On the heterophase liquid thermodynamics and cooperative dynamics
The thermodynamics and cooperative dynamics of heterophase liquid states is considered taking into account frustration and volumetric interaction of the solid-like fluctuons. It is found that the glass transition temperature range is scaled by difference of the frustration parameter and mean energy...
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| Zitieren: | On the heterophase liquid thermodynamics and cooperative dynamics / O.S. Bakai // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23604: 1-9. — Бібліогр.: 25 назв. — англ. |
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| citation_txt | On the heterophase liquid thermodynamics and cooperative dynamics / O.S. Bakai // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23604: 1-9. — Бібліогр.: 25 назв. — англ. |
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| description | The thermodynamics and cooperative dynamics of heterophase liquid states is considered taking into account frustration and volumetric interaction of the solid-like fluctuons. It is found that the glass transition temperature range is scaled by difference of the frustration parameter and mean energy of the volumetric interaction. A model of the cooperative relaxation with finite cooperatively rearranging domains is considered. The fictitious Kauzmann and Vogel-Fulcher temperatures are determined. It is found that they are close to a proper accuracy. Correlation of the temperatures of glass transition, Vogel-Fulcher and "ideal" glass transition (as it is determined in the mode coupling model) is considered too.
Розглянуто термодинаміку та кооперативну динаміку гетерофазної рідини з урахуванням фрустрації й об'ємної взаємодії твердотільних флуктуонів. Установлено, що ширина температурного інтервалу перетворення рідини в скло є пропорційною різниці параметра фрустрації та енергії об'ємної взаємодії. Знайдено уявні температури Кауцмана і Фогеля - Фулчера. Вони виявились близькими з гарною точністю. Розглянуто кореляцію температури склування, Фогеля - Фулчера та температури "ідеального" перетворення рідина - скло, яке описано в моделі взаємодіючих мод.
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Condensed Matter Physics 2010, Vol. 13, No 2, 23604: 1–9
http://www.icmp.lviv.ua/journal
On the heterophase liquid thermodynamics and
cooperative dynamics
O.S. Bakai
National Scientific Center “Kharkiv Institute of Physics and Technology”,
1 Akademicheskaya Str., 61108, Kharkiv, Ukraine
Received February 22, 2010
The thermodynamics and cooperative dynamics of heterophase liquid states is considered taking into account
frustration and volumetric interaction of the solid-like fluctuons. It is found that the glass transition temperature
range is scaled by difference of the frustration parameter and mean energy of the volumetric interaction. A
model of the cooperative relaxation with finite cooperatively rearranging domains is considered. The fictitious
Kauzmann and Vogel-Fulcher temperatures are determined. It is found that they are close to a proper ac-
curacy. Correlation of the temperatures of glass transition, Vogel-Fulcher and “ideal” glass transition (as it is
determined in the mode coupling model) is considered too.
Key words: heterophase liquid states, glass transition, Kauzmann and Vogel-Fulcher temperatures
PACS: 64.70.Ja, 64.70.Pf, 61.20.Gy
1. Introduction
The heterophase fluctuations model (HPFM) is based on the idea of heterophase structure of
glass forming liquids [1–5]. Consecutive consideration shows that in a heterophase state (HPS),
molecular clusters (fluctuons) of specified short-range order (SRO) are statistically significant en-
tities rather than single molecules. As a result, the supramolecular interactions determine the
thermodynamics and dynamics of the glass forming liquids. Equation for the free energy of HPS is
deduced within the framework of HPFM taking into account frustration and volumetric interaction
of the fluctuons [5, 6]. The model of the cooperative relaxation dynamics of HPS developed in [2, 3]
describes the α-relaxation rate in terms of the free energy. The developed theory is a proper base
for the analysis of experimental data on thermodynamics, structure and cooperative relaxation of
the glass-forming liquids [2–4].
