On the thermodynamic theory of colloidal suspensions

Physica A xxx (xxxx) xxx

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On the thermodynamic theory of colloidal suspensions ∗

I.N. Cherepanov a , , P.V. Krauzin b a b

Perm State University, Department of Radio Electronics and Information Security, Russia Perm State University, Department of Physics of Phase Transitions, Russia

article

info

Article history: Received 4 February 2019 Received in revised form 31 May 2019 Available online xxxx

a b s t r a c t Colloidal suspensions can be considered as continuous media, like molecular solutions, in the main area of theoretical research. However, they have a number of features that distinguish them from true solutions and coarse suspensions. The paper considers the thermodynamic theory of weak solutions as applied to colloidal suspensions that are in a gravity field. This problem is not clear because the colloidal suspensions are multi component systems, and, strictly, the mutual influence of the suspended particles and the liquid carrier molecules should be taken into account in thermodynamic models. It is shown that the distribution of nanoparticles by height is determined by the density of the mixture as a whole, and not by the density of the dispersion medium. Equations describing the equilibrium distribution of nanoparticles in a colloidal solution have been obtained. Investigation of the dependence of the density on the impurity concentration in a weak solution has shown that if the additivity properties of the component volumes are preserved, then the quadratic coefficient in the Gibbs energy expansion does not depend on pressure. It is shown that the theory of a weak solution gives an equation for the equilibrium distribution of the dispersed phase, which has almost identical solutions for a large range of parameters of a problem with a well-known in the literature equation obtained in the framework of the hard sphere gas model. However, the individual contributions to the thermodynamic potential of the solution (specialized by the field of gravity and interparticle interaction) in the models under consideration are different. These differences can occur in non-equilibrium processes or in the presence of additional flows of a substance (magnetophoresis, thermal diffusion, etc.). © 2019 Elsevier B.V. All rights reserved.

1. Introduction Colloidal suspensions occupy an intermediate position between true (molecular) solutions and coarse suspensions. They exhibit both the properties of solutions (diffusion, thermodiffusion) and coarse suspensions (sedimentation, light scattering, particle aggregation, and polydispersity, magnetophoresis, and others). The nanometer size of the particles allows them to participate in the thermal motion of the carrier medium molecules, which leads to manifestation of special properties of colloids. Thus, solid particles of an impurity, whose density is five times greater than the density of the carrier medium, do not settle out, and can always be maintained in a suspended state. However, due to gravitational stratification, the height distribution of particles will be non-uniform. The kinetics of the process of stratification of an impurity in a magnetic fluid was experimentally studied in [1], and in active colloidal suspensions in [2]. Measurements of the concentration profile showed that the gravitational separation ∗ Corresponding author. E-mail addresses: [email protected] (I.N. Cherepanov), [email protected] (P.V. Krauzin). https://doi.org/10.1016/j.physa.2019.123247 0378-4371/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: I.N. Cherepanov and P.V. Krauzin, On the thermodynamic theory of colloidal suspensions, Physica A (2019) 123247, https://doi.org/10.1016/j.physa.2019.123247.

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in a ferrocolloid occurs under the influence on slow molecular processes, rather than fast hydrodynamic processes typical of coarse suspensions. A theoretical study of the sedimentation process was carried out in [3]. There, exact solutions of the diffusion equation with account of the gravitational stratification were obtained, as well as the typical time of the equilibrium distribution establishment was determined. Note that in [3] the constancy of the diffusion coefficient was assumed, which is valid only for highly diluted solutions. Another manifestation of nanofluids’ special properties is the concentration dependence of viscosity. Einstein’s law [4, 5], which matches the viscosity of suspensions well, is not applicable even to weakly concentrated colloidal solutions. Viscosity in nanofluids can behave in a more complicated way [6]. The occurrence of such effects seems to be related to the following. When reducing the size of dispersed particles, their number increases in a volume unit (with a fixed volume fraction ϕ ), while the average distance between the particles decreases. To estimate the number of particles N in some volume V will assume that they have the same size and spherical shape, in this case N =

3V ϕ 4π r 3

.

(1)

R For colloid Ludox⃝ (silica particles in water) with particle radius r = 20 nm get order N = 1016 particles of impurity on one mole of water. Average distance between particles L is rated as:

( )1/3 L=

V

N

( =

4π 3ϕ

)1/3

r.

