Hydrodynamic Analysis of Particles and Macromolecules
Towards a global analysis of hybrid nanomaterial composition distributions
Dissertation submitted for the degree of Doctor of Natural Sciences
Presented by Marius Schmid
at the
Faculty of Sciences Department of Chemistry
Date of the oral examination: 13.10.2015
First referee: Professor Dr. Cölfen
Second referee: Professor Dr. Wittemann
Kurzreferat In den letzten Jahren fanden Nanopartikel ein immer breiters Anwen- dungsspektrum, sowohl in Industrie als auch in Wissenschaft. Während die Zahl von potenziellen Anwendungsgebieten von Jahr zu Jahr steigt, bleiben die analytischen Möglichkeiten immer weiter zurück. Im letzten Jahrhundert lag der Fokus von ana- lytischen Techniken mit Nanometerauösung auf biologischen Proben. Mit dem stark gestiegen Interesse an Nanopartikeln wurden auch Fortschritte in Bezug auf deren Ana- lyse gemacht, aber die heutigen Anwendungen bleiben immer noch weit hinter den Möglichkeiten zurück. Der Fokus dieser Arbeit lag auf der Entwicklung neuer Metho- den zur Analyse von Nanopartikeln und anderen Hybridmaterialien.
Das gröÿte Potential wurde in der Kombination aus verschiedenen Messtechniken ge- funden, aber es wurden auch neue Methoden für einzelne Messtechniken entwickelt.
Es wurde eine Methode zur Analyse von Nanopartikeln mittels Simulation von Tensid- schichten entwickelt. Hierfür wurden sowohl ein Ansatz, der eine sphärische Struktur voraussetzt, als auch einer für nicht- sphärische Materialien entwickelt. Auf diesem Weg können die genauen Achsenverhältnisse von stäbchenförmigen Spezies und deren Zu- samensetzung berechnet werden. Die zweite Methode verwendet ein neu entwickeltes Dichtegradientenmaterial für die Erweiterung des möglichen Anwendungsbereiches von Dichtegradienten von Dichten zwischen 1- 2,5 g/ml zu 1- 5 g/ml. Die Hafniumoxid Na- nopartikel erlauben sowohl eine analytische, als auch eine präparative Verwendung. Als dritter Ansatz wurde eine Methode entwickelt, die die Daten von mehreren verschie- dene analytischen Techniken verwendet, um einen Globalt durchzuführen. Mit dieser Methode können nicht nur hydrodynamische Parameter und Informationen über die Zusammensetzung, sondern auch Lichtstreuparameter des Materials bestimmt werden.
Es wurde ein Programm zur Analyse von AF4 Daten entwickelt, das eine annahmefreie Analyse von Daten aus Asymmetrischer Fluÿ Felduÿfraktionierung gewährleistet. Zu- dem wurde eine Diusionskorrektur für die Auswertung von AF4 Daten entwickelt. Mit Hilfe der Diusionskorrektur kann die analytische Trennung von Spezies soweit erhöht werden, dass selbst Spezies mit einem Unterschied von nur 3 % im Molekulargewicht getrennt werden konnten.
Abstract In the last decades, nanoparticles have found a lot of possible applications in both science and industry. While the synthesis of nanoparticles and number of applications have increased signicantly, the development of analytical methods has stagnated. In the last century, most of the nanometre range research was focused on biopolymers. Therefore, the analysis methods with nanometre resolution strongly fo- cus on biopolymers as well. With the growth of nanoparticle research, improvements regarding the analysis of nanoparticles were made, but there is still a huge gap between the possible applications and the ones that already exist. The major interest behind this thesis is to close this gap. The focus of this thesis was on the development of new approaches for the analysis of nanoparticles and other solutes. The biggest potential was found in a global analysis approach. Several dierent approaches were designed.
First, a surfactant layer simulation was used on spherical and non-spherical species to obtain hydrodynamic and composition information about the nanoparticles. A second approach used a newly designed density gradient material to extend the possible range of density gradient materials from 1- 2 g/ml to a new range of 1- 5 g/ml. This allows the density determination and the preparative separation of nanoparticles in density gradients. A global analysis approach was designed to determine not only the hydrody- namic parameters and the composition of particles, but information for light scattering as well.
A new program for the evaluation of AF4 data was developed. With this program gen- eral assumptions for the calculation of diusion coecient distribution can be avoided.
The program is able to apply a diusion correction, which enlarges the resolution of the analytical application of AF4. By using the diusion correction even species with a dierence of only 3%in molecular weight could be baseline separated. Furthermore, small signals that are overlayed by other signals could be observed. Dierent theories for adsorption of species on nanoparticles were developed. The approach can be used for either spherical or non-spherical hybrid materials.
Danksagung Diese Arbeit wurde zwischen Dezember 2010 und August 2014 im Ar- beitskreis von Prof. Dr. Cölfen angefertigt. Ich danke ihm für die freundliche Aufnahme in seiner Arbeitsgruppe und für seine exzellente Betreuung. Insbesondere möchte ich ihm für die interessante Themenstellung danken.
Im Weiteren möchte ich Prof. Dr. Wittemann für die Übernahme des Zweitgutach- tens danken.
Bei Benedikt Häusele möchte ich mich für die gemeinsame Arbeit zur Diusionskor- rektur und vielen anderen Themen bedanken.
Emre Brooks möchte ich für die Zusammenarbeit im Bereich der Globalanalyse und der Diusionskorrektur danken.
Cornelia Völkle möchte ich für die Zusammenarbeit im Bereich der Globalanalyse dan- ken.
Bei Maria Maier und Christan Häge bedanke ich mich für die Zusamenarbeit im Be- reich der Goldnanopartikel Analyse
Bei Holger Reiner möchte ich für die Hilfe bei den TEM- Messungen und den Gold- synthesen bedanken.
Bei Maria Helminger möchte ich mich für die TEM- Messungen und der Hilfe bei der Untersuchung der Glasbildung der Hafniumoxidpartikel bedanken.
Zudem möchte ich mich bei meinen Masterstudenten, Bachelorstudenten, Mitarbeiter- praktikanten und HIWIs bedanken, die mich bei meiner Forschung unterstützt haben.
Prof. Dr. Junk möchte ich für die Hilfe bei der Diusionskorrektur danken.
Ich möchte zudem Dirk Hake und Rose Rosenberg für die vielen schönen Stunden, die wir gemeinsam im UZ- Labor verbracht haben, danken.
Der ganzen Arbeitsgruppe möchte ich für das angenehme Arbeitsklima und die schöne Zeit danken.
Zuletzt möchte ich noch meiner Familie und vor allem Katharina danken, die mich in allen Lebenslagen immer unterstützt haben.
