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Viscosity of liquid 3He-4He mixtures in the helium II region above 1°K
Physica 26 669-686
Staas, F. A. Taconis, K. \Y. Fokkens, K. 1960
VISCOSITY
OF LIQUID 3He-4He MIXTURES HELIUM II REGION ABOVE 1°K
by F. A. STAAS, Communication
K. W. TACONIS
No. 323~ from the Kamerlingh
IN THE
and K. FOKKENS
Onnes Laboratorium,
Leiden,
Nederland
Synopsis The viscosity of liquid sHe-4He mixtures has been derived from isothermal flow through a capillary 75.6 p in diameter. The apparatus consists of two identical glass vessels, connected to each other by the glass capillary. The positions of both levels were measured as a function of time. It appeared that the logarithm of the level difference always decreased linearly with time, which means that Poiseuille’s law is obeyed. The analysis of the flow has been given. The values of the viscosity extrapolated to zero concentration (pure 4He) are in agreement with the measurements with the rotating cylinder by Heik kila and Hollis Hallett. There is also satisfactory agreement in the low concentration region with the theory given by Zharkov.
1. Introduction. Several investigators have measured the viscosity of the normal fluid Q, of pure 4He below the A-point. One can distinguish between several methods, the oldest method being the observation of the damping of an oscillating disk used by Keesom and MacWood 1) and later on followed by Andronikashvilli 2); De Troyer, Van Itterbeek and Van den Berg 3); Hollis Hallett 4) and Dash and Taylor 6)7). From these observations one can calculate the product pnqn, pn and qn being respectively the density and the viscosity of the normal fluid. The value of the viscosity of the normal fluid is obtained from an independent measurement of the normal fluid density. Independent of the knowledge of the normal fluid density H eikkila and Hollis Halle tt 8) were able to measure the normal fluid viscosities by using The results of the torsion pendulum the rotating cylinder viscometer. method and the rotating cylinder viscometer are still in quantitative disagreement. Zi n ov’ e v a 9) has derived values of the viscosity of the normal fluid from the attenuation of second sound in a resonant cavity which are in better agreement with those values found by Heikkila and Hollis Hallett. Capillary methods have been used in heat conduction experiments. Since in pure 4He below the &point heat is transferred only by the normal fluid, the heat current is related to the normal fluid velocity un. The normal Physica
26
669 -
fluid viscosity corresponding
may bc calculated from laminnr flow if the prt’ssurc gradient with the velocity of th(, normal fluid is known. ‘I%~~normal
fluid flow is accompanied by a superfluid counterflow in such a way that thcx resultant flow is zero. Now thcx prcxssure gradient can only be easily calculated from the temperature gradient if a critical velocity, abovc~ which mutual friction comes into play is not c~sct~c&~tl. Tli~ critical \rvlocity depends on the capillary diameter and is lower for a widclr capillary. This makes measurements with narrow capillaries necessary unless one is able to measure very small temperature gradients. On the other hand the choice of the capillary or slit is limited by mean free path effects. The mean fret path of phonons increases with decreasing temperatures and is in the casc~ of pure 4He at lower tcmpcraturc~s comparable with the slit width. Heat conduction methods were cmployc~l by I? r o f’s c’ van (; r o c II o u, P o 11, Dclsing and Gortcr 10) and recently by 13rt~wt~r and E d \va r cl s 11). From their heat conduction method with widca capillarit~s of 52.0 p and 107.6 ,LLdiameter Rrewer and Edwards derived values for the viscosity which arc in good agreement with those of the rotating cylinder viscometer. Until now viscosity measurements on 3Hc--1He dilute mixtures have been reported by Pellam lz), STansink and Taconis 13), Dash and Taylor 14). Wansink and Taconis wet-c able to deduce the viscosity from flow experiments through a narrow slit of 0.3 ,u, while Pellam and I>ash and Taylor used the torsion I~endulun~ method. The latter measurc~d the viscosity of three samples with liquid concentrations of about 3”1, 5(,‘;, and 1O$,, “He. We have determined the viscosity of the normal fluid straightforward using isothermal flow in a capillary 75.6 ,LLin diameter, making use of the relation between the mean velocity and the pressure gradient according to Poiseuille’s law. In liquid 3He-“He mixtures below the 3,-point a kind of osmotic prcssurc exists between two vessels containing mixtures of different concentrations separated by a narrow slit 15)lfi). In the case where one connects two vessels containing mixtures of different compositions by a capillary, superfluid will flow to the, high concentration side until the osmotic pressure is balanced by a level difference. In our cast’ using a capillary 76 ,u in diameter, this difference adjusts itself within a few seconds. A concentration difference of 10-d corresponds to an osmotic pressure of about 2 cm helium column at a temperature near 1“I<. Subsequent to the establishment of the level difference one can expect a flow of the normal fluid under influence of the hydrostatic pressure head, according to Poiseuille’s law, while the superfluid flow obeys the condition of permanent equilibrium between the hydrostatic pressure head and the osmotic pressure difference. Therefore in general at not too low concentrations where the concentration difference is relatively small superfluid flows with almost the same velocity as do the normal and 3He atoms. As wt’ shall
___~ show later
VISCOSITY
on diffusion
easily from Poiseuille’s
OF LIQUID
is negligible
sHe-4He
MIXTURES
and we can calculate
671 the viscosity
law.
2. The apparatus. The apparatus consists of two identical thin-walled glass vessels Vi and Va connected to each other by a glass capillary. A filter of fine nylon fabric N is used for protecting the capillary against the entrance of impurities. The vessels Vi and VZ are connected to the manometer system by means of two identical glass capillaries C, which are
Fig. 1. The apparatus.
partially wrapped with cotton cord K to prevent the increase of heat leaks with the falling of the surrounding helium bath. During the experiment the helium bath temperature was kept constant within 10-5 degrees monitored by a carbon resistance thermometer. It was necessary to purify the aHe-4He mixtures before using them. We purified the gaseous mixtures by leading them through a spiral capillary at a temperature of about 4°K in a separate cryostat. Measurements of the vapour pressure of the liquid mixtures were carried out with the aid
672
F. A. STAAS,
of oil manometers
K. W.
TACONIS
AND
K. FOKKEXS
Mr and M2 filled with octoil S. We deduced the concen-
tration of the solution from the known temperatures of the helium bath and vapour pressure differences with respect to the bath pressure rT)is)r9). Both parts of the apparatus could be connected to a mercury filled Toepler pump system. 3. The
method.
The measurements
of the viscosity
were preceded
by
the creation of an initial level difference of the required extent (about 2 cm). This could be done in several ways depending on the temperature and the concentration of the solution. By transferring vapour from one side to the other, one creates a concentration difference between the mixtures in both vessels due to the much higher concentration of 3He in the vapour in comparison with the liquid. We observed a level difference with the higher level at the high concentration side. In consequence of the superfluid properties within a few seconds 4He flowed from the low concentration to the high concentration vessel until the level difference corresponded to the osmotic pressure. At that moment no vapour pressure difference could be observed with any precision on oil manometer Mi showing that the concentrations of both levels were nearly equalised by the superfluid flow. The vapour pressure difference due to the very small concentration difference diminished by the weight of a small vapour column should have been observed, but this is a hardly measurable effect. The positions of both levels as a function of time were measured using a cathetometer and a stopwatch. In general the levels were followed during about half an hour in which the level difference decreased by a factor of ten. At temperatures below 1.3”K and low concentrations the creation of a level difference was more difficult. It was accomplished by condensing on one side only solutions of higher concentration than already present in the apparatus. Condensation of pure 4He in different quantities or a mixture of much lower concentration did not produce any level difference. It was also possible to create a level difference by suddenly admitting gas so that the liquid is forced through the capillary by the overpressure. This was done at the higher vapour pressures. 4. The calculation of the flow. Whe shall use the following symbols: Molar concentration of the liquid or vapour phase, being xv: the ratio of the number of 3He atoms to the totalnumber of atoms. Molar density of the liquid and the vapour phase. Pl> pv: Molar density of aHe in the liquid and in the vapour p31, p3v: phase. LW: Molar weight of the mixture. Radius and total length of the capillary. Y, 1: Xl,
VISCOSITY
Radius
R, L: A=*:
hr, hs:
h=h2--1: dX = Xi, -
Pr and k’s.
of the apparatus.
Acceleration due to the gravity. Level heights in vessels I’1 andvs. Level difference. The small concentration difference
g:
Xi,:
673
MIXTURES
and length of the vessels
A constant
41R2
sHe-4He
OF LIQUID
between
the vessels.
