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The observation of viscous waves in a He I film
THE OBSERVATION
3 June 1985
PHYSICS LETTERS
Volume 109A, number 6
OF VISCOUS
WAVES IN A He I FILM
D.S. SPENCER, M.J. LEA and P. FOZOONI Department of Physics, Bedford College, University of London, Regent’s Park, London NW1 4NS, UK
Received 14 January 1985; revised manuscript received 18 March 1985; accepted for publication 29 March 1985
The transverse acoustic impedance of a 4He film has been measured at 3.11 K and 20.5 MHz and shows the cccurrence of standing shear waves in the film. This is in good agreement with calculations using transmission line theory and gives a new method of measuring the thickness of liquid films.
In fluids, the transverse mode of sound propagation is a heavily damped viscous wave. Measurements of the transverse acoustic impedance, 2 = R - iX therefore provide a useful probe of the fluid’s properties on a length scale comparable with the penetration depth of the wave (8 = 20 nm in liquid 4He). Recently, measurements of Z have been made for bulk liquid 4He [ 121, liquid 3He [3] and for liquid 3He4He mixtures [4]. We present here measurements of the transverse acoustic impedance at 20.5 MHz for films of 4He above the X-point. The impedance was found from changes in the quality factor Q and resonant frequency f of a quartz crystal resonator vibrating in shear and covered with the helium film. The effect of the helium is to decrease the Q and resonant frequency of the crystal by [l]
VAPOUI?
t---------i
(2)
Fig. 1. The profde of the 4He film on the crystal, showing the local&d atomic layer and the region of enhanced density liquid caused by van der WaaJs forces. II is the thickness of the liquid film.
where n is the harmonic number of the resonance, is the acoustic impedance of the quartz, u is the adsorbed solid mass per unit area, and R and X are the real and imaginary parts of the impedance of the liquid film. The profile of the helium film on the crystal is shown in fig. 1. Van der Waals forces cause a localised atomic layer to be adsorbed on to the crystal, giving, by the first term of eq. (2), a temperature indepen-
dent frequency shift relative to the vacuum resonant frequency. The van der Waals forces also enhance the density of the liquid close to the crystal; since the density enhanced region is small, this too produces a shift in the crystal’s resonant frequency with negligible effect on Q [ 11. As the thickness of the film, II, is increased from u = 0, this mass loading effect is essen-
A(Q-l) = 4R/nnR,
,
Af = -4f 2u/nRq - 2fXlnlrRq ,
(1)
R, (%R,X)
0.375-9601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
295
Volume 109A, number 6
PHYSICS LETTERS
3 June 198.5
tially complete before the viscous losses of the film, given by eq. (1) and the second term of eq. (2), decrease both f and Q. The film has the properties of the bulk liquid and has its free surface in contact with 4He vapour with an impedance Zv. For a homogeneous bulk fluid, Z = (1 - i)(vpti/2)lj2
= (1 - i)r ,
(3)
with penetration depth 6 = (2~/po)l/~, where 1) is the viscosity, p is the density and w = 2nf. For a film
4 Fig. 2. The real and imaginary parts of the acoustic impedance of a helium film, thickness a, in a vacuum. The bulk Liquid has an impedance (1 - i)r and 6 is the penetration depth.
IMPEDANCE
VAPOUR 0
I
0
so
IMPEDANCE
I
I
100 150 R (kg m-2. -1 1
I 200
250
Fig. 3. The measured acoustic impedance of a 4He film at 3.11 K as u/6, the ratio of the film thickness to the penetration depth, is changed. The curve calculated from transmission line theory, eq. (4), is shown for films of bulk impedance 172 kg mm2s-t with a vapour impedance of zero (dashed line) and of 16 kg m* s-l (solid line).
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Volume 109A, number 6
PHYSICS LETTERS
of thickness (I, from analogy with transmission line theory, the impedance measured by the crystal is [S] 2 = (1 - i)r tanh(J/ t ru) ,
(4)
where y = (1 - i)/S is the propagation constant of the viscous wave, and tanh I/J= Z,/( 1 - i)r is the ratio of the vapour impedance to the liquid impedance. For a/6 4 1, the impedance measured is that of the vapour, Z,; as a increases, tanh(J/ t ~a) -+ 1 and Z tends to the bulk value. For a film in a vacuum, eq. (4) becomes Z= (1 - i)rtanh[(l
- i)a/s]
.