Contents
1 Introduction 1
2 Theory 2
2.1 Nanoparticles . . . 2
2.1.1 Nanoparticle properties . . . 2
2.1.2 Hybrid materials . . . 2
2.1.3 Gold particles . . . 4
2.1.3.1 Gold rods . . . 4
2.1.3.2 Surface plasmon resonance . . . 5
2.2 Analytical ultracentrifugation . . . 9
2.2.1 Basic Theory . . . 10
2.2.2 Velocity measurements . . . 14
2.2.3 Equilibrium measurements . . . 15
2.2.4 Density gradient centrifugation . . . 16
2.2.5 Synthetic boundary crystallization centrifugation . . . 18
2.2.6 Multiwavelength analytical ultracentrifugation . . . 19
2.3 Field Flow Fractionation . . . 20
2.3.1 General Theory . . . 21
2.3.2 Broadening Eects . . . 21
2.3.3 Assymetrical Field Flow Fractionation . . . 23
2.4 Static light scattering . . . 23
2.4.1 Zimm Plot . . . 25
2.4.2 Multi Angle Laser Light Scattering (MALLS) . . . 26
Contents Contents
2.5 Global analysis . . . 27
2.5.1 Density determination . . . 28
3 Aim of study 29 4 Results and discussion 31 4.1 Density determination . . . 31
4.1.1 Aim . . . 32
4.1.2 Surfactant layer calculation . . . 33
4.1.2.1 Spheres . . . 33
4.1.2.2 Non-spherical particles . . . 42
4.1.2.3 Summary surfactant calculation . . . 54
4.1.3 Density gradient . . . 57
4.1.3.1 Analytical density gradient . . . 58
4.1.3.2 Preparative density gradients . . . 59
4.1.3.3 Summary (density gradient materials) . . . 63
4.1.4 Global analysis . . . 64
4.1.4.1 Data combination . . . 65
4.1.4.2 Analysis via molecular weight . . . 65
4.1.4.3 Analysis via parallel evaluation of light scattering data 69 4.1.4.4 Analysis via parallel evaluation of light scattering data with mass- balance . . . 70
4.1.4.5 Summary . . . 73
4.1.5 Summary (density determination) . . . 73
4.2 Method development . . . 75
4.2.1 Aim . . . 75
4.2.2 Evaluation program for AF4 . . . 76
4.2.2.1 Channel calibration . . . 77
4.2.2.2 Diusion coecient evaluation . . . 78
4.2.2.3 Static light scattering evaluation . . . 79
Contents Contents
4.2.3 Diusion correction . . . 80
4.2.3.1 Theoretical approach . . . 81
4.2.3.2 Iterative relaxation method . . . 82
4.2.3.3 Richardson Lucy . . . 89
4.2.3.4 Summary . . . 101
4.2.4 Molecule addition to nanoparticles . . . 102
4.2.4.1 Theoretical approach . . . 102
4.2.4.2 Examples . . . 103
4.2.4.3 Summary . . . 105
4.2.5 Protein hybrid particle . . . 105
4.2.5.1 Theoretical approach . . . 106
4.2.5.2 Examples . . . 107
4.2.5.3 Summary . . . 110
4.2.6 Summary . . . 110
4.3 Nanoparticles . . . 112
4.3.1 Hafniumoxide . . . 112
4.3.1.1 Aim . . . 112
4.3.1.2 Hafniumoxide nanoparticles in aqueous solution . . . . 113
4.3.1.3 Size tuning with pH change . . . 113
4.3.1.4 Hafniumoxid glas forming behavior . . . 114
4.3.1.5 Hafniumoxide nanoparticles in organic solution . . . . 118
4.3.2 Gold nucleation . . . 119
4.3.2.1 Aim . . . 119
4.3.2.2 Synthesis . . . 119
4.3.2.3 Nucleation without further growth throughout the AUC experiment . . . 120
4.3.2.4 Nucleation with further growth through the AUC ex- periment . . . 130
4.3.2.5 Summary gold nucleation . . . 133
4.3.3 Summary nanoparticles . . . 135
Contents Contents
5 Summary and Outlook 136
6 Experimental Data 138
6.1 Chemicals . . . 138
6.2 Methods . . . 138
6.2.1 Analytical Ultracentrifugation . . . 138
6.2.1.1 XLI/XLA . . . 138
6.2.1.2 MWL AUC . . . 138
6.2.2 AF4 . . . 139
6.3 Synthesis . . . 139
6.3.1 Hafniumoxid nanoparticles . . . 139
6.3.1.1 Aqueous synthesis . . . 139
6.3.1.2 Organic synthesis . . . 139
6.3.1.3 Phase transfer . . . 141
6.3.2 Gold nanoparticles . . . 141
6.3.2.1 Gold nucleation . . . 141
6.3.2.2 Gold spherical nanoparticles . . . 141
6.3.2.3 Gold nanorods . . . 142
7 Attachment 143 7.1 Pseudocode representation . . . 143
7.1.1 Shapsim . . . 143
7.1.1.1 Algorithm, example for prolates . . . 143
7.1.2 AF4 program . . . 145
7.1.2.1 Calibration . . . 145
7.1.2.2 Diusion coecient evaluation . . . 146
7.1.2.3 Iterative relaxation method . . . 147
7.1.2.4 Richardson Lucy . . . 151
7.1.2.5 Partial Zimm Plot . . . 152
7.2 Figures . . . 155
7.3 Tables . . . 164
Contents Contents
Bibliography 166
List of Figures
2.1 schematic construction of a hybrid particle[23] . . . 3
2.2 Transverse and longitudinal modes of plasmon resonance in rod-like par- ticles . . . 5
2.3 Size dependent localized plasmon resonance in metal spheres. . . 6
2.4 Absorption spectra simulated for goldrods . . . 8
2.5 Schematic diagram of a sector-shaped ultracentrifuge cell . . . 13
2.6 Raw Data of a BSA velocity measurement . . . 15
2.7 picture of a synthetic boundary centrepiece . . . 18
2.8 Schematic representation of a synthetic boundary crystallization ultra centrifugation . . . 19
2.9 gure of MWL-AUC Measurements on β carotin . . . 20
2.10 schematic picture of a trapezoidal channel . . . 21
2.11 Schemata of the separation in a AF4 channel . . . 24
2.12 Zimm-Plot . . . 25
2.13 Setup of a typical SLS Experiment . . . 26
2.14 Partial Zimm-Plot . . . 27
4.1 Graph of a spherical hybrid particle with a single shell . . . 33
4.2 Simulation of citrate . . . 37
4.3 AF4 Measurement of tannic acid . . . 37
4.4 TEM measurements of Gold sphere dispersion 1 . . . 40
4.5 TEM measurements of Gold sphere dispersion 2 . . . 41
4.6 Graph of a prolate like hybrid particle with a single shell . . . 42
4.7 User interface of the shapesim program . . . 46
List of Figures List of Figures
4.8 Figure of the the prolate-ellipsoid ts for s and D andλmax . . . 47
4.9 Material Studio simulation of CTAB conformation and length. . . 48
4.10 UV/Vis spectra of the 3 rod dispersions . . . 49
4.11 TEM measurements for all 3 rod species . . . 49
4.12 Figure of the comparison of the prolate-ellipsoid ts for s and D for all 3 rod-species . . . 50
4.13 Figure of the comparison of the prolate-ellipsoid ts for D and λmax for all 3 rod-species . . . 52
4.14 Figure of the comparison of the prolate-ellipsoid ts for s and λmax for all three rod-species . . . 53
4.15 Figure of the comparison of the prolate-ellipsoid ts for s,D and λmax for all three rod-species. . . 55
4.16 Comparison of the prolate-ellipsoid ts for s,D and λmax with all error borders . . . 56
4.17 Density gradient scheme . . . 57
4.18 Analytical density gradient with hafniumoxide nanoparticles . . . 60
4.19 Results preparative density gradient . . . 61
4.20 Results preparative density gradient with synthetic boundary . . . 63
4.21 Alignment t without smoothing . . . 66
4.22 Results for BSA and PS 100 nm after global analysis. D,C,vbar distri- butions . . . 68
