Critical sound in fluids and mixtures
The behaviour of sound is explained within the dynamical renormalization group theory. Non-asymptotic effects are due to the deviation of renormalized couplings from their fixed point values. We calculate the temperature and frequency dependence of the sound velocity and absorption near the consol...
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| Zitieren: | Critical sound in fluids and mixtures / R. Folk, G. Moser // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 243-254. — Бібліогр.: 25 назв. — англ. |
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Folk, R. Moser, G. 2017-06-12T06:39:35Z 2017-06-12T06:39:35Z 1999 Critical sound in fluids and mixtures / R. Folk, G. Moser // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 243-254. — Бібліогр.: 25 назв. — англ. 1607-324X DOI:10.5488/CMP.2.2.243 PACS: 05.70.Jk, 64.60.Ht, 64.70.Fx, 64.70.Ja https://nasplib.isofts.kiev.ua/handle/123456789/120382 The behaviour of sound is explained within the dynamical renormalization group theory. Non-asymptotic effects are due to the deviation of renormalized couplings from their fixed point values. We calculate the temperature and frequency dependence of the sound velocity and absorption near the consolute point and the gas liquid critical point in pure fluids and mixtures. The critical non-asymptotic time scale in mixtures is different from the pure fluid case and set by an effective order parameter Onsager coefficient containing the dynamical parameter related to the enhancement of the thermal conductivity. We discuss the relation to the phenomenological theory of Ferrell and Bhattacharjee for the consolute point and compare with experiments in pure ³He, ⁴He and ³He-⁴He mixtures near the plait point. У теорії динамічної ренормалізаційної групи пропонується пояснення особливостей поведінки критичного звуку. Показано, що неасимптотичні ефекти обумовлені відхиленням ренормованих параметрів взаємодії від їх фіксованих значень. Для ряду чистих рідин та сумішей розраховані температурна й частотна залежності швидкості звуку і коефіцієнта поглинання поблизу точки розшарування і критичної точки рідина-газ. Знайдено, що неасимптотичний критичний часовий масштаб у сумішах відрізняється від того, що отримується для випадку чистих плинів, і обумовлено це зростанням ролі термопровідності. Обговорюється зв’язок з феноменологічною теорією Феррелла і Бхаттачарі для точки розшарування та проводиться порівняння з експериментами для чистих плинів ³He і ⁴He та їх сумішей поблизу критичної точки. We acknowledge support by the Fonds zur Foerderung der Wissenschaftlichen Forschung under project 12422-TPH. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Critical sound in fluids and mixtures Критичний звук у рідинах та сумішах published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Critical sound in fluids and mixtures |
| spellingShingle |
Critical sound in fluids and mixtures Folk, R. Moser, G. |
| title_short |
Critical sound in fluids and mixtures |
| title_full |
Critical sound in fluids and mixtures |
| title_fullStr |
Critical sound in fluids and mixtures |
| title_full_unstemmed |
Critical sound in fluids and mixtures |
| title_sort |
critical sound in fluids and mixtures |
| author |
Folk, R. Moser, G. |
| author_facet |
Folk, R. Moser, G. |
| publishDate |
1999 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| title_alt |
Критичний звук у рідинах та сумішах |
| description |
The behaviour of sound is explained within the dynamical renormalization
group theory. Non-asymptotic effects are due to the deviation of renormalized couplings from their fixed point values. We calculate the temperature
and frequency dependence of the sound velocity and absorption near the
consolute point and the gas liquid critical point in pure fluids and mixtures.
The critical non-asymptotic time scale in mixtures is different from the pure
fluid case and set by an effective order parameter Onsager coefficient containing the dynamical parameter related to the enhancement of the thermal conductivity. We discuss the relation to the phenomenological theory
of Ferrell and Bhattacharjee for the consolute point and compare with experiments in pure ³He, ⁴He and ³He-⁴He mixtures near the plait point.
