rus
Issue #5/2022
O.P.Posnansky
METHOD OF RENORMALIZATION OF THE RANDOM THREE-COMPONENT SYSTEM: DYNAMIC CHARACTERISTICS OF THE EFFECTIVE SUSCEPTIBILITY FUNCTION
METHOD OF RENORMALIZATION OF THE RANDOM THREE-COMPONENT SYSTEM: DYNAMIC CHARACTERISTICS OF THE EFFECTIVE SUSCEPTIBILITY FUNCTION
DOI: https://doi.org/10.22184/1993-8578.2022.15.5.272.289
In this work we investigated dynamic characteristics of the effective susceptibility of random three-component system. We have shown that in the case of large discrepancy of the static local susceptibilities effective dynamic properties are similar to the two-component system. If static coefficients of local dynamic susceptibilities of the components approach each other keeping relaxation parts different, then peculiarities of the three-component system become apparent. In this case the effective active part of the susceptibility possesses two plateaus and the relaxing part demonstrates two maximums. Amplitudes of the maximums for relaxing part depend on the dominating component. Also we investigated a case of double percolation showing that the effective properties can change two times during variation of the fraction of one of the components. In the first case the change is associated with creation of the percolation cluster built from the component (2), the second change is linked to the extrusion of the component (2) and (3) by the component (1) which builds a secondary percolation cluster.
In this work we investigated dynamic characteristics of the effective susceptibility of random three-component system. We have shown that in the case of large discrepancy of the static local susceptibilities effective dynamic properties are similar to the two-component system. If static coefficients of local dynamic susceptibilities of the components approach each other keeping relaxation parts different, then peculiarities of the three-component system become apparent. In this case the effective active part of the susceptibility possesses two plateaus and the relaxing part demonstrates two maximums. Amplitudes of the maximums for relaxing part depend on the dominating component. Also we investigated a case of double percolation showing that the effective properties can change two times during variation of the fraction of one of the components. In the first case the change is associated with creation of the percolation cluster built from the component (2), the second change is linked to the extrusion of the component (2) and (3) by the component (1) which builds a secondary percolation cluster.
Теги: double percolation hierarchical model polychromatic percolation renormalization method static and dynamic effective susceptibility двойная перколяция иерархическая модель метод ренормализации полихроматическая перколяция статическая и динамическая эффективная восприимчивость
INTRODUCTION
Percolation theory [1, 2] has applications in such various research fields as metal-insulator transitions [3, 4], gelation processes [5], fluid flow in porous media [6], and viscoelasticity [7–9]. For the modelling of these diverse physical phenomena, as a rule every element of lattice representing a two-component system, is randomly assigned to be in one of two possible states. For example, in regular bond percolation, a bond is "black" with probabilityor p ∈ [0,1] "white" with probability q = 1 – p .
Zallen [10] generalized the conventional two-component percolation problem to a multi-component which he named polychromatic percolation and provided an in-depth study of its characteristics. Physical properties of polychromatic percolation were studied by Kogut and Straley [11] who applied dual lattice and Monte-Carlo approaches in the investigation of a bicritical exponent of conductivity. Later, Halley et al. [12] used finite-lattice simulations and effective medium theory to understand the conductivities of disordered lattices in which there were up to four species. Subsequently, Halley et al. [13] provided a general review of the work carried out so far on polychromatic percolation models. Developed mathematical tools have been successfully applied to explain experiments with hopping conductivity [14–17], polydisperse granula materials [18, 19], sodium-ammonia mixtures and charge-transfer salts [12, 13], high-temperature superconducting [20].
The motives for the current work are, first, to begin a systematic investigation of properties of effective dynamic susceptibility for the case of polychromatic percolation which can be expected ultimately to find a wide range of application in real multi-component systems of scientific and technical interest [30]. Second, we explore the idea of hierarchical averaging and renormalization in multi-component percolation and present results for a wide range of concentrations of constituent components.
In this work we will show that the effective dynamic susceptibility of the random three-component system is fundamentally different than for the case of the regular two-component system in several important respects. The difference can be explained from the point of view of microgeometry, namely, in three-component systems there are two interfaces analogous to the hull. In addition to the external surfaces, there are regions where all three components combine, affecting the dynamic of effective susceptibility properties.
Measuring effective dynamic susceptibility is widely used for characterizing highly disordered composite materials and based on relations between macroscopic properties and structure at the microscopic scale commensurate with the minimal length of the inhomogeneity. Establishing such relations remains a fundamental challenge due to lack of perturbation parameter. For the first time, we show that the effective dynamic susceptibility distinguishes between the composites with different fractions of components in a broadband frequency range. Our approach enables the interpretation of measurements by objectively selecting and modelling the most relevant features of parameters of constitutive components, and allows predicting properties of metamaterials.
THEORY
A hierarchical model of composite consisting of three components
We consider composite as a three component random structure which can be mapped onto two-dimensional (2D) square lattice and investigated by percolation theory methods. Assume that each bond of the lattice is coloured "black" with probability p, "white" with probability q, and "grey" with probability r, representing components of composite and satisfying the condition:
, (1)
where D~(p, q, r) is a probability domain. Geometrically D~(p, q, r) depicts an equiangular polygon (triangle) and probabilities p, q, and r can be interpreted as components of a vector. Of note, the regular two-component composite in Eq.(1) is recovered if either p, q, or r is zero.