Time in seconds. Osmotic pressure with respect to pure 4He expressed in cm liquid mixture column. Vapour pressure of the mixture expressed in cm liquid mixture column.
;e: p,:
6: Average velocity. From this experiment it is known that the time in which the superfluid flow adjusts the equilibrium between osmotic pressure and hydrostatic pressure is very small compared with the time concerning the Poiseuille flow of the normal fluid. In our calculation we therefore assume instantaneous equilibrium between the hydrostatic and osmotic pressure :
AX, neglecting the influence of the vapour pressure difference due to the small concentration difference AX. First we consider the case in which the concentration difference between the vessels, AX, is small compared with the mean concentration of the liquid x. In this case we expect a homogeneous flow, that is, superfluid and normal fluid containing sHe are moving with the same velocity, since the composition of the liquid does not alter appreciably during the flow from vessel vs to vessel vr. Since the increase of the number of moles in vessel Iri is equal to the amount
passing through
g
Mp12gh = ;
Neglecting two vapour ferential equation :
iz
-=h
the capillary,
corrections
we get for laminar nR2pl
(nR2plhl) = - T mentioned
with
AME r
below
A =
flow:
h. we obtain
y4g 41R2’
the
dif-
(3)
which expresses the linear relation between velocity (cc h) and pressure head (cc h). Conversely a straight line dependence of log Jz on the time obtained as information from the experiment attests to the laminar flow
674
F. A.
STAAS,
I<. W.
of the liquid, and the viscosity straight line.
TACONIS
_4ND K. FOKKEXS
can be calculated
easily from the slop” of the
The vapour corrections are of two kinds: a. The correction dealing with the replacement
of vapour
by liquid on
one side (condensation) and the replacement of liquid by vapour on the other side (evaporation). This can be considered as a correction of the measured value of the flow iz this being too large by a factor (1 - paV;pa,)-i. This will be derived more accurately later on. b. The correction arising from a difference in vapour pressure corresponding to a concentration difference at both sides. This makes the driving force no longer equal to the level difference h but increases it by a factor
After taking formed into : il
these vapour corrections
~= h
_A;\f_C
~~ YI (
1-
into account
‘- ~, .~ ;‘P” /i $b() >( ;;y/’ -pi ‘L !I
1-
--’
equation
2. p&pa
(3) is trans-
)
In the case where the concentration difference is comparable with the mean concentration x, the dilution of the highest concentration becomes important, since now the decrease of h with time and the corresponding decrease of the concentration difference gives rise to a non-negligible difference in flow velocity for superfluid and normal fluid containing 3He. This is so because the concentration in the vessel having the higher level must decrease and the concentration in the other one must increase. This means that a measurement of iz no longer correctly accounts for the velocity of the normal fluid and the 3He atoms. Moreover, we have taken into consideration the variation of the concentration along the capillary according to the equilibrium between the driving force and the osmotic pressure:
(5) According
to the relation
for laminar
273 =
-
---
Q
flow in which 3He participates
pgizf
grad 9,
: (6)
in which the pressure p is expressed in cm He column After substitution of (5) this becomes:
(7)
VISCOSITY
As an approximation
aHe-4He
OF LIQUID
we assume a stationary
of moles of aHe per second passing through the capillary is then: nr2piXz?3 = c
675
MIXTURES
flow. The constant an arbitrary
T (X+X12).
pr2gM
1%
number
cross-section
of
(8)
T
The total number of moles of sHe already present in vessel I is nR2{hX1,p1
The aHe balance
For simplicity
+
(L -
h)Xvlp"J.
gives:
we introduce
htot and X: J&t = hl + h2
(a measure of the total experimental accuracy).
quantity
of liquid which remains
k&l,
xz
(10)
+
constant
h2X12 -___)
within
(11)
hot
the mean molar concentration of the liquid. Equilibrium between osmotic pressure and the driving for the vapour pressure difference gives:
(g.)T (Xl,
- Xl,) = h +
(-f$),(Xl,
force corrected
(12)
- Xl,).
With (lo), (11) and (12) we are able to express Xi, and X1, and hl in terms of the constants ht,t, X and the level difference h. (hot ______--__
x1*=x+
2ht,t -
Xl,
=
x -
hl=-----
and after substitution (1 +c+6(h));=
h)h
(‘3)
__ - __ (hot +
hot -
h)h
(14)
h
(15)
2
of (13), (14) and (15) in (9) we arrive at the equation: -AM$-(
_____(16) 1
1 -
p3vlp3
F.
676
A. STAAS,
K. W.
TACONIS
AND
K. FOKKENS
in which: A -;$ 2L wit11 p = .~~___.~.~~_ ,
htot + p
E== 2
(
8Po
ap,
8X
8X
T
>
P3,/P3”
i
(2 b(h) = - .~_..____ ape 2htot ( c?X
-
1
2,Uuh _____ i‘pv mm.-) z_hZ’
E)h2
-
BX
assuming pr and ~3,/~3~constant. The factor (1 + F + d(h)) in the left hand side of (16) deals with the fact that the highest concentration is diluted during the flow. However, the subsequent delivery of aHe from the vapour phase and the effect of a concentration gradient along the capillary also affects this factor. From our experiments we found a linear dependence of log h on the time. This indicates that (1 + E + d(h)) d oes not depend strongly on the level difference h. We regarded 6(h) as a correction of (1 + E). To get an idea of the magnitude of these corrections we assume: htot = 6 cm, L = 8 cm, lz = 1 cm and the temperature is 1°K. 6(h) is given in the following table : Calculated
x 10 .4
from
II’& I 2.511
5 Y 10-a
1.302
,O-” 5 % IO-3
1.036
1.151 /
(16)
: S(h)
-6.39
Y 10 :3
18.3 pi IO-” .~4.8 k, , 0 3 f 1.07 :: 10~-3
This shows, that 6(h) is of no significance since b(h) is small compared with 1 + E. E = 0 means that we are dealing with a homogeneous flow. In other words: the superfluid is moving with the same velocity as normal fluid and 3He atoms. The amount flowing through the capillary per unit area and per second being pv,. In the case where F # 0 the amount flowing through the capillary per unit area and per second equals pnvn + psvs with pn + ps = p. We can easily conclude that (pnun + psvs) (1 + E) = pun yielding:
v, -
vn
=
1 -~I+&
____1___
_
1 vn
1 --x with x = pn/p, expressing
vS < vn as we expected.
VISCOSITY OF LIQUID sHe-4He
In our experiment
the velocity
1 cm/s. Corresponding with this value following table :
MIXTURES
of the normal (vs -
677
fluid is of the order of
un) is calculated
and given in the
Since we considered the quantity of sHe transferred by the laminar flow, we also have to regard the transfer caused by diffusion under influence of the concentration gradient. B e en a k k e r and Taco n is measured concentration diffusion coefficients 20). According to these measurements the diffusion coefficient is of the order of 10-l at T = 1.1 “K. Our minimum values of the concentration gradient in the capillary are of the order of 10-S. The number of moles sHe transported by diffusion per second and per unit area is equal to pD grad X(of the order of 10-T). The ratio of transfer by diffusion and transfer by the laminar flow is: D grad X 53x
.
Using (6) this turns out to be 8YDIx+PlgJ4
(of order 10-a in the meaning that diffusion
( > aP0
ax
case where X = 1O-d), hence is negligible.
D grad X/63X Q 1,
5. Calibration of the apparatus. Calibration of the apparatus was performed in two different ways. Since the constant A of the apparatus contains the radius of the vessel R and the capillary dimensions, a gas flow experiment made it necessary to make independent measurements of both. The laminar flow constant r4/1, depending on cross-section and length of the capillary was determined by a helium gas flow experiment at room temperature and atmospheric pressure, using a small initial pressure head of about 15 cm oil. One end of the capillary was therefore connected to a thermostated and calibrated volume filled with helium gas at 15 cm oil pressure difference with respect to atmospheric pressure while the other end was maintained at constant atmospheric pressure. The decrease of the pressure difference was measured at the differential oil manometer Mr. However, a flow experiment with mercury at constant temperature supplied us with the complete constant A in one measurement according to (3).
The constants
A derived
from
--
=
1 the
radius
of vessels
both
methods
4.93 x
10-l’
X = 0.1690 cm and
were
equal
to within
cm3, the
capillary
7‘
length
2 is about
13.0 cm.
