- Email: [email protected]
Quantum nucleation on a wall
Physica B 329–333 (2003) 384–385
Quantum nucleation on a wall Seishi Kondo 1-29-12, Chofugaoka, Chofu-shi, Tokyo 182-0021, Japan
Abstract Quantum nucleation of solid 4 He on a wall is discussed on a phenomenological model which describes incomplete wetting. While this model behaves as well-known thin wall nucleation model near the equilibrium point, it describes the nucleation as softening of ripplon on the adsorbed film near the critical point. Since the corresponding experimental researches were done near the critical point, the nucleation rate W of the model was estimated in the the manner developed by Lifshitz and Kagan, near the critical point. The result indicates a pressure dependence ln W pðPc PÞ5=8 ; where Pc is a critical pressure. The assumption that the origin of metastability is the structural mismatch between the adsorbed film and the bulk could allow a calculation of Pc : r 2003 Elsevier Science B.V. All rights reserved. Keywords: Quantum nucleation; Solid 4 He; Wetting
Quantum solid nucleation from superfluid 4 He is an ideal phenomenon for macroscopic quantum nucleation (MQN) studies, because it is not limited by dissipation due to large solid/liquid mobility. The experimental studies [1,2] of the system have reported temperature independent nucleation in the copper cell below 0.5–1 K: When P ¼ 3–30 mbars above the melting pressure Pm ; the nucleation rate W indicates the pressure dependence ln W pðPc PÞg ;
ð1Þ
where Pc is a critical pressure which corresponds to the instability point, and gE1: Chavanne et al. [3] have also observed temperature independent nucleation below 0:3 K using a focused acoustic wave and obtained 2–3 orders of magnitude large overpressure. As for theoretical aspects, Balibar et al. [4] pointed out that a remnant crystal in the cavity of the wall has too high energy barrier if the value of Pc of this model were fitted on the experimental one. Uwaha [5] discussed the instability of film on the wall two decades before. But analysis was done near P ¼ Pm and the discussion near P ¼ Pc has never been tried. The aim of this paper is to try it. Dash [6] and Peierls [7] developed the phenomenological theory to treat the instability of flat film. They introduced Helmholtz free energy per unit area F as E-mail address: [email protected] (Seishi Kondo).
continual function of film thickness d and classified adsorption into three classes in the form of F: In this paper, we construct our model based on the class-II category which corresponds to the incomplete wetting. Here we define the potential energy of nucleation: Z U ¼ dSf12 sls ðr> dÞ2 þ fðd; d1 Þg; ð2Þ where r> ¼ ðq=qx; q=qyÞ; sls is the solid/liquid interfacial tension, and S is the area of flat wall plane. f is the thermodynamic potential density constructed by F: On P ¼ Pm ; dU ¼ 0 reproduces the Young relation and on slight overpressure, U becomes the potential energy of nucleation discussed by Uwaha (see Fig. 1(a)). The above two points are arguments that this model is reliable generic one. We analyze this model near Pc ; (see Fig. 1(c)) in the manner developed by Lifshitz and Kagan [8]. Using the Gibbs–Duhem relation, the dimensionless parameter that decides the stability of the system becomes as follows: x0 pðPc PÞ1=2 :
ð3Þ
At high temperature region, the Arrhenius law gives the nucleation rate. Therefore the dimensional analysis of U determines the pressure dependence of ln W : U ¼ U0 Jfwg;
0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)02134-8
ð4Þ
Seishi Kondo / Physica B 329–333 (2003) 384–385
φ a b d1
d c
Fig. 1. Thermodynamic potential density classified in class-II against film thickness d: d1 denotes the thickness of metastable state. Curve (a) and (c) correspond to P ¼ Pm and P ¼ Pc ; respectively. Curve (b) is the intermediate case between (a) and (c).
