Issue #7/2017

Measurement of thickness of polymer shell on surface of submicron filler particles for polymer composite materials

**A.Kurangyshev, N.Shilov, V.Kurangyshev**Measurement of thickness of polymer shell on surface of submicron filler particles for polymer composite materials

A method has been developed and optimum parameters of the optical system for measuring the thickness of the polymer shell on the surface of submicron zinc oxide particles with a resolution of 20 nm have been determined.

Теги: nephelometry polymer composite material two-layer particle двухслойная частица нефелометрия полимерный композиционный материал

To improve the physical-mechanical properties of polymer composite materials (PCM) fillers based on submicron particles are used [1]. However, the reproducibility of the properties of the final material remains low for various reasons, for example, due to poor wetting of the filler particles by the polymer matrix [2]. One way to increase the wettability of the filler particles is to cover their surface with a polymer shell with obtaining so-called "two-layer particles". The formation of such a shell can be carried out, for example, in the gas phase by depositing finely dispersed monomer droplets on the surface of submicron particles followed by their polymerization under the action of UV radiation. The thickness of the polymer shell being formed is required to be controlled, since it affects the complex of physical-mechanical characteristics of the PCM [2, 3].

Among a large number of nanodispersed fillers of polymer matrices in the production of composite materials, particular attention should be paid to inorganic components: titanium dioxide TiO2, zinc oxide ZnO [4], alumina Al2O3, boron nitride BN, etc. When encapsulating such particles with polymers, it is very difficult to measure the thickness of the formed envelope by known methods (microscopic, sedimentation, sieve analysis), especially when it comes to controlling submicron particles (100–1000 nm) in gas streams.
Among the vast number of devices for measuring particle sizes, the most interesting instruments are those based on the nephelometry method [5–10]. However, when using such a device to control the thickness of the polymer shell during its formation on the surface of particles in gas streams, or more precisely in determining the characteristic size (diameter) of bilayer particles by measuring the intensity of the scattered radiation, a number of uncontrollable problems arise, related both to the parameters of the aerosol environment, and the particles themselves.

In general, the intensity of the scattered radiation I(s) is a function of the following main characteristics of the gas flows and the particles themselves [5, 8, 10, 11]:

I(s) = I0 F (Vизм,С, f (d), n (λ, d*), λ, θ, r),

where Vизм is the measuring volume from which scattered radiation is collected; C is the concentration of scattering particles with a diameter d with a size distribution function f (d); n (λ, d) is the refractive index of the particle matter; λ is the wavelength of the probing radiation; d* is the diameter of the two-layer particle; θ is the angle of observation of radiation scattered by the particle at a distance r.

In the above formula, the wavelength of the radiation source, the counting volume (controlled by adjusting the apertures of the radiation source and receiver) and other parameters are controlled, with the exception of the following characteristics of the medium under consideration:

changes in the average concentration of particles C due to the deposition of particles of all types on the channel walls, and also because of possible escapes through technological openings in the mixing chamber;

changes in the particle distribution function with respect to the diameter f(d) due to, for example, shell growth on the surfaces of submicron particles, different deposition rates and escapes of particles with different characteristic dimensions;

change in the refractive index n of the measured particles during the formation of the polymeric shell on their surfaces [12].

In general, when controlling the characteristic diameter of two-layer particles, the measured physical quantity is a function of a set of independent parameters, the main ones being concentration, size distribution function (diameter/radius), refractive index. In this connection, a number of questions arise about the choice of parameters of an optical system based on nephelometry to measure the characteristic diameter of particles with high resolution. The aim of this work is to determine the optimum parameters of an optical system for measuring the thickness of a polymer shell on the surface of submicron zinc oxide particles with a resolution of 20 nm in real time.

Let's consider submicron particles of zinc oxide ZnO, on the surface of which a polymeric shell is formed in the gas stream. It is known that the characteristic dimensions of zinc oxide particles used as fillers in polymer matrices are usually about 100 nm [1, 2]. We assume that the thickness of the polymer shell formed on the surface of such particles can be from 10 to 100% of their radius.

