rus
Issue #6/2020
P.G.Kudryavtsev
Modeling of the properties of individual colloidal silicon oxide nanoparticles
Modeling of the properties of individual colloidal silicon oxide nanoparticles
DOI: 10.22184/1993-8578.2020.13.6.372.382
This article describes interaction between dispersed phase particles in colloidal solution and, particularly, nanoparticles of silicon oxide contained in an electrolyte solution. The proposed theoretical analysis of this interaction develops the DLVO theory. In the frame of the theory it was assumed that silicon oxide nanoparticles in the electrolyte solution are the assembly of the charged particles but, at the same time, they are electrically neutral. It was assumed that charged particle sizes are much smaller in comparison with the distance between them. This paper attempts to describe interaction of colloidal particles in the framework of the point charges model interaction. We suggest a simple model describing interaction of two systems of electric charges placed at arbitrary distances from each other. The conducted analysis of the obtained potential dependencies versus "rigid" colloidal particles revealed presence of three singular points. These points are associated with a qualitative change in the type of particle interaction with each other, which makes it possible to distinguish four types of colloidal particles found in electrolyte solutions. The calculations indicate a possibility of ultrasoft colloidal systems existence with long-range forces of interaction of particles with each other. In such systems formation of long-range hyper structures, in other words, formation of aggregates without short-range interaction between particles is possible. A system of colloidal particles classification is proposed on the basis of analysis of obtained results. A formula that describes viscosity of a colloidal solution depending on the volume part of the disperse phase and the parameter characterizing the law of attraction in the interaction of colloidal particles in case of "rigid" colloidal systems was obtained.
This article describes interaction between dispersed phase particles in colloidal solution and, particularly, nanoparticles of silicon oxide contained in an electrolyte solution. The proposed theoretical analysis of this interaction develops the DLVO theory. In the frame of the theory it was assumed that silicon oxide nanoparticles in the electrolyte solution are the assembly of the charged particles but, at the same time, they are electrically neutral. It was assumed that charged particle sizes are much smaller in comparison with the distance between them. This paper attempts to describe interaction of colloidal particles in the framework of the point charges model interaction. We suggest a simple model describing interaction of two systems of electric charges placed at arbitrary distances from each other. The conducted analysis of the obtained potential dependencies versus "rigid" colloidal particles revealed presence of three singular points. These points are associated with a qualitative change in the type of particle interaction with each other, which makes it possible to distinguish four types of colloidal particles found in electrolyte solutions. The calculations indicate a possibility of ultrasoft colloidal systems existence with long-range forces of interaction of particles with each other. In such systems formation of long-range hyper structures, in other words, formation of aggregates without short-range interaction between particles is possible. A system of colloidal particles classification is proposed on the basis of analysis of obtained results. A formula that describes viscosity of a colloidal solution depending on the volume part of the disperse phase and the parameter characterizing the law of attraction in the interaction of colloidal particles in case of "rigid" colloidal systems was obtained.
Теги: colloidal solutions dlvo theory electrical double layer potential energy of interacting colloidal nanoparticles viscosity of dispersed systems теория длфо вязкость дисперсных систем двойной электрический слой коллоидные растворы потенциальная энергия взаимодействующих коллоидных наночастиц
INTRODUCTION
Silicon oxide nanoparticles in an electrolyte solution form an aggregate of charged particles but, like all micelles, they are electrically neutral particles. When describing the interaction of neutral systems of electric charges, the basic approach is an assumption that the sizes of charged particles are much smaller than a distance from each other [1]. This assumption is strict for dilute gases only.
A similar theory was developed in 1937–1943 by B.V. Deryagin and L.D. Landau in the USSR and E. Vervay and J.T. Overback in Holland independently of each other. Accordingly to first letters of authors’ surnames the theory is called DLVO [7–10].
According to this theory, colloidal particles in a solution, due to the Brownian motion, can freely move closer to each other until the moment when their diffuse shells or layers touch. In so doing, no interaction forces arise between them. For further convergence, the particles must deform their diffuse shells so that they overlap (or penetrate into each other). But liquids are poorly compressed and, in response to deformation, the so-called wedging pressure forces appear, which impedes implementation of this process. Moreover, the larger the size of the diffuse layer, the greater the force of the wedging pressure.
