Liquid ³He–⁴He mixture phase diagram in restricted geometry
The influence of van der Waals forces on the decay of liquid solutions of helium isotopes is studied theoretically and the conditions for the phase co-existence in a confined geometry are investigated. As the models to account the influence of van der Waals forces on the helium isotope solution th...
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irk-123456789-1167712017-05-16T03:02:42Z Liquid ³He–⁴He mixture phase diagram in restricted geometry Degtiarov, I.A. Sokolov, S.S. Квантовые жидкости и квантовые кpисталлы The influence of van der Waals forces on the decay of liquid solutions of helium isotopes is studied theoretically and the conditions for the phase co-existence in a confined geometry are investigated. As the models to account the influence of van der Waals forces on the helium isotope solution the gap between two parallel planes and the cylindrical channel are considered. For each of the models we calculated the concentration profile inside the channel depending on the van der Waals constant, the initial solution concentration and the size of the channel. The phase diagrams of liquid solutions are constructed. The obtained liquid mixture phase diagrams are compared with the “bulk” phase diagram. A rather good agreement between theoretical and experimental phase diagrams is obtained for liquid helium mixture in aerogel. 2012 Article Liquid ³He–⁴He mixture phase diagram in restricted geometry / I.A. Degtiarov, S.S. Sokolov // Физика низких температур. — 2012. — Т. 38, № 1. — С. 20-27. — Бібліогр.: 15 назв. — англ. 0132-6414 http://dspace.nbuv.gov.ua/handle/123456789/116771 PACS: 67.10.–j, 67.60.–g, 67.60.gj en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы |
| spellingShingle |
Квантовые жидкости и квантовые кpисталлы Квантовые жидкости и квантовые кpисталлы Degtiarov, I.A. Sokolov, S.S. Liquid ³He–⁴He mixture phase diagram in restricted geometry Физика низких температур |
| description |
The influence of van der Waals forces on the decay of liquid solutions of helium isotopes is studied theoretically
and the conditions for the phase co-existence in a confined geometry are investigated. As the models to account
the influence of van der Waals forces on the helium isotope solution the gap between two parallel planes
and the cylindrical channel are considered. For each of the models we calculated the concentration profile inside
the channel depending on the van der Waals constant, the initial solution concentration and the size of the channel.
The phase diagrams of liquid solutions are constructed. The obtained liquid mixture phase diagrams are
compared with the “bulk” phase diagram. A rather good agreement between theoretical and experimental phase
diagrams is obtained for liquid helium mixture in aerogel. |
| format |
Article |
| author |
Degtiarov, I.A. Sokolov, S.S. |
| author_facet |
Degtiarov, I.A. Sokolov, S.S. |
| author_sort |
Degtiarov, I.A. |
| title |
Liquid ³He–⁴He mixture phase diagram in restricted geometry |
| title_short |
Liquid ³He–⁴He mixture phase diagram in restricted geometry |
| title_full |
Liquid ³He–⁴He mixture phase diagram in restricted geometry |
| title_fullStr |
Liquid ³He–⁴He mixture phase diagram in restricted geometry |
| title_full_unstemmed |
Liquid ³He–⁴He mixture phase diagram in restricted geometry |
| title_sort |
liquid ³he–⁴he mixture phase diagram in restricted geometry |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2012 |
| topic_facet |
Квантовые жидкости и квантовые кpисталлы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/116771 |
| citation_txt |
Liquid ³He–⁴He mixture phase diagram in restricted geometry / I.A. Degtiarov, S.S. Sokolov // Физика низких температур. — 2012. — Т. 38, № 1. — С. 20-27. — Бібліогр.: 15 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT degtiarovia liquid3he4hemixturephasediagraminrestrictedgeometry AT sokolovss liquid3he4hemixturephasediagraminrestrictedgeometry |
| first_indexed |
2025-07-08T11:01:18Z |
| last_indexed |
2025-07-08T11:01:18Z |
| _version_ |
1837076293967937536 |
| fulltext |
© I.A. Degtiarov and S.S. Sokolov, 2012
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1, pp. 20–27
Liquid 3He–4He mixture phase diagram in restricted
geometry
I.A. Degtiarov and S.S. Sokolov
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: degtiarov.igor@gmail.com
Received June 24, 2011
The influence of van der Waals forces on the decay of liquid solutions of helium isotopes is studied theoreti-
cally and the conditions for the phase co-existence in a confined geometry are investigated. As the models to ac-
count the influence of van der Waals forces on the helium isotope solution the gap between two parallel planes
and the cylindrical channel are considered. For each of the models we calculated the concentration profile inside
the channel depending on the van der Waals constant, the initial solution concentration and the size of the chan-
nel. The phase diagrams of liquid solutions are constructed. The obtained liquid mixture phase diagrams are
compared with the “bulk” phase diagram. A rather good agreement between theoretical and experimental phase
diagrams is obtained for liquid helium mixture in aerogel.