(2)

For example, at the concentration of ϕ = 0.1 the distance between the particles is about two of their diameters, which should lead to a significant interparticle interaction both between impurity particles and between nanoparticles and molecules of the carrier medium. Due to the fact that the internal processes of concentration inhomogeneity formation, namely gravitational separation, thermodiffusion, magnetophoresis, proceed very slowly, they are often neglected, and the colloid is considered as a homogeneous liquid. However, in a number of experimental works [7–9] the effects that do not occur in homogeneous liquids and may be caused by an inhomogeneous impurity distribution have been found. The effect of small concentration inhomogeneities with a constant diffusion coefficient on convective flows in colloidal suspensions was studied theoretically in [10–12]. A theoretical study of solid particles diffusion was carried out in [13]. In this paper, it is assumed that the additional contribution to the Gibbs energy is equivalent to the interaction energy of particles in the vacuum. Thus, it can be calculated for hard sphere gas. This theory was extended on case of magnetophoresis and polydispersity particle in works [14,15]. The model of hard spheres is also used to describe the magnetic properties of ferrofluids [1,16]. But this model does not take into account the interaction of particles with molecules of the carrier liquid in an explicit form. In numerical calculations using molecular dynamics methods, the carrier molecules are not taken into account too. One possible approach is to develop the thermodynamics of the system of suspended particles in some effective medium, which provides the constant temperature and the Brownian motion of particles. In the framework of such approach the presence of liquid carrier is included in the peculiarities of interparticle interaction, and the thermodynamics is developed in NVT ensemble. Here the free energy of the suspended particle system is calculated using some method of statistical mechanics. A special case of a colloidal suspensions are the magnetic colloids, and for this colloid the NVT approach is very useful and results in theoretical description of equilibrium magnetic response [1,16–18]. In this paper, a slightly different approach is considered, particularly, NPT ensemble is used providing the isothermal– isobaric conditions. The most general type of Gibbs energy is expanded (without specifying the interparticle interaction forces) in powers of the numerical impurity concentration. Accounting for the second coefficient in the Gibbs energy expansion leads to the distribution of particles close to the Carnahan–Starling models. Note that this does not take into account either the form or the specific forces of the interparticle interactions. 2. The density of the colloidal suspension Consider a colloidal system from a thermodynamic point of view as a solution of two components. According to [19] the Gibbs energy for a weak solution with an accuracy of a square term by number concentration has the form

Φ = Ns µ(0) s (P , T ) + Np T ln

Np eNs

+ Np µ(0) p (P , T ) +

Np2 2Ns

β (P , T ),

(3)

where the values with the index p (particle) refer to particles of an impurity, and with the index s (solvent) — to the solvent, Ns is the number of solvent molecules, Np is the number of particles of impurity, T is the temperature in (0) (0) energy units, µs is the chemical potential of pure solvent, µp is the chemical potential of single impurity particles, β is the second expansion coefficient in powers of the relative proportion of impurity particles, e is the base of the natural logarithm. The chemical potentials of the components in solution are determined by the expressions:

µs =

∂Φ n2 = µ(0) , s − Tn − β ∂ Ns 2

(4)

Please cite this article as: I.N. Cherepanov and P.V. Krauzin, On the thermodynamic theory of colloidal suspensions, Physica A (2019) 123247, https://doi.org/10.1016/j.physa.2019.123247.

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∂Φ = µ(0) p + T ln n + β n. ∂ Np

µp =

3

(5)

where n = Np /Ns is a relative proportion of impurity particles. We assume that the properties of solvent substances and impurities weakly depend on pressure, which is typical for (0) (0) liquids and solids. Then the derivatives of chemical potentials µs , µp in pressure are constant values, and have the meaning of volumes per molecule in a pure substance:

∂µ(0) s , ∂P

vs(0) ≡

vp(0) ≡

∂µ(0) p . ∂P

(6)

The volume of the entire solution is determined by the expression:

( ) ( ) ∂Φ 1 n2 ∂β n ∂β (0) (0) V = = Ns vs 1 − + Np vp 1 + (0) . ∂P 2 vs(0) ∂ P vp ∂ P

(7)

Herewith, the volume per one component of the solution component is determined as a derivative of the corresponding chemical potential with respect to pressure:

( ) ∂µs 1 n2 ∂β (0) vs = = vs 1 − , ∂P 2 vs(0) ∂ P ( ) n ∂β ∂µp = vp(0) 1 + (0) . vp = ∂P vp ∂ P

(8)

(9)

∂β

It can be seen that when ∂ P ̸ = 0 the additivity property of the volume for a solution is violated: the volume of the system is not equal to the sum of the volumes of the components before mixing. This effect is well known for molecular solutions (for example, ethanol and water are incompressible, but after mixing, the volume of the system is less than the sum of the volumes of the individual components) and is completely absent in coarse suspensions with a large size of impurity particles. The value of this parameter for colloidal suspensions is an open question. Preliminary ∂β R experimental measurements of the density of a Ludox⃝ colloidal suspension have shown that the coefficient ∂ P is close to zero. However, for fine solutions, it can have significant values. Consider the dependence of density on the volume ϕ and mass C fraction of nanoparticles. Np vp

ϕ=

V

=

)−1 ( vs 1 , = 1+ Np vp + Ns vs vp n Np vp

Np mp

C =

Np mp + Ns ms

.

