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Synthesis and Characterization of Nanoparticles

Diplomarbeit 2009 171 Seiten

Chemie - Analytische Chemie


Table of Contents

I. Introduction

II. Fundamentals
1. Plasma oscillation and Mie’s theory
1.1 Principles of plasma oscillation [20]
1.2 Scattering and absorption of small particles [23]
2. L(+)-ascorbic acid and its derivates
2.1 Chemical properties of ascorbic acid
2.2 AscX surfactants and their aggregates [28]
2.3 The formation of gel and coagel
3. Zirconia and its different structures and species
3.1 Zirconyl chloride in aqueous solutions
3.2 Structures of zirconium oxide
3.2.1 Crystal structures and martensic phase transformation
3.2.2 Tetragonal zirconia and critical crystal size
4. Structural investigation techniques
4.1 X-ray based methods
4.1.1 The nature of X-rays [48]
4.1.2 Small angle X-ray scattering (SAXS) Scattering by one electron The scattering vector [54] The electron density [55] The scattering intensity [58] The auto correlation and invariant [58] Scattering of spherical particles [58] The Guinier approximation [58] Correlation length and Porod’s law of scattering [58] Scattering of particles with non-uniform electron density [58]
4.1.3 X-ray diffraction (XRD) Phase shift and intensity [65] Bragg’s Law of diffraction [65], [67] The reciprocal lattice and the system of Miller indices [64], [65] The Scherrer equation [68]
4.2 Electron based method: scanning electron microscopy (SEM)
4.2.1 Principal setup [72]
4.2.2 The scanning process [73]

III. Experimental
5. Chemicals
5.1 Preparation of Gold Nanoparticles
5.2 Preparation of zirconium-based nanoparticles
6. Analytical Methods
6.1 Thermogravimetric analysis (TGA)
6.2 Differential scanning calorimetry (DSC)
6.3 Bright field and phase contrast microscopy
6.4 UV/Vis absorption
6.5 Raman measurements
6.6 Small angle X-ray scattering (SAXS)
6.7 X-ray diffraction (XRD)
6.8 Scanning electron microscopy (SEM)
7. Synthesis of L(+)ascorbyl stearate (Asc18)
8. Synthesis of Gold Nanoparticles
8.1 Preparation with Asc18 surfactant
8.2 Preparation with Asc12 surfactant
8.3 Preparation with Asc14 Asc10 and Asc8 surfactants
9. Preparation of ZrO2- nanoparticles
9.1 Synthesis of zirconium hydroxide by coprecipitation in homogeneous phase (sol)
9.2 Preparation of hydrous zirconia gel

IV. Results and Discussion
10. Gold nanoparticles
10.1 Determination of the cmt of Asc18
10.2 Synthesized nanoparticles and their colors
10.2.1 Influence of reaction temperature
10.2.2 Comparison of different concentrations
10.3 UV-Vis characterization
10.3.1 Comparison of different concentrations
10.3.2 Comparison of different reaction temperatures Asc10 Asc12 Asc14 Asc18
10.4 SAXS characterization
10.4.1 The Schulz Spheres fitting model [79]
10.4.2 Comparison of reactions above and below the cmc
10.4.3 Comparison of different reaction temperatures
10.4.4 Comparison of AscX surfactants with different chain lengths Asc12 Asc14 Asac18
10.5 Conclusion
11. 3. Zirconium hydroxide and oxide nanoparticles
11.1 Raman characterization
11.2 Dialysis of the sol and an aqueous ZrOCl2 solution
11.2.1 Conductivity of the sol and ZrOCl2 solution
11.2.2 Progress of pH and conductivity during gel-formation
11.3 Characterization with microscopic methods
11.3.1 LM-micrographs of untreated gel
11.3.2 LM-micrographs of squeezed gel
11.3.3 LM-micrographs of air-dried gel
11.3.4 LM-micrographs of collapsed gel
11.3.5 LM-micrographs of a freeze-dried gel
11.4 DSC measurements of the gel
11.5 Characterization by SEM
11.6 SAXS characterization of the gel
11.6.1 The unified fit model [80]-[82]
11.6.2 Structural parameters of the gel
11.7 TGA and DTG measurements
11.8 XRD characterization of calcined samples
11.8.1 Diffractogram of the gel-synthesized particles
11.8.2 Diffractogram of the sol-synthesized particles Samples containing NaCl Samples without NaCl
11.9 Conclusion

V. Annex

List of Figures

List of Tables


Part I

I. Introduction

The development of small and smallest particle is one of today’s key features in modern science. The goal is to form materials with improved properties than their “classical” ancestors with just a fractional amount of raw material.

Another key feature of nanoparticles is their different, and sometimes unexpected, behavior concerning reactivity, compared with their bulk materials. Because of this, nanoparticles have a wide range of applications, especially in the field of catalysis.

Here, characteristics of nanoparticles - more edges, corners, defects or oxygen vacancies – are used to obtain a high performance of the catalysts. Nanoscaled particles also exhibit larger surface area and higher metal dispersion, which further contributes to the catalytic possibilities.

To gain such particles, two different pathways are given: first, there is the so-called “top down” pathway, considered as further developments of micro technology, where physical preparation methods like lithography are used. The second way is the “bottom up” method where self-assembling systems, formed by surfactants, are used.

Concerning gold nanoparticles, it is reported [1] that the use of C16TAB at specified conditions, gives gold nanorods with a sharp size-distribution because the direction of growth is predetermined. Being a cationic surfactant, C16TAB affects the electrochemical potentials and introduces bromide-ions as an additional species to the reaction. To achieve gold nanoparticles from aqueous HAuCl4-solutions, the above-mentioned method needs a separate reducing agent such as ascorbic acid (Asc0), NaBH4 or N2H4.

A way of synthesizing spherical gold nanoparticles is the use of Nd:YAG laser with a salt induced agglomeration [2]. By modifying the formulation of the salt solution, different sizes are obtained. This way of synthesis, a combination of physical top-down and self-assembling bottom-up processes, can be modified by adding surfactants, like PEG, to optimize size distribution and physical characteristics, like UV-Vis absorption [3].

This method is an elegant way of synthesis; however, problems may occur by functionalizing the particles, because of a high salt content. Here, a high influence of purity, concentration and composition to the size and shape of gold nanoparticles might be given. Therefore, a route of synthesis is needed, which shows high efficiency in producing gold nanoparticles and in stabilizing them with a manageable amount of parameters. As mentioned before, good results are obtained by adding surfactants, which limit growth. In the case of gold nanoparticles, this control can be performed by gel-forming agents. To avoid impurities and to limit the parameters of the system, an optimal gel-forming specie is also the reducing agent and can influence size and shape by varying only physical parameters such as temperature.

It is reported [4]-[5], that Asc0 can be used as a reactant to reduce Au3+ to gold nanoparticles due to a low redox-potential of Asc0 (E0 versus NHE = +0.08 V) and a high redox-potential of Au3+ (E0 versus NHE = +1.498 V).

