The scattering pattern

for the incident wave, while the scattered wave is described by

with the unit vectors ( ) in -direction.

The absolutes of these vectors are given by

Considering the angle of the X-rays , the wave vectors can be combined to construct the scattering vector which is given by

The geometric representation of this relation is shown in Fig. 16. The phase shift can be described by the multiplication of this vector with .

The scattering curve is received by plotting the measured intensity against the scattering vector’s absolute value which is given by

The absolute value of can also be expressed as which is based on the following equation where the relation between , and has a similarity to the Bragg equation.

3.4.4 The scattering pattern

The coherent scattered X-ray waves interfere with each other and the amplitude of the resulting wave is given by

with the scattering length . For SAXS the scattering length of an electron is given by²⁵⁵

with the elementary charge , the electron’s mass and the speed of light . Consequently, the scattering length of an atom with the ordering number is given by

The cumulative scattering of a sample is the sum of all scattered waves. Therefore, the collective scattering amplitude is received by integration and the scattering length of a single pair of scattering centers is replaced by the density distribution of all scattering centers . This is described by the following formula.

Since this equation’s mathematical form is a Fourier transformation, the scattering amplitude and the scattering center density distribution are a pair of Fourier transforms.

Further, this equation links the real space (with the vector ) to the reciprocal space (with the scattering vector ). However, the scattering amplitude is experimentally not accessible because only the scattering intensity is measured. Their relation is defined as

In other words, the scattering intensity is the time-averaged square of the absolute value of the scattering amplitude . It is time-averaged, as indicated by the pointed brackets , because the measurement is long compared to the system’s dynamics.

Another relevant mathematical operation is the convolution of two functions and it is given by

It is commonly known that a convolution of and in the real space corresponds to the multiplication of their Fourier transforms and in the reciprocal space: ²⁶³

The application of this convolution theorem on the complex scattering amplitude and the scattering intensity yields

As briefly described above, the scattering amplitude and the scattering center density are a pair of Fourier transforms. The above convolution theorem also shows that this is also true for the scattering intensity and the pair correlation function . The scattering intensity results from the square of the absolute value of the scattering amplitude while the pair distribution function results from the self-convolution of the scattering center density .

In the form of a Fourier transform can also be written as²⁶⁴

which is the spatially averaged intensity of a statistically isotropic system without any long range order, such as dilute particles. Here, is the auto-correlation function which is given by

and the following equation highlights the relation between the pair distribution function and the auto-correlation

The following Fig. 17 illustrates these relations between real and reciprocal space graphically.

Fig. 17 Graphical representation of the mathematical operations that link the scattering amplitude , the scattering center density , the scattering intensity and the pair correlation function . (Image adapted from ²⁵⁸)

The Fourier-transformation is fully reversible in both directions while the square of the absolute value and the self-convolution are not. The wanted scattering center density is not extractable from the experimentally measured scattering intensity because the phase information is missing; this is also known as the phase problem.

This is why two different approaches have been developed to receive the scattering center density . The so-called indirect method is based on the modeling of the scattering center density using spline functions. ^265-270 The splines are transformed to the measurement space and fitted to the scattering curve. The desired scattering center density is finally received by the deconvolution of these fitted spline functions. ^271,272

Another approach is the model-based or direct method. This approach uses a given structure with a known scattering center density that is Fourier-transformed to receive the

scattering amplitude . This function’s square yields an analytical expression that can be fitted to the experimentally obtained scattering curve to receive the desired parameters. ²⁷³ 3.4.5 Form factor

The model-based or direct method describes the scattering curve as a function in the basic form of

with as the intensity of the incident beam at an angle of 0° and the form factor .²⁷⁴ Corresponding to its name, the form factor describes the particle’s form and shape. One of these simpler equations is the form factor of spherical particles with the radius and homogenous shapes that have a constant scatting length distribution. These can be described by an analytical expression that is given by273,275,276

An example of simple anisotropic particles, cylinders can be described by the following formula which uses the parameter R to describe the cylinder radius and the parameter for the cylinder length L252,277,278

with

This formula is valid for rigid cylinders, but it has to be modified to describe flexible wormlike micelles which have been investigated in this thesis. The wormlike chain model by Kratky and Porod introduced the following form factor²⁷⁹

with

In this sequence of equations is the Kuhn length and is the persistence length. The variables and indicate the direct distance between two segments and their dividing contour length. The contour length of the worm-chain is .