to consider explicit form of F: Since the van der Waals potential makes the film stable, the mechanism to make film unstable is needed. The lattice spacing of adsorbed film on a wall may be somehow different from the bulk solid, because atomic structure of wall affects the film. Huse [9] considered the structural mismatch between the adsorbed film and bulk solid, and introduced film elastic energy parallel to the plane in phenomenological manner to discuss incomplete wetting. Applying his theory to ours, we rewrite F: ! G dm 1 dm 2 F ¼ sN þ 2 þ dm 2 d d þ ðrs rl Þmm d;
Jfwg ¼
Z
ln W ¼
( ) 2 1 q 2 w þw ð1 wÞ ; d x 2 qx 2
U0 Jfwg px20 pðPc PÞ1 ; kB T
‘ g ¼ 1:
385
ð9Þ
ð5Þ
where sN ¼ sws þ sls ; dm is film thickness on Pm ; and G is the strength of the van der Waals potential. Using the above F and the Young relation,
ð6Þ
sls ð1 cos ycy Þ ¼
Here, U0 is the typical energy scale of the system and w is a dimensionless variable instead of d; and Jfwg is the dimensionless potential. At low temperature, thermal process of nucleation is suppressed and quantum process becomes dominant. Next, we introduce the inertial term K to discuss the quantum process. The flow around the nucleus on the film leads to Z Z ’ dðr ’ 0Þ dðrÞ m K¼ reff dS dS 0 : ð7Þ 4p jr r0 j Harmonic approximation of the Lagrangian L ¼ K U gives the ripplon dispersion relation o2 ¼ ak; where a is a constant which becomes zero at Pc : From this viewpoint one may say that this nucleation is the softening of ripplon at the solid/liquid interface. If the dimensionless effective Planck constant _eff determined by U and K is small enough for WKB approximation, sðtÞ 5 ‘ g¼ : ð8Þ ln W ¼ pðPc PÞ5=8 ; _eff 8 Here, sðtÞ is dimensionless action of inverted potential U: As the above mentioned, the values of g in each process are not so far from experimental results. The experimental determination of g requires measurements in a much larger pressure range, while the pressure range was always small. It follows from what has been said that experimental values must have large deviation and only the orders of magnitude of g is significant. Thus, we conclude that the values of ours are consistent with the experimental ones in view of orders of magnitude. Although the above discussion is extremely general, it could not lead to the value of Pc : To estimate it, we have
G ; 2 2dm
ð10Þ
here, ycy is Young’s static contact angle. Moreover, Pc Pm ¼
4 G rL 3 : 27 dm
ð11Þ
Eqs. (10) and (11) lead to the conclusion that Pc is increasing with ycy : These relations (10) and (11) are important to predict the value of Pc but I left the problem untouched. The author thanks Dr. T. Minoguchi and Dr. Y. Sasaki for various helpful suggestions, and Dr. K.M. Ohtsuki for providing computers. This research was supported by the Ministry of Education, Science, and Sports and Culture, Grant-in-Aid for Scientific Research (C),11640337, 1999.
References [1] J.P. Ruutu, P.J. Hakonen, J.S. Penttila, A.V. Saramaki, E.B. Sonin, Phys. Rev. Lett. 77 (1996) 2514. [2] Y. Sasaki, T. Mizusaki, J. Low Temp. Phys. 110 (1998) 49. [3] X. Chavanne, S. Balibar, F. Caupin, J. Low Temp. Phys. 125 (2001) 155. [4] S. Balibar, T. Mizusaki, Y. Sasaki, J. Low Temp. Phys. 120 (2000) 293. [5] M. Uwaha, J. Low Temp. Phys. 52 (1983) 15. [6] J.G. Dash, Phys. Rev. B 15 (1977) 3136. [7] R. Peierls, Phys. Rev. B 18 (1978) 2013. [8] I.M. Lifshitz, Yu. Kagan, Zh. Eksp. Teor. Fiz. 62 (1972) 385 [Sov. Phys. JETP 35 (1972) 206]. [9] D.A. Huse, Phys. Rev. B 29 (1984) 6985.