The Deryagin – Landau – Verwey – Overbeck (DLVO) theory is based on the assumption that, due to the thermodynamic instability of lyophobic sols, their aggregate stability can only be kinetic, and the steady state should be interpreted as a "frozen" state with a low coagulation rate. The reason for this stability is that in colloidal solutions, in contrast to ordinary molecular solutions, long-range forces are capable, under certain conditions, of creating a sufficiently high potential barrier, which sharply reduces the likelihood of particles approaching. Therefore, the DLVO theory assigns the most important place in solving the problem of the stability of a lyophobic sol to the analysis of force and potential curves obtained by the superposition of electrostatic repulsion and molecular attraction.
In this work, an attempt was made to describe the interaction of colloidal (within the framework of the interaction model) point charges. For this end, a simple model is proposed that describes interaction of two systems of electric charges located at arbitrary distances from each other.
We will consider colloidal particles as spherical, with a charge of the same sign uniformly distributed over the surface, while the charge of the opposite sign is considered to be concentrated in the center of the ion which forms a double electric layer. The particles are assumed to be neutral, and let the surface charge density (or electric potential) decrease exponentially.
We will consider the process of interaction of two colloidal particles according to Fig.1. The potential energy of particles interacting according to the Coulomb’s law is [1]:
, (1)
here ρi, ρj – charge density at points i and j of the interacting particles, ri,j is the distance between the interacting elements of the particle charges, ΔV is the volume element.
Consider, for simplicity, the one-dimensional model shown in Fig.2.
The potential energy for this case can be written as:
(2)
Since colloidal particles are considered neutral, it is possible to write:
. (3)
Dimensionless variables m and ε, according to Fig.2, are related by the condition:
1 – m = ε. (4)
Substituting e_ from (3) into (2) with account taken of (4) with the help of simple transformations we obtain:
(5)
(5)
here is the relative distance between particles.
The amount of surface charge can be defined as:
, (6)
where σ is surface charge density.
Figure 3 shows variants of the dependence of function (5) on x. Presence of the second "weak" minimum can be obtained by introducing the interaction of the colloidal particle with the medium. The parameters of the model under consideration are the value of the positive surface charge е+ and the dimensionless constant ε characterizing the colloidal particle and the charge distribution density in the region between two interacting particles. We called the value ε the rigidity of the given colloidal system. When ε → 0, the system is considered to be maximally "rigid", and when ε → 1 it is considered maximally "super soft".
Depending on the "rigidity" parameter of the colloidal particle, four regions are distinguished, characterized by a different nature of the interaction of colloidal particles. The first region A exists in the range 0 < ε <0.08425 and is characterized by complete repulsion of interacting particles. Colloidal solutions with such "rigidity" parameters are completely aggregation stable.
The second section B in the presented diagram refers to the classical version of colloidal solutions, characterized by presence of a potential barrier at their interaction. This potential barrier prevents their aggregation, but its overcoming leads to falling into the potential well and sticking of interacting particles. The potential barrier position changes depending on the "rigidity" parameter of the colloidal particle, and it changes in the range of the relative distance between particles 4.0606 < x < ∞. The position the potential well minimum also shifts and changes in the range 2 < x < 3.98493. This area exists in the range of changes in the "rigidity" of colloidal particles 0.08425 < ε < 0.111.
The third section, which is present in the calculated diagram, is characterized by the range of "rigidity" of colloidal particles 0.111 < ε < 0.7966. This range is characterized by complete attraction of colloidal particles. Colloidal solutions with such characteristics are completely aggregation unstable. The last range 0.7966 < ε < 1 is very interesting. In this range there is a potential barrier, which position changes in the range of the relative distance between particles 2.0741 < x < 7.665. This barrier prevents the aggregation of colloidal particles very strongly and stabilizes the entire dispersed system. However, in this case, such particles also have a small local minimum in the range of the relative distance between particles 7.7989 < x < ∞. The presence of this minimum indicates a possible long-range interaction of the forces of interaction between particles and, as a consequence, a possibility of the formation of long-range structures, that is, the formation of aggregates without short-range interaction between particles. A possibility of such long-range action was pointed out by E.Vervey and J.T.Overbeck in their work [11]. However, they saw such long-range action at much shorter distances.
The analysis of the obtained potential dependences on the "rigidity" of colloidal particles showed a presence of three singular points. These points are associated with a qualitative change in the type of interaction of particles with each other, which makes it possible to distinguish 4 types of colloidal particles found in the electrolyte solutions. The classification of colloidal particles proposed on the basis of the calculated data is presented in Table 1.
