Heterophase fluctuations in glass-forming liquids and random field Ising model
The liquid-to-glass transition is a process of supercooled liquid solidification. Rather large density fluctuations are revealed experimentally in many of the glass-forming liquids above the glass transition temperature while no phase transitions are identified [1,2]. In [3–5], the inhomogeneities ar...
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| Cite this: | Heterophase fluctuations in glass-forming liquids and random field Ising model / A.S. Bakai // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 675-681. — Бібліогр.: 14 назв. — англ. |
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| author | Bakai, A.S. |
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| citation_txt | Heterophase fluctuations in glass-forming liquids and random field Ising model / A.S. Bakai // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 675-681. — Бібліогр.: 14 назв. — англ. |
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| description | The liquid-to-glass transition is a process of supercooled liquid solidification. Rather large density fluctuations are revealed experimentally in many
of the glass-forming liquids above the glass transition temperature while
no phase transitions are identified [1,2]. In [3–5], the inhomogeneities are
treated as heterophase fluctuations (HPF). The process of glass formation gets there a natural description as a continuous phase transformation.
The theory of strong HPF was developed in a mean field approximation
which ignores the mesoscopic structure of the inhomogeneities which is
an issue of extensive experimental investigations and discussions [1,2,5].
In the present communication the HPF are considered in the model of interpercolating heterophase states and in Ginzburg-Landau (GL) approach.
It is shown that the GL approach results in the random field Ising model
(RFIM) for HPF. It permits to get a description of the medium range and
long-range correlations of the HPF. RFIM is very useful in studying the spin
systems with a frozen-in disorder. Therefore the theory developed makes
it possible to compare the phase states with frozen-in (spin systems) and
self-consistent (heterophase liquids) disorders. In particular, it turns out
that the heterophase liquids are similar (but not identical) to Griffiths phase
of disordered spin systems. It is seen that the developed model bridges the
theories of disordered spin systems and glass-forming liquids.
Утворення скла є процесом твердiння переохолодженої рідини. Експериментально виявлено досить великі флуктуації густини в багатьох
склоутворюючих рідинах вище температури вітрифікації, але, разом
з тим, жодних ознак фазового переходу не спостерігається [1,2].
В [3–5] ці неоднорідності розглядаються як гетерофазні флуктуації
(ГФФ), процес утворення скла описується як неперервне фазове перетворення. Теорію сильних ГФФ тут розвинуто в наближенні середнього поля, в якому ігнорується мезоскопічна структура неоднорорідностей, які є предметом інтенсивних експериментальних досліджень та обговорювань [1,2,5]. В цьому повідомленні ГФФ розглянуто в межах моделі інтерперкольованих кластерів та в підході
Гінзбурга-Ландау (ГЛ). Показано, що наближення ГЛ приводить до
моделі Ізінга у випадковому полі (МІВП). Це дозволяє розглядати кореляції ГФФ на проміжних та довгих масштабах. МІВП широко використовується при розгляді спінових систем з вмороженим безладом. Через це розроблена теорія дає можливість порівнювати фазові
стани систем з вмороженим (спінові системи) та самоузгодженим
(гетерофазні рідини) безладами. Зокрема виявляється, що гетерофазні рідини є подібні (але не ідентичні) до фази Гріфіца невпорядкованої спінової системи. Як бачимо, розроблена модель встановлює
зв’язок між теоріями спінових систем та склоутворюючих рідин.
|
| first_indexed | 2025-11-24T04:21:38Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 3(23), pp. 675–681
Heterophase fluctuations in
glass-forming liquids and random field
Ising model
A.S.Bakai
National Science Center “Kharkiv Institute of Physics and Technology”,
61108 Kharkiv, Ukraine
Received May 29, 2000
The liquid-to-glass transition is a process of supercooled liquid solidifica-
tion. Rather large density fluctuations are revealed experimentally in many
of the glass-forming liquids above the glass transition temperature while
no phase transitions are identified [1,2]. In [3–5], the inhomogeneities are
treated as heterophase fluctuations (HPF). The process of glass forma-
tion gets there a natural description as a continuous phase transformation.
