Article Cite This: J. Phys. Chem. C 2018, 122, 5537−5544
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Temperature Dependence of the Longitudinal Modulus of Liquid Argon in Nanopores Klaus Schappert* and Rolf Pelster FR Physik, Universität des Saarlandes, 66123 Saarbrücken, Germany ABSTRACT: In this paper, we study the influence of temperature on the longitudinal modulus βAr,ads and the density of liquid argon adsorbed in nanoporous Vycor glass. For this purpose, we have carried out systematic ultrasonic measurements for temperatures between boiling and freezing of the adsorbate. The study shows a continuous almost linear decrease of the longitudinal modulus of the pore condensate with increasing temperature. At all temperatures, the absolute values of the adsorbate’s modulus are enhanced in comparison to bulk argon as a result of the adsorption stress. Nevertheless, the modulus’s dependence on the temperature is similar to that for bulk argon. Furthermore, the temperature dependence of the adsorbate’s density is similar to that of bulk argon. Summarizing, we have shown that for a given pore size the enhancement of the longitudinal modulus of the adsorbed liquid argon in comparison to liquid bulk argon does not noticeably depend on temperature. Consequently, nanoconfinement of a material can be used to achieve an enhancement of the longitudinal modulus; however, at least for a simple adsorbate like argon, a variation of temperature cannot be exploited to modify the strength of the enhancement.
1. INTRODUCTION Adsorption in nanoporous media is a topic of intensive research, which can be understood as a result of a wealth of possible applications for such materials, e.g., as storage media in the automobile industry or for the geological storage of CO2.1−5 Elastic properties are among the decisive physical quantities for the behavior of such nanoscale media. Thus, also for the development of new materials it is crucial to obtain an understanding of the mechanisms influencing the elasticity of nanomaterials.6−11 Recently, some insights into the factors influencing the elastic behavior of nanoconfined materials have been achieved.12−19 Thus, experimental measurements during sorption of hexane and argon in Vycor revealed the influence of the Laplace pressure on the longitudinal modulus of the adsorbate.13,20,21 At vapor pressures below that of the bulk liquid, p0, the pore condensate exhibits menisci at the pore ends. Their curvature is related to a high negative Laplace pressure that reduces the adsorbate’s longitudinal modulus below the value at the bulk vapor pressure.13,21 Theoretical calculations and simulations of Gor and co-workers16,17 have shown that the Laplace pressure is not the sole pressure effect influencing the adsorbate’s elasticity. Thus, also for complete filling of the pores (at the bulk vapor pressure) nanoconfinement leads to an enhancement of the adsorbate’s modulus compared to bulk. This enhancement of the longitudinal modulus is caused by the so-called adsorption stress (or solvation pressure),16,17,22 which is also the origin of the experimentally observed sorption-induced deformation of porous media.22−34 A detailed comparison between the theoretically expected elastic behavior and the experimentally © 2018 American Chemical Society
observed elasticity is still needed. The molecular interaction between the adsorbed substance and the pore surface plays a significant role for the elastic behavior of nanoconfined material.12,15 Such an interaction may possibly also influence the formation of a molecular order that is absent for bulk materials.14 In spite of some recent progress in this field, experimental data on the elasticity of adsorbates in nanopores are still very rare, in particular for the longitudinal modulus. In addition to our previous data for liquid argon at two single temperatures (see refs 13 and 35) Page et al.20 also studied the longitudinal modulus during sorption of liquid hexane at one temperature. For a better understanding of the basic factors influencing the elasticity on the nanocale obviously additional measurements are necessary. Regrettably, sorption experiments can be very time-consuming because of the slow equilibration during sorption.36 Nevertheless, they are worth the effort providing answers to the following fundamental questions: Do liquid adsorbates confined in mesopores exhibit an enhanced longitudinal modulus compared to bulk fluids and does it go along with an altered density? Is the temperature dependence of the above quantities bulklike? Is the change of elastic properties mainly caused by the adsorption stress (solvation pressure)? The smaller the pore radius the stronger the effects we can expect. However, they should be observable already at pore Received: December 31, 2017 Revised: February 12, 2018 Published: February 12, 2018 5537
DOI: 10.1021/acs.jpcc.7b12839 J. Phys. Chem. C 2018, 122, 5537−5544
Article
The Journal of Physical Chemistry C diameters of 7−10 nm,16,43 which are typical for porous Vycor glass. Because of the simplicity of the molecular interactions, this mesoporous glass and the adsorbate argon serves us as a model system to gain insights into the physics underlying possible deviations from the elastic behavior of bulk substances. We focus in this paper on the influence of temperature on the enhancement of the longitudinal modulus of liquid argon in the nanopores of Vycor glass; i.e., we consider the properties of completely filled pores. In our analysis, we also compare our experimental results with the theoretically expected elastic moduli.
