rus
Issue #2/2021
H. S. M. R. Hussein, V. A. Kuklin, M. Sh. Salakhutdinov, I. I. Nureev
Determination of the Density of Nanosized Particles by Sedimentation Method
Determination of the Density of Nanosized Particles by Sedimentation Method
DOI: 10.22184/1993-7296.FRos.2021.15.2.176.185
Determination of the Density of Nanosized Particles by Sedimentation Method
H. S. M.R. Hussein1, V. A. Kuklin2, M. Sh. Salakhutdinov3, I. I. Nureev3
University of Karbala, Karbala, Iraq
Kazan Federal University, Kazan, Republic of Tatarstan, Russia
A. N. Tupolev Kazan National Research Technical University, Kazan, Republic of Tatarstan, Russia
The article proposes a mathematical model for measuring the density of nanosized particles. The mathematical model describes the relationship between the sedimentation time constant, density, fluid viscosity with density and diameter of nanosized particles. The model takes into account the influence of gravity forces, hydrostatic lifting force, forces of resistance to motion. The concentration of nanosized particles is estimated based on Rayleigh scattering simulations. Analytical dependences have been obtained that make it possible to estimate the average density of individual nanosized particles at their known sizes or the characteristic diameter of nanosized particles at their known density based on the experimentally determined sedimentation time constant.
Keywords: sedimentation, sedimentation of particles in a liquid, measurement of molecular weight, measurement of the diameter of nanosized particles, concentration measurement, Rayleigh scattering
Received: 21.01.2021
Accepted: 15.02.2021
Introduction
The widespread use of nanosized particles as fillers for dispersion-reinforced polymer composite materials is due to a significant increase in the mechanical characteristics of composites at a low (by weight) concentration of filler particles [1, 2, 3]. One of the effective methods for finding new compositions with the required mechanical characteristics is based on mathematical modeling [4, 5]. In general, there are two approaches to the construction of mathematical models of dispersion-reinforced polymer composite materials. The first approach, macroscopic, is based on the use of the traditional apparatus of polymer mechanics [4, 6, 7], the second is microscopic, which is based on the use of the apparatus of molecular dynamics [8, 9]. The use of the apparatus of polymer mechanics is the most expedient, despite a number of limitations.
Thus, the use of this approach requires information on the mechanical characteristics (e. g., elastic moduli, Poisson’s ratio) of nanosized particles [10]. In effect, it is not always possible to determine the mechanical characteristics of nanosized particles. This is due to the fact that measurement of these characteristics is sometimes impossible due to the small diameter of nanosized particles. Mathematical methods for their prediction are often associated with the solution of inverse problems that require strict proof of the uniqueness of their solution, which is aggravated by the lack of a priori information about their mechanical characteristics. It should also be noted that the structure of nanosized particles may differ from the structure of microparticles of the same type, therefore, and the mechanical properties of these particles will also be different. Attempts to estimate the density of a mixture of particles and liquid were made on the basis of an analysis of the dependence of the refractive index on concentration [11, 12, 13], but they are based on extremely expensive instruments for measuring the central wavelength of a fiber Bragg grating. In this regard, the use of indirect methods for determining the approximate values of the physical and technical characteristics of nanosized particles, which in practice will limit the search area for solving inverse problems, is relevant.
One of the important parameters of nanosized particles is their density, knowing which one can determine the range of some of their mechanical characteristics. The purpose of this work is to develop a mathematical model for determining the density of nanosized particles based on modeling the process of their sedimentation in solution [14] and empirical information about the characteristic time of their sedimentation.
Mathematical model
Figure 1, a shows a diagram of the measuring system, which shows a laser radiation source (Laser) passing through a quartz container with liquid and particles suspended in it (highlighted in green), the radiation of which is received by a photodetector (PD). Let’s denote the height of the container as H, the diameter of the laser beam as R, and the vertical position of the laser beam as h. The output current of the photodetector is proportional to the distance traveled through the container, the optical density of the liquid with particles, and the power of the initial laser radiation. In order to exclude the influence of the power of the initial laser radiation and the distance traveled by the beam through the container, as well as losses caused by the passage of radiation through the walls of the container, the optical radiation in front of the container is divided into two parts, one of which is directed into the container, and the second past him. The measurements are carried out on a differential photodetector that measures the relative attenuation of the laser beam. The optical layout for differential measurements is, for simplicity, shown in Figure 1, a not shown. Hereinafter, it is assumed that all measurements are carried out according to a differential measurement scheme for relative attenuation. Based on the measurements of the relative attenuation of the light flux, it is possible to draw a conclusion about the concentration of particles in the liquid [14, 15] and determine the time constant of sedimentation during the deposition of particles.