4.23 results obtained by global t, with t on light scattering raw data for 100 nm PS standard. . . 71
4.24 user interface of the channel calibration step . . . 78
4.25 user interface of the diusion coecient calculation step . . . 79
4.26 User interface of the static light scattering calculation . . . 79
4.27 User interface of the diusion correction program . . . 81
4.28 Picture of the three dierent CdTe dispersions . . . 84
4.29 raw data of a CdTe QD dispersion . . . 84
List of Figures List of Figures
4.30 Results of a direct application of the iterative relaxation algorithm on
raw data for a CdTe QD dispersion . . . 85
4.31 Gaussian Fit on the raw data for a CdTe QD dispersion . . . 87
4.32 Comparison of the results after deconvolution with raw data for a CdTe QD dispersion . . . 87
4.33 Structure of the Hairpin DNA . . . 88
4.34 Comparison of the results after deconvolution with raw data for DNA hairpins . . . 88
4.35 Structure of BSA and Hemoglobin . . . 90
4.36 Raw data of mixture of HEM and BSA . . . 91
4.37 Overlay of the deconvoluted and raw data of mixture of HEM and BSA 92 4.38 Results of the measurements of dierent mixtures of HEM and BSA . . 93
4.39 Test of reproducibility of mixtures of HEM and BSA . . . 94
4.40 Comparison of Peak broadening with and without deconvolution . . . . 95
4.41 Comparison of crossow dependence . . . 98
4.42 Overlay of crossow dependence . . . 99
4.43 Comparison of molecular weight distribution before and after deconvo- lution . . . 100
4.44 Molecules tested for molecular addition on silica particles . . . 104
4.45 Results obtained by AUC and AF4 for a mixture of HCP1Cys3 and 6 nm gold particle in a ratio of 11:1 . . . 108
4.46 Results obtained by AUC and AF4 for a mixture of HCP1Cys3 and 6 nm Goldparticle in a ratio of 11:1 . . . 109
4.47 acetate stabilized Hf O2 particle . . . 113
4.48 PH dependence of the sedimentation coecient of hafniumoxide . . . . 115
4.49 TEM measurement of microtome cuts of a hafniumoxide glass . . . 116
4.50 TEM measurement of a hafniumoxide nanoparticle dispersion . . . 116
4.51 TEM measurement of a hafniumoxide nanoparticle gel . . . 117
4.52 3D fractogram of a synthetic boundary measurement 1 . . . 121
4.53 Materials Studio simulation of CTAB . . . 121
List of Figures List of Figures
4.54 Results of synthetic boundary measurement with growing at dierent
wavelength for measurement three . . . 124
4.55 Pseudo 3-D plot of measurement 1 at a wavelength of 500 nm . . . 126
4.56 Radial absorption spectra for measurement two . . . 128
4.57 Comparison of the t quality of the Sedt and the Ultrascan t. . . 129
4.58 3D fractogram of a syntetic boundary measurement with growth . . . . 131
4.59 List of species obtained by synthetic boundary centrifugation according to their sedimentation coecient. . . 132
4.60 List of species obtained by synthetic boundary centrifugation according to the apparent core radius. . . 132
4.61 Normalized integral of the absorption over the wavelength . . . 134
6.1 Picture of AF4 equipment . . . 140
7.1 PS peak broadness at a crossow of 0.5 ml/min . . . 155
7.2 PS peak broadness at a crossow of 0.35 ml/min . . . 156
7.3 Alinement t with 3 point smoothing . . . 157
7.4 Alinement t with 5 point smoothing . . . 158
7.5 Alinement t with reverse point smoothing . . . 159
7.6 Results of synthetic boundary with growing at dierent wavelength for measurement one . . . 160
7.7 Results of synthetic boundary with growing at dierent wavelength for measurement two . . . 161
7.8 Results of dierent synthetic boundary measurements with growing at the 520 nm . . . 162
7.9 Iteration inuence on molecular weight . . . 163
List of Tables
4.1 Results of surfactant layer calculation using a tannic acid layer. . . 38
4.2 Results of surfactant layer calculation using a tannic acid and a citrate layer. . . 38
4.3 Results of surfactant layer calculation using a citrate layer. . . 39
4.4 Frictional coecient ratios[75] . . . 43
4.5 List of gold simulation parameters . . . 44
4.6 Results by combination the simulations of s and D . . . 49
4.7 Results by combination the simulations of D andλmax . . . 51
4.8 Results by combination the simulations of s and λmax . . . 51
4.9 Results by combination the simulations of s and λmax . . . 54
4.10 Density simulations according to density gradient measurements . . . . 59
4.11 Comparison of the AF4 results at dierent crossows . . . 96
4.12 Comparison of the deconvoluted AF4 results at dierent crossows . . . 97
4.13 Results of molecule addition on silica nanoparticle . . . 104
4.14 Simulation results for rst species of Hcp1Cys3 . . . 108
4.15 Simulation results for second species of Hcp1Cys3 . . . 110
4.16 Results obtained for the three dierent protein gold nanoparticle mixtures.110 4.17 PH dependence of hafniumoxide nanoparticles . . . 114
4.18 Results the calculation according to Sedt at 520 nm . . . 123
4.19 Results obtained by ultrascan for measurement 1 at a wavelength of 500 nm . . . 125
4.20 Apparent values for the species obtained for gold nucleation measure- ments with growth for measurement 2 . . . 133
List of Tables List of Tables
7.1 Species obtained for gold nucleation measurements with growing for mea- surement one . . . 164 7.2 Species obtained for gold nucleation measurements with growing for mea-
surement three . . . 164 7.3 Simulated values for goldcluster[96]. . . 165
List of Abbreviations
• AUC: Analytical Ultracentrifugation
• AF4: Asymmetrical Flow Field Flow Fractionation
• BSA: Bovine Serum Albumine
• DLS: Dynamic Light Scattering
• FFF: Field-ow fractionation
• HPLC: High-Performance Liquid Chromatography
• LS: Light scattering
• MALLS: Multi-Angle Laser Light Scattering
• 2DSA: 2- Dimensional Spectrum Analysis
• RI: Refractive Index
• SDS: Sodiumdodecylsulfate
• CTAB: Hexadecyltrimethylammoniumbromid
• SEC: Size-Exclusion Chromatography
• SLS: Static Light Scattering
• UV: Ultraviolet (light)
• UV/VIS: Ultraviolet and Visible (light)
• IR: Infrared
• MWL: Multi wavelength
• TOPO: Trioctylphosphine oxide
• DNA: Deoxyribonucleic acid
• RNA: Ribonucleic acid
• GSD: Gold sphere dispersion
• Rod: Gold rod dispersion
• SAXS: Small angle x-ray scattering
1 Introduction
Nanoparticles are of great importance for science and industry. However, detailed anal- ysis is still a challenge. While the number of possible applications[13] has increased during the last decade, the development of analytical possibilities nearly stagnated.