The problem of this reaction is agglomeration and growth control, because of the absence of a surfactant. However, Asc0 is not only a reducing agent, it is also a slightly weak acid (pka1 = 4.25 [6]) with two alcohol-groups, and therefore, it can react under acid conditions with fatty acids to form an alkyl-ester. This ester still has the reducing-ability of Asc0 and can form aggregates, because of their hydrophobic tail. Therefore, a surfactant with reducing properties and the ability of forming self-assembling structures is achieved. Using different carboxylic acids, different surfactants with different aggregation patterns can be produced, thus different sizes of nanoparticles can be gained. With the use of biodegradable agents, ascorbic acid and fatty acids, a step in the direction of “green chemistry” is taken. Therefore, the reaction-matrix, the gel formed by the different surfactants, is fully biodegradable.

In addition, gold nanoparticles are reported to exhibit a small degree of toxicity at low concentrations [7] and are widely used in medical applications like biomedical imaging and diagnostic tests [8]. Recent development shows a possible application of gold nanoparticles as active agent in cancer-therapy, where the nanoparticles are absorbed by the tumor cells eight times more than by normal cells. The excitement of these particles by X-rays destroys the tumor-cells due to gold’s significant high-Z X-ray absorption. [9]

Technical applications of gold nanoparticles can be found at very small particles with diameters below 10 nm. The deposition on metal oxides or activated carbon is connected to catalytic properties, especially at low temperatures, for many reactions such as CO oxidation and propylene epoxidation. [10]

A major problem of gold nanoparticles is the high price of gold. Therefore, other materials with good catalytic properties had to be found. With excellent thermal stability, catalytic properties and a comparable low-cost synthesis, zirconium dioxide is one of them. For example, the production of synthesis gas, the carbon dioxide reforming of methane, and the hydrogenation of carbon dioxide for the production of methanol, widely used as a feedstock for chemical industries and the use as an alternative fuel, which is cleaner and more efficient in fuel cells, are nowadays used applications for zircon nanoparticles. [11]

There exist plenty of possible applications for zirconia nanoparticles, like advanced structural transformation-toughened ceramics for wear parts, engine and machine components, cutting and abrasive tools, sensors for oxygen transport and detection, solid electrolytes for solid oxide fuel cells and high-temperature water-vapor electrolysis cells, catalysts for automotive exhaust cleaning and the partial oxidation of hydrocarbons, pigments and much more. [12]

However, there exist three different crystalline structures of zirconia, monoclinic, tetragonal and cubic, where the first one is stable at room temperature. The high temperature phases, which have potential applications as oxygen sensors, solid fuel cells, and several ceramic components, cannot be retained at room temperature because the transformation is reversible. The addition of small amounts of different oxides (such as yttria) can stabilize partially or fully these modifications. [13]

However, unexpected metastable tetragonal zirconia my be present at room temperature in un-doped crystals.

It has been supposed by several works ([14]-[16]) that either the occurrence of specific precursors or the control of crystal size can lead to tetragonal crystals, which are stable at room temperature. Several efforts have been made to explain this extraordinary behavior. In particular, it has been suggested that small particles are more stable by forming tetragonal particles instead of monoclinic ones because of surface energy-effects.

This means, the rational use of the preparation procedure gives particles of different sizes and structures.

Part II

II. Fundamentals

1. Plasma oscillation and Mie’s theory

illustration not visible in this excerpt

Fig. 1.1: Triangle of chemical bonds

The nature of the chemical bond is a very complex matter. In the past, it was believed, there are only three types of bonds: high-energy covalent bonds, ionic, electrostatic bonds and metallic bonds. With better-developed analytical equipments and by means of computer simulations, nowadays opinion is that the latter three types are only extreme cases of bonds. Literature [17] says every bond has certain parts of the above-mentioned ones. Nevertheless, a classification is appropriate due to an overweight of a certain bond-type. The criteria of this is the difference of electronegativity (EN) of the elements, the bond is composed of. For instance, a component forming a metallic bond has a small sum of electronegativity and a small difference in electronegativity. In case of metals, this criterion is perfectly fulfilled.

However, metallic materials, especially metals, have very interesting properties. Due to the nature of the metallic bond, electrons of the outer valance shell are delocalized and form a quasi continuum. This can be described as plasma, a state of matter “with equal concentration of positive and negative charges, of which at least one charge type is mobile. In a solid the negative charges of the conduction electrons are balanced by an equal concentration of positive charge of the ion core.” [18]

In consequence of this, many interesting behaviors of metals can be explained, for instance electrical conductivity and unique optical phenomenon.

As is well known from everyday experience, for frequencies up to the visible part of the spectrum metals, in their bulk-phase, are highly reflective, which means all light will be absorbed and reemitted without a significant change in intensity – that is the principle of a mirror - and do not allow electromagnetic waves to propagate through them.

However, metallic nanoparticles, which usually have a dimension in the order of 5 – 100 nm, show a completely different behavior. For instance, in literature [19] it is reported that elementary physical parameters, such as the melting point of gold, strongly depend on the particle size. In addition, also optical parameters, as the reflection and absorption of light, are linked to the size.

1.1 Principles of plasma oscillation [20]

The so-called quasi-static approximation can be used to describe the interaction of a particle of size d with an electromagnetic wave of a wavelength l, if

Abbildung in dieser Leseprobe nicht enthalten

is valid. With a wavelength of 300 – 800 nm the spectrum of visible light fulfils equation , if the particles are smaller than 100 nm. With this assumption, the phase of the harmonically oscillating electromagnetic field is practically constant over the particle volume.

In the case of metal nanoparticles dispersed in water, the assumption of a small, homogeneous spherical particle of radius a in an isotropic, non-absorbing medium with a dielectric constant εm is conformable. This particle is exposed to an electric field with

Abbildung in dieser Leseprobe nicht enthalten

The dielectric response of the sphere is further described by the complex dielectric function ε(ω). In the case of nanoparticles, l is significantly longer than all characteristic particle-dimensions such as the size of the unit cell or the mean free path of the electrons. So the dielectric function can be written as

Abbildung in dieser Leseprobe nicht enthalten

with ω as the angular frequency (ω = 2π·c/l). Following the mathematical path of [20], the result gives the relation of the dipole moment p and the applied field E0:

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In other words, a dipole moment is induced inside the sphere by an external field, which is proportional to | E0 |. The proportionality factor α is defined as

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For small sub-wavelength diameter spheres, α is the complex polarisability in the dielectric approximation. For a small sphere with a << λ, its representation as an ideal dipole is valid in the quasi-static regime. Under plane-wave illumination, E can be described as a function of time t and ω:

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As a consequence of equations and , the induced dipole moment p is a function of t and ω. With equation , an oscillating dipole moment is given as:

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Fig. 1.2 illustrates the shift of the electron density of a spherical metal-nanoparticle as a result of the interaction with electromagnetic radiation.

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Fig. 1.2: Schematic drawing of the interaction of an electromagnetic radiation with a metal nanosphere. A dipole is induced, which oscillates in phase with the electric field of the incoming light.

1.2 Scattering and absorption of small particles [23]

According to equation α is a function of ω, which leads to a resonance-wavelength and an enhancement of α. Because of this, resonantly enhanced polarization α is a concomitant enhancement in the efficiency with which a metal nanoparticle scatters and absorbs light.