The Vogel-Fulcher and Kauzmann temperatures (TVF and TK) [7–9] and the temperature of
the “ideal” glass transition determined within the framework of the mode-coupling model (TMCT)
[10, 11] are in use at the analysis and rationalization of experimental data. At TK the extrapolated
entropy of liquid becomes equal to that of crystal or, in other words, the configurational entropy
of liquid is going to zero. In [9, 12] it is assumed that a critical point is located at Kauzmann
temperature, TK < Tg while TVF is the assumed singular point of the cooperative relaxation [7, 8].
The relaxation time at this point is ∼ exp (TK/(T − TK)). The empiric Vogel-Fulcher formula
was obtained by Adam and Gibbs in a theoretical model [13]. They connected the size of the
cooperatively rearranging domain with the configurational entropy of liquid. The size increases
with the configurational entropy decrease while at TK it goes to infinity. As a result, TK is a singular
point of the cooperative relaxation and TK = TVF. Though TK and TVF are not accessible for a
direct observation, the formulas obtained in Gibbs-DiMarzio and Adam-Gibbs (AG) models (along
with the empiric definition by Kauzmann) make it possible to rationalize and fit the experimental
data on the liquid configurational entropy and relaxation cooperative rates.
Determination of TMCT is given in terms of the basic relations of thermodynamic and dynamic
properties of glass-forming liquids under special assumptions [10, 11]. The predicted “ideal” glass
transition has not been observed yet but this model and its consequences have been under consid-
c© O.S. Bakai 23604-1
http://www.icmp.lviv.ua/journal
O.S. Bakai
eration for several decades. In Odagaki’s model of the cooperative relaxation [14] and in [15], where
this model is reconsidered without resource of the configurational entropy arguments, the following
correlations of the singular temperatures are found: TK = TVF and TMCT − TVF = 2(Tg − TVF).
By means of experimental data fittings, the mentioned temperatures are determined for many
glass-forming liquids. It is found that TMCT > Tg and correlations TK ≈ TVF and TMCT − TVF ≈
2 (Tg − TVF) are true with an acceptable accuracy (see e.g. [15–17]). For this reason, it is desirable
to check whether these correlations are consistent with the HPFM predictions. This consideration
is performed below. Furthermore, some peculiarities of the HPFM including volumetric interaction
and frustration of solid-like fluctuons are considered.
In section 2, the thermodynamics of HPS is considered. Section 3 is devoted to a description
of cooperative relaxation dynamics in terms of thermodynamic properties of glass-forming liquid.
Consideration of the mentioned correlations and discussion is placed in section 4.
2. Glass transition in HPFM model
To get the characteristic temperatures TK and TVF, whose definitions have no connection with
microstructure of glass-forming liquid, the coarsened HPFM without specification of the short
range order of solid-like fluctuons has to be used. In this formulation of the HPFM [1, 3, 6], solid-
like fluctuons having the averaged SRO and thermodynamic properties are considered. In this case,
the order parameters of the liquid are fractions of the solid-like and fluid-like fluctuons, cs and cf ,
respectively. Due to the evident relation,
cs + cf = 1 (1)
only one of these parameters is independent.
For simplification, the number of molecules per the solid-like and fluid-like fluctuon are put to
be equal, ks = kf = k0. In this case the free energy in the mean-field approximation is
G(P, T ) =
∫
g(x; P, T )d3x. (2)
Its density can be presented as follows:
g (x, P, T ) = cf(x)g0
f + cs(x)g0
s + cs(x)cf(x)gsf +
1
2
gssc
2
s
+
1
2
∫
cs(x)Φ(x, x′)cs(x
′)d3x′ + T (cf ln cf + cs ln cs) . (3)
Here g0
i (P, T ) is the part of free energy of i-th fluctuon independent of fractions {ck}; g0
sf (P, T ) > 0
is the interfacial energy; gss > 0 is the frustration parameter; Φ(x, x′) is the attractive potential of
the solid-like fluctuons which we take in the Yukawa form
Φ(x, x′) = −
r0ϕ
r
exp(−r/R0), (4)
where r0 is the fluctuon size; ϕ > 0 and R0 > r0 is the strength and range of the potential,
respectively. The last term in (3) describes the contribution of the mixing entropy of solid-like and
fluid-like fluctuons.
