Critical transport and critical scattering in fluids
We consider the critical properties of fluids induced by the critical fluctuations in the order parameter. The theory describes the crossover from the analytic background behaviour to the universal asymptotic behaviour of several dynamic quantities. Розглядаються критичні властивості плинів, що інду...
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Folk, R. 2017-06-13T13:12:03Z 2017-06-13T13:12:03Z 2000 Critical transport and critical scattering in fluids / R. Folk // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 359-370. — Бібліогр.: 18 назв. — англ. 1607-324X DOI:10.5488/CMP.3.2.359 PACS: 05.70.Jk, 64.60.Ht, 64.70.Fx, 64.70.Ja https://nasplib.isofts.kiev.ua/handle/123456789/121027 We consider the critical properties of fluids induced by the critical fluctuations in the order parameter. The theory describes the crossover from the analytic background behaviour to the universal asymptotic behaviour of several dynamic quantities. Розглядаються критичні властивості плинів, що індуковані критичними флюктуаціями параметра порядку. Теорія описує кросовер від аналітичної фонової поведінки до асимптотичної поведінки для кількох динамічних величин. I thank G.Flossmann and G.Moser, the authors of joint papers, for collaboration in this research. The research is supported by the Fonds zur Foerderung der Wissenschaftlichen Forschung under project 12422–TPH. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Critical transport and critical scattering in fluids Критичний перенос і критичне розсіяння у плинах Article published earlier |
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Critical transport and critical scattering in fluids |
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Critical transport and critical scattering in fluids Folk, R. |
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Critical transport and critical scattering in fluids |
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Critical transport and critical scattering in fluids |
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Critical transport and critical scattering in fluids |
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critical transport and critical scattering in fluids |
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Folk, R. |
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Folk, R. |
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2000 |
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Критичний перенос і критичне розсіяння у плинах |
| description |
We consider the critical properties of fluids induced by the critical fluctuations in the order parameter. The theory describes the crossover from
the analytic background behaviour to the universal asymptotic behaviour
of several dynamic quantities.
Розглядаються критичні властивості плинів, що індуковані критичними флюктуаціями параметра порядку. Теорія описує кросовер від
аналітичної фонової поведінки до асимптотичної поведінки для кількох динамічних величин.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121027 |
| citation_txt |
Critical transport and critical scattering in fluids / R. Folk // Condensed Matter Physics. — 2000. — Т. 3, № 2(22). — С. 359-370. — Бібліогр.: 18 назв. — англ. |
| work_keys_str_mv |
AT folkr criticaltransportandcriticalscatteringinfluids AT folkr kritičniiperenosíkritičnerozsíânnâuplinah |
| first_indexed |
2025-11-25T20:30:25Z |
| last_indexed |
2025-11-25T20:30:25Z |
| _version_ |
1850523707514028032 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 2(22), pp. 359–370
Critical transport and critical scattering
in fluids
R.Folk
Institute for Theoretical Physics, University of Linz, Austria
Received March 1, 2000
We consider the critical properties of fluids induced by the critical fluctu-
ations in the order parameter. The theory describes the crossover from
the analytic background behaviour to the universal asymptotic behaviour
of several dynamic quantities.
Key words: gas-liquid phase transition, critical dynamics, light scattering,
transport coefficients
PACS: 05.70.Jk, 64.60.Ht, 64.70.Fx, 64.70.Ja
1. Introduction
Collective effects dominate near the phase transition while there arises a critical
behaviour quite different from the behaviour further away from the critical point.