Physical properties of such material can be described by the generalized ternary local probability density function:
, (2)
where δ(x) is a Dirac delta function. In Eq.(2) every bond is bounded with the closest neighbor nodes (km) in the lattices of N inner and NГ contact boundary nodes. It can either belong to component (1), (2) or (3) characterized by physical parameters G(1), G(2) and G(3) correspondingly.
We model global (or effective) properties of a material with randomly distributed local properties according to a real space renormalization group method introduced by Reynolds et al. [21–23]. The iterative transformation of generalized probability density function of Eq.(2) is defined as [8]:
, (3)
Where
. (4)
S is a surface covering N inner and NГ nodes on Г boundary, and S\SГ is a surface where nodes NГ are excluded. is the effective susceptibility function of a lattice, delineated via a basic element bounded by nodes (km), which was averaged over all configurations E in a way to preserve the invariant form of Eq.(2) for every nth scale of recursive building of hierarchical lattice. In Eq.(4) the set E is a unification, ∪, of connected, E(1) and E(2), and any (i.e. connected and disconnected), E(3), clusters. What can be noted is that the number of elements in a set E is equal 35 = 243 for the renormalization unit self-dual cell presented in Fig.1a.
The transformation of p, q, and r in Eq.(3) corresponds to polynomial connectedness functions, P(p, q, r), Q(p, q, r) and R(p, q, r), where
, (5a)
, (5b)
, (5c)
if we perform renormalization of a part of a lattice in Fig.1a.
Of note, the sum of all coefficients in Eq.(5) is equal to the number of elements in a set E. Thus the iterative recursion of a set of parameters {pn, qn, rn} is defined as
, (6)
and Eq.(3) after simplification can be written as
, (7a)
where the effective susceptibility function, , is separated into components (1), (2) and (3) depending on the connectedness of bonds in set E.
Solutions of Eq.(6) can be classified by unstable and stable roots, (p*, q*, r*). If perturbation of input parameters leads subsequently to one of two components we term such roots as critical and unstable. In turn, if the same perturbation leads to one of three components we term these roots as unstable and bicritical. Stable roots represent a lattice fully occupied by one of the components (1), (2), or (3). Interestingly unstable critical points can be located on lines as well, separating probability domain on subregions. These lines can be described by equations in a parametric form:
, (7b)
, (7c)
where . The analysis of such peculiarities of Eq.(6) is carried out in the section Results of numerical simulations.
If we perform iterative recursion in Eqs.(3, 6) approaching n → ∞, the three components become indistinguishable yielding a unification:
, (8a)
which leads to a convergence of ternary properties of Eq.(7a) to solitary
, (8b)
with effective properties corresponding to component (1), (2) or (3) depending on selection of initial probability (p0, q0, r0) at the first iteration step n = 0. Thus the probability domain, D~(p, q, r), can be divided onto three regions where the renormalization process leads to one of the corresponding stable roots:
, (8c)
. (8d)
We notice that at (p0 = p*, q0 = q*, r0 = r*) properties of components are similar. These points can be critical or bicritical and excluded from the probability domain, D~(p, q, r), according Eq.(8d).
In our calculations of the properties of the composite we suppose that the following ratio between initial probabilities of components is given:
, (9a)
. (9b)
Of note, if α → 0 then q → 0 and three-component composite approaches two-component with property p + r = 1. On the other hand, if α → ∞, then r → 0, which leads in turn to p + q = 1. Geometrical interpretation of the condition given in Eq.(9) means that properties are investigated along a line embedded in domain D(p, q, r).
In a special case we consider a situation when arc, covering domains, D(1,0,0) (p, q, r), D(0,1,0) (p, q, r), and D(0,0,1) (p, q, r), represents a trajectory of initial probabilities satisfying Eq.(1). Such a pathway is necessary to describe a phenomenon termed as a double percolation [24–26]. This phenomenon has been found to arise from the formation of an irregular, highly conducting layer following the boundaries between the conducting and the insulating components. Effectively, the system thus contains three components. It was experimentally found [24] that after achieving a connected cluster composed from one of the components afterwards it is substituted by another component. As a consequence this leads to a sharp change of effective properties twice in a course of variation of fraction of one of the components (Fig.1d).
The arc trajectory can be modelled by a part of a generalized superellipse curve [27] yielded in a parametric form:
, (9c)
where , –π/6 ≤ и
≤ π/6, and a = b = 1. We chose n1 = 6 to create a superellipse embedded in hexagon. Other parameters, n2 = 2 and n3 = n4 = 12, were used to identify a double percolation effect and create a maximum amplitude and curvature of a cusp of a parametric curve in a selected segment. Increasing n2 and decreasing n3 leads to the reduction of curvature and amplitude, and as a consequence, double percolation may disappear. Parametric curve Eq.(9c) was normalized and rotated to be included in a polygon Eq.(1).