The results obtained can be used to describe the rheological behavior of the colloidal solution, namely, to understand the dependence of the colloidal solution viscosity on its concentration. In view of the complexity of the application of the obtained model to all types of the described colloidal systems, we chose only the variant of elastic colloidal particles to describe the viscosity of a colloidal solution.
Viscosity (η) of a colloidal solution is called the internal friction between its layers moving relative to each other, which reflects the resistance to flow under the influence of external forces. Viscosity results from intermolecular interaction, and it is the higher, the greater the forces of intermolecular attraction. The viscosity of a colloidal solution varies greatly compared to the solvent in which it is prepared. In addition, it additionally changes as a result of the presence of substances dissolved in it. Viscosity of most hydrophobic sols and suspensions at low concentrations differs little from viscosity of a pure solvent. As the concentration of the dispersed phase increases, the viscosity of the colloidal solution increases. This is due to the fact that the particles of the dispersed phase interact with each other and with the molecules of the dispersion medium. They themselves and their solvation shells provide hydrodynamic resistance to the layers of the moving liquid phase. As a result of the flow around the particles, the trajectory of the liquid phase flow curves. This effect is enhanced if the particles have a shape other than spherical, because they can rotate around their axes under the action of the flowing liquid phase.
Einstein established a dependence of the viscosity of a colloidal solution on the concentration of suspended particles. The following assumptions were made in creating this equation:
particles of the dispersed phase are distanced from each other;
particles of the dispersed phase have the same size and shape;
there are no interactions between the particles of the dispersed phase;
the particle size is much larger than the solvent molecules.
, (7)
here α – is a coefficient that depends on the shape of the particles. For round particles α = 2,5; for elongated particles α > 2,5. φ – is the volume fraction of the dispersed phase, η0 – is the dispersion medium viscosity.
Therefore, Einstein’s equation is applicable for sols and dilute suspensions in which particles of the dispersed phase do not interact with the dispersion medium (i.e., for lyophobic systems) [5].
As is known, the colloidal solution viscosity of a at values of the volume fraction of the solid phase φ < 0.25, taking into account the interparticle interaction, is described by the modified Einstein equation [6]:
, (8)
where the values of the coefficients are respectively equal to α = 2,5 and β = 10,05. Here β – is a coefficient that takes into account the effect of interparticle interaction in a colloidal solution.
A colloidal solution is a system of spherical colloidal particles of radius r, uniformly distributed in a dispersed medium [4]. In highly diluted solutions, we can assume that particles perform translational and orientational Brownian motions. The average time between two successive reorientations can be written in the form [2]:
, (9)
where τ0 is the half-period of rotational vibrations, U is the rotation activation barrier characterizing the potential energy of interaction of a colloidal particle with a dispersion medium and a dispersion phase, T is the temperature.
The half-period of rotational oscillations, in its turn, is equal to:
, (10)
where I is the moment of inertia of the particle.
For a spherical particle of mass m:
. (11)
According to [2], the time between two successive rotational reorientations of a Brownian particle can also be related to the coefficient of viscous friction:
. (12)
Equating (9) and (12), we obtain, taking into account (10), the formula (11):
, (13)
The viscous friction coefficient for spherical particles is related to the fluid viscosity η by the relation [2]:
. (14)
Substituting (13) into (14), we get:
. (15)
The activation barrier is:
, (16)
where U1 characterizes the potential energy of interaction of a colloidal particle with a dispersion medium, U2 is the potential energy of interaction of particles of a dispersed phase with each other. For highly diluted solutions U1>>U2, and it is possible to write:
, (17)
where R is the average distance between colloidal particles in solution, R0 is the equilibrium distance between colloidal particles during coagulation, i.e., it corresponds to the minimum on the dependence of the interaction potential of two particles on the distance between them; n is a positive integer characterizing the law of attraction of colloidal particles to each other.
Expanding the exponential term of expression (15) in a series in and, leaving the terms of the series up to the second order in x inclusive, we get:
(18)
After that we introduce the notation:
(19)
Assuming that φ = (r / R)3, we substitute (19) into (18) and obtain the final expression for the viscosity of the colloidal solution depending on the volume fraction of the dispersed phase – φ and on the parameter characterizing the law of attraction in the interaction of colloidal particles – n:
(20)
Hence, on assumption that, n = 3 we obtain: α = 2,5; β = 9,3.
This result is in satisfactory agreement with the conclusions made in [3].
CONCLUSIONS
This article is devoted to description of the interaction of dispersed phase nanoparticles in a colloidal solution and, in particular, silicon oxide nanoparticles in an electrolyte solution. The proposed theoretical consideration of this interaction is a development of the DLVO theory.
