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Surface tension of dilute solutions of4He in3He
PHYSICA
Physica B 194-196 (1994) 873-874 North-Holland
SURFACE TENSION
OF DILUTE SOLUTIONS
O F 4HE I N 3HE
Masaru Suzuki ~, Masayuki Yanaanobe b, Natsuki Ohtani b and Akira Sato b ~Division of Natural Sciences, Univ. of Electro-Communications, Chofu, Tokyo 182, Japan bDepartment of Applied Physics, Univ. of Electro-Communications, Chofu, Tokyo 182, Japan The surface tension of dilute solutions of 4He in 3Ite was measured by means of surface wave resonance in order to examine the effect of adding small amounts of "tHe to pure 3He. \¥e found that compared with pure 3He, the surface tension of the 5.2% 4He solution increases by 6 m d y n / c m at 1K. The increase of the surface tension means that the 4He atoms are excluded from the surface of dilute solutions. The data below 1.6K were compared with Saam's model. Assuming that the 4He behaves like an ideal quasiparticle gas and the 4He quasiparticle surface scattering length is proportional to the amount of surface adsorption, the temperature dependence of the surface tension is well explained.
1. I N T R O D U C T I O N Although little attention has been paid to the free surface of dilute solutions of 4He in 3He, the surface tension of dilute solutions may give us the important information about the 4He behavior in ZHe. The surface tension of dilute solutions of 4He in 3He was measured ill order to examine the effect of adding small amounts of 4He to pure 3He. The surface tension of pure 3He down to 0.3K is considered to consist of two parts, a 3He quasiparticle and a ripplon contribution.[1] In tile case of the dilute solutions, a 4He quasiparticle contribution may add to these two contributions. According to Edwards and Santa's model[2], the difference of the 4He contribution from OK, An43 is obtained as follow, Ao'43 = nakBT(b4a + 1)~T)
(1)
where n4 is the number density, b43 is the 4He quasiparticle surface scattering length, and AT is the thermal wavelength. Here it is assumed that the 4He behaves like all ideal quasiparticle gas with effective mass m~. EXPERIMENTAL Tile surface tension was measured using the surface-wave resonance technique.[1] The dispersion relation of the surface wave is given by 2.
w2 =
(PL -- py )gk + ak 3 PL coth(kdL) + py c o t h ( k d v ) '
(2)
where g is tile gravitaional acceleration, flL and pv are the mass densities of liquid and vapor, and dL and dv are the depths of liquid and vapor, respectively. Surface waves in a cylindrical cavity were excited and detected capacitively. The frequency of the excitation was locked to a resonace frequency of tile surface wave by tile PLL technique. Iu addition to the resonace frequency, the liquid density was determined separately by means of the m e ~ u r e m e n t of dielectric constant. The vapor density was calculated on tile ~ s u m p tion that the dilute solutions are regular solutio,as. 3. R E S U L T S A N D D I S C U S S I O N Figure 1 shows the surface tension of the 5.2% 4He solution. The absolute accuracy of each point is better than 1 mdyn/cm. Compared with pure 3He, the surface tension increases by 6 m d y n / c m at 1K. The surface adsorption of 4He with respect to tile 3He surface, F43 is calculated from
F43
=
-
x4 1 d~ 2,t . . . . kBT 1 - ~7~x4LI - x4) dx4
(3)
where x 4 is tile mole fraction of 4He, and A/t:g is a constant taken to be 1.37K[3]. Since ~da > 0, the surface adsorption of 4IIe is negative, which means that the 4He atoms are excluded from the free surface of dilute solutions. This behavior may come from the difference of zero point ellergy. The surface adsorption is also considered
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E1017-G
874 to be the difference of the Gibbs dividing surface between the 3He and 4He component. This value F43/n4 is estimated to be about -6.5/~ at 1K.
150
Eo 25
:-~-~..
....,. t--
"O
~..
o--ElO0-
10
<175501
",:,-
•
presentdata
~.
0
i
i 1
i %"""
Temperature(K)
2
Figure 2. Comparlsiou between the present data and the calculated curve below 1.6K. A~r3, A~. and Aaaa denote the a Ile quasiparticle, the ripplon and the 41Ie quasiparticle contribution, respectively.
