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Correlation of Morphology and Mechanical Properties of Silica using Light Scattering Techniques.

Scattering

Energy transmitted by waves through a medium is called as radiation or radiant energy. These include sound waves, cosmic rays, X-rays, light, heat, microwaves, radio waves. This continuum of energy is called the electromagnetic spectrum. Energy residing in a  specific and narrow band of this spectrum is referred to as ‘light’. The phenomenon of light scattering may be defined as the alteration of direction and intensity of light beam that strikes an object. Scattering occurs when the bound electron of the object removes energy from the light beam (photon by photon), and then re-emits the energy (photon by photon) without otherwise altering it.

Scattering of light is classified into two types as elastic scattering and inelastic scattering. When the wavelength of incident light is preserved it is elastic scattering. When there is a difference in the wavelength of scattered light and incident light, it is inelastic scattering. The definition of light scattering as given above introduces a juncture in light scattering theory at which static light scattering (SLS) and dynamic light scattering (DLS) are separated. By the static light scattering technique, particle size information is extracted from intensity characteristics of the scattering pattern at various angles. With dynamic light scattering, particle size is determined by correlating variations in light intensity to the Brownian movement of the particles. Values obtained by the latter technique vary widely depending on the concentration and condition of the sample, as well as environmental factors. With both techniques, however, the scattered light has undergone no alteration in wavelength (elastic scattering).

Electromagnetic waves are the best known and the most commonly encountered forms of radiation that undergo scattering. Rayleigh scattering (named after Lord Rayleigh) theory explains the scattering phenomenon of light or other electromagnetic radiation, by particles that are much smaller than the wavelength of light. According to the Rayleigh Scattering Law, the intensity (I) of the scattered light varies inversely as the fourth power of its wavelength (λ-4). The Mie theory (named after Gustav Mie) on the other hand not only provides a complete analytical solution of electromagnetic radiation by spherical particles but in contrast to Rayleigh scattering, Mie theory encompasses all possible ratios of diameters of spheres to wavelengths. It assumes a homogeneous, isotropic, optically linear material that is irradiated by an infinitely extending plane wave.

Rayleigh-Gans-Debye theory

The theoretical foundation laid out by Bruno Zimm [B. H. Zimm, "The scattering of light and the radial distribution function of high polymer solutions," J. Chem. Phys. 16, 1093 (1948)] makes it possible to condense the results of the Rayleigh-Debye-Gans theory of light scattering into a simple equation as:

K* is the optical constant described by Lord Rayleigh as

is the vacuum wavelength of the incident light

dn/dc is the refractive index increment of the solution

R(θ,c) is the excess Rayleigh ratio of the solution as a function of scattering angle θ and concentration c. It is directly proportional to the intensity of the scattered light in excess of the light scattered by the pure solvent.

c is the solute concentration in g/ml.

Mw is the weight-averaged solute molar mass.

A2 is the second virial coefficient.

Na is Avogadro's number. This number always appears when concentration is measured in g/ml and molar mass in g/mol.

P(θ) describes the angular dependence of the scattered light, and can be related to the rms radius.

(The 1/λ04 scattering dependence was derived originally by Rayleigh who showed that the blueness of the sky was due to the preferential scattering of the shorter wavelengths)

The expansion of P(θ) is as follows

where n0 is the index of refraction of the solvent, θ 0 is the vacuum wavelength of the laser, and rg is the radius of gyration. Here, the relation between the size and angular dependence of the scattered light is clear. For larger sizes (rg greater than approximately 50 nm) it is necessary to include higher moments in the expansion of P().

The radius of gyration,, may be calculated immediately from the slope at θ = 0 of the measured ratios 1/R(θ,c) with respect to sin2(θ /2). If the macromolecule of mass M is made up of elements mi, it may be shown that

where rmean is the mean distance of the particles in a sample of N particles given by the following relation:

It is important to note from the Rayleigh-Gans-Debye (RGD) approximation that the incident wave is considered essentially unaffected by the scattering molecule. In other terms, the theory is only valid when

and where r is the characteristic size of the molecule, n is the refractive index of the solvated molecule relative to that of the solvent and k=(2πn0)/λ0, These two inequalities correspond, respectively, to the conditions that the molecular refractive index is almost indistinguishable from the refractive index of the solvent and the total phase shift of the incident light wave as it passes through the molecule is almost negligible On this basis, each element of the molecule is treated as a single dipole scatterer whose excitation and scattering is independent of any other element of the molecule.

