Open access peer-reviewed chapter

A Colloidal Self-Organization of Impurities in a Liquid by Density Fluctuations

By Alexander Shimkevich

Submitted: April 12th 2017Reviewed: July 24th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.70459

Downloaded: 133

Abstract

The two-parametric functional for weakly interacting fluctuations of liquid density and composition is studied within the theory based on Landau potential for these fluctuations in the kind of ensemble of phonons and compound clusters. Using the standard diagram technique, the task for weak-interacting phonons and clusters is reduced to solving the equations of proper-energetic functions of quasi-particle interaction by Neumann iterations of Feynman diagrams in “bootstrapping” of Fourier images (propagators) for correlation of the composition of liquid and its topological structure. It is shown that composition fluctuations as clusters are induced by phonons when impurity atoms being initially outside the dense part of liquid (introduction solution) become inherent constituents of the dense part (addition solution). By renormalizing parameters of the model, we have transformed weakly interacting fluctuations to free “dressed” phonons and clusters whose autocorrelation functions are characterized by various behaviors in small and large scales in comparison with the atomic spacing. In the first case, density fluctuations of liquid do not feel impurities. In the intermediate scale, the liquid matrix is inhomogeneous in the form of colloids, which is not observed at the large scales. Dynamics of such liquid is characterized by diffusion modes of solvent and oscillations of impurities.

Keywords

  • liquid
  • density and composition fluctuations
  • Feynman diagram
  • bootstrap
  • phonon
  • cluster
  • renormalization

1. Introduction

It is known that any liquid is characterized by a random chaotic packing of atoms. They are easily rearranged by little thermal fluctuations in contrast to a crystal whose topological structure is stable under any thermal fluctuations below the melting point [1].

At the same time, the topological structure of instantaneous dense part of any condensed matter (liquid, crystal, and amorphous) is represented as configurations of closely packed particles in Delaunay simplexes (dense triangular pyramids with particles in their vertices) that are connected by faces into ramified short-living tetrahedral clusters of density fluctuations [2, 3]. Using the topological criterion [3] in molecular-dynamic (MD) simulation of deterministic nonlinear system of many particles, one can exactly select these simplexes by defining a maximal length of their edges over the maximal number of obtained simplex clusters in the MD cell. The statistics of these clusters is gotten for any condensed matter [4] as their two-dimensional (2D) discrete distribution on cardinality (number of simplexes in the cluster) and on connectivity (number of their vertexes belonging also to other clusters).

For any crystal, these clusters consist of one and only one simplex, that is, their cardinality is equal to 1, but their connectivity is distributed normally in the interval of 7–23 (15 on average). In contrast to the crystal heated, the cluster cardinality of amorphous dense part achieves 10, and the connectivity of such clusters is more than 3 but less than 20 (11 on average). It means that the solid state (crystal and amorphous) is characterized by percolation of tetrahedral dense-part clusters of structural fluctuations.

The topological features of a liquid: (1) the cardinality of liquid dense-part clusters reaches 37, that is, almost four times more than the solid ones, and (2) there are almost 5% of dense-part clusters with zero connectivity sufficient for breaking off the percolation of solid dense-part clusters, providing a fluidity of liquid and forming long chains of liquid dense part. These clusters as dense configurations of particles are dynamically changed but statistically preserve the multifractal structure [3].

The existence in liquid metal of such chains with the fractal gyration radius of ~100 nm is confirmed by the experiments [5] on small-angle-scattering of neutrons. These data are obtained on the contrast of liquid-dense and nondense parts, which amount to 10–15% from the contrast of liquid boundary in vacuum.

Thus, a liquid is characterized by existence of dense-part clusters with zero connectivity in contrast to crystal and amorphous solid which have not such clusters. Moreover, the cardinality of dense-part clusters in any crystal is equal to 1, while amorphous solid occupies the intermediate position between crystal and liquid on the discrete 2D distribution of dense-part clusters [4]. At the same time, the tetrahedral clusters of dense-part open for impurity in principle two topologically differing positions in liquid and amorphous solid: (1) outside the dense-part simplexes and (2) in their clusters as compound constituents [6, 7]. The induced by density-fluctuations polymorphic transition of impurity between these positions is the subject of given theoretical consideration.

