Francis M. Gasparini

Superfluid Transition of 4He for Two-Dimensional Crossover, Heat Capacity, and Finite-Size Scaling

We report heat capacity measurements of confined films of4He. These studies were undertaken to test predictions of correlation-length scaling. They are the first measurements for completely confined films over a range of confinements, and represent a geometry where criticality changes from 3-dimensions (3D) to 2D. The finite system is realized with a4He film confined between two, 2″ diameter, silicon wafers, which are separated by a small gap. A new technique was developed to bond these wafers at a uniform separation. The gap size, which determines the film thickness, ranges from 0.05 to 0.7 μm in the present work, and has better than 1% uniformity. The bonded cells are used to conduct high precision heat capacity measurements using a modified ac technique. This involves oscillating the sample temperature, as in conventional ac calorimetry, but with simultaneous dc regulation of the average temperature. The data are analyzed using a modified Sullivan–Seidel equation, which takes into account in an empirical way the finite conductivity of the cell. This procedure yields heat capacity data with good absolute accuracy and high resolution. Scaling analysis of the data both above and below the bulk transition temperature shows collapse onto universal curves determined only by the ratio of the correlation length to the confinement size. This is true everywhere except near the heat capacity maximum. Here, and into the superfluid side there is lack of scaling which might be associated with 2D crossover. We compare this result with calculations of scaling functions and find that these tend to underestimate the effect of confinement. Comparison with earlier results for cylindrical confinement shows differences which are most striking in the region of the specific heat maximum. The cylindrical and planar confinement data follow similar trends above the superfluid transition of bulk helium. Below the transition, however, the present data show much more structure. Fits of the scaled planar data above the transition to an empirical scaling function yield a correlation length exponent of νeff=0.674±0.001.

Si wafers uniformly spaced; bonding and diagnostics

A new Si‐SiO2 bonding process has been developed to achieve a uniform spacing between two silicon wafers of 2 and 3 in. diam. Spacings between 0.1 and 3.9 μm have been obtained so far. Hydrostatic pressure is used to force the two wafers into intimate contact at points where bonding is desired. The bonding is performed at a temperature of ∼1150 °C. The uniformity of bonding and spacing between the wafers is checked by a Fabry–Perot interferometer technique at room temperature and by measurements of superfluid density of He II at low temperature. These results are compared with ellipsometer and stylus measurements of the oxide thickness which is designed to govern the wafers’ spacing. We find that these different techniques yield consistent results.

Specific heat near the superfluid transition of a 0.9869 μm 4He film

We report measurements of the specific heat of 4He near the superfluid transition while confined between silicon wafers at 0.9869 μm separation. These data are analyzed to check on the behavior expected from correlation-length scaling. Comparison is also made with other data for planar confinement, as well as data for cylindrical confinement. These represent different lower-dimensional crossovers. We find that the present data scale very well above the bulk transition temperature, and in the region immediately below it. Near the specific heat maximum however, the data for planar confinement do not collapse on a universal curve. We compare these results with specific theoretical scaling functions. In particular we find that on the normal side, and for large enough values of the scaling variable, one can describe the data well using the concept of the surface specific heat. The locus of the data in this region agrees well with the most recent theoretical calculations.

Chapter 1: Critical Behavior and Scaling of Confined 4He

Publisher Summary This chapter discusses the expected scaling behavior of helium confined in a uniform, well-defined geometry. By way of example two models are discussed—the two-dimensional (2D) Ising model and the ideal Bose gas model—to see how scaling is actually realized. A description of various experiments designed to test scaling theory, both in the case of films with a free surface and in the case of complete confinement is given. These experiments to other studies of confined helium in powders and porous glasses are compared. With liquid helium, it is easy to realize a situation of a homogeneous film consisting of a fraction of an atomic layer to a few atomic layers. Such a film becomes superfluid at a temperature well below T λ and displays critical behavior quite different from the bulk system. The heat capacity has a power-law behavior in three dimensions with an exponent close to zero. In two dimensions, the heat capacity is expected to have a broad maximum at a temperature above the transition into the superfluid state.

