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Introduction to BET (Brunauer, Emmett and Teller)

By BET (Brunauer, Emmett and Teller) the specific surface area of a sample is measured – including the pore size distribution. This information is used to predict the dissolution rate, as this rate is proportional to the specific surface area. Thus, the surface area can be used to predict bioavailability. Further it is useful in evaluation of product performance and manufacturing consistency.

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The specific surface determined by BET relates to the total surface area (reactive surface) as all porous structures adsorb the small gas molecules. The surface area determined by BET is thus normally larger than the surface area determined by air permeability. The method used complies with Ph. Eu.2.9.26 Method II.

Instrument and measuring principle, BET 

The BET instrument applied by Particle Analytical (Micromeritics Gemini 2375 and Gemini V) determines the specific surface area (m²/g) of pharmaceutical samples. The samples are dried with nitrogen purging or in a vacuum applying elevated temperatures. Unless otherwise instructed we use P/P0  of 0,1, 0,2 and 0,3 as standard measurement points. The volume of gas adsorbed to the surface of the particles is measured at the boiling point of nitrogen (-196°C). The amount of adsorbed gas is correlated to the total surface area of the particles including pores in the surface. The calculation is based on the BET theory. Traditionally nitrogen is used as adsorbate gas. Gas adsorption also enables the determination of size and volume distribution of micropores (0.35 – 2.0 nm)..

BET theory

The specific surface area of a powder is determined by physical adsorption of a gas on the surface of the solid and by calculating the amount of adsorbate gas corresponding to a monomolecular layer on the surface. Physical adsorption results from relatively weak forces (van der Waals forces) between the adsorbate gas molecules and the adsorbent surface area of the test powder. The determination is usually carried out at the temperature of liquid nitrogen. The amount of gas adsorbed can be measured by a volumetric or continuous flow procedure.

Multi-point measurements

The data are treated according to the Brunauer, Emmett and Teller (BET) adsorption isotherm equation:

A value of Va is measured at each of not less than 3 values of P/Po. Then the BET value:

is plotted against P/P_o according to equation (1). This plot should yield a straight line usually in the approximate relative pressure range 0.05 to 0.3. The data are considered acceptable if the correlation coefficient, r, of the linear regression is not less than 0.9975; that is, r² is not less than 0.995. From the resulting linear plot, the slope, which is equal to (C − 1)/V_mC, and the intercept, which is equal to 1/V_mC, are evaluated by linear regression analysis. From these values, V_m is calculated as 1/(slope + intercept), while Cis calculated as (slope/intercept) + 1. From the value of V_m so determined, the specific surface area, S, in m²·g^–1, is calculated by the equation:

A minimum of 3 data points is required. Additional measurements may be carried out, especially when non-linearity is obtained at a P/Po value close to 0.3. Because non-linearity is often obtained at a P/Po value below 0.05, values in this region are not recommended. The test for linearity, the treatment of the data, and the calculation of the specific surface area of the sample are described above.

Single point measurement

Normally, at least 3 measurements of Va each at different values of P/Po are required for the determination of specific surface area by the dynamic flow gas adsorption technique (Method I) or by volumetric gas adsorption (Method II). However, under certain circumstances described below, it may be acceptable to determine the specific surface area of a powder from a single value of Va measured at a single value of P/Po such as 0.300 (corresponding to 0.300 mole of nitrogen or 0.001038 mole fraction of krypton), using the following equation for calculating Vm:

The specific surface area is then calculated from the value of V_m by equation (2) given above.

The single-point method may be employed directly for a series of powder samples of a given material for which the material constant C is much greater than unity. These circumstances may be verified by comparing values of specific surface area determined by the single-point method with that determined by the multiple-point method for the series of powder samples. Close similarity between the single-point values and multiple-point values suggests that 1/C approaches zero.

The single-point method may be employed indirectly for a series of very similar powder samples of a given material for which the material constant Cis not infinite but may be assumed to be invariant. Under these circumstances, the error associated with the single-point method can be reduced or eliminated by using the multi-point method to evaluate C for one of the samples of the series from the BET plot, from which C is calculated as (1 + slope/intercept). Then V_m is calculated from the single value of V_a measured at a single value of P/P_o by the equation:

The specific surface area is calculated from V_m by equation (2) given above.

The following section describes the methods to be used for the sample preparation, the dynamic flow gas adsorption technique (Method I) and the volumetric gas adsorption technique (Method II).