Before formulating the basic relations of the mathematical model, let us formulate a number of assumptions and assumptions. The emergence of internal flows in a liquid entrains nanosized particles and significantly affects their sedimentation. In order to exclude the influence of internal flows in the liquid on the deposition of nanosized particles, we assume that all measurements are carried out at a stationary temperature, that is, the same and constant temperature is maintained at all external boundaries of the cell. This assumption was made in order to exclude the occurrence of internal flows in the liquid caused by the density difference due to the temperature gradient. The sedimentation rate of particles can also be affected by the diffusion of particles caused by the gradient of their concentration in the liquid. Let us assume that the concentration of nanosized particles in the liquid is sufficiently low, the container with the mixture is sufficiently deep, the laser control beam is located near the upper border of the cuvette, and the diameter of the laser beam and its distance to the upper border of the cuvette are much less than the height of the cuvette (H – h << H and R << H). Under the conditions of this assumption, it can be assumed that the concentration of nanosized particles in the region of the control beam before the beginning of deposition coincides with their concentration directly under the control beam at each moment of time. Thus, the effect of diffusion on the sedimentation time constant is taken into account only at the upper boundary of the deposition of nanosized particles, where the concentration gradient is noticeable. Let us also assume that all nanoscale particles in a liquid have the same size and mass, and the movement of nanosized particles in a liquid occurs under the action of gravitational forces, hydrostatic lift and hydrodynamic drag force of the liquid.
Using Newton’s second law, we write the equation of motion of nanosized particles in vector form:
, (1)
where vectors are the hydrostatic lifting force of Archimedes, are the forces of gravity, are the hydrodynamic drag force of the fluid, are the velocities, and M is the mass of the particle, T is the time. Let’s choose a right-handed coordinate system, pointing the oz axis up, the ox axis to the right, and write out the system of equations, defining the forces.
The force of gravity in (1) depends on the mass of the particle P = –Mg, where g is the acceleration of gravity, the minus sign in the force of gravity is due to the choice of the direction of the oz axis.
Archimedes’ hydrostatic lift is directed along the oz axis, and oppositely directed to the gravity vector. According to Archimedes’ law, it is equal to the weight of the liquid displaced by the particle:
, (2)
where V is the volume of a particle, D is its diameter, and ρ0 is the density of the liquid.
The force of resistance to motion is directed against the direction of the particle’s velocity, depending on the square of its velocity, the density of the liquid, the cross-sectional area of the particle and the Reynolds number, which determines the viscosity of the liquid. At low Reynolds numbers (for the deposition of nanosized particles in a liquid, Re ≈ 300), the force of resistance to motion has the form [14]:
, (3)
where DG is the hydrodynamic diameter of a particle, and U is the diffusion component of the velocity caused by the concentration gradient of particles in the solution. The minus sign in (3) determines that the force of hydrodynamic resistance is directed against the direction of the particle velocity.
Substituting (2) and (3) in (1), we obtain the equation of motion of a particle in natural variables:
. (4)
We transform (4), writing it down in dimensionless form, introducing dimensionless w – velocity and τ – time, determining the characteristic parameters of the problem L0 – size and T0 – time.
After transformations, we obtain the equation of motion (4) in dimensionless variables:
, (5)
where u = U · T0 / L0 is the dimensionless diffusion component of the velocity, and the designations are introduced for the dimensionless complexes α and β:
, . (6)
Dimensionless complexes α and β completely determine the problem, here ρ0 is the density of the liquid, ρ is the density and D is the diameter and DG is the hydrodynamic diameter of the particle, and L0 and T0 are the characteristic distance and time. The dimensionless complex α determines the influence of the forces of gravity and hydrostatic lift, and β is the influence of the forces of hydrodynamic resistance.
