Liquid Argon in Nanopores: The Impact of Confinement on the

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Liquid Argon in Nanopores: The Impact of Confinement on the Pressure Dependence of the Adiabatic Longitudinal Modulus Klaus Schappert, and Rolf Pelster J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b08136 • Publication Date (Web): 06 Nov 2018 Downloaded from http://pubs.acs.org on November 7, 2018

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The Journal of Physical Chemistry

Liquid Argon in Nanopores: The Impact of Confinement on the Pressure Dependence of the Adiabatic Longitudinal Modulus Klaus Schappert∗ and Rolf Pelster∗ FR Physik, Universität des Saarlandes, 66123 Saarbr¨ ucken, Germany E-mail: [email protected]; [email protected]

∗

To whom correspondence should be addressed

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ACS Paragon Plus Environment

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Abstract Elastic properties of nanoconfined argon show deviations from the behavior of bulk argon. In this paper we study the pressure dependence of the adiabatic longitudinal modulus βAr,ads of the adsorbate in nanoporous Vycor glass. Adsorption induces a normal pressure ∆pS acting on the pore surface. As it is well-known, this pressure leads to a deformation of the porous sample, but it also influences the elastic properties of the adsorbate. Ultrasonic measurements reveal a linear relation between βAr,ads and the Laplace pressure, pL , which is a major contribution to ∆pS . For the whole temperature range of the confined liquid argon (76 − 86.9 K) the proportionality constant, αAr,ads = dβAr,ads /dpL , is higher than for bulk argon, i.e. the adsorbed argon exhibits a stronger dependence on pressure than bulk argon. In addition, we observe no temperature dependence of αAr,ads , which contrasts to the clear increase of αAr,bulk with increasing temperature. Further measurements and simulations can show, how the interaction strength between adsorbate and pore surface contributes to the observed effects.

1

Introduction

Elastic properties of porous materials are of significance for different fields of research. Porous media offer a multitude of applications as a result of their possibility to adsorb fluids. 1,2 For example, the potential of porous matrices for the storage of methane or hydrogen is interesting for the automobile industry. 3,4 But also in nature we find a great number of porous rocks (e.g., shales and porous carbons), which exhibit pores of different dimensions including nanopores. 5 Such heterogeneous systems are generally described by their effective elastic properties. However, these properties depend on the elastic moduli of the constituents of the material as well as on the microstructure. 6–8 Thus, also the exploration of geological formations using sonic techniques requires a thorough understanding of the behavior of material confined in pores. A priori it is not clear, whether material in nanopores behaves elastically like the 2


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bulk substance. There are many factors that can influence the confined material’s properties. In nanopores the proportion between surface and volume is considerably increased in comparison to bulk, which can lead to a strong impact of the pore surface. It is known that the interaction between the adsorbate and the pore surface affects the structure or order of adsorbed molecules. 1 Some recent observations of differences for the elastic behavior of nanoconfined material can certainly be attributed to such a surface effect. 9–12 Adsorption causes a pressure on the pore surfaces, which results in the well-known effect of sorption-induced deformation. 13–27 This pressure depends amongst others on the surface tension between the fluid and the pore surface as well as on the pore radius (the smaller the pores, the higher the adsorption pressure). 14,15,28 For argon in nanopores Gor et al. 29,30 have shown that this surface tension is related to an enhancement of the elastic moduli of the adsorbate. Previously, ultrasonic measurements had revealed that the Laplace pressure, which is another contribution to the deforming pressure, influences the longitudinal modulus of confined argon, βAr,ads . 31 These measurements indicated a linear relation between the longitudinal modulus and the Laplace pressure. For many bulk substances it is wellknown that the pressure dependence of elastic moduli can be approximated by a linear relationship (Tait-Murnaghan equation). 32–34 Relying on computer simulations the authors in Ref. 34 conclude that the slope of the isothermal modulus versus pressure is not markedly affected by nanoconfinement (provided it is not solvophobic). They even propose to use this feature to estimate the elastic moduli of unknown porous media. 34 But until now an experimental verification is owing. In this paper we study the influence of nanoconfinement on the pressure dependence of the adsorbate’s adiabatic elastic properties via systematic ultrasonic measurements. Liquid argon in nanoporous Vycor glass serves us as a simple model system to identify the basic factors influencing the elasticity in nanopores. Isothermal sorption measurements at different temperatures give us the possibility to investigate the impact of temperature on the pressure dependence.