Local model of susceptibility function and its effective value
We use the Maxwell model of a local dynamic susceptibility (or response) function, [28]:
, (10)
where τ is a relaxation time, and ω is a frequency of an external field, G is a constant of an active component (or local static susceptibility). The characteristic frequency, ω0, can be defined as ω0 = 1/ τ. On the initial level (n=0) of iterative procedure the following unification
(11)
of dimensionless response functions represents parameters of three components in a generalized probability density function in Eqs.(2, 7a). In Eq.(11) parameters of the active component were normalized to G1 in the far frequency region, ω >> ω0. For example, if we impose inequalities G1> G2 > G3 and τ1 > τ2 > τ3, component (1) is slowly relaxing with the very large constant of active part, and the fast relaxing and lowest constant of active parts represent component (3).
The balance of all forces at any location within composite assumes a conservation law:
, (12)
if uk is a displacement on kth node. The distribution of local generalized susceptibilities, G*(km), in Eq.(12) is governed by Eq.(7a) and at each step n of renormalization should depend on the index, i.e. G*(km), n, in a general case.
The Dirichlet boundary condition in Eq.(12) corresponds and on the left, Г+, and right, Г–, sides of the unit cell respectively (Fig.1a). In Eq.(12) we assume lattice constant a = 1. Using the Kronecker delta symbol, δkm, it is possible to introduce a matrix of response functions, , which the matrix element, , is given as:
, (13)
and to rewrite Eq.(12) as a linear equation:
, (14)
where ck denotes the external force which is zero but on the right side boundary.
After solving Eq.(14) relatively to unknown um, we find the equivalent susceptibility of the lattice in Eq.(3) expressed in a form
, (15)
where nodes k are taken only for the left contacted boundary Г+. From the structure of values in Eqs.(10,11,15) it is clear that the equivalent susceptibility functions, , are complex-valued with active real and relaxing imaging parts. The statistical averaging of Eq.(15) should be performed according to distribution in Eq.(2) on a whole set E yielding effective susceptibilities, , for connected and disconnected clusters. If we carry out the iterative procedure we approach stable effective susceptibilities according to Eq.(8a) which are analyzed in the next section.
Results of numerical simulations
to perform a numerical analysis of the effective susceptibility function we, first of all, analyze a renormalization flux of fractions of components during the renormalization procedure. For this purpose we built a grid with coordinates of nodes (p0, q0, r0) which satisfy Eq.(1). Then we perform a transformation according to Eqs.(5, 6). This transformation provides a new grid with nodes (p1, q1, r1). The vector field of the renormalization flux can be constructed as (p1 – p0, q1 – q0, r1 – r0) and then assigned to the initial grid. The maximal norm of any vector cannot exceed 1 and minimal norm is bounded by 0. In Fig.1b we demonstrate a triangle described by Eq.(1) seen from the top of its normal vector, . Arrows represent a renormalization flux projected onto the polygon. The flux achieves minimal norm at the corners of the triangle at vertices A, B, and C in Fig.(1b). These are stable points of renormalization because the vector field converges to them from any direction.
Eqs.5, 6 allow the introduction of a red-green-blue color-coding scheme where the color red corresponds to a stable vertex A =(1, 0, 0), green B=(0, 1, 0) and blue C=(0, 0, 1). The mixture of colors is represented by a vector (P(p0, q0, r0), Q(p0, q0, r0), R(p0, q0, r0)) from Eq.(5), which is not aligned to any of the Cartesian axis in a general case. This color-coded background is demonstrated in Fig.1b as well.
At certain points of the polygon in Fig.1b the vector field diverges to two or three points of a set {A, B, C}. The minimal norm of the flux can be found in the vicinity of points L5 = (0.5, 0.5, 0.0), L4 = (0.5, 0.0, 0.5), L2 = (0.0,0.5,0.5) on the edges of the triangle. These are unstable points of renormalization because the vector field diverges there. Between them we can discriminate critical points, L4 and L5, characterized by a possibility of divergence in the direction of one of the two components. Bicritical point, L2 , is characterized by a possibility of divergence in the direction of one of the three components. Interestingly, we can find two lines, L2L4 and L2L5, where unstable critical points are located. These lines partition probability domain, D(p, q, r), as it is pointed in Eq.(8c).
Renormalization along edges AB, BC, AC corresponds to a two-component composite. Functions P(p, q, r), Q(p, q, r), and R(p, q, r) are plotted in Fig.1c according to Eqs.(5) if r, p or q are set to 0 respectively. A zigzag pathway represents an evolution of pn during the execution of Eq.(6). The number of iterations for building a hierarchical lattice during the renormalization procedure is defined by the constraint |pn – pn+1| ≤ 1–3 and presented at the same Fig.1c.
We vary parameter p keeping the ratio, α, between the other components unchanged (Eq.(9a,b)). Such a procedure, for example, corresponds to lines AB, AL1, AL2, AL3, AC depicted in Fig.1b as outgoing from vertex A. Test values of parameter α are shown in Table 1. We also use arc (Eq.9c) as a pathway of the variation parameter p represented as a part of a superellipse to model a double percolation effect.
For the description of the external load the frequency window, ω = [10–3; 102] rad/sec, was selected in our calculation. The real (i.e. active) and imaginary (i.e. relaxing) parts of the effective susceptibility can be demonstrated as surfaces in logarithmic scale as functions of frequency (in logarithmic scale) and fraction p. Here logarithmic scale assists to compare values with a large diversity. To evaluate sensitivities of effective values to input parameters we overlay a contour curve onto the surfaces with level –4 in all cases. We also projected this curve onto a plane (log 10(ωτ), p. The surfaces of real and imaginary parts of effective susceptibility are color-coded, where the colors red and blue correspond respectively to the minimum and maximum distance of the point on the surface relative to the 0 level. The color-coded scheme of the surfaces is projected onto a plane to facilitate a quantitative evaluation of parameters.