Sample preparation: Outgassing: Before the specific surface area of the sample can be determined, it is necessary to remove gases and vapours that may have become physically adsorbed onto the surface after manufacture and during treatment, handling and storage. If outgassing is not achieved, the specific surface area may be reduced or may be variable because an intermediate area of the surface is covered with molecules of the previously adsorbed gases or vapours. The outgassing conditions are critical for obtaining the required precision and accuracy of specific surface area measurements on pharmaceuticals because of the sensitivity of the surface of the materials.

Conditions: The outgassing conditions must be demonstrated to yield reproducible BET plots, a constant weight of test powder, and no detectable physical or chemical changes in the test powder. The outgassing conditions defined by the temperature, pressure and time should be chosen so that the original surface of the solid is reproduced as closely as possible. Outgassing of many substances is often achieved by applying a vacuum, by purging the sample in a flowing stream of a non-reactive, dry gas, or by applying a desorption-adsorption cycling method. In either case, elevated temperatures are sometimes applied to increase the rate at which the contaminants leave the surface. Caution should be exercised when outgassing powder samples using elevated temperatures to avoid affecting the nature of the surface and the integrity of the sample.

If heating is employed, the recommended temperature and time of outgassing are as low as possible to achieve reproducible measurement of specific surface area in an acceptable time. For outgassing sensitive samples, other outgassing methods such as the desorption-adsorption cycling method may be employed.

The volumetric method (Ph. Eu.2.9.26 Method II)

Principle: In the volumetric method (see Figure 2.9.26.-2), the recommended adsorbate gas is nitrogen which is admitted into the evacuated space above the previously outgassed powder sample to give a defined equilibrium pressure, P, of the gas. The use of a diluent gas, such as helium, is therefore unnecessary, although helium may be employed for other purposes, such as to measure the dead volume.

Since only pure adsorbate gas, instead of a gas mixture, is employed, interfering effects of thermal diffusion are avoided in this method.

Procedure: Admit a small amount of dry nitrogen into the sample tube to prevent contamination of the clean surface, remove the sample tube, insert the stopper, and weigh it. Calculate the weight of the sample. Attach the sample tube to the volumetric apparatus. Cautiously evacuate the sample down to the specified pressure (e.g. between 2 Pa and 10 Pa). Alternatively, some instruments operate by evacuating to a defined rate of pressure change (e.g. less than 13 Pa/30 s) and holding for a defined period of time before commencing the next step.

If the principle of operation of the instrument requires the determination of the dead volume in the sample tube, for example, by the admission of a non-adsorbed gas, such as helium, this procedure is carried out at this point, followed by evacuation of the sample. The determination of dead volume may be avoided using difference measurements, that is, by means of reference and sample tubes connected by a differential transducer. The adsorption of nitrogen gas is then measured as described below.

Raise a Dewar vessel containing liquid nitrogen at 77.4 K up to a defined point on the sample cell. Admit a sufficient volume of adsorbate gas to give the lowest desired relative pressure. Measure the volume adsorbed, V_a. For multi-point measurements, repeat the measurement of V_a at successively higher P/P_o values. When nitrogen is used as the adsorbate gas, P/P_o values of 0.10, 0.20, and 0.30 are often suitable.

Reference materials: Periodically verify the functioning of the apparatus using appropriate reference materials of known surface area, such as α-alumina, which should have a specific surface area similar to that of the sample to be examined.

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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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Introduction

In the past few years, nanotechnology research has expanded out of the chemistry department and into the fields of medicine, energy, aerospace and even computing and information technology. With bulk materials, the surface area to volume is insignificant in relation to the number of atoms in the bulk, however when the particles are only 1 to 100 nm across, different properties begin to arise. For example, commercial grade zinc oxide has a surface area range of 2.5 to 12 m²/g while nanoparticle zinc oxide can have surface areas as high as 54 m²/g . The nanoparticles have superior UV blocking properties when compared to the bulk material, making them useful in applications such as sunscreen. Many useful properties of nanoparticles rise from their small size, making it very important to be able to determine their surface area.

Overview of BET Theory

The BET theory was developed by Stephen Brunauer (Figure 2.3.12.3.1 ), Paul Emmett (Figure 2.3.22.3.2 ), and Edward Teller (Figure 2.3.32.3.3 ) in 1938. The first letter of each publisher’s surname was taken to name this theory. The BET theory was an extension of the Langmuir theory, developed by Irving Langmuir (Figure 2.3.42.3.4 ) in 1916.