PACS: 67.10.–j Quantum fluids: general properties;
67.60.–g Mixtures of 3He and 4He;
67.60.gj Restricted geometries.
Keywords: helium isotopes, liquid solutions, phase diagram of liquid solutions 3He–4He.
Introduction
The phase diagram of liquid solutions 3 4He – He [1]
and, in particular, the finite solubility of 3 He in a phase
riched in 4 He which arises under stratification of homo-
geneous solution (the corresponding concentration 6.7%x
at = 0T ), lead to very interesting thermodynamic and
kinetic properties of such solutions. The superfluid solu-
tions are paid the special attention. Superfluid liquid mix-
ture of 3 4He – He can penetrate into the small pores and
channels, and there is a unique opportunity to study the
thermodynamic and other characteristics of solutions of
quantum liquids 3 He and 4 He in a “narrow” geometry.
Such studies are very popular in recent decades, particular-
ly by the example of aerogels, which is a highly porous
medium from a glassy material that is filled with helium
[2–7]. Another example of a porous medium, where such
research is provided, is vycor [8,9], glassy material, much
more dense than aerogel, in which the cavity filled with
helium, form a system of regular channels located far
enough from each other. The characteristic channel size in
vycor varies in 20–100 Å. It was established experimental-
ly that the phase diagram of a solution 3 4He – He in a por-
ous substance is different of that in bulk case (see Fig. 1).
It turned out that the phase diagram depends strongly on
the porosity of the material, which is filled by a solution.
One should consider separately aerogels with a porosity
of 95% and 95% (according to [7] at a porosity
95%∼ deviation of the phase diagram of bulk case practi-
Fig. 1. Phase x T− diagram of liquid solutions 3 4He – He. Solid
line — bulk phase diagram. The dotted curve is qualitatively
shown deviation from the bulk phase diagram in aerogel with a
porosity of more than 95% [2,3], dash-dotted line is a deviation
of the phase diagram from the bulk case, observed at a porosity of
less than 95% [4–6].
T
,
K
x
0 0.25 0.5 0.75 1.0
0.5
1.0
1.5
2.0
xtp
Superfluid mixtures
3 4
He– He
Normal Fluid
He– He
3 4
Liquid 3He–4He mixture phase diagram in restricted geometry
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1 21
cally was not observed.) Note that, as evidenced by acous-
tic studies [10], carried out for pure 4 He , the structure of
aerogels is different in these ranges of porosity, as it fol-
lows from the change in temperature dependence of the
sound velocity. If the aerogel is dense (porosity less than
95%), such a dependence corresponds to a velocity of
fourth sound in a narrow capillary, whereas to a velocity of
first sound in a bulk liquid in the opposite limit of a more
rarefied medium. One can guess that a dense aerogels is
similar to vycor being an environment that can be approx-
imated by a set of loosely coupled individual channels in a
dense glass whereas the rarefied aerogel is almost hollow
medium filled with helium and has interwoven glass fiber
located relatively far from each other. The difference in the
structure of aerogels affects the form of the phase diagram
of liquid helium isotopes solution.