Further form factors for various particle shapes are compiled in the articles of Pedersen and Förster. ^252,275

Homogeneous particles have a constant density and their scattering length can be described using Heaviside functions where the step corresponds to the particle’s radius. This is not the case for inhomogenous particles, such as self-assembled micelles. However, it is possible to model these systems using core-shell-models which combine the form factors of a homogenous sphere with a homogenous hollow sphere which are blended using density profiles. The approach by Förster and Burger describes the scattering amplitudes using hypergeometric functions, which can be solved analytically in many cases. ²⁸⁰ A typical core-shell-structure with a -dependent density profile yields the following expression

with

and the hypergeometric functions

with for spheres, for cylinders and for lamellae.

3.4.6 Structure factor

So far, this chapter has only covered the diffusive scattering of diluted systems. With raising concentrations, one also has to consider interparticular relations which complicate the mathematical description of the system. ^281,282 This leads to the introduction of the structure factor which appears as an additional factor in the scattering intensity formula

with

This expression is also valid for isotropic systems that form cubic lattices such as spheres.

For increasing concentrations and more ordered systems, this factor generates peaks that decay exponentially. This is indicated by an intensity drop at low q-values along with the raising of a first peak. The peak intensity increases with higher concentrations and higher order of the scattering system. However, strongly diluted or disordered systems will be described solely by the form factor because the structure factor becomes 1. This trend is also shown in the graph of Fig. 18.

Fig. 18 Simulated scattering curves for spherical particles of varying concentrations. These particles are disordered at the lowest concentration and are solely described by the form factor. Raising concentrations lead to increasing particle interactions and the resulting scattering of weakly ordered particles (without long-range order) is described by the structure factor. Highly concentrated particles lead to defined scattering peaks from the long-range ordered lattice that is body-centered cubic in this example. (Image from ²⁵⁸, Copyright M. Konrad)

3.4.7 Bragg reflexes

When a scattering system starts to form long-range ordered structures that are similar to lattices in crystals, like in the case of liquid crystalline or lyotropic samples, Bragg peaks can be observed in the scattering pattern. Then the principles of classical crystallography can be applied for this system’s description while the above-described scattering principles are also valid for these systems of higher order. ²⁸³ In this context, it can be beneficial to use as the scattering vector’s absolute value because a peak position’s reciprocal value ( ) directly equals the corresponding lattice plane distance. The particle scattering that has been outlined so far can be applied to extended, periodic structures because they can be viewed as a successive array of interfering scattering centers with the periodic distance . The density

profile can be expressed using a corresponding delta function. ²⁵⁵

The Fourier transformation yields the scattering amplitude

which shows that a delta function is also obtained in the reciprocal space. The intensity yields discrete lines with a distance of and spatially limited periodicity results in peaks with finite width. The delta function approach is also valid for the structure factor while applying the principles of classical crystallography. Peaks are only generated for discrete values of

with

where , and are the Miller indices of the crystal planes and , and are the reciprocal vectors of the unit cell.

3.4.8 Crystallographic description of lyotropic mesophases

Similar to the formalism of crystallography, lyotropic phases and their periodic structures can be described using unit cells. The translation of a unit cell reconstructs the whole lattice and a unit cell is defined by its edge lengths ( , and ) and the corresponding angles ( ,

and ). A general unit cell is illustrated in Fig. 19.

Fig. 19 Unit cell as parallelepiped with its corresponding coordinate system. Both are defined by the edge lengths ( , and ) and the corresponding angles ( , and ). (Image adapted from ²⁵⁸)

The planes of the crystal lattice which is defined by the unit cell can be uniquely identified by the Miller indices , and . The indices denote the planes orthogonal to a direction in the basis of the reciprocal lattice vectors while their greatest common divisor should be 1. The unit cell coordinate system of a cubic lattice is orthogonal with as illustrated in

Fig. 20. This figure shows examples of lyotropic mesophases: a body-centered cubic (bcc) lattice of long-range ordered spherical micelles (A), closest packed cylinders (B) and a lamellar phase (C).