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2

Experimental Details

In this study we have used a nanoporous Vycor glass sample with an average pore radius of rp = 3.8 nm and a porosity of φ = 27.6%. Porous Vycor possesses a three-dimensional structure of interconnected pores and its matrix is nearly pure quartz glass. 35 The sample and the experimental setup is identical to the one used in Ref. 36. The porous glass was filled at different temperatures between 76 and 86.9 K with liquid argon via the gas phase. 37 For this, the pressure p in the sample cell was successively increased up to the value of the bulk saturation vapor pressure p0 at the corresponding temperature. Similarly, the sample was drained by a stepwise decrease of the relative pressure p/p0 . The volume filling fraction of the pores, f = VAr,ads /Vpores , is determined via measurements of the velocity ct of ultrasonic shear waves propagating through the sample (for details see Refs. 36,38). This method for the determination of f requires that the adsorbate is liquid and possesses a vanishing shear modulus (and in this case it yields the same results as volumetric measurements). 36,38 From previous measurements we know that solid argon is only present for temperatures below T ≈ 75.8 K. 11 Furthermore, the variation of the relative pressure p/p0 during a sorption isotherm causes no significant change of the effective shear modulus, i.e. in the temperature range of the measurements the adsorbed argon behaves basically like an ideal liquid (see Fig. 1 in Ref. 39). Simultaneously to the velocity of shear waves, ct , we have measured the velocity of longitudinal ultrasonic waves, cl , which allows the calculation of the effective longitudinal modulus β = ρ0 (cl · ct,0 /ct )2 , where ρ0 denotes the density of the unfilled sample and ct,0 the velocity of the shear waves propagating through the unfilled sample (see Ref. 36). As we have pointed out in Ref. 36, the determination of the modulus does not depend on the density of the adsorbed liquid (that may be unknown and differ from the bulk density). As a result of the ultrasonic method, all elastic quantities presented in this paper represent the adiabatic values. For the determination of the longitudinal modulus of the adsorbed argon, βAr,ads , we have used our effective medium analysis that relates the modulus of the pore filling, βp , to 4


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the longitudinal moduli of the empty and (partly) filled sample, β0 and β, and that of the matrix material, βm (in the high contrast limit βp βm ): 28,36,40

β0 β = β0 + 1 − βp βm | {z }

(1)

1/C

For completely filled pores βp is equal to the modulus of the adsorbed argon, i.e. βp = βAr,ads . In this paper we study the adsorbate’s modulus only for pore fillings near f = 1 (for relative pressures p/p0 & 0.7). This means that we consider a range where all pores are filled with capillary condensate, but the liquid-vapor menisci at the pore ends vary their curvature as a consequence of the variation of p/p0 . This influences the filling fraction very slightly, which can be taken into account by replacing βp = f βAr,ads (see Ref. 28). Generally, this method supplies values for the modulus of adsorbate in Vycor that are in accordance with theory (see, e.g., Refs. 28,36) using the value of nonporous Vycor glass for the modulus of the matrix material, i.e. for βm = βnpV = 72.85 GPa. 36 41 However, Eq. 1 is not the only effective medium formula predicting a relation between measured properties of a sample (β, β0 ) and that of the adsorbate, βAr,ads . Under the assumption that the pore fluid has enough time for the equilibration of wave-induced pressure gradients, Gassmann developed in 1951 a different relation between the effective modulus and the elastic moduli of the constituents of the porous system. 6,7,42,43 In the high contrast limit Gassmann’s equation has the following definition for the factor C in Eq. 1: C = φ/ (1 − K0 /Km )2 (with the bulk modulus of the empty sample, K0 , and that of the matrix material, Km , see Refs. 36,44). For ultrasonic waves propagating through a porous system with a carrier frequency in the range of some MHz a pressure equilibration is not necessarily possible. 6,7 A low pore connectivity can prevent the required equilibration. 45 It would be also the case in materials that are homogeneous on the scale of the wavelength λ (here about 300 µm) but heterogeneous on a smaller scale exceeding the fluid diffusion length, 46 up to which pressure equilibration can take place. Using the value of the bulk modulus of nonporous Vycor glass, KnpV = 35.58 GPa, for the modulus of the

