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Introduction

The physical properties of colloids (nanoparticles) and suspensions are strongly dependent on the nature and extent of the particle-liquid interface. The behavior of aqueous dispersions between particles and liquid is especially sensitive to the ionic and electrical structure of the interface.

Zeta potential is a parameter that measures the electrochemical equilibrium at the particle-liquid interface. It measures the magnitude of electrostatic repulsion/attraction between particles and thus, it has become one of the fundamental parameters known to affect stability of colloidal particles. It should be noted that that term stability, when applied to colloidal dispersions, generally means the resistance to change of the dispersion with time. Figure 2.5.12.5.1 illustrates the basic concept of zeta potential.

From the fundamental theory’s perspective, zeta potential is the electrical potential in the interfacial double layer (DL) at the location of the slipping plane (shown in Figure 2.5.12.5.1 ). We can regard zeta potential as the potential difference between the dispersion medium and the stationary layer of the fluid attached to the particle layer. Therefore, in experimental concerns, zeta potential is key factor in processes such as the preparation of colloidal dispersions, utilization of colloidal phenomena and the destruction of unwanted colloidal dispersions. Moreover, zeta potential analysis and measurements nowadays have a lot of real-world applications. In the field of biomedical research, zeta potential measurement, in contrast to chemical methods of analysis which can disrupt the organism, has the particular merit of providing information referring to the outermost regions of an organism. It is also largely utilized in water purification and treatment. Zeta potential analysis has established optimum coagulation conditions for removal of particulate matter and organic dyestuffs from aqueous waste products.

Brief History and Development of Zeta Potential

Zeta potential is a scientific term for electrokinetic potential in colloidal dispersions. In prior literature, it is usually denoted using the Greek letter zeta, Ζ, hence it has obtained the name zeta potential as Ζ-potential. The earliest theory for calculating Zeta potential from experimental data was developed by Marian Smoluchowski in 1903 (Figure 2.5.22.5.2 ). Even till today, this theory is still the most well-known and widely used method for calculating zeta potential.

Interestingly, this theory was originally developed for electrophoresis. Later on, people started to apply his theory in calculation of zeta potential. The main reason that this theory is powerful is because of its universality and validity for dispersed particles of any shape and any concentration. However, there still some limitations to this early theory as it was mainly determined experimentally. The main limitations are that Smoluchowski’s theory neglects the contribution of surface conductivity and only works for particles which have sizes much larger than the interface layer, denoted as κ_a (1/κ is called Debye length and a is the particle radius).

Overbeek and Booth as early pioneers in this direction started to develop more theoretical and rigorous electrokinetic theories that were able to incorporate surface conductivity for electrokinetic applications. Modern rigorous electrokinetic theories that are valid almost any κa mostly are generated from Ukrainian (Dukhin) and Australian (O’Brien) scientists.

Principle of Zeta Potential Analysis

Electrokinetic Phenomena

Because an electric double-layer (EDL) exists between a surface and solution, then any relative motion between the rigid and mobile parts of the EDL will result in the generation of an electrokinetic potential. As described above, zeta potential is essentially a electrokinetic potential which rises from electrokinetic phenomena. So it is important to understand different situations where electrokinetic potential can be produced. There are generally four fundamental ways which zeta potential can be produced, via electrophoresis, electro-osmosis, streaming potential, and sedimentation potential as shown from Figure 2.5.32.5.3 .

Calculations of Zeta Potential

There are many different ways of calculating zeta potential . In this section, the methods of calculating zeta potential in electrophoresis and electroosmosis will be introduced.

Zeta Potential in Electrophoresis

Electrophoresis is the movement of charged colloidal particles or polyelectrolytes, immersed in a liquid, under the influence of an external electric field. In such case, the electrophoretic velocity, v_e (ms^-1) is the velocity during electrophoresis and the electrophoretic mobility, u­­_e (m ² V ^-1 s ^-1 ) is the magnitude of the velocity divided by the magnitude of the electric field strength. The mobility is counted positive if the particles move toward lower potential and negative in the opposite case. And therefore, we have the relationship v_e­= u_eE, where E is the externally applied field.

Thus, the formula accounted for zeta potential in electrophoresis case is given in EQ, where ε_rs is the relative permittivity of the electrolyte solution, ε₀ is the electric permittivity of vacuum and η is the viscosity.ue =εrsε0ζη(2.5.1)(2.5.1)ue =εrsε0ζηve =εrsε0ζηE(2.5.2)(2.5.2)ve =εrsε0ζηE

There are two cases regarding the size of κa:

1. κa < 1: the formula is similar, 2.5.32.5.3 .
2. κa > 1: the formula is rather complicated and we need to solve equation for zeta potential, 2.5.42.5.4 , where yeζ= eζ/kTyeζ= eζ/kT , m is about 0.15 for aqueous solution.

ue=23εrsε0ζη(2.5.3)(2.5.3)ue=23εrsε0ζη32ηeεrsε0kTue=32yek−6[yek2−ln 2ζ{1−e−ζyek}]2+ka1+3m/ζ2e−ζyek2(2.5.4)(2.5.4)32ηeεrsε0kTue=32yek−6[yek2−ln 2ζ{1−e−ζyek}]2+ka1+3m/ζ2e−ζyek2

Zeta Potential in Electroosmosis

Electroosmosis is the motion of a liquid through an immobilized set of particles, a porous plug, a capillary, or a membrane, in response to an applied electric field. Similar to electrophoresis, it has the electroosmotic velocity, v_eo (ms ^-1 ) as the uniform velocity of the liquid far from the charged interface. Usually, the measured quantity is the volume flow rate of liquid divided by electric field strength, Q_eo,E (m ⁴ V ^-1 s ^-1 ) or diveided by the electric current, Q_eo,I (m ³ C ^-1 ). Therefore, the relationship is given by 2.5.52.5.5 .Qeo= ∫∫veodS(2.5.5)(2.5.5)Qeo= ∫∫veodS

Thus the formula accounted for Zeta potential in electroosmosis is given in EQ.