2. EXPERIMENTAL DETAILS For the study of the elasticity of liquid argon in nanoconfinement we have used the well-known porous Vycor glass, which has a spongy structure of interconnected pores. The matrix material consists of a nearly pure quartz glass (SiO2: 96%)37,38 and our sample has an average pore radius of 3.8 nm. The pore radius was determined from the desorption branch of an isotherm with argon at 86 K using the Kelvin equation for the calculation of the radius of the withdrawing menisci and the Halsey equation for the thickness of the argon layer that initially remains on the pore surface during draining (for details see refs 39 and 40). The effective elastic moduli of the empty or filled sample were determined from ultrasonic measurements. The longitudinal or transveral waves were generated by applying voltage pulses to piezoelectric crystals that are glued onto the sample [LiNbO3: 36◦ Y-cut for longitudinal waves (carrier freqency ≈ 11 MHz) and 41° X-cut for transversal waves (carrier freqency ≈ 7 MHz)]. A schematic sketch of the experimental setup with the sample cell is shown in Figure 1. The pulse echo method allows the determination of the ultrasonic velocity (cl for longitudinal waves; ct for transversal waves) of the waves traveling forth and back through the sample of thickness d with the aid of the measured transit times: ci = 2d/Δti (see Figure 1 and, e.g., refs 36 and 39). From these velocities the effective moduli can by calculated via the relation β = ρc2l for the longitudinal modulus and via G = ρc2t for the shear modulus, with the effective density of the sample, ρ = msample/Vsample. The calculation of the density of the empty sample, ρ0 = msample,empty/Vsample, was performed with values of the mass of the empty sample, msample,empty, and of its volume Vsample, which were determined at room temperature. Because of the very small thermal contraction of porous Vycor glass (typical thermal expansion coefficient of porous glass ≈8 × 10−7 K−1),41 we can use the same value for ρ0 at all temperatures. The determination of the effective density of the filled sample (with n0 mol of argon) within the sample cell requires a different method (e.g., the ultrasonic method described below) as the filled sample cannot be weighed. The effective shear modulus of the nanoporous glass sample is not affected by the adsorption of liquid argon as liquids cannot sustain shear stress. Thus, the effective shear modulus G = ρc2t of the filled sample is equal to the effective shear modulus of the empty sample, G0 = ρ0c2t,0, i.e., G = G0, and thus ⎛ ct ,0 ⎞2 ρ = ρ0 ⎜ ⎟ ⎝ ct ⎠
Figure 1. Schematic sketch of the experimental setup with the sample cell. The porous sample is placed in a closed metallic container on top of the cold head of a cryostat. For the filling and draining of the sample the cell is connected to a gas distribution system with a pressure gauge and a vacuum pump via a capillary (see Figure 2 in ref 39). For the generation of longitudinal and shear waves, two piezoelectric crystals are glued on the sample. The crystals are connected to a generator of voltage pulses (Matec 6600). The generated ultrasonic pulses propagate through the sample, are reflected at its bottom, and return to the piezoelectric crystals (pulse echo method). The resulting voltage signals are displayed on a digital oscilloscope, and the ultrasonic velocities ci = 2d/Δti are calculated from the transit times of the signals thorough the sample, Δti, and the thickness of the sample, d.
