THE MODEL OF MISCIBILITY PHASE TRANSITIONS IN MIXED JAHNTELLER SYSTEMS
1
Fishman A.Ya., 2Ivanov M.A., 1Shunyaev K.Yu., 3Tkachev N.K.
1
Institute of Metallurgy, RAS (Ural Division), Ekaterinburg, Russian Federation
2
Institute of Metal Physics, NAS of Ukraine, Kiev, Ukraine
3
Institute of High Temperature Electrochemistry, RAS (Ural Division), Ekaterinburg,
Russian Federation
ABSTRACT
Miscibility in crystal and liquid ionic systems with Jahn-Teller (JT) ions is
studied theoretically. Dependence of the phase diagram topology upon dominated
mechanism of degeneration removal (either cooperative interaction or random crystal
fields) is established. Possible types of phase equilibria in mixed JT crystals are
discussed within mean-field approach to the JT interactions. It is shown that
satisfactory agreement of the developed theory with the experimental phase diagrams
of Mn3-x AxO4 solid solutions takes place. In case of ionic melts random fields (caused
by topological disorder) are the only reason of miscibility of JT nature. Both the
boundaries of possible phase instabilities and the thermodynamics of homogeneous
liquid mixture turned out to be very sensitive to the concentration dependence of the
random fields' dispersion.
I. STUCTURAL PHASE TRANSFORMATION AND
RESTRICTED SOLUBILITY IN JAHN-TELLER SOLIDS
MODEL
OF
The substitution of orbitally nondegenerate ions for JT ions in such mixed
systems can lead to a separation into phases with a higher and lower concentrations of
JT ions at low temperatures. Structural phase transformations of the displacement type
are related to a certain extent to phase transitions of separation type due to the fact that
in both cases the system has a tendency to maximum possible splitting of degenerate
states. This also applies to JT crystals with cooperative structural phase transition of
the first and second order. Peculiarities of the phenomena under consideration are
associated to a considerable extent with random crystal fields which are always present
in disordered systems. Like cooperative interactions between JT ions, such random
fields can be the main mechanism of splitting of degenerate states, and hence can
considerably affect the phase equilibrium in binary mixtures under investigation.
Structural phase transformations (PT) and other properties of JT systems in random
crystal fields were investigated by many authors (see, for example, [1]). It was proved
that such fields reduce the transition temperature and the order parameter and can
suppress the PT completely in the case of enough high intensities. Obviously, random
crystal fields must also affect the position of the boundaries of immiscibility regions in
systems with structural PT. Besides, such fields themselves can lead to thermodynamic
instability of some phase states. The interest to the influence of random crystal fields
on the thermodynamic properties and phase transformations in JT systems is
considerably stimulated by the advances in experimental investigations of crystals with
anomalous magnetoresistance, doped fullerites, HTSC oxides, etc., i.e., the materials in
which centers with a degenerate or pseudo-degenerate ground state strongly affect their
properties.
271
Model Of Structural PT Associated With The Cooperative JT Effect
Let us consider for simplicity structural transformations of the “ferro”-type in
cubic crystals with doubly degenerate JT ions. In the case of the structural phase
transition of the second order the Hamiltonian of the JT subsystem in the molecularfield approximation can be written in the form


ˆ    λ  h jz σ jz  h jx σ jx  ,
Η
j
(1)
1 0 
0 1
 , σ x  
 ,
σ z  
 0  1
1 
where  is the parameter of the cooperative interaction between JT ions, hjz and hjx are
the components of random crystal field at the j-th JT ion, jz and jx are the orbital
operators defined in the space of wave functions of an orbital doublet. Angle brackets
denote the quantum-statistical average (<…>T) and the average over the configuration
of random fields (<…>c) of the operator z (  <<z >T>c). A nonzero order
parameter in the model under investigation corresponds to the emergence of tetragonal
deformation of the lattice eJT = ezz – (exx + eyy)/2. Energy spectrum of a single JT
center, the partition function (ZJT ), free energy (FJT ) and the order parameter are
defined by the following equations:
2
E ( , h)    hz   hx2 , Z JT  , h   2 cosh  E ( , h)  ,
kT 

