Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature
Altitude and time dependence of nonequilibrium values of refractive index gradient for inhomogeneous methanol-hexane solution under gravity near consolute critical temperature was investigated in the work as system approaches equilibrium state. Based on these data there have been suggested scali...
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| Zitieren: | Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature / A.D. Alekhin, Yu.L. Ostapchuk // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 449-457. — Бібліогр.: 19 назв. — англ. |
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irk-123456789-1204212017-06-13T03:04:26Z Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature Alekhin, A.D. Ostapchuk, Yu.L. Altitude and time dependence of nonequilibrium values of refractive index gradient for inhomogeneous methanol-hexane solution under gravity near consolute critical temperature was investigated in the work as system approaches equilibrium state. Based on these data there have been suggested scaling equations of nonequilibrium liquid under gravity for temperatures above consolute critical temperature. It has been shown that the scaling hypothesis is also valid for nonequilibrium solution close to the critical point for small values of order parameter В роботі досліджена висотна та часова залежність нерівноважних значень градієнта показника заломлення неоднорідного розчину метанол-гексан у гравітаційному полі поблизу критичної температури розшарування при прямуванні системи до стану рівноваги. На основі цих даних запропоновані масштабні рівняння нерівноважної рідини в гравітаційному полі для температур вищих від критичної температури розшарування. Показано, що і для нерівноважного розчину поблизу критичної точки для малих значень параметра порядку системи виконується масштабна гіпотеза. 2001 Article Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature / A.D. Alekhin, Yu.L. Ostapchuk // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 449-457. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 05.70.Jk, 64.30.+t, 64.60.Ht, 64.70.Ja DOI:10.5488/CMP.4.3.449 http://dspace.nbuv.gov.ua/handle/123456789/120421 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Altitude and time dependence of nonequilibrium values of refractive index
gradient for inhomogeneous methanol-hexane solution under gravity near
consolute critical temperature was investigated in the work as system approaches
equilibrium state. Based on these data there have been suggested
scaling equations of nonequilibrium liquid under gravity for temperatures
above consolute critical temperature. It has been shown that the scaling hypothesis
is also valid for nonequilibrium solution close to the critical point
for small values of order parameter |
| format |
Article |
| author |
Alekhin, A.D. Ostapchuk, Yu.L. |
| spellingShingle |
Alekhin, A.D. Ostapchuk, Yu.L. Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature Condensed Matter Physics |
| author_facet |
Alekhin, A.D. Ostapchuk, Yu.L. |
| author_sort |
Alekhin, A.D. |
| title |
Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature |
| title_short |
Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature |
| title_full |
Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature |
| title_fullStr |
Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature |
| title_full_unstemmed |
Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature |
| title_sort |
dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120421 |
| citation_txt |
Dynamic scaling equation of state for nonequilibrium solution under gravity above consolute critical temperature / A.D. Alekhin, Yu.L. Ostapchuk // Condensed Matter Physics. — 2001. — Т. 4, № 3(27). — С. 449-457. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT alekhinad dynamicscalingequationofstatefornonequilibriumsolutionundergravityaboveconsolutecriticaltemperature AT ostapchukyul dynamicscalingequationofstatefornonequilibriumsolutionundergravityaboveconsolutecriticaltemperature |
| first_indexed |
2025-07-08T17:51:18Z |
| last_indexed |
2025-07-08T17:51:18Z |
| _version_ |
1837102089883353088 |
| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 3(27), pp. 449–457
Dynamic scaling equation of state for
nonequilibrium solution under gravity
above consolute critical temperature
A.D.Alekhin, Yu.L.Ostapchuk
Physics Department, Taras Shevchenko National University of Kyiv,
6 Glushkova Ave., 03680 Kyiv, Ukraine
Received August 11, 2001, in final form June 26, 2001
Altitude and time dependence of nonequilibrium values of refractive index
gradient for inhomogeneous methanol-hexane solution under gravity near
consolute critical temperature was investigated in the work as system ap-
proaches equilibrium state. Based on these data there have been suggest-
ed scaling equations of nonequilibrium liquid under gravity for temperatures
above consolute critical temperature. It has been shown that the scaling hy-
pothesis is also valid for nonequilibrium solution close to the critical point
for small values of order parameter.