Revealing a mechanism of such self-organization of impurities in liquids will allow to have found an approach to their structural modification over chosen attributes by impurities.

2. The method of Green function

The method of Green function used in physics of phase transitions allows so to have formulated and disposed questions of theory that one can obtain topologically exact answers without knowing an explicit kind of the state equation [8]. This method bases on Landau potential [9] which usually is represented by a functional of generalized variables expressing parameters of the local order. Then, structural and phase changes are described by calculus variations of these parameters [8]. They mean by topological and compound (chemical) order. The first is understood as ordering of atoms regardless of the particles nature. The second is characterized by spatial correlation of different atoms and is responsible for the microstratification and clustering of the particles.

Besides the compound parameter of order (CPO), the two-parametrical fluctuation model of liquid alloy includes the topological parameter of order (TPO) which can induce by density fluctuations the clustering of impurity atoms far off from the phase change [10].

We consider the double system, A1xBx, where x=n2/n1+n2¯is the average concentration of impurity component, B; ni is the density of i-particles number (i = 1, 2) for representing Landau potential, ΔF, of this system by the functional of two parameters n1n2[10]:

ΔF=VfΔi,Δi,n,xd3r+ΔF0nxE1

Here, d3ris the differential of 3D space, f is the density of Helmholtz free energy, Δi=nin¯iis the density fluctuation of i-particles number, is the gradient, n is the average density of particles, ΔF0=ΔFΔi=0is the free energy of homogeneous system, and V is its volume.

The Δi(r)fluctuations are averaged in the neighborhood of point, r, in a small volume which however contains sufficiently great number of particles as well as a distance, where Δi(r)function changes is appreciably more than the interatomic spacing, r0[8]. In this case, the other degrees of freedom (electronic, vibration et al) require a time far less than the configuration field, Δi(r), for reaching equilibrium. Therefore, one can apply the adiabatic approximation for describing the fluctuations fields of CPO and TPO in double system.

Then, one can limit Taylor expansion of fi) as a function of small parameter, Δi, by the members of third-order infinitesimal: Δ13, Δ12Δ2, and Δ1Δ22, which correct the second and third approximation of perturbation theory for F0. One can also neglect the members of fourth-order infinitesimal: Δi4, because the coefficients of Δi2in Taylor expansion of fi) are positive, and the Δi proportional members of Taylor series are equal to zero in (1) owing to the constant number of particles in the system.

Further for the isotropic liquid, the first derivatives, Δi, can come into Taylor expansion of f(Δi,Δi)only in the scalar combination (ΔiΔk), and the second ones can be as products: const2Δiand Δi2Δk. The first of them gives the insignificant addition into the integral (1.1), which for the second is transformed into integral of (ΔiΔk)[11].

Thus, without limiting a task generality for liquid, one can present f as [10]

f=12(μ1n1Δ12+2μ1n2Δ1Δ2+μ2n2Δ22)+16(2μ1n12Δ13+32μ1n1n2Δ12Δ2+32μ1n22Δ1Δ22)+K112n(Δ1)2+K12n(Δ1Δ2)+K222n(Δ2)2E2

Here, μi (nxT) ≡ (∂f/∂ni)TV is the chemical potential of i-component, T is Kelvin temperature, Kik=2Uikr0r02z, Uikr0is the pair-interaction potential of nearest particles of kind: i and k, and z is the average coordination number. Considering the homogeneous liquid of double system by the model of ideal solution, one can present the chemical potential, μi (i = 1, 2), in the form

μ1=μ10Tn+Tln1xμ2=μ20Tn+TlnxE3

which, obviously, satisfies to Gibbs-Duhem relation

1xμ1x+xμ2x=0E4

Then, we will obtain [7]

μ10nμ20n=1ndPdnTμ1n1=Tnβ+x1xμ1n2=Tnβ1μ2n2=Tnβ+1xx2μ1n12=Tn2βx2x1x22μ1n22=2μ1n1n2=Tn2β+1E5

at the condition that the first bracket in (2) is the quadratic form positively defined. Here, β=n/Tμ10/n, β'=n2/T2μ10/n2, and P is the static pressure.