У теорії динамічної ренормалізаційної групи пропонується пояснення особливостей поведінки критичного звуку. Показано, що неасимптотичні ефекти обумовлені відхиленням ренормованих параметрів
взаємодії від їх фіксованих значень. Для ряду чистих рідин та сумішей
розраховані температурна й частотна залежності швидкості звуку і
коефіцієнта поглинання поблизу точки розшарування і критичної точки рідина-газ. Знайдено, що неасимптотичний критичний часовий
масштаб у сумішах відрізняється від того, що отримується для випадку чистих плинів, і обумовлено це зростанням ролі термопровідності. Обговорюється зв’язок з феноменологічною теорією Феррелла і Бхаттачарі для точки розшарування та проводиться порівняння
з експериментами для чистих плинів ³He і ⁴He та їх сумішей поблизу
критичної точки.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120382 |
| citation_txt |
Critical sound in fluids and mixtures / R. Folk, G. Moser // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 243-254. — Бібліогр.: 25 назв. — англ. |
| work_keys_str_mv |
AT folkr criticalsoundinfluidsandmixtures AT moserg criticalsoundinfluidsandmixtures AT folkr kritičniizvukurídinahtasumíšah AT moserg kritičniizvukurídinahtasumíšah |
| first_indexed |
2025-11-25T21:44:47Z |
| last_indexed |
2025-11-25T21:44:47Z |
| _version_ |
1850560326432456704 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No 2(18), pp. 243–254
Critical sound in fluids and mixtures
R.Folk 1 , G.Moser 2
1 Institute for Theoretical Physics, University of Linz, Austria
2 Institute for Physics and Biophysics, University of Salzburg, Austria
Received June 26, 1998
The behaviour of sound is explained within the dynamical renormalization
group theory. Non-asymptotic effects are due to the deviation of renormal-
ized couplings from their fixed point values. We calculate the temperature
and frequency dependence of the sound velocity and absorption near the
consolute point and the gas liquid critical point in pure fluids and mixtures.
The critical non-asymptotic time scale in mixtures is different from the pure
fluid case and set by an effective order parameter Onsager coefficient con-
taining the dynamical parameter related to the enhancement of the ther-
mal conductivity. We discuss the relation to the phenomenological theory
of Ferrell and Bhattacharjee for the consolute point and compare with ex-
periments in pure 3He, 4He and 3He-4He mixtures near the plait point.
Key words: critical dynamics, sound propagation, consolute point, plait
point, renormalization group theory, transport properties
PACS: 05.70.Jk, 64.60.Ht, 64.70.Fx, 64.70.Ja
1. Introduction
Fluids and mixtures at their critical points, like the gas-liquid critical point
(plait point in mixtures) and the demixing (or consolute) point, belong to the same
static and dynamic universality class. All static critical effects can be related to the
singular behaviour of the Ising model universality class; all dynamic critical effects
can be related to the singularities of model H [1]. This is true for critical effects in
the sound propagation too.
Non-asymptotic effects describing the crossover from the non-critical non-uni-
versal background to the universal asymptotics depend on the system considered
and is different for pure fluids and mixtures. This is in general due to the fact that
the fluid parameters entering the physical quantity considered have not reached
their asymptotic (fixed point) values. One striking example is the thermal conduc-
tivity near the consolute point, which turns out to be finite at the critical point. Its
non-universal enhancement is related to the non-asymptotic behaviour of a certain
dynamical parameter with fixed point value of zero. Thus the parameter drops out
c© R.Folk, G.Moser 243
R.Folk, G.Moser
of the expression for the thermal conductivity in the asymptotics, but determines its
(lower) background value [2]. This enhancement at the consolute point has indeed
been observed in 2-butoxy-ethanol-water mixture at the critical concentration [3,4].
In this review we present recent progress in calculating the non-asymptotic
crossover functions for the sound velocity and sound attenuation as a function of
temperature and frequency (see [5] for pure fluids and [6] for mixtures).
The critical sound propagation in pure fluids and near a consolute point has been
described by Ferrell and Bahttacharjee within a phenomenological theory based on
a generalization of the specific heat to finite frequencies taking into account the
causal and scaling properties of the dynamic functions involved [7,8]. An earlier
application of the RG theory to critical sound at the consolute point used a reduced
model neglecting e.g. the thermal diffusion ratio [9]. Recently Onuki used the bulk
viscosity [10] for an intuitive derivation of the critical sound behaviour [11] which is
in agreement with our RG calculations in asymptotics and is also valid at the plait
point.
