Structure of inert layer 4He adsorbed on a mesoporous silica

TL;DR

This study investigates the structure and phase behavior of inert layer 4He adsorbed on mesoporous silica, revealing coexistence of localized solid and excited fluid phases and suggesting solidification near superfluid transition density.

Contribution

It provides a detailed analysis of the phase coexistence and structural properties of 4He adsorbed on mesoporous silica using vapor pressure and heat capacity measurements.

Findings

Heat capacity exhibits a Schottky-like peak indicating localized solid to fluid excitation.

Localized solid and excited fluid coexist at high temperatures.

Inert layer likely solidifies just below the superfluid transition density.

Abstract

We have studied the structure of inert layer 4He adsorbed on mesoporous silica (FSM-16), by the vapor pressure and heat capacity measurements. The heat capacity shows a Schottky-like peak due to the excitation of a part of localized solid to fluid. We analyzed the heat capacity over a wide temperature region based on the model including the contribution of the localized solid and excited fluid and clarified that the excited fluid coexists with the localized solid at high temperature. As the areal density approaches the value at which superfluid appears (n_C), the fluid amount is likely to go to zero, suggesting a possibility that the inert layer is solidified just below n_C.

Equations

$${\kappa }_T=\frac{1}{n^2{k}_{B}T}\left(\frac{\partial n}{\partial \ln P}\right),$$

$$q_{st}=-k_B\frac{d\ln P}{d(1/T)}.$$

$$C$$

$$A_{\mathrm{exc}}$$

$$D_0={\left(\frac{\Delta q_{st}^0}{\Delta n}\right)}^{-1}.$$

Full text

††preprint: APS/123-QED

I Introduction

4He films adsorbed on various substrates have been extensively studied, since it can be a stage of novel superfluidity. In fact, two-dimensional Kosterlitz-Thouless transitionBishop_1978 , its size effectShirahama_1990 , dimensional crossover in superfluidityMatsushita_2017 , superfluidity intertwined with a density wave orderNyeki_2017 were reported. On the other hand, the film structure itself has also attracted researchers’ interests. On graphite, which is known by homogeneous adsorption potential, layer-by-layer growth of films were confirmed up to 6-atomic layersZimmerli_1992 , and very complicated phase diagram for the first and the second layer was clarifiedShick_1980 ; Greywall . On other substrates, such complicated phase diagram was not reported, due to the heterogeneous adsorption potential compared with graphite. On these substrates, it is thought that an inert layer is formed below the areal density at which a superfluid layer appears ($n_{\mathrm{c}}$).

Among various heterogeneous substrates, on a vycor glass, the film structure has been studied in detail based on heat capacity measurementsTait_1979 . Tait and Reppy explained the observed bend in heat capacity by the excitation of a part of 4He from localized solid islands to delocalized gas area. They insisted that, on the high temperature side of the bend, the excited gas coexists with solid islands, and covers only a fraction of the total surface area, due to a lateral pressure caused by the distribution of long-range adsorption potential.

Recently, Toda et al. studied the film structure of 4He adsorbed on a mesoporous silica called HMM-2 based on vapor pressure and heat capacity measurementsToda_2009 . They observed the similar heat capacity for the inert layer. They insisted that above the areal density of the first layer completion, the heat capacity on the high-temperature side of the bend can be explained as a normal fluid which shows an amorphous-solid like temperature dependenceAndreev_1978 . To confirm this, they first qualitatively evaluated the density state of two-level systems (TLS) in the amorphous solid state from the vapor pressure data.

In the previous work, the structure of the inert layer was discussed based on the heat capacity of limited temperature or areal density regions. Our motivation is to understand its structure more comprehensively and quantitatively. Thus, we chose the mesoporous silica called FSM-16, whose distribution of adsorption potential is theoretically studiedRossi_2006 , and performed the heat capacity and the vapor pressure measurements. We analyzed the heat capacity of the inert layer in a wide temperature region (0.18-4.5 K), based on the model including the contribution of the localized solid and excited fluid. The results support the picture that the excited fluid coexists with the localized solid at high temperature. On the other hand, its amount is likely to go to zero as the areal density approaches $n_{\mathrm{C}}$, suggesting a new possibility that the inert layer is solidified just below $n_{\mathrm{C}}$.