Key words: scaling equation of state, nonequilibrium solution,
gravitational effect, consolute critical point
PACS: 05.70.Jk, 64.30.+t, 64.60.Ht, 64.70.Ja
Several behaviour features of inhomogeneous liquids which are not observed in
homogeneous systems had been detected for the first time earlier while studying
the kinetics of equilibrium establishment for inhomogeneous liquid system under
gravity close to the critical point for temperatures above the critical one (T > T c).
There has been discovered nonmonotonous temperature dependence of equilibrium
time te(∆T ) for inhomogeneous substance under gravity [1–4]. The greatest time
te = max corresponds to a temperature ∆T > 0 – not to the critical temperature
∆T = T − Tc = 0. It was proved for the first time that time te depends not only
on relaxation time τ(ρc) ∼ D−1 ∼ ∆T−ν [5–7] but also on the thickness of the
layer ∆z of liquid with critical density ρc which varies with temperature as ∆z ∼
∆T βδ(te ∼ τ∆z ∼ (∆T )βδ−ν [1,2]. There has been obtained the nonmonotonous
altitude dependence of relaxation time τ(z) of refractive index gradient dn/dz of
inhomogeneous liquid under gravity [8,9]. Maximum value of the τ(z) does not
correspond to the level z = 0 with critical density ρc of the investigated substance
but to a height z 6= 0, in the vicinity of which isotherms of refractive index gradient
dn/dz intersect each other. It is close to this intersection point that magnitudes of
c© A.D.Alekhin, Yu.L.Ostapchuk 449
A.D.Alekhin, Yu.L.Ostapchuk
nonequilibrium values dn/dz slowly vary with time when the system is passing to
the equilibrium state. It was also detected for the first time [8,9] that the relaxation
properties of inhomogeneous liquid at a certain height z are determined not by a
single relaxation time but by the spectrum of times, which characterizes the whole
inhomogeneous system under gravity.
The purpose of the present paper is to continue these experimental investigations
for detection of kinetics peculiarities of equilibrium establishment in spatially inho-
mogeneous liquid systems close to the critical point; to develop the scaling equation
of state for such a nonequilibrium system in external gravitational field.
Kinetics of equilibrium establishment for inhomogeneous binary methanol-hex-
ane solution under gravity has been studied in the work by using refractometry
technique at different temperatures above consolute critical temperature. Routine
of the experiment and the experimental equipment were described in detail in the
works [8,10].
The binary methanol-hexane solution with critical mass fraction of methanol
c1 = 0.314 [11] was poured into temperature-controlled optical cell with parallel
glass. The mass of the solution was such that the vapour-gas phase was held over
the investigated substance at all temperatures under study. The height of liquid in
the cell was 2 cm. Solution was quickly heated for 1.5–2 hour initially from double-
phase state at room temperature T = 293 K to the consolute critical temperature
Tc = 307.1 K. The temperature, at which the phase interface of the solution com-
ponents disappeared was taken for the critical one. Refractive index gradient and
intensity of scattered light at this temperature at the height z = 0, at which critical
values of inhomogeneous substance of density and concentration are realized, attain
the maximum value. The investigated solution had been thermostated at this tem-
perature with an accuracy ±0.01 K for a long time, almost a day, until the refractive
index gradient dn/dz ceased to vary at all heights of inhomogeneous solution. Ther-
mostating system allowed to maintain the height gradient of temperature not higher
than dT/dz = (1 ÷ 2) · 10−3 K/cm [12,13], which were measured by two resistance
thermometers set in the top end and the bottom of the cell.