The theory of strong HPF was developed in a mean field approximation
which ignores the mesoscopic structure of the inhomogeneities which is
an issue of extensive experimental investigations and discussions [1,2,5].
In the present communication the HPF are considered in the model of in-
terpercolating heterophase states and in Ginzburg-Landau (GL) approach.
It is shown that the GL approach results in the random field Ising model
(RFIM) for HPF. It permits to get a description of the medium range and
long-range correlations of the HPF. RFIM is very useful in studying the spin
systems with a frozen-in disorder. Therefore the theory developed makes
it possible to compare the phase states with frozen-in (spin systems) and
self-consistent (heterophase liquids) disorders. In particular, it turns out
that the heterophase liquids are similar (but not identical) to Griffiths phase
of disordered spin systems. It is seen that the developed model bridges the
theories of disordered spin systems and glass-forming liquids.
Key words: glass-forming liquids, heterophase fluctuations, continuous
phase transformation, random field Ising model
PACS: 64.70.Pf, 61.20.Lc
1. Introduction
Phase transformations and phase transitions belong to the most challenging prob-
lems in the condensed matter physics (see e.g. [6]). The liquid-to-glass transitions
have been investigated for a long time. A lot of fascinating ideas were formulated
on this way though none had a decisive success. In this paper the physics of HPF in
c© A.S.Bakai 675
A.S.Bakai
glass-forming liquids is considered. It is believed that the HPF are responsible for
the known properties of the glass-forming liquids and that they play a decisive role
in the glass forming process.
In a macroscopically homogeneous phase, heterophase fluctuations do exist. Usu-
ally they are weak due to a comparatively high free energy of formation. Neverthe-
less, as it is shown by Frenkel [7], they are not negligible in the vicinity of the phase
coexisting curve on the (P,T)-plane. Investigations of thermodynamics and phase
transitions in the droplet approach [7–9] show that the critical point (end point on
the phase coexisting curve) is determined by the condition that the interfacial free
energy equals to zero. In this case, in the supercritical region no phase separation is
possible and no heterophase fluctuations can occur. It means that the phase is mi-
croscopically homogeneous in the supercritical region in accordance with the droplet
models.
Recently [3–5,9] the thermodynamics of heterophase states was reconsidered
within the framework of the interpercolating heterophase state model. This model
was developed to be applicable to the studies of the states where heterophase fluc-
tuations are not weak and their volume fraction exceeds a percolation threshold. It
turns out that the interfacial free energy is positive at the critical point and that
there exists a domain of the supercritical region where phase separation is possi-
ble. Therefore, a mesoscopically heterophase state does exist in this region though
the phase is macroscopically homogeneous and no phase transitions take place with
crossing of the coexisting curve. Therefore if P and T are changing to pass around the
critical point, from one side of the coexisting curve to another, a continuous phase
transformation takes place. The mesoscopic heterophase substructure is changing
with that transformation but no phase transitions occur. The fraction of a “pure”
phase is a natural order parameter of the heterophase state. In the vicinity of the
critical point, the free energy of the heterophase state can be presented in the stan-
dard Landau series on the degrees of the order parameter. The GL approach can
also be applied. A break through the frame of the mean field approximation leads
to RFIM model as it will be seen.
2. HPF of glass-forming liquids in the mean field
approximation
Above Tg the heterophase liquid (HPL) is assumed to consist of a fluid fraction
and non-crystalline solid clusters. The HPF can be described using the model of
interpercolating heterophase states. This model in the mean field approximation
gives the following expression for the free energy per molecule:
µ(P, T ) = nsµs(P, T ) + (1− ns)µf(P, T ) + ns(1− ns)∆µint(P, T )
+
kBT
k0
[ns lnns + (1− ns) ln(1− ns)], (1)
∆µint(P, T ) = µint(P, T )− [µs(P, T ) + µf(P, T )/2]. (2)
676
Fluctuations in glass-forming liquids and RFIM
Here ns is the fraction of molecules belonging to solid clusters; µ s, µf , and µint are
chemical potentials of molecules in the “pure” solid, fluid, and within the interfa-
cial layer respectively; k0 is the associativity (or the mixing ability parameter) of
molecules; kB is Boltzmann’s constant.