The attractive volumetric interaction of fluctuons can lead to long-range correlations of solid-
like fluctuons (the Fischer cluster formation) [4, 6]. The contribution of long-range correlations of
solid-like fluctuons into the free energy is comparatively small [6] and can be neglected for a while.
Therefore, we can replace cs(x) by its mean value cs in (3). As result, equation (3) reads
g(x, P, T ) ⇒ g(P, T ) = cfg
0
f + csg
0
s + cscfgsf + 1
2 (gss − ϕ0)c
2
s + T (cf ln cf + cs ln cs) , (5)
ϕ0 = 4π
∫
Φ(r)r2dr′ = −4πR2
0ϕ. (6)
23604-2
On the heterophase liquid thermodynamics and cooperative dynamics
The equation determining the liquid equilibrium state, ∂g/∂cs = 0, reads
(1 − 2cs)g̃sf + T ln
cs
1 − cs
= hsf , (7)
g̃sf = gsf − (gss − ϕ0) /2, hsf = g0
f − g0
s − (gss − ϕ0) /2.
It follows from (7) that the phase coexistence temperature, Te, (at which the solution cs = 1/2
exists) is determined by the following equation
hsf = 0. (8)
To show the effect of the frustration and volumetric interaction on the equilibrium and glass
transition, let us introduce the phase equilibrium temperature without fluctuon interactions as
follows:
T 0
e = ∆εsf/∆ssf . (9)
Here ∆εsf and ∆ssf is the difference of energies and entropies of the non-interacting solid-like and
fluid-like fluctuons at the point where g0
s (P, T ) = g0
f (P, T ). Then, as it follows from (8), the phase
coexistence temperature is,
Te = T 0
e − (gss − ϕ0)/2∆sf,s . (10)
The frustration decreases while the volumetric interaction increases the phase coexistence temper-
ature.
Apparently, for the majority of substances the glass transition proved to be continuous. It
follows from equation (7) that the continuous transformation takes place if
g̃sf 6 2Te (11)
or
gsf < 2[T 0
e + (gss − ϕ0)(1 − 1/∆sf,s)/2]. (12)
Taking into account that the difference of entropies of fluid and solid is ∼1 per molecule and that
k0 ∼ 10, we have ∆sf,s � 1. Therefore, with gss ∼ gsf the condition (10) can be easily fulfilled.
It is of particular interest to consider the heterophase fluctuations in Frenkel’s limit [18], at
cs → 1 and cs → 0, i.e., for almost pure solid and fluid phases with small amount of heterophase
fluctuations. In these cases, the solutions of equation (7) take the following form
c(2)
s (T ) = 1 − exp
{
[∆sf,s(T − T 1
e ) − gsf ]/T 1
e
}
, (13)
c(3)
s (T ) = exp
{
[∆sf,s(T
0
e − T ) − gsf ]/T 0
e
}
(14)
where
T 1
e = T 0
e − gss/∆sf,s = T 0
e − ∆Tg . (15)
Evidently, ∆Tg = gss/∆sf,s is the glass transition temperature range of the equilibrium liquid. We
remind that the glass transition temperature Tg depends on the cooling rate because in the vicinity
of T 1
e the relaxation dynamics dramatically slows down, the liquid becomes non-equilibrium and
non-ergodic. Therefore, calorimetrically determined Tg is above T 1
e .
From equations (13)–(15) we see that the frustration, apart from decreasing the phase coex-
istence temperature Te, causes the appearance of two characteristic temperatures, T 0
e and T 1
e .
The physical meaning of the temperatures T 0
e and T 1
e is clear. At T 0
e , the transformation of an
almost homogeneous fluid into the heterophase liquid state takes place. The non-Frenkelian het-
erophase liquid state exists within the temperature range
[
T 0
e , T 1
e
]
. The fraction cs increases with
the temperature decrease and at T 1
e the heterophase liquid transforms into a glass with weak fluid-
like heterophase fluctuations. Therefore, T 1
e may be regarded as thermodynamic glass transition
temperature.