II Experiments

The mesoporous silica called FSM was first synthesized by Inagaki et al. in Toyota Central R&D Labs., Inc. Japan.Inagaki_1993 It forms a honeycomb structure of a 1D uniform nanometer-size straight channel without interconnection. Using an organic molecule as a template, the diameter of the channel was precisely controlled. Its homogeneity was confirmed by the transmission electron micrograph and the X-ray diffraction (XRD) pattern.Inagaki_1996 The sample we used is the one with 2.8-nm channel.

For the heat capacity measurements, we used the cell which was used in the previous heat capacity measurements for pressurized liquid 4He.Taniguchi_2011 Due to aging, the surface area $S$ was reduced to 145 m2, in the present measurements. Unlike the liquid measurements, the capillary is plumbed directly to the cell, since the thermal conductivity is not seriously high.

The heat capacity was measured by a quasi-adiabatic heat-pulse technique up to 14 atoms/nm2, which corresponds to $n_{\mathrm{C}}$Demura_2017 . The temperature of the sample cell was monitored with RuO2, which were glued onto the bottom face of the cell. These were calibrated against a commercial calibrated RuO2 thermometer. The thermal relaxation time from the cell to the stage was 360-5000 s, which was more than one order of magnitude larger than that in the cell, 40-240 s.

For the vapor pressure measurements, we adopted the cell which was used in the previous double torsional oscillator (DTO) measurementsTaniguchi_2013 . As in the previous workIkegami_2003 , the pressure was measured by means of the capacitive strain gauge with a membrane 100$\mu$m in thickness whose one side was Au sputtered as an electrode. The pressure was calibrated against the 4He saturated vapor pressure. Its accuracy was 2$\times 10^{-3}$ mbar. This gauge was attached directly to the cell, which was mounted on the 1K pot of the refrigerator, in order to avoid the temperature difference.

III Results and Discussion

III.1 Isothermal compressibility and isosteric heat of sorption

Vapor pressure ($P$) of the adsorbed 4He gives us the isothermal compressibility $\kappa_{T}$ and the isosteric heat of sorption $q_{st}$, which are useful to examine the change in film structure. Since $\kappa_{T}$ becomes small as the density is enhanced near the layer completion, its minimum is often used as a criterion of the completion of layers. $\kappa_{T}$ is deduced from the $P$- isotherm as

[TABLE]

where $n$ is the areal density, $k_{B}$ is the Boltzmann constant, and $T$ is the temperature.

Figure 1 shows the obtained $\kappa_{T}$ as a function of areal density up to 16.4 atoms/nm2, where the capillary condensation occurs. $\kappa_{T}$ shows a minimum at around 9$\pm 1$ atoms/nm2, which we determine as the areal density of the first layer completion ($n_{1}$). Above $n_{1}$, $\kappa_{T}$ increases with increasing areal density, and turns to decrease at around 12.7$\pm 0.3$ atoms/nm2, and then has a small minimum at $n_{i}=$14 atoms/nm2, which coincides with $n_{\mathrm{C}}$. Such a small minimum of $\kappa_{T}$ was also reported for 4He in 4.7- and 2.8-nm channels of FSM series by Ikegami et al.Ikegami_2003 ; Com0 Just above the areal density of the small minimum, the superfluid transition is commonly observed in both the present and the previous work. It indicates that the inert layer is slightly compressed as the areal density approaches $n_{\mathrm{C}}$.

On the other hand, from the temperature dependence of vapor pressure, we calculated $q_{st}$ as

[TABLE]

By subtracting the heat capacity of gas phase 4He, we estimate the isosteric heat of sorption at absolute zero, $q_{st}^{0}=q_{st}-(5/2)k_{B}T$, which corresponds to the depth of the adsorption potentialToda_2009 . The obtained $q_{st}^{0}$ is shown as a function of $n$ in Fig. 1(b). It is $\sim$140 K at 1.8 atoms/nm2 and decreases monotonously with increasing areal density. It reaches $\sim$40 K at $n_{1}$, above around which its decrease becomes slow. The obtained value of $q_{st}^{0}$ is close to that of 4.7-nm channels of FSM, up to $n_{1}$Ikegami_2003 . It indicates that the difference in the channel size between 2.8 and 4.7 nm does not affect strongly on the adsorption potential in the submonolayer region.