After that the given inhomogeneous but equilibrium system was quickly heat-
ed for 10–15 min from the critical temperature to various temperatures T i > Tc
and thermostated for a long time until it turned into another equilibrium state
at temperature Ti. The period of time, after which the value dn/dz(z) practically
didn’t change, was taken for the equilibrium time te [14]. These temperature jumps
∆T = Ti − Tc differed from each other and varied in the range of ∆T from 0.1 K to
20 K.
The altitude dependence of the refractive index gradient dn/dz(z) of the inves-
tigated solution was continuously varying with the time t while thermostating. This
change of dn/dz(z) with time t at the temperature ∆T = Ti − Tc = 3.96 K is given
as an example in figure 1.
Figure 1 shows kinetics of change of symmetrized values of refractive index
gradient dn/dz(z, t) = 1/2 · (dn/dz(z > 0) + dn/dz(z < 0) at different heights
z of the cell with inhomogeneous methanol-hexane solution at the temperature
450
Dynamic scaling equation of state for nonequilibrium solution. . .
(a) (b)
Figure 1. Kinetics of change of altitude dependence of symmetrized values of
refractive index gradient dn/dz and concentration gradient dc/dz for inhomoge-
neous methanol-hexane solution at temperature ∆T = 3.96 K: (a) 3-dimensional
surface dn/dz(z, t); (b) sections of the dn/dz(z, t) surface by planes t = const for
different times t after the beginning of thermostating.
∆T = Ti − Tc = 3.96 K as the system approaches equilibrium state. The values
dn/dz(z) were taken at heights z symmetric with respect to the level z = 0, where
the critical values of density and concentration of solution are realized. The action of
the mentioned temperature gradient dT/dz = (1÷2)·10−3 K/cm at this temperature
cannot considerably influence the magnitude of refractive index gradient and the ki-
netics of its change with time according to calculations [13]. As it is obvious from
the figure, with time t increasing, the value of refractive index gradient at the height
z = 0, where critical values of density and concentration are realized, decreases; and
at the heights z > 0.2 cm it increases on the contrary. This results in intersection of
altitude dependencies dn/dz(z, t) in the vicinity of the height z ≈ 0.2 cm.
By means of Lorentz-Lorenz formula [15]
n2 − 1
n2 + 2
= ρs(c1∆r + r2) (1)
these data (figure 1) were used to analyze kinetics behaviour of establishment of
equilibrium concentration gradient values of the investigated solution (figure 1):
dc(z)
dz
=
1
ρs∆r
·
6n
(n2 + 2)2
·
dn(z)
dz
. (2)
Here c1 is mass concentration of methanol in the solution; ∆r = r1 − r2; r1, r2 are
specific refractions of the components; ρs is density of the solution.
451
A.D.Alekhin, Yu.L.Ostapchuk
(a) (b)
Figure 2. Kinetics of change of altitude dependence of concentration for
methanol-hexane solution at temperature ∆T = 3.96 K: (a) 3-dimensional sur-
face (c− cc)/cc(z, t); (b) sections of the (c− cc)/cc(z, t) surface by planes t =
const for different times t after the beginning of thermostating.
Concentration deviations from critical value (value of order parameter of solu-
tion) ∆c∗(z) = (c− cc)/cc = 1/cc ·
z
∫
0
dc/dz dz were obtained by integrating the
derivative dc/dz(z, t) over height z. Therefore, altitude distribution of the concen-
tration c(z, t) = cc±∆c(z, t) was obtained as well. These data are shown in figure 2.
The above presented experimental data of kinetics of equilibrium establishment
of concentration gradient values dc/dz(z, t) and concentrations ∆c(z, t) were used to
build-up dynamic scaling equations of state for substance under gravity close to the
critical point for temperatures above the critical one T > Tc. The analysis of alti-
tude and time dependencies behaviour of the obtained data dc/dz(z, t) and ∆c(z, t)
has allowed to suppose that similar equilibrium properties of substance for certain
temperatures θ = (T − Tc)/Tc correspond to these nonequilibrium characteristics of
solution at different times t.