The parameters of the nephelometric systems, for example, of the photodetector part (the angles of observation of the scattered radiation, the aperture and the sensitivity of the photodetector) and the parameters of the probing radiation source (radiation power, wavelength, monochromaticity, beam width and its divergence, etc.) can be different and depend mainly on the parameters of the medium containing the particles. As shown in the review [7, 8, 13–15], the most common are the schemes for measuring the sizes and concentrations of submicron particles with small observation angles (0–15°) and in the lateral scattering region with respect to the direction of propagating radiation (90°). However, measurements of the characteristic size of an ensemble of particles and other physical parameters of the medium are not limited to recording the intensity of scattered radiation in the zone of small and lateral angular directions. For example, there are schemes in which scattered radiation is observed over the entire range of discrete angles by a set of receivers. The latter systems are more informative in the study of various parameters of a dispersed medium than schemes, in which only a few angles are used to observe the scattering indicatrix (the most common is the simultaneous use of 2–3 receivers installed at different angles relative to the principal axis of the propagating probe beam) [7, 14, 15].

When it is required to control the thickness of the polymer shell on the surface of submicron particles (more precisely, the characteristic diameter of bilayer particles) in aerosol flows by nephelometric systems, the use of traditional measurement methods, for example, of the small-angle indicatrix method (1–15°) or the normal scattering method (80–100°) is inappropriate. As it was said above, this is explained by the fact that the measurement of the characteristic size of submicron two-layer particles is simultaneously affected by changes in concentration, distribution function in size, refractive index, etc. Therefore, a reasonable study of the ideology of measurement and a special approach to the choice of parameters of the optical system are required.

The effect of particle concentration can be eliminated by creating a small countable volume in which only one particle will be present at the time of measurement [7, 16]. This can be achieved, for example, by using a "sampling duct" with a small aperture, where the particles will flow in sequence one at a time (the distance between the particles is larger than the size of the counting volume). However, when the objects of investigation are submicron-sized particles in a gas stream, such a solution is not advisable. In practice, the particles have a non-spherical shape, complex morphology, composition, heterogeneity of dimensions, etc. To determine the characteristic diameter of particle in a polydisperse medium using the above-described method, many measurements will be required. However, in a gas stream, when encapsulating submicron particles with a polymer shell, instantaneous measurements are required.

From the Mie theory of light scattering by dispersed weakly absorbing submicron particles it is known that with increasing particle size, including in a polydisperse medium, the total intensity of the scattered radiation increases in the region of small angles and decreases in the region of the inverse angles with a constant wavelength of the source of probing radiation in the visible region of the spectrum. That is, with an increase in the diffraction parameter, the intensity of the scattered radiation rises rapidly in the region of small angles along the axis of propagation of the probe radiation (forward scattering) and decreases in the opposite direction (backward scattering) [5,6]. Therefore, it is possible to determine the change in the characteristic size of submicron particles from the ratio of the radiation intensities scattered at different angles and with different wavelengths. Such a solution will make it possible to get rid not only of the influence of fluctuations in the concentration of submicron particles in the measuring volume, but also of the instability of the radiation source, which may prove to be important in the practical implementation of this approach to the measurement of the characteristic size of particles in gas flows. This approach can be implemented in several ways, for example, by measuring the ratio of scattering intensities by a dispersed medium at two different angles at the same wavelength, or at certain angles at different wavelengths. Each of these approaches requires laborious computational processes, so we will consider only the first of them.

The ideology of measurement is as follows. A ray of light at a wavelength λ emitted by the source shines through the measuring volume V. The detectors located in different angular directions relative to the principal axis of propagation of the probe radiation collect information on the magnitude of the scattering intensities I1 and I2, respectively. Software processing of the ratio of the received signals from the outputs of detectors allows us to estimate the characteristic size of submicron particles without taking into account the effect of their concentration in the countable volume under consideration [17].

Thus, the effect of particle concentration on the measurement of their characteristic diameter can be eliminated by estimating the ratio of the two scattering intensities at different angles. However, the use of this method requires the determination of the most appropriate parameters of the measuring system in which the influence of the time-varying refractive index and the particle size distribution function will be weak, which will allow controlling the thickness of the polymer shell on the surface of submicron particles (characteristic diameters of bilayer particles) with the required resolution.