In Fig.2 we demonstrate effective susceptibility along lines AB and AC from Fig.1b corresponding to a two-component composite with parameters taken from Table 1. Formally the variation of p along these lines is defined by α → 0 and α → ∞ (in our numerical calculations α = 1000). In Fig.2a and Fig.2c we can see that in the high frequency limit, ωτ → ∞, the real part of the dynamic effective susceptibility function approach its static values for components (3) or (2) from Table 1 if p → 0. On the other hand the active part of effective susceptibility is frequency independent if p → 1 taking the value corresponding to the component (1) from Table 1. In a low frequency limit, ωτ → 0, the real part of effective susceptibility demonstrates a linear dependence which transits into plateau near characteristic frequencies ω2 → 1/τ2 (Fig.2c) and ω3 → 1/τ3 (Fig.2a) at p < p*. The transition corresponds to the maximum of the imaginary part of the effective susceptibility at the same characteristic frequencies (Figs.2b,d). The characteristic frequency is shifted to the lower limit if we increase the fraction p → p*. At the point p = p* we observe a jump of the real and maximum of the imaginary parts of the effective susceptibility. Besides, the imaginary part of the effective susceptibility achieves its global maximum at the characteristic frequency at p = p*.
In Fig.3 we present results of the calculation of effective susceptibility for a three-component system (parameters are given in Table 1). The trajectory follows lines AL1 and AL3 (Fig.1b). Due to the large difference in initial static susceptibilities of components (2) and (3), i.e. G(2),0 >> G(3),0, the behaviour of the effective susceptibility is very similar to the two-component system described earlier. Contrarily to the previous situation, if initial static susceptibilities of components (2) and (3) approach each other, we can observe affects which are specific to the three-component system. For the simulation reason we keep the same ratio of initial relaxation times, but assume G(1),0 > G(2),0 = G(3),0 (see Table 1) and follow lines AL1, AL2 and AL3 (Fig.1b).
In this case the effective active part is not very sensitive to α although we observe a slight folding of plateau near ω2 and ω3 (Fig.4a,c,e). This can be evaluated by the iso-contour curve projected onto plane(log 10(ωτ), p). For the case of the effective active part we see two shifts and three plateaus if we consider a projection iso-contour curve. For the better estimation of the effect in Fig.4h we plotted the effective active part for p → 0 as in logarithmic scale (grey colored curve) as without (black colored curve). There we observe two transitions near characteristic frequencies, ω2 and ω3 (thin dash lines).
At the same time, the effective relaxation component exhibits high sensitivity to the parameter α. We observe two distinct maximums of effective susceptibility at percolation line p = p* (Fig.4b-f). If α = 0,5, the global maximum is shifted to the high frequencies (Fig.4b). In case of α = 1 maximums at high and low frequencies are nearly the same (Fig.4d). For α = 2, maximum is shifted to the low frequencies (Fig.4f).
We can also demonstrate a double percolation phenomenon described in Refs.[24–26]. For this purpose the trajectory of initial concentrations p corresponds to arc (Fig.1b and Eq.(9c)). Following this trajectory means achieving a percolation of component (2) near the edge BC, then simultaneously reducing of the fractions of the components (2) and (3), and increasing the fraction of the component (1) we approach the stable point A on the edge AC. Such a transformation assumes that the percolation cluster generated from component (2), afterwards is substituted by the percolating cluster composed from component (1). The result is demonstrated in Fig.5 for different initial values of active components collected in Table 1. For simplicity it is assumed that the relaxation rate is the same for all compounds.
What can be noticed in Fig.5 is that the effective susceptibility significantly changes twice near p = pI ≈ 0,1 and p = pII ≈ 0,8. Depending on the initial active values we model three possible situations of the variation of effective susceptibility as a function of the concentration p, namely, effective susceptibility can be concave (Fig.5a), increasing (Fig.5c) and convex (Fig.5e) function along the parameter p. To estimate the extent of the effect we plotted the real part of effective susceptibility with a limit ωτ → ∞ (Fig.5h) in logarithmic scale and without.
CONCLUSIONS
In this work we investigated susceptibility properties of a random three-component system using newly developed hierarchical scale averaging approach. We have shown that in the case of a large discrepancy of the static local susceptibilities dynamic effective properties are similar to a two-component system. If static susceptibilities of the components (2) and (3) approach each other keeping relaxation parts different, then peculiarities of the three-component system become apparent. In this case the effective active part possesses two plateaus and the relaxing part demonstrates two maximums. Amplitudes of maximums for the relaxing part depend on the dominating component.
Also for the first time we investigated a case of double percolation showing that the dynamic effective properties of susceptibility change two times. In the first case change is associated with creation of the percolation cluster built from the component (2), the second change is linked to the extrusion of the component (2) and (3) by the component (1) which builds a secondary percolation cluster.