(2)
FJT
2
FJT   / 2  kT ln Z JT  , h  ,
0 .
c

In the case of the structural phase transition of the first order it is possible to
consider limits of weak and strong JT anharmonic coupling. The energy spectrum and
the partition function must be described by different expressions for these asymptotics.
For systems with weak mentioned interaction the two level model of JT ion can be
saved and the extra terms ~p3 must be added into the splitting energy E(,h), where
parameter of anharmonicity p<<1. For systems with strong anharmonic interaction the
triplet model of vibronic JT states has to be considered:
ˆ    λ  h jz U jz  h jx U jx  ,
Η


j
0 
 0
0 0 0
(3)



3
U z   0 1/ 2 0  , U x 
0 1 0 ,
 
0

0 1 / 2 

0 0 1
where Uz and Ux are the orbital operators, defined in the space of the vibronic triplet
wave functions. The corresponding expressions for partition function ZJT, free energy
FJT and order parameter << Uz >T>c can be found easily.
Here and below, we use for configuration averaging the random-field in
distribution function f(h)f(hz, hx) of the Gaussian type:
 h2  h2 
1
(4)
f (h)  2 exp  z 2 x  ,

 

where  is the dispersion of random crystal fields.
For example, an analysis of the dependencies of the order parameter and the
temperature TD of the second order PT from the “para”- to the “ferro”-phase, on the
dispersion can be carried out using the expressions [2]
272
    2  
 2 y 
 y 
2
  2 exp  
   y exp(  y ) I 1 
 tanh 
dy ,
  
T 
     0
(5)

 y 
(6)
dy  1 ,
0
TD 
where I1(x) is the Bessel function of the imaginary argument.
In the absence of random fields, when   0, the structural phase transformation
temperature TD is equal to 0 in the case of the second order PT and 0.5470 in case of
the first order PT in systems with strong anharmonic interactions. Intermediate cases
can be considered as well. Besides, within this simplest microscopical approach the
Gaussian random fields do not influence the type of the structural PT’s.
However in the region of critical values of  for which the PT disappears in the
system (TD()  0), the solutions of Eqs. (5) and (6) can be written in the form:
2
2
2
y exp(  y ) tanh 


2    

 TD 
 ,
(7)
  
1    ,  0 
0
3 (3) 
2
  
3
2
   T  
 2  
2
(8)
    1   1     ,
     0    TD  
where (y) is the Riemann zeta-function. Phase transformation in the JT system is
completely suppressed by random crystal fields when  > 0 = (/2)0 , i.e., when
their dispersion becomes larger than the energy of the cooperative interaction between
degenerate centers.
The above models allow us to analyze the phase equilibrium of systems with JT
ions for various types of substitutions in the crystal lattice [3]. We must only specify
the dependencies of the molecular field parameters and random-field dispersion on the
concentration of the replaced centers.
3
Model Of The Mixture
Let us consider the model of a mixture in which the free energy is determined by
the splitting of degenerate centers, and the configuration entropy corresponds to a
random distribution of JT ions in the mixture. In this case, the free energy of a quasibinary system in the mean- field approximation can be written in the form:
F  c JT FJT  Fid
(9)
Fid  kT c JT ln c JT  1  c JT  ln 1  c JT  ,
where the parameters  and  depend on the concentration cJT of the JT centers.
Expressions (9) allow us to calculate the chemical potentials a (of components
with JT ions), b (of component with orbitally nondegenerate ions), and the exchange
chemical potential  [4-6] of the quasi-binary system under investigation:
 c

 FJT
F
(10)

  a   b  FJT  c JT
 T ln  JT  ,
 c JT
 c JT
1  c JT 
F
 a  F  1  c JT    FJT  c JT 1  c JT  JT  T ln c JT .
(11)
 c JT
Expressions (10) and (11) make possible to calculate binodals, i.e., temperature
dependencies of concentrations of components in two equilibrium phases (with the
273
same values of  and a ). The boundaries of the region of absolute instability of the
solution to phase separation are determined by the condition
2F

(12)