When creating the model, it was assumed that silicon oxide nanoparticles in an electrolyte solution present a collection of charged nanoparticles, but at the same time they are electrically neutral nanoparticles. When describing interaction of these systems, the approximation is used that the size of charged nanoparticles is small compared to the distance between them. In this work, an attempt is made to describe interaction of colloidal nanoparticles within the framework of the model of interaction of point charges. For this end, a simple model is proposed that describes interaction of two systems of electric charges located at arbitrary distances from each other. Analysis of the obtained potential dependences on the "elasticity" of colloidal nanoparticles showed a presence of three singular points. These moments are associated with a qualitative change in the type of interaction of nanoparticles with each other, which makes it possible to distinguish between 4 types of colloidal nanoparticles in electrolyte solutions.
Calculations indicate a possibility of existence of super soft colloidal systems with long-range interaction forces between nanoparticles. In such systems, the formation of long-range hyperstructures is possible, that is, the formation of aggregates without short-range interaction between nanoparticles. Based on the analysis of the obtained results, a classification system for colloidal nanoparticles is proposed. For rigid colloidal systems, an expression is obtained for the viscosity of a colloidal solution depending on the volume fraction of the dispersed phase and a parameter characterizing the law of attraction at interaction of colloidal nanoparticles. ■
The author expresses his gratitude to Mr. Omer Yagel, VP Business Development and Training DigiSec Ltd., the official representative of Maplesoft, a division of Waterloo Maple Inc. in Israel, for the opportunity to use the software product of this company – the program of analytical calculations Maple.
Silicon oxide nanoparticles in an electrolyte solution form an aggregate of charged particles but, like all micelles, they are electrically neutral particles. When describing the interaction of neutral systems of electric charges, the basic approach is an assumption that the sizes of charged particles are much smaller than a distance from each other [1]. This assumption is strict for dilute gases only.
A similar theory was developed in 1937–1943 by B.V. Deryagin and L.D. Landau in the USSR and E. Vervay and J.T. Overback in Holland independently of each other. Accordingly to first letters of authors’ surnames the theory is called DLVO [7–10].
According to this theory, colloidal particles in a solution, due to the Brownian motion, can freely move closer to each other until the moment when their diffuse shells or layers touch. In so doing, no interaction forces arise between them. For further convergence, the particles must deform their diffuse shells so that they overlap (or penetrate into each other). But liquids are poorly compressed and, in response to deformation, the so-called wedging pressure forces appear, which impedes implementation of this process. Moreover, the larger the size of the diffuse layer, the greater the force of the wedging pressure.
The Deryagin – Landau – Verwey – Overbeck (DLVO) theory is based on the assumption that, due to the thermodynamic instability of lyophobic sols, their aggregate stability can only be kinetic, and the steady state should be interpreted as a "frozen" state with a low coagulation rate. The reason for this stability is that in colloidal solutions, in contrast to ordinary molecular solutions, long-range forces are capable, under certain conditions, of creating a sufficiently high potential barrier, which sharply reduces the likelihood of particles approaching. Therefore, the DLVO theory assigns the most important place in solving the problem of the stability of a lyophobic sol to the analysis of force and potential curves obtained by the superposition of electrostatic repulsion and molecular attraction.
In this work, an attempt was made to describe the interaction of colloidal (within the framework of the interaction model) point charges. For this end, a simple model is proposed that describes interaction of two systems of electric charges located at arbitrary distances from each other.
We will consider colloidal particles as spherical, with a charge of the same sign uniformly distributed over the surface, while the charge of the opposite sign is considered to be concentrated in the center of the ion which forms a double electric layer. The particles are assumed to be neutral, and let the surface charge density (or electric potential) decrease exponentially.
We will consider the process of interaction of two colloidal particles according to Fig.1. The potential energy of particles interacting according to the Coulomb’s law is [1]:
, (1)
here ρi, ρj – charge density at points i and j of the interacting particles, ri,j is the distance between the interacting elements of the particle charges, ΔV is the volume element.
Consider, for simplicity, the one-dimensional model shown in Fig.2.
The potential energy for this case can be written as:
(2)
Since colloidal particles are considered neutral, it is possible to write:
. (3)
Dimensionless variables m and ε, according to Fig.2, are related by the condition:
1 – m = ε. (4)
Substituting e_ from (3) into (2) with account taken of (4) with the help of simple transformations we obtain:
(5)
(5)
here is the relative distance between particles.
