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Heat capacity of liquid helium near criticality in mesoscale planar confinement
Journal of Molecular Liquids 127 (2006) 151 – 152 www.elsevier.com/locate/molliq
Heat capacity of liquid helium near criticality in mesoscale planar confinement K.A. Chalyy a,⁎, L.A. Bulavin a , A.V. Chalyi b a
Faculty of Physics, Kiev Taras Shevchenko National University, 6, Glushkov Prosp., 03127 Kiev, Ukraine b Department of Biophysics, National Medical University, 13, Shevchenko Blvd., 01601 Kiev, Ukraine Available online 19 June 2006
Abstract Influence of finite-size effect upon the liquid helium heat capacity and the shift of the transition temperature are theoretically studied for the case of planar confinement. The analytical results are found to be in fair agreement with the number of experiments where the 4He films were ranged in thickness from 48 nm to 57 μm that could be referred to as the mesoscale. The contributions to the shift of transition temperature caused by the gravitation effect and by the finite-size effect are examined. © 2006 Elsevier B.V. All rights reserved. Keywords: Confined helium; Heat capacity; Transition temperature; Mesoscale
Many systems of experimental and theoretical interest, such as thin films, silicon wafers, interfaces, biomembranes, synaptic clefts, etc., have a reduced planar geometry. Experimental studies of such systems that could exhibit size-dependent second order phase transitions are providing principal results for finite-size theory verification. In this paper the properties of confined liquid helium in the vicinity of λ-transition point will be examined in terms of the heat capacity temperature dependence. The system to be considered hereafter has a reduced geometry in form of plane-parallel layer with the typical thickness of a few hundred angstroms and more. Such range of sizes could be associated with the upper border of nanoscale and substantial part of microscale and could be referred to as the mesoscale. Let us consider the geometry of thin liquid systems in the form of a plane-parallel layer D × D × H. The thickness of the layer H is supposed to be much smaller than distance D in x–y directions. We will study the case that corresponds to the situation when the correlation length ξ becomes comparable or even larger than H but still much smaller than D. It is possible to examine the characteristics of the liquid systems with well-
⁎ Corresponding author. E-mail address: [email protected] (K.A. Chalyy). 0167-7322/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2006.03.040
defined reduced geometry (film, cylinder, bar or sphere and cube) in terms of the pair-correlation function G2 and associated correlation length ξ. The pair correlation function G2 for planar geometry was deduced by applying the Helmholtz operator procedure (see, for example, [1]). One can get the expression for G2 in the form: l X 1 G2 ð x; y; zÞ ¼ pH ð1−ð−1Þn Þ 2 n¼0 " # np pffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2 n2 p2 1=2 2 2 z : K0 x þy n þ 2 cos H H
Here K0(u) is the cylindrical McDonald function. Because of a non-exponential shape of G2, the correlation length of density fluctuation could be defined as ξ = (M2)1/2, where M2 denotes the normalized second spatial moment of G2. Thus, in the case of planar geometry the heat capacity Cplan(τ, H ) 1 1 −a pn0 v pn0 v Cplan ðs; H Þf þ1 sþ : ð1Þ H H Here τ = |T − Tc|/Tc is the temperature variable, α = − 0.0127 and ν = 0.6705 are the critical exponents for 4He.
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K.A. Chalyy et al. / Journal of Molecular Liquids 127 (2006) 151–152
Table 1 Experimental data versus theoretical results (according to Eq. (2)) for the temperature shifts of the heat capacity maximum Tm from its bulk value Tλ in the confined liquid 4He Film thickness H, Shift ΔτE , nm Experiment 48.3 107.4 211.3 318.9 503.9 691.8 986.9 19 000 57 000
9.356 × 10− 4 2.860 × 10− 4 1.273 × 10− 4 5.3 × 10− 5 2.920 × 10− 5 1.808 × 10− 5 1.278 × 10− 5 1.2 × 10− 7 2.5 × 10− 8
Shift ΔτT , Theory
Experimental points [Ref.]