It turns out that the non-asymptotic time scale appearing in the frequency de-
pendent sound velocity and attenuation in a mixture is not set by the order param-
eter Onsager coefficient alone but by an effective one which contains the dynamic
parameter responsible for the critical enhancement of the thermal conductivity.
2. The dynamical equations
The starting point of the theoretical calculations is a stochastic model fulfilling
the constraints. In equilibrium it is described by the usual φ4-model including in a
quadratic form the additional dynamic densities and a coupling between the order
parameter and the other degrees of freedom. The static functional for a mixture
reads (the reduction to the pure fluid model is easily performed)
H =
∫
ddx
{ 1
2
o
τ φ2
0(x) +
1
2
(∇φ0(x))
2 +
1
2
~q T
0 (x)
↔
A ~q0(x) +
1
2
ajj
2(x)
+
o
ũ
4!
φ4
0(x) +
1
2
o
~γ T
q ~q0(x)φ
2
0(x)−
o
~hT
q ~q0(x)
}
, (1)
where the order parameter φ is the entropy density in the case of a gas-liquid phase
transition and the concentration fluctuation c in the case of the demixing transition.
The upright o at the parameters and the subscript 0 on the densities denote unrenor-
malized quantities. The static functional has the same structure as in pure liquids
[5] but now with two respective secondary densities ~q0
T = (
o
q1,
o
q2). As a consequence
the parameters
↔
A=
(
a11 a12
a12 a22
)
,
o
~γq=
o
γ1
o
γ2
,
o
~hq=
o
h1
o
h2
(2)
are matrices and vectors instead of scalars. The momentum density appearing in (1)
is also rescaled by the Avogadro number j =
√
NA △ j ′. The parameter aj follows
244
Critical sound
from the expression for the kinetic energy and reads aj = 1/(RTρ). The coefficients
of the matrix
↔
A are the lowest order contributions to the static two-point vertex
functions and therefore do not contain critical effects. They are related to background
values of thermodynamic derivatives at the plait and the consolute point respectively.
The dynamical model equations are an extension of model H [1]
∂φ0
∂t
=
o
Γ ∇2 δH
δφ0
+
o
~LT
q ∇2 δH
δ~q0
− o
g (∇φ0)
(
δH
δjl
+
δH
δjt
)
+Θφ , (3)
∂~q0
∂t
=
o
~Lq ∇2 δH
δφ0
+
o
↔
Λq ∇2 δH
δ~q0
− o
g (∇~q0)
(
δH
δjl
+
δH
δjt
)
− (
o
~cq +
o
~gl φ0+
o
↔
g q ~q0)∇
δH
δjl
+ ~Θq , (4)
∂j l
∂t
=
o
λl ∇2 δH
δjl
−
o
~cTq ∇
δH
δ~q0
−
o
~g T
l ∇
(
φ0
δH
δ~q0
)
−∇
(
~q T
0
o
↔
g q
δH
δ~q0
)
+
o
g (1− T )
[
(∇φ0)
δH
δφ0
+ (∇~q0)
T δH
δ~q0
]
− o
g (1− T )
∑
k
[
jk∇
δH
δjk
−∇kj
δH
δjk
]
+Θl , (5)
∂jt
∂t
=
o
λt ∇2 δH
δjt
+
o
g T
[
(∇φ0)
δH
δφ0
+ (∇~q0)
T δH
δ~q0
]
− o
g T
∑
k
[
jk∇
δH
δjk
−∇kj
δH
δjk
]
+Θt , (6)
where T is the projector to the direction of the transverse momentum density. The
static functional H is given by (1). Assuming a Markovian process the fluctuating
forces [Θi]
T = (Θφ, ~Θ
T
q ,Θl,Θl) fullfill the Einstein relations
〈[Θi](x, t)⊗ [Θj ](x
′, t′)〉 = 2[Λij ]δ(t− t′)δ(x− x′) (7)
with the matrix
[Λij] =
−
o
Γ∇2 −
o
~LT
q ∇2 0 0
−
o
~Lq∇2 −
o
↔
Λq∇2 0 0
0 0 −
o
λl∇2 0
0 0 0 −
o
λt∇2
. (8)
The Onsager coefficient vector and the Onsager coefficient matrix in (3) and (4) are
o
~Lq=
o
L
o
Lφ
,
o
↔
Λq=
o
µ
o
L12
o
L12
o
λ
. (9)
245
R.Folk, G.Moser
The mode coupling vectors and the mode coupling matrix introduced in (4) and (5)