Static Light Scattering

In static light scattering, the intensity of the scattered light is measured as a function of the angle. The course of scattered intensity as a function of the detector angle depends upon the size as well as the particle structure. As described above, Rayleigh developed a theory that can be used to calculate the angular intensity distribution of light scattered by molecules much smaller than the wavelength of light (size<λ /20). If the above condition is satisfied, the intensity of scattered is a function of only the particle size and not on their structure or their concentration. If the incident light is not polarized, then we have the Rayleigh ratio as:

The angular distribution of Rayleigh scattering is governed by the, is symmetric, in the plane orthogonal to the direction of incident light which means that the forward scatter equals the backward scatter. (The terms indicated in the above equation have been fully explained in RGD section)

For larger particles, the influence of particle structure and its concentration are to be considered. If the particle concentration is high enough to influence and increase intraparticle optical interference and interparticle interactions, the angular dependency of scattered light is described by:

Zimm discovered that in the limit of low scatter vectors and for low concentrations, the angular distribution of the scattered intensity becomes independent of the particle shape. By the extrapolation of / to zero concentration and zero angle the average molecular weight, the radius of gyration and the second virial coefficient can be estimated.

Dynamic Light Scattering

Introduction

The use of dynamic light scattering as an investigative tool began around 1914 when Leon Brillouin published theoretical findings on the light scattered from an excited isotropic body. Experimental results of frequency shifts in the light scattered from a liquid followed shortly after the theoretical predictions, but it was the advent of the laser in the 1960’s that allowed small frequency changes to be measured and thus gave an impetus for the application of light scattering to a wide variety of systems including solids, gels, membrane vesicles and colloidal dispersions. Structural properties of a material, such as size and shape, can be determined from a light scattering experiment. In 1964, Robert Pecora developed a technique called photon correlation spectroscopy that was particularly useful in the study of macromolecular systems.

Theory

When a beam of light passes through a colloidal dispersion, the particles suspended in the medium scatter the incident light in all directions. When one studies this phenomenon in depth, it is the electric field of the incident light that imparts an oscillating polarization to the suspended particles in the medium. When the polarity of the particles differs from the polarity of the surroundings, light gets scattered. When the particles are small in comparison to the wavelength of light, the scattering is uniform in all directions (Rayleigh scattering). For particles that are larger than the wavelength of light, the intensity is dependent upon the angle upon the angle of the incident beam (Mie Scattering).

Brownian motion

The term Brownian motion (named after Robert Brown) refers to the physical phenomenon where minute particles, immersed in a fluid or floating on its surface, move about randomly (stochastic process). The time evolution of the position of the Brownian particle may be described approximately by the Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvent on the Brownian particle.

In physics, a Langevin equation is a stochastic differential equation describing Brownian motion:

where a is the acceleration of a Brownian particle of mass m, expressed as the sum of a viscous force which is proportional to the particles velocity v (Stokes' law), a noise term representing the effect of a continuous series of collisions with the atoms of the underlying fluid and F(x) which is the interaction force due to the intramolecular and intermolecular interactions.

Considering that the light source is monochromatic and coherent as in a laser, we observe time dependent fluctuations in the intensity of scattered light and this can be carefully studied using a suitable detector as a photomultiplier. It is important to use a monochromatic source of light as it becomes extremely complicated to study the scattering wave vector that is a function of incident wavelength (λ0) if there are multiple wavelengths as in the case of white light. Within the sample, it is to be noted that the particles are in constant Brownian motion that is random. Owing to this condition the distance between particles in the medium is continuously varying with time. It is the random movement of these particles that cause fluctuations in the intensity of the detected signal (constructive and destructive interference of light). Analysis of this time dependence of this intensity fluctuation can yield the diffusion coefficient of the particle via the Stokes-Einstein equation.

