Nanoparticle size determination by X-ray diffraction technique and dynamic light scattering method for colloidal nanoparticles

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Nanoparticle size determination by X-ray diffraction technique and dynamic light scattering method for colloidal nanoparticles

1.         X-ray Powder Diffraction (XRD)

X-ray powder diffraction is a non-destructive technique widely applied for the characterization of crystalline materials. The method is normally applied to data collected under ambient conditions, but in situ diffraction as a function of an external constraints (temperature, pressure, stress, electric field, atmosphere, etc.) is important for the interpretation of solid state transformations and materials behaviour. Various kinds of micro- and nano-crystalline materials can be characterised from X-ray powder diffraction, including inorganics, organics, drugs, minerals, zeolites, catalysts, metals and ceramics. The physical states of the materials can be loose powders, thin films, polycrystalline and bulk materials. For most applications, the amount of information which is possible to extract depends on the nature of the sample microstructure (crystallinity, structure imperfections, crystallite size, texture), the complexity of the crystal structure (number of atoms in the asymmetric unit cell, unit cell volume) and the quality of the experimental data (instrument performances, counting statistics).

1.1.      Fundamental Principles of X-ray Powder Diffraction (XRD)

Max von Laue, in 1912, discovered that crystalline substances act as three-dimensional diffraction gratings for X-ray wavelengths similar to the spacing of planes in a crystal lattice. X-ray diffraction is now a common technique for the study of crystal structures and atomic spacing.

X-ray diffraction is based on constructive interference of monochromatic X-rays and a crystalline sample. These X-rays are generated by a cathode ray tube, filtered to produce monochromatic radiation, collimated to concentrate, and directed toward the sample. The interaction of the incident rays with the sample produces constructive interference (and a diffracted ray) when conditions satisfy Bragg's Law (nλ=2d sin θ). This law relates the wavelength of electromagnetic radiation to the diffraction angle and the lattice spacing in a crystalline sample. These diffracted X-rays are then detected, processed and counted. By scanning the sample through a range of 2θ angles, all possible diffraction directions of the lattice should be attained due to the random orientation of the powdered material. Conversion of the diffraction peaks to d-spacings allows identification of the mineral because each mineral has a set of unique d-spacings. Typically, this is achieved by comparison of d-spacings with standard reference patterns.

All diffraction methods are based on generation of X-rays in an X-ray tube. These X-rays are directed at the sample, and the diffracted rays are collected. A key component of all diffraction is the angle between the incident and diffracted rays. Powder and single crystal diffraction vary in instrumentation beyond this.

1.2.      X-ray Powder Diffraction (XRD) Instrumentation - How Does It Work?

X-ray diffractometer consist of three basic elements: an X-ray tube, a sample holder, and an X-ray detector.

X-rays are generated in a cathode ray tube by heating a filament to produce electrons, accelerating the electrons toward a target by applying a voltage, and bombarding the target material with electrons. When electrons have sufficient energy to dislodge inner shell electrons of the target material, characteristic X-ray spectra are produced. These spectra consist of several components, the most common being K_α and K_β. K_α consists, in part, of K_α1 and K_α2. K_α1 has a slightly shorter wavelength and twice the intensity as K_α2. The specific wavelengths are characteristic of the target material (Cu, Fe, Mo, Cr). Filtering, by foils or crystal monochrometers, is required to produce monochromatic X-rays needed for diffraction. K_α1and K_α2 are sufficiently close in wavelength such that a weighted average of the K_α1and K_α2 is used. Copper is the most common target material for single-crystal diffraction, with CuK_α radiation with l = 1.5418Å. These X-rays are collimated and directed onto the sample. As the sample and detector are rotated, the intensity of the reflected X-rays is recorded. When the geometry of the incident X-rays impinging the sample satisfies the Bragg Equation, constructive interference occurs and a peak in intensity occurs. A detector records and processes this X-ray signal and converts the signal to a count rate which is then output to a device such as a printer or computer monitor.

Table 1. Applications of Powder Diffraction

The observed diffraction line profiles in a powder diffraction pattern are distributions of intensities I(2θ) defined by several parameters: (i) the reflection angle position 2θ at the maximum intensity (related to the lattice spacing d of the diffracting hkl plane and the wavelength λ by Bragg’s law, λ = 2d sinθ), (ii) the dispersion of the distribution, full-width at half-maximum and integral breadth, (iii) the line shape factor, and (iv) the integrated intensity (proportional to the square of the structure factor amplitude). The specific applications derived from each of them have been listed in Table 1.