(5)
As a increases, X and R increase and pass through a series of heavily damped maxima and minima corresponding to standing wave resonances in the film, as shown in fig. 2. Fig. 3 shows X versus R as a/s varies for films of bulk impedance r = 172 kg m-2,-1, both in a vacuum (Zv = 0) and for a vapour in contact with the film (Zv > 0). The locus of Z is a spiral starting at Z, for a/s < 1 and rapidly converging to Z = (1 - i)r. The quartz crystal was mounted horizontally in a copper cell, a tube attached to the cell being partially filled with liquid helium. The thickness of the film on the crystal at a temperature T is determined by [6] Ak/a3 = -kT ln(P/Po)
,
(6)
where the van der Waals constant A is 1.986 X 1O-27 K mm3 for our crystals [ 11, k is the Boltzmann constant, PO is the saturated vapour pressure of the film and P is the actual pressure on the crystal. For an isothermal cell, the pressure P, and hence the film thickness will be determined by the crystal’s height h above the liquid level and Akla3 = mgh
(7)
giving, for h = 5.8 cm, a = 19 nm. The film thickness was swept by creating a small temperature difference (-0.5 mK) between the crystal and the liquid in the tube. This changed the SVP of the film on the crystal and hence the film thickness could be swept from 1.5 nm to around 60 nm. Fig. 3 shows data for Xversus R at 3.11 K. The start of the spiral is the impedance of the vapour in contact with the film, measured when u/S 4 1, with 6 = 19 nm. Assuming the vapour is in the hydrodynamic region, Z, = (1 - i)(q,p,o/2)1/2, and taking
3 June 1985
the vapour viscosity to be qv = 7.9 PP [7], the real part of the vapour’s impedance was calculated to be 17 kg mm2se1. For thick films, a/S > 1, the measured film impedance tends towards that of the bulk liquid. Previous data [8], taken in the bulk liquid under SW gives the real part of the impedance to be 172 kg m -2s-1. The fit of eq. (2) to the data is shown in fig. 3, where the real part of the vapour impedance is taken to be 16 kg rnm2sm1, the bulk impedance is 172 kg m-2 s-l and the X = 0 line is chosen so that X = R for a/& % 1, thus allowing for the effects of mass loading. The R = 0 line here corresponds to the temperature independent value (T < 0.6 K) of Q-l where the superfluid helium exerts no viscous forces on the crystal. As shown in fig. 3, the transverse acoustic impedance of a helium film is described well by transmission line theory, giving a method for determining a which should be applicable to fluids other than helium. This experiment is the transverse analogue at 20 MHz of that of Sabisky and Anderson [9] who used longitudinal waves at around 30 GHz to detect standing wave resonances in a 4He film; for a transverse wave however, the standing waves are much more heavily damped. We would like to thank Professor E.R. Dobbs for his encouragement and support, A.K. Betts, F. Greenough and A. King for technical assistance and the SERC (UK) for a Research Grant and a Studentship (for D.S.S). References [l] M.J. Lea, P. Fozooni and P.W. Retz, J. Low Temp. Phys. 54 (1984) 303. [2] M.J. Lea and P. Fozooni, J. Low Temp. Phys. 56 (1984) 25. [ 31 F.P. Millikan, R.W. Richardson and S.J. Williamson, J. Low Temp. Phys. 45 (1981) 409. [4] M.J. Lea and P.W. Retz, Physica 107B (1981) 225. [S] MJ. Lea and P. Fozooni, Ultrasonics, to be published. [6 ] D.F. Brewer, in: The physics of liquid and solid helium, VoL II, eds. K.H. Bennemann and J.B. Ketterson (Wiley, New York, 1978) p. 573. [7] D.S. Betts, Cryogenics 16 (1976) 3. [ 81 MJ. Lea and P. Fozooni, unpublished. [9] E.S. Sabisky and C.H. Anderson, Phys. Rev. A7 (1973) 790.