These critical phenomena are seen in the experiments in a more or less wide region
round the critical point. E.g., the thermal conductivity of a pure liquid diverges
strongly, with an exponent of O(1), whereas the shear viscosity diverges weakly,
with an exponent of O(0.1), as a function of the relative temperature distance from
the critical point with some power law in the region asymptotically near the critical
temperature Tc. Outside this region, a normal analytic behaviour is observed. A
delicate point is the strength of the critical behaviour compared to the noncritical
or background behaviour and determining the background values of the transport
properties. Depending on this strength, both types of behaviour can be separated
with more or less accuracy. In a series of papers, the crossover behaviour from criti-
cality to the background behaviour has been studied [1,2] and extended to mixtures
[3]. The goal of these studies was the description of all dynamical quantities with-
in the formalism of the field theoretic formulation of renormalization group (RG)
theory [4]. One advantage of the nonasymptotic RG-theory is that the calculations
lead directly to the complete value (without the analytic temperature dependence)
of the transport coefficients in the background and no separation of the fluctuation
contribution as in the mode coupling theory has to be performed [5].
An important issue in the description of the critical behaviour is the appropriate
choice of the collective variables [7] connected to the so-called order parameter (OP).
c© R.Folk 359
R.Folk
In statics near the liquid-gas critical point in pure liquids these are the deviations
of the density from the critical density, in dynamics according to model H [8] it is
the entropy density fluctuation ∆σ, and we introduce the order parameter field φ0
given by
φ0(x) =
√
NA(△σ(x)− 〈△σ(x)〉) . (1)
In addition, one has to consider the tranverse momentum current j t which is dynam-
ically coupled to the OP. The Hamiltonian describing thermodynamic equilibrium
is
H =
∫
ddx
{1
2
o
τ φ2
0(x) +
1
2
(∇φ0(x))
2 +
o
ũ
4!
φ4
0(x) +
1
2
ajj
2
t (x)
}
, (2)
and the dynamic equations are
∂φ0
∂t
=
o
Γ ∇2 δH
δφ0
− o
g (∇φ0)
δH
δj
+Θφ , (3)
∂jt
∂t
=
o
λt ∇2 δH
δjt
+
o
g T
{
(∇φ0)
δH
δφ0
}
− o
g T
{
∑
k
[
jk∇
δH
δjk
+∇kj
δH
δjk
]}
+Θt . (4)
T is the projector to the direction of the transverse momentum density, which cor-
responds to a projection orthogonal to the wave vector in Fourier space. In the fast
fluctuating forces Θi(x, t) (i = φ, t) memory effects are irrelevant and their Gaussian
spectrum fulfils the Einstein relations
〈Θi(x, t) Θj(x
′, t′)〉 = 2Lij(x)δ(t− t′)δ(x− x′), (5)
where the matrix [Lij ] of the diffusive modes is given by
[Lij ] =
(
−
o
Γ ∇2
0 −
o
λt ∇2
)
. (6)
The nonrenormalized mode coupling is defined as
o
g= RT/
√
NA with the gas constant
R and the Avogadro number NA.
Within this model all static and dynamic critical properties can be calculated.
One obtains especially the correlation functions, and from their half width the char-
acteristic frequencies. In the hydrodynamic region the transport coefficients as func-
tions of temperature result from these frequencies [1]. Our theory yields all these
dynamical quantities as functions of the measurable correlation length ξ and the dy-
namical parameters entering the model equations (3) and (4). The most important
is the renormalized mode coupling ft (its nonrenormalized counterpart is
o
g /
√
o
Γ
o
λt ).
It’s value for a certain temperature, wave vector or frequency, is determined by
the solution (see equation (9) below) of a renormalization group equation and an
appropriate matching condition.
360
Critical transport
10-7 10-6 10-5 10-4 10-3 10-2 10-1
500
microgravity 2 Hz
ground (h=0.7mm)
ground (h=4.0mm)
700
600
t0 = 0.001
f0 = 0.959
Γ0 = 8.82 x 10-18
xη = 0.069
A = 5
R
e(
η)
[µ
P
oi
se
]
t
Figure 1. Shear viscosity of xenon [9] with and without gravity compared with
RG theory.