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matrix material (i.e., Km = KnpV ), Gassmann’s equation yields values for the adsorbate’s modulus that are far too low (see Fig. 6 in Ref. 36). There are some hints that the moduli of the matrix material are lower than those of nonporous Vycor glass. 47–50 Our previous study on the system of argon in nanoporous Vycor glass revealed that acceptable values for the modulus of the adsorbed argon (i.e., values that are in accordance to theory) can either be obtained with the Gassmann equation using Km /KnpV ≈ 0.405 ± 0.01 or with our equation for 0.8 . βm /βnpV ≤ 1 (see also Fig. 6 in Ref. 36). Accordingly, a recent study suggested that the effective modulus of porous Vycor filled with liquid adsorbates (argon or hexane) can be estimated with Gassmann’s equation. 50 However, the modulus of the matrix material, Km , has to be known with a high precision to obtain reasonable values for the modulus of the adsorbate, βAr,ads (see above and Ref. 36). At present it is not yet possible to determine Km with the required precision. Different techniques for the determination of the elastic modulus of the matrix material yielded varying values. 47–50 The ratio Km /KnpV reported for porous Vycor glass samples varies significantly between 0.40 and 0.74. 48–50 Thus, the usage of Gassmann’s equation for the evaluation of the adsorbate’s modulus remains currently problematic. In contrast, our factor C is less sensitive to errors in the modulus of the matrix material (see also Ref. 36). A definite clarification of the appropriate form of the factor C is still required, but the value of the factor C that we use is appropriate (see also Ref. 36). In order to make sure that our results are not forged by an error in the factor C, we also shall perform a consistency check. As an alternative evaluation we will put the method proposed in Ref. 34 to the test on our system of argon in nanoporous Vycor glass (see below).

3

Measurements and Analysis

We have measured the effective longitudinal modulus of the nanoporous system consisting of argon in Vycor glass as a function of the relative pressure, p/p0 , at different temperature between 76 and 86.9 K. The temperatures studied cover the complete existence range for

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Figure 1: (a) Filling fraction f and (b) effective longitudinal modulus β of the porous Vycor glass sample as a function of the relative pressure p/p0 during sorption of liquid argon at 80 K. Only for nearly completely filled pores (f ≈ 1) the effective modulus β differs from the modulus of the empty sample β0 . In the corresponding pressure range the effective longitudinal modulus β exhibits a nearly linear dependence on the relative pressure [see red marking in (b)].

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liquid argon in the nanopores of Vycor glass with a pore diameter of ≈ 8 nm. 11,51 For partial pore fillings liquid adsorbate and vapor coexist in the pores. As a result of the high compressibility of the vapor the effective longitudinal modulus is not noticeably influenced by the liquid adsorbate as long as vapor remains in the pores (see Fig. 1 for T = 80 K and for details Refs. 40,44,52). For filling fractions near f = VAr,ads /Vpores = 1 the effective longitudinal modulus β differs from the modulus of the empty sample, β0 . In accordance with our previous results in Ref. 31, the longitudinal modulus exhibits for f ≈ 1 an almost linear dependence on p/p0 [see Fig. 1(b) above p/p0 ≈ 0.94 on adsorption as well as above p/p0 ≈ 0.75 on desorption]. During desorption the intrusion of some vapor in the pore system near p/p0 = 0.75 leads again to a deviation from this linear behavior and a strong reduction of β (see Fig. 1). At all temperatures of our measurements between the freezing and boiling points of the confined argon we observe a similar behavior of the effective longitudinal modulus β as the one for 80 K presented in Fig. 1. In the pressure range where f ≈ 1 holds, the experimental data reveal a continuous decrease of ∆β = β − β0 (i.e., of the adsorbate’s contribution to the effective longitudinal modulus β) with decreasing relative pressure p/p0 [see Fig. 2(a)]. At different temperatures the values for ∆β vary for a given relative pressure, which is basically a result of the temperature dependence of the adsorbate’s longitudinal modulus. Previously, we could attribute the nearly linear dependence of the effective longitudinal modulus β on the relative pressure p/p0 to a change of the longitudinal modulus of the adsorbate, βAr,ads , which is again a consequence of a change of pore pressure. 31 For bulk materials it is known that elastic moduli generally depend on pressure, i.e. the higher the pressure the higher the moduli. In a limited pressure range the dependence between the moduli of bulk substances and the applied pressure pa can be approximated by a linear function (Tait-Murnaghan equation), 32–34,53 which holds also for bulk argon (see below and Supporting Information):