As with electrophoresis there are two cases regarding the size of κa:

• κa >>1 and there is no surface conduction, where Ac is the cross-section area and KL is the bulk conductivity of particle.
• κa < 1, 2.5.82.5.8 , where Δu =KσKLΔu =KσKL is the Dukhin number account for surface conductivity, KσKσ is the surface conductivity of the particle.

Qeo,E=−εrsε0ζηAc(2.5.6)(2.5.6)Qeo,E=−εrsε0ζηAcQeo,I=−εrsε0ζη1KL(2.5.7)(2.5.7)Qeo,I=−εrsε0ζη1KLQeo,I=−εrsε0ζη1KL(1+2Δu)(2.5.8)(2.5.8)Qeo,I=−εrsε0ζη1KL(1+2Δu)

Relationship Between Zeta Potential and Particle Stability in Electrophoresis

Using the above theoretical methods, we can calculate zeta potential for particles in electrophoresis. The following table summarizes the stability behavior of the colloid particles with respect to zeta potential. Thus, we can use zeta potential to predict the stability of colloidal particles in the electrokinetic phenomena of electrophoresis.

Instrumentation

In this section, a market-available zeta potential analyzer will be used as an example of how experimentally zeta potential is analyzed. Figure 2.5.42.5.4 shows an example of a typical zeta potential analyzer for electrophoresis.

The inside measuring principle is described in the following diagram, which shows the detailed mechanism of zeta potential analyzer (Figure 2.5.52.5.5 ).

When a voltage is applied to the solution in which particles are dispersed, particles are attracted to the electrode of the opposite polarity, accompanied by the fixed layer and part of the diffuse double layer, or internal side of the “sliding surface”. Using the following formula below of this specific Analyzer and the computer program, we can obtain the zeta potential for electrophoresis using this typical zeta potential analyzer (Figure 2.5.62.5.6 .

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Dynamic light scattering (DLS), which is also known as photon correlation spectroscopy (PCS) or quasi-elastic light scattering (QLS), is a spectroscopy method used in the fields of chemistry, biochemistry, and physics to determine the size distribution of particles (polymers, proteins, colloids, etc.) in solution or suspension. In the DLS experiment, normally a laser provides the monochromatic incident light, which impinges onto a solution with small particles in Brownian motion.

And then through the Rayleigh scattering process, particles whose sizes are sufficiently small compared to the wavelength of the incident light will diffract the incident light in all direction with different wavelengths and intensities as a function of time. Since the scattering pattern of the light is highly correlated to the size distribution of the analyzed particles, the size-related information of the sample could be then acquired by mathematically processing the spectral characteristics of the scattered light.

Herein a brief introduction of basic theories of DLS will be demonstrated, followed by descriptions and guidance on the instrument itself and the sample preparation and measurement process. Finally, data analysis of the DLS measurement, and the applications of DLS as well as the comparison against other size-determine techniques will be shown and summarized.

DLS Theory

The theory of DLS can be introduced utilizing a model system of spherical particles in solution. According to the Rayleigh scattering (Figure 2.4.12.4.1), when a sample of particles with diameter smaller than the wavelength of the incident light, each particle will diffract the incident light in all directions, while the intensity II is determined by 2.4.12.4.1 , where I0I0 and λλ is the intensity and wavelength of the unpolarized incident light, RR is the distance to the particle, θθ is the scattering angel, nnis the refractive index of the particle, and rr is the radius of the particle.

I = I01 +cos2θ2R2(2πλ)4(n2 − 1n2 + 2)2r6(2.4.1)(2.4.1)I = I01 +cos2⁡θ2R2(2πλ)4(n2 − 1n2 + 2)2r6

If that diffracted light is projected as an image onto a screen, it will generate a “speckle” pattern (Figure 2.4.22.4.2 ); the dark areas represent regions where the diffracted light from the particles arrives out of phase interfering destructively and the bright area represent regions where the diffracted light arrives in phase interfering constructively.

In practice, particle samples are normally not stationary but moving randomly due to collisions with solvent molecules as described by the Brownian motion, 2.4.22.4.2, where (Δx)2¯¯¯¯¯¯¯¯¯¯¯¯¯(Δx)2¯ is the mean squared displacement in time t, and D is the diffusion constant, which is related to the hydrodynamic radius a of the particle according to the Stokes-Einstein equation, 2.4.32.4.3 , where kB is Boltzmann constant, T is the temperature, and μ is viscosity of the solution. Importantly, for a system undergoing Brownian motion, small particles should diffuse faster than large ones.(Δx)2¯¯¯¯¯¯¯¯¯¯¯¯¯ = 2Δt(2.4.2)(2.4.2)(Δx)2¯ = 2ΔtD =kBT6πμa(2.4.3)(2.4.3)D =kBT6πμa

As a result of the Brownian motion, the distance between particles is constantly changing and this results in a Doppler shift between the frequency of the incident light and the frequency of the scattered light. Since the distance between particles also affects the phase overlap/interfering of the diffracted light, the brightness and darkness of the spots in the “speckle” pattern will in turn fluctuate in intensity as a function of time when the particles change position with respect to each other. Then, as the rate of these intensity fluctuations depends on how fast the particles are moving (smaller particles diffuse faster), information about the size distribution of particles in the solution could be acquired by processing the fluctuations of the intensity of scattered light. Figure 2.4.32.4.3 shows the hypothetical fluctuation of scattering intensity of larger particles and smaller particles.