2. Shear viscosity
The temperature and density dependent shear viscosity at zero frequency calcu-
lated within model H reads [2]
η̄(t,∆ρ) = η̄0
1− f 2
t (t,∆ρ)/36
1− f 2
0 /36
(
f 2
0 ξ(t,∆ρ)
f 2
t (t,∆ρ)ξ(t0)
)xη
≡ η̄0 exp(xηH(t,∆ρ)) , (7)
with the amplitude
η̄0 =
kBT
4π
ξ(t0)
f 2
0Γ0
(
1− f 2
0
36
)
, (8)
and the mode coupling
f 2
t (t,∆ρ) =
24
19
[
1 +
ξ(t0)
ξ(t,∆ρ)
(
24
19f 2
0
− 1
)]
−1
. (9)
The dependence on the relative temperature distance from the critical tempera-
ture Tc and the relative density distance from the critical density ρc enters via the
correlation length ξ(t,∆ρ) which is obtained from the equations
t =
T − Tc
Tc
= (1− b2θ2) r , (10)
∆ρ =
ρ− ρc
ρc
= k (θ + c θ3) rβ , (11)
ξ(t,∆ρ) = ξ0 (1 + 0.16θ2) r−ν = ξ0 t
−ν (1 + 0.16θ2)(1− b2θ2)ν , (12)
of the cubic model [6].
361
R.Folk
1x10-7 10-6 10-5 10-4 10-3
550
600
650
700
10-6 10-5
-0.04
-0.03
-0.02
-0.01
0.00
0.01
-0.04
-0.03
-0.02
-0.01
0.00
0.01
RNG
t
0
= 0.001
f
0
= 0.959
Γ
0
= 8.82 x 10-18
x
η
= 0.069
A = 1
MC
q
C
= 2.77x106 cm-1
q
D
= 8.70x106 cm-1
τ
0
= 2.31x10-12
x
η
= 0.069
A = 1
2 Hz
3 Hz
5 Hz
8 Hz
12 Hz
R
e(
η)
[µ
P
oi
se
]
t
RNG
t
0
= 0.001
f
0
= 0.959
Γ
0
= 8.82 x 10-18
x
η
= 0.069
A = 1
MC
q
C
= 2.77x106 cm-1
q
D
= 8.70x106 cm-1
τ
0
= 2.31x10-12
x
η
= 0.069
A = 1
2 Hz
3 Hz
5 Hz
8 Hz
12 Hz
-I
m
(η
)
/ R
e(
η)
t
Figure 2. Comparison of the mode coupling theory [10] (light gray) and the RG
theory (dark). Data from [9].
362
Critical transport
It is necessary to know the density dependence to calculate the gravity effects on
the shear viscosity measurements on earth. Because of the finite height of the cell,
a density gradient over the cell height is caused by gravity and depending on the
positions of the oscillating disks by which the shear viscosity is measured a mean
value of shear viscosities at different densities is measured. Since the gravity field
is conjugated to the order parameter, the gradient over the cell increases when one
approaches the critical temperature and this drives the shear viscosity away from the
critical point at t = ∆ρ = 0. Therefore it reaches a finite value instead of diverging
near the critical point. This is demonstrated in figure 1 where two measurements
on earth in different cells and the measurements on board the “Discovery” [9] are
shown together with our calculations.
In the low gravity experiment, the frequency dependence of the shear viscosity
can be observed. This is not possible on earth since the gravity effects already cover
up the frequency effects. The frequency dependence of the shear viscosity was also
calculated in one loop order [1,2] and we compare our result with the low gravity
experiments [9] and mode coupling theory [10] in figure 2. No agreement is found in
both theories. Introducing a phenomenological parameter into the frequency scale
[9] agreement can be achieved in mode coupling theory for both the real part and
the ratio of the real part to the imaginary part of the viscosity. This also holds
for the RG-result for the real part of the viscosity (with a value different [11] from
the mode coupling theory). The ratio of the real part to the imaginary part of the
viscosity cannot be improved by this procedure since its limiting value at T c is given
by the universal one loop order result [11] (f ∗2
t = 24/19)
lim
T→Tc
ℑ(η)
ℜ(η) =
f ∗2
t
96
π
2
[
1− f ∗2
t
96
{3 ln(1/4)− 1/3}
]
−1
≃ 0.0195. (13)
Within the mode coupling theory of [10,9] the value
lim
T→Tc
ℑ(η)
ℜ(η) = tan
πxη
2(3 + xη)
≃ 0.0353 (14)
is obtained (the experimental value xη = 0.069 of the shear viscosity exponent is
used) and seems to be in agreement with the data. As a result, one might conclude
that the one loop theory is not sufficient to describe the critical frequency dependence
of the shear viscosity. A complete two loop calculation is in preparation [12].
