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Liquids and Liquid Crystals LIQUIDS AND LIQUID CRYSTALS FEATURES OF THE CONFINED HELIUM 4He SPECIFIC HEAT NEAR THE ¸-POINT IN PLANE MESOSCALE PORES K.A. CHALYY UDC 536.764 Taras Shevchenko Kyiv National University, Faculty of Physics °c 2006 (6, Academician Glushkov Prosp., Kyiv 03022, Ukraine; e-mail: [email protected]) The influence of the finite-size effect on the liquid helium specific be reached without the analysis of the fundamental heat and a shift of the transition temperature are theoretically experiments with confined helium. Amongst the pioneer examined for the case of planar confinement. A considerable studies of the spatial-limitation effect, we mention the growth of the specific heat manifests itself at a new transition temperature which is calculated in the context of the present helium experiment conducted by Chen and Gasparini geometric conditions. The analytic results are found to be in fair [2] more than the quarter of a century ago. More agreement with a number of experiments where 4He films were sophisticated measurements for films were reported later ranged in thickness from 48 nm to 57 ¹m that can be referred as the mesoscale values. The contributions to the shift of the on in (see, e.g., [3]). transition temperature caused by the gravitation effect and by the Conducting the experiments with confined helium finite-size effect are examined. is rather difficult due to a number of complicated circumstances. In such Earth-based measurements, the characteristic size of a sample is supposed to be small enough in order to neglect the influence of gravity. 1. Introduction On the Earth orbit, such a limitation vanishes. On Many systems of experimental and theoretical the Earth in the measurements of specific heat at the interest, such as thin films, silicon wafers, interfaces, helium lambda point, the resolving power is limited due biomembranes, synaptic clefts, etc., have a reduced to the competition between the finite-size effect and planar geometry. The physical properties of the systems the pressure dependence of the transition temperature undergoing critical phenomena and phase transitions (the gravity effect). Therefore, it is important to be are essentially influenced by spatial limitations. sure that just the finite-size effects cause the observed Experimental studies of such systems that can exhibit rounding and shift of the specific heat maximum. A size-dependent second-order phase transitions provide strong desire of researchers to eliminate the influence principal results for the verification of the theory of of the gravity effect became a reason of performing the finite-size effects. The various peculiarities of confined extremely precise experiment on the near-Earth orbit. systems represent a long-standing interest in the physics Most reliable data on the confined helium heat capacity of liquids. Despite the active experimental studies, were reported by the group of J. Lipa during the last there is a steady demand for more precise tests of decade [4, 5]. the existing theoretical predictions. New horizons in In this paper, we are going to present the this field would be opened upon conducting the new comparative study that is focused on the consideration series of experiments in fundamental physics [1] that are of the temperature dependence of the liquid helium heat developed for the realization on the International Space capacity near the ¸-point. The system to be considered Station (ISS). hereafter has a reduced geometry in the form of a plane- A better understanding of the effect of spatial parallel layer with the typical thickness from a few limitation and finite-size-induced phenomena cannot hundreds of angstroms and over. Such a range of sizes ISSN 0503-1265. Ukr. J. Phys. 2006. V. 51, N 9 863 K.A. CHALYY " µ ¶ # can be associated with the upper border of the nanoscale p n2¼2 1=2 ³n¼ ´ £K x2 + y2 »¡2 + cos z : (2) and a substantial part of the microscale and can be 0 H2 H referred to as the mesoscale. Films represent an example of the so-called “well-defined geometry” that allows one Here, K0(u) is the cylindrical McDonald function. to perform direct analytic calculations. The verification Because of a non-exponential shape of G2, the of the validity of the theoretical results proposed here correlation length of density fluctuations can be defined is conducted by comparing them with high-resolution 1=2 as » = (M2) , where M2 denotes the normalized experimental data [36]. second spatial moment of G2 . Following the method described elsewhere, it is possible to get the dominating 2. Temperature and Size Dependence of component of the correlation length in the XY -plane: Specific Heat "õ ¶ 1 ! µ ¶ 1 #¡º ¼» º ¼» º »¤ = » 0 + 1 ¿ + 0 : (3) The fluctuation effects must be taken into account 0 H H in order to