In order to mathematically process the fluctuation of intensity, there are several principles/terms to be understood. First, the intensity correlation function is used to describe the rate of change in scattering intensity by comparing the intensity I(t) at time t to the intensity I(t + τ) at a later time (t + τ), and is quantified and normalized by 2.4.42.4.4 and 2.4.52.4.5 , where braces indicate averaging over t.G2(τ)= ⟨I(t)I(t + τ)⟩(2.4.4)(2.4.4)G2(τ)= ⟨I(t)I(t + τ)⟩g2(τ)=⟨I(t)I(t + τ)⟩⟨I(t)⟩2(2.4.5)(2.4.5)g2(τ)=⟨I(t)I(t + τ)⟩⟨I(t)⟩2

Second, since it is not possible to know how each particle moves from the fluctuation, the electric field correlation function is instead used to correlate the motion of the particles relative to each other, and is defined by 2.4.62.4.6 and 2.4.72.4.7 , where E(t) and E(t + τ) are the scattered electric fields at times t and t+ τ.G1(τ)= ⟨E(t)E(t + τ)⟩(2.4.6)(2.4.6)G1(τ)= ⟨E(t)E(t + τ)⟩g1(τ)=⟨E(t)E(t + τ)⟩⟨E(t)E(t)⟩(2.4.7)(2.4.7)g1(τ)=⟨E(t)E(t + τ)⟩⟨E(t)E(t)⟩

For a monodisperse system undergoing Brownian motion, g₁(τ) will decay exponentially with a decay rate Γ which is related by Brownian motion to the diffusivity by 2.4.82.4.8 , 2.4.92.4.9 , and 2.4.102.4.10 , where q is the magnitude of the scattering wave vector and q² reflects the distance the particle travels, n is the refraction index of the solution and θ is angle at which the detector is located.g1(τ)= e−Γτ(2.4.8)(2.4.8)g1(τ)= e−ΓτΓ = −Dq2(2.4.9)(2.4.9)Γ = −Dq2q=4πnλsinΘ2(2.4.10)(2.4.10)q=4πnλsinΘ2

For a polydisperse system however, g₁(τ) can no longer be represented as a single exponential decay and must be represented as a intensity-weighed integral over a distribution of decay rates G(Γ) by 2.4.112.4.11 where G(Γ) is normalized, 2.4.122.4.12 .g1(τ)=∫∞0G(Γ)e−ΓτdΓ(2.4.11)(2.4.11)g1(τ)=∫0∞G(Γ)e−ΓτdΓ∫∞0G(Γ)dΓ = 1(2.4.12)(2.4.12)∫0∞G(Γ)dΓ = 1

Third, the two correlation functions above can be equated using the Seigert relationship based on the principles of Gaussian random processes (which the scattering light usually is), and can be expressed as 2.4.132.4.13 , where β is a factor that depends on the experimental geometry, and B is the long-time value of g₂(τ), which is referred to as the baseline and is normally equal to 1. Figure 2.4.42.4.4 shows the decay of g₂(τ) for small size sample and large size sample.g2(τ)= B + β[g1(τ)]2(2.4.13)(2.4.13)g2(τ)= B + β[g1(τ)]2

When determining the size of particles in solution using DLS, g₂(τ) is calculated based on the time-dependent scattering intensity, and is converted through the Seigert relationship to g₁(τ) which usually is an exponential decay or a sum of exponential decays. The decay rate Γ is then mathematically determined (will be discussed in section ) from the g₁(τ) curve, and the value of diffusion constant D and hydrodynamic radius a can be easily calculated afterwards.

Experimental

Instrument of DLS

In a typical DLS experiment, light from a laser passes through a polarizer to define the polarization of the incident beam and then shines on the scattering medium. When the sizes of the analyzed particles are sufficiently small compared to the wavelength of the incident light, the incident light will scatters in all directions known as the Rayleigh scattering. The scattered light then passes through an analyzer, which selects a given polarization and finally enters a detector, where the position of the detector defines the scattering angle θ. In addition, the intersection of the incident beam and the beam intercepted by the detector defines a scattering region of volume V. As for the detector used in these experiments, a phototube is normally used whose dc output is proportional to the intensity of the scattered light beam. Figure 2.4.52.4.5 shows a schematic representation of the light-scattering experiment.

In modern DLS experiments, the scattered light spectral distribution is also measured. In these cases, a photomultiplier is the main detector, but the pre- and postphotomultiplier systems differ depending on the frequency change of the scattered light. The three different methods used are filter (f > 1 MHz), homodyne (f > 10 GHz), and heterodyne methods (f < 1 MHz), as schematically illustrated in Figure 2.4.62.4.6 . Note that that homodyne and heterodyne methods use no monochromator of “filter” between the scattering cell and the photomultiplier, and optical mixing techniques are used for heterodyne method. shows the schematic illustration of the various techniques used in light-scattering experiments.