The Langmuir theory relates the monolayer adsorption of gas molecules (Figure 2.3.52.3.5 ), also called adsorbates, onto a solid surface to the gas pressure of a medium above the solid surface at a fixed temperature to 2.3.12.3.1 , where θ is the fractional cover of the surface, P is the gas pressure and α is a constant.Θ = α⋅P1 + (α⋅P)(2.3.1)(2.3.1)Θ = α⋅P1 + (α⋅P)

The Langmuir theory is based on the following assumptions:

• All surface sites have the same adsorption energy for the adsorbate, which is usually argon, krypton or nitrogen gas. The surface site is defined as the area on the sample where one molecule can adsorb onto.
• Adsorption of the solvent at one site occurs independently of adsorption at neighboring sites.
• Activity of adsorbate is directly proportional to its concentration.
• Adsorbates form a monolayer.
• Each active site can be occupied only by one particle.

The Langmuir theory has a few flaws that are addressed by the BET theory. The BET theory extends the Langmuir theory to multilayer adsorption (Figure 2.3.12.3.1 ) with three additional assumptions:

• Gas molecules will physically adsorb on a solid in layers infinitely.
• The different adsorption layers do not interact.
• The theory can be applied to each layer.

How does BET Work?

Adsorption is defined as the adhesion of atoms or molecules of gas to a surface. It should be noted that adsorption is not confused with absorption, in which a fluid permeates a liquid or solid. The amount of gas adsorbed depends on the exposed surface are but also on the temperature, gas pressure and strength of interaction between the gas and solid. In BET surface area analysis, nitrogen is usually used because of its availability in high purity and its strong interaction with most solids. Because the interaction between gaseous and solid phases is usually weak, the surface is cooled using liquid N₂ to obtain detectable amounts of adsorption. Known amounts of nitrogen gas are then released stepwise into the sample cell. Relative pressures less than atmospheric pressure is achieved by creating conditions of partial vacuum. After the saturation pressure, no more adsorption occurs regardless of any further increase in pressure. Highly precise and accurate pressure transducers monitor the pressure changes due to the adsorption process. After the adsorption layers are formed, the sample is removed from the nitrogen atmosphere and heated to cause the adsorbed nitrogen to be released from the material and quantified. The data collected is displayed in the form of a BET isotherm, which plots the amount of gas adsorbed as a function of the relative pressure. There are five types of adsorption isotherms possible.

Type I Isotherm

Type I is a pseudo-Langmuir isotherm because it depicts monolayer adsorption (Figure 2.3.62.3.6 ). A type I isotherm is obtained when P/P_o < 1 and c > 1 in the BET equation, where P/P_o is the partial pressure value and c is the BET constant, which is related to the adsorption energy of the first monolayer and varies from solid to solid. The characterization of microporous materials, those with pore diameters less than 2 nm, gives this type of isotherm.

Type II Isotherm

A type II isotherm (Figure 2.3.72.3.7 ) is very different than the Langmuir model. The flatter region in the middle represents the formation of a monolayer. A type II isotherm is obtained when c > 1 in the BET equation. This is the most common isotherm obtained when using the BET technique. At very low pressures, the micropores fill with nitrogen gas. At the knee, monolayer formation is beginning and multilayer formation occurs at medium pressure. At the higher pressures, capillary condensation occurs.

Type III Isotherm

A type III isotherm (Figure 2.3.82.3.8 ) is obtained when the c < 1 and shows the formation of a multilayer. Because there is no asymptote in the curve, no monolayer is formed and BET is not applicable.

Type IV Isotherm

Type IV isotherms (Figure 2.3.92.3.9 ) occur when capillary condensation occurs. Gases condense in the tiny capillary pores of the solid at pressures below the saturation pressure of the gas. At the lower pressure regions, it shows the formation of a monolayer followed by a formation of multilayers. BET surface area characterization of mesoporous materials, which are materials with pore diameters between 2 – 50 nm, gives this type of isotherm.

Type V Isotherm

Type V isotherms (Figure 2.3.102.3.10 ) are very similar to type IV isotherms and are not applicable to BET.

Calculations

The BET Equation, 2.3.22.3.2 , uses the information from the isotherm to determine the surface area of the sample, where X is the weight of nitrogen adsorbed at a given relative pressure (P/Po), X_m is monolayer capacity, which is the volume of gas adsorbed at standard temperature and pressure (STP), and C is constant. STP is defined as 273 K and 1 atm.1X[(P0/P)−1]=1XmC+C−1XmC(PP0)(2.3.2)(2.3.2)1X[(P0/P)−1]=1XmC+C−1XmC(PP0)

Multi-point BET

Ideally five data points, with a minimum of three data points, in the P/P₀ range 0.025 to 0.30 should be used to successfully determine the surface area using the BET equation. At relative pressures higher than 0.5, there is the onset of capillary condensation, and at relative pressures that are too low, only monolayer formation is occurring. When the BET equation is plotted, the graph should be of linear with a positive slope. If such a graph is not obtained, then the BET method was insufficient in obtaining the surface area.