Temperature (K) Figure 1. Surface tension of the 5.2% 4He dilute solution as a flmction of temperature. The solid curve shows the result of pure 3Ite.[1] Next let us consider the temperature dependence. Assuming that the 4He contribution is additive and the 4He behaves like an ideal quasiparticle with effective mass m~ = 4.5m414], we can calculate the 4He contribution by using Eq. (1). Regarding the the scattering length b43 , it is reasonable that the 4He quasiparticle reflects not at the 3He but at the 4He Gibbs dividing surface. The scattering length b43 may be expressed as b43 = F43/n4 -I- b0,
eo
(4)
where b0 is constant. ( In the case of pure 3He in Ref. 1, it was assumed that the 3He quasiparticle scattering length b3 is constant becatme the scattering length is considered to depend on the density profile of the surface and the profile may not change drastically at low temperatures.) The comparision between the experimental data and the calculated curve (b0 = 0.9.~ ) is shown in Fig. 2. The data below 1.6K are in good agreement with the calculated curve.
In conclusion, the surface tension of dilute solutions of 4He in 3He was measured in order to examine the 4He behavior in 3He. We found that the 4He atoms are excluded from the free surface of dilute solutions and the 4Ite contribution to the surface tension was well explained by Edwards and Santa's model. AKNOWLEDGEMENTS This work was partly supported by a Grantin-Aid from the Ministry of Education of Japan. One of the authors (A.S.) is greatly indebted to the Japan Society for the Promotion of Science for Japanese Junior Scientists. REFERENCES
1. M. Iino, hi. Suzuki, A.J. Ikushim~t and Y. Okuda, J. Low Temp. Phys. 59, 291(1985). 2. D.O. Edwards and W.F. Saam, in Prog. Low Temp. Phys., D.F. Brewer, ed. (North Holland, Amsterdam 1978) vol. VII A, p.283. 3. E. Mendoza, in Helium Three, J.G. Daunt, ed. (Ohio Univ. Press, Columbus, 1960), p.150. 4. W.F. Saam and J.P. Laheute, Phys. Rev., A4, 1170(1971).
Physica B 194-196 (1994) 873-874 North-Holland
SURFACE TENSION
OF DILUTE SOLUTIONS
O F 4HE I N 3HE
Masaru Suzuki ~, Masayuki Yanaanobe b, Natsuki Ohtani b and Akira Sato b ~Division of Natural Sciences, Univ. of Electro-Communications, Chofu, Tokyo 182, Japan bDepartment of Applied Physics, Univ. of Electro-Communications, Chofu, Tokyo 182, Japan The surface tension of dilute solutions of 4He in 3Ite was measured by means of surface wave resonance in order to examine the effect of adding small amounts of "tHe to pure 3He. \¥e found that compared with pure 3He, the surface tension of the 5.2% 4He solution increases by 6 m d y n / c m at 1K. The increase of the surface tension means that the 4He atoms are excluded from the surface of dilute solutions. The data below 1.6K were compared with Saam's model. Assuming that the 4He behaves like an ideal quasiparticle gas and the 4He quasiparticle surface scattering length is proportional to the amount of surface adsorption, the temperature dependence of the surface tension is well explained.
1. I N T R O D U C T I O N Although little attention has been paid to the free surface of dilute solutions of 4He in 3He, the surface tension of dilute solutions may give us the important information about the 4He behavior in ZHe. The surface tension of dilute solutions of 4He in 3He was measured ill order to examine the effect of adding small amounts of 4He to pure 3He. The surface tension of pure 3He down to 0.3K is considered to consist of two parts, a 3He quasiparticle and a ripplon contribution.[1] In tile case of the dilute solutions, a 4He quasiparticle contribution may add to these two contributions. According to Edwards and Santa's model[2], the difference of the 4He contribution from OK, An43 is obtained as follow, Ao'43 = nakBT(b4a + 1)~T)
(1)
where n4 is the number density, b43 is the 4He quasiparticle surface scattering length, and AT is the thermal wavelength. Here it is assumed that the 4He behaves like all ideal quasiparticle gas with effective mass m~. EXPERIMENTAL Tile surface tension was measured using the surface-wave resonance technique.[1] The dispersion relation of the surface wave is given by 2.