As most of the solution analytics with nanometre resolution were historically used for biopolymers, analysis programs have also been developed with the focus on biopoly- mers. Nanoparticles have some more complex properties, which makes the data eval- uation challenging. Dispersions of nanoparticles are in most cases polydisperse or a mixture of dierent species. Also, most nanoparticles need a stabilizing shell and are therefore hybrid materials that have a size and density distribution. Due to these facts it is challenging to obtain coherent information about the distribution of size, shape and composition/density of the nanoparticles. As their chemical,[4] electronic,[5]
magnetic,[6] optical,[4,7,8] catalytic[5,9] and self assembling[10] properties are mainly de- ned by their size and composition, the composition is a property of major interest.
As nanoparticles are polydisperse, a fractionation method should be used to anal- yse the density distribution. Analytical Ultracentrifugation (AUC) and Asymmetrical Field Flow Fractionation (AF4) are two methods which allow fractionation and the calculation of hydrodynamic parameters for nanoparticle dispersions. Nanotechnology becomes increasingly important in all aspects of life, such as medicine, development of new building materials or food science. Thus, mankind is more exposed to nanoparti- cles than before.[11]Consequently, plenty of studies with a focus on nano toxicology are executed nowadays.[12] For these kind of systems the AUC and AF4 as fractionation methods are suitable as well. The challenge is the data evaluation. Therefore, the focus of this thesis is the development of data evaluation techniques for nanoparticle systems.
2 Theory
2.1 Nanoparticles
2.1.1 Nanoparticle properties
Nanoparticles are dispersed particles with at least one dimension in the nanometre range, regardless whether the dispersion consists of inorganic or organic particles. Ex- amples for inorganic materials are gold or silica substances, while the main fraction for organic nanoparticles are polymer based. Nanoparticle research is one of the biggest ar- eas of scientic interest due to a wide range of potential uses, for example in optical,[1]
biomedical[2] and catalytical[3] applications. The huge scientic interest can be ex- plained as nanoparticles are a bridge between atomic and bulk materials. While bulk materials have constant physical properties, those of nanoparticles show a strong size dependence. For bulk materials, the number of atoms on the surface is insignicant in relation to the atoms in bulk. In contrast, for nanoparticles the number of atoms on the surface is signicant. Resulting from this high surface area several interesting physical properties can be observed. For example, the optical properties of gold nanoparticles have been known since 1857.[13]The size also has an inuence on stability, which results for example in the change of the melting point.[14]
2.1.2 Hybrid materials
Hybrid materials consist of a minimum of two dierent materials. The most com- mon hybrid nanomaterial is an inorganic or organic core that is coated with a sur- factant layer. As most nanoparticles are stabilized by surfactants, they are hybrid materials. Their chemical,[4] electronic,[5] magnetic,[6] optical,[4,7,8] catalytic[5,9] and self
2. THEORY 2.1. NANOPARTICLES
assembly[10] properties are mainly dened by their size and composition. Therefore, a wide range of applications was found for hybrid nanoparticles. They can be applied in drug delivery,[2] in solar cells,[15] in optoelectronics,[7,16] nanophotonics,[1]bioimaging[17]
and in catalysis.[18,19] In the case of organic-inorganic hybrid materials, strong syner- gistic eects that combine the properties of both organic and inorganic components are often possible.[20,21] Hybrid organic-inorganic materials in general represent the natu- ral interface between two worlds of chemistry each with very signicant contributions to the eld of material science, and each with characteristic properties that result in distinct advantages and limitations.[22]
Figure 2.1: schematic construction of a hybrid particle[23]
2. THEORY 2.1. NANOPARTICLES
2.1.3 Gold particles
The history of gold nanoparticles can be tracked back to the 5th century. Since that time gold was used as a colorant in ruby glass and ceramics. However it needed more than 1000 years for the discovery of the reason for the optical eects that made gold nanoparticles such a popular material.[24] Faraday discovered that the color is caused by metallic gold in colloidal form. He prepared a deep red solution of colloidal gold by reducing an aqueous solution of chloroaurate using phosphorus in CS2.[13] During the 20th century, various methods for gold nanoparticle preparation were introduced.
The most popular one has been reduction of chloroauric acid in water by citrate, which was introduced by Turkevich in 1951.[25] Since then, colloidal nanoparticles have received widespread attention, due to both their unusual properties and promising application. For example, spherical colloidal gold particle dispersions with a radius of around 10 nm show a ruby red color while particles of a size around 100 nm have a blueish color. The color does not only result from the size of particles, the shape has an inuence as well. The color is a consequence of the surface plasmon resonance described in chapter 2.1.3.2. Due to their widespread properties, gold nanoparticles have turned out to play an important role in nanoscience.[7] By suitable choice of experimental conditions and additives, anisotropic shapes of nanoparticles such as rods, wires, tubes and concentric core-shell structures can be produced.[8] Nanorods have drawn the biggest attention as the synthesis uses relatively simple processes such as wet chemical methods. In addition a rational control over the aspect ratio is possible.
Because of their interesting behaviour, a closer look on gold nanorods is interesting.