• The slope and y-intercept can be obtained using least squares regression.
• The monolayer capacity X_m can be calculated with 2.3.32.3.3 .
• Once X_m is determined, the total surface area S_t can be calculated with the following equation, where L_av is Avogadro’s number, A_m is the cross sectional area of the adsorbate and equals 0.162 nm² for an absorbed nitrogen molecule, and M_v is the molar volume and equals 22414 mL, 2.3.42.3.4 .

Xm =1s + i=C−1Cs(2.3.3)(2.3.3)Xm =1s + i=C−1CsS =XmLavAmMv(2.3.4)(2.3.4)S =XmLavAmMv

Single point BET can also be used by setting the intercept to 0 and ignoring the value of C. The data point at the relative pressure of 0.3 will match up the best with a multipoint BET. Single point BET can be used over the more accurate multipoint BET to determine the appropriate relative pressure range for multi-point BET.

Sample Preparation and Experimental Setup

Prior to any measurement the sample must be degassed to remove water and other contaminants before the surface area can be accurately measured. Samples are degassed in a vacuum at high temperatures. The highest temperature possible that will not damage the sample’s structure is usually chosen in order to shorten the degassing time. IUPAC recommends that samples be degassed for at least 16 hours to ensure that unwanted vapors and gases are removed from the surface of the sample. Generally, samples that can withstand higher temperatures without structural changes have smaller degassing times. A minimum of 0.5 g of sample is required for the BET to successfully determine the surface area.

Samples are placed in glass cells to be degassed and analyzed by the BET machine. Glass rods are placed within the cell to minimize the dead space in the cell. Sample cells typically come in sizes of 6, 9 and 12 mm and come in different shapes. 6 mm cells are usually used for fine powders, 9 mm cells for larger particles and small pellets and 12 mm are used for large pieces that cannot be further reduced. The cells are placed into heating mantles and connected to the outgas port of the machine.

After the sample is degassed, the cell is moved to the analysis port (Figure 2.3.112.3.11 ). Dewars of liquid nitrogen are used to cool the sample and maintain it at a constant temperature. A low temperature must be maintained so that the interaction between the gas molecules and the surface of the sample will be strong enough for measurable amounts of adsorption to occur. The adsorbate, nitrogen gas in this case, is injected into the sample cell with a calibrated piston. The dead volume in the sample cell must be calibrated before and after each measurement. To do that, helium gas is used for a blank run, because helium does not adsorb onto the sample.

Shortcomings of BET

The BET technique has some disadvantages when compared to NMR, which can also be used to measure the surface area of nanoparticles. BET measurements can only be used to determine the surface area of dry powders. This technique requires a lot of time for the adsorption of gas molecules to occur. A lot of manual preparation is required.

The Surface Area Determination of Metal-Organic Frameworks

The BET technique was used to determine the surface areas of metal-organic frameworks (MOFs), which are crystalline compounds of metal ions coordinated to organic molecules. Possible applications of MOFs, which are porous, include gas purification and catalysis. An isoreticular MOF (IRMOF) with the chemical formula Zn₄O(pyrene-1,2-dicarboxylate)₃ (Figure 2.3.122.3.12 ) was used as an example to see if BET could accurately determine the surface area of microporous materials. The predicted surface area was calculated directly from the geometry of the crystals and agreed with the data obtained from the BET isotherms. Data was collected at a constant temperature of 77 K and a type II isotherm (Figure 2.3.132.3.13 ) was obtained.

The isotherm data obtained from partial pressure range of 0.05 to 0.3 is plugged into the BET equation, 2.3.22.3.2 , to obtain the BET plot (Figure 2.3.142.3.14 ).

Using 2.3.52.3.5 , the monolayer capactiy is determined to be 391.2 cm³/g.Xm =1(2.66E − 3) + (−5.212E − 0.05)(2.3.5)(2.3.5)Xm =1(2.66E − 3) + (−5.212E − 0.05)

Now that X_m is known, then 2.3.62.3.6 can be used to determine that the surface area is 1702.3 m²/g.S =391.2cm2∗0.162nm2∗6.02∗102322.414:L(2.3.6)(2.3.6)S =391.2cm2∗0.162nm2∗6.02∗102322.414:L