Appendix A
Rotations in three dimensions can be described equivalently by either the SO (3) or SU (2) representations [29]. The first representation uses 3 × 3 rotation matrices and 3 × 1 vectors. The matrices are orthonormal and their matrix elements are real. In the SU (2) representation, a general rotation matrix U can be written:
(A1)
which acts on a function z = f(x, y) in Eq.(9c) yielded as
(A2)
Elements of matrix U are complex numbers known as the Cayley-Klein parameters and the asterisk represents complex conjugation. The matrix U is unitary and Hermitian. The general rotation can be written:
(A3)
where symbol H is a Hermitian conjugate. In terms of Euler angles the general rotation matrix U can be decomposed:
(A4)
and
. (A5)
The mathematical simplification afforded by the SU(2) representation is a consequence of the fewer number of constraints. This situation is analogous to solving a system of linear equations, where it is nearly always easier to solve fewer equations with fewer constraints. For our application we combined rotations and translations
, (A6)
where ψ = 0, Ѳ1 = –90°, Ѳ2 = –45°, ϕ = 125,26 and
. (A7)
The curve described in Eq.(9c) can be transferred according Eq.(A6) to be embedded in a probability domain.
PEER REVIEW INFO
Editorial board thanks the anonymous reviewer(s) for their contribution to the peer review of this work. It is also grateful for their consent to publish papers on the journal’s website and SEL eLibrary eLIBRARY.RU.
Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Percolation theory [1, 2] has applications in such various research fields as metal-insulator transitions [3, 4], gelation processes [5], fluid flow in porous media [6], and viscoelasticity [7–9]. For the modelling of these diverse physical phenomena, as a rule every element of lattice representing a two-component system, is randomly assigned to be in one of two possible states. For example, in regular bond percolation, a bond is "black" with probabilityor p ∈ [0,1] "white" with probability q = 1 – p .
Zallen [10] generalized the conventional two-component percolation problem to a multi-component which he named polychromatic percolation and provided an in-depth study of its characteristics. Physical properties of polychromatic percolation were studied by Kogut and Straley [11] who applied dual lattice and Monte-Carlo approaches in the investigation of a bicritical exponent of conductivity. Later, Halley et al. [12] used finite-lattice simulations and effective medium theory to understand the conductivities of disordered lattices in which there were up to four species. Subsequently, Halley et al. [13] provided a general review of the work carried out so far on polychromatic percolation models. Developed mathematical tools have been successfully applied to explain experiments with hopping conductivity [14–17], polydisperse granula materials [18, 19], sodium-ammonia mixtures and charge-transfer salts [12, 13], high-temperature superconducting [20].
The motives for the current work are, first, to begin a systematic investigation of properties of effective dynamic susceptibility for the case of polychromatic percolation which can be expected ultimately to find a wide range of application in real multi-component systems of scientific and technical interest [30]. Second, we explore the idea of hierarchical averaging and renormalization in multi-component percolation and present results for a wide range of concentrations of constituent components.
In this work we will show that the effective dynamic susceptibility of the random three-component system is fundamentally different than for the case of the regular two-component system in several important respects. The difference can be explained from the point of view of microgeometry, namely, in three-component systems there are two interfaces analogous to the hull. In addition to the external surfaces, there are regions where all three components combine, affecting the dynamic of effective susceptibility properties.
Measuring effective dynamic susceptibility is widely used for characterizing highly disordered composite materials and based on relations between macroscopic properties and structure at the microscopic scale commensurate with the minimal length of the inhomogeneity. Establishing such relations remains a fundamental challenge due to lack of perturbation parameter. For the first time, we show that the effective dynamic susceptibility distinguishes between the composites with different fractions of components in a broadband frequency range. Our approach enables the interpretation of measurements by objectively selecting and modelling the most relevant features of parameters of constitutive components, and allows predicting properties of metamaterials.
THEORY
A hierarchical model of composite consisting of three components
We consider composite as a three component random structure which can be mapped onto two-dimensional (2D) square lattice and investigated by percolation theory methods. Assume that each bond of the lattice is coloured "black" with probability p, "white" with probability q, and "grey" with probability r, representing components of composite and satisfying the condition:
, (1)
where D~(p, q, r) is a probability domain. Geometrically D~(p, q, r) depicts an equiangular polygon (triangle) and probabilities p, q, and r can be interpreted as components of a vector. Of note, the regular two-component composite in Eq.(1) is recovered if either p, q, or r is zero.
Physical properties of such material can be described by the generalized ternary local probability density function:
, (2)
where δ(x) is a Dirac delta function. In Eq.(2) every bond is bounded with the closest neighbor nodes (km) in the lattices of N inner and NГ contact boundary nodes. It can either belong to component (1), (2) or (3) characterized by physical parameters G(1), G(2) and G(3) correspondingly.
We model global (or effective) properties of a material with randomly distributed local properties according to a real space renormalization group method introduced by Reynolds et al. [21–23]. The iterative transformation of generalized probability density function of Eq.(2) is defined as [8]:
, (3)
Where
. (4)
S is a surface covering N inner and NГ nodes on Г boundary, and S\SГ is a surface where nodes NГ are excluded. is the effective susceptibility function of a lattice, delineated via a basic element bounded by nodes (km), which was averaged over all configurations E in a way to preserve the invariant form of Eq.(2) for every nth scale of recursive building of hierarchical lattice. In Eq.(4) the set E is a unification, ∪, of connected, E(1) and E(2), and any (i.e. connected and disconnected), E(3), clusters. What can be noted is that the number of elements in a set E is equal 35 = 243 for the renormalization unit self-dual cell presented in Fig.1a.