The characteristic parameters of problem (6) can be chosen arbitrarily. At the same time, one can require their choice so that the dimensionless complexes α and β would be of the same order, e. g., α = β = 1 [16]:
, . (7)
The solution of equation (5) makes it possible to completely determine the dependence of the particle sedimentation rate on time under the assumption that the diffusion component of the velocity U does not depend on time, but depends only on the concentration gradient of nanosized particles in the solution near the control region, where the particle concentration is low:
. (8)
It can be shown that the last term in (8) with increasing dimensionless time is a very rapidly decaying function, and the deposition rate almost instantly takes on a maximum value:
. (9)
Therefore, we can assume that all nanosized particles in a liquid are deposited at a constant rate wMax, determined by relation (9).
Let us define the sedimentation time constant τ0 as the time passed by the particles from the upper boundary of the cuvette to the middle of the laser beam, which can be written based on the geometry of the problem and the deposition rate:
(10)
where α and β are functions of the physical parameters of the problem.
Empirical estimates can be used to assess the effect of the diffusion component of the velocity on the sedimentation rate. Microscopic studies allow determining the geometric radius of the particles. The estimates of the diameter of nanosized particles for aluminum oxide by the method of dynamic light scattering on an electron microscope showed that the diameter of nanosized particles of aluminum oxide is estimated at ~53.1 nm [17, 18]. The hydrodynamic radius can be estimated by the method of dynamic light scattering, and the effective hydrodynamic radius can be calculated based on the analysis of the diffusion coefficient according to the Stokes-Einstein equation for spherical particles R = k · T / (6πhD), where k is the Boltzmann constant, T is temperature, h is dynamic viscosity of the liquid. The most probable value of the hydrodynamic diameter of nanosized aluminum oxide particles in experiments is estimated at ~284 nm [17], the diffusion coefficient is estimated at 8.6 · 10–13 m2 / s, which gives an estimate of the order of the diffusion velocity component ~10–11–10–11 m / s, when assessing the deposition rate of nanosized particles in a stationary liquid ~10–7–10–8 m / s.
Therefore, the influence of the diffusion component of the velocity on the deposition time constant can be neglected, since it is approximately three orders of magnitude less than the deposition rate. The same work [17] gives the estimates for the ratio of the hydrodynamic and geometric diameters of nanosized particle, i. e. 5.3.
The relative concentration of particles in the area of the monitoring laser beam can be related to the cross-sectional area of the beam covered by the particles. Assuming the cross section of the laser beam to be a circle, and connecting the upper boundary of the deposited particles with the maximum deposition rate wMax, we obtain
, (11)
where, c(wτ) is the relative intensity of the light flux, H is the height of the cuvette, h is the height of the position of the laser beam, R is its radius, wτ is the upper boundary of the deposited particles. Dependence (11) for the relative concentration of particles in the area of the control beam (11) is given under normalization conditions, when the maximum concentration of particles (before sedimentation) is taken as a unit, and the value when the upper boundary of particles has completely passed area of the control beam. Integral (11) can be taken analytically, then for c(wτ) we obtain:
. (12)
Expression (12) allows you to construct the relative change in concentration as a function of time. Figure 2 shows the dependence of the curve of the relative change in the concentration of particles in the zone of the laser beam in dimensionless variables. The data are given for a cell with a height of 0.05 m, a laser beam with a diameter of 0.01 m, located at a height of 0.03 m.
Figure 3 shows the curves of the predicted values of the sedimentation time constant for polymer particles (size 50, 100, 150 nm) depending on their density. On the auxiliary axis, the dotted line of the same color shows the maximum deposition rate.
As can be seen from the data shown in Figure 3, the deposition rate of particles linearly depends on their density and increases with increasing particle size. At the same time, the sedimentation time constant decreases nonlinearly with increasing particle density, and for particles of larger diameter, the sedimentation time constant is less. However, in both cases, the sedimentation time constant is measured in days.
The obtained dependences were partially confirmed by the experimental data obtained at the A. N. Tupolev Kazan National Research Technical University at the Department of Electronic and Quantum Means of Information Transmission.