Quantum nucleation on a wall Seishi Kondo 1-29-12, Chofugaoka, Chofu-shi, Tokyo 182-0021, Japan
Abstract Quantum nucleation of solid 4 He on a wall is discussed on a phenomenological model which describes incomplete wetting. While this model behaves as well-known thin wall nucleation model near the equilibrium point, it describes the nucleation as softening of ripplon on the adsorbed film near the critical point. Since the corresponding experimental researches were done near the critical point, the nucleation rate W of the model was estimated in the the manner developed by Lifshitz and Kagan, near the critical point. The result indicates a pressure dependence ln W pðPc PÞ5=8 ; where Pc is a critical pressure. The assumption that the origin of metastability is the structural mismatch between the adsorbed film and the bulk could allow a calculation of Pc : r 2003 Elsevier Science B.V. All rights reserved. Keywords: Quantum nucleation; Solid 4 He; Wetting
Quantum solid nucleation from superfluid 4 He is an ideal phenomenon for macroscopic quantum nucleation (MQN) studies, because it is not limited by dissipation due to large solid/liquid mobility. The experimental studies [1,2] of the system have reported temperature independent nucleation in the copper cell below 0.5–1 K: When P ¼ 3–30 mbars above the melting pressure Pm ; the nucleation rate W indicates the pressure dependence ln W pðPc PÞg ;
ð1Þ
where Pc is a critical pressure which corresponds to the instability point, and gE1: Chavanne et al. [3] have also observed temperature independent nucleation below 0:3 K using a focused acoustic wave and obtained 2–3 orders of magnitude large overpressure. As for theoretical aspects, Balibar et al. [4] pointed out that a remnant crystal in the cavity of the wall has too high energy barrier if the value of Pc of this model were fitted on the experimental one. Uwaha [5] discussed the instability of film on the wall two decades before. But analysis was done near P ¼ Pm and the discussion near P ¼ Pc has never been tried. The aim of this paper is to try it. Dash [6] and Peierls [7] developed the phenomenological theory to treat the instability of flat film. They introduced Helmholtz free energy per unit area F as E-mail address: [email protected] (Seishi Kondo).
continual function of film thickness d and classified adsorption into three classes in the form of F: In this paper, we construct our model based on the class-II category which corresponds to the incomplete wetting. Here we define the potential energy of nucleation: Z U ¼ dSf12 sls ðr> dÞ2 þ fðd; d1 Þg; ð2Þ where r> ¼ ðq=qx; q=qyÞ; sls is the solid/liquid interfacial tension, and S is the area of flat wall plane. f is the thermodynamic potential density constructed by F: On P ¼ Pm ; dU ¼ 0 reproduces the Young relation and on slight overpressure, U becomes the potential energy of nucleation discussed by Uwaha (see Fig. 1(a)). The above two points are arguments that this model is reliable generic one. We analyze this model near Pc ; (see Fig. 1(c)) in the manner developed by Lifshitz and Kagan [8]. Using the Gibbs–Duhem relation, the dimensionless parameter that decides the stability of the system becomes as follows: x0 pðPc PÞ1=2 :
ð3Þ
At high temperature region, the Arrhenius law gives the nucleation rate. Therefore the dimensional analysis of U determines the pressure dependence of ln W : U ¼ U0 Jfwg;
0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)02134-8
ð4Þ
Seishi Kondo / Physica B 329–333 (2003) 384–385
φ a b d1
d c
Fig. 1. Thermodynamic potential density classified in class-II against film thickness d: d1 denotes the thickness of metastable state. Curve (a) and (c) correspond to P ¼ Pm and P ¼ Pc ; respectively. Curve (b) is the intermediate case between (a) and (c).