As for an actual DLS instrument, take the Zetasizer Nano (Malvern Instruments Ltd.) as an example (Figure 2.4.72.4.7), it actually looks like nothing other than a big box, with components of power supply, optical unit (light source and detector), computer connection, sample holder, and accessories. The detailed procedure of how to use the DLS instrument will be introduced afterwards.

Sample Preparation

Although different DLS instruments may have different analysis ranges, we are usually looking at particles with a size range of nm to μm in solution. For several kinds of samples, DLS can give results with rather high confidence, such as monodisperse suspensions of unaggregated nanoparticles that have radius > 20 nm, or polydisperse nanoparticle solutions or stable solutions of aggregated nanoparticles that have radius in the 100 – 300 nm range with a polydispersity index of 0.3 or below. For other more challenging samples such as solutions containing large aggregates, bimodal solutions, very dilute samples, very small nanoparticles, heterogeneous samples, or unknown samples, the results given by DLS could not be really reliable, and one must be aware of the strengths and weaknesses of this analytical technique.

Then, for the sample preparation procedure, one important question is how much materials should be submit, or what is the optimal concentration of the solution. Generally, when doing the DLS measurement, it is important to submit enough amount of material in order to obtain sufficient signal, but if the sample is overly concentrated, then light scattered by one particle might be again scattered by another (known as multiple scattering), and make the data processing less accurate. An ideal sample submission for DLS analysis has a volume of 1 – 2 mL and is sufficiently concentrated as to have strong color hues, or opaqueness/turbidity in the case of a white or black sample. Alternatively, 100 – 200 μL of highly concentrated sample can be diluted to 1 mL or analyzed in a low-volume microcuvette.

In order to get high quality DLS data, there are also other issues to be concerned with. First is to minimize particulate contaminants, as it is common for a single particle contaminant to scatter a million times more than a suspended nanoparticle, by using ultra high purity water or solvents, extensively rinsing pipettes and containers, and sealing sample tightly. Second is to filter the sample through a 0.2 or 0.45 μm filter to get away of the visible particulates within the sample solution. Third is to avoid probe sonication to prevent the particulates ejected from the sonication tip, and use the bath sonication in stead.

Measurement

Now that the sample is readily prepared and put into the sample holder of the instrument, the next step is to actually do the DLS measurement. Generally the DLS instrument will be provided with software that can help you to do the measurement rather easily, but it is still worthwhile to understand the important parameters used during the measurement.

Firstly, the laser light source with an appropriate wavelength should be selected. As for the Zetasizer Nano series (Malvern Instruments Ltd.), either a 633 nm “red” laser or a 532 nm “green” laser is available. One should keep in mind that the 633 nm laser is least suitable for blue samples, while the 532 nm laser is least suitable for red samples, since otherwise the sample will just absorb a large portion of the incident light.

Then, for the measurement itself, one has to select the appropriate stabilization time and the duration time. Normally, longer striation/duration time can results in more stable signal with less noises, but the time cost should also be considered. Another important parameter is the temperature of the sample, as many DLS instruments are equipped with the temperature-controllable sample holders, one can actually measure the size distribution of the data at different temperature, and get extra information about the thermal stability of the sample analyzed.

Next, as is used in the calculation of particle size from the light scattering data, the viscosity and refraction index of the solution are also needed. Normally, for solutions with low concentration, the viscosity and refraction index of the solvent/water could be used as an approximation.

Finally, to get data with better reliability, the DLS measurement on the same sample will normally be conducted multiple times, which can help eliminate unexpected results and also provide additional error bar of the size distribution data.

Data Analysis

Although size distribution data could be readily acquired from the software of the DLS instrument, it is still worthwhile to know about the details about the data analysis process.

Cumulant method

As is mentioned in the Theory portion above, the decay rate Γ is mathematically determined from the g₁(τ) curve; if the sample solution is monodispersed, g₁(τ) could be regard as a single exponential decay function e^-Γτ, and the decay rate Γ can be in turn easily calculated. However, in most of the practical cases, the sample solution is always polydispersed, g₁(τ) will be the sum of many single exponential decay functions with different decay rates, and then it becomes significantly difficult to conduct the fitting process.

There are however, a few methods developed to meet this mathematical challenge: linear fit and cumulant expansion for mono-modal distribution, exponential sampling and CONTIN regularization for non-monomodal distribution. Among all these approaches, cumulant expansion is most common method and will be illustrated in detail in this section.

Generally, the cumulant expansion method is based on two relations: one between g₁(τ) and the moment-generating function of the distribution, and one between the logarithm of g₁(τ) and the cumulant-generating function of the distribution.

To start with, the form of g₁(τ) is equivalent to the definition of the moment-generating function M(-τ, Γ) of the distribution G(Γ), 2.4.142.4.14 .g1(τ)= ∫∞0G(Γ)e−ΓτdΓ = M(−τ,Γ)(2.4.14)(2.4.14)g1(τ)= ∫0∞G(Γ)e−ΓτdΓ = M(−τ,Γ)

The mth moment of the distribution mm(Γ)mm(Γ) is given by the mth derivative of M(-τ, Γ) with respect to τ, 2.4.152.4.15 .mm(Γ)= ∫∞0G(Γ)Γme−ΓτdΓ∣−τ=0(2.4.15)(2.4.15)mm(Γ)= ∫0∞G(Γ)Γme−ΓτdΓ∣−τ=0