9.337 × 10− 4 2.835 × 10− 4 1.033 × 10− 4 5.594 × 10− 5 2.827 × 10− 5 1.762 × 10− 5 1.037 × 10− 5 1.260 × 10− 7 2.447 × 10− 8
F1 [3] F2 [3] F3 [3] F4 [5] F5 [3] F6 [3] F7 [3] F8 [2] F9 [4]
especially exciting to compare the theoretical evaluation ΔτT with the result F9 of the most advanced Confined Helium Experiment (CHEX) [4]. CHEX has performed a highresolution measurement (± 1 nK) of the finite size effect using a sample of liquid helium, confined to a planar geometry with thickness 57 μm aboard of the Space Shuttle. For this size it appears that ΔτT is about 2.1% lower than ΔτE ≈ 2.5·10− 8, that seems to be a good result. It is possible to compare the contributions to the shift of transition temperature caused by the gravity effect (δTg) and by the finite-size effect (δTfs) in terms of direct calculations. Let us introduce the “effect's ratio” Ω(H) = (δTg/δTfs) × 100% that gives an opportunity to check which effect dominates for any given film thickness H: g
mþ1 m
100%;
Let us examine the limiting cases, namely H→∞ and τ = 0. Eq. (1) appears reasonable and leads to the expected expressions:
Xð H Þ ¼
Cplan ðs; HYlÞfs−a ;
where coefficient γ = 1.273 μK/cm [6]. It is assumed [7] that data are unaffected by gravity if |T − Tλ| ≥ 10·δTλ. According to this statement, the confined helium experimental results could be considered as unaltered by the gravity if Ω(H) ≤ 10%. It gives an exact value of the maximum film thickness for earthbound experiments as follows: HL = 50.29 μm. Thus, the results which were obtained with the current theoretical approach are reasonably matched to confined 4He heat capacity experimental data over the wide range of system's sizes from 50 nm up to about 60 μm for planar type of geometry. It is worth noticing that the analysis of cylindrically confined helium [8] gave the comparable results in terms of the agreement with the correspondent experimental data.
which obviously demonstrates a classic bulk behavior, and −a
a
Cplan ðs ¼ 0; H Þfðpn0 Þ v H v ; which shows that the heat capacity remains finite at bulk Tλ (τ = 0) if H b ∞. A new transition temperature Tc(H) indicating the location of the heat capacity maximum in the liquid film system is described by the following formula: Tc⁎(H) = Tc[1 + (πξ0/H)1/ν]− 1. The liquid helium film's experimental data [2– 5] and the current analytical estimations are combined in the Table 1. Here, ΔτE = (Tλ − Tm)/Tm are the experimental temperature shifts of the helium heat capacity maximum Tm from its bulk value Tλ, taking into account that Tλ N Tm. The shifts ΔτT are calculated in accordance with the proposed theoretical expression: DsT ¼ ðpn0 =HÞ
1=m
ð2Þ
Eq. (2) demonstrates an agreement with the finite-size scaling theory predictions: Δτ = aL−1/ν , where a is a constant depending on the geometry. In the case considered above, the linear size of a system L is treated as the film thickness H and, as a result, the expression for the scaling coefficient a reads: a = (πξ0)1/ν . The comparison shows that the theoretical values ΔτT in most cases (F1, F2, F5, F6 and F9) underestimate the shift of the new transition temperature on 1.8% in average. It was
ðpn0 Þ
1=m
Tk
H
References [1] K.A. Chalyy, K. Hamano, A.V. Chalyi, J. Mol. Liq. 92 (2001) 153. [2] J.A. Nissen, T.C.P. Chui, J.A. Lipa, J. Low Temp. Phys. 92 (1993) 353. [3] S. Mehta, M.O. Kimball, F.M. Gasparini, J. Low Temp. Phys. 114 (1999) 467. [4] J.A. Lipa, D.R. Swanson, J.A. Nissen, Z.K. Geng, P.R. Williamson, D.A. Stricker, T.C.P. Chui, U.E. Israelsson, M. Larson, Phys. Rev. Lett. 84 (2000) 4894. [5] M. Diaz-Avila, F.M. Gasparini, M.O. Kimball, J. Low Temp. Phys. 134 (2004) 613. [6] G. Ahlers, Phys. Rev. 171 (1968) 275. [7] P.B. Weichman, A.W. Harter, D.L. Goodstein, Rev. Mod. Phys. 73 (2001) 1. [8] K.A. Chalyy, Low Temp. Phys. 30 (2004) 686.