The phase coexisting curve is determined by the equation
µs(P, T ) = µf(P, T ). (3)
Solution of this equation gives the temperature of the phase equilibrium, T e(P ).
It is shown [4] that a critical point on the coexisting curve is determined by the
following equation
∆µint(P, T ) = 2Te/k0 . (4)
In the region where
0 < µint 6 2Te/k0 (5)
no phase transitions take place on the coexisting curve. In the vicinity of this curve,
with 0, 15 < ns < 0, 85, infinite interpercolating solid and fluid clusters do coexist.
If the liquid is cooled down passing the region (5), the continuous (without phase
transitions) transformation of the fluid in non-crystalline solid state takes place.
The local order in amorphous states changes from site to site. The above used
chemical potentials of the solid and fluid fractions are averaged on the local (short
range) disorder. HPF have a coarse grained mesoscopic structure which can be stud-
ied using the GL approach. This approach is valid on the scales which are larger
than the intermolecular distance, a. Let us consider the classic field of the order
parameter ns(x) and include the contribution of the field gradients into the free
energy. Then instead of (1) one has the following equation for the free energy:
G(P, T ) = v−1
∫
[A(∇α)2 +Bα2 + Cα4 + hα]dx. (6)
Here
α(P, T ) = ns(P, T )− 1/2, (7)
B = 2k−1
0 Tc −∆µint
∼= 2k−1
0 Tc −∆µint(Te), (8)
C = 2k0Te/3, (9)
h = µs − µf
∼= (sf − ss)(T − Tc). (10)
In (10) sf and ss are entropies per molecule of the fluid and solid phases respectively,
v is the specific volume. Coefficient A in (6) is proportional to interfacial energy.
Note that the “external field”, h, depends on the temperature. Properties of the
system (6) can be analyzed using the approaches of the theory of critical phenom-
ena. Results of the analysis will be published elsewhere [10]. Here we give just the
expression determining the correlation length, Rc, which is a natural scale of the
mesoscopic inhomogeneities of HPL:
Rc ≈ |B/A|−ν, h≪ hc(B); Rc ≈ |h|−µ, h≪ hc(B), (11)
677
A.S.Bakai
where hc(B) = B(B/C)1/2, and ν, µ are critical exponents.
The parameter τ = B/Te determines how close to a critical point the system is
in the vicinity of the coexisting curve. Because the thermodynamic quantities of the
glass-forming liquids as a rule possess no singularities of above T g, one can conclude
that the parameter τ is not very small. In the estimations we can put τ ≈ 10−1.
For this reason the correlation length Rc is usually comparatively small, Rc ∼ 10a
(a is the intermolecular distance). Therefore the HPL possesses the coarse grained
heterophase structure and Rc is the characteristic scale of the grains.
3. Mesoscopic heterogeneity
Investigations of the thermodynamics of the solid clusters in liquid [11] show that
the potential relief minima of molecules being random obey a Gaussian distribution.
Because the potential relief formation is a cooperative phenomenon, it is natural to
assume that the depths of the minima are highly correlated at least on the scale
∼ Rc which is a consequence of the coarse graining. With this assumption, the
distribution found in [11] is as follows:
ψs(ε,N) =
1
(πNsδ2s )
1/2
exp[Nsζs −Ns(ε− εs)
2/δ2s ], (12)
δ2s = (Rc/a)
3δ20s , (13)
where ζs is the configurational entropy, δ
2
0s is the variance with Rc = a, and εs is the
average depth of the potential well. The index s marks the quantities related to a
solid cluster.
Simple calculations give the following expression of the averaged with (12) µ s:
Gs(Ns, P, T ) = Ns[εs − δ2s /T − σsT + µs,vib(P, T )]. (14)
Here µs,vib(P, T ) is the contribution of the vibrational motions. The solid cluster in
the liquid is assumed to be ergodic.