Now the corresponding cross sections for scattering and absorption Csca and Cabs can be calculated [23] (k = 2π/l):

Abbildung in dieser Leseprobe nicht enthalten

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For small values of a, the efficiency of absorption (~ a 3) dominates over the efficiency of scattering (~ a 6). De facto equation demonstrates one of the urgent problems in scattering. With a being the dominant factor of the scattering efficiency, big particles are the dominant species in a scattering experiment, while smaller particle have a much lower efficiency. This means, small objects hardly can be picked out from the background of larger scatteres. Equations and show, for metallic nanoparticles, both, absorption and scattering (and thus extinction), are resonantly enhanced at the dipole particle plasmon resonance. The result is the extinction cross section Cext for a small sphere of radius a assuming the quasi-static approximation.

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In general, Cext is defined for spherical particle with a complex, ω -depended dielectric function (see equation ), embedded in a medium of dielectric function εm, as:

Abbildung in dieser Leseprobe nicht enthalten

an and bn are scattering coefficients in terms of Ricatti-Bessel functions. Discussed by the German physicist Gustav Mie in 1908 [25], in the case small particles with k·a <<1, only the first terms of equation are important to calculate Cext.

Abbildung in dieser Leseprobe nicht enthalten

This means the dipole-term is predominant for the absorption, which depending on the whole metal-concentration in the solution not on the particle size. Therefore, the “classical” Lambert-Beer equation Abbildung in dieser Leseprobe nicht enthalten (with σ as the molar absorption-coefficient and c as the molar concentration) is not strictly valid if the concentration of a particle with a special size is concerned.

This approximation is adequate if the particle size is above 5 nm, because of a change of the dielectric properties of the material at very small particle-sizes.

Consequently, all nano-scaled metals have specific absorption-bands. The resonance wavelength lp strongly depends on the electron density N and the effective mass of electrons µ of the material. As mentioned before, this phenomenon is known as plasmon-resonance. In the case of spherical metallic nanoparticles, the term l ocalized s urface p lasmon r esonance (LSPR) is used.

Per definition, a plasmon is a quantum of a plasma oscillation, while a plasma oscillation is a collective longitudinal excitation of the conduction electron gas in a metal. [18] lp, also known as the plasma wavelength, is characteristic for each metal and can be calculated:

Abbildung in dieser Leseprobe nicht enthalten,

with e0 as the elementary-charge and cls as the speed of light in vacuum (2.997·108 m/s).

With and the Lambert-Beer-equation, for a dilute colloidal solution containing M particles per unit volume, the measured attenuation of light of intensity I0 over a path length of x is given in the differential form as:

Abbildung in dieser Leseprobe nicht enthalten

Equation shows, that the loss of light is directly proportional to the number of light extinguishing particles. By integration, the absorbance A can be expressed as:

Abbildung in dieser Leseprobe nicht enthalten

With a useful relation between theory and experimental data is given, that is similar to the classical absorption-equation.

It is therefore possible to simulate an absorption spectrum for nanoparticles of a certain size. Eadon and Creighton [24] calculated the absorption spectra of several metallic nanoparticles. For red-colored spherical gold nanoparticle with a = 5 nm, there are two major regions of absorption (Fig. 1.3), a small UV-active area at 260 nm and a red-shifted absorption at 525 nm.

Abbildung in dieser Leseprobe nicht enthalten

Fig. 1.3: Calculated absorption spectra of spherical gold nanoparticles with a diameter of 10 nm, continuous line was calculated with a surrounding medium equal to water (ε = 1.33), dashed line was calculated for gold nanoparticles in vacuum. [24]

In conclusion, nanoparticles hardly effect the reflection of visible light. Light is scattered (equation ) and absorbed (equation ). If the nanoparticles are colorful, the resonance is in the visible part of the spectrum, if they are colorless, the absorption-band is localized in the UV-region of the light. Basic equations of light-absorption are valid, if the total metal-concentration is concerned (equation ).

2. L(+)-ascorbic acid and its derivates

Vitamin-C (Asc0) is one of the most important biochemical agents for humans and animals. Its main function in the body is being a cofactor at the production of Collagen, the most abundant protein in mammals and the main protein of connective tissue. The name “ascorbic” is derived from Latin (“a” = “no” and “scorbuticus” = “scurvy”) and illustrates the main purpose of natural Asc0 the prevention of scurvy. Despite its importance, the molecule was not fully discovered and identified as an essential substance until 1933.

Both, the discovery of the structure and the discovery of the medical importance, were honored in 1937 with the Noble-Prize for Chemistry (Walter Haworth) and Physiology or Medicine (Albert Szent-Györgyi), respectively.

2.1 Chemical properties of ascorbic acid

Besides its use for the assembly of Collagen, Asc0 can also react as a scavenger. This ability gives Asc0 the importance as natural antioxidant in many industrial applications. Fig. 2.1 gives the basic reaction of Asc0 to dehydroascorbic acid (DHA), which is one of the key-feature of this acid.

Abbildung in dieser Leseprobe nicht enthalten

Fig. 2.1: Oxidation-reaction of Asc0 to the oxidized form (Asc) and finally to dehydroascorbic acid (DHA)

At standard conditions, Asc0 has a potential, E0 (versus NHE), of +0.08 V. Despite the fact Asc0 has no “classical” acidic groups, such as carboxyl-groups or phosphoric-groups, it exhibit a comparably low pKa1-value of 4.25 [6] (acidic acid: pKa= 4.75) of the hydroxyl-group at position 3. This arises from the enediol-structure of Asc0 and the neighbor carbonyl group. A second reason for the high acidity is the mesomeric stabilization of the resulting negative charge.

There is also a second acidic group, the hydroxyl-group of position 2, with a much higher pKa2-value of 11.79 [6]. This weaker acidity is a result of the already negative charged mesomeric system. Besides that, the other two -OH groups of position 5 and 6 behave as a secondary and a primary alcoholic residue, respectively.

Because of that, Asc0 can behave as a diprotic acid or as an alcohol. This latter property is utilized at the syntheses of AscX-surfactants (Fig. 2.2), where X is the chain length of the used carbon acid. The primary hydroxyl group at position 6 is used as nucleophile to esterify the carboxylic acid. For this purpose, ascorbic acid is dissolved in concentrated sulfuric acid, to prevent the deprotonation-reaction and to scavenge the resulting H2O formed during the synthesis.

Abbildung in dieser Leseprobe nicht enthalten

Fig. 2.2: General reaction of Asc0 to AscX

The used carbon acid then determines the dimension of the hydrophobic tail of the surfactant and thus the hydrophilic-lipophilic balance of the molecule. Other results of the esterification are changes of chemical properties, like the solubility in water, and physical parameters, like the melting point, while other characteristics such as the absorbance-wavelength of visible light or the electrochemical potential E0 are almost identical to Asc0. In general, AscX-surfactants are anionic surfactants in aqueous solutions.

2.2 AscX surfactants and their aggregates [28]

The solubility of AscX is decreased with the alkyl chain length and increased with temperature. Above the critical micellar concentration (cmc) and the critical micellar temperature (cmt), an aggregation takes place and, according to their hydrophilic character, different microstructures can be formed.