Similarly, the logarithm of g₁(τ) is equivalent to the definition of the cumulant-generating function K(-τ, Γ), EQ, and the mth cumulant of the distribution km(Γ) is given by the mth derivative of K(-τ, Γ) with respect to τ, 2.4.162.4.16 and 2.4.172.4.17 .ln g1(τ)=ln M(−τ,Γ) = K(−τ,Γ)(2.4.16)(2.4.16)ln g1(τ)=ln M(−τ,Γ) = K(−τ,Γ)km(Γ)=dmK(−τ,Γ)d(−τ)m∣−τ=0(2.4.17)(2.4.17)km(Γ)=dmK(−τ,Γ)d(−τ)m∣−τ=0

By making use of that the cumulants, except for the first, are invariant under a change of origin, the km(Γ) could be rewritten in terms of the moments about the mean as 2.4.182.4.18 , 2.4.192.4.19 , 2.4.202.4.20 , and 2.4.212.4.21 where here μm are the moments about the mean, defined as given in 2.4.222.4.22 .k1(τ)k2(τ)k3(τ)k4(τ)= ∫∞0G(Γ)ΓdΓ=Γ¯= μ2= μ3= μ4−3μ22⋯(2.4.18)(2.4.19)(2.4.20)(2.4.21)(2.4.18)k1(τ)= ∫0∞G(Γ)ΓdΓ=Γ¯(2.4.19)k2(τ)= μ2(2.4.20)k3(τ)= μ3(2.4.21)k4(τ)= μ4−3μ22⋯μm = ∫∞0G(Γ)(Γ − Γ¯)mdΓ(2.4.22)(2.4.22)μm = ∫0∞G(Γ)(Γ − Γ¯)mdΓ

Based on the Taylor expansion of K(-τ, Γ) about τ = 0, the logarithm of g₁(τ) is given as 2.4.232.4.23 .ln g1(τ)= K(−τ,Γ)= −Γ¯τ +k22!τ2 −k33!τ3 +k44!τ4⋯(2.4.23)(2.4.23)ln g1(τ)= K(−τ,Γ)= −Γ¯τ +k22!τ2 −k33!τ3 +k44!τ4⋯

Importantly, if look back at the Seigert relationship in the logarithmic form, 2.4.242.4.24 .ln(g2(τ)−B)=lnβ + 2ln g1(τ)(2.4.24)(2.4.24)ln(g2(τ)−B)=lnβ + 2ln g1(τ)

The measured data of g₂(τ) could be fitted with the parameters of km using the relationship of 2.4.252.4.25 , where Γ¯Γ¯ (k₁), k₂, and k₃ describes the average, variance, and skewness (or asymmetry) of the decay rates of the distribution, and polydispersity index γ = k2Γ¯2γ = k2Γ¯2 is used to indicate the width of the distribution. And parameters beyond k₃ are seldom used to prevent overfitting the data. Finally, the size distribution can be easily calculated from the decay rate distribution as described in theory section previously. Figure 2.4.62.4.6 shows an example of data fitting using the cumulant method.ln(g2(τ)−B)=]lnβ + 2(−Γ¯τ +k22!τ2 −k33!τ3⋯)(2.4.25)(2.4.25)ln(g2(τ)−B)=]lnβ + 2(−Γ¯τ +k22!τ2 −k33!τ3⋯)

When using the cumulant expansion method however, one should keep in mind that it is only suitable for monomodal distributions (Gaussian-like distribution centered about the mean), and for non-monomodal distributions, other methods like exponential sampling and CONTIN regularization should be applied instead.

Three Index of Size Distribution

Now that the size distribution is able to be acquired from the fluctuation data of the scattered light using cumulant expansion or other methods, it is worthwhile to understand the three kinds of distribution index usually used in size analysis: number weighted distribution, volume weighted distribution, and intensity weighted distribution.

First of all, based on all the theories discussed above, it should be clear that the size distribution given by DLS experiments is the intensity weighted distribution, as it is always the intensity of the scattering that is being analyzed. So for intensity weighted distribution, the contribution of each particle is related to the intensity of light scattered by that particle. For example, using Rayleigh approximation, the relative contribution for very small particles will be proportional to a⁶.

For number weighted distribution, given by image analysis as an example, each particle is given equal weighting irrespective of its size, which means proportional to a⁰. This index is most useful where the absolute number of particles is important, or where high resolution (particle by particle) is required.

For volume weighted distribution, given by laser diffraction as an example, the contribution of each particle is related to the volume of that particle, which is proportional to a³. This is often extremely useful from a commercial perspective as the distribution represents the composition of the sample in terms of its volume/mass, and therefore its potential money value.

When comparing particle size data for the same sample represented using different distribution index, it is important to know that the results could be very different from number weighted distribution to intensity weighted distribution. This is clearly illustrated in the example below (Figure 2.4.92.4.9 ), for a sample consisting of equal numbers of particles with diameters of 5 nm and 50 nm. The number weighted distribution gives equal weighting to both types of particles, emphasizing the presence of the finer 5 nm particles, whereas the intensity weighted distribution has a signal one million times higher for the coarser 50 nm particles. The volume weighted distribution is intermediate between the two.

Furthermore, based on the different orders of correlation between the particle contribution and the particle size a, it is possible to convert particle size data from one type of distribution to another type of distribution, and that is also why the DLS software can also give size distributions in three different forms (number, volume, and intensity), where the first two kinds are actually deducted from the raw data of intensity weighted distribution.

An Example of an Application

As the DLS method could be used in many areas towards size distribution such as polymers, proteins, metal nanoparticles, or carbon nanomaterials, here gives an example about the application of DLS in size-controlled synthesis of monodisperse gold nanoparticles.