develop a reliable prediction of system’s behavior near a second-order phase transition. A The correlation length »¤ of a thin liquid film depends, singular growth of the correlation length » can naturally, not only on the temperature variable ¿ but be realized only for spatially infinite systems. The also on the thickness H of a layer. One can suggest that expression that is predicted by the fluctuation (scaling) the expression theory [7] reads as õ ¶ 1 ! µ ¶ 1 º º ¡º ¼»0 ¼»0 » = »0¿ : (1) ¿ ¤ = + 1 ¿ + H H Here, »0 is the amplitude of the correlation length, ¿ = jT ¡Tcj=Tc is the temperature variable, and º is the represents the new temperature variable that governs correlation length exponent. To use Eq. (1) for systems the near-critical behavior of the liquid film system. of experimental interest, it is necessary to satisfy the Based on the current results, it would be possible to condition that assumes that the linear size of a system, estimate the linear size of ordered clusters, which is, L, is much larger than the correlation length ». Physical obviously, appears to be a function of ¿ and H: Near the properties and, in particular, correlation properties (see, phase-transition point, this size can essentially exceed e.g., [8, 9]) of finite-size systems are different from those the distance associated with the direct intermolecular of the bulk systems. A system can be considered as finite- interaction and become macroscopic. The film geometry, sized near the critical point or the phase transition point as well as the cylindrical one [10], makes the correlation if its characteristic linear size becomes comparable with properties anisotropic. Consequently, it is believed that, the correlation length ». It is possible to examine the at the achievement of the new critical temperature of a ¤ characteristics of the liquid systems with well-defined finite-size liquid Tc (H) < Tc in a sample of the planar reduced geometry (film, cylinder, bar, sphere, or cube) geometry, a considerable growth of the correlation length ¤ in terms of the pair-correlation function G2 and the » along the unrestricted direction of XY -plane might associated correlation length ». exist. Let us consider the geometry of a thin liquid system In order to verify the validity of the theoretical in the form of a plane-parallel layer D £ D £ H. The results proposed here, it is important to have a layer thickness H is supposed to be much smaller than possibility to compare them with high-resolution the distance D in XY directions. We will study the experimental data. In view of the fact that most of the region where the value of correlation length » becomes data [36], which are available up to this date, were comparable or even larger than H but still much derived from the studies of the heat capacity of confined 4 smaller than D. The pair correlation function G2 for the He, it is necessary to obtain an analytic expression planar geometry was deduced by applying the Helmholtz particularly for this property. operator procedure (see, e.g., [9, 10]). One can get the Regardless of the geometry, the relationship between expression for G2 in the form: the heat capacity C and correlation length » is given by the expression (see, for example, [11]): X1 1 n G (x; y; z) = ¼H (1 ¡ (¡1) )£ ®=º 2 2 C » » : (4) n=0 864 ISSN 0503-1265. Ukr. J. Phys. 2006. V. 51, N 9 FEATURES OF THE CONFINED HELIUM 4HE In fact, near the lambda point of liquid helium, a small negative value of ® ¼ ¡0:026 was observed in early experiments. Later on, in the recent revision [12] of the most precise microgravity experiment [5] on the near- Earth orbit, it was confirmed to be ® = ¡0:0127§0:0003. However, it is known that the exponent ® appears to be positive and close to 0.1 near the liquid-gas critical point. On the other hand, for liquid 4He, the hyperscaling relation ® = 2 ¡ 3º with º=0.6705 [13] yields ®=º = ¡0:0172. Thus, adopting Eq. (3), by assuming » = »¤, and making an ansatz for Eq. (4), we can write the formula for heat capacity Cplan(¿; H) in case of planar geometry as "õ ¶ 1 ! µ ¶ 1 #¡® ¼» º ¼» º C (¿; H) » 0 + 1 ¿ + 0 ; (5) Fig. 1. Dependence of the confined helium heat capacity plan H H Cplan(¿,H) (arbitrary units) on the reduced temperature ¿ for film samples on the semilogarithmic scale [according to Eq. (5)]. The with H being the film thickness. The amplitude of the higher curve represents the 5039-A˚ film, and the lower one does correlation length » for helium is equal to 0.36 nm below 0 the 1074-A˚ one and is by 0.143 nm above the transition temperature. Next, we will examine the limiting cases, namely H ! 1 and ¿=0. Equation (5) appears reasonable and distinctive lambda-shape. However, in the actual leads to the expected expressions experiments, the heat capacity peak is considerably rounded.
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