1.6’Ii
\,I
7‘
1.7 li
n-- 7‘
2.045
“,
7‘=1.81<
2.1 ‘ii
A
_ 1.
v
/I
2.61 4.03 6.36 7.69 0.855 0.23 7.99 30.35 34.99 0.018 0.085
_
’
_~ 1
‘I<
1.9 I< I‘;\111.1:
I/
2.0 ‘Ii
-I . 0
-
I
/A’ 13.73 13.09 13.21 13.21 14.84
1.019 I.018
1.023 1.023
~
I.018
1.023
~
1.018 I.019
1.023 1.023
17.56
1.015 l.Old
I.023 1.023
13.26 14.00 14.28 26.30 18.60
L
1.007 1.006 1.018 1.015 I.018 1.018
I
1”;~:
1.014 1.013 1.023 I .023 1.023 I .023
1.003
::z 1.002
i
1.008 1.031 1.001 1.001
14.2~3 13.59 13.75 13.75 1S.JS 17.74 lj.79
1.001 1.243
14.29 14.52 22.18
I.051 1.331 I .026
18.41 21.25 19.00
VISCOSITY
sHe-4He
OF LIQUID
679
MIXTURES
6. Results and discussions of the results. The results of the measurements are listed in tables I and II and are given in figures 2, 3 and 4 in which we
”
P /
6
1.3
,l.O
---o-t.1
0
10
x
Fig. 3. The viscosity l
0 .
as a function
20
30
of the concentration
"/o
at temperatures
T =
l.O”K.
o
T ~7 1.5”K
T =
l.l”K
0
T =
T =
1.3”K
40
below
1.6”K
1.6”K
drew the curves for constant temperatures. 7’ is calculated according to equation (3). The corrected value 7 is calculated from equation (16). The factors 1
and 1 1 -
P3vlP3
depend slightly on temperature and even much less on the concentration. In order to give an idea of the magnitude of the corrections we give the following table (see also table I):
F. A. STAAS, K. U’. TACONIS AND K. FOKKENS
680
T =
T = 21°K
l.O’K
1.05
1.01 ;--~ +7j;
=
ax
ax
1 1(@o/aX)~ De Bruyn $0 =
1.08
1.Ol
P3vlP31
was calculated from the formula for regular Ouboter, Beenakker and Taconis: -pro
RT In (1 -
XL) -
WXLZ
solutions
PP7jK =
with
given
1.54”K.
by
21)
With this formula the correction can be calculated sufficiently accurately in the whole temperature and concentration region of our experiment.
m 6
Fig. 4. The viscosity
as a function of the concentration at temperatures (concentrations below 8%)
.
T -= l.O”K
n
T :
0
‘f = 7‘ =
o
T
.
l.l”K 1.3”K
1.5”K : 1.6”K
below
1.6”K.
VISCOSITY
OF
LIQUID
3He-4He
TABLE X % T =
l.O”K
2.44
r-
7’ PP 10.38 11.45
681
MIXTURES
I
T-
77
x
PP
%
11.15 12.21
0.14
12.05
12.57 12.31
I
7’ /“p
I T =
I
r)
I rup
1.8”K
1.025
17.18
17.61
0.62
1.475 2.41
16.45
lb.92
10.80
11.60
6.11
11.21
15.41
15.90
1.2 35.15
18.11
18.73
5.24
11.03
12.12
4.12
15.10 14.30
32.1
17.20
17.88
4.57
10.79
11.86
6.14
14.63 13.85
30.7
16.60
17.26
4.03
10.73
11.81
8.39
14.11
14.53
29.15
16.09
16.77
T =
1.9” K
1.635
15.70
16.15
27.6
15.60
16.30
8.03
13.09
14.36
0.352
20.78 23.00
21.00
24.3
14.46
6.09
12.00
13.21
23.20
21.5
4.96
4.89 1.472
14.91
15.40
18.2
13.69 12.79
15.13 14.36 13.50
3.63
11.85 11.50
13.09 12.70
15.71
16.20
12.75
12.52
11.40
12.14
12.08
13.30
8.49
10.74
13.87
11.38
0.14 1.89
12.89 11.18
12.34
31.6
15.59
15.81
9.37 7.76
10.84 10.61
11.57
22.6
12.84 14.20
14.19 13.22
2.6 1 6.75
11.37
13.70
15.52 12.81
12.02
7.03
11.08
5.71
13.41
31.10
6.76
12.19
11.10
15.64
26.30
5.75
10.30
11.07
10.52 15.6
14.30
0.084
33.21 26.82
10.34 10.36
lb.46
17.88
0.134
24.00
23.98
5.11
10.18
10.84
0.178
23.50 = 1.301
23.70
0.072
14.26 T = 1.6” K
13.30
0.275
0.051
.
14.43
1.599 2.39
11.69
12.31
0.17
12.20
12.57
11.43
12.08
35.08
18.70
19.41
4.25
11.19
11.81
33.2
7.21
10.35 14.14
10.94
30.85 25.6
19.10 17.25
19.83 18.00
0.145 0.029 0.038
16.85
14.97 16.18
15.20
15.95
21.1
13.51
14.29
12.20 11.28
13.00 12.09
T = 2.0” K 7.06
14.89
16.41
7.11
15.70
17.35
5.40
14.10
15.63
4.58
13.84 13.31
15.37 14.79
12.58 12.25
13.92 13.58
11.79
13.04
3.61 2.57 1.37 0.74
16.60 13.21
16.05
17.0
13.80
13.35
12.30
12.95
11.51
5.52
15.20
16.88
12.60
10.34
11.19
4.54
15.10
18.25
12.80
13.14 13.38
10.35 7.94
10.70
17.2
10.31
11.18
3.44
13.75
10.05
10.89
2.49
33.0
lb.59
14.29 17.08
14.50 13.50
16.77 16.12
22.8
6.59 5.46 4.76
9.85
10.68
9.94 13.81
11.65
12.28
4.04
10.09
10.92
2.06 1.35
13.89 13.81
11.92
12.53
3.31
9.95
10.78
0.35 15.55
T = 2.045OK
15.02 15.47 15.39
T = 2.l”K
lb.2
12.49
13.07
2.86
9.83
10.63
4.45
18.81
20.93
21.0
13.50
14.03
2.57
9.91
10.73
16.92
28.4
15.91 17.26
2.24
10.21
11.06
la.19 17.76
34.0
15.40 16.80
2.55 1.81
lb.25
17.57
13.00 T ’ = 1.5”K
13.59
7.96
10.97 10.75
4.19
18.75
20.86
7.67 7.19
10.20
10.95
7.08 5.28
11.90 11.70
15.10 14.24
16.58
18.3
0.53 0.12 1.61
14.47
17.10
10.14 1.701
10.97
34.6
9.90
10.62
4.30
17.91 16.71
19.98 18.62
6.20 4.68
9.60 9.50 10.00
10.36 10.22 10.43
3.08 0.14
15.25 15.08 15.58
17.01
3.37
1.375 I
0.09
-
1=
10.01 10.15
10.91
4.10
11.08
10.19 12.02 12.41
10.77 12.45 12.43
3.08 2.36 1.84 1.25
i 5.89
15.96
lb.82 17.41
682
F. ~1.STAAS,
~3” was calculated
K. W.
from ~3” =
TACONIS
p ~ fQ( 1 -
AND
K. FOKKESS
SI)
pzv(’
P3(’
with the value of psvO measured by Kerr 22). The spread of the data amounts to less than 1‘t;, in most of the casts with the exception of some data showing a greater deviation from the curvc’s. In the neighbourhood of the i-point the deviation is of the order of 61,,,. From figures 2,3 and 4 we derived figure 5 : “The bchaviour of the viscosity Dash and Taylor 14) foutid a minimum as a function of the temperature”. in the viscosity for pm-c “He: 14.75 $C’ at a temperature 7’ = 1.8% and
Jcig. 5. l‘hc
viscosity
as a function
of the tcmpcrature.
Concrntrations:
0, 0.2, 0.5,
1, 5, 10, 15, 20, 25, 30, loo),,.
for a concentration of about 10 91”3He : 14.45,u.P at II‘ = 1.66”K, whereas our values are 12.65,uP extrapolated at T = 1.8”K and for IOq, “He: 1 1.40,uP at T = 1.55”K. Our minimum shifts to lower temperatures for increasing concentration. This shift is about 3 x 1OW degree per percent 3He while going from 0 to 5% 3He the viscosity minimum decreases from 12.6pP to 10.3 PP and above X = 5!/, increases again by about 0.14 PUPper percent 3He. A very striking result of this study is found in extending the measurements to zero concentration. If one dilutes the mixture and extrapolates the determined viscosity to X = 0 it appears very essential to apply the c: correction in order to arrive at the value found by Heikkila and Hollis
VISCOSITYOF LIQUID sHe-4He Hallet
MIXTURES
683
t. The E’S however grow to values so large, that they can no longer
be considered as corrections. From the measurement of the level heights and the mean concentration one is able to calculate with (lo), (11) and (12) the total quantity of 3He present in one of the vessels at any instant during the flow. The time derivative of this quantity is proportional to the flow velocity of sHe in the capillary and was found to be proportional to the level difference within experimental accuracy and we could conclude the validity of 5s = -C grad p. In the case of pure 4He the flow of the normal fluid through a capillary with circular cross section according to the two fluid model is given by 6,
zzz
-
-
r2 8%
grad p.