An expression similar to (12) can also be obtained for the chemical potential of
a liquid cluster:
Gf(Ns, P, T ) = Nf [εf − δ2f /T − ζfT + µf,vib(P, T )]. (15)
The index f marks the quantities related to solid cluster.
4. The RFIM of the HPF
When substituted in (1) the chemical potentials (12) and (15) determine the free
energy in the mean field approximation. Our goal is to take into account spatial
fluctuations of the chemical potentials:
δµs,f(x) = µs,f(x)− µs,f . (16)
678
Fluctuations in glass-forming liquids and RFIM
To this end, we have to include them in the external field term of (6):
h = µs − µf
∼= (sf − ss)(T − Tc). (17)
Because of (12) and (15), the random field δh obeys to the Gaussian distribution:
P (δτ) = p0 exp
−
∫
|δτ(x)|2d3x
4ξ3hδ
2
, (18)
δ2 = δ20s + δ20f . (19)
The quantity ξh is the correlation length of the random field h. Evidently ξh > Rc.
The equations (6), (17)–(18) describe a RFIM with the Gaussian random field.
Systems of this type are under investigation in the theory of disordered spin sys-
tems (DSS) [12,13]. An important difference between the HPF and a DSS has to
be pointed out. In the DSS the disorder is frozen in, it is static and is determined
by a history of the system preparation. The random field of the HPF model is
self-consistent. Because of the ergodicity of the liquid it is not static. The distri-
bution (18) describes the HPF in a fixed time. A more general description should
include equations for correlation functions of h(x, t). They have to be considered
elsewhere [10]. One can expect that the correlation time of h(x, t), τh, is propor-
tional to ξ2hτα; τα is the α- relaxation time. With t ≪ τh one can use the static
approximation (18).
It is known from the physics of RFIM (see e.g. [14]) that the critical temperature
is effected by the random field:
Tc(δ) = Tc(0)− c1δ
2/ϕ − c2δ
2, (20)
δ is the variance of the random field, c1, c2 are constants. For spin systems ϕ ≈ 1.42.
In accordance with (20) the critical temperature is depressed by the field disorder.
Considering the HPF we have to replace the criterion (4) by (20) putting Tc(0) =
k0µint/2 > 0. Thus
Tc(δ) = k0µint/2− c1δ
2/ϕ − c2δ
2 < Te (21)
no phase transitions take place on the coexisting curve.
It is seen that not only weak interfacial energy but also strong random field
fluctuations exclude phase transitions in the HPL and lead to the continuous phase
transformation in the vicinity of Te. Because too many liquids and polymers are good
glassformers, we have no reason to expect that in all of them the interfacial free en-
ergy, µint, is negligible while the multiplicity of the local ordering of molecules with a
wide variance of energies looks like their common feature. Therefore we can conclude
that the multiplicity and correlation of the low energy molecular configuration in
solid clusters are important factors of glass formation.
RFIM developed for the HPF in the glass-forming liquids gives good grounds
for theoretical investigations of heterogeneity and dynamics in the vicinity of T g.
679
A.S.Bakai
Because the structure of a liquid above Tg is inherited by glass below Tg, we have
also good grounds for dealing with mesoscopic heterogeneity of glass.
In disordered spin systems the Griffiths phase exists in the temperature region
Tc(0) > T > Tc(δ). (22)
HPF exist in the form of droplets with µint > 0 but the interpercolating het-
erophase state appears just in the vicinity of the coexisting curve. It is worth noting
that the glass-forming liquids possess the dynamic properties (characteristic relax-
ation laws) similar to those of Griffiths phase. It is considered to be an important
similarity.
5. Conclusions
RFIM developed for the HPF permits to reveal the impact of disorder on the
structure of the glass-forming liquids as well as on the phase transitions and phase
transformations within them. Besides, it directly bridges up the theories of the
disordered spin systems and glass-forming liquids.
6. Acknowledgements
It is my big pleasure to acknowledge E.W.Fischer for extensive helpful discus-
sions.