If the porosity of aerogels is 95 % the deviation of
the phase diagram from that in bulk liquid is observed in
right branch with < < 1tpx x [2,3] (see Fig. 1). This devia-
tion of the phase diagram is qualitatively explained in [11],
where the phase diagram was studied by the Monte Carlo,
taking into account changes in the chemical potential near
randomly administered impurity molecules 2SiO . For the
porosity of aerogels 95 %, the whole phase diagram of
liquid solutions “shifted” [4–6], this effect is associated
with a change in concentration of the solution by forming a
solid layer of pure 4 He on the glass surface. However the
detailed analysis of this fact accounting for a solid mono-
layer of 4 He on the wall, demonstrated that the deviation
of the phase diagram is observed only on the left branch
(with 0 < < tpx x ) of the phase diagram describing the
stratification of solutions (see Fig. 1). The same idea was
proposed to explain qualitatively the deviation of observed
phase diagram from that in bulk observed for liquid solu-
tions of helium isotopes in vycor [8]. However, quantita-
tive calculation of this effect, unfortunately, was not pre-
sented.
Thus the question with associated features of the ther-
modynamic properties of solutions of helium isotopes
placed in the porous structure remains valid.
The aim of present paper is to study the conditions of
phase co-existence of liquid solutions of helium isotopes in
a confined geometry, depending on the characteristic size
of the channel filled with liquid, as well as for various sub-
stances that form a porous medium. We believe that, in the
narrow geometry, the determining influence on the condi-
tion of co-existence have the van der Waals forces, be-
cause of what one should investigate the conditions of
thermodynamic equilibrium in a field of such forces. We
study two geometric models of the porous medium and
investigate the effect of the van der Waals constant on the
thermodynamic characteristics of the solution 3 4He – He .
The results obtained allow to construct the phase diagrams
being in qualitative agreement with those observed expe-
rimentally.
2. The influence of van der Waals forces on liquid
solutions of 3He–4He
We model the porous structure filled with liquid solu-
tion of the helium isotopes as a closed surface formed by
strands of 2SiO , which form channels and pores. Liquid
solutions of helium isotopes fill the void. The key point of
the analysis is the chemical potential of the liquid. Here we
restrict ourselves to the approach of ideal solutions taking
in mind mainly qualitative consideration of the problem
and write the chemical potentials of each component of a
“bulk” solution as [12]:
0= ln ;j
j j
N
T e
N
μ μ + (1)
index =j 3 or 4 stands for 3 He and 4 He , respectively;
0 jμ is uniform chemical potential of pure helium isotopes;
jN is partial number of atoms of each isotope and N the
total number of atoms. Concentration of 3 He in the solu-
tion is 3= /x N N . To account for the influence of the
channel walls, we should supplement the (1) by additional
term associated with a potential U of van der Waals
forces acting between helium and the walls of the channel,
which leads to the following conditions [12,13]:
3
4
( , , ) = const,
( , , ) = const.
x T P U
x T P U
μ +⎧
⎨μ +⎩
(2)
Taking into account that 0 = ,j jd dPμ v where jv are
atomic volumes and P is the pressure of (2) we arrive to
the following equations:
3
4
( ) 0,
(1 ) ( ) 0;
1
TdP dx dU
x
TdP d x dU
x
⎧ + + ξ =⎪⎪
⎨
⎪ + − + ξ =
⎪ −⎩
v
v
where ξ being the direction normal to the channel walls.
We multiply the first equation by (1 ),x− and then second
by x, and then add them. Also we subtract one equation
from other and finally obtain a system of equations defin-
ing the conditions of thermodynamic equilibrium:
= ( ),
ln = .
1
dP dU
xd dP
x T
− ξ⎧
⎪
Δ⎨ ⎛ ⎞
⎜ ⎟⎪ −⎝ ⎠⎩
v
v (3)
where 3 4= .Δ −v v v From the first equation we find the
differential pressure
3 4
( ) ( )= = .
(1 )
dU dUdP
x x
ξ ξ
− −
+ −v v v
(4)
Substituting this expression into the second equation, we
obtain:
3 4
( )ln = .