Fig. 20 Examples of common lyotropic mesophases: body-centered cubic (A), hexagonally closest packed (B) and lamellar (C). (Image from ²⁵⁸, Copyright M. Konrad)

The lattice plane distance for orthogonal unit cells is given by

where , and are the so-called reciprocal vectors which are defined as

, ,

with as the unit cells volume.

The simple cubic lattice ( ) represents a special case where the lattice plane distance is given by

with the peak positions of the lattice

In case of a hexagonal lattice with and , only the planes that are parallel to the c-axis (= ) are of interest. This allows the reduction of the three-dimensional problem to a two-dimensional one and the point lattice can be visualized two-dimensionally as shown in Fig. 21.

Fig. 21 Point lattice projection ( ) of the hexagonal packing. The six-fold symmetry and the (10) and (11) lattice planes are indicated by the lines. (Image from ²⁵⁸, Copyright M. Konrad)

The calculation of lattice plane distance of this two-dimensional lattice is given by

with the corresponding peak positions

3.4.9 Model-based order analysis

The model-based analysis of experimental scattering data from ordered systems requires additional parameters. The complete structure factor for ordered systems is given by²⁵⁸

with the number of structure elements (spheres, cylinders or lamellae) per unit cell, the dimensionality (3 for spheres, 2 for cylinders and 1 for lamellae) and the dimension-dependent volume . The parameter has a value of 1 for lamellae, for cylinders and

for spheres. The factor considers the multiplicity of peaks that stem from lattice plane multitudes with identical peak positions while the factor contains any extinction rules.

The profile , which determines the peaks shapes and locations, is normalized:

and its general equation is given by²⁸⁰

with

This profile shape smoothly transitions from a Lorentz peak shape at very small -values

to a Gauß shape at large values of

The peak width is determined by the domain size which is given by

The correlation function is given by

which also describes the deviation, known as the Debye-Waller factor, from the ideal lattice position based on the following equation

In this formula is the mean square deviation and is the nearest neighbor distance.

3.4.10 Particle orientation distribution

In the above discussion, the orientation of anisotropic particles, such as cylindrical micelles or wormlike micelles, has been assumed to be isotropic. This results in ring-like, isotropic scattering patterns on two-dimensional detectors which can simply be represented as radially averaged scattering curves. If the particles in a sample are oriented however, the resulting scattering patterns are also anisotropic and both components of the scattering vector ( and

) need to be considered in the analysis.

A good experimental example for anisotropic scattering patterns are small angle X-ray scattering studies of shear-oriented wormlike micelles in small microfluidic channel geometries, as illustrated in Fig. 22.

Fig. 22 (A) The sample is flowing though the microchannel geometry and passes a narrow section (red box). (B) 3D illustration of the X-ray microbeam that passes the microchannel of flowing wormlike micelles (hexagonal closest packing, determined by SAXS) and the resulting SAXS-pattern which is captured using a digital detector (Pilatus 300K, Dectris).

The parallel or perpendicular orientation of the wormlike micelles is controlled by varying the experimental parameters, such as the channel geometry, the flow rate or the sample concentration.(Adapted from ¹³, Copyright PNAS)

The wormlike micelles in the microchannel are oriented according to the flow-induced shear and extensional forces which are controlled by the experimental parameters such as microchannel geometry, flow speed and particle concentration. Due to the highly reproducible flow conditions, this setup enables the correlation between structural- and orientation information from SAXS studies with fluid dynamic studies from other methods like high speed video analysis, particle image velocimetry or polarization microscopy.

Instead of splitting the scattering vector into its components ( and ) it is more beneficial to use the polar coordinate system. ²⁵² Here, the scattering vector is described by its absolute value and the angle between cylinder axis and the scattering vector. This results in the following equation for the scattering intensity of oriented cylindrical micelles²⁵⁸

with the form factor , the structure factor , the fraction of micelles with the angle and the number of micelles . The form factor is further defined by

The distribution function describes the cylindrical micelle orientation with the angle between the cylinder axis with the base vector and the director which defines the direction of the shear field.