are defined as
o
~cq=
(
0
o
c
)
,
o
~gl=
(
0
o
gl
)
,
o
↔
g q=
(
0 0
0
o
g
)
(10)
with the parameters
o
c = RTρ,
o
g = RT/
√
NA and
o
gl = RTQ2/
√
NA with Q2
given by thermodynamic derivatives [6]. The Onsager coefficients in the momentum
density equations (5) and (6) are related to the background values of the shear
viscosity η̄(0) and the bulk viscosity ζ (0) by
o
λl = RT (ζ (0) + 4
3
η̄(0)) and
o
λt = RT η̄(0).
In hydrodynamics only three transport coefficients, which are the thermal con-
ductivity κT, the thermal diffusion coefficient kT and the mass diffusion coefficient
D, appear in the equations for the entropy and the concentration. The hydrody-
namic equation for the mass density is the continuity equation, which does not
involve any dissipation. As a consequence only three (
o
Γ,
o
L and
o
µ) of the six Onsager
coefficients in equations (3)–(6) are independent. The remaining three coefficients
are determined by
o
Lφ= −Q2
o
Γ,
o
λ= Q2
2
o
Γ,
o
L12= −Q2
o
L . (11)
Consequently the fluctuating forces in (7) are not independent. Nevertheless a dy-
namic functional analogue to [12] may be derived [5].
We now apply the renormalization group procedure in its field theoretic ver-
sion (dimensional regularization, minimal subtraction) to this dynamic functional.
Comparing the dynamic vertex functions of the model in the hydrodynamic limit
with the results from the usual hydrodynamic equations for mixtures [13] leads to
expressions for the transport coefficients. Comparing one of them (e.g. the shear
viscosity) allows determinating the non-universal background parameters entering
the expression of the transport coefficients. In this way all dynamic parameters are
fixed and predictions become possible.
3. Sound velocity and absorption
By the above mentioned procedure we also find the sound velocity cs and the
sound attenuation αs at finite frequency ω and temperature distance t = (T−Tc)/Tc.
Both expressions are obtained from the complex sound velocity Cs
c2s (t, ω) = ℜ[C2
s (t, ω)] αs(t, ω) = −ωℑ[C2
s (t, ω)]
2c3s(t, ω)
, (12)
where we have neglected subleading and constant background terms restoring the
complete hydrodynamic result in the background. It is convenient to eliminate one
γ-coupling in (1) by introducing rotated secondary densities; then only one of them
is coupled to the order parameter. After this transformation the structure of the
complex sound velocity Cs turns out to be the same for all critical points in pure
246
Critical sound
liquids and mixtures. It agrees in its structure with [11], however our expression
contains the non-asymptotic crossover effects. The complex sound velocity reads
C2
s (t, ω) = c2s(0, 0) +
{
c2s(t̄, 0)− c2s (0, 0)
} 1 + γ2(t̄)F
(s)
+
(
u(t̄)
)
1 + γ2(t̄)F+
(
v(t, t̄), w(t̄)
) . (13)
This expression reduces at zero frequency to the real static sound velocity c2s(t, 0),
which for both phase transitions (gas-liquid and demixing) is given by c 2s(t, 0) =
(
∂P
∂ρ
)
σ,c
. For mixtures the value of the sound velocity at Tc (t = 0) is nonzero.
The total attenuation reduces to its non-critical background value, thus the above
expression for the attenuation is zero.