Conclusions
Based on the results of the work carried out, the following conclusions can be formulated. The task of plotting the dependence of the sedimentation time constant on the density of the liquid, the coefficient of its dynamic viscosity, the density and size of the precipitated particles was set and solved. The possibility of measuring the sedimentation time constant during the experiment allows one to unambiguously determine the density, and with it, the molecular weight of the particles, provided that their sizes are known. At the same time, the size of the particles can be determined, provided that their density is known. These data should be useful for experimenters to verify the conditions and results of experiments using the data of the mathematical model.
The practical significance of the work is to clarify the conditions for conducting experiments on long-term sedimentation. The proposed mathematical model makes it possible to reduce the time for conducting experiments to estimate the density of nanosized particles without waiting for their complete deposition.
Funding
I. I. Nureev was funded by Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075–03–2020–051 / 3, topic No. fzsu‑2020–0021) in part of physical task formulation, mathematical model construction. V. A. Kuklin was funded by Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075–03–2020–051, topic No. fzsu‑2020–0020) in part of data and results verification, training and tuning, and realization.
About Authors
Hussein Safaa Mohammed Ridha Hussein, University of Karbala, Karbala, 56001, Iraq, safaa_m333@yahoo.com.
ORCID:0000-0001-6022-0548
Kuklin Vladimir Alexanrovich, Kazan Federal University, 18 Kremlyovskaya street, Kazan 420008, Russian Federation, iamkvova@gmail.com.
Salakhutdinov Marat Shamilevich, Kazan National Research Technical University n. a. A. N. Tupolev, 10, K. Marx St., Kazan, Tatarstan 420111, Russia, alladin76@yandex.ru – corresponding author.
ORCID 0000-0001-5176-1334
Nureev Ilnur Ildarovich, Kazan National Research Technical University n. a. A. N. Tupolev, 10, K. Marx St., Kazan, Tatarstan 420111, Russia, n2i2@mail.ru.
Author Contributions
Conceptualization Nureev I. I. and Hussein S. M. R. H.; Formal analysis Nureev I. I., and Salakhutdinov M. Sh.; Funding acquisition Nureev I. I., Kuklin V. A.; Investigation Hussein S. M. R. H; Methodology, Kuklin V. A., Hussein S. M. R. H; Supervision Nureev I. I.; Writing-review & editing Salakhutdinov M. Sh, Hussein S. M. R. H. All authors have read and agreed to the published version of the manuscript.
H. S. M.R. Hussein1, V. A. Kuklin2, M. Sh. Salakhutdinov3, I. I. Nureev3
University of Karbala, Karbala, Iraq
Kazan Federal University, Kazan, Republic of Tatarstan, Russia
A. N. Tupolev Kazan National Research Technical University, Kazan, Republic of Tatarstan, Russia
The article proposes a mathematical model for measuring the density of nanosized particles. The mathematical model describes the relationship between the sedimentation time constant, density, fluid viscosity with density and diameter of nanosized particles. The model takes into account the influence of gravity forces, hydrostatic lifting force, forces of resistance to motion. The concentration of nanosized particles is estimated based on Rayleigh scattering simulations. Analytical dependences have been obtained that make it possible to estimate the average density of individual nanosized particles at their known sizes or the characteristic diameter of nanosized particles at their known density based on the experimentally determined sedimentation time constant.
Keywords: sedimentation, sedimentation of particles in a liquid, measurement of molecular weight, measurement of the diameter of nanosized particles, concentration measurement, Rayleigh scattering
Received: 21.01.2021
Accepted: 15.02.2021
Introduction
The widespread use of nanosized particles as fillers for dispersion-reinforced polymer composite materials is due to a significant increase in the mechanical characteristics of composites at a low (by weight) concentration of filler particles [1, 2, 3]. One of the effective methods for finding new compositions with the required mechanical characteristics is based on mathematical modeling [4, 5]. In general, there are two approaches to the construction of mathematical models of dispersion-reinforced polymer composite materials. The first approach, macroscopic, is based on the use of the traditional apparatus of polymer mechanics [4, 6, 7], the second is microscopic, which is based on the use of the apparatus of molecular dynamics [8, 9]. The use of the apparatus of polymer mechanics is the most expedient, despite a number of limitations.