For simple liquids, β >> 1 and (∂P/∂n)T weakly depends on the number density, n, of particles. Therefore, one can accept β’ ~β [7].

Transforming the quadratic forms in (2) to diagonal ones, one can present Landau potential as a sum of free-field Hamiltonians and the weak-interaction potential. Then, we will have the almost ideal Bose gas of two components [8].

Using relations (3)–(5), one can do (2) by diagonal square form by means of linear transformation

Δ1=na11φ+a12χΔ2=na21φ+a22χE6

Substituting (6) into (2), we will find parameters

a11=1a22=xa12=xα11xα1α2/α1γa21=x1+x1α1γ1α2βγE7

for x < 1/γ and zero coefficients at (φ ⋅ χ) and (φχ)[10]. Here, α1=K12/K111, α2=K22/K111, and γ=11α1βare the alternating-sign factor. As a result, Eq. (2) to x2 becomes

f/βnT=1+xγ22φ2+1+xα1γ22K11βTφ2+x1+x1α12β2βχ2+x2α2α122K11βTχ21+xγ36φ3x1α1+x21α2γ2φ2χx21α122φχ2E8

Labeling a0=1 + xγ, b0=1+xα1γ, c=1 + x1α12β/β, λ=1α1+x1α2γ, and ρ=rβT/K11, we will obtain

ΔF(φ,χ)=nK11K11/βTV(βTK11)3/2d3ρ[a022φ2+b022(φ)2a036φ3+x(c2χ2+xα2α122(χ)2λ2φ2χx(1α1)22φχ2)]E9

What sense have the parameters of order, φ and χ? We obtain φΔ1+Δ2/nand χΔ2/xΔ1/nout of (6) when αi1and x < 1. Then, φ is the reduced TPO of liquid, and χ expresses the reduced CPO for clustering the initially homogeneous liquid alloy to microregions of different composition, that is, the parameter, χ, describes the compound fluctuation field as opposed to the parameter, φ, which describes the topological fluctuation field.

Each of these fields can be presented as a set of oscillations of averaged corresponding collective modes that are Fourier images of topological and compound fluctuations of the liquid alloy. They are defined by Green functions, G(φ) и G(χ) [12].

In the integral (9), Hamiltonian (2) defines the change of free energy of weak-interacting long-wave phonons and clusters in the double alloy. In the adiabatic approximation, one can take into account only the given ordering (φ, χ) without caring of other variables of the system. Then, we will define the equilibrium fields, φ(ρ)and χ(ρ), in the minimum of ΔF(φχ) [8]. This condition looks like Euler variation equation which for the entered parameters of order gives equations [10].

b022φ+a02φa03φ2/2=xλφχ+x1α12χ2/2xα12α22χ+cχλφ2/2=x1α12φχE10

Using the standard diagram techniques [11] for averaged collective variables, one can reduce the task for weak-interacting phonons and clusters to solve the equations of proper-energetic functions of interacting quasi-particles [10]. For this, we use an averaged correlator ϕ(ρ)ϕ(ρ)which is Green function at ρ=0[8]:

Gφρ=φρφ0E11

In such case, one can present the effects of alloy fluctuation nonhomogeneity as the integrals containing correlation functions, Gφ(ρ)and Gχ(ρ)or their spectral densities

φkφk=Gkφδkkχkχk=GkχδkkE12

obtained by Fourier conversion:

φρ=eikρφkd3k/2π3χρ=eikρχkd3k/2π3E13

where φ(k)=φ(k)and χ(k)=χ(k). According to Wiener-Khinchin theorem, Gi(ρ)and Gki(i = φ, χ) are equivalent functions because they are connected by Fourier conversion

Giρ=Gkieikρd3k/2π3E14

Thanking δ-normalization of φ(k)and χ(k), one can change the differential equations (10) to the algebraic ones for Fourier-images of Green functions: Gk(φ)=|φ(k)|2. We will calculate them in approximation of the perturbation theory by means of iterations and Neumann series of Feynman diagrams [13].