This suggestion follows from qualitatively the same time behaviour of nonequi-
librium values dn/dz(z, t) (figure 1) and temperature dependencies of equilibrium
values dn/dz(z, θ) [16,17]. The analysis of the obtained data has showed that magni-
tude dn/dz(z, t) at the level z = 0 decreases by power relation dn/dz(z = 0, t) ∼ t−x
(x = 0.543) with time increasing. At the levels z > z0 ≈ 0.2 cm with time t increas-
ing the value dn/dz also increases. The same temperature dependence of equilibrium
values of derivative dn/dz(θ) and intensity of scattered light I ∼ dρ/dµ(θ) at the
different heights of inhomogeneous system is confirmed by all existent experimen-
tal data [16,17] and theoretical accounts of gravity effect [17] based on the modern
theory of phase transition.
452
Dynamic scaling equation of state for nonequilibrium solution. . .
Figure 3. Relationship between time, for which altitude distributions dn/dz(z, t)
of nonequilibrium liquid correspond, and respective temperature of equilibrium
states of inhomogeneous liquid.
Proceeding from these qualitatively same time dn/dz(z, t) ∼ t−x and temper-
ature dependencies dn/dz(z, θ) ∼ θ−γ for the case dn/dz(z, ti) = dn/dz(z, θi) it
is possible to propose the following relation between temperature θ i of equilibrium
value dn/dz(z, θi) and time ti of nonequilibrium values dn/dz(z, ti):
dn/dz(z = 0, ti)
dn/dz(z = 0, te)
=
(
ti
te
)
−x
=
dn/dz(z = 0, θi)
dn/dz(z = 0, θe)
=
(
θi
θe
)
−γ
. (3)
Here dn/dz(z = 0, te) ≡ dn/dz(z = 0, θe) are experimentally measured equilibrium
values of refractive index gradient at the temperature ∆T = 3.96 K, where time
te = 31.25 hour. From expression (3) the relation between ti and θi follows:
θi(ti) = θe
(
ti
te
)x/γ
= C · tni . (4)
This dependence is shown in figure 3. It results from these data that coefficient
C = θe/t
x/γ
e = 2.82 · 10−3 hour−n and exponent n = x/γ = 0.434.
Given this, based on the received data dn/dz(z, t) there were suggested the
scaling equations of a nonequilibrium fluid under gravity in a differential and integral
view:
dc
dz
= θ−γ · f1(z
∗) = t−nγ · f ′
1(z
∗′), (5)
∆c∗ = θβ · f2(z
∗) = tnβ · f ′
2(z
∗′). (6)
Here f1(z
∗) and f ′
1(z
∗′); f2(z
∗) and f ′
2(z
∗′) are scaling functions of the scaling argu-
ments z∗ = z/θβδ and z∗′ = z/tnβδ respectively (here γ ≈ 5/4; β ≈ 1/3; δ ≈ 5 are
critical exponents of the fluctuation theory [5–7].
453
A.D.Alekhin, Yu.L.Ostapchuk
(a) (b)
Figure 4. The scaling functions of concentration gradient (a) and concentration
(b) for all investigated values of concentrations (∆c∗ = 0÷ 0.8).
If the scaling hypothesis [5–7] exists for nonequilibrium systems close to critical
point, the transition to the scaling equation of state should transform 3-dimensional
surfaces dc/dz(z, t) and ∆c∗(z, t) to the scaling lines f ′
1(z
∗′) and f ′
2(z
∗′). The view
of these scaling functions is shown in figure 4a,b.