Let's conduct numerical calculations to determine and refine the most important parameters of the measuring system.

The investigation of light scattering, including absorption by dispersed particles, is carried out by approximate and laborious mathematical calculations on the basis of various theories. Each theory is characterized by certain restrictions on the morphology of the object under consideration, its refractive index and size relative to the wavelength of light [18]. The use of approximate calculation methods does not guarantee the correctness of calculations, since a number of physical quantities is replaced by model parameters.

Depending on the properties of the particles and the requirements, for the analysis, basically three theories are used, each of which is tied to specific ranges of particle size and wavelength ratios:

Rayleigh theory for the smallest particles, the dimensions of which are much smaller than the wavelength of the probing source of radiation;

Mie theory for particles whose dimensions are close to the wavelength of the radiation source (the model requires knowledge of the values of the optical parameters);

Fraunhofer theory for large particles whose exact optical parameters are unknown.

In this paper, the Mie theory will be used to study scattered radiation with a polydisperse medium containing zinc oxide particles on the surface of which a polymeric shell is formed. It is well suited for studying scattering by bilayer particles with some assumptions and limitations [19].

The mathematical dependences of Mie theory are rather cumbersome, so it is advisable to simplify them with various assumptions, which, however, should not subsequently affect the results of calculating the scattering intensities of light by the medium under consideration. We assume the followings:

particles in a dispersed medium are spherical and homogeneous;

particle size distribution is described by the log-normal distribution law;

shell on the particles grows evenly during encapsulation;

multiple scattering of radiation by a dispersed medium is insignificant;

particles are non-absorbent and non-reflective;

particles are not charged.

The above assumptions facilitate calculations, practically without changing the results of calculations of the scattering intensity for the range of problems of interest to us. In fact, since the particles in the dispersed medium (in the aerosol flow) are in a chaotic order or, in any case, their mutual arrangement changes over times comparable to the observation time, each instantaneous intensity of the scattered radiation will be averaged and regarded as scattering by an ensemble of spherical particles [8, 20]. And the intensity of total scattered radiation is the sum of the scattering intensities for each particle (under the condition of a single scattering). Thus, the scattering intensity at any instant of time can be characterized as scattering by spherical dispersed particles.

For an ensemble of particles of different sizes, the calculation formulas for the scattering intensity at all solid angles are complicated, and the results depend on the distribution law used. As it was established by A.N.Kolmogorov, the distribution of the sizes of many naturally or industrially created particles is described by a log-normal law [21–23].

The assumption that the shell on the particles grows evenly is not always true. In the technological process, different cases of polymerization are possible, for example, on larger particles, the shell can grow at one speed, and on smaller particles on the other. Ultimately, it is possible that the results of the mathematical calculation will be different from practical measurements of the intensity of scattered radiation. However, we will assume that the technological process allows to build up a shell on the surface of particles of different sizes with equal speed.

With regard to the fourth assumption, an excessive increase in the concentration of particles in the aerosol flow and a decrease in the distance between them may lead to the impossibility of encapsulating the particles by a shell. We assume that the gas flow is controlled, the concentration is set by the system in such a way that the distances between the particles are larger than their sizes and the light is scattered by the dispersed medium once.

If the particles are non-absorbent and non-reflective, then their complex refractive indices are zero. Indeed, the refractive indices for the particles of zinc oxide and polymer (which makes up the shell on the particle surface) have only real components, that is, their complex components are zero.

The assumption that the particles are not charged is based on the fact that the charge present on the particle surfaces during the encapsulation does not affect the intensity of the scattered light. As is known, charged and uncharged particles scatter light equally [8].

It was assumed in the calculations that the characteristic size of submicron particles (the core) on the surface of which a shell of a polymeric material is formed is 100 nm. The polymeric shell on the surface of the nuclei grows evenly with a step of 10% of their characteristic radius. In this case, the particle size distribution function is described by a log-normal law (dispersion of 10 nm), and its form does not change during the growth of the shell on the particle surface.