References
1. Sillescu H. // J. Non-Cryst. Solids, 1999, vol. 243, p. 81.
2. Fischer E.W. // Physica A, 1993, vol. 201, p. 183.
3. Bakai A.S. // Low Temp. Phys., 1996, vol. 22, p. 733.
4. Bakai A.S. – In: The Polyclaster Concept of Amorphous Solids in Glassy Metals. III.
(ed. by H. Beck and H.-J. Guntherodt) Heidelberg, Springer, 1994, p. 209.
5. Fischer E.W., Bakai A.S. Slow dynamics in complex systems. – In: AIP Conf. Proc.,
469, (ed. M. Tokuyama, I. Oppenheim), 1999, p. 325
6. Yukhnovskii I.R. Phase Transitions of the Second Order. Collective Variables Method.
Singapore, World Sci. Publ. Co. Ltd., 1987.
7. Frenkel J. // J. Chem. Phys., 1938, vol. 7, p. 200, p. 538.
8. Fisher M.E. // Physics, 1967, vol. 3, p. 255.
9. Bakai A.S. // Low Temp. Phys., 1998, vol. 24, p. 20.
10. Bakai A.S. // (in preparation).
11. Bakai A.S. // Low Temp. Phys., 1994, vol. 20, p. 373, p. 379.
12. Dotsenko V.S. // Uspekhi Fis. Nauk, 1995, vol. 165, p. 481.
13. Spin Glasses and Random Fields. (ed. A.P. Young) Singapore, World Sci. Publ. Co.
Ltd., 1998.
14. Belanger D.P. – In: Spin Glasses and Random Fields. (ed. A.P. Young) Singapore,
World Sci. Publ. Co. Ltd., 1998.
680
Fluctuations in glass-forming liquids and RFIM
Гетерофазні флуктуації у склоутворюючих рідинах і
модель Ізінга у випадковому полі
О.С.Бакай
Національний Науковий Центр
“Харківський фізико-технічний інститут”,
61108 Харків
Отримано 29 травня 2000 р.
Утворення скла є процесом твердiння переохолодженої рідини. Екс-
периментально виявлено досить великі флуктуації густини в багатьох
склоутворюючих рідинах вище температури вітрифікації, але, разом
з тим, жодних ознак фазового переходу не спостерігається [1,2].
В [3–5] ці неоднорідності розглядаються як гетерофазні флуктуації
(ГФФ), процес утворення скла описується як неперервне фазове пе-
ретворення. Теорію сильних ГФФ тут розвинуто в наближенні се-
реднього поля, в якому ігнорується мезоскопічна структура неодно-
рорідностей, які є предметом інтенсивних експериментальних до-
сліджень та обговорювань [1,2,5]. В цьому повідомленні ГФФ роз-
глянуто в межах моделі інтерперкольованих кластерів та в підході
Гінзбурга-Ландау (ГЛ). Показано, що наближення ГЛ приводить до
моделі Ізінга у випадковому полі (МІВП). Це дозволяє розглядати ко-
реляції ГФФ на проміжних та довгих масштабах. МІВП широко ви-
користовується при розгляді спінових систем з вмороженим безла-
дом. Через це розроблена теорія дає можливість порівнювати фазові
стани систем з вмороженим (спінові системи) та самоузгодженим
(гетерофазні рідини) безладами. Зокрема виявляється, що гетеро-
фазні рідини є подібні (але не ідентичні) до фази Гріфіца невпорядко-
ваної спінової системи. Як бачимо, розроблена модель встановлює
зв’язок між теоріями спінових систем та склоутворюючих рідин.