3. The formalism of Feynman diagrams

For “bare” phonon propagator Gk0φ, determined by the first equation of system (10) without the member on the right (x = 0), we have [10]

a02+b02k2Gk0φ=a03/2FkGφ0ρE15

where Fk[Gφ0(ρ)]is the iteration procedure presented by the chain

E16

which is converted into the recurrence form [12]

E17

and has the analytic solution

Gk0φ=a02+b02k2Σ0k1>0E18

under the condition: |Σ0(k)|/(a02+b02k2)<1. Here, Σ0(k)is the proper-energetic function expressed by the equation [10]

Σ0k=a068d3p2π3gpφgkpφE19

and gpφ=a02+b02p21as the solution of (15) with unit on the right. From the second equation of system (10) without the member on the right, we obtain the equation for “bare” cluster propagator, Gk0χ[10]:

c+xα2α12k2Gk0χ=λ2FkGφ0ρE20

The solution of this equation converted into the recurrence form has the graphic form

E21

and the analytic one under the conditions, |Π0(k)|/[c+x(α2α12)k2]<1and |k|<1:

Gk0χ=c+xα2α12k2Π0k1>0E22

Here, Π0(k)is the phonon-proper-energetic function determined by the equation [10]

Π0k=λ28d3p2π3Gp0φGkp0φE23

The solution (22) of the Eq. (20) defines the propagator of induced compound field entering in Hamiltonian (9), that is, the clusters are generated forcedly by phonons unlike their free field with the propagator, Gk0φ, whose fluctuations are formed spontaneously.

The natural development of this idea is the “bootstrap” hypothesis [14] which consists in the following. The fluctuations of CPO, χ, arising at the interaction of phonons deform partially the density-fluctuations field, φ, “dressing” the propagator, Gk0φ, by the proper-energetic function

Σ1k=x2λ2d3p2π3Gp0φGkp0χ+x41α148d3p2π3Gp0χGkp0χE24

defined by the members of the first equation of system (10) on the right. The graphic and analytic solution of this equation is [10]

E25

and

Gkφ=1/Gk0φΣ1k1>0E26

This formula makes sense under the obvious condition |Σ1(k)|Gk0(φ)<1.

Now, one can analytically express the first (topological) bootstrapping of deformed CPO field by replacing function, Gk0φ, in (23) by “dressed” phonon propagator, Gkφ:

Π1k=λ28d3p2π3GpφGkpφE27

and its substitution in the formula (22) instead of Π0(k). It is possible under the condition: |Π1(k)|/[c+x(α2α12)k2]<1. Taking into account the member in the second equation of system (10) on the right gives for propagator, Gk0(χ), the proper-energetic function in the final form

Π2k=x21α14d3p2π3Gp0χGkpφE28

This is expressed in graphic and analytic forms by

E29

and

Gkχ=1/Gk0χΠ2k1>0E30

under the condition |Π2(k)|Gk0(χ)<1.

Thus, one can find the fluctuation fields of the liquid density and compound in the form of autocorrelation functions of impurity concentration, x, and the parameters α1α2βby means of the graphic, algebraic, and integral Eqs. (17)–(19) and (21)–(30).

4. The coherent propagators of phonons and clusters

One can find the solutions of the Eq. (10) in the form of phonons and clusters that are averaged on ensemble of the casual states defined by Hamiltonian (9). The representation of own functions of this Hamiltonian by flat waves with k = |k|<1is a good approximation for the impurity content far from the saturation of liquid alloy.

For dilute solutions (x << 1), one can restrict the proper-energetic functions (19), (24), (27), (28) by the second degree of k and present the propagators (18), (22), (26), (30) in the form [10].