From the viscosity measurements by Heikkila and Hollis Hallett we are able to calculate values of cn as a function of grad 9. These are in good
Fig. 6. The viscosity calculated from Zharkov’s theory. ___ T = l.O”K _____ T = 1.6”K agreement with our values of da extrapolated to zero concentration. All of this allows the conclusion that the normal fluid part of the 4He and the aHe together behave as one viscous fluid. The application of the two fluid model is therefore strongly confirmed.
684
F. A. STAAS,
K. W.
TACONIS
AND
K. FOKKEiYS
7. Theoretical consideration. The behaviour of the viscosity for pure 4He at temperatures below 1.6”K is described by the theory of Landau and Khalatnikov 25)26). Zharkov 27) has extended of We impurities. According 1) 2) 3) the
to this theory
the theory
of Khalatnikov
one can split up the viscosity
for the case into three
parts:
the roton viscosity: Q the phonon viscosity: ‘ljph the impurity viscosity: 7~3, measured viscosity being the sum of these parts: 17 = qr +
qph
+
T’3.
This is a consequence of the basic relation of proportionality of transfer of momentum and drift velocity gradient in case phonon gas, roton gas and
Fig. 7. Comparison
of our results with
. 0
(Concentrations ?‘ -: 1,O”K T -: l.l”K
.
T =
below
Zharkov’s
formula.
X 7: 1%) q
0
T = 1.5”K 7‘ -:: 1.6”K
1.3”K
impurity gas obey this relation and travel with the same drift velocity at all points in the liquid. The rotons and impurity excitations can be regarded as heavy particles and thus phonons are obstructed in their motion by rotons and impurities. Transfer of momentum by phonons is therefore strongly
VISCOSITY
OF LIQUID
sHe-4He
affected by the density of rotons and impurities.
MIXTURES
685
The decrease of the viscosity
with increasing temperature in the case of pure 4He below T = 1.6”K is due to the rapid increase of the roton density. We calculated the three partial viscosities according to the theory of Zharkov. They are shown in fig. 6 in the case of T = l.O”K (full curve) and T = 1.6”K (dotted curve). At 1°K the phonon viscosity decreases strongly with increasing concentration
due to the obstruction
of the phonons by sHe impurities.
The same
1.3 1.5 1.6
Fig. 8. Comparison of our results with (Concentrations below X = * q T = l.O”K 0 T = l.l”K o . I‘ = 1.3”K
Zharkov’s formula. 10%) T-1.5”K T = 1.6%
thing occurs to the roton viscosity. On the other hand the transfer of momentum by the sHe impurities increases with increasing concentration and one can expect an increase of the total viscosity for the higher concentrations in case the decrease of ?j$h and qr is smaller than the increase of 73 (see fig. 3). At higher temperatures (T = 1.6”K) the phonon viscosity does not change appreciably with variable sHe concentration. This is due to the very high roton density and transfer of momentum by phonons is mainly limited by rotons. At this temperatures we see that the transfer of momentum by sHe excitations is also less than at 1°K because of the density of the rotons. Physica 26
686
aHe-4He
OF LIQUID
VISCOSITY
MIXTURES
Though the theory given by Zharkov is only valid for concentrations smaller than IO-4 we compared it with our measurements. The constants 6 and a in Z h ar k o v’s formula depending on phonon-impurity and impurityimpurity interaction respectively are found to be B = 1.86 and a = 1.50. The results of the calculation are compared with experimental data in fig. 7, low concentrations and in fig. 8, high concentrations. The spread of the data in fig. 7 is mainly due to the insensitive determination of the concentration. The authors are very indebted to Mr. C. le Pair jr. Acknowledgements. and Mrs. A. Staas-Sluysmans for their help during the measurements and to Mr. C. J. N. van den Meydenberg for valuable discussions. They wish to express their thanks to Mr. I. Ncu t eboom for the technical assistance given. lieccived
17-6-60.
‘1
Krrsorn,
2)
I’hrsica 5 (1938) 737. Andronikash\ illi, I:. L., J. cxp.
\V. II. and M;rvM’ood,
G. I<., COIIIIII~III.Katr~cl-li~~~h Onncs LA)., I.eitlwl 11). 254~;
3)
I)r Tro”er,
4)
(1951) 50. Hollis Hallett,
5)
Hollis
thcor.
I’hys.
I‘.S.S.II.
.I. ;111d\‘a11 ClCll lierg,
A., \‘a~ Ittrrheek,
IH (1948) 429. G. J:, C~~n~mun. No. 264~; l%yhiC:\
17
Hallett,
A. C., Prw.
North
Holla~~d Publ. Camp.,
6)
I)ash,
J. G. and Taylor,
7)
l)ash,
J. G. and Taylor,
8)
Heikkila, Ziliov’eva,
9)
n>!_. Sw.
A “10
(1952)
A. C., Progrw~ ill I.ow ‘l‘cmpcratuw Aneterdan~
\V. J. and Hollii K. N., J. exp.
I, ~tlit~d 11~C‘. J. (;oI-ter, Ch. I\.,
1955.
Ii. I)., Phys. Rev. Ii. I)., Phys.
404. Physics
Rev.
I0.i (1957) 7. JOG (1957) 398.
Hallctt, A. C., Canad. .J. I’hys. XI (1955) 420. thror. Phy<. Y.S.S.R. :%I (1956) 31; So\?ct I’hy\iv
(1957) 36. van Groeuou, A., Poll, 10) Ijroese 304f; Physica ‘2 (1956) 905.
.J. I)., I)c,liin
g, .i. 11. c;. ;LII(IGortvr,
J.l:.T.P.
C. J., COlll1llllll. No.
1’) 12)
Brewer, Pcllam,
‘3) 14)
Wansink,
15)
Daunt, .I. G., Probst, R. 1:. and Johnston, H. L., Phy~. lie\,. 7:! (1948) 638. Wansink, D. H. N. and Tacorris, K. \\.., Comrnun. So. 305~; Physica Z:1 (1957)
125.
Wansink,
Physica
‘6) ‘7)
Dash,
(1956)
I). 1:. and I~~d\vards, I). O., Proc my. Sw. A 131 (1959) 247. J, I<., Confer. de Phys. dcs basses Temp. (1955) 24. Phys. Rev. !#I) (1955) I). H. S. and Taconis,
J, (;. and Taylor,
Ii. \lr., Corrm~un. No. 306b;
Ii. I)., Phys.
I). H. N,, Taconis,
Rev.
I07
K. LY. and Stash,
(1957)
4
1327.
Phyiic;L %I (1957) 273.
1228.
1:. A., COIIIU~UII.N11.304n;
ZL’
449.
20)
Roberts, T. R. and Sydoriak, S. G., I’hys. Iicv. letters .4 (1960) 321. Roberts, T. K. and Sydoriak, S. G., l’bys. Rev. 102 (1956) 304. Beenakker, J. J. M. and Taronis, K. \I:., Progress in Low l‘ctnperaturc
21)
by C. J. Garter, Ch. VI, North Holland De Uruyn Oubotcr, Ii., Ueenakker,
22) 23)
116~; Physica @5 (1959) 1162. Kerr, E. C., Phys. Rev. ‘JB (1954) 551. Kerr, E. C., Proc. fifth internat. Calf. 011low Temp. Phys. and Chem., Madison 1957 (L:niversity
‘8) 19)
Physics
I, edited
Publ. Curnp., Arusterdatu 1955. J. J. M. and Tacoui?, Ii. B’., Conmu~~. Suppl.
No.
of Wisconsin Press, Madison 1956) p. 158. L. D. and Khalatnikov, 1. RI., J. exp. theor. Phys. IY.S.S.R. I!) (1949) 637, 709. 24) Landau, I. M., Uspekhi fiz. Nauk. 3!) (1956) 673. 25) Khalatnikov, V. N., J. exp. them. Phys. U.S.S.R. :I:1 (1957) 929; Sovxt Physics J.E.T.P. (i (1958) 26) Zharkoo, 7 14.