8


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Figure 2: (a) Contribution ∆β = β − β0 of the adsorbed liquid argon to the effective longitudinal modulus as a function of the relative pressure p/p0 (for f ≈ 1) at different temperatures covering the complete liquid regime of argon in porous Vycor glass. (b) Longitudinal modulus for the adsorbed liquid argon, βAr,ads (calculated from the ultrasonic data via Eq. 1), as a function of the Laplace pressure pL (Eq. 3). The data reveal a continuous decrease of the effective modulus β with decreasing relative pressure. The almost linear reduction of the adsorbate’s longitudinal modulus with increasing (negative) Laplace pressure reveals the pressure dependence of βAr,ads .

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βAr,bulk = βAr,bulk (p0 ) + αAr,bulk (pa − p0 )

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(2)

During a sorption isotherm the change of the relative pressure causes a significant Laplace pressure because of the curvature of the interface between the liquid and its vapor. 31 A reduction of the relative pressure p/p0 (in a range where f ≈ 1) results in an increase of the curvature of the menisci at the pore ends. This curvature is related to a Laplace pressure, pL , that can be expressed as 54 Rg T ln pL = Vm

p p0

,

(3)

with the gas constant Rg , the temperature T , and the molar volume of the adsorbate, Vm [calculated via Vm = Mm /ρAr,bulk (with the molar mass of argon, Mm = 39.948 g/mol) using a linear fit to the data for the density reported by Goldman and Scrase 55 , see solid line in Fig. S1(b) in the Supporting Information]. 56 The above equation reveals that a reduction of the relative pressure p/p0 increases the negative Laplace pressure. Thus, the reduction of the relative pressure from p/p0 = 1 to p/p0 ≈ 0.8 increases the negative Laplace pressure from 0 to ≈ −6 MPa (cp. area marked in Fig. 1(b) for f ≈ 1 at 80 K). The simultaneous change of the pressure p in the sample cell, p ≤ p0 , is generally negligible in comparison to the change of the Laplace pressure (p0 < 0.1 MPa for all temperatures). From our measurements of the effective longitudinal modulus we can calculate the modulus of the adsorbed liquid argon via our effective medium analysis (see Eq. 1). The adsorbate’s modulus βAr,ads exhibits at all temperatures a continuous decrease with increasing (negative) Laplace pressure, pL [see Fig. 2(b)]. The absolute values of βAr,ads increase with decreasing temperature, which corresponds to the temperature dependence of bulk argon (see Ref. 36). Thus, the analysis of our ultrasonic data at different temperatures shows that a linear relation between the longitudinal modulus of the adsorbed liquid argon, βAr,ads , and the Laplace pressure pL can be observed [analogous to Eq. 2 for bulk argon]. At all temperatures studied

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the slope αAr,ads = dβAr,ads /dpL seems to be similar. Linear fits to the data reveal that αAr,ads varies between ≈ 13.6 and ≈ 17 for different temperatures (see Fig. 3) and we obtain an average value of α = 15.3 (see solid blue line). Within the limits of the accuracy of the measurements we do not observe any clear temperature dependence. Note that the detection of the pressure dependence requires a high resolution of the ultrasonic measurements. For example, at 80 K the transit time for the ultrasonic waves changes by only ≈ 0.2% between pL = 0 and pL ≈ −6 MPa (cp. Fig. 2 for 80 K).