Gk0φ=a0+b0k21Gkφ=a+bk21Gk0χ=u0+v0k21Gkχ=u+vk21E31

At such restriction, it is easy to find all the proper energetic functions. For this, we will substitute (31) into (24), (27), and (28) and transform these multiple integrals to the kind

Ilmk=0σl+τlp21σm+τm|kp|21d3p/2π3=arctgΛlΛmkΛl+Λm/4πτlτmkE32

where Λl=τl/σl. Under the condition of k<1/Λl+1/Λm, one can transform (32) into Taylor expansion on k up to the second member [10]:

IlmkΛlΛm4πτlτmΛl+Λm1ΛlΛmΛl+Λm2k23E33

Substituting (33) into (19), (24), (27), and (28), we will obtain

Σ0k=a0564πb031b02k212a02E34
Σ1k=x24πv0λ2b0a0b0+u0v0+x21α1216u0v0x2k212πv0λ2b0a0b0+u0v03+x2v01α1264u03v0E35
Π1k=λ264πab31bk212aE36
Π2k=x21α124πbv0ab+u0v01k23ab+u0v02E37

Now using formulas (18), (31), and (34), we will obtain

a0=a021a0/b0364πb0=b021+a0/b03768πE38

The parameters u0v0can be obtained by means of (22), (31), (36), and the comment to (27)

u0=cλ264πab3v0=xα2α12+λ2768πa3bE39

At last, the mutual solution of (26) and (30) gives the parameters of “dressed” phonon and cluster propagators, Gkφand Gkχ:

a=a0x24πv0λ2/b0a0b0+u0v0+x21α1216u0v0b=b0+x212πv0λ2/b0a0b0+u0v03+x21α1264u03/v0E40
u=u0x21α124πbv0ab+u0v0v=v0+x21α1212πbv0ab+u0v03E41

It means that the renormalization procedure of the model (9) parameters carries out isomorphic transformation of weak-interacting fields of TPO and CPO into the ensemble of free “dressed” phonons and clusters with Hamiltonian

ΔFφχ=nK11K11/βTVβTK113/2d3ρ[a2φ2+b2φ2+xu2χ2+v2χ2]E42

under the condition a > 0 and u > 0. In this representation, the correlation functions for TPO and CPO functions look like:

Gφρ=βTK113/2expρΛφ4πβρnbGχρ=βTK113/2expρΛχ4πβρnvE43

where Λφ=b/aand Λχ=v/u. It is easy to see that Giρρ1at ρ < Λi, and this function exponentially works for zero, when ρ > Λi.

It is clear that the Eq. (40) is obtained under the condition: k<1/Λφ0=a0/b0, that is equivalent to |ρ|>Λφ0, that is, the relation, Gφρ|ρ|1, is valid for the interval, Λφ0<|ρ|<Λφ.

Under the condition: |ρ|<Λφ0, it is necessary to replace the correlator, Gkφ, by the “bare” propagator, Gk0φ, with the parameters (39).

Thus, the TPO fluctuations in the liquid alloy are characterized by various behaviors in small and large scales in comparison with Λφ0. In the case of |ρ|<Λφ0, density fluctuations of liquid do not feel impurities. When Λφ0<|ρ|<Λφ, the liquid matrix is inhomogeneous in the form of impurity colloids, and for |ρ|>Λφ, such heterogeneity is not observed at all [10].

5. Stratification of impurity by density fluctuations of liquid alloy

The structural modification of the liquid alloy at varying the system parameters xα1α2βis characterized by changing the correlation radii Λϕ and Λχ of Green functions (43). They define the characteristic ranges of observed TPO and CPO fluctuations [8]. Therefore, the concentration dependence, Λi(x), is interested to consider for different xα1α2βof the model (9). At the same time, one should remember that this model is applied only in Taylor expansion (2) of f(Δi,Δi)under the conditions [11]: x<1/4|1+αi1β|and |i|2Λi<<1that are reduced to: b3/a,v3/u>>e2βT/2z|J11|3/e1+z[10].

The solutions of Eqs. (38)–(41) obtained under these conditions are illustrated in Figures 14 by the graphs of functions, Λϕ(x) and Λχ(x), in logarithmic coordinates for the ranges: 0.095 < α12< α2≤ 1.4 and 10 ≤ β ≤ 150. The last one characterizes liquid metals where the alloy components have a tendency for demixing at α12<α2in contrast to clustering at α12>α2. The structural features of such alloy are discussed below.

Figure 1.

The graphs of lg Λφ(1) and lg Λχ(2) as functions of the impurity concentration, lgx, at α1=0.31, α2=0.1, and β = 10.

Figure 2.