The latest gold rush is happening in the eld of biosensing and chemical sensing, using highly environmentally sensitive gold nanoparticles. The gold particles can be detected using the plasmon resonance as well as other optical eects, such as surface- enhanced Raman or uorescence spectroscopy.[26]
2.1.3.1 Gold rods
Since 1992, when rod-like particles were observed as a by-product of spherical nanopar- ticles, many eorts have been made to gain control of the shape of gold nanoparticles.[27]
2. THEORY 2.1. NANOPARTICLES
The rst approaches included alkyltrimethylammonium chlorides as stabilizers, and UV light was used as a reduction source.[28,29] Unfortunately, under these conditions, gold rods are only formed as a side product. The introduction of CTAB (hexadecyltrimethy- lammonium bromid) as a stabilizer was the next major step as, with CTAB, a controlled rod synthesis was possible.[30] Using this stabilizer and varying the AgNO3 concentra- tion, the aspect ratio can be adjusted in the range between 1 and 5. Gold rods show a dierent color depending on the aspect ratio, which is caused by the two intense surface plasmon resonance peaks. The longitudinal surface plasmon peak and the transverse surface plasmon peak correspond to the oscillation of the free electrons along and per- pendicular to the long axis of the rods.[7] The color change provides the opportunity to use gold nanorods for optical applications. There have been many applications utilizing this intense color and its tunablity.[8]
Figure 2.2: Transverse and longitudinal modes of plasmon resonance in rod-like particles[26]
2.1.3.2 Surface plasmon resonance
The optical properties of noble metal particles originate from localized surface plas- mons. This phenomenon is based on electromagnetic eld interactions with conduction band electrons and induces a coherent oscillation of electrons.[31] As a result, a strong absorption band appear in the electromagnetic spectrum in regions that depend on the size of the particles. These surface plasmon polarisations are electromagentic excita- tions on the interface and evanescently conned in perpendicular direction. But when a metal nanoparticle is exposed to a eld, non-propagating excitations of conduction electrons create size dependent localized surface plasmons. When a conductor or metal
2. THEORY 2.1. NANOPARTICLES
is placed in an oscillating eld of incoming radiation, the electron cloud is driven into oscillations as can be seen in gure 2.3. In the case of a subwavelength conductive nanoparticle, the curved surface of the particle exerts an eective restoring force on these driven electrons (analogy with a damped, driven harmonic oscillator). Like any driven oscillator system, in the case of nanoparticles, a resonance can arise, leading to a eld amplication both inside and outside the particle. For gold and silver parti- cles this resonance lies in the visible region of the electromagnetic radiation, which is responsible for their bright color.[32]
2.1.3.2.1 Surface plasmon resonance for spherical particles In the plasma model, the electrons in metals can move freely like a gas against a xed background of positive ion cores. This model leads to the relation between the dielectric function of the metal, (ω) and the absorption coecient k and refractive index n. In the plasma model, when electrons oscillate in response to the applied electromagnetic eld, their motion is damped via collisions that occur with a frequency of γ1d
Figure 2.3: Size dependent localized plasmon resonance in metal spheres.[26]
D(ω) = 1− ωp2
ω2+iγdω (2.1)
Here γd is a phenomenological dampening constant, ωp2 is the plasmon frequency of the free electron gas and is described according to equation 2.2
ωp2 = n0e2
0mef f (2.2)
2. THEORY 2.1. NANOPARTICLES
mef f is the eective mass of the electron, where the dampening constant can be ex- plained due to
γD(r) = γd0+ AνF
r (2.3)
The rst part is γd0 which is the bulk dampening constant, νF is the velocity of the conduction electrons at Fermi energy. Aincludes the details of scattering processes. ris the size of the gold nanoparticle. So the size dependent shift of the plasmon resonance can be explained due to the size dependence of the damping constant. According to Mie Theory it is dicult to distinguish smaller particles from a background of larger scatterers because the scattering cross-section scales with r6. Anyhow, for smaller particles r << λ the absorbance dominates over scattering and the scattering can be neglected. The expression for the extinction for spherical particles can be written as:
γ
NPV = 18πm32 λ
2
(1+ 2m)2+22 (2.4)
Np represents the number concentration of particles, V the volume of a single particle, λ the wavelength of light in vacuum, m the dielectric constant of the solvent and 1 and 2 are the real and imaginary parts of the complex dielectric function of the nanoparticle.
2.1.3.2.2 Surface plasmon resonance for rod like particles For gold nanorods, two dierent plasmon resonances can be found, corresponding to the oscillation of the free electrons along and perpendicular to the long axis of the rods. While the transverse mode ( perpendicular to the long axis) shows a resonance in the range of the seed particles, the longitudinal oscillation occurs at a higher wavelength depending on the aspect ratio.[33]As the aspect ratio is increased, the longitudinal peak is red shifted.
for the optical properties the gold rods have been treated as ellipsoids, which allows the Gans formula to be applied.[34] The Gans formula for randomly oriented elongated ellipsoids in the dipole approximation can be described as
γ
NpV = 2πm32 3λ
C
X
i=A
1 Pj2
2 h
1+(1−P
j) P
mi2
+22
(2.5)
2. THEORY 2.1. NANOPARTICLES
The geometrical factor Pj can be explained for elongated ellipsoids along the dierent axes (A,B and C) according to
PA= 1−e2 e2
1 2eln
1 +e 1−e
−1
(2.6)
PB =PC = 1−PA
2 (2.7)
While e can be expressed through e=
L2−d2 L2
12
(2.8) Lis the length of the longitudinal axis anddis the length of the transversal axis. Using these equations the results shown in gure 2.4 have been calculated. The shift in the absorption spectra induced by axial ratio can be directly observed.
Figure 2.4: Absorption spectra calculated with the expression of Gans for elongated ellipsoids using the bulk optical data for gold. The number on the spectral curve indicates the aspect ratio of the rod[26]
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
2.2 Analytical ultracentrifugation
Since the AUC has been developed in the 20's of the last century several important discoveries have been enabled, such as that of polymers by Staudinger. The discoveries of Theodor Svedberg, about disperse systems, were rewarded with the Nobel prize.
Although the AUC was originally developed for the analysis of nanoparticle size distri- butions, the focus quickly shifted toward biological applications such as biopolymers.
Due to time consuming photographic data acquisition in the early instruments and the upcoming of new methods for biopolymer analysis, e.g. HPLC, AUC lost its impor- tance as one of the leading analytical techniques for biopolymers during the middle of the last century. Since the introduction of a new generation of analytical ultra- centrifuges, namely the Optima XL-A by Beckman Instruments (Palo Alto, USA)in the 1990's, a renaissance of analytical ultracentrifugation took place. The AUC allows several dierent approaches to analyse samples. The basic experiments used in AUC are:
• Sedimentation velocity high rotation speed standard cell type
measurement of sedimentation coecient disribution, diusion coecient and frictional ratio
• Sedimentation equilibrium
low to moderate rotation speed standard cell
measurement of Molar mass, equilibrium constants and stoichiometry of interacting systems
• Synthetic boundary high rotation speed synthetic boundary cell
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
measurement of diusion coecient and sedimentation coecient
• Density gradient
moderate to high rotation speed standard cell
measurement of density
2.2.1 Basic Theory
In AUC experiments a gravitational eld, induced by the spinning of a centrifuge rotor, is applied on a sample that has been dissolved or dispersed. The gravitational eld results in a sedimentation of the solutes. The centrifugal eld can be varied within a range from approximately 73 to 260000 g, for a radial position of 6.5 cm and with g = 9.81 ms−2. The maximal centrifugal eld is high enough to even measure small molecules and ions. Depending on the type of experiment, dierent data is collected: In sedimentation velocity experiments (chapter 2.2.2) the concentration change with time and radius c(r, t) is measured. In the case of sedimentation equilibrium experiments only the the radial concentration prole c(r) is measured after equilibrium between diusion and sedimentation is reached. The concentration can be detected by dierent detectors. The most common detector systems are interference and absorption optics.