This formation is driven by thermodynamic forces. The traditional picture of micelle formation thermodynamics is based on the Gibbs–Helmholtz equation at isothermal (Δ T = 0) and isomolar (Δ n = 0) conditions:

Abbildung in dieser Leseprobe nicht enthalten

At room temperature, the process is characterized by a small, positive enthalpy, ΔHm, and large, positive entropy of micellisation, ΔSm. The latter is considered as the main contribution to the negative ΔGm value, and has led to the idea that micellization is an entropy-driven process. High positive values of ΔSm are surprising since aggregation, in terms of configurational entropy, should result in a negative contribution. In addition, large values of ΔHm would have been expected since hydrocarbon groups have very little solubility in water, and consequently a high enthalpy of solution.

The formation of micelles and other aggregates of surfactant molecules partly results from the tendency of the hydrophobic groups to minimize contacts with water by forming oily micro-domains within the solvent. There, alkyl–alkyl interactions are maximized, while hydrophilic headgroups remain surrounded by water. This aggregation gives maximum negative values of ΔGm and therefore, provides the most stable energetic configuration of the system.

As mentioned before, different surfactants have specific values of cmc and cmt. Homologous series of AscX surfactant therefore, show a general tendency of these characteristic values. Longer hydrophilic tails imply a decrease of cmc and an increase of cmt (Table 2.1).

illustration not visible in this excerpt

Table 2.1: cmc and cmt of different AscX-surfactants. cmc measurements were performed at 30 °C, cmt measurements were carried out on aqueous solutions at a concentration of 10% (w/w). [26]

By plotting log(cmc) of different AscX against their chain length (Fig. 2.3) and calculating the equation of the resulting straight line, the cmc of other ascorbic acid derivates can be calculated as an estimate.

Abbildung in dieser Leseprobe nicht enthalten

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Fig. 2.3: Plot of log(cmc) against the chain length of different AscX-surfactants

As a good approximation, the cmc of Asc18 is calculated to be about 3.23·10-4 mmol/l. In literature [27] it is reported that Vitamin C derivates with alkyl-chains longer than 12 carbon atoms (X > 12) form bilayers. The ability of forming this kind of aggregates is due to long AscX in general, and of ASC18 in particular, molecular structure.

To determine the expected microstructure, the surfactant parameter, Ns, was introduced [28]:

Abbildung in dieser Leseprobe nicht enthalten,

with ν as the volume of the hydrocarbon core, in nm³, l as the maximum length of the hydrocarbon chain in nm, and a 0 as the effective area per polar headgroup in the units of nm². ν and l can be estimated by [28]:

Abbildung in dieser Leseprobe nicht enthalten.

In equation nc is the total number of carbon atoms per chain and nMe the number of methyl groups respectively. It is worth mentioning, that in this model a methyl-group is twice the size of a CH2-group. The value 0.15 nm is derived from the van der Waals radius of the terminal methyl group (0.21 nm) minus half the bond length of the first atom not contained in the hydrocarbon core (0.06 nm) and 0.127 nm is the carbon-carbon bond length (0.154 nm) projected onto the direction of the chain in the all-trans conformation. [28]

In equation , the most problematic quantity is the effective area per polar headgroup, a 0, which cannot be calculated a priori with such easy methods l and ν can be calculated. The reason for this is the diversity of surfactants and their different dependency on electrostatic and external conditions, such as temperature, etc. a 0 of ionic surfactants depends on electrolyte and surfactant concentration, while in the case of non-ionic and zwitterionic surfactants a 0 is influenced by external conditions. [28]

In the case of AscX, a 0 ranges from 65 Å2 (Asc8) to 21 Å2 (Asc14 and longer chain length). [27] In this case, the structure formed by surfactants determines chemical parameters such as pka-value.

According to literature [28], the optimal stability of aggregates is given by following Ns values:

Spherical micelles Ns ≤ 0.33

Infinite cylinders 0.33 < Ns ≤ 0.5

Planar bilayers 0.5 < Ns ≤ 1

Inverted cylinders and micelles Ns > 1

Abbildung in dieser Leseprobe nicht enthalten

Fig. 2.4: Schematic drawings of possible aggregates for surfactants in water. a) spherical micelles, b) infinite cylinders, c) planar bilayers

Nevertheless, the above-mentioned “borders” for different structures are just guidelines. For example, a surfactant with an Ns value of 0.45 can form spherical micelles or already infinite cylinders.

For before mentioned surfactants (Asc8 – Asc18), the following Ns -parameters have been calculated:

illustration not visible in this excerpt

Table 2.2: Calculated Ns -parameters and suggested structures of different AscX [27]

Theory shows that for 0.33< Ns < 0.5, hexagonal phases are the optimal state, if the hydrocarbon chains are truly fluid. As Ns increases above 0.5 (with increasing chain length), two possibilities occur: If the chains are stiff, the system jumps immediately to lamellar (or cubic) phases. If the chains are not stiff and very fluid, it passes through unilamellar vesicles of increasing size to multiwalled vesicles and, eventually, to lamellar phases as the surfactant parameter increases toward unity. [29]

2.3 The formation of gel and coagel

It is known that AscX-surfactants form a so-called “coagel”. To obtain this coagel phase, an aqueous solution of AscX has to be heated up above the surfactant’s cmt. By doing so, AscX forms clear solutions with different viscosity containing microscopic aggregates, which, depending on the surfactant, can be micelles and bilayers. After cooling down to room temperature, the clear solution becomes opaque again, but no precipitation occurs and a white paste is produced. The coagel-phase has been formed.

In particular, a solution of Asc8 and Asc10 form liquid micellar phases upon heating. By contrast, Asc12 to Asc18 show a coagel-to-gel phase transition with the absence of micellar solutions, at least below 80°C (see Fig. 2.5). The gel-phase is a high viscous, clear solution.

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Fig. 2.5: Schematic structure of the coagel, micellar aggregate and gel. Depending on the surfactant, different aggregates can be formed. [27]

As Fig. 2.5 shows, in the coagel-phase Vitamin C –headgroups are surrounded by strongly bound water molecules, while in the gel-phase additional water, generally referred to as intermediate water, is able to enter the bilayer-structure. As a result, the total amount of water between the ascorbic acid-headgroups is increased and thus the distance of two bilayers.

This inter-phase leads to a most interesting application for AscX-surfactants. As previously mentioned, the electrochemical properties of AscX-surfactants are mostly the same than those of Asc0, and the surfactant can still react as an acid.

If water-soluble substances, such as salts, are mixed with an aqueous AscX-solution, those substances can invade the interlayer. Electrochemical-active ions, such as AuCl4-, can perform electrochemical reactions. Because of the bilayer-structure, a limitation in space is given and the particles can only grow up to a specific diameter. As an anionic surfactant, AscX is also able to stabilize the gained particle because of adsorbing on the metal surface. This adsorption process stabilizes colloidal metal particles by electrostatic repulsion due to the negative charged ascorbic headgroups and by steric stabilization due to the hydrophobic tails. This leads to very stable, aqueous colloidal solutions, where attractive van der Waals forces are outweighed by electrostatic and steric repulsion. [30]

3. Zirconia and its different structures and species

Zirconium was fist discovered by M.H. Klaproth in 1789, as one of the main compound of the silicate ZrSiO4 (“Zirkon”). This mineral is one of the most common sources for the production of metallic zirconium and its derivatives. Due to its high resistance to corrosion, zirconium is a well-established material in the construction of chemical instruments with characteristics comparable to titanium. Further, zirconium is used for the construction of nuclear power plants and as casing-material for nuclear fuel elements. In the field of metallurgy, the ability of adsorbing nitrogen, oxygen and sulfur on the surface gives zirconium great importance in the production of high-quality steel. [31]