Staas, F. A. Taconis, K. \Y. Fokkens, K. 1960
VISCOSITY
OF LIQUID 3He-4He MIXTURES HELIUM II REGION ABOVE 1°K
by F. A. STAAS, Communication
K. W. TACONIS
No. 323~ from the Kamerlingh
IN THE
and K. FOKKENS
Onnes Laboratorium,
Leiden,
Nederland
Synopsis The viscosity of liquid sHe-4He mixtures has been derived from isothermal flow through a capillary 75.6 p in diameter. The apparatus consists of two identical glass vessels, connected to each other by the glass capillary. The positions of both levels were measured as a function of time. It appeared that the logarithm of the level difference always decreased linearly with time, which means that Poiseuille’s law is obeyed. The analysis of the flow has been given. The values of the viscosity extrapolated to zero concentration (pure 4He) are in agreement with the measurements with the rotating cylinder by Heik kila and Hollis Hallett. There is also satisfactory agreement in the low concentration region with the theory given by Zharkov.
1. Introduction. Several investigators have measured the viscosity of the normal fluid Q, of pure 4He below the A-point. One can distinguish between several methods, the oldest method being the observation of the damping of an oscillating disk used by Keesom and MacWood 1) and later on followed by Andronikashvilli 2); De Troyer, Van Itterbeek and Van den Berg 3); Hollis Hallett 4) and Dash and Taylor 6)7). From these observations one can calculate the product pnqn, pn and qn being respectively the density and the viscosity of the normal fluid. The value of the viscosity of the normal fluid is obtained from an independent measurement of the normal fluid density. Independent of the knowledge of the normal fluid density H eikkila and Hollis Halle tt 8) were able to measure the normal fluid viscosities by using The results of the torsion pendulum the rotating cylinder viscometer. method and the rotating cylinder viscometer are still in quantitative disagreement. Zi n ov’ e v a 9) has derived values of the viscosity of the normal fluid from the attenuation of second sound in a resonant cavity which are in better agreement with those values found by Heikkila and Hollis Hallett. Capillary methods have been used in heat conduction experiments. Since in pure 4He below the &point heat is transferred only by the normal fluid, the heat current is related to the normal fluid velocity un. The normal Physica
26
669 -
fluid viscosity corresponding
may bc calculated from laminnr flow if the prt’ssurc gradient with the velocity of th(, normal fluid is known. ‘I%~~normal
fluid flow is accompanied by a superfluid counterflow in such a way that thcx resultant flow is zero. Now thcx prcxssure gradient can only be easily calculated from the temperature gradient if a critical velocity, abovc~ which mutual friction comes into play is not c~sct~c&~tl. Tli~ critical \rvlocity depends on the capillary diameter and is lower for a widclr capillary. This makes measurements with narrow capillaries necessary unless one is able to measure very small temperature gradients. On the other hand the choice of the capillary or slit is limited by mean free path effects. The mean fret path of phonons increases with decreasing temperatures and is in the casc~ of pure 4He at lower tcmpcraturc~s comparable with the slit width. Heat conduction methods were cmployc~l by I? r o f’s c’ van (; r o c II o u, P o 11, Dclsing and Gortcr 10) and recently by 13rt~wt~r and E d \va r cl s 11). From their heat conduction method with widca capillarit~s of 52.0 p and 107.6 ,LLdiameter Rrewer and Edwards derived values for the viscosity which arc in good agreement with those of the rotating cylinder viscometer. Until now viscosity measurements on 3Hc--1He dilute mixtures have been reported by Pellam lz), STansink and Taconis 13), Dash and Taylor 14). Wansink and Taconis wet-c able to deduce the viscosity from flow experiments through a narrow slit of 0.3 ,u, while Pellam and I>ash and Taylor used the torsion I~endulun~ method. The latter measurc~d the viscosity of three samples with liquid concentrations of about 3”1, 5(,‘;, and 1O$,, “He. We have determined the viscosity of the normal fluid straightforward using isothermal flow in a capillary 75.6 ,LLin diameter, making use of the relation between the mean velocity and the pressure gradient according to Poiseuille’s law. In liquid 3He-“He mixtures below the 3,-point a kind of osmotic prcssurc exists between two vessels containing mixtures of different concentrations separated by a narrow slit 15)lfi). In the case where one connects two vessels containing mixtures of different compositions by a capillary, superfluid will flow to the, high concentration side until the osmotic pressure is balanced by a level difference. In our cast’ using a capillary 76 ,u in diameter, this difference adjusts itself within a few seconds. A concentration difference of 10-d corresponds to an osmotic pressure of about 2 cm helium column at a temperature near 1“I<. Subsequent to the establishment of the level difference one can expect a flow of the normal fluid under influence of the hydrostatic pressure head, according to Poiseuille’s law, while the superfluid flow obeys the condition of permanent equilibrium between the hydrostatic pressure head and the osmotic pressure difference. Therefore in general at not too low concentrations where the concentration difference is relatively small superfluid flows with almost the same velocity as do the normal and 3He atoms. As wt’ shall
___~ show later
VISCOSITY
on diffusion
easily from Poiseuille’s
OF LIQUID
is negligible
sHe-4He
MIXTURES
and we can calculate
671 the viscosity
law.
2. The apparatus. The apparatus consists of two identical thin-walled glass vessels Vi and Va connected to each other by a glass capillary. A filter of fine nylon fabric N is used for protecting the capillary against the entrance of impurities. The vessels Vi and VZ are connected to the manometer system by means of two identical glass capillaries C, which are
Fig. 1. The apparatus.
partially wrapped with cotton cord K to prevent the increase of heat leaks with the falling of the surrounding helium bath. During the experiment the helium bath temperature was kept constant within 10-5 degrees monitored by a carbon resistance thermometer. It was necessary to purify the aHe-4He mixtures before using them. We purified the gaseous mixtures by leading them through a spiral capillary at a temperature of about 4°K in a separate cryostat. Measurements of the vapour pressure of the liquid mixtures were carried out with the aid
672
F. A. STAAS,
of oil manometers
K. W.
TACONIS
AND
K. FOKKEXS
Mr and M2 filled with octoil S. We deduced the concen-
tration of the solution from the known temperatures of the helium bath and vapour pressure differences with respect to the bath pressure rT)is)r9). Both parts of the apparatus could be connected to a mercury filled Toepler pump system. 3. The
method.
The measurements
of the viscosity
were preceded
by
the creation of an initial level difference of the required extent (about 2 cm). This could be done in several ways depending on the temperature and the concentration of the solution. By transferring vapour from one side to the other, one creates a concentration difference between the mixtures in both vessels due to the much higher concentration of 3He in the vapour in comparison with the liquid. We observed a level difference with the higher level at the high concentration side. In consequence of the superfluid properties within a few seconds 4He flowed from the low concentration to the high concentration vessel until the level difference corresponded to the osmotic pressure. At that moment no vapour pressure difference could be observed with any precision on oil manometer Mi showing that the concentrations of both levels were nearly equalised by the superfluid flow. The vapour pressure difference due to the very small concentration difference diminished by the weight of a small vapour column should have been observed, but this is a hardly measurable effect. The positions of both levels as a function of time were measured using a cathetometer and a stopwatch. In general the levels were followed during about half an hour in which the level difference decreased by a factor of ten. At temperatures below 1.3”K and low concentrations the creation of a level difference was more difficult. It was accomplished by condensing on one side only solutions of higher concentration than already present in the apparatus. Condensation of pure 4He in different quantities or a mixture of much lower concentration did not produce any level difference. It was also possible to create a level difference by suddenly admitting gas so that the liquid is forced through the capillary by the overpressure. This was done at the higher vapour pressures. 4. The calculation of the flow. Whe shall use the following symbols: Molar concentration of the liquid or vapour phase, being xv: the ratio of the number of 3He atoms to the totalnumber of atoms. Molar density of the liquid and the vapour phase. Pl> pv: Molar density of aHe in the liquid and in the vapour p31, p3v: phase. LW: Molar weight of the mixture. Radius and total length of the capillary. Y, 1: Xl,
VISCOSITY
Radius
R, L: A=*:
hr, hs:
h=h2--1: dX = Xi, -
Pr and k’s.
of the apparatus.
Acceleration due to the gravity. Level heights in vessels I’1 andvs. Level difference. The small concentration difference
g:
Xi,:
673
MIXTURES
and length of the vessels
A constant
41R2
sHe-4He
OF LIQUID
between
the vessels.