Figure 3: Pressure dependence of the longitudinal modulus of the confined liquid argon as a function of temperature. The values for αAr,ads = dβAr,ads /dpL show no clear temperature dependence and we observe an average value of αAr,ads = 15.3 (solid line). The error bars are based on the standard errors in the slopes of the linear fits to the data shown in Fig. 2(b). These observations raise the question, whether the pressure dependence differs from that for bulk argon. For this comparison we consider the pressure dependence studied during ultrasonic measurements by Thoen et al. 57 and Carome et al. 58 as well as the data from the NIST 59 for bulk argon. 60 Carome et al. 58 have measured the ultrasonic velocity (of longitudinal waves) up to a pressure of ≈ 10 MPa. Our calculation of βAr,bulk (p) = cl (p)2 ρ (p) using these data shows an almost linear relation with the pressure p (see Supporting Information). The proportionality constant αAr,bulk (see Eq. 2) increases clearly with the temperature (see turquois symbols in Fig. 4). The analysis of the measurements by Thoen et al. 57 and of the data from the NIST 59 reveals similar values for αAr,bulk , if we consider pressures up to 10 MPa 11



(see red solid circles and purple solid stars in Fig. 4). However, the relation between βAr,bulk and the pressure is not absolutely linear but somewhat weaker for higher pressures, which is shown by the values of αAr,bulk resulting from a linear fit up to a pressure of ≈ 25 MPa (see red open circles and open purple stars in Fig. 4). Nevertheless, for bulk argon a clear temperature dependence of αAr,bulk can be observed experimentally, which contrasts to our observation for argon confined to the nanopores of porous Vycor glass. Furthermore, the pressure dependence for the confined argon seems to be stronger than for bulk argon (on the average more than 20%, see the blue squares in Fig. 4). The observed pressure dependences for the nanoconfined argon as well as for bulk argon (presented in Fig. 4) are for the adiabatic longitudinal modulus (βAr,ads or βAr,bulk ) as the ultrasonic experiments can be regarded as an adiabatic process. In contrast to these observations for the adiabatic modulus, Monte Carlo simulations for the isothermal modulus could not reveal a distinct difference between the pressure dependence of bulk argon and argon confined in spherical silica pores with smooth walls. 34

Figure 4: Comparison between the pressure dependence of the longitudinal modulus of the nanoconfined liquid argon (blue squares) and liquid bulk argon (red, turquois, and purple symbols). The confined argon exhibits a stronger pressure dependence than bulk argon. Furthermore, we observe no temperature dependence, in contrast to bulk argon. The values for bulk argon are calculated from ultrasonic measurements by Thoen et al. 57 and Carome et al. 58 as well as from data from the NIST 59 (see text and Supporting Information). Our observation of a (roughly) temperature independent pressure dependence of the 12


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adsorbate’s modulus has also an implication for the sorption-induced deformation of the nanoporous matrix. It is well-known that adsorption causes a pressure on the pore surface that becomes noticeable in a macroscopic deformation/expansion of the porous matrix. 14,15 The Laplace pressure discussed above contributes to this pressure but it is not the sole contributing quantity. Gor and Neimark 14,15 have shown that the so-called solvation pressure includes also a surface term (and the external pressure p). Thus, the deformation can be described by the following normal pressure ∆pS , which is exerted by the adsorbate, 28

∆pS =

∆psat | {zS }

+pL + (p0 − p) ,

(4)

−γSL /rP −σ0

with the solid-liquid surface tension γSL (pore surface-adsorbate), the pore radius rP , and the pre-stress σ0 in a reference state. Note that the 1/rP scaling of ∆psat S is valid for pore diameters above a few nanometers. 14 At the bulk vapor pressure p = p0 the Laplace pressure, pL , as well as the term in the brackets, (p − p0 ), vanish, and ∆pS = ∆psat S , i.e. the normal pressure at saturation. As a result of this normal pressure the porous system is deformed also at saturation (cp. Refs. 14,15,28). During a sorption measurement with our sample the main variation of ∆pS can be attributed to the change of the Laplace pressure. The pressure 15,34,61 A recent contribution ∆psat S is generally regarded as constant for a given temperature.