23604-3
O.S. Bakai
To complete the description of cs(T ) we find a solution of equation (6) in the vicinity of Te. It
can be presented in the form of a series in T − Te [1]
cs(T ) =
1
2
+
hs,f(T )
2(2T − g̃s,f)
[
1 −
2Teh
2
s,f(T )
3(2T − g̃s,f)3
]
=
1
2
+
Te − T
2δT
[
1 −
2Te(Te − T )2
3(δT )3
]
, (16)
where
δT =
2T − g̃s,f
∆ss,f
. (17)
To estimate the role of correlation of the solid like fluctuons, let us take the pair correlation function
wss(r) = 〈cs(x)cs(x
′)〉 = c
S
ωss(r) + c2
s = cs
(
r0
r
)3−D
exp(−r/ξ) + c2
s (1 − exp(−r/ξ)) , (18)
r > r0, r = |x − x′| ,
where D 6 3 and ξ > R0 is the fractal dimension and correlation length, respectively. The last
term in (18) describes the input from non-correlated fluctuons. The exponent exp(−r/ξ) describes
a smooth decrease of the pair correlation on the scale r ∼ ξ. It follows from (18) that
ωss =
[
(r0
r
)3−D
− cs
]
exp(−r/ξ), r > r0 . (19)
Neglecting the long-range correlations, we can take for estimations ξ ∼ R0. Then the correlation
energy is [6],
εcorr(cs; D, ξ) ≈ 2π
∞
∫
r0
Φ(r)ωss(r)r
2dr′ ∼ −c̄2
s
(
ξ
R0 + ξ
)D−1
|ϕ0| ∼ −c̄2
s
(
1 − (D − 1)
R0
ξ
)
|ϕ0| ,
ξ (cs, D) ≈ (cs)
1
D−3 r0 . (20)
One can see from this equation that εcorr is proportional to c̄2
s , ϕ0. It increases with the temperature
decrease due to an increase of D, ξ and c̄2
s . Formally, the correlation of the solid-like fluctuons can
be taken into account by substitution ϕ0 → ϕ̃0(T ) ≈ ϕ0 (1 + β(Te − T )) in equations (7), (13)–(17).
Here β is the logarithmic derivative of ϕ̃0(T ) at Te.
3. The cooperative relaxational dynamics of the heterophase liquid
(α-relaxation)
A model of the cooperative relaxational dynamics of the heterophase liquids was proposed in
[2, 3]. It is based on a simple physical idea. The orientation and location of fluctuons are correlated
due to the volumetric interaction on a length ξ which is comparable to the interaction potential
range R0. Single jumps of molecules do not change the correlations within the correlated domain
(CD). However, cooperative rearrangement, involving many molecules within CD can destroy and
change the local correlation. Let us denote the size of the cooperatively rearranging domain (CRD)
by RCRD. Evidently the cooperative rearrangement is an activated process. The number of the
fluctuons involved in the rearrangement zC is ∼ (RCRD/r0)
3 and the number of molecules within
CRD is ∼ zCk0. To get an idea on the activation free energy let us consider the coarse-grained on
the scale r ∼ RCRD free energy landscape of CRD’s. It looks like this is shown in figure 1. The
deepest minima of gCRD belong to states with cs (x) nearly equal to an equilibrium value, c̄s, found
in the previous section. The maxima of gCRD belong to states with a small amount of solid-like
fluctuons, which are responsible for the correlation settling, cs (x) � 1. In accordance with this
picture we can determine the activation free energy of α-relaxation at constant pressure as follows:
gac (T ) = E + zC
[
g0
f (T ) − g (T ) + z
−1/3
C cs (T ) gsf
]
, (21)
23604-4
On the heterophase liquid thermodynamics and cooperative dynamics
Figure 1. Schematic pattern of the coarse-grained free energy landscape of the heterophase
liquid. The free energy of CRD, gCRD, vs a configurational coordinate is shown. zCg
0
f is the free
energy of “molten” CRD; zCg is the equilibrium free energy of CRD.