III.2 Heat capacity of inert layer 4He

Figure 2 shows the heat capacity ($C$) for various areal densities as a function of $T$. Here, the heat capacity of the empty cell is subtracted. For 3.5 atoms/nm2, a broad peak appears at around 1.1 K, in addition to the slope a little smaller than $T^{2}$. With increasing areal density, this peak shifts to the low-temperature side, with its height shrinking. Finally, it becomes unclear above 11.4 atoms/nm2. Since the peak is very broad, we define $T_{\mathrm{p}}$ as the temperature where $C/T$ starts to deviate from the extrapolation of the low-temperature side, which gives us the lower limit of peak temperature. (see the inset of Fig. 2.) On the other hand, the temperature dependence at the high-temperature side keeps a little smaller than $T^{2}$ up to around 10.2 atoms/nm2, and above that its slope slightly decreases.

As is clear from the inset of Fig. 2, the heat capacity at the temperature fully higher than $T_{\mathrm{p}}$ can be explained by the sum of $T$-linear ($y$-intercept) and $T$-squared (slope) terms. In addition, there is a peak at around $T_{\mathrm{p}}$. It should be noted that the $y$-intercept of the extrapolation from the heat capacity near the lowest temperature is lower than that of the extrapolation from the high-temperature side of $T_{\mathrm{p}}$ (dashed line in the inset of Fig. 2.). There is a possibility that the difference of $T$-linear term comes from the contribution of the excited atoms at around $T_{\mathrm{p}}$.

Therefore, we set the heat capacity model as

[TABLE]

where $\beta=1/k_{B}T$. Here, the first parenthesis, the second and the third terms correspond to the heat capacity of the localized solid, the excited fluid 4He, and a Schottky peak due to the excitation, respectively. Regarding their origin, Tait and Reppy suggested that $A_{l}$ and $B$ come from the amorphous property and 2D phonon of the localized solid, and that $A_{\mathrm{exc}}$, from 2D Bose gas. $A_{\mathrm{exc0}}$ corresponds to the slope when 4He atoms are fully excited. $2\Delta E$ is the energy gap between the localized solid and the 2D gas states. Based on this model, we discuss the film structure later.

Figure 3 shows the fitted curves of 5.5 atoms/nm2 as an example. The heat capacity is well reproduced over the entire temperature range. At around the lowest temperature, $T$-linear term becomes dominant, while near the highest temperature $T^{2}$-term is dominant. At around $T_{\mathrm{p}}$, the contribution of Schottky term becomes large, and the slope of $T$-linear term increases due to the contribution of $A_{\mathrm{exc}}T$. The heat capacity for all areal densities between 3.5 and 13.8 atoms/nm2 are fitted well to the eq. (3).

III.3 Property of the localized solid

It is well known that the amorphous property of the localized solid is characterized by the $T$-linear heat capacity due to the quantum tunneling between different states in the two-level systems (TLS). Figure 4(a) shows the fitting results of $A_{l}$ as a function of areal density. It is quite small up to around 9 atoms/nm2 and then soars with increasing areal density.

The coefficient of $T$-linear term for amorphous solid is described as $\pi^{2}/6\cdot D_{0}k_{B}$, where $D_{0}$ is the density of states. When the TLS is generated by the distribution of adsorption potential, $D_{0}$ is often approximated as Tait_1979 ; Toda_2009

[TABLE]

Using $q^{0}_{st}$ in Fig. 1(b), we estimated the coefficient of $T$-linear heat capacity, $A_{qst}$, which is shown in Fig. 4(a) for comparison. The calculated $A_{qst}$ shows the same areal density dependence as that of $A_{l}$, except that it is slightly larger than $A_{l}$. The semiquantitative agreement means that $A_{l}T$ term is well explained by the contribution of amorphous solid.

Next, we consider the $T^{2}$ term. From the fitting results of $B$, we calculate the Debye temperature $\Theta_{\mathrm{Dc}}$ as $\Theta_{\mathrm{Dc}}=(28.8Nk_{B}/B)^{0.5}$, where $N$ is the number of solid 4He. Here, we approximate $N$ by $n$ multiplied by the surface area $S$, neglecting the number of excited 4He. The areal density dependence of $\Theta_{\mathrm{Dc}}$ is shown in Fig. 4(b). $\Theta_{\mathrm{Dc}}$ is 43 K for 3.5 atoms/nm2, and drops to $\sim 35$ K at 5.5 atoms/nm2 and above that decreases slightly with increasing areal density. The value at around $n_{1}$ (34 K) is close to those for other substrates such as Cu (32 K)Roy_1971 and graphite (33.0 K at 9.67 atoms/nm2)Bretz_1973 .