3.1 Zirconyl chloride in aqueous solutions

Despite above-mentioned benefits, the production of pure zirconium is problematic because of strong chemical and physical relationships to hafnium. One method is the transformation to ZrCl4 and following fractioned distillation. Hydrolysis of this product, even in high acidic aqueous solutions, such as concentrated hydrochloric acid, leads to zirconyl chloride ZrOCl2·8H2O, a water-soluble zirconium-derivate, which contains the tetrameric cation [Zr4(OH)8(H2O)16]8+ . The crystal structure is tetragonal. [32]

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Fig. 3.1: Structure of the tetramer [Zr4(OH)8(H2O)16]8+in the crystals of ZrOCl2·8H2O

As shown in Fig. 3.1, the four Zr atoms are cross-linked by two hydroxo bridges. The attachment of four additional H2O molecules to each zirconium atom leads to a coordination polyhedron of a distorted dodecahedron. Structural studies showed the four H2O molecules per Zr atom are specifiable in two inert and two labile water-species. [33]

By heating an aqueous solution of ZrOCl2·8H2O for at least 24 h at 105 °C, further hydrolyzing-processes take place and the formation of octamers occurs. [34] The same effect can be observed by titration with dilute solutions of NH4OH [34] or NaOH [35].

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Fig. 3.2: Simplified principal reaction of two tetramers to the octamer form.

Clearfield [34] confirmed this mechanism by pH measurement performed in 1964. He adjusted the pH value of 1 M zirconyl chloride solutions with ammonia to values between 2.5 and 1. The more basic solution (above a pH value of 2) showed a small amount of precipitation, which was re-dissolved by heating. The clear solutions were refluxed for 48 h and samples were removed at various time intervals. A continuous decrease in pH of the solution was detected.

However, also octamers are hydrolyzed thus forming bigger aggregates. This can be achieved by adding higher amounts of base or heating for more than 48 h, so polymorph structures are gained. Here, the same mechanism as for the formation of octamers is proposed.

Tetramers, octamers and polymeric species in aqueous solutions are very similar in their optical, physical and chemical parameters, which makes a determination of single species almost impossible. Nevertheless, a qualitative identifications of tetrameric and octameric species with NMR- and Raman-spectroscopy and small angle X-ray scattering (SAXS) have been published ([33]-[37]).

A next step of hydrolyzing takes place, if the amount of base is further increased: the clear polymeric solution is getting opaque and a white solid precipitates. According to literature [38], this effect can be observed when the pH of the solution is raised over a value of 2. The hydrolysis reaction gives, in a first step, zirconium tetrahydroxid hydrate Zr(OH)4·aq. By writing this specie in a “coordination-sphere-correct” way, [Zr4(OH)8(OH)8 (H2O)8], shows a structural relationship to [Zr4(OH)8(H2O)16]8+ (see Fig. 3.1). After aging, this species is transformed to ZrO(OH)2·aq. The calcination of this gives ZrO2. [31]

3.2 Structures of zirconium oxide

Zirconium(IV) forms many different compounds in aqueous solutions. However, structures of solid zirconium derivates are even more versatile than their liquid colleagues. An interesting fact here is a dramatically change of properties of compounds distinguishable just in structure. Zirconium(IV)-oxide is an excellent example of this behavior and is in focus of scientists for many years.

3.2.1 Crystal structures and martensic phase transformation

The most naturally occurring crystalline modification of ZrO2 is the monoclinic structure, mainly in the mineral “baddeleyite”. At room temperature, this structure is the most stable one. The coordination number (CN) of zirconium is 7, i.e. one zirconium atom is coordinated by 7 oxygen atoms (see Fig. 3.3).

Abbildung in dieser Leseprobe nicht enthalten

Fig. 3.3: Structure of monoclinic ZrO2, projected onto the (010) plane.

Depending on given parameters, such as temperature, doping or crystal size, other modifications are known. At about 1000-1200 °C [31], a phase transition to tetragonal crystals occurs; the volume of the crystal is decreased while the coordination number is increased (CN = 8, see Fig. 3.4).

Abbildung in dieser Leseprobe nicht enthalten

Fig. 3.4: Structure of tetragonal ZrO2, projected onto the (110) plane.

At temperatures above 2370 °C [31], these crystals transform to cubic zirconia (CN = 8, fcc = face centered cubic). Because of high temperature stability (melting point 2680 °C [31]) and the volume decrease at the transition from monoclinic to tetragonal [31], the latter two symmetries are focused for technical applications.

The problem here is the stabilization of the desired crystalline structure. At standard conditions, tetragonal or cubic zirconia is instable and a phase transition back to the less dense monoclinic crystals is inevitable. A way to stabilize tetragonal and cubic zirconia is doping with metal oxides, for example CaO, MgO or Y2O3. The first two oxides are used for the production of artificial gemstones; by further doping with metal oxides, also colorful species of cubic zirconia are known. Because of this, mono-crystalline cubic zirconia is a frequently used as imitation of diamond, with comparable optical and mechanical properties, like refractive index and hardness.

Y2O2-doped ZrO2 is used for high-temperature applications, for example in process monitoring. Hence, tetragonal-cubic zirconium dioxide has become one of the most important materials in environmental technologies, as it is the key-feature in the so-called l-sensor, an oxygen-selective sensor based on cubic zirconia doped with Y2O3. ZrO2 is here the crystalline matrix, while the doping with Y2O3 provides oxygen defect sites, which are responsible for the conduction of O2- ions. Compared to above-mentioned cubic zirconia species, this material is white. Despite of doping, a mixture of 85% of ZrO2 stabilized by 15% of Y2O3 is used as resistance element due to its high electric conductivity.

With its excellent mechanical behavior, tetragonal ZrO2 is a well-used additive in car paints to obtain higher scratch resistance and is used as top coating for tiles.

As written before, there are three major crystalline structures for ZrO2, among which the monoclinic is stable without further treatment. Nevertheless, it undergoes a reversible martensitic phase transformation at about 1200 °C to a tetragonal structure.

A martensitic phase transition is a diffusionless, co-operative phenomenon, where a large number of atoms are slightly moved by a shear mechanism, for example mechanical twinning in steel production. Thus, change in crystal structure is achieved by a homogeneous deformation of the parent phase.

The velocity of this movement is comparable to that of sound waves in crystals and is not associated with changes in composition. In general, the high-temperature phase cannot be retained by quenching. Martensitic phase transition differs from the classical phase-transition process. In the latter, a first step of nucleation is followed by a growth process. Both steps are thermally activated and the amount of transformation depends on time and temperature. In the case of martensitic transformations, instead, pre-existing nuclei grow without thermal activation. This justifies the adoption of an athermal nature of the transformation and leads to the assumption that the transition itself is independent from temperature; by contrast, the amount of transformation is dependent on temperature and independent of time. However, in some systems, it was possible to observe isothermal and athermal or just isothermal kinetics. Therefore, “classical” kinetic considerations are not valid in the case of martensitic phase transformations.

Usually these transformations are accompanied by a shape change, where the surface tilts on a polished surface caused by lattice shear. This generates crystal defects and imperfections, which themselves are responsible for improved hardness due to higher sliding-resistance of single lattices.