The expressions and are yielded from the factorization of the scattering amplitude into its cross section- and length-contribution. The pointed brackets indicate the averaging across the corresponding size distributions of cylinders and radii. As described above, it is possible to express for typical bock copolymers, that have a core-shells-structure and a density-profile of , by using hypergeometric functions. is given by

with and the ratios of the radii ( ) and densities ( ) compared to the shell.

These are given by

The form factor calculation requires a definition for the relation between the angles and , as shown in Fig. 23.

Fig. 23 Three-dimensional illustration of the vector sphere. (Image from ^252,258, Copyright Elsevier).

In this figure the director is determined by the angles and . The base vectors of the cylinder axes are positioned on a cone that is directed towards . Therefore, the range of angles of is given by and the integration of the function leads to

The vector is calculated by using the rotation matrix based on the following equation

The vectors and are given by

and

The rotation matrix is defined by

with

Lastly, is given by

The orientation of anisotropic particles can be described by a range of distribution functions that are given by

with the parameter which can take on values between zero and infinity. The graphs of these different distribution functions are shown in Fig. 24.

Fig. 24 The graphs of different distribution functions. (Image from ²⁵², Copyright Elsevier).

Among these functions, the Onsager and Maier-Saupe distributions can be pointed out because they are very important for the description of particle orientations in lyotropic and thermotropic liquid-crystalline systems. The distribution functions are normalized using the following factor

The resulting mean deviation angle between cylinders and the director, which can take on values between 0° and 90°, is given by

A very general function is the Laguerre distribution which changes from a Gauß function (at

) to a Heaviside function (at very large ).

The order parameter can take on values between 0 and 1. For a known distribution function it is given by

The above equations are valid for diluted systems and with raising concentrations it becomes necessary to take the structure factor into account as described by van-der-Schoot²⁸⁴

with the cylinder concentration and with

4 Methods and Techniques

4.1 Photolithography

Early materials that have been used for the fabrication of microfluidic devices include glass or silicon. ⁷⁸ These materials were used because a wide range of microstructuring techniques were readily available from the field of semiconductor technology and because these materials offer resistance against a wide range of solvents. While glass is additionally optically transparent, silicon is opaque which limits microscopic applications for the latter material. A downside of both materials is the need for expensive clean room environments during their processing and the use of aggressive chemicals which makes it an expensive and resource intensive process. ^57,285

The fast and effective fabrication of microfluidic devices in a short time became possible through the progress in the areas of photo- and soft lithography. ⁵⁶ Polymer-based materials offer more application friendly properties which is why they have gained importance during the past years. ^230,286 One example for these are photoresists like SU-8 which are used in photolithography. ²⁸⁷ This technology enables the creation of microstructures based on specific technical drawings which are created using computer aided design (CAD) software.

By using the computer-designed photo masks, photo resists are selectively exposed for the generation of microstructures. ⁵⁶ These microstructures are then replicated using soft materials such as polymers to create microstructured stamps or microfluidic devices which is why this technology is called soft lithography. ⁵⁵ Due to the precise control over design features and the shortened system optimization feedback loop, this process enables (simulation-based) rapid protoyping. ⁵⁷ Furthermore, the replication templates, also called masters, can be re-used multiple times without quality loss enabling mass production and low fabrication costs. ²⁸⁸

The microstructuring of the photo resist happens by selective exposure using photo masks. ⁵⁷ In case of a negative photoresist like SU-8, the non-cross-linked areas of the photoresist remain soluble and will be removed in the development stage of the lithographic process. The development bath typically contains 80 to 100% 2-methoxy-1-methylethyl acetate solution and is, similar to the photo resists, commercially available (mr-DEV 600, Microchem). Only the insoluble cross-linked areas of the photoresist remain on the substrate, typically a polished flat silicon wafer, and will be used as a master template for the subsequent replication steps.

55,56 The resolution of soft lithography is limited by diffraction and, hence, dependent on the wave length of the light source. ⁵⁶ The range of available photo resist materials is only small

55,56 The resolution of soft lithography is limited by diffraction and, hence, dependent on the wave length of the light source. ⁵⁶ The range of available photo resist materials is only small

Im Dokument Microfluidics at high-intensity X-ray sources: from microflow chips to microfluidic liquid jet systems (Seite 55-0)