3. Thermal diffusivity
For a complete description of the dynamic critical behaviour it is also necessary
to consider the thermal conductivity or thermal diffusivity. Once the values of the
nondynamical universal parameters such as Γ0 and f0 are determined, all quantities
can be calculated from one theory. E.g., the thermal diffusivity reads
DT (t,∆ρ) = D0
ξ2(t0)
ξ2(t,∆ρ)
(
1− f2
t (t,∆ρ)
16
)
(
1− f2
0
16
)
(
f 2
0 ξ(t.∆ρ)
f 2
t (t,∆ρ)ξ(t0)
)xκ
, (15)
363
R.Folk
100 150 200 250 300
0
5
10
15
20
25
30
∆η
[µ
P
oi
se
]
103 ρ [g/cm3]
100 150 200 250 300
10-4
10-3
10-2
D
T
[c
m
2 /s
]
103 ρ [g/cm3]
Figure 3. (a) Shear viscosity in C2H6 along various isotherms. The plot contains
our results (thick curves) as well as experimental data [15] and theoretical results
of the mode coupling theory (thin dashed curves) [14]. The curves were shifted
by 5, 10 or 15µPoise respectively for better clearness. (b) Thermal diffusivity
in C2H6 along various isotherms. The plot contains our results (thick curves) as
well as experimental data [16] and theoretical results of the mode coupling theory
(thin curves) [14]. We have used the analytic background expressions of [17] (from
[2]).
As an example the comparison of a calculation of the shear viscosity and the thermal
diffusivity is shown in figure 3. Only the shear viscosity data at ∆ρ = 0 are used to
determine the nonuniversal parameters. It should be mentioned that an extension
of such calculations to sound propagation is possible and it has also been considered
for pure fluids in [1] and for mixtures in [3].
4. Scaling for pure fluids
The dynamic scaling assumption states that the dynamic correlation function
χdyn of the OP in the asymptotic region is a homogeneous function of its variables
and can be written in the form (see [13])
χdyn(ξ, k, ω) =
χst(ξ, k)
ωc(ξ, k)
F
(
ω
ωc(ξ, k)
, kξ
)
. (16)
364
Critical transport
The characteristic frequency ωc (we define the half width at half height) is also a
homogeneous function
ωc(ξ, k) = kzf(kξ), (17)
where the frequency is measured in an appropriate time scale. The static correlation
function χst also scales as
χst(ξ, k) = k−2+ηg(kξ) , (18)
and the shape function F (y, x) (y = ω/ωc(ξ, k) and x = kξ) fulfils the relations
∫
dyF (y, x) = 2π, F (1, x) =
1
2
F (0, x). (19)
Since the OP is conserved for the cases considered here, the correlation function
has to be proportional to k2 in the limit k → 0. Neglecting static and dynamic
interactions of fluctuations, conventional theory leads to the exponents η = 0 and
z = 4 (for a nonconserved OP the exponent would be z = 2). Moreover, the shape
function F (y, x) is independent of x. RGT calculates the values of the exponents as
η ∼ 0.04 and z = 4− η − xλ ∼ 3 since xλ ∼ 0.916 [8].
In addition to this considerable change of the exponent’s values, the shape func-
tion F (y, x) will depend on x. Let us now consider several regions in the (ξ−1, k, ω)-
space.