0 .
2
 c JT  c JT
Eq. (12) defines the spinodal curve Ts(cJT), with the peak corresponding to the
critical (or consolute) point Tc , where the spinodal and binodal are coinciding.
Estimated equilibrium phase diagrams (binodals) are shown in Fig. 1 - 2. In case
of the second order PT’s and   cJT (Fig. 1a and b) in the temperature range below
the point of 1/3 phase separation into the “para” -phase and a more concentrated
“ferro” -phase of JT ions takes place. Random fields can be able to change
significantly the phase diagram in case of second order structural phase transition [2,3].
In particular, at strong fields the miscibility gap drops onto binary-peak configuration.
Fig. 1 b. shows how it happens at strong Gaussian fields. The value of parameter d can
be associated either with the difference in sizes of the substituting ion and the ion being
replaced, or with the difference in their valencies.
Fig. 1. Configurations of miscibility gaps at different intensities of random fields:
d/0 =0.5(a), 1(b).
It can be seen from formula (7) that the structural phase transformation in a
system with a random distribution of JT ions for the given concentration dependence of


dispersion occurs only for concentrations cJT > c0 , where c0  1  20 4d 2  .
Consequently, the boundaries of the region of absolute instability of the mixture
considered above (the region of spinodal phase separation in the “ferro”-phase) must
also be displaced towards higher values of cJT
Thus, the quasi-binary system is thermodynamically unstable in the entire
concentration range at low temperatures, i.e., it is advantageous from the energy point
of view for a system with JT ions in random crystal fields to undergo phase separation
(at T  0 K ) into phases with relatively high and low concentrations of degenerate
centers. The difference from a similar tendency for systems with cooperative JT effect
lies in the fact that both states mentioned above are highly symmetric, i.e., “para”phases.
Similar behavior is also observed for a wide class of JT systems with a structural
first order PT (see, for example, [7]). In this case the miscibility gap takes place over
the entire range of temperature below TD for cJT =1. In contrast to the system
274
1
considered above, such systems display the coincidence of the critical point of mixing
with the PT temperature in a pure compound with JT ions [8]. The typical phase
diagram for JT system with structural first order PT and strong anharmonical
interaction is presented on Fig. 2.
Fig. 2. Miscibility region for JT system with the first order structural PT.
Transformation of the spinodal shape by switching anharmonical interactions
(p0) is illustrated Fig. 3.
Fig. 3. Evolution of the spinodal decomposition boundaries due to anharmonical
contribution to the JT Hamiltonian. Numbers near curves denote the value of p
parameter.
It can be easily seen that with the increasing of anharmonical contribution into
Hamiltonian the spinodal decomposition region becomes broader with the slight
simultaneous increase of TD.
275
The influence of possible non-JT interaction with the energy of mixing Emix =
Wc(1-c) (where W={½ (EAA+EBB) – EAB}) on the shape of miscibility gap is shown on
Fig. 4.
Fig. 4. The influence of essential non-JT interaction on the shape of miscibility
gap in JT system with first order structural PT (W/0 = -0.5 (), 0.5 (), 1.2 ()).
For real systems such as Mn3-x AxO4 solid solutions, where A=Al, Cr, Co, Cu,
Mg, Zn quasibinary version of our model requires certain modifications. In particular,
it is necessary to take into account actual cation distribution upon tetrahedral and
octahedral sites. As a result the ideal contribution into the free energy must reflect the
form of this cation distribution. On Fig. 5 peculiar dependencies of two-phase region
from the distribution character are presented. It was proved that “sigar” – type of the
two phase region can occur. The content boundary of this region is determined by
concentration for which full occupation of tetrahedral positions by the only sort of a
cation takes place. Our results are in good agreement with the experimental data for the
following solid solutions of Mn3-x AxO4, where A=Al, Cr, Co, Cu, Mg, Zn and other
[7].
276
1
T/T D (0)
0.8
0.6
0.4
0.2
0
0.5
1
X Mg
1.5
Fig. 5. “Sigar”-like and “usual” form of the two-phase regions for
Solid curve and dashed curve
(Mn12-x Mg2x )[Mg 2x(1 ) Mn 4x(1 ) Mn32(1x x) ] O 4 .
correspond to  = 0.566 and 0.5.
It is necessary to note that the proposed models in their simplest form have no
any adjustable parameters. Only the temperature of pure Mn3O4 structural PT and the
data on cation distribution (if needed) were used.
II. THE EFFECT OF DEGENERATION IN THE THERMODYNAMICS
OF BINARY IONIC MELTS
In this section we review some of our recent results on the disordered systems,
where the above-mentioned mechanism is absent and the random fields is the only
mechanism of the degeneration removal [9]. One of the important examples of such
systems might be ionic liquids like molten salts or molten oxides with some types of
degenerate ions. The random fields in ionic liquids are related with the topological
disorder intrinsic for the liquid state. Nature of degeneration or pseudo-degeneration
can be widely diversified. In particular, the role of the degenerate centers in ionic
liquids might be played by some 3d ions like the orbital degenerate cations Mn3+, Fe2+
etc in solids.
Let us consider quasi-binary system AcB1-c , where each of the components is the
molten electrolyte of the type of MX. We shall restrict our consideration by the type of
binary system where degenerate ions D are presented only in one component of the
mixture DX-MX.
For the sake of simplicity we can further limit our analysis by the simplest case of
the degenerate doublet. Thus, the Helmholtz free energy of this mixture can be
presented as follows:
277
F  Fid  c Fd , (DX) c (MX)1c
Fid  c ln c  (1  c) ln (1  c) ,
,
(13)
(14)
 E2 
  2  dE ,
(15)