Ключові слова: склоутворююча рідина, гетерофазні флуктуації,
фазові перетворення, модель Ізінга з випадковими полями
PACS: 64.70.Pf, 61.20.Lc
681
682
|
| id | nasplib_isofts_kiev_ua-123456789-120998 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-11-24T04:21:38Z |
| publishDate | 2000 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Bakai, A.S. 2017-06-13T12:23:40Z 2017-06-13T12:23:40Z 2000 Heterophase fluctuations in glass-forming liquids and random field Ising model / A.S. Bakai // Condensed Matter Physics. — 2000. — Т. 3, № 3(23). — С. 675-681. — Бібліогр.: 14 назв. — англ. 1607-324X DOI:10.5488/CMP.3.3.675 PACS: 64.70.Pf, 61.20.Lc https://nasplib.isofts.kiev.ua/handle/123456789/120998 The liquid-to-glass transition is a process of supercooled liquid solidification. Rather large density fluctuations are revealed experimentally in many of the glass-forming liquids above the glass transition temperature while no phase transitions are identified [1,2]. In [3–5], the inhomogeneities are treated as heterophase fluctuations (HPF). The process of glass formation gets there a natural description as a continuous phase transformation. The theory of strong HPF was developed in a mean field approximation which ignores the mesoscopic structure of the inhomogeneities which is an issue of extensive experimental investigations and discussions [1,2,5]. In the present communication the HPF are considered in the model of interpercolating heterophase states and in Ginzburg-Landau (GL) approach. It is shown that the GL approach results in the random field Ising model (RFIM) for HPF. It permits to get a description of the medium range and long-range correlations of the HPF. RFIM is very useful in studying the spin systems with a frozen-in disorder. Therefore the theory developed makes it possible to compare the phase states with frozen-in (spin systems) and self-consistent (heterophase liquids) disorders. In particular, it turns out that the heterophase liquids are similar (but not identical) to Griffiths phase of disordered spin systems. It is seen that the developed model bridges the theories of disordered spin systems and glass-forming liquids. Утворення скла є процесом твердiння переохолодженої рідини. Експериментально виявлено досить великі флуктуації густини в багатьох склоутворюючих рідинах вище температури вітрифікації, але, разом з тим, жодних ознак фазового переходу не спостерігається [1,2]. В [3–5] ці неоднорідності розглядаються як гетерофазні флуктуації (ГФФ), процес утворення скла описується як неперервне фазове перетворення. Теорію сильних ГФФ тут розвинуто в наближенні середнього поля, в якому ігнорується мезоскопічна структура неоднорорідностей, які є предметом інтенсивних експериментальних досліджень та обговорювань [1,2,5]. В цьому повідомленні ГФФ розглянуто в межах моделі інтерперкольованих кластерів та в підході Гінзбурга-Ландау (ГЛ). Показано, що наближення ГЛ приводить до моделі Ізінга у випадковому полі (МІВП). Це дозволяє розглядати кореляції ГФФ на проміжних та довгих масштабах. МІВП широко використовується при розгляді спінових систем з вмороженим безладом. Через це розроблена теорія дає можливість порівнювати фазові стани систем з вмороженим (спінові системи) та самоузгодженим (гетерофазні рідини) безладами. Зокрема виявляється, що гетерофазні рідини є подібні (але не ідентичні) до фази Гріфіца невпорядкованої спінової системи. Як бачимо, розроблена модель встановлює зв’язок між теоріями спінових систем та склоутворюючих рідин. It is my big pleasure to acknowledge E.W.Fischer for extensive helpful discussions. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Heterophase fluctuations in glass-forming liquids and random field Ising model Гетерофазні флуктуації у склоутворюючих рідинах і модель Ізінга у випадковому полі Article published earlier |
| spellingShingle | Heterophase fluctuations in glass-forming liquids and random field Ising model Bakai, A.S. |
| title | Heterophase fluctuations in glass-forming liquids and random field Ising model |
| title_alt | Гетерофазні флуктуації у склоутворюючих рідинах і модель Ізінга у випадковому полі |
| title_full | Heterophase fluctuations in glass-forming liquids and random field Ising model |
| title_fullStr | Heterophase fluctuations in glass-forming liquids and random field Ising model |
| title_full_unstemmed | Heterophase fluctuations in glass-forming liquids and random field Ising model |
| title_short | Heterophase fluctuations in glass-forming liquids and random field Ising model |
| title_sort | heterophase fluctuations in glass-forming liquids and random field ising model |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120998 |
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