1.3.      State-of-the art in powder diffraction

1.3.1.   Instrumentation and standard reference materials

Improved instrumental resolution has contributed to reduce the line-profile overlap problem. Powder diffraction with conventional X-ray sources has benefited from the development of new instruments, particularly the introduction of parallel beams with graded multilayer reflective mirrors and polycapillary optics, from which new applications with in-house diffractometers will emerge. Determination of peak positions with accuracy, on a routine basis, of 0.01°(2θ) can be obtained. Even higher accuracy is achievable with synchrotron radiation, as a consequence of parallel beams. Further developments are multi-detectors for faster data collection, as well as 2-dimensional detectors for samples with low diffracting power and image plates in combination with Guinier-de Wolff cameras.

Standard reference materials (SRM) are available for instrument calibration, evaluation of instrument characteristics, quantitative analysis, intensity sensitivity, etc.

1.3.2.   The analytical tools: phase identification and quantitative analysis

Phase identification is traditionally based on a comparison of observed data with interplanar spacings d and relative intensities I compiled for crystalline materials. The Powder Diffraction File, edited by the International Centre for Diffraction Data (USA), contains powder data for more than 130000 substances. The powder diffraction pattern of a known phase should act as a “fingerprint” which can be used to identify the phase. The computer “search-match” algorithms are used to compare experimental pattern with ICDD database of known compounds.

Quantitative phase analysis involves the determination of the amounts of different phases present in a multi-component mixture. The powder method is widely used to determine the abundance of distinct crystalline phases, e.g. in rocks and in mixtures of polymorphs, such as zirconia ceramics. Modern approaches are based on the Rietveld method (using a simple relationship between individual scale factors). Errors due to residual preferred orientation effects are reduced by considering the whole pattern. Because of the potential health hazard of respirable crystalline silica, X-ray powder diffraction is also used for identifying and quantifying silica of all types of samples from airborne dusts to bulk commercial products.

1.3.3.   Crystal structure analysis

Among the most noteworthy advances of the powder method is the determination ab- initio of crystal structures from powder diffraction data (to date more than 400 successful examples have been reported). It is an exciting application for which the resolution of the pattern is of prime importance. In addition to X-ray sources, neutrons also play an important role in powder diffraction structure analysis, e.g. in case of too low atomic contrasts for X-rays and for precise refinement of atomic coordinates. Neutron powder diffraction was used to determine the oxygen content and position in high T_c cuprates.

Powerful methods are available for pattern indexing, extraction of integrated intensities, structure solution and refinement of the structure model with the Rietveld method. Efficient Rietveld programs are available, among them the most popular are FULLPROF from Rodriguez-Carvajal (F), GSAS from Von Dreele (USA) and RIETAN from Izumi (J). Materials studied successively during the last ten years were inorganics, organometallics, minerals, organics and more recently pharmaceutical compounds. For these last remarkable applications, the traditional methods (e.g. direct methods) are generally not efficient. New direct space methods have been introduced, e.g. Monte Carlo simulated annealing, genetic algorithm, crystal structure prediction by energy minimisation and molecular modelling, etc. These techniques are promising approaches, even with molecules with several torsion angles. A promising approach is based on combining NMR, electron diffraction and powder diffraction.

1.3.4    Microstructure of real materials

Microstructural imperfections, such as (i) lattice distortions, stacking and twin faults, dislocations, (ii) the small size of crystallites (i.e. nanoscale domains over which diffraction is coherent) and crystallite size distributions, are usually extracted from the integral breadth or a Fourier analysis of individual diffraction lines. Crystallite sizes well determinable by line broadening analysis are in the range 20-1500 Å. Crystallite shape anisotropy has been determined in strain-free nanocrystalline materials (e.g. ZnO, CeO₂). Stacking faults may occur in close-packed or layer structures, e.g. hexagonal Co and ZnO. Lattice distortion (microstrain/stress) represents variable displacements of atoms from their sites in the idealized crystal structure. Anisotropic microstrains have been successfully interpreted by the presence of (specific) dislocation distributions in various materials. The structure around a dislocation is nicely revealed by HREM, but understanding materials behavior requires taking into account the dislocation distribution (type and spatial) in the whole sample. Only powder diffraction can provide such a statistical representation. Recently, profile modelling (synthesis) techniques (whole powder pattern fitting approach), instead of the more usual profile decomposition techniques, have extended the frontiers of microstructural investigations (three-dimensional microstructural properties have already been reported). They include the effect of crystallite size distributions, the effect of lattice distortion parameters and the effect of strain fields in crystalline materials. The analysis of macro-stress (engineering stress) on the basis of the so-called sin²ψ method has become more and more important for materials science. New methods have been developed to analyse the state of stress in thin films. Materials behavior is largely determined by the presence of macro- and microstresses (e.g. thin films). Powder diffraction is the only experimental method available for both a description of micro- and macro-stresses.