The transformations are always reversible and show thermal hysteresis in the range of just a few degrees up to several hundred degrees [39]. In the case of zirconia, the phase transformation of monoclinic to tetragonal phase shows a hysteresis gap of 208 °C (width of hysteresis loop at midpoint; see Fig. 3.5).

Abbildung in dieser Leseprobe nicht enthalten

Fig. 3.5: Monoclinic to tetragonal phase transformation in zirconia [39]

In newer literature [42] the mechanism of this phase transition is also explained by the occurrence of “centaur nanoparticles”. The name centaur is derived from Greek mythology, where a “Kentauros” is a creature half-man half-horse. Based on this image, a centaur nanoparticle is a particle partially describable by methods of quantum mechanics (micro-cosmos) and by methods of classical mechanics (macro-cosmos). [43] In particular, zirconia centaurs involve fragments of monoclinic and tetragonal structures coexisting within the same nanoparticle.

In this mixture of classical and quantum states centaur particles are areas in which the coherence of wave packets can be annihilated and created again. Quantum mechanical phenomenon, for example the superposition principle, which says the sum of two different solutions of Schrödinger’s equation resulting a third one, are valid.

However, in the nanoworld, the decoherence limits this state of existence to a very short time. For example, during the formation of coherent boundary between regions of different crystalline structures in a centaur nanoparticle (see also Fig. 3.6), the nanosized system is in a pure state Abbildung in dieser Leseprobe nicht enthalten described by a wave function Abbildung in dieser Leseprobe nicht enthalten; using the superposition and the coherence of Abbildung in dieser Leseprobe nicht enthalten, it is possible to write [43]:

Abbildung in dieser Leseprobe nicht enthalten

Here, Abbildung in dieser Leseprobe nicht enthaltenare classical distinguishable states with G being the number of these states and Ci are constants. The hypothesis of spontaneous decoherence says, every now and than, a spontaneous wave function collapse due external influences takes place, so that a coherent system is driven to an incoherent system [43]:

Abbildung in dieser Leseprobe nicht enthalten

The creation of a centaur particle can be treated like this. The probability of this transformation can be calculated by:

Abbildung in dieser Leseprobe nicht enthalten

The sum of all probabilities over G therefore, is 1:

Abbildung in dieser Leseprobe nicht enthalten

The occurrence of this particle by contrast is only possible, if the relationships between the coefficients Ci defined in equations and are valid. [43]

In the case of zirconia nanoparticles of approximately the same shape, equation describes, due to the superposition of states, the system as follows [43]:

Abbildung in dieser Leseprobe nicht enthalten

Here, m indexes the monoclinic, t the tetragonal, c the cubic and cent the centauric homogeneous structure, respectively, and Abbildung in dieser Leseprobe nicht enthalten. With and the estimate

Abbildung in dieser Leseprobe nicht enthalten

the probability of transition P in which the nanosized system transforms into one of the centaur particles, can be expressed by:

Abbildung in dieser Leseprobe nicht enthalten

Two classical distinguishable states Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten are located in different spatial regions. Their wave functions φi and φk overlap slightly which means, the matrix element Abbildung in dieser Leseprobe nicht enthalten is small in magnitude. This matrix element is strongly dependent on the number of atoms M in the system under consideration:

Abbildung in dieser Leseprobe nicht enthalten

Here, q is a nonnegative parameter, which cannot exceed unity and is defined by equation , a consequence from the Cauchy-Schwarz-Bunyakovskii inequation [44]:

Abbildung in dieser Leseprobe nicht enthalten

If the states Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten are different, then q < 1; with increasing numbers of atoms, at a fixed q-value, the matrix element monotonically decreases (see equation ). Therefore, a minimum particle size for centaur particles can be calculated. According to literature [43], centaur nanoparticles of ZrO2 have a minimum size of approximately 10 nm. As mentioned above, the mechanism of monoclinic to tetragonal phase transition in zirconia nanoparticles is based on the formation of centaur particles.

Abbildung in dieser Leseprobe nicht enthalten

Fig. 3.6: Structural model of a ZrO2 centaur particle at the phase transition of monoclinic to tetragonal structure. [43]

Fig. 3.6 shows the principle mechanism of such a phase transition. As mentioned above, the structure of centaur nanoparticles is not specified and is supposed to be a quantum mechanical phenomenon.

3.2.2 Tetragonal zirconia and critical crystal size

As written before, stabilization of metastable crystal structures can be achieved by doping with other metal oxides. In case of cubic zirconia, this is quite successful. However, pure ZrO2 with a metastable crystalline structure, especially tetragonal zirconia, has to be stabilized in a different way. One way is to form the tetramer species by means of precipitation from alkaline aqueous solutions or by calcining salts of zirconium such as ZrCl4 [40]. Another method is the preparation of “active powders”. These materials are called active because they possess a very large specific surface and therefore, excess energy, compared to a large single crystal. As a result of this, active materials show different reactivity and chemical stability compared to bulk materials. For example, these powders tend to adsorb polar molecules such as isobutyl alcohol [15].

Observations of Garvie show that the formation of the precipitate leads to a stabilization of the tetragonal phase. X-ray diffraction studies have shown that the tetragonal phase is stabilized by parameters such as crystal size and hydrostatic stress. [16]

The so-called “Crystallite Size Effect” can be explained by following the formalism of literature [41].

The transformation from monoclinic to tetragonal ZrO2 is accompanied by a change in the Gibbs free energy. Assuming a spherical particle, this free energy change can be described as:

Abbildung in dieser Leseprobe nicht enthalten

Here, r is the particle radius, NA the Avogadro number, V0 = abc·sinβ is the unit cell volume of monoclinic zirconia, gt and gm are the free surface energies of the tetragonal and monoclinic modification, respectively. ΔfGt0 and ΔfGm0 are the molar formation energies of tetragonal and monoclinic ZrO2. In addition, equation accounts for the fact that one unit cell contains four formula units of ZrO2 (see also Fig. 3.3).

Further, the phase transition of monoclinic to tetragonal zirconia is accompanied with a volume change at constant ZrO2 amount. Therefore, the change in the Gibbs free energy is normalized to the initial volume of m -ZrO2. The transformation can proceed spontaneously, if Abbildung in dieser Leseprobe nicht enthalten. This is achieved for particles smaller than a critical diameter Abbildung in dieser Leseprobe nicht enthalten.

As a result of this, the critical particle diameter can be expressed as:

Abbildung in dieser Leseprobe nicht enthalten

As equation shows, the critical size is determined by the surface-to-volume energy ratio of zirconia particles of the corresponding modifications.

Since the surface energies of different zirconia modifications are close to each other ([45], [46]), an independence of the difference gt - gm of temperature is assumed. In the presence of phase transition, the Gibbs energy, enthalpy and entropy are defined as [21]:

Abbildung in dieser Leseprobe nicht enthalten.

Following literature [42], by assuming the Ulrich approximation, a useful expression of the difference in free Gibbs energy of monoclinic and tetragonal modification can be described as:

Abbildung in dieser Leseprobe nicht enthalten

In this equation, ΔtrH is the phase transition enthalpy of monoclinic to tetragonal crystal structure at the transition temperature Ttr. With equations and the critical particle diameter can be expressed as:

Abbildung in dieser Leseprobe nicht enthalten

Equation gives a useful expression that connects the critical particle size d c with temperature T, the enthalpy of the phase transition and the difference of surface energy. The latter two parameters depend on the focused material.