Thus, the use of this approach requires information on the mechanical characteristics (e. g., elastic moduli, Poisson’s ratio) of nanosized particles [10]. In effect, it is not always possible to determine the mechanical characteristics of nanosized particles. This is due to the fact that measurement of these characteristics is sometimes impossible due to the small diameter of nanosized particles. Mathematical methods for their prediction are often associated with the solution of inverse problems that require strict proof of the uniqueness of their solution, which is aggravated by the lack of a priori information about their mechanical characteristics. It should also be noted that the structure of nanosized particles may differ from the structure of microparticles of the same type, therefore, and the mechanical properties of these particles will also be different. Attempts to estimate the density of a mixture of particles and liquid were made on the basis of an analysis of the dependence of the refractive index on concentration [11, 12, 13], but they are based on extremely expensive instruments for measuring the central wavelength of a fiber Bragg grating. In this regard, the use of indirect methods for determining the approximate values of the physical and technical characteristics of nanosized particles, which in practice will limit the search area for solving inverse problems, is relevant.
One of the important parameters of nanosized particles is their density, knowing which one can determine the range of some of their mechanical characteristics. The purpose of this work is to develop a mathematical model for determining the density of nanosized particles based on modeling the process of their sedimentation in solution [14] and empirical information about the characteristic time of their sedimentation.
Mathematical model
Figure 1, a shows a diagram of the measuring system, which shows a laser radiation source (Laser) passing through a quartz container with liquid and particles suspended in it (highlighted in green), the radiation of which is received by a photodetector (PD). Let’s denote the height of the container as H, the diameter of the laser beam as R, and the vertical position of the laser beam as h. The output current of the photodetector is proportional to the distance traveled through the container, the optical density of the liquid with particles, and the power of the initial laser radiation. In order to exclude the influence of the power of the initial laser radiation and the distance traveled by the beam through the container, as well as losses caused by the passage of radiation through the walls of the container, the optical radiation in front of the container is divided into two parts, one of which is directed into the container, and the second past him. The measurements are carried out on a differential photodetector that measures the relative attenuation of the laser beam. The optical layout for differential measurements is, for simplicity, shown in Figure 1, a not shown. Hereinafter, it is assumed that all measurements are carried out according to a differential measurement scheme for relative attenuation. Based on the measurements of the relative attenuation of the light flux, it is possible to draw a conclusion about the concentration of particles in the liquid [14, 15] and determine the time constant of sedimentation during the deposition of particles.
Before formulating the basic relations of the mathematical model, let us formulate a number of assumptions and assumptions. The emergence of internal flows in a liquid entrains nanosized particles and significantly affects their sedimentation. In order to exclude the influence of internal flows in the liquid on the deposition of nanosized particles, we assume that all measurements are carried out at a stationary temperature, that is, the same and constant temperature is maintained at all external boundaries of the cell. This assumption was made in order to exclude the occurrence of internal flows in the liquid caused by the density difference due to the temperature gradient. The sedimentation rate of particles can also be affected by the diffusion of particles caused by the gradient of their concentration in the liquid. Let us assume that the concentration of nanosized particles in the liquid is sufficiently low, the container with the mixture is sufficiently deep, the laser control beam is located near the upper border of the cuvette, and the diameter of the laser beam and its distance to the upper border of the cuvette are much less than the height of the cuvette (H – h << H and R << H). Under the conditions of this assumption, it can be assumed that the concentration of nanosized particles in the region of the control beam before the beginning of deposition coincides with their concentration directly under the control beam at each moment of time. Thus, the effect of diffusion on the sedimentation time constant is taken into account only at the upper boundary of the deposition of nanosized particles, where the concentration gradient is noticeable. Let us also assume that all nanoscale particles in a liquid have the same size and mass, and the movement of nanosized particles in a liquid occurs under the action of gravitational forces, hydrostatic lift and hydrodynamic drag force of the liquid.
Using Newton’s second law, we write the equation of motion of nanosized particles in vector form:
, (1)
where vectors are the hydrostatic lifting force of Archimedes, are the forces of gravity, are the hydrodynamic drag force of the fluid, are the velocities, and M is the mass of the particle, T is the time. Let’s choose a right-handed coordinate system, pointing the oz axis up, the ox axis to the right, and write out the system of equations, defining the forces.