w2 =
(PL -- py )gk + ak 3 PL coth(kdL) + py c o t h ( k d v ) '
(2)
where g is tile gravitaional acceleration, flL and pv are the mass densities of liquid and vapor, and dL and dv are the depths of liquid and vapor, respectively. Surface waves in a cylindrical cavity were excited and detected capacitively. The frequency of the excitation was locked to a resonace frequency of tile surface wave by tile PLL technique. Iu addition to the resonace frequency, the liquid density was determined separately by means of the m e ~ u r e m e n t of dielectric constant. The vapor density was calculated on tile ~ s u m p tion that the dilute solutions are regular solutio,as. 3. R E S U L T S A N D D I S C U S S I O N Figure 1 shows the surface tension of the 5.2% 4He solution. The absolute accuracy of each point is better than 1 mdyn/cm. Compared with pure 3He, the surface tension increases by 6 m d y n / c m at 1K. The surface adsorption of 4He with respect to tile 3He surface, F43 is calculated from
F43
=
-
x4 1 d~ 2,t . . . . kBT 1 - ~7~x4LI - x4) dx4
(3)
where x 4 is tile mole fraction of 4He, and A/t:g is a constant taken to be 1.37K[3]. Since ~da > 0, the surface adsorption of 4IIe is negative, which means that the 4He atoms are excluded from the free surface of dilute solutions. This behavior may come from the difference of zero point ellergy. The surface adsorption is also considered
0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved SSDI 0921-4526(93)E1017-G
874 to be the difference of the Gibbs dividing surface between the 3He and 4He component. This value F43/n4 is estimated to be about -6.5/~ at 1K.
150
Eo 25
:-~-~..
....,. t--
"O
~..
o--ElO0-
10
<175501
",:,-
•
presentdata
~.
0
i
i 1
i %"""
Temperature(K)
2
Figure 2. Comparlsiou between the present data and the calculated curve below 1.6K. A~r3, A~. and Aaaa denote the a Ile quasiparticle, the ripplon and the 41Ie quasiparticle contribution, respectively.
Temperature (K) Figure 1. Surface tension of the 5.2% 4He dilute solution as a flmction of temperature. The solid curve shows the result of pure 3Ite.[1] Next let us consider the temperature dependence. Assuming that the 4He contribution is additive and the 4He behaves like an ideal quasiparticle with effective mass m~ = 4.5m414], we can calculate the 4He contribution by using Eq. (1). Regarding the the scattering length b43 , it is reasonable that the 4He quasiparticle reflects not at the 3He but at the 4He Gibbs dividing surface. The scattering length b43 may be expressed as b43 = F43/n4 -I- b0,
eo
(4)
where b0 is constant. ( In the case of pure 3He in Ref. 1, it was assumed that the 3He quasiparticle scattering length b3 is constant becatme the scattering length is considered to depend on the density profile of the surface and the profile may not change drastically at low temperatures.) The comparision between the experimental data and the calculated curve (b0 = 0.9.~ ) is shown in Fig. 2. The data below 1.6K are in good agreement with the calculated curve.
In conclusion, the surface tension of dilute solutions of 4He in 3He was measured in order to examine the 4He behavior in 3He. We found that the 4He atoms are excluded from the free surface of dilute solutions and the 4Ite contribution to the surface tension was well explained by Edwards and Santa's model. AKNOWLEDGEMENTS This work was partly supported by a Grantin-Aid from the Ministry of Education of Japan. One of the authors (A.S.) is greatly indebted to the Japan Society for the Promotion of Science for Japanese Junior Scientists. REFERENCES
1. M. Iino, hi. Suzuki, A.J. Ikushim~t and Y. Okuda, J. Low Temp. Phys. 59, 291(1985). 2. D.O. Edwards and W.F. Saam, in Prog. Low Temp. Phys., D.F. Brewer, ed. (North Holland, Amsterdam 1978) vol. VII A, p.283. 3. E. Mendoza, in Helium Three, J.G. Daunt, ed. (Ohio Univ. Press, Columbus, 1960), p.150. 4. W.F. Saam and J.P. Laheute, Phys. Rev., A4, 1170(1971).