The transformation of p, q, and r in Eq.(3) corresponds to polynomial connectedness functions, P(p, q, r), Q(p, q, r) and R(p, q, r), where
, (5a)
, (5b)
, (5c)
if we perform renormalization of a part of a lattice in Fig.1a.
Of note, the sum of all coefficients in Eq.(5) is equal to the number of elements in a set E. Thus the iterative recursion of a set of parameters {pn, qn, rn} is defined as
, (6)
and Eq.(3) after simplification can be written as
, (7a)
where the effective susceptibility function, , is separated into components (1), (2) and (3) depending on the connectedness of bonds in set E.
Solutions of Eq.(6) can be classified by unstable and stable roots, (p*, q*, r*). If perturbation of input parameters leads subsequently to one of two components we term such roots as critical and unstable. In turn, if the same perturbation leads to one of three components we term these roots as unstable and bicritical. Stable roots represent a lattice fully occupied by one of the components (1), (2), or (3). Interestingly unstable critical points can be located on lines as well, separating probability domain on subregions. These lines can be described by equations in a parametric form:
, (7b)
, (7c)
where . The analysis of such peculiarities of Eq.(6) is carried out in the section Results of numerical simulations.
If we perform iterative recursion in Eqs.(3, 6) approaching n → ∞, the three components become indistinguishable yielding a unification:
, (8a)
which leads to a convergence of ternary properties of Eq.(7a) to solitary
, (8b)
with effective properties corresponding to component (1), (2) or (3) depending on selection of initial probability (p0, q0, r0) at the first iteration step n = 0. Thus the probability domain, D~(p, q, r), can be divided onto three regions where the renormalization process leads to one of the corresponding stable roots:
, (8c)
. (8d)
We notice that at (p0 = p*, q0 = q*, r0 = r*) properties of components are similar. These points can be critical or bicritical and excluded from the probability domain, D~(p, q, r), according Eq.(8d).
In our calculations of the properties of the composite we suppose that the following ratio between initial probabilities of components is given:
, (9a)
. (9b)
Of note, if α → 0 then q → 0 and three-component composite approaches two-component with property p + r = 1. On the other hand, if α → ∞, then r → 0, which leads in turn to p + q = 1. Geometrical interpretation of the condition given in Eq.(9) means that properties are investigated along a line embedded in domain D(p, q, r).
In a special case we consider a situation when arc, covering domains, D(1,0,0) (p, q, r), D(0,1,0) (p, q, r), and D(0,0,1) (p, q, r), represents a trajectory of initial probabilities satisfying Eq.(1). Such a pathway is necessary to describe a phenomenon termed as a double percolation [24–26]. This phenomenon has been found to arise from the formation of an irregular, highly conducting layer following the boundaries between the conducting and the insulating components. Effectively, the system thus contains three components. It was experimentally found [24] that after achieving a connected cluster composed from one of the components afterwards it is substituted by another component. As a consequence this leads to a sharp change of effective properties twice in a course of variation of fraction of one of the components (Fig.1d).
The arc trajectory can be modelled by a part of a generalized superellipse curve [27] yielded in a parametric form:
, (9c)
where , –π/6 ≤ и
≤ π/6, and a = b = 1. We chose n1 = 6 to create a superellipse embedded in hexagon. Other parameters, n2 = 2 and n3 = n4 = 12, were used to identify a double percolation effect and create a maximum amplitude and curvature of a cusp of a parametric curve in a selected segment. Increasing n2 and decreasing n3 leads to the reduction of curvature and amplitude, and as a consequence, double percolation may disappear. Parametric curve Eq.(9c) was normalized and rotated to be included in a polygon Eq.(1).
Local model of susceptibility function and its effective value
We use the Maxwell model of a local dynamic susceptibility (or response) function, [28]:
, (10)
where τ is a relaxation time, and ω is a frequency of an external field, G is a constant of an active component (or local static susceptibility). The characteristic frequency, ω0, can be defined as ω0 = 1/ τ. On the initial level (n=0) of iterative procedure the following unification
(11)
of dimensionless response functions represents parameters of three components in a generalized probability density function in Eqs.(2, 7a). In Eq.(11) parameters of the active component were normalized to G1 in the far frequency region, ω >> ω0. For example, if we impose inequalities G1> G2 > G3 and τ1 > τ2 > τ3, component (1) is slowly relaxing with the very large constant of active part, and the fast relaxing and lowest constant of active parts represent component (3).
The balance of all forces at any location within composite assumes a conservation law:
, (12)
if uk is a displacement on kth node. The distribution of local generalized susceptibilities, G*(km), in Eq.(12) is governed by Eq.(7a) and at each step n of renormalization should depend on the index, i.e. G*(km), n, in a general case.
The Dirichlet boundary condition in Eq.(12) corresponds and on the left, Г+, and right, Г–, sides of the unit cell respectively (Fig.1a). In Eq.(12) we assume lattice constant a = 1. Using the Kronecker delta symbol, δkm, it is possible to introduce a matrix of response functions, , which the matrix element, , is given as:
, (13)
and to rewrite Eq.(12) as a linear equation:
, (14)
where ck denotes the external force which is zero but on the right side boundary.