The size and size distribution of gold particles are controlled by subtle variation of the structure of the polymer, which is used to stabilize the gold nanoparticles during the reaction. These variations include monomer type, polymer molecular weight, end-group hydrophobicity, end-group denticity, and polymer concentration; a total number of 88 different trials have been conducted based on these variations. By using the DLS method, the authors are able to determine the gold particle size distribution for all these trials rather easily, and the correlation between polymer structure and particle size can also be plotted without further processing the data. Although other sizing techniques such as UV-V spectroscopy and TEM are also used in this paper, it is the DLS measurement that provides a much easier and reliable approach towards the size distribution analysis.

Comparison with TEM and AFM

Since DLS is not the only method available to determine the size distribution of particles, it is also necessary to compare DLS with the other common-used general sizing techniques, especially TEM and AFM.

First of all, it has to be made clear that both TEM and AFM measure particles that are deposited on a substrate (Cu grid for TEM, mica for AFM), while DLS measures particles that are dispersed in a solution. In this way, DLS will be measuring the bulk phase properties and give a more comprehensive information about the size distribution of the sample. And for AFM or TEM, it is very common that a relatively small sampling area is analyzed, and the size distribution on the sampling area may not be the same as the size distribution of the original sample depending on how the particles are deposited.

On the other hand however, for DLS, the calculating process is highly dependent on the mathematical and physical assumptions and models, which is, monomodal distribution (cumulant method) and spherical shape for the particles, the results could be inaccurate when analyzing non-monomodal distributions or non-spherical particles. Yet, since the size determining process for AFM or TEM is nothing more than measuring the size from the image and then using the statistic, these two methods can provide much more reliable data when dealing with “irregular” samples.

Another important issue to consider is the time cost and complication of size measurement. Generally speaking, the DLS measurement should be a much easier technique, which requires less operation time and also cheaper equipment. And it could be really troublesome to analysis the size distribution data coming out from TEM or AFM images without specially programmed software.

In addition, there are some special issues to consider when choosing size analysis techniques. For example, if the originally sample is already on a substrate (synthesized by the CVD method), or the particles could not be stably dispersed within solution, apparently the DLS method is not suitable. Also, when the particles tend to have a similar imaging contrast against the substrate (carbon nanomaterials on TEM grid), or tend to self-assemble and aggregate on the surface of the substrate, the DLS approach might be a better choice.

In general research work however, the best way to do size distribution analysis is to combine these analyzing methods, and get complimentary information from different aspects. One thing to keep in mind, since the DLS actually measures the hydrodynamic radius of the particles, the size from DLS measurement is always larger than the size from AFM or TEM measurement. As a conclusion, the comparison between DLS and AFM/TEM is shown in Table 2.4.12.4.1 .

Conclusion

In general, relying on the fluctuating Rayleigh scattering of small particles that randomly moves in solution, DLS is a very useful and rapid technique used in the size distribution of particles in the fields of physics, chemistry, and bio-chemistry, especially for monomodally dispersed spherical particles, and by combining with other techniques such as AFM and TEM, a comprehensive understanding of the size distribution of the analyte can be readily acquired.

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Light interacts with matter in different ways, transmitting through some materials, while reflecting or scattering off others. Both the material and the colour (wavelength) of the light affect this interaction. We call the study of this light ‘spectroscopy’. Which parts of the visible spectrum enter our eyes determines which colours we perceive.

A substance might appear blue, for example, if it absorbs the red parts of the spectrum of light falling upon it, only reflecting (or scattering) the blue parts into our eyes.

Raman spectroscopy looks at the scattered light

If you were to shine blue light—from just one part of the spectrum—onto the material, you might expect to just see blue light reflected from it, or no light at all if it is completely absorbed (i.e. a black material).

However, by using a Raman spectrometer, you can see that often a very tiny fraction of the scattered light has a different colour. It has changed frequency because, during the scattering process, its energy changed by interacting with molecular vibrations. This is the Raman scattering process, named after its discoverer, the famous Indian physicist C.V. Raman. He was awarded the 1930 physics Nobel Prize for this great discovery.

By studying the vibration of the atoms we can discover the chemical composition and other useful information about the material.

The Raman effect is very weak; only about 1 part in 10 million of the scattered light has a shifted colour. This is too weak to see with the naked eye, so we analyse the light with a highly sensitive spectrometer.

Raman spectrometers

These systems consist of:

• one or more single coloured light sources (lasers)
• lenses (both to focus the light onto the sample and to collect the scattered light)
• filters (to purify the reflected and scattered light so that only the Raman light is collected)
• a means of splitting the light into its constituent colours (normally a diffraction grating or prism)
• a very sensitive detector (to detect the weak light)
• a device such as a computer to control the whole system, display the spectrum and enable this information to be analysed

Raman scattering offers significant advantages for the investigation of materials over other analytical techniques, such as x-raying them or seeing how they absorb light (e.g. infrared absorption or ultraviolet absorption).

aman spectroscopy reveals the chemical and structural composition of samples. Generally, all materials produce Raman spectra, with the exception of pure metals.

Raman scattering

Raman scattering occurs when light interacts with molecular vibrations. This is similar to the more widely known infrared absorption spectroscopy, but different rules apply. A change in molecular polarisability is required during the vibration for the Raman effect to occur.

You will see some vibrations in the Raman spectrum that are not visible in the infrared spectrum, and vice-versa, because of the different selection rules. For example, Raman spectroscopy is superb for studying the carbon atoms that make up the structure of diamond, unlike infrared absorption spectroscopy.