It was necessary to qualitatively evaluate the parameters of the optical system for control of the characteristic diameter of two-layer particles, which realize the ratio of two scattering intensities at different angles at one wavelength and provide a resolution of 20 nm. The wavelengths of the radiation source were chosen on the basis of the size of the particles under study according to the Mie theory. Monochrome radiation sources with wavelengths of 430, 530 and 630 nm are widely used to study particles of about 100 nm in size, and viewing angles are ranged from 5° to 170°. The refractive indices of the submicron particle and polystyrene (shells on the surface of the particle) for the wavelengths of the source of the probe radiation were chosen according to [24, 25].

Computer analysis of various combinations of the wavelengths of the radiation source and the angles of observation of scattered light has shown that it is advisable to use angles of 10° and 90° and a source of visible light. Fig. shows the operating characteristics of the ratio of two scattering intensities at different wavelengths (430, 530, 630 nm) recorded at angles of 10° and 90° respectively, depending on the diameter of the two-layer particle (if d = 100 nm, the ZnO particle does not have a polymer shell). The refractive indices of zinc oxide (n1) and polystyrene (n2) for different wavelengths are shown to the right of the graphs.

We note the following. Initially, when the thickness of the polymer is small relative to the core diameter (of zinc oxide particle), the refractive index of the particle in the shell is equal to the refractive index of the core. With an increase in the shell thickness, at some point, the refractive index begins to change rapidly and eventually becomes equal to the refractive index of polystyrene [12, 23].

As shown by the calculations, to control the thickness of the polymer shell on the surface of submicron particles of zinc oxide with a diameter of 100 nm, it is expedient to use a ratio of two scattering intensities at angles of 10° and 90° at a wavelength of 430 nm. The wavelengths of 530 and 630 nm do not allow measuring the size of the polymer shell on the surface of particles of this diameter with a resolution of 20 nm. In other words, the Mie parameter (q = πd/λ) should not be less than 0.75. This means that, for example, using observation angles of 10° and 90° to control the thickness of the shell on the surface of zinc oxide particles with a diameter of 200 nm, the wavelength of the radiation source should not exceed 860 nm.

Thus, to monitor the thickness of the polymer shell on the surface of zinc oxide particles (with characteristic diameter of 100 nm) with a resolution of 20 nm, the following parameters of the optical measurement system are required, realizing a ratio of two intensities: viewing angles of 10° and 90°; the wavelength of the radiation source is 430 nm. And it is possible to measure the thickness of the polymer shell on the surface of submicron particles of zinc oxide without affecting their concentration in the counting volume (under the condition of a single scattering).

Concluding the article, we would like to add that the correctness of software calculations was verified by solving several test problems, formulated on the basis of published data of experimental and theoretical calculations of the scattering intensities of particles with different parameters (size, shape, refractive index) depending on the viewing angle. In addition, we used proven mathematical programs designed to calculate various particle parameters on the basis of the Mie equations, in particular, MieScattering, MiePlot4600, etc. [26, 27]. ■

Among a large number of nanodispersed fillers of polymer matrices in the production of composite materials, particular attention should be paid to inorganic components: titanium dioxide TiO2, zinc oxide ZnO [4], alumina Al2O3, boron nitride BN, etc. When encapsulating such particles with polymers, it is very difficult to measure the thickness of the formed envelope by known methods (microscopic, sedimentation, sieve analysis), especially when it comes to controlling submicron particles (100–1000 nm) in gas streams.

In general, the intensity of the scattered radiation I(s) is a function of the following main characteristics of the gas flows and the particles themselves [5, 8, 10, 11]:

I(s) = I0 F (Vизм,С, f (d), n (λ, d*), λ, θ, r),

where Vизм is the measuring volume from which scattered radiation is collected; C is the concentration of scattering particles with a diameter d with a size distribution function f (d); n (λ, d) is the refractive index of the particle matter; λ is the wavelength of the probing radiation; d* is the diameter of the two-layer particle; θ is the angle of observation of radiation scattered by the particle at a distance r.