297
3 June 1985
PHYSICS LETTERS
Volume 109A, number 6
OF VISCOUS
WAVES IN A He I FILM
D.S. SPENCER, M.J. LEA and P. FOZOONI Department of Physics, Bedford College, University of London, Regent’s Park, London NW1 4NS, UK
Received 14 January 1985; revised manuscript received 18 March 1985; accepted for publication 29 March 1985
The transverse acoustic impedance of a 4He film has been measured at 3.11 K and 20.5 MHz and shows the cccurrence of standing shear waves in the film. This is in good agreement with calculations using transmission line theory and gives a new method of measuring the thickness of liquid films.
In fluids, the transverse mode of sound propagation is a heavily damped viscous wave. Measurements of the transverse acoustic impedance, 2 = R - iX therefore provide a useful probe of the fluid’s properties on a length scale comparable with the penetration depth of the wave (8 = 20 nm in liquid 4He). Recently, measurements of Z have been made for bulk liquid 4He [ 121, liquid 3He [3] and for liquid 3He4He mixtures [4]. We present here measurements of the transverse acoustic impedance at 20.5 MHz for films of 4He above the X-point. The impedance was found from changes in the quality factor Q and resonant frequency f of a quartz crystal resonator vibrating in shear and covered with the helium film. The effect of the helium is to decrease the Q and resonant frequency of the crystal by [l]
VAPOUI?
t---------i
(2)
Fig. 1. The profde of the 4He film on the crystal, showing the local&d atomic layer and the region of enhanced density liquid caused by van der WaaJs forces. II is the thickness of the liquid film.
where n is the harmonic number of the resonance, is the acoustic impedance of the quartz, u is the adsorbed solid mass per unit area, and R and X are the real and imaginary parts of the impedance of the liquid film. The profile of the helium film on the crystal is shown in fig. 1. Van der Waals forces cause a localised atomic layer to be adsorbed on to the crystal, giving, by the first term of eq. (2), a temperature indepen-
dent frequency shift relative to the vacuum resonant frequency. The van der Waals forces also enhance the density of the liquid close to the crystal; since the density enhanced region is small, this too produces a shift in the crystal’s resonant frequency with negligible effect on Q [ 11. As the thickness of the film, II, is increased from u = 0, this mass loading effect is essen-
A(Q-l) = 4R/nnR,
,
Af = -4f 2u/nRq - 2fXlnlrRq ,
(1)
R, (%R,X)
0.375-9601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
295
Volume 109A, number 6
PHYSICS LETTERS
3 June 198.5
tially complete before the viscous losses of the film, given by eq. (1) and the second term of eq. (2), decrease both f and Q. The film has the properties of the bulk liquid and has its free surface in contact with 4He vapour with an impedance Zv. For a homogeneous bulk fluid, Z = (1 - i)(vpti/2)lj2
= (1 - i)r ,
(3)
with penetration depth 6 = (2~/po)l/~, where 1) is the viscosity, p is the density and w = 2nf. For a film
4 Fig. 2. The real and imaginary parts of the acoustic impedance of a helium film, thickness a, in a vacuum. The bulk Liquid has an impedance (1 - i)r and 6 is the penetration depth.
IMPEDANCE
VAPOUR 0
I
0
so
IMPEDANCE
I
I
100 150 R (kg m-2. -1 1
I 200
250
Fig. 3. The measured acoustic impedance of a 4He film at 3.11 K as u/6, the ratio of the film thickness to the penetration depth, is changed. The curve calculated from transmission line theory, eq. (4), is shown for films of bulk impedance 172 kg mm2s-t with a vapour impedance of zero (dashed line) and of 16 kg m* s-l (solid line).
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Volume 109A, number 6
PHYSICS LETTERS
of thickness (I, from analogy with transmission line theory, the impedance measured by the crystal is [S] 2 = (1 - i)r tanh(J/ t ru) ,
(4)
where y = (1 - i)/S is the propagation constant of the viscous wave, and tanh I/J= Z,/( 1 - i)r is the ratio of the vapour impedance to the liquid impedance. For a/6 4 1, the impedance measured is that of the vapour, Z,; as a increases, tanh(J/ t ~a) -+ 1 and Z tends to the bulk value. For a film in a vacuum, eq. (4) becomes Z= (1 - i)rtanh[(l
- i)a/s]
.