The diffusivity of a particle suspended in a liquid is opposed by the liquid’s viscosity. So particles spread slowly throughout the liquid with increase in viscosity of the liquid. For small motion, we can assume a laminar or a steady state flow. The Stokes-Einstein equation for the diffusivity of particles in a liquid defines the diffusion coefficient, D, of the particles as

where kB is Boltzmann’s constant, T is the temperature in Kelvin, η is the viscosity of the liquid in which the particles are suspended and r is the particle size.

Since the fluctuation of the intensity signal is caused by the motion of particles, those particles that diffuse more quickly are characterized by a smaller fluctuation time. As seen in the above equation, the diffusion coefficient is inversely proportional to the size of the particles, and, consequently, the smaller the particle, the shorter the fluctuation time.

Autocorrelation Function

A correlation equation is used to determine the relationship between the measurements of a fluctuating signal. The autocorrelation function is used to measure the correlation of an intensity signal with a delayed version of itself. Performing an autocorrelation G(τ) on the light intensity function I(t) gives a proper representation of the time scale of the light intensity fluctuations. The autocorrelation is mathematically defined as:

where  indicates a time average in x, I(t) is the intensity detected at time t, I(t+τ) is the intensity detected at time t+τ, where τ is the delay time – the autocorrelation variable. The autocorrelation function is an elegant means to compare the value of the intensity at a given time t with that after an interval of time τ. Since the intensity is related to the pattern of particle positions, a high correlation between the intensity at two different times indicates that the particles have not diffused very far in the interval between the measurements. Thus an autocorrelation that remains at high magnitude for a long time interval τ indicates large, slowly moving particles. The converse logic is true for small particles. Via computation of the autocorrelation for a large range of τ, a quantitative measure of the rapidity of light fluctuations may be found.

For particles that are monodisperse the autocorrelation of the scattered light intensity is a single decaying exponential:

G(τ) = exp(−Γτ )

For polydisperse samples, the autocorrelation function is the sum of the exponentials of each component size. Since the motion of Brownian particles is random, and the autocorrelation function of a random variable is a decaying exponential, G(τ) may be described as above. It is common to consider the inverse Fourier transform of the correlation function; in our case it happens to be the Lorentzian transform, that is known as the power spectrum:

Scattering vector (q)

The scattering vector (q) may be defined as the difference between the scattered wave vector and the incident wave vector. The line width of the spectrum Γ is related to the diffusion coefficient by Γ = Dq2.

The scattering vector (q) is mathematically defined as follows:

where n is the index of refraction of the liquid medium, θ is the scattering angle and θo is the incident wavelength in air. The scattering vector (q) may be defined as the difference between the scattered wave vector and the incident wave vector. By combining the Stokes-Einstein relation and the equation for the linewidth, , an equation for the size (r) of the scattering particle is obtained as:

Flowchart of the working of DLS setup

Monochromatic Light Source (Laser) Particles in Brownian Motion scatter light Autocorrelation Function [G(τ )]

&

Scattering vector (q)

Γ = Dq2

Stokes- Einstein Equation

Particle size (r)

Experiment Procedure

Once the instrument connections are all set and made, the sample is loaded in a disposable cuvette. The ALV software is used to fix the initial parameters as the minimum angle, maximum angle and the angular step. Once these parameters are set and accepted the set up is ready.

The standard used is toluene. The solvent used is distilled water Then the solution (solute and solvent) of a fixed concentration is measured.

It is to be noted that utmost care needs to be taken while performing the measurements to prevent any contamination by dust particles that could affect the outcome of the experiment significantly. Filtration is recommended to prevent interference by dust or other unwanted airborne constituents. Toluene is generally used as the standard as the Rayleigh ratios for toluene are well documented. To prevent excess scattering by solvent molecule, the solvent is measured prior to measuring the solution. After the solution is measured, the scattering due to the solvent is subtracted to yield greater accuracy. Since the solvent used is water (n=1.33), care needs to be taken to choose solute particles whose size is to be measured have a relatively different refractive index (say n=1.38). The built in correlator software calculates the correlation function. A typical correlation function plotted as a function of log time looks as follows:

The particle is calculated from the correlation function as discussed in detail at the outset. The graph below indicates a single size distribution of about 50 nm

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