices offer stability at high flowrates, allow highthroughput applications and enable highly controllable operational conditions, as the flow behaviour is the only influential parameter that needs to be considered for

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

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

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 (turbu-

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

<sup>4</sup>*η<sup>L</sup> <sup>a</sup>*<sup>2</sup> � *<sup>r</sup>*

With pressure *p* and viscosity η over the channel length L and channel radius a.

*<sup>R</sup>* <sup>¼</sup> <sup>8</sup>*η<sup>L</sup>*

For rectangular shaped channels with height *h*, width *w* and small aspect ratio

*<sup>n</sup>*<sup>3</sup> <sup>1</sup> � *cosh n<sup>π</sup> <sup>y</sup>*

X∞ *n*¼1, 3, 5

*cosh nπ <sup>w</sup>* 2*h* � � " # *sin n<sup>π</sup> <sup>z</sup>*

1

� � " # ! �<sup>1</sup>

*h* � �

*<sup>n</sup>*<sup>5</sup> *tanh <sup>n</sup>π<sup>w</sup>*

2*h*

<sup>2</sup> � � (13)

*<sup>π</sup>a*<sup>4</sup> (14)

*h*

� � (15)

(16)

*<sup>ν</sup>x*ð Þ¼� *<sup>r</sup>*, *<sup>ϕ</sup> <sup>Δ</sup><sup>p</sup>*

time-resolved studies.

can be derived as:

**17**

*2.1.1 Flow field considerations*

lent) mixing needs to be considered.

The hydraulic resistance *R* results then as:

*<sup>ν</sup><sup>x</sup>* <sup>¼</sup> <sup>4</sup>*h*<sup>2</sup>

*<sup>R</sup>* <sup>¼</sup> <sup>12</sup>*η<sup>L</sup>*

and the hydraulic resistance R is then [14]:

(*w* > *h*) the velocity field over the coordinates *x, y, z* is:

*Δp π*<sup>3</sup>*ηL*

*wh*<sup>3</sup> <sup>1</sup> � *<sup>h</sup>*

X∞ *n*, *odd*

*w*

192 *π*5

1

#### **1.3 Microfluidic devices and X-rays**

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

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.