1 (1 )
x dUd
x T x x
Δ ξ⎛ ⎞ −⎜ ⎟− + −⎝ ⎠
v
v v
(5)
I.A. Degtiarov and S.S. Sokolov
22 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1
Rewrite (5) in the form
3 4 = ( ),
1
dx dU
x x T
Δ⎡ ⎤+ ξ⎢ ⎥−⎣ ⎦
v v v (6)
and integrating (6), we obtain, for ( )x ξ :
[ ] 3 4 4
( ) = exp ( ) ./1 ( )
Cx U
Tx
⎡ ⎤ξ Δ
ξ⎢ ⎥
⎣ ⎦− ξ v v
v
v
(7)
For further consideration we will need to use the expli-
cit form of the potential of van der Waals force acting be-
tween the walls of the channel and the helium isotopes, and
to choose the geometry of the channel filled with fluid. The
real geometry of aerogel and vycor is too complicated and
irregular making intractable the description of van der
Waals forces in it. For this reason we consider, as “limit-
ing” geometries, two models:
— model of plane channel, bounded on two sides by
two parallel planes;
— model of a hollow cylinder.
As we will see the results for these two geometries dif-
fer only qualitatively guessing that the results for real geo-
metry would be close to those of simple geometries we
consider.
2.1. The model of a narrow plane-parallel channel
The potential of the van der Waals forces for the plane
is well known, it can be written as:
3( ) = ( ) = ,aU U z
z
ξ − (8)
where a being the van der Waals constant of substance of
the channel walls, the axis z is supposed to be perpen-
dicular to the plane which forms the channel.
If we have a channel, bounded on two sides, one should
take into account the influence of van der Waals forces
from each of the planes. Then the interaction potential
( )U z becomes
3 3( ) = ,
( )
a aU z
z D z
− −
−
(9)
where D is the distance between the planes forming the
channel.
To calculate ( )x z using (7) one has to determine the
constant of integration. We determine the value of C from
the condition that at = / 2z D
[ ] 3 4
4
/1 ( / 2)
= exp ( / 2) ,
( / 2)
x D
C U D
x D T
− ⎡ ⎤Δ
⎢ ⎥
⎣ ⎦
v v v
v
(10)
that is, to determine the integration constants C we need
to find the value of the concentration ( / 2)x D in the cen-
ter of channel. To set the value of ( / 2)x D , we use an ad-
ditional condition which is imposed on the function ( )x z .
The solution in a narrow channel is characterized by a re-
distribution of concentrations. The concentration profile is
shown schematically in Fig. 2,a. The initial “bulk” concen-
tration of the solution corresponds to 0 ,x shown in Fig. 2.
The Maxwell rule (law of conservation of matter) should
be satisfied, according to which areas of 1S and 2S must
be equal (Fig. 2,b). From the Figure one, can see
0
1 0 0= ( ) ( ) ;
z
s
ds
S z d x x z dz− − ∫
/2
2 0 0
0
= ( ) ( / 2 ) ,
D
z
S x z dz D z x− −∫
Fig. 2. Concentration profile in the two-dimensional narrow channel. sd — the thickness of the solid 4He at the channel walls; D —
the channel width; 0x — the initial concentration of solution (a); the law of conservation of matter, which requires that the space 1S
and 2S are equal to each other (b).
ds
0
D
z
D/2
D/20
x0
x0
z0
z0
a b
xD/2
x
x
z
D d– sds
S1
S2
Liquid 3He–4He mixture phase diagram in restricted geometry
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1 23
where 5sd Å is a width of the solid inert layer of pure
4 He [13], 0z is determined by condition 0 0( ) =x z x . To
estimate the initial concentration of the mixture we take
into account the solid layer of pure 4 He , that is formed
on the walls of the channels. In this case the concentra-
tion inside the channel is greater than in bulk. The initial
concentration inside channels with = 20D Å increases
in 2 times in comparison with bulk case, in channels
= 100 ÅD increases 1.1 times, and in channels 500 Å
increases only 1.02 times. Under further increase of chan-
nels width the initial concentration does not change practi-
cally.
Equating the areas 1S and 2S , we obtain the following
conditions which ( )x z must satisfy:
/2
0 0 /2( ) = ( / 2 ) ; < < 1.
