1D Crossover, Universality and Finite-Size Scaling of the Specific Heat

We report measurements of the specific heat of He-He mixtures near the superfluid transition when confined to channels of 1 μm square cross section. These data test the universality of finite-size scaling as function of He concentration for 1D crossover. The analysis of these data requires that data measured at fixed concentration be converted to a specific heat at constant chemical potential difference φ = μ3 − μ4. This is carried out according to a procedure performed for planar mixtures by Kimball and Gasparini. We find that, in the most self-consistent analysis of the data, the mixtures define a separate scaling locus from that of pure He, both above and below Tλ. An analysis whereby the exponent α is forced to have the same universal value—as opposed to the best-fit value—yields a good collapse of all the data. This is achieved, however, at a cost of self-consistency. These results mirror very closely those obtained for finite-size scaling of confined planar mixtures, i.e. for 2D crossover. Helium-4 becomes superfluid at a temperature Tλ(P, x) which depends both on He concentration x and pressure P . According to universality, the same critical exponents and ratios of critical amplitudes should be realized for the transition at any Tλ(P, x). The same universality should hold for helium uniformly confined. This was tested for the specific heat by Kimball and Gasparini for 2 Dimensional (2D) crossover with films of 48.3 and 986.9 nm [1]. Data for seven different concentrations were used. The conclusion from these data was that 7 different mixtures do determine a universal locus, however, this locus is different from that of pure He. This difficulty could be traced to the fact that the bulk specific heat data, away from Tλ(PSat, 0), yield a different value for the critical exponent α. In this paper we discuss universality of data taken for 1D crossover as realized with channels of 1 μm cross section and 4 mm length. These were formed using a combination of lithography and direct silicon-wafer bonding [2; 3]. Data for both pure He and 2 mixtures were obtained. To analyze the mixture data one must convert the data measured at constant concentration to that at constant chemical potential difference φ = μ3 − μ4. This was done following a procedure outlined in reference [1]. The bulk specific heat, corresponding to the two mixtures for which data were obtained, x = 0.1485, 0.2195, was calculated from existing bulk data [4; 5]. These data were parametrized and interpolated to the desired concentrations. In figure 1 we show the converted data as CPφ for x = 0 and the two concentrations. The behavior of the bulk data for x = 0 and both 1 Present address: Jet Propulsion Laboratory, Pasadena CA 91109 θ C p φ ( J m ol −1 K −1 ) 10 10 10 10 10 10 20 40 60 80 10 0 x 0.0 0.1485 0.2195 Figure 1. Data converted from the measured specific heat at constant He-He concentration to a specific heat at constant chemical potential difference between He and He CPφ. The variable θ is the dimensionless temperature difference from Tλ appropriate for CPφ. The bulk data is shown as lines. concentrations are shown as the lines. The temperature variable is θ = |1 − T/Tλ| along the thermodynamic path of φ = φλ. Finite-size scaling of these data can be tested using the following functional form [1]