At finite frequency the fraction on the right hand side of equation 13 becomes
different from one, since the dynamic function F+ becomes different from its static
counterpart F
(s)
+ . Moreover the effective temperature t̄ = t̄(t, ω), resulting from a
matching condition defined below, becomes different from t. Now in the full expres-
sion the static coupling γ2(t̄) enters.
The static velocity cs may be expressed by thermodynamic derivatives parallel
to the second order phase transition line with negligible temperature dependence
in the critical region and the inverse of a weak diverging (∼ t−α) susceptibility. At
the consolute point this weak diverging susceptibility is the specific heat at constant
pressure and concentration [14],
c2s (0, 0) =
T
ρ2
(
∂P
∂T
)2
c,Tc
C−1
c,Tc
,
c2s(t̄, 0)− c2s (0, 0) =
c2s(0, 0)
(
∂T
∂σ
)2
c,Tc
Cc,Tc
/T −
(
∂T
∂σ
)
c,P
(
∂T
∂σ
)
c,P
(14)
with Cc,Tc
= T
(
∂σ
∂T
)
c,Tc
+ T
ρ2
(
∂P
∂T
)
c,Tc
(
∂ρ
∂T
)
c,Tc
. From these expressions we can derive
the coupling constant used in the ansatz of the phenomenological theory of Ferrell
and Bhattacharjee [15,8] for pure fluids and mixtures near the consolute point.
For the plait point the weak diverging susceptibility is the concentration suscep-
tibility at constant pressure and entropy,
c2s(0, 0) =
1
ρ2
(
∂P
∂∆
)2
σ,Tc
χ−1
σ,Tc
,
c2s(t̄, 0)− c2s(0, 0) =
c2s (0, 0)
(
∂∆
∂c
)2
σ,Tc
χσ,Tc
−
(
∂∆
∂c
)
σ,P
(
∂∆
∂c
)
σ,P
(15)
with χσ,Tc
=
(
∂c
∂∆
)
σ,Tc
+ 1
ρ2
(
∂P
∂∆
)
σ,Tc
(
∂ρ
∂∆
)
σ,Tc
. Using thermodynamic relations one
may rewrite these expressions involving other ‘weak’ susceptibilities on the right
hand side. The non-universal values of the static quantities, both of cs(0, 0) (large for
the consolute point, small for the plait point and zero for the pure liquid) and of the
247
R.Folk, G.Moser
Figure 1. Curves of ‘equal distance from criticallity’ in the frequency-temperature
plane. Full lines calculated from the temperature dependence of Γ(t) shown in
figure 2b, dotted lines calculated from the asymptotic behaviour Γ(t)
weak diverging susceptibility (dominated by its background term for the consolute
point, pure liquid like behaviour for the plait point) might be quite different and
in consequence lead to a different frequency dependence of the sound velocity and
absorption.
This different behaviour of the ‘weak’ susceptibility has also consequences for
the static coupling because γ is related to the logarithmic derivatives of the singular
part of the sound velocity and may be approximated by
γ2(t̄) ≈ −2
ν
d ln{c2s(t̄, 0)− c2s (0, 0)}
d ln t̄
. (16)
In the asymptotic limit it reaches its fixed point value γ ⋆2 = 2α
ν
whereas in the
background it goes to zero. Depending on the sign of the Wegner correction in
the weak susceptibility γ reaches zero monotonically (at consolute points) or has a
maximum larger than its critical value (gas-liquid critical points).
The function F+ is an amplitude function, related to the so-called ‘frequency
dependent specific heat’, which may also be calculated within model H’ and reads
in one loop order
F+(v, w) = −1
4
{
v2
v+v−
ln v +
1
v+ − v−
[v2
−
v+
ln v− − v2+
v−
ln v+
]
}
(17)
with v± = v
2
±
√
(
v
2
)2
+ iw̄, v(t, t̄) = ξ−2(t)/ξ−2(t̄) and w̄(t̄) = ω/(2Γeff(t̄)ξ
−4(t̄)).