D
s D
ds
x z dz D d x x x−∫ (11)
Solving, in a self-consistent way, Eqs. (7) and (11) and
taking into account (10), we can determine simultaneously
the concentration profile of ( )x z and the value of ( /2),x D
set C in (7). To realize such a scenario, we established an
algorithm of numerical calculation, which allows to find
( )x z depending on the van der Waals constant and the
channel width D . It should be noted that when the width
of the channel is small enough, due to the action of van der
Waals forces, the entire channel is overgrown with a solid
4 He and one cannot speak on a liquid solution in the
channel. Thus, there is a natural limiting value of the width
of the channel 10D Å, for which our analysis remains
valid.
Concentration profiles in the narrow channels for dif-
ferent a are shown in Fig. 3, and, depending on D for
some 0x in Fig. 4.
The Figure demonstrates that the stronger interaction of
the solution with the channel walls, the greater the concen-
tration varies with coordinate z. For narrow channels
(Fig. 4,a) the concentration in the center can highly exceed
the initial concentration of solution. For example, in the
channels of a width 20D ∼ Å the initial concentration is
exceeded about twice for not very large 0.x In wide chan-
nels (Fig. 4,b), the essential difference from the bulk con-
centration can be observed only near the walls, in the
channel center the solution concentration is almost exactly
the same as the bulk concentration of the solution.
It should be emphasized that the channel center concen-
tration is always above the initial concentration corres-
ponding to the bulk case. In this connection the question
arises how can change the phase diagram of the solution,
placed in a narrow channel. Using the calculated concen-
tration profile, we estimate the change in the phase dia-
gram for different values of the width of the channels in
the glass medium. As the criterion for choosing the solu-
tion concentration at which the stratification starts in a nar-
row channel, we took the value of concentration at the cen-
ter of the channel, which is equal to the concentration of
the stratification in a “bulk” solution where one disregards
Fig. 3. Concentration profile of a narrow parallel channel for a
solution of 3He in the 4He with an initial concentration of
0 = 0.1.x Van der Waals constants are taken for neon — dotted
line, glass — solid line, metal — dashed line [14]. The width of
the channel is = 20D Å.
0
0.05
0.10
0.15
0.20
x
6 8 10 12 14
z, 10
–8
cm
Fig. 4. Concentration profile of liquid solutions of helium isotopes across the channel width and for the some initial
concentration of the solution. The width of the channel = 20D Å (a) and 100 Å (b). The initial concentration of 0x is
equal to 0.1 — dotted line, 0.45 — solid line, and 0.8 — dashed line. The substance of the walls — glass.
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
x x
bа
6 68 810 12 214 4
z, 10
–8
cm z, 10
–7
cm
I.A. Degtiarov and S.S. Sokolov
24 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1
the influence of van der Waals forces, vol ,sx that is, we
took vol
/2 =D sx x for a given temperature T . Figure 5
shows the phase diagram for the bulk case and for the solu-
tion placed in a narrow plane-parallel channel for the dif-
ferent channel widths. It should be noted that the distribu-
tion of solution concentration in the channel will make
changes only in the left branch of the phase diagram of
stratification, when the bulk stratification concentration
will be achieved in the center of channel. In the right
branch of the phase diagram, such a distribution always
leads to the fact that at a given temperature inside the
channel, the stratification concentration is always achiev-
ed. Probably, for this reason only normal–superfluid transi-
tion was observed in the right branch of phase diagram
whereas the phase transition of stratification was not ob-
served. As can be seen from Fig. 5, the narrower is the
channel and stronger interaction with the walls of the solu-
tion, the greater is the deviation of the phase diagram of
the bulk case. With increasing D the deviation decreases
and at 1000D Å the influence of the walls on the phase
diagram becomes negligible.
The applied method also allows, knowing the concen-
tration distribution in a narrow channel and integrating the
Eq. (4), find the pressure distribution in the channel:
4
4
( )
1= ( ) ( ) = .
1 ( )
z
s
ds
dU z
dzP P z P d dz
x z
Δ − −
⎡ ⎤Δ
+⎢ ⎥
⎣ ⎦
∫v v
v
(12)
An example of numerical calculation of the pressure
profile is shown in Fig. 6.