(10) (11)

The relative number of impurity particles can be expressed as ms C vs ϕ = . vp 1 − ϕ mp 1 − C

n= With

∂β ∂P

ρ= ρ= With

(12)

= 0 the mixture density, expressed through volumetric and mass concentrations, has the form Np mp + Ns ms Np vp + Ns vs

(

C

ρp

+

1−C

ρs

= ϕρp + (1 − ϕ )ρs ,

)−1

.

(13) (14)

∂β ∂P

̸= 0 the dependence of the mixture density on the mass concentration is determined by the expression: ( ) )−1 ( )−1 ( (0) C 1−C C q C ρs 1−C ρ= + = 1+ + (0) , mp /vp ms /vs 2 1 − C ρp(0) ρp(0) ρs (0)

(15)

(0)

where ρs , ρp are the densities of the pure solvent and the substance of the impurity, and the designation is introduced

vs(0) ∂β . )2 ∂P vp(0)

q≡ (

(16)

Since the volume per mixture particle depends on n (see Eqs. (8), (9)), the explicit expression of the density by the volume fraction of the impurity is rather complicated. Using the condition of smallness n, we obtain the dependence of the number Please cite this article as: I.N. Cherepanov and P.V. Krauzin, On the thermodynamic theory of colloidal suspensions, Physica A (2019) 123247, https://doi.org/10.1016/j.physa.2019.123247.

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of particles on the volume fraction n + O(n2 ) =

vs(0)

ϕ . vp(0) 1 − ϕ + qϕ

(17)

Then the dependence of the density of the system on the volume fraction will be

( ρ ≈ 1−

qϕ

)

1 − ϕ + qϕ

ρp(0) ϕ + ρs(0) (1 − ϕ ).

(18)

From Eqs. (15), (18) it is clear that the presence of an impurity leads to a decrease in the effective density of the solution. 3. Equations of the dispersed phase equilibrium distribution Consider the change in concentration with height for a solution in a gravity field. The presence of a gravity field leads to the addition of the potential energy of the mixture particles [19]. To begin with, we consider a linear expansion in the number of particles (β = 0). The chemical potentials of the components of the solution will take the form:

µs = µ(0) s − Tn + ms gz ,

(19)

µp = µ

(20)

(0) p

+ T ln n + mp gz .

where z is the vertical coordinate. In the state of thermodynamic equilibrium, the chemical potentials and the temperature of the system are constant. Let us differentiate the resulting expressions by height:

∂µs ∂P ∂n = vs(0) −T + ms g = 0, ∂z ∂z ∂z ∂P 1 ∂n ∂µp = vp(0) +T + mp g = 0. ∂z ∂z n ∂z

(21) (22)

In [19] the addendum was supposed to be small T ∂∂ nz . In this case, the pressure gradient is ∂∂Pz = −ρs g, and the concentration of particles in height is described by the classical barometric distribution, with account the Archimedes force acting on the impurity particles ( z) , (23) n = n0 exp − l where l ≡ (ρ −ρT )v g is sedimentation length. More generally (when T ∂∂ nz ̸ = 0) the pressure gradient, as follows from (22), p s p (22) is described by the expression Np mp + Ns ms ∂P = g = −ρ g . ∂z Np vp + Ns vs

(24)

Note that the pressure gradient is determined by ρ — the density of the mixture as a whole (13), and not by the density of the dispersion medium ρs . The Eq. (24) is reduced to (23) under the following conditions ms vs n ≪ 1, n ≪ 1. (25) mp vp In colloidal suspension, the impurity particles are much larger and heavier than the molecules of the carrier medium, and these requirements may be violated. For solids, the volume ratio

γ ≡

vp(0)

(26)

vs(0)

can be estimated with the formula

γ =

π d3 ρs 6 Ms

NA ,

(27)

where d is the diameter of impurity particles, Ms is the molar mass of the solvent, NA is the Avogadro constant. The values γ are significantly different for colloids of different nature. We give an estimate of this parameter for two colloidal R suspensions: a non-magnetic colloid Ludox⃝ with a particle size of d = 22 nm [8] (denoted as γL ), a kerosene based magnetic fluid with a particle size of d = 10 nm [15] (denoted as γFC )

γL = 1.8 × 105 ,

γFC = 1.5 × 104 .