In the above formula, the wavelength of the radiation source, the counting volume (controlled by adjusting the apertures of the radiation source and receiver) and other parameters are controlled, with the exception of the following characteristics of the medium under consideration:

changes in the average concentration of particles C due to the deposition of particles of all types on the channel walls, and also because of possible escapes through technological openings in the mixing chamber;

changes in the particle distribution function with respect to the diameter f(d) due to, for example, shell growth on the surfaces of submicron particles, different deposition rates and escapes of particles with different characteristic dimensions;

change in the refractive index n of the measured particles during the formation of the polymeric shell on their surfaces [12].

In general, when controlling the characteristic diameter of two-layer particles, the measured physical quantity is a function of a set of independent parameters, the main ones being concentration, size distribution function (diameter/radius), refractive index. In this connection, a number of questions arise about the choice of parameters of an optical system based on nephelometry to measure the characteristic diameter of particles with high resolution. The aim of this work is to determine the optimum parameters of an optical system for measuring the thickness of a polymer shell on the surface of submicron zinc oxide particles with a resolution of 20 nm in real time.

Let's consider submicron particles of zinc oxide ZnO, on the surface of which a polymeric shell is formed in the gas stream. It is known that the characteristic dimensions of zinc oxide particles used as fillers in polymer matrices are usually about 100 nm [1, 2]. We assume that the thickness of the polymer shell formed on the surface of such particles can be from 10 to 100% of their radius.

The parameters of the nephelometric systems, for example, of the photodetector part (the angles of observation of the scattered radiation, the aperture and the sensitivity of the photodetector) and the parameters of the probing radiation source (radiation power, wavelength, monochromaticity, beam width and its divergence, etc.) can be different and depend mainly on the parameters of the medium containing the particles. As shown in the review [7, 8, 13–15], the most common are the schemes for measuring the sizes and concentrations of submicron particles with small observation angles (0–15°) and in the lateral scattering region with respect to the direction of propagating radiation (90°). However, measurements of the characteristic size of an ensemble of particles and other physical parameters of the medium are not limited to recording the intensity of scattered radiation in the zone of small and lateral angular directions. For example, there are schemes in which scattered radiation is observed over the entire range of discrete angles by a set of receivers. The latter systems are more informative in the study of various parameters of a dispersed medium than schemes, in which only a few angles are used to observe the scattering indicatrix (the most common is the simultaneous use of 2–3 receivers installed at different angles relative to the principal axis of the propagating probe beam) [7, 14, 15].

When it is required to control the thickness of the polymer shell on the surface of submicron particles (more precisely, the characteristic diameter of bilayer particles) in aerosol flows by nephelometric systems, the use of traditional measurement methods, for example, of the small-angle indicatrix method (1–15°) or the normal scattering method (80–100°) is inappropriate. As it was said above, this is explained by the fact that the measurement of the characteristic size of submicron two-layer particles is simultaneously affected by changes in concentration, distribution function in size, refractive index, etc. Therefore, a reasonable study of the ideology of measurement and a special approach to the choice of parameters of the optical system are required.

The effect of particle concentration can be eliminated by creating a small countable volume in which only one particle will be present at the time of measurement [7, 16]. This can be achieved, for example, by using a "sampling duct" with a small aperture, where the particles will flow in sequence one at a time (the distance between the particles is larger than the size of the counting volume). However, when the objects of investigation are submicron-sized particles in a gas stream, such a solution is not advisable. In practice, the particles have a non-spherical shape, complex morphology, composition, heterogeneity of dimensions, etc. To determine the characteristic diameter of particle in a polydisperse medium using the above-described method, many measurements will be required. However, in a gas stream, when encapsulating submicron particles with a polymer shell, instantaneous measurements are required.

From the Mie theory of light scattering by dispersed weakly absorbing submicron particles it is known that with increasing particle size, including in a polydisperse medium, the total intensity of the scattered radiation increases in the region of small angles and decreases in the region of the inverse angles with a constant wavelength of the source of probing radiation in the visible region of the spectrum. That is, with an increase in the diffraction parameter, the intensity of the scattered radiation rises rapidly in the region of small angles along the axis of propagation of the probe radiation (forward scattering) and decreases in the opposite direction (backward scattering) [5,6]. Therefore, it is possible to determine the change in the characteristic size of submicron particles from the ratio of the radiation intensities scattered at different angles and with different wavelengths. Such a solution will make it possible to get rid not only of the influence of fluctuations in the concentration of submicron particles in the measuring volume, but also of the instability of the radiation source, which may prove to be important in the practical implementation of this approach to the measurement of the characteristic size of particles in gas flows. This approach can be implemented in several ways, for example, by measuring the ratio of scattering intensities by a dispersed medium at two different angles at the same wavelength, or at certain angles at different wavelengths. Each of these approaches requires laborious computational processes, so we will consider only the first of them.