E is the activation energy of CRD in the fluid-like state. The term ∼ z2/3 is the interfacial free
energy.
The cooperative relaxation plays a key role in the heterophase liquid existing at T < T 0
e . The
relaxation rate is
Γα (T ) = Γ0
α exp [−gac (T )β] , (22)
Γ0
α = Γ
(
T 0
e
)
is a constant; β = 1/T . The Boltzmann constant is put equal to 1.
Considering Γα (T ) in the vicinities of the characteristic temperatures T 0
e , T 1
e and Te where
dependence of cs on T is given by expressions (13)–(17). At
∣
∣T − T 1
e
∣
∣� T 1
e :
Γα (T ) = Γα1 = Γ0
α exp
(
−
E
T
− cs
zC∆sf,s + z
2/3
C gsf
T
)
. (23)
At |T − Te| � Te:
Γα (T ) = Γ(
αTe) exp
{
−
E
T
−
zC∆sf,s
2
[
1 +
z
−1/3
C gsf
2∆sf,s (2Te − g̃sf)
]
T − Te
T
}
, (24)
Γα (Te) = Γ0
α2−zC exp
(
−zC
−g̃sf + z
−1/3
C gsf
Te
)
. (25)
At
∣
∣T − T 0
e
∣
∣� T 0
e :
Γα (T ) = Γα0 (T ) = Γ0
α exp
[
zC∆s
T − T 0
e
T
]
. (26)
In [2, 3] these equations were successfully used to fit the relaxation rate and its derivatives of
salol in the glass transition temperature range. Reliability of the developed α-relaxation model
was checked by fitting the experimental data for many other liquids. These results have not been
published yet.
4. Correlations of characteristic temperatures and discussion
Let us refer to the models of a heterophase state with and without accounting the frustration
and volumetric interaction of fluctuons as Model 1 and Model 2 respectively. The models differ in
the thermodynamic coefficients. The substitution
g0
s → g̃0
s = g0
s + (gss − ϕ0) /2, gs,f → g̃ss = gs,f − (gss − ϕ0) /2 (27)
leads to the transformation Model 1 → Model 2.
Due to the revealed independence of mathematical formulation of the Models on details of the
pair interaction of solid-like fluctuons, we can expect that they are applicable for a description
23604-5
O.S. Bakai
of glass-forming liquids of many types. It has to be noted that a variety of SRO of the solid-like
fluctuons in some cases has to be taken into account like in [5] where two types of the solid-like
fluctuons were taken in consideration to describe the liquid-liquid 1st order phase transition.
The Model 2 has evident advantages. It accounts for important components of the solid-like
fluctuon interaction – i.e., the frustration and volumetric interaction. Hence, we have got the
determination of the glass transition temperature range
[
T 1
e , T 0
e
]
.
Further, in Model 1 the condition gsf < 2Te has to be fulfilled to explain and describe the
continuous liquid-to-glass transformation. This condition looks like unphysical, since gsf is compa-
rable with liquid – crystalline fluctuon interfacial free energy, gcr,f , which is apparently larger than
2Tm (Tm is crystallization temperature) because crystallization is a discontinuous 1st order phase
transition. Model 2 is free of this shortcoming.
Thus, the HPFM is general enough to be applied to the examination of correlations of charac-
teristic temperatures TVF, TK, TMCT and Tg.
The found expressions for g(x, P, T ) and its derivatives show no singularities at all temperatures.
The relaxation rate (18) also has no singularities when the condition (11) is fulfilled. However, we
can determine the Kauzmann temperature, TK, by extrapolation of the configurational entropy of
liquid below Te.
To determine TVF we have to compare the found α-relaxation rate with the Vogel-Fulcher
formula.