(28)

Even with a small relative number of impurity particles n in considered media, condition (25) may be violated due to a large difference in the mass and volume of the particles of the impurity and the carrier liquid. Therefore, to calculate the Please cite this article as: I.N. Cherepanov and P.V. Krauzin, On the thermodynamic theory of colloidal suspensions, Physica A (2019) 123247, https://doi.org/10.1016/j.physa.2019.123247.

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particle height distribution, it is necessary to use a full equation for pressure (24). On the other hand, this allows us to consider large volume concentrations of impurities: the condition of smallness n ≪ 1 in terms of volume fraction

ϕ ≪ γ (1 − ϕ )

(29)

true up to ϕ = 1. We substitute this expression into the equation for the gradient of the chemical potential (22) and obtain 1 ∂n

vp

+ (ρp − ρ )g = 0. (30) n ∂z T For a more convenient analysis, let us move on from the relative number of particles to the volume concentration, and considering that ρp − ρ = (1 − ϕ )(ρp − ρs ), the equation for the gradient of chemical potential take the form 1

∂ϕ

1 − ϕ ∂z

1

+ ϕ (1 − ϕ ) = 0.

(31)

l

We note a number of features of the solution obtained. Sedimentation term is different from the classical expression by an additional factor (1 − ϕ ), which is caused by considering the addend T ∂∂ nz in Eq. (22). This factor cannot be obtained in the model of a single-component hard sphere gas in vacuum. This equation is valid only under the condition that the number of impurity particles is much smaller than the number of solvent molecules n ≪ 1. However, due to a large value γ the volume fraction of particles can be quite large. However, there are no physically incorrect decisions when ϕ > 1. Consider the influence of higher order members. Taking into account quadratic in the number of particles, the terms of the Gibbs energy of the solution (β ̸ = 0) chemical potentials of the components will take the form:

( )2 β ϕ ϕ − + ms gz , 1−ϕ 2 1−ϕ ϕ ϕ µp = µ(0) +β + mp gz . p + T ln 1−ϕ 1−ϕ µs = µ(0) s −T

(32) (33)

The equation for the pressure gradient (24) is preserved, and the equilibrium condition for impurity particles will be written as 1 + ϕ (p − 1)

{ ( )} 1 qϕ ∂ϕ ϕ 1 ρp − ρ 1 + + × = 0, 1 − ϕ (1 − q/2) 1 − ϕ ∂ z l ρp − ρs 2 (1 − ϕ ) 1

∂µp ∂z

=0 (34)

where the following designation is introduced p≡

β . γT

(35)

If we assume that the interaction energy of particles weakly depends on pressure (q = 0), the equation takes the form 1 + ϕ (p − 1) ∂ϕ (1 − ϕ )2

∂z

+

ϕ l

(1 − ϕ ) = 0.

(36)

The parameter p can be determined from the maximum concentration of the impurity nc . According to [19], the increment of the thermodynamic potential of the solution due to the transition δ Np of particles from the pure substance into the solution has the form

δ Φ = δ Np (µp − µ′p ),

(37)

where µp is the chemical potential of a pure solute. The maximum concentration is determined from the ratio δ Φ = 0. ′

(0)

Assuming that the chemical potential of the solid particle outside the solvent is equal to µ′p = µp , and substituting decomposition (5) into condition (37), we obtain T ln nc + nc β = 0.

(38)

We turn to volume concentration, using (12) p=

1 − ϕc

ϕc

(

ln γ − ln

ϕc 1 − ϕc

)

.

(39)

Consider the behavior of the parameter p or two values of ultimate concentration: the closest arrangement of hard spheres ϕc = 0.74, and random close arrangement ϕc = 0.60. The parameter dependency p on the ratio of the volume of impurity particles and carrier liquid γ is shown in Fig. 1. The vertical lines correspond to the above estimates γL , γFC . Please cite this article as: I.N. Cherepanov and P.V. Krauzin, On the thermodynamic theory of colloidal suspensions, Physica A (2019) 123247, https://doi.org/10.1016/j.physa.2019.123247.

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Fig. 1. Parameter dependency p on the ratio of volumes γ .