ln
2
cosh

E
exp

  0
  
where =1/kT is the inverse temperature and E is the splitting energy of degenerate
state,  is the dispersion of the random fields. For definiteness sake Gaussian’s
distribution has been used.
The proposed model must take into account the concentration dependence of the
dispersion as well. Let us choose it by means of traditional approximation:
2
Fd  
(c) =


2
2
2
A c   B (1  c)  
c(1  c) ,
2
2

  A  B
2
2

(16)
where parameter  describes the deviations of the dispersion from the additive law.
This choice should be offered to the ideal mixture when A=B and =0.
For considered systems the exchange chemical potential ex = F/c is the sum
of two contributions (idex and dex ). The ideal part is equal to

idex  ln c1  c

.
(17)
The corresponding contribution from the effects of the degeneration removal (dex =
Fd/c) is described by the expressions
F

2 
2
(18)
 dex  d  c
I (  ) , I ( ) 
 tanh  x exp  x x dx .
c
c
 0
Numerical estimations have proved that the higher level of doublet is mostly
negligible, at least for the miscibility problem. However, in our calculations
everywhere below (in figures) we kept general expression (15).
For the large variety of situations where we need the explicit description we can
simply use the expansion of the hyperbolic functions in the integrals. As an example,
the expression for the exchange chemical potential can be rewritten
2
2 
 F 
 dex   d   
1  2 e  1  erf ( ) .
(19)
 c
 c T ,V
So in further analysis the expression for the total exchange potential will be
given in the asymptotic limit within the first term of (19), which admits the explicit
consideration. In the case kT <<  the exchange chemical potential ex has the
following form for the mixtures under consideration






 ex   idex     3 c  A2   B2   2 B2   2 c 3  4 c  .
(20)
Negative deviations from ideality take place for the case when dex/c is
positive. In contrast, for negative values of this derivative the positive deviations can
be noticed, which is responsible for the miscibility at low temperatures. Boundaries of
absolute instability of the mixture, with respect to the phase separation onto two liquids
of different concentrations, must be determined by the following equation
2F/c2 = ex/c = 0 .
(21)
If the miscibility is present in some system then the equation (21) defines the
concentration dependence of the spinodal Ts(c). The maximum of the latter coincides
with the consolute or critical point Tc .
In the supposed model of mixture the character of deviations from ideality of the
thermodynamic properties is controlled by two parameters: the difference of the
1
278
dispersions in pure components A - B and the magnitude of the dispersion deviations
from the additive law (). To understand the effects of these both responsibilities on
the thermodynamics our analysis was carried out mainly for the following limiting
cases: 1)  = 0, i.e., the deviations of the dispersion are absent and 2) A  B , i.e.,
only these deviations are meaningful.
Associated spinodals are obtained by the equations
 3  A2   B2  4 2

 B  c  A2   B2 
(  0 )

3


kTs
 4   3

.
(22)
c (1  c)   2
2
2
 A   B 
4 B 3 c  1   c 3  4 c 1  2 c 
 4 3 

For the case of equal dispersions in both of the pure components the miscibility
can be realized only if the sign before 2 in the concentration dependence (16) is
negative. The displayed asymptotic formulae in (10) has suggested namely this choice
of the sign and the fulfillment of the condition (/) < 2.
It is clearly seen that both of the considered features of the concentration
dependence can be responsible for the miscibility. However, it causes to the unlike
shapes of the spinodals, and obviously, to the unlike shapes of the miscibility gaps
(binodals).