The dependence of the function on its variables is the same for pure fluids and
248
Critical sound
10-5 10-4 10-3 10-2 10-1
0,3
0,4
0,5
0,6
3He (X=1)
4He (X=0)
X=0.8
X=0.66
X=0.45
t
γ2
10-5
4He (X=0)
3He (X=1)
X=0.45
X=0.8
3He-4He mixture
(d
ρ/
dP
) σ,
c
[g
/(
cm
3 T
or
r)
]
10-5 10-4 10-3 10-2 10-1
0,0
0,2
0,4
0,6
0,8
t
w
3
0,4
0,6
0,8
1,0
3He (X=1)
4He (X=0)
X=0.8
X=0.65
X=0.45
f t
10-18
10-17
10-16
4He (X=0)
3He (X=1)
3He-4He mixtures
Γ
[c
m
4 /s
)]
Figure 2. Crossover behaviour of (a) the static coupling γ(t) between the order
parameter and sound calculated from the adiabatic compressibility and (b) for
the dynamic parameters for pure 3He and 4He and mixtures (from [6]).
mixtures. This is a manifestation of universality for all phase transitions considered
here. The only difference between pure fluids and mixtures lies in the definition of
the effective Onsager coefficient Γ̄ in the non-asymptotic region.
The temperature distance t and the frequency ω enter explicitly but also via the
effective temperature distance t̄, which is finite at finite frequency in the limit t = 0
and which is found from the matching condition t8ν + (2ξ40ω/Γeff(t̄))
2
= t̄8ν . This
effective temperature appears also in c2s (t̄, 0) − c2s (0, 0) and leads to the respective
asymptotic power law behaviour in frequency or temperature. Due to the matching
condition our expression for F+ remains finite in the asymptotic limits (t → 0 and
or ω → 0). No exponentiation is performed. Expanding the static part down to the
denominator one gets agreement with [16,17]. Comparing with the result in [11] in
the asymptotic region shows that Onuki’s function F is represented in our theory
by the second part of equation (13) divided by c2s (t, 0) − c2s (0, 0). The quantitative
differences between the asymptotic limit of our theory and Onuki’s results for pure
fluids have been presented recently [18] in a reconsideration of the sound data of
3He and 4He.
The effective OC of the order parameter, which appears in the frequency vari-
able w̄ and the matching condition, contains for the mixtures the dynamic param-
249
R.Folk, G.Moser
10-5 10-4 10-3 10-2 10-1
0
1
2
3
α ad
1.5MHz
1MHz
0.5MHz
4He
0
1
2
3
4
5MHz
3MHz
1.5MHz
1MHz
0.5MHz
3He
α ad
10-5 10-4 10-3 10-2 10-1
0,0
0,2
0,4
0,6
0,8
1,0 X=0.8
X=0.66
X=0.45
f=1MHzX=0.8
X=0.45
f=1MHz
f=3MHz
f=5MHz
α λ
t
70
80
90
100
110
120
3He-4He mixture
0MHz
X=0.45 f=1MHz
X=0.8 f=1MHz
f=3MHz
3MHz 1MHz
X=0.8
X=0.45
c s
[
m
/s
]
Figure 3. Sound attenuation for (a) pure He (from [5]) and sound velocity and
sound attenuation for mixtures (from [6]).
eter w, Γeff = Γ (1− w). As has been mentioned in the introduction this parameter
determines the enhancement of the thermal conductivity (it is genuine of dynami-
cal origin; another more complicated and dominating effect determines the critical
enhancement [19,6] near the plait point). Since the fixed point value of w⋆ = 0
asymptotically there is no difference in the asymptotic region to the the pure fluid
case. In fact the asymptotic behaviour is completely determined by the pure fluid
exponents and scaling functions. Differences between pure fluids and mixtures arise
only because of the nonzero value of the corresponding sound velocity at Tc in the
mixtures. Observing this, the scaling properties of the attenuation agree with those
mentioned in [20] for the consolute point.