Heat capacity of liquid helium near criticality in mesoscale planar confinement K.A. Chalyy a,⁎, L.A. Bulavin a , A.V. Chalyi b a
Faculty of Physics, Kiev Taras Shevchenko National University, 6, Glushkov Prosp., 03127 Kiev, Ukraine b Department of Biophysics, National Medical University, 13, Shevchenko Blvd., 01601 Kiev, Ukraine Available online 19 June 2006
Abstract Influence of finite-size effect upon the liquid helium heat capacity and the shift of the transition temperature are theoretically studied for the case of planar confinement. The analytical results are found to be in fair agreement with the number of experiments where the 4He films were ranged in thickness from 48 nm to 57 μm that could be referred to as the mesoscale. The contributions to the shift of transition temperature caused by the gravitation effect and by the finite-size effect are examined. © 2006 Elsevier B.V. All rights reserved. Keywords: Confined helium; Heat capacity; Transition temperature; Mesoscale
Many systems of experimental and theoretical interest, such as thin films, silicon wafers, interfaces, biomembranes, synaptic clefts, etc., have a reduced planar geometry. Experimental studies of such systems that could exhibit size-dependent second order phase transitions are providing principal results for finite-size theory verification. In this paper the properties of confined liquid helium in the vicinity of λ-transition point will be examined in terms of the heat capacity temperature dependence. The system to be considered hereafter has a reduced geometry in form of plane-parallel layer with the typical thickness of a few hundred angstroms and more. Such range of sizes could be associated with the upper border of nanoscale and substantial part of microscale and could be referred to as the mesoscale. Let us consider the geometry of thin liquid systems in the form of a plane-parallel layer D × D × H. The thickness of the layer H is supposed to be much smaller than distance D in x–y directions. We will study the case that corresponds to the situation when the correlation length ξ becomes comparable or even larger than H but still much smaller than D. It is possible to examine the characteristics of the liquid systems with well-
⁎ Corresponding author. E-mail address: [email protected] (K.A. Chalyy). 0167-7322/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2006.03.040
defined reduced geometry (film, cylinder, bar or sphere and cube) in terms of the pair-correlation function G2 and associated correlation length ξ. The pair correlation function G2 for planar geometry was deduced by applying the Helmholtz operator procedure (see, for example, [1]). One can get the expression for G2 in the form: l X 1 G2 ð x; y; zÞ ¼ pH ð1−ð−1Þn Þ 2 n¼0 " # np pffiffiffiffiffiffiffiffiffiffiffiffiffiffi −2 n2 p2 1=2 2 2 z : K0 x þy n þ 2 cos H H
Here K0(u) is the cylindrical McDonald function. Because of a non-exponential shape of G2, the correlation length of density fluctuation could be defined as ξ = (M2)1/2, where M2 denotes the normalized second spatial moment of G2. Thus, in the case of planar geometry the heat capacity Cplan(τ, H ) 1 1 −a pn0 v pn0 v Cplan ðs; H Þf þ1 sþ : ð1Þ H H Here τ = |T − Tc|/Tc is the temperature variable, α = − 0.0127 and ν = 0.6705 are the critical exponents for 4He.
152
K.A. Chalyy et al. / Journal of Molecular Liquids 127 (2006) 151–152
Table 1 Experimental data versus theoretical results (according to Eq. (2)) for the temperature shifts of the heat capacity maximum Tm from its bulk value Tλ in the confined liquid 4He Film thickness H, Shift ΔτE , nm Experiment 48.3 107.4 211.3 318.9 503.9 691.8 986.9 19 000 57 000
9.356 × 10− 4 2.860 × 10− 4 1.273 × 10− 4 5.3 × 10− 5 2.920 × 10− 5 1.808 × 10− 5 1.278 × 10− 5 1.2 × 10− 7 2.5 × 10− 8
Shift ΔτT , Theory
Experimental points [Ref.]