The Debye temperature can be also deduced from the phonon velocity $v_{\mathrm{p}}$ as $\Theta_{\mathrm{D}\kappa}=(hv_{\mathrm{p}}/k_{B})(n/\pi)^{0.5}$. The phonon velocity is related to the adiabatic compressibility $\kappa_{S}$ as $v_{\mathrm{p}}=\sqrt{1/(mn\kappa_{S})}$. Here, we evaluate $\Theta_{\mathrm{D}\kappa}$ by assuming that $\kappa_{T}$ approximately equals $\kappa_{S}$.Toda_2009 $\Theta_{\mathrm{D}\kappa}$ increases with increasing areal density and then turns to decrease at $\sim$8 atoms/nm2, above which $\Theta_{\mathrm{D}\kappa}$ agrees well with $\Theta_{\mathrm{Dc}}$. The reason why $\Theta_{\mathrm{D}\kappa}$ is smaller than $\Theta_{\mathrm{Dc}}$ below 8 atoms/nm2 is left as a question. However, one possibility is that in this areal density region, the solid islands are considered to be independent of each other. In this situation, $\kappa_{T}$ may not reflect the compressibility of the solid 4He itself.

The monotonous decrease of $\Theta_{\mathrm{Dc}}$ with increasing areal density is not intuitive. Naively, when the number of atoms increases, the film is expected to be compressed, making the film stiff, i.e. raising the Debye temperature. In fact, for monolayer 4He adsorbed on graphite, the increase of the Debye temprature is reportedBretz_1973 . To understand the $\Theta_{\mathrm{Dc}}$ behavior, it is necessary to consider the distribution of adsorption potential, which generates the lateral pressure to compress the film. The calculation by Rossi et al. shows that the adsorption potential depends on the azimuthal direction in the cross section, and that its distribution decreases with increasing the film thickness i.e. the areal density. It means that with increasing areal density, the lateral pressure lowers, leading to the decrease of $\Theta_{\mathrm{Dc}}$.

III.4 Excitation to the 2D gas state

First, we focus on the Schottky peak term. In Fig. 5(a) and (b), the fitting parameters $\Delta E$, and $D$ divided by $k_{B}S$ are plotted as a function of $n$, respectively, up to 11.4 atoms/nm2, above which the peak cannot be identified. Here, $D/(k_{B}S)$ corresponds to the number of atoms that can be excited in the unit of areal density. $\Delta E$ is 2.2 K for 3.5 atoms/nm2, and decreases with increasing areal density, at a decreasing rate. The areal density dependence and the order of magnitude of $\Delta E$ agree with those for 4He on vycor glass, supporting that the adsorbed 4He on the present substrate are also excited to the gas state.

On the other hand, $D/(k_{B}S)$ is at most $\sim$10 % for 3.5 atoms/nm2, and decreases with increasing areal density. Above $n_{1}$, it decreases at an accelerated rate and seems to go to zero at around 13 atoms/nm2. The fitting results indicate that 2D gas disappears just before the inert layer completes, contrary to the conventional understanding that the 2D gas phase below $n_{\mathrm{C}}$ is continuous to the normal phase above the superfluid transition temperature above $n_{\mathrm{C}}$.

To examine the relation between the amount of 2D gas 4He and the magnitude of $T$-linear heat capacity, we plot $A_{\mathrm{exc0}}$ against $n$ in Fig. 5(c). It is 0.06 mJ/K2 for 3.5 atoms/nm2, and increases with increasing areal density at first. Then, it becomes almost constant between 8 and 10 atoms/nm2, and turns to decrease. The important thing is that $A_{\mathrm{exc0}}$ is not proportional to $D/k_{B}S$, i.e. the amount of 2D gas at a fully high temperature.

In the case of an ideal 2D Bose gas, the heat capacity is $T$-linear and depends not on the number of atoms but on the surface area that the gas covers ($S_{g}$). The coefficient is described as $(mS_{g})/(2\pi\hbar^{2})\cdot k_{B}^{2}$, where $m$ is the mass of 4He atomDaunt_1972 . When the entire surface area is covered, it becomes 2.6 mJ/K2, which is much larger than the obtained $A_{\mathrm{exc}}$, suggesting that only a part of surface is occupied by the excited gas.