(5)
As a increases, X and R increase and pass through a series of heavily damped maxima and minima corresponding to standing wave resonances in the film, as shown in fig. 2. Fig. 3 shows X versus R as a/s varies for films of bulk impedance r = 172 kg m-2,-1, both in a vacuum (Zv = 0) and for a vapour in contact with the film (Zv > 0). The locus of Z is a spiral starting at Z, for a/s < 1 and rapidly converging to Z = (1 - i)r. The quartz crystal was mounted horizontally in a copper cell, a tube attached to the cell being partially filled with liquid helium. The thickness of the film on the crystal at a temperature T is determined by [6] Ak/a3 = -kT ln(P/Po)
,
(6)
where the van der Waals constant A is 1.986 X 1O-27 K mm3 for our crystals [ 11, k is the Boltzmann constant, PO is the saturated vapour pressure of the film and P is the actual pressure on the crystal. For an isothermal cell, the pressure P, and hence the film thickness will be determined by the crystal’s height h above the liquid level and Akla3 = mgh
(7)
giving, for h = 5.8 cm, a = 19 nm. The film thickness was swept by creating a small temperature difference (-0.5 mK) between the crystal and the liquid in the tube. This changed the SVP of the film on the crystal and hence the film thickness could be swept from 1.5 nm to around 60 nm. Fig. 3 shows data for Xversus R at 3.11 K. The start of the spiral is the impedance of the vapour in contact with the film, measured when u/S 4 1, with 6 = 19 nm. Assuming the vapour is in the hydrodynamic region, Z, = (1 - i)(q,p,o/2)1/2, and taking
3 June 1985
the vapour viscosity to be qv = 7.9 PP [7], the real part of the vapour’s impedance was calculated to be 17 kg mm2se1. For thick films, a/S > 1, the measured film impedance tends towards that of the bulk liquid. Previous data [8], taken in the bulk liquid under SW gives the real part of the impedance to be 172 kg m -2s-1. The fit of eq. (2) to the data is shown in fig. 3, where the real part of the vapour impedance is taken to be 16 kg rnm2sm1, the bulk impedance is 172 kg m-2 s-l and the X = 0 line is chosen so that X = R for a/& % 1, thus allowing for the effects of mass loading. The R = 0 line here corresponds to the temperature independent value (T < 0.6 K) of Q-l where the superfluid helium exerts no viscous forces on the crystal. As shown in fig. 3, the transverse acoustic impedance of a helium film is described well by transmission line theory, giving a method for determining a which should be applicable to fluids other than helium. This experiment is the transverse analogue at 20 MHz of that of Sabisky and Anderson [9] who used longitudinal waves at around 30 GHz to detect standing wave resonances in a 4He film; for a transverse wave however, the standing waves are much more heavily damped. We would like to thank Professor E.R. Dobbs for his encouragement and support, A.K. Betts, F. Greenough and A. King for technical assistance and the SERC (UK) for a Research Grant and a Studentship (for D.S.S). References [l] M.J. Lea, P. Fozooni and P.W. Retz, J. Low Temp. Phys. 54 (1984) 303. [2] M.J. Lea and P. Fozooni, J. Low Temp. Phys. 56 (1984) 25. [ 31 F.P. Millikan, R.W. Richardson and S.J. Williamson, J. Low Temp. Phys. 45 (1981) 409. [4] M.J. Lea and P.W. Retz, Physica 107B (1981) 225. [S] MJ. Lea and P. Fozooni, Ultrasonics, to be published. [6 ] D.F. Brewer, in: The physics of liquid and solid helium, VoL II, eds. K.H. Bennemann and J.B. Ketterson (Wiley, New York, 1978) p. 573. [7] D.S. Betts, Cryogenics 16 (1976) 3. [ 81 MJ. Lea and P. Fozooni, unpublished. [9] E.S. Sabisky and C.H. Anderson, Phys. Rev. A7 (1973) 790.
297