While interference optics measure the refractive index dierence, the absorption optics measure the absorption of light by of the sample. For the explanation of the sedimen- tation, the rst and simplied approach invented by Svedberg can be used. A particle with a mass mp and density ρp is dispersed in a solvent with density ρs and viscosity ηs. When the particle is exposed to a gravitational eldω2rat a radial distancer from the center of rotation, it is exposed to 3 dierent forces.[35],[36]
• The centrifugal force, induced by the rotation of the spinning AUC rotor:
Fs =mpω2r = Mp NA
ω2r (2.9)
NA is the Avogardro number and Mp is the molar mass of the particle.
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
• The buoyant force Fb which is needed to displace the mass of solvent ms. The buoyant force works in the opposite direction of the centrifugal force.
Fb =−msω2r=−mpνρsω2r =−Mp
NAνρsω2r (2.10) ν is the partial specic volume of the particle.
• The frictional force Ff is induced by the resistance encountered by the particle moving through the solvent. Also, in the opposite direction of the centrifugal force
Ff =−f ∗u (2.11)
f is the frictional coecient andu is the sedimentation velocity of the solution.
As the measured movement of the particle with a constant sedimentation velocity u results from the superposition of all 3 forces shown above, the following relation can be used:
Fs+Fb+Ff = 0 = Mp NA
ω2r− Mp NA
νρsω2r−f∗u (2.12) Rearrangement leads to:
M(1−νρs)
N f = u
ω2r ≡s (2.13)
Due to equation 2.13 the denition of the sedimentation coecient is given. The frictional coecient f is well described by hydrodynamic theory. It is depending on shape and size of the particle and the viscosityηof the solvent. The molecular denition of the diusion coecient was given by Albert Einstein as:[37]
D= kT
f = RT
NAf (2.14)
Where k is the Boltzmann constant andR is the gas constant. For the simplest case, a spherical particle of a radius rp, the frictional coecient can be described due to Stokes's Law according to:[38]
f0 = 6πηrp (2.15)
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
By the combination of equations 2.14 and 2.13 the important Svedberg equation can be obtained.
M = sRT
D−(1−νρs) (2.16)
The Svedberg equation is the basis for several theoretical approaches described in this thesis. For spherical particles, a simple equation for the calculation of particle sizes from sedimentation coecients can be obtained by combination of equations 2.15, 2.14 and 2.13.
dp =
18ηs ρp−ρs
1/2
(2.17) The second and more accurate approach for the description of sedimentation processes is the Lamm equation.[39],[40] This approach reduces the system to two basic transport mechanisms for the sample particles. The rst is the mass transport via sedimentation and the second is the transport via diusion. Consider a volume elementdV in a sector shaped ultracentrifugation cell, which is shown in gure 2.5. If this cell is rotating at an angular velocityω, the volume element will move from radius point r to the radial point r +dr. As the magnitude of the centrifugal eld is dened as ω2r, the mass transport rate according to sedimentationdms/dt is proportional to the concentration of the particle at the surfacec, the surface area and the sedimentation velocity u. The area of the sector shaped cell is known as:[41]
Acell =φar (2.18)
Whereφis the angle of the sector andais the thickness of the cell. The volume element can be expressed as:
dV =φardr (2.19)
Using equations 2.13 and 2.18, for the denition of sedimentation velocity, a description for the mass transport according to sedimentation can be proposed.
dms
=cφaru=cφarsω2r (2.20)
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
Figure 2.5: Schematic diagram of a sector-shaped ultracentrifuge cell
The mass transport according to diusion dmD/dt is described by Fick's law:[42]
dmD
dt =−Dφarδc
δr (2.21)
δc
δr is the concentration gradient along the radius. As the mass transport dm/dt is dened as the sum of both eects, the mass transfer according to sedimentation and diusion, the following relation can be found:
dm
dt =φar
csω2r−Dδc δr
(2.22) A similar equation exists for the net transport across the surface at the radial distance r+dr. By the subtraction of this equation from equation 2.22 the net accumulation of mass in the volume element per time unit can be obtained. The quotient of the mass changedm/dtwith the volume elementdV gives the concentration change in the volume element per time unit.
δc
δt = dm
dt /dV = dm dt
1
φarδr (2.23)
The combination with equation 2.22 leads to:
δc δt = 1
r δ δr
Dδc
δr −sω2r
(2.24)
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
With the restriction thatsandDdoes not change with the concentration, the equation can be transformed into the general dierential Lamm equation.
δc δt =D
δ2c δr2 + 1
r δc δr
−ω2s
rδc δr + 2c
(2.25) The Lamm equation consists of two dierent terms. The rst term describes the diu- sion and the second term the sedimentation. In most of the commonly used programs, the Lamm equation is used to simulate data, to calculate sedimentation coecients for measured data.
2.2.2 Velocity measurements
The sedimentation velocity measurement is the most important AUC experiment. It is carried out at high centrifugal elds and provides the easiest way to obtain sedi- mentation coecient distributions. Therefore, it is also the main approach to get size distributions of nanoparticles. In a sedimentation velocity experiment the particles sediment according to their size, density and shape. With a high rotation speed, the back-diusion along the generated concentration gradient can be minimized. There- fore, the diusion broadening can be reduced. As the sedimentation velocity is size dependent, a fractionation according to the particle size is possible, as long as the density and shape are similar.[36]
The centrifugal force is not constant within the AUC cell. The force is proportional to the radius as equation 2.26 shows. Therefore, the velocityu is radius dependent as well and increases with the radius. To take this behaviour into account a dierential expression can be created:
s= u
ω2r = drbnd/dt
ω2r (2.26)
Where rbnd is the radius of the boundary. By integration the following correlation can be made:
ln(rbnd/rm) = sω2t=s
t
Z
0
ω2dt (2.27)
According to equation 2.27 the easiest way to distinguish the sedimentation coecient would be the plotting ofln(rbnd/rm)versus the time. The slope would be equal tosω2.
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
Figure 2.6: Raw Data of a BSA velocity measurement
As most measured samples are not monodisperse and consist in this case of multiple components or a continuous distribution, another approach is used nowadays. If the approach explained above were used for mixtures the result would be the weight average sedimentation coecient.However, it would be of interest to distinguish between the dierent species. Hence, an approach that uses the complete dierential sedimentation coecient distributiong(s) was invented. This approach is dened as:
g(s) = d(c(s)/c0) ds
r rm
2
= δG(s)
δs (2.28)
2.2.3 Equilibrium measurements
In sedimentation equilibrium experiment the sample is exposed to a centrifugal eld as long as it needs to balance the two terms in Lamm equation 2.25 to get to equilibrium.