Since we have used the model of cooperatively rearranging domains, our approach has to be
compared with the AG model of the α-relaxtion controlled by cooperative rearrangements. In AG
model, the configurational entropy per molecule, sC, is taken as a controlling parameter of the
glass transition. A domain consisting of z molecules has W (z, sC) ≈ exp(zsC) structure states of
the lowest energy. Structure rearrangement of the domain is possible if it has at least two different
structure states, W (z, sC) > 2. This condition gives lower boundary of CRD, z > ln 2/sC. One can
see that with sC → 0 the size of CRD increases and the transformation time increases too,
τα ∼ exp
(
z∆µ
T
)
= exp
(
Constant∆µ
TsC
)
. (28)
Here ∆µ is the conventional free energy barrier per molecule to rearrangement. The Constant
is ∼1.
Taking into account that for glass-forming liquids
sC
T
<
∂sC
∂T
(29)
we see that with linear extrapolation of sC(T )
sC (T ) = 0 at TK = Tg −
sC
∂ss/∂T
∣
∣
∣
∣
T>Tg
> 0. (30)
At TK, as equation (28) shows, τα → ∞ as it should be at the 2nd order phase transition.
In contrast to AG-model, we are considering CRDs with finite number of molecules, z = zC ·k0.
It was revealed experimentally [19–22] that in the vicinity of Tg the CRDs are finite. This property
of cooperative relaxation was used to explain the dynamic heterogeneity of supercooled liquids
[21, 23]. Due to this property τα is finite below Tg, as in the HPFM.
In a glass-forming liquid, the activation energy of rearrangement increases with the temper-
ature decrease mainly due to the growth of cs and supercooling, gac ∼ zC∆ss,f(Te − T )cs. This
quantity is large but finite at Tg and at the thermodynamic glass transition temperature T 1
e . The
configurational entropy is finite too at T 1
e . It is equal to the residual entropy of glass at T → 0.
Thus, both temperatures TVF and TK are fictitious.
The AG ideal glass possesses a single structure state of minimal energy which has no translati-
onal invariance and other symmetries. In HPFM, glass is formed by non-crystalline clusters (frozen
fluctuons) consisting of k0 bids. The configurational entropy of glass is ssc = ss − scr; scr is the
entropy of crystalline state; ssc is the residual entropy of glass at T → 0.
23604-6
On the heterophase liquid thermodynamics and cooperative dynamics
To determine TK in HPFM, let us find the liquid entropy,
s (T ; cs) = −
∂g (T ; cs)
∂T
= csss (T ) + (1 − cs)sf (T ) + smix(cs) = csss + ∆sf,scf + smix(cs),
smix(cs) = − [cs ln cs + (1 − cs) ln(1 − cs)] , (31)
smix with T → T 1
e is negligibly small and ss → scr + ssc. Therefore,
s (T ; cs) ≈ ssc + ∆sf,scf ≈ sC + ∆sf,s (1 − α) . (32)
As it follows from equation (16), in linear on T − Te approximation
α =
hs,f(T )
2(2T − g̃s,f)
≈
∆ss,f(Te − T )
2(2Te − g̃s,f)
. (33)
Since at the TK extrapolated value of s (T ; cs) is zero, we have
α (TK) = 1 +
ssc
∆sf,s
(34)
or
TK = Te −
2 (2Te − g̃s,f)
∆sf,s
(
1 +
ssc
∆sf,s
)
. (35)
Calorimetric data for liquids and glasses show that usually ∆sf,s � ssc. Therefore,
TK ≈ Te −
2 (2Te − g̃s,f)
∆sf,s
(36)
and α(TK) ≈ 1.
The empiric three parameter Vogel-Fulcher (VF) formula reads
ln Γα = A +
B
T − TVF
. (37)
Here A, B and TVF are fitting parameters.
To determine TVF in the HPFM, let us use the method proposed by Stickel at al. [24, 25]. This
method of experimental data presentation is known as the Stickel plot.
Noting that the quantity
γ−1 =
d ln Γα
dT
= −
B
(T − TVF)2
(38)
depends on two parameters, B and TVF, and that γ = 0 at T = TVF, Stickel has used an ex-
tensive set of experimental data on relaxation in supercooled liquids using γ to determine both
parameters [25].