Parameter p (due to its logarithmic dependence on the ratio of γ ) in the area under consideration varies slightly and for the random close arrangement is equal to p ≈ 7. An alternative approach to obtaining the chemical potential of a colloidal suspension was proposed in [13]. Instead of the Gibbs energy expansion by the number of impurity particles, the interparticle interaction has been estimated by adding the contribution determined with the configuration integral. The equation for the equilibrium concentration distribution in the gravity field in the Carnahan–Starling approximation, obtained in [20], has the form

( 1+

2ϕ (4 − ϕ ) (1 − ϕ )4

)

∂ϕ 1 + ϕ = 0. ∂z l

(40)

4. Comparative analysis of models Compare the equilibrium concentration distributions in various models: barometric distribution (23), ideal solution (31), weakly nonideal solution (36) with p = 7, and Carnahan–Starling approximation (40). The equations were solved numerically. The distributions of the volume fraction by height for the models under consideration are shown in Fig. 2. In a large range of task parameters (h/l and the average volume fraction ϕ¯ ) the Eqs. (36), (40) give almost identical results. However, Eq. (36) is derived from the most general considerations, without taking into account the specific interparticle interaction function. Thus, various interparticle interactions, including those of non-steric nature, should lead only to a change in the parameters p, q. In practice, of interest are liquids with a volume fraction of the solid phase of approximately 10%–20%, therefore, we consider the expansion of equations accurate within square terms. Expansion of Eq. (40) takes the form

(1 + 8ϕ)

1 ∂ϕ + ϕ = 0, ∂z l

(41)

and of Eq. (36)

(1 + (p + 1)ϕ)

∂ϕ 1 + ϕ (1 − ϕ ) = 0. ∂z l

(42)

To compare the solutions, we consider the standard deviation of the particle distribution profile

√ ε=

1 h

h

∫ 0

(

ϕ − ϕCS ϕ¯

)2

dz ,

(43)

where ϕCS is the solution of Eq. (40). The dependence of ε on ϕ¯ is shown in Fig. 3. From this it follows that with an average volume concentration ϕ¯ = 0.2 Eq. (42) can be used even for a fairly small sedimentation length. 5. Conclusion In this paper, on the basis of the theory of weak solutions, the equations that describe the equilibrium distributions of nanoparticles in a colloidal suspension in a gravity field have been obtained. It has been shown that the consistent application of the theory of weak solutions without additional approximation leads to the dependence of the sedimentation flow on the local density of the mixture, and not on the density of the dispersion medium, which is reflected in Eq. (30). Please cite this article as: I.N. Cherepanov and P.V. Krauzin, On the thermodynamic theory of colloidal suspensions, Physica A (2019) 123247, https://doi.org/10.1016/j.physa.2019.123247.

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Fig. 2. The dependence of the volume concentration on the height for some values ϕ¯ and relationships h/l: (a) ϕ¯ = 0.1, h/l = 1, (b) ϕ¯ = 0.1, h/l = 10, (c) ϕ¯ = 0.2, h/l = 10. The solid line is the barometric distribution, the dashed line is the approximation of the ideal solution (p = 0), the dash-dotted line is the approximation of a weakly nonideal solution with p = 7, the dotted line is Carnahan–Starling approximation.

It has been established that taking into account quadratic in concentration terms in the Gibbs energy for solutions, in the general case leads to a violation of the additivity property of volumes (the volume of the mixture is not equal to the sum of the volumes of the components before mixing). The accurate satisfaction of the additivity property for a particular solution leads to the absence of a dependence on the pressure of the quadratic in concentration contribution to the Gibbs energy. In this case, the equation of equilibrium distribution of particles contains only one additional (in relation to the ideal solution) parameter p, determined by the ratio of the volume of particle to the volume of one molecule of the dispersion medium, as well as the maximum concentration of the dispersed phase. For colloidal suspensions the characteristic value is p ≈ 7. Acknowledgment The studies were carried out with the financial support of the Russian Foundation for Basic Research (No. 16-31-60074). Please cite this article as: I.N. Cherepanov and P.V. Krauzin, On the thermodynamic theory of colloidal suspensions, Physica A (2019) 123247, https://doi.org/10.1016/j.physa.2019.123247.

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Fig. 3. Dependence of the standard deviation ε on the average concentration ϕ¯ for l/h = 0.5: the dashed line is the solution approximation for p = 7, the dash-dotted line is the approximation (42), the solid line is (41), the dotted line is a barometric distribution (23).

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Please cite this article as: I.N. Cherepanov and P.V. Krauzin, On the thermodynamic theory of colloidal suspensions, Physica A (2019) 123247, https://doi.org/10.1016/j.physa.2019.123247.