In the hydrodynamic region kξ ≪ 1 in fluids and mixtures the dynamic behaviour
is described by diffusive modes, e.g. ω = Dk2 for the thermal diffusion, that means
for the temperature dependence of the OP diffusion D
D(ξ) = Γ(ξ)/χst(ξ) ∼ ξ2−z, Γ(ξ) ∼ ξ2−z+γ/ν (20)
with Γ the OP Onsager coefficient. In the conventional theory Γ is noncritical,
whereas RGT predicts Γ ∼ ξxλ (we have used the static scaling law γ = ν(2 − η)).
In the background both Γ, χ and D are temperature independent (apart from a
weak analytic temperature dependence outside the scope of our considerations).
Throughout the hydrodynamic regime, the shape of the correlation function is of
Lorentzian form
χdyn(ξ, k, ω) = χst(ξ, k)2 ℜ 1
−iω +D(ξ)k2
(21)
thus
F (y, x << 1) = 2 ℜ 1
iy + 1
. (22)
In order to consider the scaling laws in the critical region kξ ≫ 1 we may directly
go to Tc. At Tc (ξ = ∞), the correlation function can be written as
χdyn(k, ω) ∼ k−z−2+ηF (y,∞) (23)
since ωc ∼ kz and χst ∼ k−2+η. Because of the conservation property of the correla-
tion function in the limit k → 0 mentioned above and because its value is finite in
365
R.Folk
-7
-5
-3
-1
1
3
5
7
-1
-3
-5
-7
f
0
= 0.01
f
0
= 0.1
f
0
= f
t
*
10
lo
g
ω
c /
ω
cvH
10 lo
g
k
[Å
-1 ]
10
log ξ -1
[Å -1
]
Figure 4. Ratio of the nonasymptotic characteristic frequency of the RG calcu-
lation to the characteristic frequency of the van Hove theory for different back-
ground values of the mode coupling f0. Upper wide mash f0 = f∗, next lower
mash f0 = 0.1, inner ‘pyramid’ f0 = 0.01. For f0 = 0 one obtains for the ratio
the bottom plane. The smaller the value of the mode coupling the smaller is the
region around the critical point k = ξ−1 = 0 where the asymptotic power laws
are seen.
the limit ω → 0, the shape function behaves as
F (y,∞) ∼ const, y → 0; (24)
F (y,∞) ∼ y−(z+4−η)/z, y → ∞. (25)
This shows that RGT predicts a non-Lorentzian shape at Tc since it decays faster
in the scaled frequency, namely roughly as y−2.3 instead of y−2. In other words this
demonstrates the non Markovian property of the OP time correlations.
5. Characteristic frequency
A complete calculation of the shape function and width in RG-theory is lacking.
A one loop calculation gives no frequency dependent perturbational contribution
to the order parameter vertex functions, which seems to be in contradiction to
the scaling arguments of the preceding section. In the following we remain with-
in the Lorentzian approximation for the correlation function, then the result of a
nonasymptotic RG calculation of the characteristic frequency yields
ωc(k, x) = Γas k
z
(
1 + x2
x2
)1−xλ/2
cxλ
na(k, x)f(k, x) (26)
366
Critical transport
10-5 10-4 10-3 10-2
10-6
10-5
10-4
t0 = 1.000
f0 = 0.251
Γ0 = 2.62 x 10-18
xη = 0.063
k = 0.414 x 105 cm-1
k = 1.555 x 105 cm-1
asymptotic
non-asymptotic
ω c /
k
2 [
cm
2 s-1
]
∆t
Figure 5. Nonasymptotic and asymptotic characteristic frequency ωc as function
of temeperature for diffenrent wave vectors k calculated in ε-expansion. Data for
CO2 from [18].