1.3.5.   Dynamic and non-ambient diffraction

Time- and temperature-dependent X-ray diffraction includes the measurement of series of diffraction patterns as a function of time and temperature. The time required for collecting data decreases considerably with the availability of fast detectors, such as position sensitive detectors, and high brightness of the X-ray source. Limitations with conventional X-rays can be overcome with synchrotron and neutron diffraction facilities. Equipment has been developed for in situ application of heating, pressure and tensile testing. In principle, line profile parameters can be extracted from each pattern (peak position and integrated intensity, breadth and line shape) and interpreted in structural and microstructural terms. Thus it is possible to establish the reaction path during solid-state transformations and to determine transformation kinetics. A representative application is the investigation of fast and self propagating solid combustion reactions on a subsecond time-scale, using synchrotron radiation for high intensity. In situ powder diffraction can also be combined with complementary techniques applied simultaneously, such as EXAFS (Extended X-ray Absorption Fine Structure). Then both long range order information (powder diffraction) and short range order information (EXAFS) is obtained (example: formation of heterogeneous catalysts).

1.3.6.   Data reduction

Results are commonly presented as peak positions at 2θ and X-ray counts (intensity) in the form of a table or an x-y plot (shown above). Intensity (I) is either reported as peak height intensity, that intensity above background, or as integrated intensity, the area under the peak. The relative intensity is recorded as the ratio of the peak intensity to that of the most intense peak ( relative intensity = I/I₁ x 100 ).

1.4.      Particle Size Effect

Broadening due to finite size of the crystallites

Scherrer (1918) first observed that small crystallite size could give rise to line broadening.  He derived a well known equation for relating the crystallite size to the broadening, which is called “Scherrer Formula”

Δ(2θ) or β = Kλ/{D cos θ}

D  = Average Crystallite Size

K  = Scherrer Constant, in the range 0.87 – 1.0, usually assume K = 0.9.

λ   = The wavelength of the radiation

β   = The integral breadth of a reflection (in radians 2q) located at 2q

Broadening due to crystallite microdistortions(strain)

Δ(2θ) = 4 ε tan θ

Where ε Is the relative deformation of the interreticular distance : ε  = Δd_hkl/d_hkl

The microdistortions and size effects can be present in the same sample. But it is not easy to distinguish and measure separately. For that more or less complex methods have been proposed by Warren in 1969 and Klug & Alexander in 1974.

For the accurate analysis for size and/or strain effects one must accurately account for instrumental broadening.  The manner of doing this differs depending upon the peak shape. If it is

Williamson and Hall (1953) proposed a method for deconvoluting size and strain broadening by looking at the peak width as a function of 2θ.  Here the Williamson-Hall relationship for the Lorentzian peak shape is presented, but it can derived in a similar manner for the Gaussian peak shape

 

To make a Williamson – Hall plot

ü  Plot {β_obs –β_inst} cosθ on the y – axis (in radians 2θ)

ü  Plot 4 sinθ on the x – axis

If we get a linear fit to the data we can extract

The crystallite size from the y-intercept of the fit

The strain from the slope of the fit

Determination of Unit Cell Dimensions

For determination of unit cell parameters, each reflection must be indexed to a specific hkl.

1.5.      Sample Collection and Preparation

Determination of an unknown requires: the material, an instrument for grinding, and a sample holder.

• Grind the sample to a fine powder, typically in a fluid to minimize inducing extra strain (surface energy) that can offset peak positions, and to randomize orientation. Powder less than ~10 μm (or 200-mesh) in size is preferred
• The sample holders are aluminum slides and steel backing clips or Glass Plates with a centre grove.
• Transfer enough sample powder into the rectangular sample "well" to slightly over-fill it.
• Distribute the sample powder evenly in the sample well using a clean spatula or glass slide. Add more if necessary, or remove any excessive amounts.
• Using a clean glass slide or the rectangular brass plunger, gently press the sample powder flat and flush with the upper surface of the aluminum slide. If the surface of the sample is not flush, significant shifts in peak positions will result.
• Wipe the aluminum/glass surfaces of the slide with your finger or a Kimwipe and alcohol to remove any excess sample dust
• The sample is now ready to be loaded into the automatic sample changer.

For unit cell determinations, a small amount of a standard with known peak positions (Eg. Si; that does not interfere with the sample) can be added and used to correct peak positions.

1.6.      Strengths and Limitations of X-ray Powder Diffraction (XRD)?

Strengths

• Powerful and rapid (< 20 min) technique for phase identification of an unknown mineral
• In most cases, it provides an unambiguous mineral determination
• Non destructive method and minimum quantity of sample is required
• XRD units are extensively available
• Data interpretation is relatively easy.

Limitations

• Homogeneous and single phase material is best for identification of an unknown.
• Must have access to a standard reference file for inorganic compounds (d-spacings, hkls)
• For unit cell determinations, indexing of patterns for non-isometric crystal systems is complicated
• Peak overlay may occur and worsens for high angle 'reflections'