2.2. The model of a narrow hollow cylinder
To describe the concentration distribution in a model of
a narrow cylinder, we still use the Eq. (7), where the coor-
dinate ξ is now the radial variable r. The main problem
now is to modify the interaction potential of the van der
Waals forces, compared with the planar case. We use the
method previously applied in a theoretical consideration of
collective hydrodynamic oscillations of stratified solution
in cylindrical geometry [15]. The potential interaction be-
tween the liquid and solid substrate can be written in the
gas approximation in the form:
He–sol sol 2 2 3
1( ) = ' ,
(| | | | )
U b n d dz
z z
′−
′ ′− + −∫ ∫r r
r r
(13)
where z is directed along the axis of the cylinder, and
He–solb is the interaction constant between helium atoms
and the particles of wall matter, soln is the volume concen-
tration of these particles. The expression (13) is written
neglecting the interaction of helium atoms with each other
compared to the interaction with the substance of the walls,
which is true for most substances (except for solidified
inert gases and hydrogen). In the case of plane geometry,
gas approximation leads to the expression (8), if one makes
the change He–sol sol6a b nπ . To obtain the dependence
of ( )U r from a we have made the similar change in
Eq. (13), which, after integration over z′ and ′φ is reduc-
ed to:
2 2 2
4 3
4( ) ( ) ( ) ( )
( , ) = 6 ,
( ) ( )R
r r E m r r K m
U r a r dr
r r r r
∞ ⎡ ⎤′ ′+ − −⎣ ⎦′ ′φ −
′ ′− +∫
(14)
where R being the channel radius, ( )E m and ( )K m are
the complete elliptic integrals, the argument of which is
equal to 2= 4 /( ) .m rr r r′ ′+
Unfortunately, the integration over r′ in the expression
(14) cannot be done analytically. However, these integrals
can be computed using numerical methods.
Substituting the Eq. (14) for the potential of van der
Waals forces in the Eq. (7) we find the integration constant
C in the center of a cylindrical channel, similarly to the
–40
–30
–20
–10
0
�
P
P
z
P
d
=
(
)–
(
),
b
ar
s
6 8 10 12 14
z, 10
–8
cm
Fig. 6. Viewing excess pressure in the narrow gap between two
plane-parallel planes for different coupling constants of van der
Waals forces: the dotted line — neon, solid line — glass; dashed
line — metal (constant Van der Waals forces of [14]). The width
of the channel is = 20D Å.
0
0.2
0.4
0.6
0.8
1.0
T
,
K
x
0.40.2 0.6 0.8 1.0
Fig. 5. Phase diagrams of solutions of 3 4He – He for the bulk
case — the solid curve and for narrow channel of width 50 Å —
dot-dashed curve, 100 Å — dashed curve, 500 Å — dotted curve.
Liquid 3He–4He mixture phase diagram in restricted geometry
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1 25
Eq. (10) (now = rξ ). Continuing to follow the above
scheme for plane-parallel channel, we write the condition
on the concentration of ( )x r inside the hollow cylinder of
radius R:
2
0 0 centre
0
( )
( ) = ; < < 1.
2
R ds
sR d
rx r dr x x x
−
−
∫ (15)
As in the case of a plane geometry, we solve self-con-
sistently (7) and (15), which makes it possible to construct
the concentration profile in a narrow hollow cylinder de-
pending on the parameters a, R and 0.x As noted above,
the initial concentration within the channel is higher than
the concentration in the bulk liquid because of the solid
layer of 4 He , on the channel walls. So, for a cylindrical
channel in the glass with a radius = 20R Å the initial con-
centration in the channel increases by 2.4 times compared
to the bulk concentration of the solution and for the chan-
nel radius = 500R Å only 1.04 times. With further in-
crease of the channel width, initial concentration remains
practically unchanged.
The results of the calculation of the concentration pro-
file in the cylindrical channel are presented in Fig. 7.