The force of gravity in (1) depends on the mass of the particle P = –Mg, where g is the acceleration of gravity, the minus sign in the force of gravity is due to the choice of the direction of the oz axis.
Archimedes’ hydrostatic lift is directed along the oz axis, and oppositely directed to the gravity vector. According to Archimedes’ law, it is equal to the weight of the liquid displaced by the particle:
, (2)
where V is the volume of a particle, D is its diameter, and ρ0 is the density of the liquid.
The force of resistance to motion is directed against the direction of the particle’s velocity, depending on the square of its velocity, the density of the liquid, the cross-sectional area of the particle and the Reynolds number, which determines the viscosity of the liquid. At low Reynolds numbers (for the deposition of nanosized particles in a liquid, Re ≈ 300), the force of resistance to motion has the form [14]:
, (3)
where DG is the hydrodynamic diameter of a particle, and U is the diffusion component of the velocity caused by the concentration gradient of particles in the solution. The minus sign in (3) determines that the force of hydrodynamic resistance is directed against the direction of the particle velocity.
Substituting (2) and (3) in (1), we obtain the equation of motion of a particle in natural variables:
. (4)
We transform (4), writing it down in dimensionless form, introducing dimensionless w – velocity and τ – time, determining the characteristic parameters of the problem L0 – size and T0 – time.
After transformations, we obtain the equation of motion (4) in dimensionless variables:
, (5)
where u = U · T0 / L0 is the dimensionless diffusion component of the velocity, and the designations are introduced for the dimensionless complexes α and β:
, . (6)
Dimensionless complexes α and β completely determine the problem, here ρ0 is the density of the liquid, ρ is the density and D is the diameter and DG is the hydrodynamic diameter of the particle, and L0 and T0 are the characteristic distance and time. The dimensionless complex α determines the influence of the forces of gravity and hydrostatic lift, and β is the influence of the forces of hydrodynamic resistance.
The characteristic parameters of problem (6) can be chosen arbitrarily. At the same time, one can require their choice so that the dimensionless complexes α and β would be of the same order, e. g., α = β = 1 [16]:
, . (7)
The solution of equation (5) makes it possible to completely determine the dependence of the particle sedimentation rate on time under the assumption that the diffusion component of the velocity U does not depend on time, but depends only on the concentration gradient of nanosized particles in the solution near the control region, where the particle concentration is low:
. (8)
It can be shown that the last term in (8) with increasing dimensionless time is a very rapidly decaying function, and the deposition rate almost instantly takes on a maximum value:
. (9)
Therefore, we can assume that all nanosized particles in a liquid are deposited at a constant rate wMax, determined by relation (9).
Let us define the sedimentation time constant τ0 as the time passed by the particles from the upper boundary of the cuvette to the middle of the laser beam, which can be written based on the geometry of the problem and the deposition rate:
(10)
where α and β are functions of the physical parameters of the problem.
Empirical estimates can be used to assess the effect of the diffusion component of the velocity on the sedimentation rate. Microscopic studies allow determining the geometric radius of the particles. The estimates of the diameter of nanosized particles for aluminum oxide by the method of dynamic light scattering on an electron microscope showed that the diameter of nanosized particles of aluminum oxide is estimated at ~53.1 nm [17, 18]. The hydrodynamic radius can be estimated by the method of dynamic light scattering, and the effective hydrodynamic radius can be calculated based on the analysis of the diffusion coefficient according to the Stokes-Einstein equation for spherical particles R = k · T / (6πhD), where k is the Boltzmann constant, T is temperature, h is dynamic viscosity of the liquid. The most probable value of the hydrodynamic diameter of nanosized aluminum oxide particles in experiments is estimated at ~284 nm [17], the diffusion coefficient is estimated at 8.6 · 10–13 m2 / s, which gives an estimate of the order of the diffusion velocity component ~10–11–10–11 m / s, when assessing the deposition rate of nanosized particles in a stationary liquid ~10–7–10–8 m / s.
Therefore, the influence of the diffusion component of the velocity on the deposition time constant can be neglected, since it is approximately three orders of magnitude less than the deposition rate. The same work [17] gives the estimates for the ratio of the hydrodynamic and geometric diameters of nanosized particle, i. e. 5.3.