In other words, the high temperature phase can exist at a specific temperature, if the particles are smaller than the critical particle size of that temperature. In the case of room temperature and ZrO2, a critical particle size of about 10 to 30 nm with a stability range from room temperature to about 600 °C is obtained [42], [16].

4. Structural investigation techniques

The knowledge of structure has become a key feature in chemical research and engineering. As shown above, it is most important for the use of chemical substances, which structure and, therefore, the material’s properties, a substance has.

With the development of high efficient computers, digital acquisition techniques (for example CCD or PSD) and further proceeding in the understanding of the nanoworld, it was possible to develop investigative methods which give information on structure and the interaction with other particles.

These techniques are well established and commonly used for analysis of nanoparticles. The kinds of instruments and their theoretical background are divers and offer abilities of investigation ranging from aggregates of dimensions of microns to single atoms of only some Angstroms in size.

4.1 X-ray based methods

4.1.1 The nature of X-rays [48]

With typical particle sizes of 10 to 100 nm, a characterization of nanoparticles, especially in solutions, requires special techniques. One type of these techniques is the scattering of neutrons and electromagnetic waves. To perform small angle neutron scattering (SANS), a nuclear reactor or synchrotron sources are needed. Today there are about 37 neutron sources and only 32 SANS instruments [47]. Therefore, scattering of electromagnetic waves, like X-rays or visible light of lasers, are more commonly used. However, principal concepts of SANS are also valid for small angle X-ray scattering (SAXS).

The generation of X-rays is, compared to neutrons, easy to realize. Traditionally, X-rays are generated by means of a filament tube, invented by Coolidge in 1913. Here, two types have been developed: sealed tube and rotating anode. The modus of generation is in both types the same, but there are differences in quality and quantity of radiation. However, X-rays are produced by focusing an electron beam on a metal target, mostly Cu or Mo. The impact of the electron on the metal surface causes an electron hole in the inner-shell of the atom, which is recombined by electrons of outer-shells (Fig. 4.1).

Abbildung in dieser Leseprobe nicht enthalten

Fig. 4.1: Sketch of possible electron junctions after an electron impact

This process is in conjunction with a change in the energy of the electron and leads, due to the law of conservation of energy, to a transmission of radiation. By adjusting the acceleration voltage of the electrons, the region of impact can be focused to electrons on the K-shell. Therefore, most likely junctions are L à K and M à K , where the first one is called Kα, the second one Kβ. Focusing on the fist transition, the L state shows three different configurations, named L1, L2 and L3. The difference lies in the occupation of different atomic orbital and therefore, the electrons possess different energies (see Fig. 4.2). As will be explained later, different energies of photons result different frequencies of the radiation.

Abbildung in dieser Leseprobe nicht enthalten

Fig. 4.2: Energy levels of involved electron states in Cu K X-ray emission

X-rays from this kind of source are therefore, polychromatic; derived from visible light, they are called “white” radiation. Just like it is the case in applications of visible light, a monochromator needs to be used, to gain a certain wavelength. In case of Cu-radiation, the strongest line is the Cu Kα1 line. This means, other part of the emission-spectra has to be terminated. In commercial units, suitable crystals are used. X-rays enter the crystal and are reflected at different crystal lattices. According to Bragg’s law, most frequencies interfere destructively and are terminated. Ideally, there is only one frequency interfering constructively and therefore, are amplified. Nevertheless, in case of the Cu Kα radiation, no distinguishing of Kα1 and Kα2 can be made by this method. With a difference in wavelength of about 1.8·10-3 Å, this incident is negligible in standard investigations, so an average value for the wavelength of Cu Kα rays is used.

With a wavelength in the order of most interatomic distances (lCu Kα = 1.542 Å), X-rays are very useful for the investigation of the arrangement of atoms in matter. A special feature of electromagnetic radiation is the wave-particle duality. This means, the radiation can be described by its frequency n and wavelength l, terms describing the wave characteristics, and by its energy E and momentum p, which accounts the particle behavior of photons. Both properties are unified by [48]:

Abbildung in dieser Leseprobe nicht enthalten

Here, h is Planck’s constant (6.626·10-34 Js). To get an expression for the momentum, the de Broglie relation is used:

Abbildung in dieser Leseprobe nicht enthalten

Using the Cu Kα radiation with a wavelength of 1.542 Å, the energy of one photon can be calculated with Planck’s equation:

Abbildung in dieser Leseprobe nicht enthalten

As a result, an X-ray photon is reaching an energy-value of 8.04 keV. As a comparison, the ionization energy of one hydrogen atom is 13.6 eV [31], the total energy released in fusion of deuterium and tritium to helium is 17.6 MeV [49]. This relative high energy level of X-ray radiation can cause heating of the sample and, especially for organic compounds and long data-acquisition times, a decomposition of the sample. It is also the main difference between SANS and SAXS, because of the low energies of neutron radiation compared to X-rays (in the range of 2.16 meV from cold neutron sources to 172 meV from hot neutron sources) [48].

4.1.2 Small angle X-ray scattering (SAXS)

The interaction of X-rays with matter is comparable with those of visible light (see also page 14, scattering and absorption of light by nanoparticles). Therefore, scattering processes can be observed. By characterizing the scattering behavior of particles in a homogeneous medium, information of size and correlation can be gained. The use of X-rays allows the characterization of particles, such as polymers and micelles, and pores in a solid of size between 0.5 to 100 nm [50]. In general, X-ray scattering depends on the number of particles, the scattering length density (sometimes-named electron contrast) between the particle and their surrounding medium and the size and shape of particles. [51]

The simplified general equation of SAXS consists of terms considering these influences [52] and expresses the scattering intensity I (q) - where q is the scattering vector (see page 51) - by the product of the number of particles per unit volume, N, the scattering function of a single isolated particle, I1( q ), and orientation averaged effective structure function, S( q ), that accounts for the short range spatial correlation between the particles:

Abbildung in dieser Leseprobe nicht enthalten

I1( q ) therefore, is given by the following equation [52]:

Abbildung in dieser Leseprobe nicht enthalten

Equation gives, combined with equation the general equation of SAXS [53]:

Abbildung in dieser Leseprobe nicht enthalten

Here, Ie is the scattering intensity of a single electron, r is the relative distance of an electron pair, Δ ρ is the electron density contrast between the particle and the solvent, V is the irradiated volume by X-ray and g(r) is the spatial correlation function of the electron density. Scattering by one electron

Diffraction is produced by the interference of waves scattered by an object. In the case of X-rays striking this object, every electron becomes the source of a scattered wave. Compared with the binding energy of an atom, the energy of an X-ray photon is very large. Thus, all electrons will behave as if they were free and emit a secondary wave of the same intensity, which can be expressed by the Thomson formula [58]:

Abbildung in dieser Leseprobe nicht enthalten

where Ipr is the primary intensity, ade is the distance of the object to the detector, rel is the classical radius of an electron (2.81·10-13 cm) and q is the scattering angle. The scattering vector [54]

The scattering of X-rays by a particle result in coherent waves. Coherence means, the amplitudes are added and the intensity is then given by the absolute square of the resulting amplitude. The amplitudes are equal in magnitude and differ only by their phase φ, which depends on the position of the electron in space [58]:

Abbildung in dieser Leseprobe nicht enthalten

Therefore, the scattering vector q, which is the modulus of the resultant between the incident and scattered wavevectors, ki and ks, respectively (see Fig. 4.3), was introduced. [54]

Abbildung in dieser Leseprobe nicht enthalten

Fig. 4.3: Principal scheme of X-ray scattering at a spherical particle.