From the viewpoint of change in $S_{g}/S$, we consider the structure of the inert layer. Here, we define $S_{g}/S$ as $A_{\mathrm{exc0}}/(2.6$ mJ/K${}^{2})$, which is shown as the right vertical axis of Fig. 5(c). On the other hand, the surface area covered by the localized solid ($S_{s}$) is approximated by $(n/n_{1})S$ in the submonolayer region, while above $n_{1}$ it is defined as $S_{s}=S-S_{g}$. $S_{g}$ increases in conjunction with $n$, i.e., $S_{s}$, at first, and then its increase stops at around 8 atoms/nm2, where $S_{g}+S_{s}$ reaches $S$. The almost constant $S_{g}$ between 8 and 10 atoms/nm2 indicates that the increase of $n$ causes promotion of the gas 4He to the overlayer. It is thought to be accompanied by the compression of the first layer, since $\kappa_{T}$ takes minimum in the same areal density region. Above that, $S_{g}/S$ turns to decrease and is likely to go to zero at around 13 atoms/nm2. It is understood as the increase of density in the overlayer raises the ratio of $S_{s}$ to $S_{g}$ and finally lets the entire overlayer solidified. It is consistent with the decrease of $\kappa_{T}$ above around 13 atoms/nm2, indicating that the compression starts at the end of the coexistence of the gas and solid.

As the origin which limits $S_{g}$, the lateral pressure due to the distribution of long-range adsorption potential is suggestedTait_1979 . In the case of vycor glass, it is thought that the distribution of pore size generates the distribution of the adsorption potential. Although the channel size of FSM is uniform, it is reported that the adsorption potential of a hexagonal channel has an azimuthal dependence in cross-sectionRossi_2006 . Its amplitude is several tens K near the pore wall, and decreases as the distance from the wall increases, and becomes zero when the film thickness reaches 0.75 atoms/nm2, at which superfluid appearsDemura_2017 . The amplitude in the submonolayer region is larger than the obtained $\Delta_{E}$, letting the excited gas locate only around the rim of the solid islands. It means that the excited gas does not spread uniformly but is condensed. Therefore, we call it fluid in the following discussion.

III.5 Phase diagram

We discuss the phase diagram of 4He film, based on the foregoing discussion. Figure 6 summarizes the typical temperatures of phase separations, $T_{\mathrm{P}}$ and $T_{\mathrm{C}}$ as a function of areal density. In the low areal density region, 4He atoms are adsorbed on the areas with a deep adsorption potential and forms solid islands at fully low temperature. At around $T_{\mathrm{P}}$, a small part of the localized 4He is excited and forms a fluid area surrounding the solid islands. With increasing areal density, $T_{\mathrm{P}}$ decreases due to the decrease of adsorption potential distribution, and finally disappears at around 13 atoms/nm2. Above this areal density, instead of the decrease in fluid areas, the compression of the inert layer composed of only solid occurs. Then above $n_{i}$, the liquid layer appears on top of the inert layer, which shows superfluidity at low temperature.

This phase diagram is similar to the one probably characteristic of adsorbed 4He on heterogeneous substrates proposed by Tait and Reppy. The only thing that is different is that the fluid and the liquid phases are not continuous. We consider that after the inert layer completion at which the lateral pressure becomes fully small, the uniform 2D liquid phase appears. This idea is consistent with the fact that the superfluid fraction is almost proportional to $(n-n_{i})$.

IV Summary

In summary, we have studied the structure of inert layer 4He in a 2.8-nm channel of FSM-16, based on the vapor pressure and heat capacity data. We analyzed the heat capacity of the inert layer based on the model including the contribution of the localized solid and excited fluid. The results support the picture that the excited fluid coexists with the localized solid at high temperature. With increasing areal density, the amount of excited fluid is decreased and likely to go to zero just below $n_{\mathrm{C}}$(=$n_{i}$). It suggests the possibility that the normal fluid phase below $n_{\mathrm{C}}$ is not continuous to the liquid phase above $n_{\mathrm{C}}$.

Acknowledgements.

The work was partly supported by JSPS KAKENHI Grant No. 26400352 and No. 18K03535. We thank S. Inagaki for the supply of the FSM-16.