9.337 × 10− 4 2.835 × 10− 4 1.033 × 10− 4 5.594 × 10− 5 2.827 × 10− 5 1.762 × 10− 5 1.037 × 10− 5 1.260 × 10− 7 2.447 × 10− 8
F1 [3] F2 [3] F3 [3] F4 [5] F5 [3] F6 [3] F7 [3] F8 [2] F9 [4]
especially exciting to compare the theoretical evaluation ΔτT with the result F9 of the most advanced Confined Helium Experiment (CHEX) [4]. CHEX has performed a highresolution measurement (± 1 nK) of the finite size effect using a sample of liquid helium, confined to a planar geometry with thickness 57 μm aboard of the Space Shuttle. For this size it appears that ΔτT is about 2.1% lower than ΔτE ≈ 2.5·10− 8, that seems to be a good result. It is possible to compare the contributions to the shift of transition temperature caused by the gravity effect (δTg) and by the finite-size effect (δTfs) in terms of direct calculations. Let us introduce the “effect's ratio” Ω(H) = (δTg/δTfs) × 100% that gives an opportunity to check which effect dominates for any given film thickness H: g
mþ1 m
100%;
Let us examine the limiting cases, namely H→∞ and τ = 0. Eq. (1) appears reasonable and leads to the expected expressions:
Xð H Þ ¼
Cplan ðs; HYlÞfs−a ;
where coefficient γ = 1.273 μK/cm [6]. It is assumed [7] that data are unaffected by gravity if |T − Tλ| ≥ 10·δTλ. According to this statement, the confined helium experimental results could be considered as unaltered by the gravity if Ω(H) ≤ 10%. It gives an exact value of the maximum film thickness for earthbound experiments as follows: HL = 50.29 μm. Thus, the results which were obtained with the current theoretical approach are reasonably matched to confined 4He heat capacity experimental data over the wide range of system's sizes from 50 nm up to about 60 μm for planar type of geometry. It is worth noticing that the analysis of cylindrically confined helium [8] gave the comparable results in terms of the agreement with the correspondent experimental data.
which obviously demonstrates a classic bulk behavior, and −a
a
Cplan ðs ¼ 0; H Þfðpn0 Þ v H v ; which shows that the heat capacity remains finite at bulk Tλ (τ = 0) if H b ∞. A new transition temperature Tc(H) indicating the location of the heat capacity maximum in the liquid film system is described by the following formula: Tc⁎(H) = Tc[1 + (πξ0/H)1/ν]− 1. The liquid helium film's experimental data [2– 5] and the current analytical estimations are combined in the Table 1. Here, ΔτE = (Tλ − Tm)/Tm are the experimental temperature shifts of the helium heat capacity maximum Tm from its bulk value Tλ, taking into account that Tλ N Tm. The shifts ΔτT are calculated in accordance with the proposed theoretical expression: DsT ¼ ðpn0 =HÞ
1=m
ð2Þ
Eq. (2) demonstrates an agreement with the finite-size scaling theory predictions: Δτ = aL−1/ν , where a is a constant depending on the geometry. In the case considered above, the linear size of a system L is treated as the film thickness H and, as a result, the expression for the scaling coefficient a reads: a = (πξ0)1/ν . The comparison shows that the theoretical values ΔτT in most cases (F1, F2, F5, F6 and F9) underestimate the shift of the new transition temperature on 1.8% in average. It was
ðpn0 Þ
1=m
Tk
H
References [1] K.A. Chalyy, K. Hamano, A.V. Chalyi, J. Mol. Liq. 92 (2001) 153. [2] J.A. Nissen, T.C.P. Chui, J.A. Lipa, J. Low Temp. Phys. 92 (1993) 353. [3] S. Mehta, M.O. Kimball, F.M. Gasparini, J. Low Temp. Phys. 114 (1999) 467. [4] J.A. Lipa, D.R. Swanson, J.A. Nissen, Z.K. Geng, P.R. Williamson, D.A. Stricker, T.C.P. Chui, U.E. Israelsson, M. Larson, Phys. Rev. Lett. 84 (2000) 4894. [5] M. Diaz-Avila, F.M. Gasparini, M.O. Kimball, J. Low Temp. Phys. 134 (2004) 613. [6] G. Ahlers, Phys. Rev. 171 (1968) 275. [7] P.B. Weichman, A.W. Harter, D.L. Goodstein, Rev. Mod. Phys. 73 (2001) 1. [8] K.A. Chalyy, Low Temp. Phys. 30 (2004) 686.