Scattered light

The first step in producing a Raman spectrum is to illuminate your sample with a monochromatic light source, such as a laser.

Most of the light that scatters off is unchanged in energy (‘Rayleigh scattered’). A minute fraction—perhaps 1 part in 10 million—has lost or gained energy (‘Raman scattered’). This Raman shift occurs because photons (particles of light) exchange part of their energy with molecular vibrations in the material.

Where energy is lost the Raman scattering is designated as ‘Stokes’; where energy is gained the Raman scattering is designated as ‘anti-Stokes’. We rarely use anti-Stokes Raman light as it is less intense than the Stokes, however it does represent equivalent vibrational information of the molecule.

Vibrating atoms

The change in energy depends on the frequency of vibration of the molecule. If it is very fast (high frequency)—light atoms held together with strong bonds—the energy change is significant. If it is very slow (low frequency)—heavy atoms held together with weak bonds—the energy change is small.

Raman spectrometers

Renishaw inVia systems consist of:

• single or multiple lasers, from UV (244 nm) to IR (1064 nm) – switch with a single click
• high quality objective lenses, from high confocal 100× to long working distance and immersion options
• custom designed motorised spectrometer lenses­ – automatically align for each configuration
• laser-line-specific Rayleigh filters with a dual filter arrangement to optimise sensitivity
• highest quality master diffraction gratings provide exceptional dispersion and longevity
• thermoelectrically cooled (- 70 ºC) CCD detector – stable and sensitive
• high specification multi-core PC for data collection and analysis

Raman spectra

We graphically depict the results of our measurements as Raman spectra. We plot the intensity of the scattered light (y-axis) for each energy (frequency) of light (x-axis). The frequency is traditionally measured in a unit called the wavenumber (number of waves per cm, cm^-1).

We plot the x-axis frequencies relative to that of the laser as it is the shift in energy of the light that is of particular interest.

How do I get the information I want from my spectrum?

You can tell a great deal about a material from its Raman spectrum, with different features relating to different aspects of the material.

The key features are:

The Raman shifts and relative intensities of all of the Raman bands of the material
With this, we can identify the material.

Individual band changes
A band may shift, narrow or broaden, or vary in intensity. These changes can reveal information about stresses in the sample, variations in crystallinity, and the amount of material respectively.

Variations in spectra with position on the sample
This will reveal changes in the uniformity (homogeneity) of the material. You can analyse at several arbitrary points, or systematically measure an array of points (enabling the production of images of composition, stress, crystallinity, etc.)

What do the Raman bands represent?

It is easy to understand the Raman spectrum of crystals with a regular array of identical atoms, all in the same configuration (such as the carbon atoms in diamond). In these cases, you often see just one dominant Raman band (because there is just one molecular environment of the crystal).

The Raman spectrum of polystyrene, however, is much more complex because the molecule is less symmetric and has hydrogen atoms in addition to carbon atoms. There are also different bond types connecting the atoms.

Vibration frequencies

The frequencies of vibration depend on the masses of the atoms involved and the strength of the bonds between them. Heavy atoms and weak bonds have low Raman shifts. Light atoms and strong bonds have high Raman shifts.

We see the high frequency carbon-hydrogen (C-H) vibrations in the polystyrene spectrum at about 3000 cm^-1. The low frequency carbon-carbon (C-C) vibrations are at around 800 cm^-1. The C-H vibrations have a higher frequency than the C-C vibrations because hydrogen is lighter than carbon.

We see the vibrations of two carbon atoms linked by strong double bonds (C=C) at around 1600 cm^-1. This is at a higher frequency than two carbon atoms lined by a weaker single bond (C-C, 800 cm^-1).

You can use these simple rules to explain many of the features of Raman spectra.

Vibrations in detail

You can see more subtle effects if you inspect spectra closely. The strength of bonds also affects their vibration rates. For example, the C-H vibrations of polystyrene appear in two bands, at approximately 2900 cm^-1 and 3050 cm^-1. The carbons in the former are part of carbon chains (‘aliphatic’), whereas the carbons in the latter form part of carbon rings (‘aromatic’).

You can view the vibrations of a complex molecule as partly consisting of many simple diatomic vibrations. However the full richness of the Raman spectrum can only be understood by considering the vibrations of larger groups of atoms (such as the expanding/contracting ‘breathing mode’ of the aromatic carbon ring that appears at 1000 cm^-1 in polystyrene).

Low frequency vibrations

You can also study Raman bands with low Raman shifts, below 100 cm^-1. These originate from very heavy atoms or very large-scale vibrations, such as the whole crystal lattice vibrating. Renishaw’s Raman instruments enable you to study these modes and explore a wide range of materials and crystals, and distinguish between different crystalline forms (polymorphs).

The big picture

A Raman spectrum therefore consists of a range of features, each associated with a vibrational mode. The spectrum is unique to the material and enables you to identify it. It is important to note that, although a full understanding of the vibrational modes is of interest, you rarely need this as you can use a reference database for identification.

When a sample is illuminated by a laser, both Raman scattering and photoluminescence (PL) can occur. The latter can be many times stronger than the former and can prevent successful Raman analysis.

PL comprises both fluorescence and phosphorescence processes and originates from an absorption/emission process between different electronic energy levels in the material. The amount and type of PL depends on which material you are studying and which laser wavelength you are using. Unwanted fluorescence interference can normally be avoided by choosing an appropriate laser wavelength.