In this state the mass ow due to sedimentation and diusion is equal. The time needed can range from 10-200 h. Equilibrium measurements allow the determination of thermodynamic parameters such as molecular weights, equilibrium constants of self- associating systems[35,40,43,44] and particle charge.[45,46] In case of equilibrium between
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
diusion and sedimentation, the radial concentration distribution is time independent.
This leads to:
s
D = dc/dr
ω2rc (2.29)
Which can be derived from Lamm equation with the information dc/dt = 0. It also takes several assumptions into account. The rst one states that the sample is a mono disperse, non-interacting solution. The second is the assumption of innite dilution, which avoids all concentration dependent eects. By combination with the Svedberg equation 2.16 the following correlation results:
M = RT
(1−νρs)ω2 dc/dr
rc (2.30)
By solvingdc/c = dlncand 2rdr= dr2 equation 2.31 results in:
M = 2RT
(1−νρs)ω2 dlnc
dr2 (2.31)
2.2.4 Density gradient centrifugation
In density gradient centrifugation experiments, a separation according to the density is achieved by the formation of a density gradient ρ(r). Heavy auxiliary additives are added in high concentrations up to25wt%to a light solvent.[47] The additive accumu- lates according to its higher density more in direction of the cell bottom. Due to the gradient a dierent composition of the solvent can be found at every radial position r. Depending on the ratio of additive to solvent, a density gradient along the radial axis can be observed, with a higher density near the bottom and lower densities at the meniscus. For the theoretical approach, two important eects have to be considered.
At rst the formation of the radial static density gradient ρs(r), and secondly, the radial distribution of the measured sample within the density gradient.[36]
2.2.4.0.3 Radial static density gradient The approach to explain the static density gradient is similar to the approach for sedimentation equilibrium measurements in chapter 2.2.3. The main dierence to the theory that is applied to sedimentation equilibrium is the very high concentration in density gradients. For this problem the
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
activitiesacand c have to be taken in account instead of simply the concentration as in highly diluted systems. For the approach in this chapter the assumption is made that the concentration still can be used to solve the problem. This will lead to an error, which can not be avoided. The gradient has formed when the diusion and sedimentation of the additive are at equilibrium. Therefore the mass ow at each radial position dmdt within the cell is zero. Starting with the equation for mass ow shown in equation 2.22 and assuming that sedimentation and diusion are at equilibrium and mass ow disappears, the following correlation can be found.[36]
csω2r=Dδc
δr (2.32)
As the density of the solution is directly proportional to the concentration of the density gradient material the concentration prole can be connected to the density gradient.
δc δr = δc
δρs
δρs
δr (2.33)
Replacings and Dby using the Svedberg equation 2.16 and combining 2.32 with 2.33, a rst explanation of the density gradient is found.
δρs
δr = δρs
δlnc(1−νρs)M ω2r
RT (2.34)
If the equation is rearranged and some parts are collected in the parameter β(ρs)
β(ρs) = δlnc δρs
RT 1−νρs
M (2.35)
δρs
δr = ω2r
β(ρs) (2.36)
As the density of the solvent solution is directly proportional to the concentration of the gradient material,β varies only with the densityρs. In cases of low concentration and low rotor speed,β(ρs)can be assumed as constant over the whole radius. An empirically
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
determined value can be used to replace it. Using this assumption a general equation for the calculation of densities according to the radial position can be derived:
ρs(r) = ρs,0+ 1
βω2(r−r0)2 (2.37)
The easiest way to measure an unknown sample would be the addition of a sample with a known densityρs,0. The radial pointr0 can be used to calculate the density for every other radial point. With the known material densities in the range of roughly 1.0-2.0 g/cm2 can be measured.[35] The most common density gradient materials are CsCl, KBr, RbBr, RbCl and sucrose.
2.2.5 Synthetic boundary crystallization centrifugation
The basic idea of this method is the performance of chemical reactions inside the AUC cell. The synthetic boundary crystallization centrifugation is carried out in a synthetic boundary cell and uses a special construction to delay the mixture until an adequate rotation speed is reached. This adds the advantage of an in situ synthesis inside the AUC cell and a direct observation of the process. At the boundary layer species with higher mass are formed and start to sediment.[36]
Figure 2.7: picture of a synthetic boundary centrepiece
2. THEORY 2.2. ANALYTICAL ULTRACENTRIFUGATION
In general, the method is known as active enzyme centrifugation. The chemical re- action takes place between an enzyme and its substrate.[48] Anyhow, there also have been studies on particle nucleation using synthetic boundary crystallisation. For the measurement of nanoparticle systems, a study on the nucleation of cadmium sulde can be made. In this study, the nucleation of cadmium sulde particles took place at the synthetic boundary between the two solutions.
Figure 2.8: Schematic representation of a synthetic boundary crystallization ultra centrifu- gation
2.2.6 Multiwavelength analytical ultracentrifugation
An exiting optical novelty for AUC is the multi-wavelength optical detection system.
This detection system is based on absorption optics, but it allows the detection of a range of wavelengths, while the standard absorption optics are limited to a small number of distinct wavelengths. Each scan can be then seen as a 3D-surface, with information on the radius, absorption and wavelength as shown in gure 2.9. Solutes in a mixture that all have their own characteristic absorbance maxima can be studied in parallel.[49]This even works when the sedimentation coecient dierence is still beyond the resolution limit of the AUC. Information on particle composition can already be obtained from visual inspection of the raw data.[50]
2. THEORY 2.3. FIELD FLOW FRACTIONATION
Figure 2.9: gure of MWL-AUC Measurements onβ carotin[49]
2.3 Field Flow Fractionation
Field Flow Fractionation is becoming an increasingly important family of methods.
For several years the technics of commercially available eld ow fractionation setups have been improving. The method is receiving an increasingly growing response from the scientic community. The history of Field Flow Fractionation goes back to the 1960s. The prototype was developed by Professor J. Calvin Giddings et al.[51] The family of FFF techniques is characterized by the use of an external force eld applied perpendicularly to the direction of sample elution ow through an thin ribbonlike channel. According to the small channel width of the FFF channel a laminar parabolic ow prole is formed. Resulting from the ow prole, the ow velocity is increasing from near zero at the channel walls to a maximum at the centre of the channel. This force drives the sample toward the accumulation wall. Depending on the force, a counteracting force develops due to the concentration build up at the wall and drives the analyte back towards the centre of the channel. When the forces are in balance, steady-state equilibrium is reached and an exponential analyte concentration prole is built up. This concentration prole is the source of separation as it is depending on the equilibrium of the forces, which can be dierent for dierent analytes. The separation
2. THEORY 2.3. FIELD FLOW FRACTIONATION
occurs because the dierent proles result in dierent positioning in the ow velocity zones.
b0
bL
Figure 2.10: schematic picture of a trapezoidal channel[52]
2.3.1 General Theory
The theoretical development of the FFF retention is based on a number of assumptions which include the parabolic ow prole between innite parallel plates, the absence of analyte-analyte and analyte-FFF wall interactions and uniformity of the applied force eld. As the eld ow fractionation is a chromatagraphic method, the signicant value for the dierent species is the retention timetr. The retention ratio is given as a ratio of velocities and can be expressed as a ratio of two times.[53]
R = t0
tr (2.38)
Where t0 is the time of the void peak. t0 can be obtained by the elution time of unretained components, which travel with the average velocity of the carrier. With a known layer thickness λ, the retention ratio can be calculated according to equation 2.39.[54,55]
R =λ
coth 1 2λ −2λ
(2.39) As the equation 2.39 can not be solved directly, a common approximation is given in equation 2.40 . This approximation is valid within 2% when R <0.06 and 5% when R
< 0.15.