To build-up these functions there have been used experimental data dc/dz(z, t)
and ∆c∗(z, t) in the whole range of the investigated concentrations: from the small
ones ∆c∗(z, t) 6 (0 ÷ 0.3) ≪ 1 to the large ones ∆c∗(z, t) 6 (0.4 ÷ 0.8). It has
had an influence on the build-up of a scaling equation of state. As it is visible from
figure 4 the surfaces dc/dz(z, t) and ∆c∗(z, t) are not transformed into single lines
in the whole range of heights z and times t. It is due to the fact that the Ginzburg
criterion Gi ≪ 1 [19] is defaulted for large concentrations ∆c∗(z, t) 6 (0.4÷0.8) 6 1
according to [5]. That is, the system falls out from fluctuation area and cannot
be described by scaling equation of state. That is why we have selected from the
experimental data dc/dz(z, t) and ∆c∗(z, t) the range of heights z and times t,
for which only small concentrations ∆c∗(z, t) 6 0.3 ≪ 1 correspond. The scaling
functions f ′
1(z
∗′) and f ′
2(z
∗′), built-up specially for this range of concentrations, are
shown in figures 5a,b.
As it is obvious, it is only for these small concentrations that the three-dimension-
al surfaces dc/dz(z, t) and ∆c∗(z, t) really converge to the single lines f ′
1(z
∗′) and
f ′
2(z
∗′) of the scaling argument z∗′ = z/tnβδ. These lines can be described by the
following scaling equations:
f ′
1(z
∗′) =
∞
∑
n=0
An(z
∗′)2n ≈ A0 −A1(z
∗′)2 + A2(z
∗′)4 + . . . , (7)
f ′
2(z
∗′) =
∞
∑
n=0
Bn(z
∗′)2n+1 ≈ B0(z
∗′)− B1(z
∗′)3 + . . . . (8)
454
Dynamic scaling equation of state for nonequilibrium solution. . .
(a) (b)
Figure 5. The scaling functions of concentration gradient (a) and concentration
(b) for small values of concentrations ∆c∗ = (0÷ 0.3) ≪ 1.
Here are
A0 = (1.31± 0.04) · 10−2 hournγ/mm;
A1 = (7.95± 0.05) · 10−2 hour2nβδ+nγ/mm3;
A2 = (0.26± 0.01) hour4nβδ+nγ/mm5;
B0 = (0.95± 0.03) hournβδ−nβ/mm;
B1 = (1.2± 0.2) hour3nβδ−nβ/mm3.
In figures 1, 2 these surfaces are marked by thick lines, which bound concentrations
∆c∗ 6 0.3 ≪ 1.
Thus, the analysis of the obtained results has shown that in the range of small
concentrations ∆c∗ = (0 ÷ 0.3) ≪ 1 kinetics of equilibrium establishment in inho-
mogeneous nonequilibrium methanol-hexane system under gravity can be described
by dynamic scaling equations of state of equilibrium systems [5]. Hence, based on
the relation (4) it is possible to predict in advance not only nonequilibrium values
dc/dz(z, t) and ∆c∗(z, t) at different times t, but also to determine scaling functions
f1(z
∗) and f2(z
∗) (5), (6) for equilibrium solution under gravity close to critical
point.
The work was supported by the Ukrainian State Fund of Fundamental research.
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Динамічне рівняння стану нерівноважного розчину в
гравітаційному полі вище критичної температури
розшарування
О.Д.Альохін, Ю.Л.Остапчук
Фізичний факультет,
Київський національний університет ім. Тараса Шевченка,
03680 Київ, просп. Глушкова, 6
Отримано 11 серпня 2001 р., в остаточному вигляді –
26 червня 2001 р.
В роботі досліджена висотна та часова залежність нерівноважних
значень градієнта показника заломлення неоднорідного розчину ме-
танол-гексан у гравітаційному полі поблизу критичної температури
розшарування при прямуванні системи до стану рівноваги. На осно-
ві цих даних запропоновані масштабні рівняння нерівноважної рідини
в гравітаційному полі для температур вищих від критичної темпера-
тури розшарування. Показано, що і для нерівноважного розчину по-
близу критичної точки для малих значень параметра порядку систе-
ми виконується масштабна гіпотеза.
Ключові слова: масштабне рівняння стану, нерівноважний розчин,
гравітаційний ефект, критична точка розшарування
PACS: 05.70.Jk, 64.30.+t, 64.60.Ht, 64.70.Ja
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