Time in seconds. Osmotic pressure with respect to pure 4He expressed in cm liquid mixture column. Vapour pressure of the mixture expressed in cm liquid mixture column.
;e: p,:
6: Average velocity. From this experiment it is known that the time in which the superfluid flow adjusts the equilibrium between osmotic pressure and hydrostatic pressure is very small compared with the time concerning the Poiseuille flow of the normal fluid. In our calculation we therefore assume instantaneous equilibrium between the hydrostatic and osmotic pressure :
AX, neglecting the influence of the vapour pressure difference due to the small concentration difference AX. First we consider the case in which the concentration difference between the vessels, AX, is small compared with the mean concentration of the liquid x. In this case we expect a homogeneous flow, that is, superfluid and normal fluid containing sHe are moving with the same velocity, since the composition of the liquid does not alter appreciably during the flow from vessel vs to vessel vr. Since the increase of the number of moles in vessel Iri is equal to the amount
passing through
g
Mp12gh = ;
Neglecting two vapour ferential equation :
iz
-=h
the capillary,
corrections
we get for laminar nR2pl
(nR2plhl) = - T mentioned
with
AME r
below
A =
flow:
h. we obtain
y4g 41R2’
the
dif-
(3)
which expresses the linear relation between velocity (cc h) and pressure head (cc h). Conversely a straight line dependence of log Jz on the time obtained as information from the experiment attests to the laminar flow
674
F. A.
STAAS,
I<. W.
of the liquid, and the viscosity straight line.
TACONIS
_4ND K. FOKKEXS
can be calculated
easily from the slop” of the
The vapour corrections are of two kinds: a. The correction dealing with the replacement
of vapour
by liquid on
one side (condensation) and the replacement of liquid by vapour on the other side (evaporation). This can be considered as a correction of the measured value of the flow iz this being too large by a factor (1 - paV;pa,)-i. This will be derived more accurately later on. b. The correction arising from a difference in vapour pressure corresponding to a concentration difference at both sides. This makes the driving force no longer equal to the level difference h but increases it by a factor
After taking formed into : il
these vapour corrections
~= h
_A;\f_C
~~ YI (
1-
into account
‘- ~, .~ ;‘P” /i $b() >( ;;y/’ -pi ‘L !I
1-
--’
equation
2. p&pa
(3) is trans-
)
In the case where the concentration difference is comparable with the mean concentration x, the dilution of the highest concentration becomes important, since now the decrease of h with time and the corresponding decrease of the concentration difference gives rise to a non-negligible difference in flow velocity for superfluid and normal fluid containing 3He. This is so because the concentration in the vessel having the higher level must decrease and the concentration in the other one must increase. This means that a measurement of iz no longer correctly accounts for the velocity of the normal fluid and the 3He atoms. Moreover, we have taken into consideration the variation of the concentration along the capillary according to the equilibrium between the driving force and the osmotic pressure:
(5) According
to the relation
for laminar
273 =
-
---
Q
flow in which 3He participates
pgizf
grad 9,
: (6)
in which the pressure p is expressed in cm He column After substitution of (5) this becomes:
(7)
VISCOSITY
As an approximation
aHe-4He
OF LIQUID
we assume a stationary
of moles of aHe per second passing through the capillary is then: nr2piXz?3 = c
675
MIXTURES
flow. The constant an arbitrary
T (X+X12).
pr2gM
1%
number
cross-section
of
(8)
T
The total number of moles of sHe already present in vessel I is nR2{hX1,p1
The aHe balance
For simplicity
+
(L -
h)Xvlp"J.
gives:
we introduce
htot and X: J&t = hl + h2
(a measure of the total experimental accuracy).
quantity
of liquid which remains
k&l,
xz
(10)
+
constant
h2X12 -___)
within
(11)
hot
the mean molar concentration of the liquid. Equilibrium between osmotic pressure and the driving for the vapour pressure difference gives:
(g.)T (Xl,
- Xl,) = h +
(-f$),(Xl,
force corrected
(12)
- Xl,).
With (lo), (11) and (12) we are able to express Xi, and X1, and hl in terms of the constants ht,t, X and the level difference h. (hot ______--__
x1*=x+
2ht,t -
Xl,
=
x -
hl=-----
and after substitution (1 +c+6(h));=
h)h
(‘3)
__ - __ (hot +
hot -
h)h
(14)
h
(15)
2
of (13), (14) and (15) in (9) we arrive at the equation: -AM$-(
_____(16) 1
1 -
p3vlp3
F.
676
A. STAAS,
K. W.
TACONIS
AND
K. FOKKENS
in which: A -;$ 2L wit11 p = .~~___.~.~~_ ,
htot + p
E== 2
(
8Po
ap,
8X
8X
T
>
P3,/P3”
i
(2 b(h) = - .~_..____ ape 2htot ( c?X
-
1
2,Uuh _____ i‘pv mm.-) z_hZ’
E)h2
-
BX
assuming pr and ~3,/~3~constant. The factor (1 + F + d(h)) in the left hand side of (16) deals with the fact that the highest concentration is diluted during the flow. However, the subsequent delivery of aHe from the vapour phase and the effect of a concentration gradient along the capillary also affects this factor. From our experiments we found a linear dependence of log h on the time. This indicates that (1 + E + d(h)) d oes not depend strongly on the level difference h. We regarded 6(h) as a correction of (1 + E). To get an idea of the magnitude of these corrections we assume: htot = 6 cm, L = 8 cm, lz = 1 cm and the temperature is 1°K. 6(h) is given in the following table : Calculated
x 10 .4
from
II’& I 2.511
5 Y 10-a
1.302
,O-” 5 % IO-3
1.036
1.151 /
(16)
: S(h)
-6.39
Y 10 :3
18.3 pi IO-” .~4.8 k, , 0 3 f 1.07 :: 10~-3
This shows, that 6(h) is of no significance since b(h) is small compared with 1 + E. E = 0 means that we are dealing with a homogeneous flow. In other words: the superfluid is moving with the same velocity as normal fluid and 3He atoms. The amount flowing through the capillary per unit area and per second being pv,. In the case where F # 0 the amount flowing through the capillary per unit area and per second equals pnvn + psvs with pn + ps = p. We can easily conclude that (pnun + psvs) (1 + E) = pun yielding:
v, -
vn
=
1 -~I+&
____1___
_
1 vn
1 --x with x = pn/p, expressing
vS < vn as we expected.
VISCOSITY OF LIQUID sHe-4He
In our experiment
the velocity
1 cm/s. Corresponding with this value following table :
MIXTURES
of the normal (vs -
677
fluid is of the order of
un) is calculated
and given in the
Since we considered the quantity of sHe transferred by the laminar flow, we also have to regard the transfer caused by diffusion under influence of the concentration gradient. B e en a k k e r and Taco n is measured concentration diffusion coefficients 20). According to these measurements the diffusion coefficient is of the order of 10-l at T = 1.1 “K. Our minimum values of the concentration gradient in the capillary are of the order of 10-S. The number of moles sHe transported by diffusion per second and per unit area is equal to pD grad X(of the order of 10-T). The ratio of transfer by diffusion and transfer by the laminar flow is: D grad X 53x
.
Using (6) this turns out to be 8YDIx+PlgJ4
(of order 10-a in the meaning that diffusion
( > aP0
ax
case where X = 1O-d), hence is negligible.
D grad X/63X Q 1,
5. Calibration of the apparatus. Calibration of the apparatus was performed in two different ways. Since the constant A of the apparatus contains the radius of the vessel R and the capillary dimensions, a gas flow experiment made it necessary to make independent measurements of both. The laminar flow constant r4/1, depending on cross-section and length of the capillary was determined by a helium gas flow experiment at room temperature and atmospheric pressure, using a small initial pressure head of about 15 cm oil. One end of the capillary was therefore connected to a thermostated and calibrated volume filled with helium gas at 15 cm oil pressure difference with respect to atmospheric pressure while the other end was maintained at constant atmospheric pressure. The decrease of the pressure difference was measured at the differential oil manometer Mr. However, a flow experiment with mercury at constant temperature supplied us with the complete constant A in one measurement according to (3).
The constants
A derived
from
--
=
1 the
radius
of vessels
both
methods
4.93 x
10-l’
X = 0.1690 cm and
were
equal
to within
cm3, the
capillary
7‘
length
2 is about
13.0 cm.
