## 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, *n*_{i} is the density of *i*-particles number (*i* = 1, 2) for representing Landau potential, Δ*F*, of this system by the functional of two parameters

Here, *f* is the density of Helmholtz free energy, *i*-particles number, *n* is the average density of particles, *V* is its volume.

The

Then, one can limit Taylor expansion of *f*(Δ_{i}) as a function of small parameter, Δ_{i}, by the members of third-order infinitesimal: *f*(Δ_{i}) are positive, and the Δ_{i} 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,

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

Here, *μ*_{i} (*n*, *x*, *T*) ≡ (∂*f*/∂*n*_{i})_{TV} is the chemical potential of *i*-component, *T* is Kelvin temperature, *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, *μ*_{i} (*i* = 1, 2), in the form

which, obviously, satisfies to Gibbs-Duhem relation

Then, we will obtain [7]

at the condition that the first bracket in (2) is the quadratic form positively defined. Here, *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

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

for *x* < 1/γ and zero coefficients at (*φ* ⋅ *χ*) and *x*^{2} becomes

Labeling

What sense have the parameters of order, *φ* and *χ?* We obtain *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, *ΔF*(*φ*, *χ*) [8]. This condition looks like Euler variation equation which for the entered parameters of order gives equations [10].

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

In such case, one can present the effects of alloy fluctuation nonhomogeneity as the integrals containing correlation functions,

obtained by Fourier conversion:

where *i* = *φ, χ*) are equivalent functions because they are connected by Fourier conversion

Thanking δ-normalization of

## 3. The formalism of Feynman diagrams

For “bare” phonon propagator *x* = 0), we have [10]

where

which is converted into the recurrence form [12]

and has the analytic solution

under the condition:

and

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

and the analytic one under the conditions,

Here,

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,

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,

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

and

This formula makes sense under the obvious condition

Now, one can analytically express the first (topological) bootstrapping of deformed CPO field by replacing function,

and its substitution in the formula (22) instead of

This is expressed in graphic and analytic forms by

and

under the condition

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

## 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* =

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].

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

where *k* up to the second member [10]:

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

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

The parameters

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

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

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

where *ρ* < Λ_{i,} and this function exponentially works for zero, when *ρ* > Λ_{i}.

It is clear that the Eq. (40) is obtained under the condition:

Under the condition:

Thus, the TPO fluctuations in the liquid alloy are characterized by various behaviors in small and large scales in comparison with

## 5. Stratification of impurity by density fluctuations of liquid alloy

The structural modification of the liquid alloy at varying the system parameters _{ϕ} and Λ_{χ} of Green functions (43). They define the characteristic ranges of observed TPO and CPO fluctuations [8]. Therefore, the concentration dependence, Λ_{i}(*x*), is interested to consider for different

The solutions of Eqs. (38)–(41) obtained under these conditions are illustrated in Figures 1–4 by the graphs of functions, Λ_{ϕ}(*x*) and Λ_{χ}(*x*), in logarithmic coordinates for the ranges: 0.095 < *β* ≤ 150. The last one characterizes liquid metals where the alloy components have a tendency for demixing at

One can see that the correlation radius of phonons (Λ_{ϕ}) is practically not changed with growing the impurity concentration as opposed to the correlation radius of impurity demixing (Λ_{χ}) which increases: the higher values of *α*_{i} at

## 6. Impurity clustering induced by alloy density fluctuations

At _{ϕ}(*x*) and lg Λ_{χ}(*x*) are shown in Figures 5–8 for *β* in the range of 10–150.

It turned out that Λ_{χ} decreases sharply at some critical point, *x*_{c}. 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, *x*_{c} does not practically change because this point is defined by the value of _{ϕ}(*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

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

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.