According to Fig. 4.3, q depends on the scattering angle q and the used X-ray wavelength l and is given by:

Abbildung in dieser Leseprobe nicht enthalten

q therefore, has the dimension of length-1, where the unit Å-1 is commonly used. The wavelength of the X-rays, l, is a constant parameter given by the used instrument.

l can be expressed by Bragg’s Law of Diffraction. This relationship connects l and the distance d, which is the distance of crystal lattice in diffractometry:

Abbildung in dieser Leseprobe nicht enthalten

where n is an integer determined by the order of scattering. This order is considered to be 1 in most SAXS-experiments, especially in the case of spherical particles.

By using equations and , a useful expression is given to get information on the probed dimension:

Abbildung in dieser Leseprobe nicht enthalten

Thus, the maximum q-value, qmax, gives information on the limit of detection, while qmin, the lowest reachable q-value, marks the detection-limit of bigger dimensions. Conventional SAXS instruments can measure aggregates from about 0.5 to 100 nm, which are q-values from 1.3 to 6.3·10-3 Å-1, respectively. Due to equation it is also possible to estimate the size of scattering body by using the position of a scattering peak in the q-space.

In the particular case of nearly homogeneous particles, equation yields a structural parameter, ds, corresponding to the average distance between particles. Therefore, considering qm as the q value corresponding to the maximum of scattering intensity, equation can be rewritten as: [52]

Abbildung in dieser Leseprobe nicht enthalten

This equation is not exact, if a system is disordered because the average distance also depends on the particular type of arrangement around each particle. [58] However, it can be applied as a useful semiquantitative estimate in order to characterize trends in structural formations. [52] The electron density [55]

Considering the enormous number of electrons and the fact, that a single electron cannot be exactly located (according to Heisenberg uncertainty principle, [56]), the concept of electron density was introduced. In case of X-rays, the scattering cross-section of an atom increases in direct proportion to the number of its electrons, which means the higher an atoms ordering number Z in the periodic table, the higher its interaction with X-rays. The other way round, scattering caused by small atoms like hydrogen can hardly be detected. Further, it is very important for SAXS-measurement, that solvent and solved component differ in their ability of scattering X-ray.

Thus, for SAXS-experiments, the difference of the electron densities, Δ ρ, of the sample and the solvent is important. If these two identities do not differ significant, hardly any scattering can be achieved. The electron contrast is given as:

Abbildung in dieser Leseprobe nicht enthalten

In equation ρP is the constant electron densities of a homogeneous particle and ρm of the surrounding medium. To account this fact, the (X-ray) s cattering l ength d ensity (SLD) was introduced as a quantity for the electron density ρ:

Abbildung in dieser Leseprobe nicht enthalten,

where rel is the classical radius of an electron and Zi is the ith atom in the molecular volume vm. [55] The scattering intensity [58]

By assuming a volume element d V and a number of electrons per unit volume ρ (r), this volume contain ρ (r)d V electrons at position r. The integration over the whole volume V irradiated by the incident beam, gives the amplitude F( q ) [58]:

Abbildung in dieser Leseprobe nicht enthalten

Therefore, F( q ) is the Fourier transform of the electron density. To obtain the scattering intensity I( q ), F( q ) has to be multiplied with its conjugated complex F*(q):

Abbildung in dieser Leseprobe nicht enthalten

In this equation, the relative distance (r1 - r2) = r = constant is introduced. The integration of equation is divided into two steps: first, all pairs with equal relative distances are summarized. This step gives the auto-correlation, Abbildung in dieser Leseprobe nicht enthalten, which is defined by:

Abbildung in dieser Leseprobe nicht enthalten

The second step of the integration gives the intensity distribution in q, a reciprocal space, which is determined by the structure of the object, as expressed by Abbildung in dieser Leseprobe nicht enthalten.

Abbildung in dieser Leseprobe nicht enthalten

By using the Debye formula [60], the phase factor (see also equation ) e-i·qr can be replaced by its average, taken over all directions of r [58]:

Abbildung in dieser Leseprobe nicht enthalten

This leads to a reduction of equation , if the system is a statistically isotropic with no long range order.

Abbildung in dieser Leseprobe nicht enthalten The auto correlation and invariant [58]

Due to the absence of long range order, the electron densities are becoming independent at large values of r and are replaced by the mean value Abbildung in dieser Leseprobe nicht enthalten. Thus, the autocorrelation Abbildung in dieser Leseprobe nicht enthalten must tend to values of Abbildung in dieser Leseprobe nicht enthalten, while the initial value Abbildung in dieser Leseprobe nicht enthalten is Abbildung in dieser Leseprobe nicht enthaltenand is the maximum of the auto correlation. This means, the structure of an object is represented by this finite region, while the region of large r doesn’t contribute any information. To account the fact, the electron density fluctuation Abbildung in dieser Leseprobe nicht enthalten instead of the electron density is considered. The auto-correlation is then redefined as [58]:

Abbildung in dieser Leseprobe nicht enthalten

Here, g(r) is the so-called correlation function [53] and can be interpreted as the average of the product of two fluctuations at a distance r:

Abbildung in dieser Leseprobe nicht enthalten

The borders here, as it is the case of the electron density, are Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten for large values of r. This means, a largest diameter D exists, where g(rD) = 0.

According to [53], equation takes the form:

Abbildung in dieser Leseprobe nicht enthalten

To get g(r) from equation , a Fourier transformation has to be done:

Abbildung in dieser Leseprobe nicht enthalten

This equation take a particularly simple form for q = 0 and r = 0, because of equation , also known as the Debye factor, has the value 1 [58]:

Abbildung in dieser Leseprobe nicht enthalten

Furthermore, equation shows that the integral of the intensity over the reciprocal space is directly related to the mean square of the fluctuation of electron density. In case of a shift or a deformation of the structure, the integral in equation remain invariant. This invariant, Q, is expressed as:

Abbildung in dieser Leseprobe nicht enthalten Scattering of spherical particles [58]

In a diluted system of identical spherical particles with a constant electron density, only the electron contrast Δ ρ (see page 53) is relevant for diffraction. The scattering particles are widely separated from each other, so it is plausible to assume, they contribute independent to the diffracted intensity. This leads to the single-particle assumption. [58]

Because of their spherical symmetry, all orientations in space are equivalent, which means, the calculation of the intensity I1( q ) can be done by squaring the amplitude F1( q ). For a sphere of radius R0 and volume V, the result is given by Rayleigh [61]:

Abbildung in dieser Leseprobe nicht enthalten

The scattering intensity for this special case shows a maximum at q = 0 (see Fig. 4.4)

Abbildung in dieser Leseprobe nicht enthalten

Fig. 4.4: Scattered intensity of a sphere [58]



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Titel: Synthesis and Characterization of Nanoparticles