1.6’Ii
\,I
7‘
1.7 li
n-- 7‘
2.045
“,
7‘=1.81<
2.1 ‘ii
A
_ 1.
v
/I
2.61 4.03 6.36 7.69 0.855 0.23 7.99 30.35 34.99 0.018 0.085
_
’
_~ 1
‘I<
1.9 I< I‘;\111.1:
I/
2.0 ‘Ii
-I . 0
-
I
/A’ 13.73 13.09 13.21 13.21 14.84
1.019 I.018
1.023 1.023
~
I.018
1.023
~
1.018 I.019
1.023 1.023
17.56
1.015 l.Old
I.023 1.023
13.26 14.00 14.28 26.30 18.60
L
1.007 1.006 1.018 1.015 I.018 1.018
I
1”;~:
1.014 1.013 1.023 I .023 1.023 I .023
1.003
::z 1.002
i
1.008 1.031 1.001 1.001
14.2~3 13.59 13.75 13.75 1S.JS 17.74 lj.79
1.001 1.243
14.29 14.52 22.18
I.051 1.331 I .026
18.41 21.25 19.00
VISCOSITY
sHe-4He
OF LIQUID
679
MIXTURES
6. Results and discussions of the results. The results of the measurements are listed in tables I and II and are given in figures 2, 3 and 4 in which we
”
P /
6
1.3
,l.O
---o-t.1
0
10
x
Fig. 3. The viscosity l
0 .
as a function
20
30
of the concentration
"/o
at temperatures
T =
l.O”K.
o
T ~7 1.5”K
T =
l.l”K
0
T =
T =
1.3”K
40
below
1.6”K
1.6”K
drew the curves for constant temperatures. 7’ is calculated according to equation (3). The corrected value 7 is calculated from equation (16). The factors 1
and 1 1 -
P3vlP3
depend slightly on temperature and even much less on the concentration. In order to give an idea of the magnitude of the corrections we give the following table (see also table I):
F. A. STAAS, K. U’. TACONIS AND K. FOKKENS
680
T =
T = 21°K
l.O’K
1.05
1.01 ;--~ +7j;
=
ax
ax
1 1(@o/aX)~ De Bruyn $0 =
1.08
1.Ol
P3vlP31
was calculated from the formula for regular Ouboter, Beenakker and Taconis: -pro
RT In (1 -
XL) -
WXLZ
solutions
PP7jK =
with
given
1.54”K.
by
21)
With this formula the correction can be calculated sufficiently accurately in the whole temperature and concentration region of our experiment.
m 6
Fig. 4. The viscosity
as a function of the concentration at temperatures (concentrations below 8%)
.
T -= l.O”K
n
T :
0
‘f = 7‘ =
o
T
.
l.l”K 1.3”K
1.5”K : 1.6”K
below
1.6”K.
VISCOSITY
OF
LIQUID
3He-4He
TABLE X % T =
l.O”K
2.44
r-
7’ PP 10.38 11.45
681
MIXTURES
I
T-
77
x
PP
%
11.15 12.21
0.14
12.05
12.57 12.31
I
7’ /“p
I T =
I
r)
I rup
1.8”K
1.025
17.18
17.61
0.62
1.475 2.41
16.45
lb.92
10.80
11.60
6.11
11.21
15.41
15.90
1.2 35.15
18.11
18.73
5.24
11.03
12.12
4.12
15.10 14.30
32.1
17.20
17.88
4.57
10.79
11.86
6.14
14.63 13.85
30.7
16.60
17.26
4.03
10.73
11.81
8.39
14.11
14.53
29.15
16.09
16.77
T =
1.9” K
1.635
15.70
16.15
27.6
15.60
16.30
8.03
13.09
14.36
0.352
20.78 23.00
21.00
24.3
14.46
6.09
12.00
13.21
23.20
21.5
4.96
4.89 1.472
14.91
15.40
18.2
13.69 12.79
15.13 14.36 13.50
3.63
11.85 11.50
13.09 12.70
15.71
16.20
12.75
12.52
11.40
12.14
12.08
13.30
8.49
10.74
13.87
11.38
0.14 1.89
12.89 11.18
12.34
31.6
15.59
15.81
9.37 7.76
10.84 10.61
11.57
22.6
12.84 14.20
14.19 13.22
2.6 1 6.75
11.37
13.70
15.52 12.81
12.02
7.03
11.08
5.71
13.41
31.10
6.76
12.19
11.10
15.64
26.30
5.75
10.30
11.07
10.52 15.6
14.30
0.084
33.21 26.82
10.34 10.36
lb.46
17.88
0.134
24.00
23.98
5.11
10.18
10.84
0.178
23.50 = 1.301
23.70
0.072
14.26 T = 1.6” K
13.30
0.275
0.051
.
14.43
1.599 2.39
11.69
12.31
0.17
12.20
12.57
11.43
12.08
35.08
18.70
19.41
4.25
11.19
11.81
33.2
7.21
10.35 14.14
10.94
30.85 25.6
19.10 17.25
19.83 18.00
0.145 0.029 0.038
16.85
14.97 16.18
15.20
15.95
21.1
13.51
14.29
12.20 11.28
13.00 12.09
T = 2.0” K 7.06
14.89
16.41
7.11
15.70
17.35
5.40
14.10
15.63
4.58
13.84 13.31
15.37 14.79
12.58 12.25
13.92 13.58
11.79
13.04
3.61 2.57 1.37 0.74
16.60 13.21
16.05
17.0
13.80
13.35
12.30
12.95
11.51
5.52
15.20
16.88
12.60
10.34
11.19
4.54
15.10
18.25
12.80
13.14 13.38
10.35 7.94
10.70
17.2
10.31
11.18
3.44
13.75
10.05
10.89
2.49
33.0
lb.59
14.29 17.08
14.50 13.50
16.77 16.12
22.8
6.59 5.46 4.76
9.85
10.68
9.94 13.81
11.65
12.28
4.04
10.09
10.92
2.06 1.35
13.89 13.81
11.92
12.53
3.31
9.95
10.78
0.35 15.55
T = 2.045OK
15.02 15.47 15.39
T = 2.l”K
lb.2
12.49
13.07
2.86
9.83
10.63
4.45
18.81
20.93
21.0
13.50
14.03
2.57
9.91
10.73
16.92
28.4
15.91 17.26
2.24
10.21
11.06
la.19 17.76
34.0
15.40 16.80
2.55 1.81
lb.25
17.57
13.00 T ’ = 1.5”K
13.59
7.96
10.97 10.75
4.19
18.75
20.86
7.67 7.19
10.20
10.95
7.08 5.28
11.90 11.70
15.10 14.24
16.58
18.3
0.53 0.12 1.61
14.47
17.10
10.14 1.701
10.97
34.6
9.90
10.62
4.30
17.91 16.71
19.98 18.62
6.20 4.68
9.60 9.50 10.00
10.36 10.22 10.43
3.08 0.14
15.25 15.08 15.58
17.01
3.37
1.375 I
0.09
-
1=
10.01 10.15
10.91
4.10
11.08
10.19 12.02 12.41
10.77 12.45 12.43
3.08 2.36 1.84 1.25
i 5.89
15.96
lb.82 17.41
682
F. ~1.STAAS,
~3” was calculated
K. W.
from ~3” =
TACONIS
p ~ fQ( 1 -
AND
K. FOKKESS
SI)
pzv(’
P3(’
with the value of psvO measured by Kerr 22). The spread of the data amounts to less than 1‘t;, in most of the casts with the exception of some data showing a greater deviation from the curvc’s. In the neighbourhood of the i-point the deviation is of the order of 61,,,. From figures 2,3 and 4 we derived figure 5 : “The bchaviour of the viscosity Dash and Taylor 14) foutid a minimum as a function of the temperature”. in the viscosity for pm-c “He: 14.75 $C’ at a temperature 7’ = 1.8% and
Jcig. 5. l‘hc
viscosity
as a function
of the tcmpcrature.
Concrntrations:
0, 0.2, 0.5,
1, 5, 10, 15, 20, 25, 30, loo),,.
for a concentration of about 10 91”3He : 14.45,u.P at II‘ = 1.66”K, whereas our values are 12.65,uP extrapolated at T = 1.8”K and for IOq, “He: 1 1.40,uP at T = 1.55”K. Our minimum shifts to lower temperatures for increasing concentration. This shift is about 3 x 1OW degree per percent 3He while going from 0 to 5% 3He the viscosity minimum decreases from 12.6pP to 10.3 PP and above X = 5!/, increases again by about 0.14 PUPper percent 3He. A very striking result of this study is found in extending the measurements to zero concentration. If one dilutes the mixture and extrapolates the determined viscosity to X = 0 it appears very essential to apply the c: correction in order to arrive at the value found by Heikkila and Hollis
VISCOSITYOF LIQUID sHe-4He Hallet
MIXTURES
683
t. The E’S however grow to values so large, that they can no longer
be considered as corrections. From the measurement of the level heights and the mean concentration one is able to calculate with (lo), (11) and (12) the total quantity of 3He present in one of the vessels at any instant during the flow. The time derivative of this quantity is proportional to the flow velocity of sHe in the capillary and was found to be proportional to the level difference within experimental accuracy and we could conclude the validity of 5s = -C grad p. In the case of pure 4He the flow of the normal fluid through a capillary with circular cross section according to the two fluid model is given by 6,
zzz
-
-
r2 8%
grad p.