The relative concentration of particles in the area of the monitoring laser beam can be related to the cross-sectional area of the beam covered by the particles. Assuming the cross section of the laser beam to be a circle, and connecting the upper boundary of the deposited particles with the maximum deposition rate wMax, we obtain
, (11)
where, c(wτ) is the relative intensity of the light flux, H is the height of the cuvette, h is the height of the position of the laser beam, R is its radius, wτ is the upper boundary of the deposited particles. Dependence (11) for the relative concentration of particles in the area of the control beam (11) is given under normalization conditions, when the maximum concentration of particles (before sedimentation) is taken as a unit, and the value when the upper boundary of particles has completely passed area of the control beam. Integral (11) can be taken analytically, then for c(wτ) we obtain:
. (12)
Expression (12) allows you to construct the relative change in concentration as a function of time. Figure 2 shows the dependence of the curve of the relative change in the concentration of particles in the zone of the laser beam in dimensionless variables. The data are given for a cell with a height of 0.05 m, a laser beam with a diameter of 0.01 m, located at a height of 0.03 m.
Figure 3 shows the curves of the predicted values of the sedimentation time constant for polymer particles (size 50, 100, 150 nm) depending on their density. On the auxiliary axis, the dotted line of the same color shows the maximum deposition rate.
As can be seen from the data shown in Figure 3, the deposition rate of particles linearly depends on their density and increases with increasing particle size. At the same time, the sedimentation time constant decreases nonlinearly with increasing particle density, and for particles of larger diameter, the sedimentation time constant is less. However, in both cases, the sedimentation time constant is measured in days.
The obtained dependences were partially confirmed by the experimental data obtained at the A. N. Tupolev Kazan National Research Technical University at the Department of Electronic and Quantum Means of Information Transmission.
Conclusions
Based on the results of the work carried out, the following conclusions can be formulated. The task of plotting the dependence of the sedimentation time constant on the density of the liquid, the coefficient of its dynamic viscosity, the density and size of the precipitated particles was set and solved. The possibility of measuring the sedimentation time constant during the experiment allows one to unambiguously determine the density, and with it, the molecular weight of the particles, provided that their sizes are known. At the same time, the size of the particles can be determined, provided that their density is known. These data should be useful for experimenters to verify the conditions and results of experiments using the data of the mathematical model.
The practical significance of the work is to clarify the conditions for conducting experiments on long-term sedimentation. The proposed mathematical model makes it possible to reduce the time for conducting experiments to estimate the density of nanosized particles without waiting for their complete deposition.
Funding
I. I. Nureev was funded by Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075–03–2020–051 / 3, topic No. fzsu‑2020–0021) in part of physical task formulation, mathematical model construction. V. A. Kuklin was funded by Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075–03–2020–051, topic No. fzsu‑2020–0020) in part of data and results verification, training and tuning, and realization.
About Authors
Hussein Safaa Mohammed Ridha Hussein, University of Karbala, Karbala, 56001, Iraq, safaa_m333@yahoo.com.
ORCID:0000-0001-6022-0548
Kuklin Vladimir Alexanrovich, Kazan Federal University, 18 Kremlyovskaya street, Kazan 420008, Russian Federation, iamkvova@gmail.com.
Salakhutdinov Marat Shamilevich, Kazan National Research Technical University n. a. A. N. Tupolev, 10, K. Marx St., Kazan, Tatarstan 420111, Russia, alladin76@yandex.ru – corresponding author.
ORCID 0000-0001-5176-1334
Nureev Ilnur Ildarovich, Kazan National Research Technical University n. a. A. N. Tupolev, 10, K. Marx St., Kazan, Tatarstan 420111, Russia, n2i2@mail.ru.
Author Contributions
Conceptualization Nureev I. I. and Hussein S. M. R. H.; Formal analysis Nureev I. I., and Salakhutdinov M. Sh.; Funding acquisition Nureev I. I., Kuklin V. A.; Investigation Hussein S. M. R. H; Methodology, Kuklin V. A., Hussein S. M. R. H; Supervision Nureev I. I.; Writing-review & editing Salakhutdinov M. Sh, Hussein S. M. R. H. All authors have read and agreed to the published version of the manuscript.
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