It follows from (16), (24) and (38) that below Te
γ−1 =
d ln Γα
dT
= −
B(α)
T 2
, B(α) = E − g̃sf/2 +
zC
2
(
∆sf,sTe + z
−1/3
C gsf
)
(1 + 2α). (39)
Noting that
γ1/2 =
T
[B(α)]
1/2
=
T
[B(0) (1 + 2Kα)]
1/2
≈
T (1− Kα)
[B(0)]
1/2
, K = 1 −
E − g̃sf/2
B(0)
, (40)
we see that the value of γ extrapolated to temperature below Te goes to zero at
α (TVF) =
1
K
= 1 +
E − g̃sf/2
B(0)
. (41)
23604-7
O.S. Bakai
The last term in brackets of r.h.s in this equation is ≈ ln
(
Γα(Te)/Γα(T 0
e )
)
� 1. Therefore,
α (TVF) ≈ 1. (42)
As it follows from (41),
TVF = Te −
2 (2Te − g̃s,f)
∆sf,s
(
1 +
E − g̃sf/2
B(0)
)
≈ Te −
2 (2Te − g̃s,f)
∆sf,s
. (43)
Comparing (34), (35) and (42), (43) we see that
TVF ≈ TK . (44)
To find correlation of TMCT, Tg and relation TVF, it has to be noted that TMCT is close to Te
as Stickel et al. have found [24, 25]. Though Tg is not a properly defined temperature, evidently
cs(Tg) ≈ 1 or α(Tg) ≈ 1/2. Thus, we have from (33) and (42) that
TMCT − TVF ≈ 2 (Tg − TVF) . (45)
It follows from (23) and (38) that at T 6 T 1
e
Γα (T ) ∼ exp
(
−
E + zC∆sf,s + z
2/3
C gsf
2T
)
(46)
and, consequently,
B = E + zC∆sf,s + z
2/3
C gsf , TVF = 0. (47)
Similarly, from (26), (38) we have that at T > T 0
e
B = E, TVF = 0. (48)
Thus, beyond the glass transition temperature range
[
T 1
e , T 0
e
]
the relaxation rate obeys the Arrhe-
nius equations with different activation energies.
5. Conclusions
Taking account for the frustration and volumetric interaction of the solid-like fluctuons in the
HPFM leads to renormalization of thermodynamic coefficients saving the mathematical form of the
equilibrium state equations. For this reason, equations (3), (7) can be taken as a minimal model of
the HPS thermodynamics. The liquid-to-glass transition takes place within the temperature range
of width scaled by (gss − ϕ0).
Cooperatively rearranging domains are formed due to the interactions of fluctuons and the
rearrangements are thermally activated events. The fact that the activation energy increases and
liquid entropy decreases with the temperature decrease lies in the base of the assumption on the
non-analytic behavior of entropy and singularity of the α-relaxation time at finite temperatures TK
and TVF which in the AG model coincide but are not accessible due to the violation of ergodicity
at Tg. In our model, the free energy derivatives have no singularities at TK and TVF but these
temperature are determined by an extrapolation procedure. As a result, it is found that TVF ≈ TK
and TMCT − TVF ≈ 2 (Tg − TVF).
Above T 0
e and below Tg the relaxation rate obeys the Arrhenius equations with different acti-
vation energies.
23604-8
On the heterophase liquid thermodynamics and cooperative dynamics
References
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M. Tokuyama, I. Oppenheim, 1999, p. 325.
3. Fischer E.W., Bakai A., Patkowski A., Steffen W., Reinhardt L., J. Non-Cryst. Solids, 2002, 307–310,
584.
4. Bakai A.S., Fischer E.W., J. Chem. Phys., 2004, 120, 5235–5252.
5. Bakai A.S., J. Chem. Phys., 2006, 125, 064503.
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20. Cicerone M.T., Ediger M.D., J. Chem. Phys., 1996, 103, 5684.
21. Silescu H., J. Non-Cryst. Solids, 1999, 243, 81.
22. Ediger M.D., Ann. Rev. Phys. Chem., 2000, 51, 99.
23. Richert R., J. Phys.: Condens. Matter, 2002, 14, R703.
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25. Stickel F. Dissertation, Mainz, 1995.