From the viscosity measurements by Heikkila and Hollis Hallett we are able to calculate values of cn as a function of grad 9. These are in good
Fig. 6. The viscosity calculated from Zharkov’s theory. ___ T = l.O”K _____ T = 1.6”K agreement with our values of da extrapolated to zero concentration. All of this allows the conclusion that the normal fluid part of the 4He and the aHe together behave as one viscous fluid. The application of the two fluid model is therefore strongly confirmed.
684
F. A. STAAS,
K. W.
TACONIS
AND
K. FOKKEiYS
7. Theoretical consideration. The behaviour of the viscosity for pure 4He at temperatures below 1.6”K is described by the theory of Landau and Khalatnikov 25)26). Zharkov 27) has extended of We impurities. According 1) 2) 3) the
to this theory
the theory
of Khalatnikov
one can split up the viscosity
for the case into three
parts:
the roton viscosity: Q the phonon viscosity: ‘ljph the impurity viscosity: 7~3, measured viscosity being the sum of these parts: 17 = qr +
qph
+
T’3.
This is a consequence of the basic relation of proportionality of transfer of momentum and drift velocity gradient in case phonon gas, roton gas and
Fig. 7. Comparison
of our results with
. 0
(Concentrations ?‘ -: 1,O”K T -: l.l”K
.
T =
below
Zharkov’s
formula.
X 7: 1%) q
0
T = 1.5”K 7‘ -:: 1.6”K
1.3”K
impurity gas obey this relation and travel with the same drift velocity at all points in the liquid. The rotons and impurity excitations can be regarded as heavy particles and thus phonons are obstructed in their motion by rotons and impurities. Transfer of momentum by phonons is therefore strongly
VISCOSITY
OF LIQUID
sHe-4He
affected by the density of rotons and impurities.
MIXTURES
685
The decrease of the viscosity
with increasing temperature in the case of pure 4He below T = 1.6”K is due to the rapid increase of the roton density. We calculated the three partial viscosities according to the theory of Zharkov. They are shown in fig. 6 in the case of T = l.O”K (full curve) and T = 1.6”K (dotted curve). At 1°K the phonon viscosity decreases strongly with increasing concentration
due to the obstruction
of the phonons by sHe impurities.
The same
1.3 1.5 1.6
Fig. 8. Comparison of our results with (Concentrations below X = * q T = l.O”K 0 T = l.l”K o . I‘ = 1.3”K
Zharkov’s formula. 10%) T-1.5”K T = 1.6%
thing occurs to the roton viscosity. On the other hand the transfer of momentum by the sHe impurities increases with increasing concentration and one can expect an increase of the total viscosity for the higher concentrations in case the decrease of ?j$h and qr is smaller than the increase of 73 (see fig. 3). At higher temperatures (T = 1.6”K) the phonon viscosity does not change appreciably with variable sHe concentration. This is due to the very high roton density and transfer of momentum by phonons is mainly limited by rotons. At this temperatures we see that the transfer of momentum by sHe excitations is also less than at 1°K because of the density of the rotons. Physica 26
686
aHe-4He
OF LIQUID
VISCOSITY
MIXTURES
Though the theory given by Zharkov is only valid for concentrations smaller than IO-4 we compared it with our measurements. The constants 6 and a in Z h ar k o v’s formula depending on phonon-impurity and impurityimpurity interaction respectively are found to be B = 1.86 and a = 1.50. The results of the calculation are compared with experimental data in fig. 7, low concentrations and in fig. 8, high concentrations. The spread of the data in fig. 7 is mainly due to the insensitive determination of the concentration. The authors are very indebted to Mr. C. le Pair jr. Acknowledgements. and Mrs. A. Staas-Sluysmans for their help during the measurements and to Mr. C. J. N. van den Meydenberg for valuable discussions. They wish to express their thanks to Mr. I. Ncu t eboom for the technical assistance given. lieccived
17-6-60.
‘1
Krrsorn,
2)
I’hrsica 5 (1938) 737. Andronikash\ illi, I:. L., J. cxp.
\V. II. and M;rvM’ood,
G. I<., COIIIIII~III.Katr~cl-li~~~h Onncs LA)., I.eitlwl 11). 254~;
3)
I)r Tro”er,
4)
(1951) 50. Hollis Hallett,
5)
Hollis
thcor.
I’hys.
I‘.S.S.II.
.I. ;111d\‘a11 ClCll lierg,
A., \‘a~ Ittrrheek,
IH (1948) 429. G. J:, C~~n~mun. No. 264~; l%yhiC:\
17
Hallett,
A. C., Prw.
North
Holla~~d Publ. Camp.,
6)
I)ash,
J. G. and Taylor,
7)
l)ash,
J. G. and Taylor,
8)
Heikkila, Ziliov’eva,
9)
n>!_. Sw.
A “10
(1952)
A. C., Progrw~ ill I.ow ‘l‘cmpcratuw Aneterdan~
\V. J. and Hollii K. N., J. exp.
I, ~tlit~d 11~C‘. J. (;oI-ter, Ch. I\.,
1955.
Ii. I)., Phys. Rev. Ii. I)., Phys.
404. Physics
Rev.
I0.i (1957) 7. JOG (1957) 398.
Hallctt, A. C., Canad. .J. I’hys. XI (1955) 420. thror. Phy<. Y.S.S.R. :%I (1956) 31; So\?ct I’hy\iv
(1957) 36. van Groeuou, A., Poll, 10) Ijroese 304f; Physica ‘2 (1956) 905.
.J. I)., I)c,liin
g, .i. 11. c;. ;LII(IGortvr,
J.l:.T.P.
C. J., COlll1llllll. No.
1’) 12)
Brewer, Pcllam,
‘3) 14)
Wansink,
15)
Daunt, .I. G., Probst, R. 1:. and Johnston, H. L., Phy~. lie\,. 7:! (1948) 638. Wansink, D. H. N. and Tacorris, K. \\.., Comrnun. So. 305~; Physica Z:1 (1957)
125.
Wansink,
Physica
‘6) ‘7)
Dash,
(1956)
I). 1:. and I~~d\vards, I). O., Proc my. Sw. A 131 (1959) 247. J, I<., Confer. de Phys. dcs basses Temp. (1955) 24. Phys. Rev. !#I) (1955) I). H. S. and Taconis,
J, (;. and Taylor,
Ii. \lr., Corrm~un. No. 306b;
Ii. I)., Phys.
I). H. N,, Taconis,
Rev.
I07
K. LY. and Stash,
(1957)
4
1327.
Phyiic;L %I (1957) 273.
1228.
1:. A., COIIIU~UII.N11.304n;
ZL’
449.
20)
Roberts, T. R. and Sydoriak, S. G., I’hys. Iicv. letters .4 (1960) 321. Roberts, T. K. and Sydoriak, S. G., l’bys. Rev. 102 (1956) 304. Beenakker, J. J. M. and Taronis, K. \I:., Progress in Low l‘ctnperaturc
21)
by C. J. Garter, Ch. VI, North Holland De Uruyn Oubotcr, Ii., Ueenakker,
22) 23)
116~; Physica @5 (1959) 1162. Kerr, E. C., Phys. Rev. ‘JB (1954) 551. Kerr, E. C., Proc. fifth internat. Calf. 011low Temp. Phys. and Chem., Madison 1957 (L:niversity
‘8) 19)
Physics
I, edited
Publ. Curnp., Arusterdatu 1955. J. J. M. and Tacoui?, Ii. B’., Conmu~~. Suppl.
No.
of Wisconsin Press, Madison 1956) p. 158. L. D. and Khalatnikov, 1. RI., J. exp. theor. Phys. IY.S.S.R. I!) (1949) 637, 709. 24) Landau, I. M., Uspekhi fiz. Nauk. 3!) (1956) 673. 25) Khalatnikov, V. N., J. exp. them. Phys. U.S.S.R. :I:1 (1957) 929; Sovxt Physics J.E.T.P. (i (1958) 26) Zharkoo, 7 14.