The amount of surface charge can be defined as:
, (6)
where σ is surface charge density.
Figure 3 shows variants of the dependence of function (5) on x. Presence of the second "weak" minimum can be obtained by introducing the interaction of the colloidal particle with the medium. The parameters of the model under consideration are the value of the positive surface charge е+ and the dimensionless constant ε characterizing the colloidal particle and the charge distribution density in the region between two interacting particles. We called the value ε the rigidity of the given colloidal system. When ε → 0, the system is considered to be maximally "rigid", and when ε → 1 it is considered maximally "super soft".
Depending on the "rigidity" parameter of the colloidal particle, four regions are distinguished, characterized by a different nature of the interaction of colloidal particles. The first region A exists in the range 0 < ε <0.08425 and is characterized by complete repulsion of interacting particles. Colloidal solutions with such "rigidity" parameters are completely aggregation stable.
The second section B in the presented diagram refers to the classical version of colloidal solutions, characterized by presence of a potential barrier at their interaction. This potential barrier prevents their aggregation, but its overcoming leads to falling into the potential well and sticking of interacting particles. The potential barrier position changes depending on the "rigidity" parameter of the colloidal particle, and it changes in the range of the relative distance between particles 4.0606 < x < ∞. The position the potential well minimum also shifts and changes in the range 2 < x < 3.98493. This area exists in the range of changes in the "rigidity" of colloidal particles 0.08425 < ε < 0.111.
The third section, which is present in the calculated diagram, is characterized by the range of "rigidity" of colloidal particles 0.111 < ε < 0.7966. This range is characterized by complete attraction of colloidal particles. Colloidal solutions with such characteristics are completely aggregation unstable. The last range 0.7966 < ε < 1 is very interesting. In this range there is a potential barrier, which position changes in the range of the relative distance between particles 2.0741 < x < 7.665. This barrier prevents the aggregation of colloidal particles very strongly and stabilizes the entire dispersed system. However, in this case, such particles also have a small local minimum in the range of the relative distance between particles 7.7989 < x < ∞. The presence of this minimum indicates a possible long-range interaction of the forces of interaction between particles and, as a consequence, a possibility of the formation of long-range structures, that is, the formation of aggregates without short-range interaction between particles. A possibility of such long-range action was pointed out by E.Vervey and J.T.Overbeck in their work [11]. However, they saw such long-range action at much shorter distances.
The analysis of the obtained potential dependences on the "rigidity" of colloidal particles showed a presence of three singular points. These points are associated with a qualitative change in the type of interaction of particles with each other, which makes it possible to distinguish 4 types of colloidal particles found in the electrolyte solutions. The classification of colloidal particles proposed on the basis of the calculated data is presented in Table 1.
The results obtained can be used to describe the rheological behavior of the colloidal solution, namely, to understand the dependence of the colloidal solution viscosity on its concentration. In view of the complexity of the application of the obtained model to all types of the described colloidal systems, we chose only the variant of elastic colloidal particles to describe the viscosity of a colloidal solution.
Viscosity (η) of a colloidal solution is called the internal friction between its layers moving relative to each other, which reflects the resistance to flow under the influence of external forces. Viscosity results from intermolecular interaction, and it is the higher, the greater the forces of intermolecular attraction. The viscosity of a colloidal solution varies greatly compared to the solvent in which it is prepared. In addition, it additionally changes as a result of the presence of substances dissolved in it. Viscosity of most hydrophobic sols and suspensions at low concentrations differs little from viscosity of a pure solvent. As the concentration of the dispersed phase increases, the viscosity of the colloidal solution increases. This is due to the fact that the particles of the dispersed phase interact with each other and with the molecules of the dispersion medium. They themselves and their solvation shells provide hydrodynamic resistance to the layers of the moving liquid phase. As a result of the flow around the particles, the trajectory of the liquid phase flow curves. This effect is enhanced if the particles have a shape other than spherical, because they can rotate around their axes under the action of the flowing liquid phase.
Einstein established a dependence of the viscosity of a colloidal solution on the concentration of suspended particles. The following assumptions were made in creating this equation:
particles of the dispersed phase are distanced from each other;
particles of the dispersed phase have the same size and shape;
there are no interactions between the particles of the dispersed phase;
the particle size is much larger than the solvent molecules.
, (7)
here α – is a coefficient that depends on the shape of the particles. For round particles α = 2,5; for elongated particles α > 2,5. φ – is the volume fraction of the dispersed phase, η0 – is the dispersion medium viscosity.