propagating through the empty sample, ct,0, or the filled sample, ct.36,42 Consequently, the simultaneous measurements of the longitudinal and transveral velocities gives us a direct access to the effective density ρ (see eq 1) and the effective longitudinal modulus of the filled sample β=
(1)
ρc l2
⎛ ct,0 ⎞2 = ρ ⎜c ⎟ ⏟1 0 ⎝ l ct ⎠ eq
(2)
without any additional assumptions. With the aid of eq 1 it is also possible to calculate the molar amount of adsorbate, n0, from the transversal velocities, as the
with the density of the empty sample, ρ0 = msample,empty/Vsample, and the temperature dependent velocities of shear waves 5538
DOI: 10.1021/acs.jpcc.7b12839 J. Phys. Chem. C 2018, 122, 5537−5544
Article
The Journal of Physical Chemistry C effective density ρ = msample,filled/Vsample depends on the mass of the pores, mpores msample,empty + mpores msample,empty + n0mM,Ar ρ= = Vsample Vsample (3)
with mpores = n0mM,Ar, where mM,Ar denotes the molar mass of argon. Hence, the molar amount of adsorbate can be calculated via eqs 1 and 3 according to the following relation: ⎤ ⎛ msample,empty ⎞⎡⎛ ct,0 ⎞2 ⎟⎟⎢⎜ ⎟ − 1⎥ n0 = ⎜⎜ ⎥⎦ ⎝ mM,Ar ⎠⎢⎣⎝ ct ⎠
(4)
The porosity of the sample ϕ = Vpores/Vsample is determined via the molar amount n0 of adsorbate that is necessary for complete filling of the pores (at the bulk vapor pressure, p0) and the adsorbate’s molar volume. As the ultrasonic measurements do not supply the latter quantity we may only assume that the adsorbed liquid argon at 86 K possesses the molar 86K volume of bulk argon (i.e., V86K M,Ar,ads = VM,Ar,bulk) or equivalently 86K the density of bulk argon (i.e., ρAr,ads = ρ86K Ar,bulk). Thus, the pore volume can be calculated using Vpores = n0VM,Ar,bulk and we obtain a porosity of 27.6%, which matches to the value of ϕ = 28% given by the producer of the glass (Corning, Inc.).38 On the other hand, for the density of argon in nanopores also slightly higher and lower values than in the bulk state have been reported (see section 3).16,18,43 For example, a deviation from the density of bulk argon by ±3% would result in a value of 27.6% ± 0.9% for the porosity ϕ. In contrast to the effective density (see eq 1), the effective modulus β (see eq 2), and the amount of adsorbate n0 (see eq 4), the determination of the absolute values of the confined argon’s density ρAr,ads =
n0mM,Ar Vpores
2 ⎤ ρ0 ⎡⎛ ct,0 ⎞ ⎢ = ⎜ ⎟ − 1⎥ ⎥⎦ ϕ ⎢⎣⎝ ct ⎠
Figure 2. Molar amount n0 of liquid argon necessary for complete filling of the nanopores of the Vycor glass sample as a function of temperature. The displayed amount of adsorbate was calculated via eq 4 using the measured velocities of transversal waves propagating through the empty and filled sample. An increase of temperature comes along with a decrease of n0 as a result of the temperature dependence of the adsorbed argon’s density. The dashed line is a linear fit to the data.
temperature (see Figure 2). Such a decrease can of course be explained by a decrease of the density ρAr,ads of the adsorbed argon. But corresponds the slope of the observed decrease to the known behavior for bulk argon? We can compare this dependence to the density of bulk argon, which is known in the literature.44 The possible temperatures for liquid bulk argon are limited to a much narrower range of 84−87 K because of the higher freezing point of bulk argon.44,45 The density of bulk argon shows a linear dependence from the temperature (see green squares and linear fit in Figure 3). At 86 K the value shown in Figure 3 for the density of the adsorbed argon is equal to that of bulk argon because we have used a measurement at this temperature for the determination of the porosity assuming 86K ρ86K Ar,bulk = ρAr,ads (see section 2). Though this appears to be a reasonable assumption (see section 2), we cannot exclude a small difference to the value for the density of bulk argon. From simulations it is known that the density of argon in nanopores may deviate from the bulk value, notably for smaller nanopores.16,18,43 For nanopores with a smaller pore radius of rP = 3.0 nm a deviation of ±2−3% (at 87.3 K) was observed, which depends decisively on the pore structure.43 In spite of the scaling of the density at 86 K, we notice in particular at the lower temperatures a small deviation from the density of bulk argon (