The ideology of measurement is as follows. A ray of light at a wavelength λ emitted by the source shines through the measuring volume V. The detectors located in different angular directions relative to the principal axis of propagation of the probe radiation collect information on the magnitude of the scattering intensities I1 and I2, respectively. Software processing of the ratio of the received signals from the outputs of detectors allows us to estimate the characteristic size of submicron particles without taking into account the effect of their concentration in the countable volume under consideration [17].

Thus, the effect of particle concentration on the measurement of their characteristic diameter can be eliminated by estimating the ratio of the two scattering intensities at different angles. However, the use of this method requires the determination of the most appropriate parameters of the measuring system in which the influence of the time-varying refractive index and the particle size distribution function will be weak, which will allow controlling the thickness of the polymer shell on the surface of submicron particles (characteristic diameters of bilayer particles) with the required resolution.

Let's conduct numerical calculations to determine and refine the most important parameters of the measuring system.

The investigation of light scattering, including absorption by dispersed particles, is carried out by approximate and laborious mathematical calculations on the basis of various theories. Each theory is characterized by certain restrictions on the morphology of the object under consideration, its refractive index and size relative to the wavelength of light [18]. The use of approximate calculation methods does not guarantee the correctness of calculations, since a number of physical quantities is replaced by model parameters.

Depending on the properties of the particles and the requirements, for the analysis, basically three theories are used, each of which is tied to specific ranges of particle size and wavelength ratios:

Rayleigh theory for the smallest particles, the dimensions of which are much smaller than the wavelength of the probing source of radiation;

Mie theory for particles whose dimensions are close to the wavelength of the radiation source (the model requires knowledge of the values of the optical parameters);

Fraunhofer theory for large particles whose exact optical parameters are unknown.

In this paper, the Mie theory will be used to study scattered radiation with a polydisperse medium containing zinc oxide particles on the surface of which a polymeric shell is formed. It is well suited for studying scattering by bilayer particles with some assumptions and limitations [19].

The mathematical dependences of Mie theory are rather cumbersome, so it is advisable to simplify them with various assumptions, which, however, should not subsequently affect the results of calculating the scattering intensities of light by the medium under consideration. We assume the followings:

particles in a dispersed medium are spherical and homogeneous;

particle size distribution is described by the log-normal distribution law;

shell on the particles grows evenly during encapsulation;

multiple scattering of radiation by a dispersed medium is insignificant;

particles are non-absorbent and non-reflective;

particles are not charged.

The above assumptions facilitate calculations, practically without changing the results of calculations of the scattering intensity for the range of problems of interest to us. In fact, since the particles in the dispersed medium (in the aerosol flow) are in a chaotic order or, in any case, their mutual arrangement changes over times comparable to the observation time, each instantaneous intensity of the scattered radiation will be averaged and regarded as scattering by an ensemble of spherical particles [8, 20]. And the intensity of total scattered radiation is the sum of the scattering intensities for each particle (under the condition of a single scattering). Thus, the scattering intensity at any instant of time can be characterized as scattering by spherical dispersed particles.

For an ensemble of particles of different sizes, the calculation formulas for the scattering intensity at all solid angles are complicated, and the results depend on the distribution law used. As it was established by A.N.Kolmogorov, the distribution of the sizes of many naturally or industrially created particles is described by a log-normal law [21–23].

The assumption that the shell on the particles grows evenly is not always true. In the technological process, different cases of polymerization are possible, for example, on larger particles, the shell can grow at one speed, and on smaller particles on the other. Ultimately, it is possible that the results of the mathematical calculation will be different from practical measurements of the intensity of scattered radiation. However, we will assume that the technological process allows to build up a shell on the surface of particles of different sizes with equal speed.