To construct the phase diagram of liquid solutions of
helium isotopes in a hollow narrow cylinder we act simi-
larly to the case of plane channel, which was reviewed in
the previous section. The constructed phase diagram is
shown in Fig. 8. As was expected, the narrower the chan-
nel, the stronger is the effect of helium interaction with the
walls and the greater is the deviation of the phase diagram
of the bulk case. It should be noted that the same characte-
ristic width of the channel, in the case of the cylinder phase
diagram, is different from the bulk case much stronger than
in the case of a plane channel. As can be seen from Fig. 8,
when the radius of the cylinder channel is about = 500 ÅR
one observes the deviation of the phase diagram from the
bulk case, for = 100R Å, and = 50R Å deviation of the
phase diagram becomes large enough, under a further de-
crease in the size of the cylinder, phase diagram of solu-
tions of helium isotopes significantly shifted to the left.
Thus, if we assume a porous substance cavity of cylindric-
al shape, then the radii 500R Å must show the shift to
the left of the phase diagram. At smaller sizes of these cav-
ities deviation of the phase diagram will become even
more pronounced.
2.3. Comparison with experimental data
To compare the results of calculations with experimen-
tal data, we took data on the phase diagram of liquid solu-
tions of helium isotopes in aerogel with porosity < 95%
[4–6]. Recall that the deviation of the phase diagram in
porous materials on the phase diagrams of solutions in the
bulk case, is due to two facts. The first is the formation of a
–2·10
–7
–2·10
–7
2·10
–7
2·10
–7
–1·10
–7
–1·10
–7
1·10
–7
1·10
–7
0.1
0.2
x b
r, 10 cm
–8
0
0.05
0.10
0.15
0.20
0.25
0.30
5 1 1.5
d, cm
a
0 0
Fig. 7. Profiles of the concentrations of 3He in solution 3 4He – He in a narrow hollow cylinder, the substance of the walls — glass.
The initial concentration of the solution 0 = 0.2x , the radius of the cylinder = 20R Å (a); solution concentration 0 = 0.1x , with a ra-
dius of the cylinder = 20R Å for the three constants of the van der Waals [14]: dotted line — neon, solid curve — glass, dashed curve
— metal (b).
0
0.2
0.4
0.6
0.8
1.0
T
,
K
x
0.40.2 0.6 0.8 1.0
Fig. 8. Phase diagram in the “bulk” (solid line); in a hollow cy-
linder ( = 500R Å — dotted line, = 100R Å — dashed line,
= 50R Å — dot-dashed curve. The substance of the walls —
glass.
I.A. Degtiarov and S.S. Sokolov
26 Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1
solid layer of 4 He at the surface of the channels, the
second is the change of the concentration profile in the
channels due to the influence of van der Waals forces act-
ing on solution from the substrate. Accounting for changes
in concentration due to formation of a solid layer imple-
mented in the papers [4,5] for aerogels of porosity 85%
and 87%, the experimental result on the deviation of the
phase diagram of the bulk case in these works is almost
identical. Therefore, we make a comparison with the re-
sults of [5]. Comparing the deviation of the phase diagram
of solutions 3 4He – He caused by the influence of van der
Waals in aerogel, we can estimate the channel size that
meets the experimental observation of a deviation, for both
considered geometries (Fig. 9) even though our geometric-
al models are oversimplified and cannot pretend to de-
scribe the real aerogel.
We found that the calculated curves adjust the experi-
mental data, in a plane model, for the width of the channel
= 50D Å and, for cylindric channel model for a channel
radius = 100R Å. The calculated phase diagrams are very
close and are indistinguishable in the Fig. 9. That is why
we restrict our estimates to a cylindrical channel. As is
seen the models applied in our calculation adjust the expe-
rimental data reasonably well. At the same time, the most
plausible, in the case of aerogels, is the result for a cylin-
drical channel with a pore diameter of 200 Å. Plane-
parallel model assumes a pore size of about 50 Å. Note
that similar dimensions of the channels in a porous sub-
stance are specific for vycor where the porosity is usually
less than 30%.
3. Conclusions
The paper analyzes the thermodynamic properties of
liquid mixtures of helium isotopes in the narrow geometry.
The influence of van der Waals forces on the properties of
the solution is considered and we show that the “offset” of
phase diagram observed in experiments on liquid mixtures
in aerogels and vycor may be due to the placement of fluid
in the narrow geometry and the influence of wall material
on a solution. The paper discusses two models: a narrow
plane-parallel channel and the channel of cylindrical shape.