After solving Eq.(14) relatively to unknown um, we find the equivalent susceptibility of the lattice in Eq.(3) expressed in a form
, (15)
where nodes k are taken only for the left contacted boundary Г+. From the structure of values in Eqs.(10,11,15) it is clear that the equivalent susceptibility functions, , are complex-valued with active real and relaxing imaging parts. The statistical averaging of Eq.(15) should be performed according to distribution in Eq.(2) on a whole set E yielding effective susceptibilities, , for connected and disconnected clusters. If we carry out the iterative procedure we approach stable effective susceptibilities according to Eq.(8a) which are analyzed in the next section.
Results of numerical simulations
to perform a numerical analysis of the effective susceptibility function we, first of all, analyze a renormalization flux of fractions of components during the renormalization procedure. For this purpose we built a grid with coordinates of nodes (p0, q0, r0) which satisfy Eq.(1). Then we perform a transformation according to Eqs.(5, 6). This transformation provides a new grid with nodes (p1, q1, r1). The vector field of the renormalization flux can be constructed as (p1 – p0, q1 – q0, r1 – r0) and then assigned to the initial grid. The maximal norm of any vector cannot exceed 1 and minimal norm is bounded by 0. In Fig.1b we demonstrate a triangle described by Eq.(1) seen from the top of its normal vector, . Arrows represent a renormalization flux projected onto the polygon. The flux achieves minimal norm at the corners of the triangle at vertices A, B, and C in Fig.(1b). These are stable points of renormalization because the vector field converges to them from any direction.
Eqs.5, 6 allow the introduction of a red-green-blue color-coding scheme where the color red corresponds to a stable vertex A =(1, 0, 0), green B=(0, 1, 0) and blue C=(0, 0, 1). The mixture of colors is represented by a vector (P(p0, q0, r0), Q(p0, q0, r0), R(p0, q0, r0)) from Eq.(5), which is not aligned to any of the Cartesian axis in a general case. This color-coded background is demonstrated in Fig.1b as well.
At certain points of the polygon in Fig.1b the vector field diverges to two or three points of a set {A, B, C}. The minimal norm of the flux can be found in the vicinity of points L5 = (0.5, 0.5, 0.0), L4 = (0.5, 0.0, 0.5), L2 = (0.0,0.5,0.5) on the edges of the triangle. These are unstable points of renormalization because the vector field diverges there. Between them we can discriminate critical points, L4 and L5, characterized by a possibility of divergence in the direction of one of the two components. Bicritical point, L2 , is characterized by a possibility of divergence in the direction of one of the three components. Interestingly, we can find two lines, L2L4 and L2L5, where unstable critical points are located. These lines partition probability domain, D(p, q, r), as it is pointed in Eq.(8c).
Renormalization along edges AB, BC, AC corresponds to a two-component composite. Functions P(p, q, r), Q(p, q, r), and R(p, q, r) are plotted in Fig.1c according to Eqs.(5) if r, p or q are set to 0 respectively. A zigzag pathway represents an evolution of pn during the execution of Eq.(6). The number of iterations for building a hierarchical lattice during the renormalization procedure is defined by the constraint |pn – pn+1| ≤ 1–3 and presented at the same Fig.1c.
We vary parameter p keeping the ratio, α, between the other components unchanged (Eq.(9a,b)). Such a procedure, for example, corresponds to lines AB, AL1, AL2, AL3, AC depicted in Fig.1b as outgoing from vertex A. Test values of parameter α are shown in Table 1. We also use arc (Eq.9c) as a pathway of the variation parameter p represented as a part of a superellipse to model a double percolation effect.
For the description of the external load the frequency window, ω = [10–3; 102] rad/sec, was selected in our calculation. The real (i.e. active) and imaginary (i.e. relaxing) parts of the effective susceptibility can be demonstrated as surfaces in logarithmic scale as functions of frequency (in logarithmic scale) and fraction p. Here logarithmic scale assists to compare values with a large diversity. To evaluate sensitivities of effective values to input parameters we overlay a contour curve onto the surfaces with level –4 in all cases. We also projected this curve onto a plane (log 10(ωτ), p. The surfaces of real and imaginary parts of effective susceptibility are color-coded, where the colors red and blue correspond respectively to the minimum and maximum distance of the point on the surface relative to the 0 level. The color-coded scheme of the surfaces is projected onto a plane to facilitate a quantitative evaluation of parameters.
In Fig.2 we demonstrate effective susceptibility along lines AB and AC from Fig.1b corresponding to a two-component composite with parameters taken from Table 1. Formally the variation of p along these lines is defined by α → 0 and α → ∞ (in our numerical calculations α = 1000). In Fig.2a and Fig.2c we can see that in the high frequency limit, ωτ → ∞, the real part of the dynamic effective susceptibility function approach its static values for components (3) or (2) from Table 1 if p → 0. On the other hand the active part of effective susceptibility is frequency independent if p → 1 taking the value corresponding to the component (1) from Table 1. In a low frequency limit, ωτ → 0, the real part of effective susceptibility demonstrates a linear dependence which transits into plateau near characteristic frequencies ω2 → 1/τ2 (Fig.2c) and ω3 → 1/τ3 (Fig.2a) at p < p*. The transition corresponds to the maximum of the imaginary part of the effective susceptibility at the same characteristic frequencies (Figs.2b,d). The characteristic frequency is shifted to the lower limit if we increase the fraction p → p*. At the point p = p* we observe a jump of the real and maximum of the imaginary parts of the effective susceptibility. Besides, the imaginary part of the effective susceptibility achieves its global maximum at the characteristic frequency at p = p*.