The graphs of lg Λφ(1) and lg Λχ(2) as functions of the impurity concentration, lgx, at α1=0.89, α2=0.8, and β = 10.

Figure 3.

The graphs of lg Λφ(1) and lg Λχ(2) as functions of the impurity concentration, lgx, at α1=1.1, α2=1.4, and β = 10.

Figure 4.

The graphs of lg Λφ(1) and lg Λχ(2) as functions of the impurity concentration, lgx, at α1=1.1, α2=1.4, and β = 150.

One can see that the correlation radius of phonons (Λϕ) is practically not changed with growing the impurity concentration as opposed to the correlation radius of impurity demixing (Λχ) which increases: the higher values of αi at α12<α2, the more is. At the same time, increasing β partially decreases this effect (compare Figures 3 and 4) [10].

6. Impurity clustering induced by alloy density fluctuations

At α12>α2, the graphs of lg Λϕ(x) and lg Λχ(x) are shown in Figures 58 for α1=0.6, α2=0.3, and for four values of β in the range of 10–150.

Figure 5.

The graphs of lg Λφ(1) and lg Λχ(2) as functions of the impurity concentration, lgx, at α1=0.6, α2=0.3, and β = 10.

Figure 6.

The graphs of lg Λφ(1) and lg Λχ(2) as functions of the impurity concentration, lgx, at α1=0.6, α2=0.3, and β = 50.

Figure 7.

The graphs of lg Λφ(1) and lg Λχ(2) as functions of the impurity concentration, lgx, at α1=0.6, α2=0.3, and β = 100.

Figure 8.

The graphs of lg Λφ(1) and lg Λχ(2) as functions of the impurity concentration, lgx, at α1=0.6, α2=0.3, and β = 150.

It turned out that Λχ decreases sharply at some critical point, xc. This indicates the decay of CPO fluctuations of double alloy into compound clusters on the background of long-wave density fluctuations of liquid. One can see that the range of impurity concentration of clusters existence decreases with growing the rigidity, β, of condensed matter.

At the same time, xc does not practically change because this point is defined by the value of α12α2which is constant. The following sharp increase of the CPO correlation radius (see Figures 7 and 8) is interpreted as aggregation of clusters [10]. The observed growing of TPO correlation radius, Λϕ(x), can be caused by impurity precipitations that do more lengthy the density fluctuations.

7. Conclusions

According to the two-parametric model represented above, density fluctuations of liquid induce mono-ordering impurity in micro-regions at α12<α2(see Figures 24) and its clustering with basic component at α12>α2(see Figures 58). Such self-organization of liquid alloy has no thermodynamic singularities of the first-order phase transition because it has continuous character without the potential jump and concerns only to change the impurity state in liquid alloy, that is, it is interpreted as a component phase transition of the first order [15].

The scale of this transition increases with growing the concentration and bond force of impurity particles and it decreases with growing the rigidity of condensed matter inclined to stratification of components (α12<α2). For opposite components inclined to clustering (α12>α2), the composition fluctuations of double alloy decay to local states in the form of quasi-molecular fluctuations.

By renormalizing parameters of this model, we have transformed weakly interacting fluctuations to free “dressed” phonons and clusters whose autocorrelation functions are characterized by various behaviors in small and large scales in comparison with the atomic spacing. In the first case, density fluctuations of liquid do not feel impurities. In the intermediate scale, the liquid matrix is inhomogeneous in the form of colloids, which is not observed at the large scales. Dynamics of such liquid is characterized by diffusion modes of solvent and oscillations of impurities.

At the same time, any liquid can be composed from two structures. The first of them represents finite and ramified clusters from almost tetrahedrons having common faces in pairs. The second is locally less dense which includes micropores as elements of free volume of liquid.

Acknowledgments

The author thanks the colleagues for helping in this work and for useful discussion of the approach to forced modification of liquids by density fluctuations.

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Alexander Shimkevich (December 20th 2017). A Colloidal Self-Organization of Impurities in a Liquid by Density Fluctuations, Chaos Theory, Kais A. Mohamedamen Al Naimee, IntechOpen, DOI: 10.5772/intechopen.70459. Available from:

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