R = 6λ (2.40)
2.3.2 Broadening Eects
As eld ow fractionation (FFF) is a chromatographic method with detection after fractionation, the zone broadening eects, caused by diusion and other eects inside
2. THEORY 2.3. FIELD FLOW FRACTIONATION
the channel are of signicant importance for the discussion of the measurement re- sults. The theory is discussed extensively in several publications.[56,57] In this chapter, only the results of the broadening theories are presented. Zone broadening in FFF is usually expressed by the plate height H. The broadening function for narrow particle distributions can be described as:
H=Hn+Hd+X
k
Hk+Hp (2.41)
Where Hn is the non-equilibrium term and is the main contributor to the measured peak width. This eect is caused by to the dierential axial movement of the zone components, which is due to their location in dierent velocity streamlines across the channel hight. The main inuences onHnare the ow rate and the channel dimensions.
It has been shown thatHn is a complex function of the retention parameterλ as shown in equation 2.41. The second eect is the axial diusionHd and it describes the eect of longitudinal diusion due to the axial concentration gradient. Hd is signicant only in the case of very low ow velocity. In the case of small diusion coecients the contribution of Hd to the plate height is negligible. P
kHk is the plate height due to instrumental contributions like the eects of relaxation, triangular ends, injection of nite sample volume and detection. In a well-designed system the eect ofP
kHk can be reduced to a negligible level. The last broadening eectHp is due to polydispersity contribution to the plate height.[57]
H = 2D
R < v > +χ < v >
D +X
k
Hk+Hp (2.42)
andχis aλdependent nonequilibrium coecient. While the the average linear velocity of the uid < v > can be described according to:[52]
< v >=
V˙in− ˙
Vinω V0
h
b0z−z2 (b02L−bL)i ω
h
b0− (b0−bLL)zi (2.43)
2. THEORY 2.4. STATIC LIGHT SCATTERING
As shown in gure 2.10,b0 and bL are the channel breadths at z = 0 and z =L. L is the channel length.[56]
χ= 24λ3
1 +e−λ1 −2λ
1−+e−λ1 28λ2+ 1
1−e−1λ
−10λ
e−1λ + 1
− 1 3λ2
−2
λ + 4− 1 λ
1−+e−λ1
4λ
1 + 1 λ
1−+e−λ1 − 1 3λ −6
!
(2.44)
2.3.3 Assymetrical Field Flow Fractionation
In Asymmetrical Flow Field Flow Fractionation (AF4) the force eld is due to a cross- ow through a membrane at the bottom of the channel. In this method particles can be separated in a size range between 1 nm and 800 nm. With bigger particles, an inversion from classical separation to a sterical mode can be observed. In sterical mode the particles roll on the surface of the membrane. This leads to an inversion of the separation. Bigger particles are faster than smaller ones. Due to their larger size, they are eected by the inner, and, therefore faster ow zones and become faster. In this chapter, only the classical mode is described. In AF4 the force is described according to 2.45. A direct approach for the determination of the diusion coecient D can be obtained,[58]
λ= DV0
V˙cω2 (2.45)
in which V0 is the void volume, ω2 is the channel thickness and V˙c is the crossow in the channel. The void volume can be calculated according to:[59]
t0 = V0
V˙cln 1 + V˙c Vout˙
!
(2.46)
2.4 Static light scattering
Static light scattering is a famous method to obtain the average molecular weight M, radius of gyrationRg and the second virial coecientA2 of macromolecules or proteins
2. THEORY 2.4. STATIC LIGHT SCATTERING
ω
Vc Jz
Jz
Ve
1
2 3
x z
Figure 2.11: Separation principle of AF4. The channel width ω is formed between a solid wall 1 , a membrane 2 and a frit 3 . Particles are moved in x-direction by the elution owVE. The ratio ofVcandJzdetermines the average position of the particle in z-direction and thereby the eective velocity that is exerted on each particle.[60]
in solution. This is done by synchronising the angle dependence of the scattered light.
The scattering vectorqis dened as the dierence between the wave vectorS0 and the wave vector of the scattered beam S(θ) atθ.[61]
q(θ) = S(θ)−S0 (2.47)
For vertically polarized light q(θ)can be described as:[62]
q(θ) = 4πn0
1 λsin
θ 2
(2.48) n0 is the refractive index of the solvent and λ the wavelength of the laser. In order to dene a measurement size which is independent from the specic setup the intensity can be described according to the Rayleigh ratioR(θ)[63]
R(θ) = I(θ)s−2(θ)
I0f V (2.49)
Here f is the polarisation parameter, V is the scattering volume , I(θ) is the angle dependent intensity, I0 is the total intensity of the primary beam and s−2(θ) is the distance between sample volume and detector. A common description of the eect according to Rayleigh scattering is the Zimm equation:[64],[65]
KC
R(θ, c) = 1 M
1 + q2R2g 3
+
∞
X
i=2
iAici−1 (2.50)
2. THEORY 2.4. STATIC LIGHT SCATTERING
Figure 2.12: For a normal Zimm-Plot, linear regressions are conducted for measured data at each angle and concentration 1. The regression lines are extrapolated to sin2 θ2
→0 2 andA2c →0 3. The parameters of the regression lines yield Mw 4,Rg andRS.[60]
Where K is the optical constant described in equation 2.52 , c is the concentration, θ is the measurement angle,R(θ, c) is the Rayleigh ratio described in equation 2.49 and q is the scattering vector.
n(c) =n0+cdn/dc (2.51)
Where dn/dc is the concentration dependence of the refractive index.[66]
K = 4π2n02(dn/dc)2
NAλ4 (2.52)
2.4.1 Zimm Plot
There are dierent approaches to solve the Zimm equation, but the most common is the Zimm plot.[65] In a Zimm plot R(θ,c)KC is plotted againstsin2(θ/2) +kc.
Due to the measurement of several angles, the approach allows the extrapolation forθ
= 0 to obtain the molecular weight ad the interception point with the y-axis and the radius of gyration as the slope. The second extrapolation is for a concentration of 0.
This yields the second viral coecient.