4. Experimental comparison
We are now ready to compare with experimental data 3He-4He mixtures and the
pure components at the gas-liquid critical point. For 3He-4He mixtures the finite
value of the sound velocity at Tc is not observable. Therefore all formulas reduce
within the experimental region to the pure fluid case. Moreover the static singular
part of the sound velocity c2s(t) behaves as tα(1 + gt0.5) with a positive amplitude
g of the Wegner correction leading to the temperature dependence of the coupling
250
Critical sound
10-3 10-2 10-1 100 101 102 103 104 105 106 107
0.0
0.2
0.4
0.6
0.8
1.0
FB
FM
FM3
FL
a l/a
lc
W
Figure 4. Comparison of different scaling functions for the normalized sound
attenuation in one wavelength (from [6]) FB: Ferrell Bhattacharjees empirical
function, FM, FM3: our result obtained from an ε-expansion and a d = 3 calcu-
lation, FL: result for pure fluids.
γ shown in figure 2a. Accordingly a maximum is to be expected but not seen in
the temperature region of the experiments. Both dynamic parameters w(t) and
Γ(t) can be determined within the dynamical equations of model H’, uncoupled
to the sound degrees of freedom. We take the theoretical shear viscosity [5,6] at
zero frequency and compare it with the experimental shear viscosity [21] in order to
find the background values of w, f and Γ. The resulting temperature dependence
of the order parameter Onsager coefficient, the mode coupling f and the parameter
w is shown in figure 2b. The remaining input in the calculation of the frequency
dependence of the sound velocity and sound absorption is the sound velocity at zero
frequency. It may be either taken from a sound velocity measurement or calculated
from thermodynamic relations. Putting everything together and inserting into (12)
and (13) we compare with the data for the attenuation in pure He [22,23] in figure 3a
and with the sound velocity and attenuation in mixtures [24] in figure 3b. Thereby
we have normalized the attenuation at t = 10−5 and 1 MHz. The differences between
the values at t = 10−5 of the 1 MHz and 3 MHz data are due to the non-asymptotic
behaviour of the static coupling γ, which has not reached its fixed point value for
these frequencies.
Concerning the consolute point we compare the result of our theory for the nor-
malized sound attenuation in one wave length with the phenomenological theory of
Ferrell and Bhattacharjee in figure 4. Quantitative differences appear only to the
empirical expression. Due to the specific expression for αλ and the approximation
allowed in the experimental region one obtains almost the same result for this quan-
251
R.Folk, G.Moser
tity as suggested in [25]. To some extent one may include non-asymptotic effects in
“scaling functions” see e.g. [5].
Ackowledgement: We acknowledge support by the Fonds zur Foerderung der
Wissenschaftlichen Forschung under project 12422-TPH.
References
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253
R.Folk, G.Moser
Критичний звук у рідинах та сумішах
Р. Фольк 1 , Г. Мозер 2
1 Інститут теоретичної фізики, університет м.Лінц, Лінц, Австрія
2 Інститут фізики і біофізики, університет м.Зальцбург,
Зальцбург, Австрія
Отримано 26 червня 1998 р.
У теорії динамічної ренормалізаційної групи пропонується пояснен-
ня особливостей поведінки критичного звуку. Показано, що неасим-
птотичні ефекти обумовлені відхиленням ренормованих параметрів
взаємодії від їх фіксованих значень. Для ряду чистих рідин та сумішей
розраховані температурна й частотна залежності швидкості звуку і
коефіцієнта поглинання поблизу точки розшарування і критичної точ-
ки рідина-газ. Знайдено, що неасимптотичний критичний часовий
масштаб у сумішах відрізняється від того, що отримується для ви-
падку чистих плинів, і обумовлено це зростанням ролі термопровід-
ності. Обговорюється зв’язок з феноменологічною теорією Феррел-
ла і Бхаттачарі для точки розшарування та проводиться порівняння
з експериментами для чистих плинів 3He і 4He та їх сумішей поблизу
критичної точки.
Ключові слова: критична динаміка, поширення звуку, точка
розшарування, критична точка газ-рідина, теорія ренормгрупи,
властивості переносу
PACS: 05.70.Jk, 64.60.Ht, 64.70.Fx, 64.70.Ja
254
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