We have shown that in the case of narrow channels co-
ordinate-dependent concentration profile of 3 He arises,
which has a minimum value at the channel wall and reach-
es a maximum at its center. Although qualitatively, in both
models, the effect of van der Waals forces on the thermo-
dynamic properties of liquid solutions is the same, the
geometry of the channel affects the effect quantitatively.
The deviation of the phase diagram from that in bulk is
well seen for the channel size 20–100 Å, and if the size of
the channel is 500 Å, the effect is practically absent. In
the case of a cylindrical channel deviation of the phase dia-
gram from the bulk is stronger than that for plane channel.
The obtained results well describe the experimentally
observed data on the phase diagram of solutions 3 4He – He
in aerogel. At the same time, the model of a hollow cylin-
drical channel seems more plausible to describe the porous
material than the model of plane-parallel channel, because
the model of a cylindrical hollow channel gives a more
realistic pore size. For a detailed agreement with the expe-
rimental data one requires more information about the
geometrical structure of pores in the sample and calcula-
tions corresponding to this geometry, taking also into ac-
count the possible mutual influence of closely spaced
channels.
Acknowledgments
The work is partially supported by STCU program
through the project 5211.
The authors are greatly indebted to E.Ya. Rudavskii,
V.K. Chagovets, and A.A. Zadorozhko for useful discus-
sions.
1. B.N. Esel'son, V.N. Grigor'ev, V.G. Ivantsov, E.Ya. Rudav-
skii, D.G. Sanikidze, and I.A. Serbin, Solutions of Quantum
Liquids 3He–4He, Moscow, Nauka (1973).
2. J. Ma, S.B. Kim, L.W. Hrubeshand, and M.H.W Chan, J.
Low Temp. Phys. 93, 945 (1993).
3. N. Mulders, J. Ma, S.B. Kim, J. Yoon, and M.H.W. Chan, J.
Low Temp. Phys. 101, 95 (1995).
4. M. Paetkau and J.R. Beamish, Czech. J. Phys. 46, 153
(1996).
5. M. Paetkau and J.R. Beamish, Phys. Rev. Lett. 80, 5591
(1998).
6. J. Yoon, N. Mulders, L.W. Hrubeshand, and M.H.W. Chan,
Czech. J. Phys. 46, 157 (1996).
7. J. Yoon, N. Mulders, and M.H.W. Chan, J. Low Temp. Phys.
110, 585 (1998).
8. T. Hohenberger, R. Konig, and F. Pobell, J. Low Temp.
Phys. 110, 579 (1998).
Fig. 9. Comparison of the experimental data on the deviation of
the phase diagram of the bulk case with the calculation. The solid
line shows the phase diagram for the bulk case. Filled circles —
experimental points [5]; dashed line — a model of a cylindrical
channel of radius = 100R Å.
0
0.2
0.4
0.6
0.8
1.0
T
,
K
x
0.40.2 0.6 0.8 1.0
Liquid 3He–4He mixture phase diagram in restricted geometry
Low Temperature Physics/Fizika Nizkikh Temperatur, 2012, v. 38, No. 1 27
9. J.R. Beamish, A. Hikata, L. Tell, and C. Elbaum, Phys. Rev.
Lett. 7, 425 (1983).
10. A.A. Zadorozhko, V.K. Chagovets, E.Ya. Rudavskii, G.A.
Sheshin, Sh.E. Kekutia, N.D. Chkhaidze, and N. Mulders,
Fiz. Nizk. Temp. 35, 962 (2009) [Low Temp. Phys. 35, 752
(2009)].
11. A. Falicov and A.N. Berker, J. Low Temp. Phys. 107, 51
(1996).
12. L.D. Landau and E.M. Lifshitz, Statistical Physics, Nauka,
Moscow (1976), P. 1, p. 309.
13. M. Chester, J.-P. Laheurte, and J.-P. Romagnan, Phys. Rev.
B14, 2812 (1976).
14. E.S. Sabisky and C.H. Andersen, Phys. Rev. A7, 790 (1973).
15. Yu.P. Monarkha and S.S. Sokolov, Fiz. Nizk. Temp. 15, 339
(1989) [Low Temp. Phys. 15, 191 (1989)].
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