with
cna(k, x) =
1 +
k
k0
√
1 + x2
x2
(27)
and
f(k, x) = 1− 3
38 cna(k, x)
[
−5 + 6 x−2 ln(1 + x2)
]
, (28)
where
x = kξ(t), Γas = Γ0
(
19
24
f 2
0 ℓ0
ξ0
)xλ
, k−1
0 =
(
24
19f 2
0
− 1
)
ξ0
ℓ0
. (29)
There are two dynamic non universal parameters. One of them, Γ0, sets the scale of
367
R.Folk
the frequency and the other one, the dimensionless mode coupling f0, achieves the
crossover from the van Hove theory (f0 = 0, and z = 4)
ωc,vH = Γ0 k
4
(
1 + x2
x2
)
(30)
to the asymptotic theory (f0 = f ∗, z ∼ 3)
ωc,as = Γ0
(
ℓ0
ξ0
)xλ
kz
(
1 + x2
x2
)1−xλ/2 (
1− 3
38
[
−5 + 6 x−2 ln(1 + x2)
]
)
. (31)
For any finite f0, a crossover from the van Hove theory in the background to the
asymptotics results near the critical point, ξ → ∞ and k → 0. In this case the
asymptotic amplitude, Γas (29), of the characteristic frequency depends on the mode
coupling f0.
We rewrite the nonasymptotic expression for the characteristic frequency in the
following form
ωc(k, ξ) = ωc,vH(k, ξ) (kξ0)
−xλ
(
19
24
f 2
0 ℓ0
)xλ
(
1 + x2
x2
)
−xλ/2
cna(k, x)
xλf(k, x). (32)
The crossover behaviour for finite mode coupling f0 can be seen in figure 4. Near
Tc the asymptotic expression (31) applies whereas in the background van Hove
behaviour is reached. Taking the nonuniversal value for Γ0 from the shear viscosity
the value of f0 is adjusted to fit the characteristic frequency data. This leads to a
value definitive different from the fixed point value as can be seen from figure 5.
Similar results are obtained for xenon [11].
I thank G.Flossmann and G.Moser, the authors of joint papers, for collabora-
tion in this research. The research is supported by the Fonds zur Foerderung der
Wissenschaftlichen Forschung under project 12422–TPH.
References
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attenuation near the critical point in liquids. I. Theoretical results. // Phys. Rev. E,
1998, vol. 57, No. 1, p. 683–704; II. Comparison with experiment. // Phys. Rev. E,
1998, vol. 57, No. 1, p. 705–719.
2. Flossmann G., Folk R., Moser G. Frequency-dependent shear viscosity, sound velocity
and sound attenuation near the critical point in liquids. III. The shear viscosity. //
Phys. Rev. E, 1999, vol. 60, No. 1, p. 779–791.
3. Folk R., Moser G. Critical dynamics in mixtures. // Phys. Rev. E, 1998, vol. 58,
No. 5, p. 6246–6274; Critical sound propagation in mixtures. // Europhys. Lett.,
1998, vol. 41, No. 2, p. 177–182.
4. Bausch R., Janssen H.K., Wagner H. Renormalized field theory of critical dynamics.
// Z. Phys. B, 1976, vol. 24, p. 113–127.
368
Critical transport
5. Sengers Jan van Effects of critical fluctuations on the thermodynamic and transport
properties of supercritical fluids. – In: Supercritical Fluids, ed. Kiran E. and Levelt-
Sengers J.M.H., p. 231–271.
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369
R.Folk
Критичний перенос і критичне розсіяння у плинах
Р.Фольк
Інститут теоретичної фізики, Університет м. Лінца, Австрія
Отримано 1 березня 2000 р.
Розглядаються критичні властивості плинів, що індуковані критич-
ними флюктуаціями параметра порядку. Теорія описує кросовер від
аналітичної фонової поведінки до асимптотичної поведінки для кіль-
кох динамічних величин.
Ключові слова: фазовий перехід газ-рідина, критична динаміка,
розсіяння світла, коефіцієнти переносу
PACS: 05.70.Jk, 64.60.Ht, 64.70.Fx, 64.70.Ja
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