• Energy diagram showing absorption of light and the processes involved in the emission of light as fluorescence and phosphorescence.

What PL can tell us

In many cases photoluminescence carries useful information that can facilitate sample analysis and augment the Raman data. inVia confocal Raman microscopes are suited to the analysis of both Raman scattering and PL.

Fluorescence imaging (a type of PL) is often employed in the biological sciences, where fluorescent tags are used to reveal the presence and distribution of molecular species. However, this approach is more invasive than Raman analysis, which is typically tag-free. Renishaw’s inVia confocal Raman microscope can be used to generate images of fluorescent tags, but more commonly provides valuable tag-free chemical information.

You can also use PL to study crystal defects, such as atomic vacancies and substitutions. This is of particular importance for materials such as diamond and silicon carbide (SiC). Not only can you identify the defect, but you can also tell if the crystal has internal stresses.

• Stress image generated from the ruby R2 PL band position

How to avoid PL backgrounds

Occasionally PL bands are strong and broad, masking Raman information. You can counter this by using a different laser wavelength. This can move the Raman bands away from the peak emission of the PL band and may even avoid generation of the PL entirely.

Ideally, a Raman instrument should be able to switch rapidly and easily between different laser wavelengths, so that you can select or avoid PL features, depending on your requirements.

Raman images (sometimes referred to as maps) depict a variation in spectral information from different points on, or in your sample. They can take the form of one-dimensional profiles, two-dimensional images, or three-dimensional rendered volumes. With them, you can rapidly see how a Raman parameter alters with position.

The parameter could be as simple as the intensity of a particular Raman band, or you could derive it from a more complicated analysis of the whole Raman spectrum.

The two main methods of collecting the spectral data to generate these images are Raman mapping and Raman imaging.

• White light and Raman images of washing powder

Raman mapping

Raman mapping collects a spectral hypercube (a Raman spectrum from each position on the sample in a single file), rather than a simple intensity image. The hypercube is analysed to produce Raman images.

There are several Raman mapping methods, such as:

• Point-by-point mapping
  The laser is focused to a spot. A motorised stage moves the sample under the laser. Spectra are sequentially acquired from an array of sample points spanning the defined region of interest. Fast versions of this are Renishaw’s StreamHR™ and StreamHR Rapide.
• Line focus mapping
  This is similar to point-by-point mapping, but the laser illuminates a line on the sample, rather than a spot. This enables you to simultaneously collect spectra from multiple positions on the sample, saving time. With this method you can use higher laser powers without damaging the sample (reducing exposure times). Renishaw’s StreamLine™ is a sophisticated modern implementation of this concept.

It is important to consider the potentially undesirable effects of undersampling when mapping. This is most clearly illustrated when point-by-point mapping: parts of the sample will be ‘missed’ if the laser spot is smaller than the spacing between acquisition points. Renishaw has solved this problem through the use of the StreamLine™ Slalom mode.

Generating Raman images from map data

Once all the Raman spectra are collected from the mapping experiment, they can be analysed to produce profiles, images or rendered volumes. Analysis options in Renishaw’s WiRE software include:

• Intensity at one frequency in the spectrum
  This produces an equivalent image to that from Raman imaging. These are quick to generate but may be misleading because it is not possible to differentiate between intensities arising from a Raman band of interest and those associated with a broad background fluorescence.
• Curve fit parameters
  All the spectra in the set have a theoretical curve fitted to one of the Raman bands. Images are then made based on the theoretical curve parameters for each spectrum. Images are often made using the centre frequency of the curve (band), or the full width at half maximum (FWHM), as this is sensitive to stresses and crystallinity within the sample respectively.
• Multivariate parameters
  Images can be generated using chemometric tools, such as generic principal component analysis (PCA), or Renishaw’s Empty Modelling™, which is optimised for Raman data. The Empty Modelling method reveals systematic variations between the Raman spectra, and highlights the distribution of these variations across the sample as an image. This is achieved without the need for prior knowledge of what is present within the sample, which greatly simplifies the analysis process. Multivariate analysis is very powerful because it uses information from the entire spectrum, not just one part of it (intensity at one frequency) or one curve-fitted band. This typically results in higher quality Raman images.

Raman imaging

Raman imaging is analogous to taking a photograph; spectral intensity values are collected simultaneously from the entire area of interest. The laser illuminates a square or circular region on the sample. The light is filtered so that the intensity of just one narrow part of the spectrum is recorded on the detector.

The single image collected contains limited information, just the intensity of the light at that frequency. However, these images can be acquired rapidly. This is especially true if you have a high power laser; because the light is spread over an area, you can use all the power without damaging your samples, with correspondingly short exposure times.

Two-dimensional images are typically produced using this method. Renishaw’s True Raman Imaging is an example of Raman imaging.

Note that it is possible to collect intensity values covering multiple points of the spectrum by using multiple and/or tuneable filters.

Spatial resolution

Point-by-point Raman mapping

Spatial resolution is determined by a combination of the laser spot size and the spacing between acquisition points on the sample.

• Laser spot size
  This is a function of the objective magnification and the laser wavelength (higher magnification and shorter wavelengths produce smaller spot sizes)
• Spacing between acquisition points on the sample (sampling)
  This is a function of the sample stage (ideally stages should have a large travel range while still enabling a step size down to 100 nm, smaller than the smallest spot size)

Raman imaging

Spatial resolution is determined by the magnification of the optics in the system and the size of the elements in the detector. Ultimately this is limited, by the inherent wavelike nature of light, to a little under a micrometre.