Термодинамiка i кооперативна динамiка гетерофазної
рiдини
О.С.Бакай
Нацiональний науковий центр “Харкiвський фiзико-технiчний iнститут” НАН України,
вул. Академiчна, 1, 61108, Харкiв, Україна
Розглянуто термодинамiку i кооперативну динамiку гетерофазної рiдини з урахуванням фрустрацiї
i об’ємної взаємодiї твердотiльних флуктуонiв. Встановлено, що ширина температурного iнтервалу
перетворення рiдини в скло є пропорцiйною рiзницi параметру фрустрацiї i енергiї об’ємної взаємо-
дiї. Знайдено уявнi температури Кауцмана i Фогеля-Фулчера. Вони виявились близькими з гарною
точнiстю. Розглянуто також кореляцiю температури склування, Фогеля-Фулчера i температури “iде-
ального” перетворення рiдина-скло, яке описано в моделi взаємодiючих мод.
Ключовi слова: гетерофазнi рiдкi стани, перехiд склування, температури Кауцманна та
Фогеля-Фулчера
23604-9
Introduction
Glass transition in HPFM model
The cooperative relaxational dynamics of the heterophase liquid (-relaxation)
Correlations of characteristic temperatures and discussion
Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-32095 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-11-24T10:06:31Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Bakai, O.S. 2012-04-08T15:51:56Z 2012-04-08T15:51:56Z 2010 On the heterophase liquid thermodynamics and cooperative dynamics / O.S. Bakai // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23604: 1-9. — Бібліогр.: 25 назв. — англ. 1607-324X PACS: 64.70.Ja, 64.70.Pf, 61.20.Gy https://nasplib.isofts.kiev.ua/handle/123456789/32095 The thermodynamics and cooperative dynamics of heterophase liquid states is considered taking into account frustration and volumetric interaction of the solid-like fluctuons. It is found that the glass transition temperature range is scaled by difference of the frustration parameter and mean energy of the volumetric interaction. A model of the cooperative relaxation with finite cooperatively rearranging domains is considered. The fictitious Kauzmann and Vogel-Fulcher temperatures are determined. It is found that they are close to a proper accuracy. Correlation of the temperatures of glass transition, Vogel-Fulcher and "ideal" glass transition (as it is determined in the mode coupling model) is considered too. Розглянуто термодинаміку та кооперативну динаміку гетерофазної рідини з урахуванням фрустрації й об'ємної взаємодії твердотільних флуктуонів. Установлено, що ширина температурного інтервалу перетворення рідини в скло є пропорційною різниці параметра фрустрації та енергії об'ємної взаємодії. Знайдено уявні температури Кауцмана і Фогеля - Фулчера. Вони виявились близькими з гарною точністю. Розглянуто кореляцію температури склування, Фогеля - Фулчера та температури "ідеального" перетворення рідина - скло, яке описано в моделі взаємодіючих мод. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics On the heterophase liquid thermodynamics and cooperative dynamics Термодинаміка і кооперативна динаміка гетерофазної рідини Article published earlier |
| spellingShingle | On the heterophase liquid thermodynamics and cooperative dynamics Bakai, O.S. |
| title | On the heterophase liquid thermodynamics and cooperative dynamics |
| title_alt | Термодинаміка і кооперативна динаміка гетерофазної рідини |
| title_full | On the heterophase liquid thermodynamics and cooperative dynamics |
| title_fullStr | On the heterophase liquid thermodynamics and cooperative dynamics |
| title_full_unstemmed | On the heterophase liquid thermodynamics and cooperative dynamics |
| title_short | On the heterophase liquid thermodynamics and cooperative dynamics |
| title_sort | on the heterophase liquid thermodynamics and cooperative dynamics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32095 |
| work_keys_str_mv | AT bakaios ontheheterophaseliquidthermodynamicsandcooperativedynamics AT bakaios termodinamíkaíkooperativnadinamíkageterofaznoírídini |