With regard to the fourth assumption, an excessive increase in the concentration of particles in the aerosol flow and a decrease in the distance between them may lead to the impossibility of encapsulating the particles by a shell. We assume that the gas flow is controlled, the concentration is set by the system in such a way that the distances between the particles are larger than their sizes and the light is scattered by the dispersed medium once.

If the particles are non-absorbent and non-reflective, then their complex refractive indices are zero. Indeed, the refractive indices for the particles of zinc oxide and polymer (which makes up the shell on the particle surface) have only real components, that is, their complex components are zero.

The assumption that the particles are not charged is based on the fact that the charge present on the particle surfaces during the encapsulation does not affect the intensity of the scattered light. As is known, charged and uncharged particles scatter light equally [8].

It was assumed in the calculations that the characteristic size of submicron particles (the core) on the surface of which a shell of a polymeric material is formed is 100 nm. The polymeric shell on the surface of the nuclei grows evenly with a step of 10% of their characteristic radius. In this case, the particle size distribution function is described by a log-normal law (dispersion of 10 nm), and its form does not change during the growth of the shell on the particle surface.

It was necessary to qualitatively evaluate the parameters of the optical system for control of the characteristic diameter of two-layer particles, which realize the ratio of two scattering intensities at different angles at one wavelength and provide a resolution of 20 nm. The wavelengths of the radiation source were chosen on the basis of the size of the particles under study according to the Mie theory. Monochrome radiation sources with wavelengths of 430, 530 and 630 nm are widely used to study particles of about 100 nm in size, and viewing angles are ranged from 5° to 170°. The refractive indices of the submicron particle and polystyrene (shells on the surface of the particle) for the wavelengths of the source of the probe radiation were chosen according to [24, 25].

Computer analysis of various combinations of the wavelengths of the radiation source and the angles of observation of scattered light has shown that it is advisable to use angles of 10° and 90° and a source of visible light. Fig. shows the operating characteristics of the ratio of two scattering intensities at different wavelengths (430, 530, 630 nm) recorded at angles of 10° and 90° respectively, depending on the diameter of the two-layer particle (if d = 100 nm, the ZnO particle does not have a polymer shell). The refractive indices of zinc oxide (n1) and polystyrene (n2) for different wavelengths are shown to the right of the graphs.

We note the following. Initially, when the thickness of the polymer is small relative to the core diameter (of zinc oxide particle), the refractive index of the particle in the shell is equal to the refractive index of the core. With an increase in the shell thickness, at some point, the refractive index begins to change rapidly and eventually becomes equal to the refractive index of polystyrene [12, 23].

As shown by the calculations, to control the thickness of the polymer shell on the surface of submicron particles of zinc oxide with a diameter of 100 nm, it is expedient to use a ratio of two scattering intensities at angles of 10° and 90° at a wavelength of 430 nm. The wavelengths of 530 and 630 nm do not allow measuring the size of the polymer shell on the surface of particles of this diameter with a resolution of 20 nm. In other words, the Mie parameter (q = πd/λ) should not be less than 0.75. This means that, for example, using observation angles of 10° and 90° to control the thickness of the shell on the surface of zinc oxide particles with a diameter of 200 nm, the wavelength of the radiation source should not exceed 860 nm.

Thus, to monitor the thickness of the polymer shell on the surface of zinc oxide particles (with characteristic diameter of 100 nm) with a resolution of 20 nm, the following parameters of the optical measurement system are required, realizing a ratio of two intensities: viewing angles of 10° and 90°; the wavelength of the radiation source is 430 nm. And it is possible to measure the thickness of the polymer shell on the surface of submicron particles of zinc oxide without affecting their concentration in the counting volume (under the condition of a single scattering).

Concluding the article, we would like to add that the correctness of software calculations was verified by solving several test problems, formulated on the basis of published data of experimental and theoretical calculations of the scattering intensities of particles with different parameters (size, shape, refractive index) depending on the viewing angle. In addition, we used proven mathematical programs designed to calculate various particle parameters on the basis of the Mie equations, in particular, MieScattering, MiePlot4600, etc. [26, 27]. ■

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