to consider explicit form of F: Since the van der Waals potential makes the film stable, the mechanism to make film unstable is needed. The lattice spacing of adsorbed film on a wall may be somehow different from the bulk solid, because atomic structure of wall affects the film. Huse [9] considered the structural mismatch between the adsorbed film and bulk solid, and introduced film elastic energy parallel to the plane in phenomenological manner to discuss incomplete wetting. Applying his theory to ours, we rewrite F: ! G dm 1 dm 2 F ¼ sN þ 2 þ dm 2 d d þ ðrs rl Þmm d;
Jfwg ¼
Z
ln W ¼
( ) 2 1 q 2 w þw ð1 wÞ ; d x 2 qx 2
U0 Jfwg px20 pðPc PÞ1 ; kB T
‘ g ¼ 1:
385
ð9Þ
ð5Þ
where sN ¼ sws þ sls ; dm is film thickness on Pm ; and G is the strength of the van der Waals potential. Using the above F and the Young relation,
ð6Þ
sls ð1 cos ycy Þ ¼
Here, U0 is the typical energy scale of the system and w is a dimensionless variable instead of d; and Jfwg is the dimensionless potential. At low temperature, thermal process of nucleation is suppressed and quantum process becomes dominant. Next, we introduce the inertial term K to discuss the quantum process. The flow around the nucleus on the film leads to Z Z ’ dðr ’ 0Þ dðrÞ m K¼ reff dS dS 0 : ð7Þ 4p jr r0 j Harmonic approximation of the Lagrangian L ¼ K U gives the ripplon dispersion relation o2 ¼ ak; where a is a constant which becomes zero at Pc : From this viewpoint one may say that this nucleation is the softening of ripplon at the solid/liquid interface. If the dimensionless effective Planck constant _eff determined by U and K is small enough for WKB approximation, sðtÞ 5 ‘ g¼ : ð8Þ ln W ¼ pðPc PÞ5=8 ; _eff 8 Here, sðtÞ is dimensionless action of inverted potential U: As the above mentioned, the values of g in each process are not so far from experimental results. The experimental determination of g requires measurements in a much larger pressure range, while the pressure range was always small. It follows from what has been said that experimental values must have large deviation and only the orders of magnitude of g is significant. Thus, we conclude that the values of ours are consistent with the experimental ones in view of orders of magnitude. Although the above discussion is extremely general, it could not lead to the value of Pc : To estimate it, we have
G ; 2 2dm
ð10Þ
here, ycy is Young’s static contact angle. Moreover, Pc Pm ¼
4 G rL 3 : 27 dm
ð11Þ
Eqs. (10) and (11) lead to the conclusion that Pc is increasing with ycy : These relations (10) and (11) are important to predict the value of Pc but I left the problem untouched. The author thanks Dr. T. Minoguchi and Dr. Y. Sasaki for various helpful suggestions, and Dr. K.M. Ohtsuki for providing computers. This research was supported by the Ministry of Education, Science, and Sports and Culture, Grant-in-Aid for Scientific Research (C),11640337, 1999.
References [1] J.P. Ruutu, P.J. Hakonen, J.S. Penttila, A.V. Saramaki, E.B. Sonin, Phys. Rev. Lett. 77 (1996) 2514. [2] Y. Sasaki, T. Mizusaki, J. Low Temp. Phys. 110 (1998) 49. [3] X. Chavanne, S. Balibar, F. Caupin, J. Low Temp. Phys. 125 (2001) 155. [4] S. Balibar, T. Mizusaki, Y. Sasaki, J. Low Temp. Phys. 120 (2000) 293. [5] M. Uwaha, J. Low Temp. Phys. 52 (1983) 15. [6] J.G. Dash, Phys. Rev. B 15 (1977) 3136. [7] R. Peierls, Phys. Rev. B 18 (1978) 2013. [8] I.M. Lifshitz, Yu. Kagan, Zh. Eksp. Teor. Fiz. 62 (1972) 385 [Sov. Phys. JETP 35 (1972) 206]. [9] D.A. Huse, Phys. Rev. B 29 (1984) 6985.