Therefore, Einstein’s equation is applicable for sols and dilute suspensions in which particles of the dispersed phase do not interact with the dispersion medium (i.e., for lyophobic systems) [5].
As is known, the colloidal solution viscosity of a at values of the volume fraction of the solid phase φ < 0.25, taking into account the interparticle interaction, is described by the modified Einstein equation [6]:
, (8)
where the values of the coefficients are respectively equal to α = 2,5 and β = 10,05. Here β – is a coefficient that takes into account the effect of interparticle interaction in a colloidal solution.
A colloidal solution is a system of spherical colloidal particles of radius r, uniformly distributed in a dispersed medium [4]. In highly diluted solutions, we can assume that particles perform translational and orientational Brownian motions. The average time between two successive reorientations can be written in the form [2]:
, (9)
where τ0 is the half-period of rotational vibrations, U is the rotation activation barrier characterizing the potential energy of interaction of a colloidal particle with a dispersion medium and a dispersion phase, T is the temperature.
The half-period of rotational oscillations, in its turn, is equal to:
, (10)
where I is the moment of inertia of the particle.
For a spherical particle of mass m:
. (11)
According to [2], the time between two successive rotational reorientations of a Brownian particle can also be related to the coefficient of viscous friction:
. (12)
Equating (9) and (12), we obtain, taking into account (10), the formula (11):
, (13)
The viscous friction coefficient for spherical particles is related to the fluid viscosity η by the relation [2]:
. (14)
Substituting (13) into (14), we get:
. (15)
The activation barrier is:
, (16)
where U1 characterizes the potential energy of interaction of a colloidal particle with a dispersion medium, U2 is the potential energy of interaction of particles of a dispersed phase with each other. For highly diluted solutions U1>>U2, and it is possible to write:
, (17)
where R is the average distance between colloidal particles in solution, R0 is the equilibrium distance between colloidal particles during coagulation, i.e., it corresponds to the minimum on the dependence of the interaction potential of two particles on the distance between them; n is a positive integer characterizing the law of attraction of colloidal particles to each other.
Expanding the exponential term of expression (15) in a series in and, leaving the terms of the series up to the second order in x inclusive, we get:
(18)
After that we introduce the notation:
(19)
Assuming that φ = (r / R)3, we substitute (19) into (18) and obtain the final expression for the viscosity of the colloidal solution depending on the volume fraction of the dispersed phase – φ and on the parameter characterizing the law of attraction in the interaction of colloidal particles – n:
(20)
Hence, on assumption that, n = 3 we obtain: α = 2,5; β = 9,3.
This result is in satisfactory agreement with the conclusions made in [3].
CONCLUSIONS
This article is devoted to description of the interaction of dispersed phase nanoparticles in a colloidal solution and, in particular, silicon oxide nanoparticles in an electrolyte solution. The proposed theoretical consideration of this interaction is a development of the DLVO theory.
When creating the model, it was assumed that silicon oxide nanoparticles in an electrolyte solution present a collection of charged nanoparticles, but at the same time they are electrically neutral nanoparticles. When describing interaction of these systems, the approximation is used that the size of charged nanoparticles is small compared to the distance between them. In this work, an attempt is made to describe interaction of colloidal nanoparticles within the framework of the model of interaction of point charges. For this end, a simple model is proposed that describes interaction of two systems of electric charges located at arbitrary distances from each other. Analysis of the obtained potential dependences on the "elasticity" of colloidal nanoparticles showed a presence of three singular points. These moments are associated with a qualitative change in the type of interaction of nanoparticles with each other, which makes it possible to distinguish between 4 types of colloidal nanoparticles in electrolyte solutions.
Calculations indicate a possibility of existence of super soft colloidal systems with long-range interaction forces between nanoparticles. In such systems, the formation of long-range hyperstructures is possible, that is, the formation of aggregates without short-range interaction between nanoparticles. Based on the analysis of the obtained results, a classification system for colloidal nanoparticles is proposed. For rigid colloidal systems, an expression is obtained for the viscosity of a colloidal solution depending on the volume fraction of the dispersed phase and a parameter characterizing the law of attraction at interaction of colloidal nanoparticles. ■
The author expresses his gratitude to Mr. Omer Yagel, VP Business Development and Training DigiSec Ltd., the official representative of Maplesoft, a division of Waterloo Maple Inc. in Israel, for the opportunity to use the software product of this company – the program of analytical calculations Maple.
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