0.09
3
kT/
1
0.08
0.07
2
4
0.06
0.05
0.1
0.3
0.5
0.7
CA
0.9
Fig. 6. Calculated miscibility gaps for systems of the first type. The deviations of
the dispersion are absent (solid curves 1 and 2 relate to A/B=1.25 and 1.3,
correspondingly). The dispersions of pure components are equal to each other (dashed
curves 3 and 4 relate to negative deviations of the dispersion with A/B=1, /B=0.7
and 0.8).
279
The sensitivity of the miscibility regions to these parameters (A  B and ) is
shown in Fig.1. The miscibility gaps were calculated by means of standard equilibrium
conditions, for chemical potentials of each component between two coexisting phases
of different concentrations. These chemical potentials can be easily found in terms of
the exchange potential and the free energy. It can be seen readily that the coexisting
curves have the classical (or mean-field like) parabolic form with the above mentioned
asymmetry effects. For the case of  = 0 this asymmetry is small and the gap is
displaced insignificantly to the component without degenerate ions. The value of this
displacement is increasing with the increasing of the dispersions mismatch (A  B).
In contrast, if this mismatch is quite small, the miscibility gap has the strong tendency
to be displaced to the liquid with the degenerate states.
Fig. 2a and 2b illustrate the concentration behavior of the thermodynamic mixing
functions for various ratios of (AB) and . For the case when A = B and negative
deviations of the dispersion are realized we have quite non-trivial depiction at the
temperature region above the consolute one. In particular, the activity of the
component B has the narrow range of negative deviations from the Raoult’s law. The
curves 3 and 4 display the existence of negative deviations of the activities even when
A = B . Therefore, this simple model of the molten salt binary (DX)с(MX)1-с
incorporating the degeneration effects gives the noticeable variety of predictions both
for the shapes of the miscibility gaps and for the concentration behavior of the
thermodynamic mixing functions in homogeneous liquid mixture.
Fig. 7. Thermodynamics of mixing for the mixtures with the only sort of
degenerate ions. Various contributions into the free energy of mixing (a) if A = B and
/B=0.7; kT/B =0.05, which is slightly above the critical one. Activities of the
components (b): curves 1 and 2 relate to Fig. 2a situation and the dashed curves 3 and
4 relate to positive deviations of the dispersion; the absolute values of  and
temperature are the same as in the case (a).
We have shown that the presence of degenerate centers in systems with
topological disorder is capable to cause the miscibility transitions and lead to quite
non-trivial behavior of the thermodynamic mixing functions. Concentration
dependence of the random fields dispersion controls the deviations of the
thermodynamic properties from ideality. The universal mechanism of the miscibility
for both considered types of the molten salt binaries with degenerate centers was
determined. This mechanism consists in the tendency to maximize the removal of
degeneration and as the consequence these ions tend to be segregated in the phase with
higher random fields. Thus the proposed model of ionic mixture provides the
description of large variety of situations on the behavior of the thermodynamic mixing
functions including the asymmetry effects and the deviations from ideality of both
signs.
CONCLUSION
The analysis carried out here proved that limited miscibility is manifested to a
certain extent in partly-disordered and topologically disordered systems with Jahn –
Teller ions. The variety of its possible manifestations must reflect different nature of
degenerate or pseudo-degenerate states and accordingly the mechanisms of their
splitting. The same mechanism operates for all types of PT. It is associated with the
tendency of JT ions at low temperatures to be in the phase with the maximum possible
splitting of degenerate levels.
280
The results of analysis of possible instability regions for one-phase states makes
it possible to predict anomalous behavior of some parameters of disordered JT mixed
systems, in which the equilibrium state of the JT subsystem is not realized in view of
kinetic limitations. This primarily refers to the formation of thermodynamic properties
typical for glass-like states.
This work was supported by Russian Foundation of Fundamental Researches,
Grant № 00-03-32335.
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