In Fig.3 we present results of the calculation of effective susceptibility for a three-component system (parameters are given in Table 1). The trajectory follows lines AL1 and AL3 (Fig.1b). Due to the large difference in initial static susceptibilities of components (2) and (3), i.e. G(2),0 >> G(3),0, the behaviour of the effective susceptibility is very similar to the two-component system described earlier. Contrarily to the previous situation, if initial static susceptibilities of components (2) and (3) approach each other, we can observe affects which are specific to the three-component system. For the simulation reason we keep the same ratio of initial relaxation times, but assume G(1),0 > G(2),0 = G(3),0 (see Table 1) and follow lines AL1, AL2 and AL3 (Fig.1b).
In this case the effective active part is not very sensitive to α although we observe a slight folding of plateau near ω2 and ω3 (Fig.4a,c,e). This can be evaluated by the iso-contour curve projected onto plane(log 10(ωτ), p). For the case of the effective active part we see two shifts and three plateaus if we consider a projection iso-contour curve. For the better estimation of the effect in Fig.4h we plotted the effective active part for p → 0 as in logarithmic scale (grey colored curve) as without (black colored curve). There we observe two transitions near characteristic frequencies, ω2 and ω3 (thin dash lines).
At the same time, the effective relaxation component exhibits high sensitivity to the parameter α. We observe two distinct maximums of effective susceptibility at percolation line p = p* (Fig.4b-f). If α = 0,5, the global maximum is shifted to the high frequencies (Fig.4b). In case of α = 1 maximums at high and low frequencies are nearly the same (Fig.4d). For α = 2, maximum is shifted to the low frequencies (Fig.4f).
We can also demonstrate a double percolation phenomenon described in Refs.[24–26]. For this purpose the trajectory of initial concentrations p corresponds to arc (Fig.1b and Eq.(9c)). Following this trajectory means achieving a percolation of component (2) near the edge BC, then simultaneously reducing of the fractions of the components (2) and (3), and increasing the fraction of the component (1) we approach the stable point A on the edge AC. Such a transformation assumes that the percolation cluster generated from component (2), afterwards is substituted by the percolating cluster composed from component (1). The result is demonstrated in Fig.5 for different initial values of active components collected in Table 1. For simplicity it is assumed that the relaxation rate is the same for all compounds.
What can be noticed in Fig.5 is that the effective susceptibility significantly changes twice near p = pI ≈ 0,1 and p = pII ≈ 0,8. Depending on the initial active values we model three possible situations of the variation of effective susceptibility as a function of the concentration p, namely, effective susceptibility can be concave (Fig.5a), increasing (Fig.5c) and convex (Fig.5e) function along the parameter p. To estimate the extent of the effect we plotted the real part of effective susceptibility with a limit ωτ → ∞ (Fig.5h) in logarithmic scale and without.
CONCLUSIONS
In this work we investigated susceptibility properties of a random three-component system using newly developed hierarchical scale averaging approach. We have shown that in the case of a large discrepancy of the static local susceptibilities dynamic effective properties are similar to a two-component system. If static susceptibilities of the components (2) and (3) approach each other keeping relaxation parts different, then peculiarities of the three-component system become apparent. In this case the effective active part possesses two plateaus and the relaxing part demonstrates two maximums. Amplitudes of maximums for the relaxing part depend on the dominating component.
Also for the first time we investigated a case of double percolation showing that the dynamic effective properties of susceptibility change two times. In the first case change is associated with creation of the percolation cluster built from the component (2), the second change is linked to the extrusion of the component (2) and (3) by the component (1) which builds a secondary percolation cluster.
Appendix A
Rotations in three dimensions can be described equivalently by either the SO (3) or SU (2) representations [29]. The first representation uses 3 × 3 rotation matrices and 3 × 1 vectors. The matrices are orthonormal and their matrix elements are real. In the SU (2) representation, a general rotation matrix U can be written:
(A1)
which acts on a function z = f(x, y) in Eq.(9c) yielded as
(A2)
Elements of matrix U are complex numbers known as the Cayley-Klein parameters and the asterisk represents complex conjugation. The matrix U is unitary and Hermitian. The general rotation can be written:
(A3)
where symbol H is a Hermitian conjugate. In terms of Euler angles the general rotation matrix U can be decomposed:
(A4)
and
. (A5)
The mathematical simplification afforded by the SU(2) representation is a consequence of the fewer number of constraints. This situation is analogous to solving a system of linear equations, where it is nearly always easier to solve fewer equations with fewer constraints. For our application we combined rotations and translations
, (A6)
where ψ = 0, Ѳ1 = –90°, Ѳ2 = –45°, ϕ = 125,26 and
. (A7)
The curve described in Eq.(9c) can be transferred according Eq.(A6) to be embedded in a probability domain.
PEER REVIEW INFO
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Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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