Sorption studies of helium and neon by crystals of C60 and C70

We present sorption measurements for3He and4He in the temperature range of 1.5 K to 4.1 K, and for20Ne in the temperature range of 22 K to 27 K by crystals of C60, C70 and crystals of the mixture of these two molecules, 80% C60, 20% C70 We analyze these data by taking into account the non-ideality of the gas in equilibrium with the adsorbate. We calculate chemical potentials and isosteric heats. We find that there is no obvious evidence of intercalation of helium in these crystals at low temperatures. At higher temperatures there are some anomalies in the helium isotherms, and indication of excess sorption. The isosteric heat shows a minimum in this region which can be interpreted as penetration of the helium into a region of repulsive potential. We also find that levels of sorption, at the same chemical potential difference from saturation, are higher for4He than for3He. They are also higher for4He on C70 than for the other crystals. For neon our work is concentrated around the triple point. We find that the isotherms indicate the formation of liquid or solid films. Below the triple point, and above a few atomic layers, the neon film does not grow uniformly.

Some properties of Nuclepore filters

We present the results of various measurements we have made to characterize Nuclepore filters. These include nitrogen and helium adsorption isotherms, heat capacity and SEM studies of pore size distribution and pore density.

Adsorption and specific heat studies of4He in 0.2-m Nuclepore filters

We have made adsorption studies and specific heat measurements for helium adsorbed on Nuclepore filters with a pore size of 0.2 µm. We have been able to identify the first and second layer completion on this substrate by a Langmuir and BET analysis of the data. For coverages near the second layer completion the data obey a Frenkel-Halsey-Hill isotherm, and we have obtained the van der Waals constant α=(2.4±0.4)×10−37 erg-cm3. We find this result in excellent agreement with a theoretical estimate of 2.17 × 10−37 erg-cm3. The interpretation of the isotherm data is confirmed by measurements of the specific heat. Data near a monolayer completion is found proportional toT2 with a characteristic two-dimensional Debye temperature of ΘD=23±1 K. Measurements of specific heat with helium samples equivalent to about 17 layers and higher show two λ transitions shifted in temperature, as is characteristic of helium confined to two different small dimensions. These data are in agreement with the thermodynamic instability for capillary condensation in a cylindrical geometry as calculated by Saam and Cole.

The superfluid transition of 4He, a test case for finite-size scaling at a second-order phase transition

The second-order phase transition of 4He from a normal fluid to a superfluid is ideally suited for studies of critical behaviour. In particular, effects of confinement have been studied recently to verify theoretical predictions of correlation-length scaling and calculations of specific scaling functions. These predictions are summarized for the specific heat and the superfluid density. The method of achieving confinement is discussed, as well as the measuring technique. The specific heat and the superfluid density in planar confinement are examined. It is found that the specific heat scales well on the normal side, and just as well on the superfluid side until the region of the specific heat maximum is reached. Here deviations from scaling are seen. It is possible that this behaviour is associated with the specific crossover in two dimensions. The superfluid fraction, which has been measured for the same type of confinement in two different ways, does not scale. Results of a calculation for the superfluid density to assess the role of the inhomogeneity induced by the van der Waals attraction at the confining walls are presented.

Adiabatic Fountain Resonance for 4He and 3He-4He Mixtures

We report an analysis of a superfluid Helmholtz resonance in the case of helium confined in a superleak. The resonance of the superfluid is achieved under nearly adiabatic conditions. Equations are derived for the resonance frequency, the temperature oscillations of the superleak and the phase relation of this signal relative to an ac heat input. The resonance frequency yields the superfluid fraction of the confined helium. Data are analyzed as function of frequency and temperature and yield parameters such as the dissipation and thermal conductivity which determine the resonance line shape. Estimates are made of the thermodynamic parameters in the resonance equation by using derivatives along the pressure-temperature-concentration lambda surface. These parameters are compared with results from the analysis of the resonance.

Lack of correlation-length scaling for an array of boxes

Finite-size scaling theory predicts that uniformly small critical systems that have the same dimensionality and belong to the same universality class will scale as a function of the ratio of the spatial length L to the correlation length ζ. This should occur for all temperatures within the critical region. Measurements of the heat capacity of liquid 4He confined to a two-dimensional (2D) planar geometry agree well with this prediction when the 4He is normal but disagree near the specific heat maximum where the confined 4He becomes superfluid. Data for 4He confined to 1D structures show a similar behavior (however the lack of data collapse is not as dramatic). Recent measurements of the heat capacity from two 0D confinements, which differ by a factor of two in size, fail to scale at any temperature within the critical region. This lack of scaling may be due to the interaction of neighboring boxes through the shallow channels used to fill them. This is quite surprising since the liquid in the channels is not superfluid at the temperatures of interest for the helium in the boxes. Furthermore, measurements of the superfluid density of the helium within the channels reveal a critical temperature that is higher than expected suggesting that the normal fluid is affected by the already superfluid regions at each end of these channels. Both of these anomalies might be explained by a proximity effect analogous to what is seen when normal metals are sandwiched between two superconductors.