estimate of the variation of γSL with the relative pressure p/p0 (for a system of argon and silica) suggests that this is a reasonable assumption (see Ref. 61, in particular footnote 132 therein). Now let us assume that we obtain bulk properties for a vanishing normal pressure (i.e. for ∆pS → 0). 28 Using the linear relation between the adsorbate’s modulus and the pressure, βAr,ads = βAr,bulk + αAr,ads ∆pS (see above), and Eq. 4, we obtain the following equation for the modulus of the adsorbate:

βAr,ads = βAr,bulk + αAr,ads ∆psat S + αAr,ads [pL + (p0 − p)]

(5)

The first two terms on the right hand side do not vary during a sorption isotherm (see 13



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assumption above) and the contribution in the round brackets, (p0 − p), is generally negligible (p ≤ p0 pL ). Consequently, βAr,ads ≈ βAr,bulk + αAr,ads ∆psat S + αAr,ads pL holds and thus dβAr,ads /d∆pS ≈ dβAr,ads /dpL . Both values for the slope differ by less than 0.4% for our measurements. As we have shown recently, the above relation (Eq. 5) allows us to evaluate the deforming pressure, ∆psat S , at the bulk vapor pressure, p0 . At the saturation vapor pressure the last term on the right hand side of Eq. 5 vanishes and ∆psat S can be calculated from the difference between the experimentally determined modulus of the adsorbate (at p = p0 ) and the modulus of bulk argon [i.e., (βAr,ads − βAr,bulk )p=p0 ] as well as the slope αAr,ads : 28

∆psat S =

(βAr,ads − βAr,bulk )p=p0 αAr,ads

(6)

Using our data (see Figs. 2 and 3) as well as Eq. 6 we obtain values for ∆psat S that are displayed in Fig. 5. On the average we observe a value of ∆psat S = 10.0 MPa (solid blue line).

Figure 5: Normal pressure at saturation ∆psat as a function of temperature (calculated S according to Eq. 6). The pressure varies around an average value of 10.0 MPa (solid line) and exhibits no significant temperature dependence. This pressure ∆psat S acts on the pore surface at the saturation vapor pressure, p0 . Thus, the analysis of the ultrasonic measurements indicates that the expansion of the porous Vycor glass sample does not significantly depend on temperature (for 76 K ≤ T ≤ 86.9 K). The error bars are based on the standard errors in αAr,ads (see Fig. 3).

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When we developed this method in a previous study, we obtained for a different Vycor ≈ 15 MPa), however, sample slightly higher values for the pressure at saturation (∆psat s the measurements had a somewhat larger error and we did not study the temperature dependence. 28 Nevertheless, this observation could be related to slightly different structures of the pores or pore surfaces. Such a difference can lead to a different interaction between adsorbate and pore surface and consequently also to a different pressure and deformation. Density functional theory yielded at 80 K a value of ∆psat S ≈ 16 MPa for argon in a smooth cylindrical silica pore with a pore radius of rP = 3.8 nm, but this study disregarded the heterogeneous pore structure of actual samples. 28,29

Figure 6: Modulus of the adsorbed liquid argon, βAr,ads , as a function of temperature at saturation (p = p0 ). Density functional theory and simulations predict an enhancement of βAr,ads in comparison to the bulk values [cp. shaded area (Eq. 5 with αAr,ads = 15 ± 2 and ∆psat S = 12 ± 4 MPa) and green solid line for βAr,bulk (data calculated from Refs. 55,62, see Supporting Information)]. Our effective medium analysis (see solid red circles, calculated via Eq. 1) supplies moduli that are in accordance to theory. The assumption that the pressure dependence of the adsorbate’s modulus, αAr,ads = dβAr,ads /dpL , is equal to that for bulk argon, αAr,bulk (using the linear fit to the values from Ref. 58, see Fig. 4), would lead to values for βAr,ads that are far too low (see text). This clearly indicates that the pressure dependence of the longitudinal modulus of nanoconfined argon is stronger than that of bulk argon. The results presented above for αAr,ads and ∆psat S depend of course significantly on the effective medium analysis, i.e. in particular on the factor C that relates the measured moduli to the modulus of the adsorbate, βAr,ads = C ·(β − β0 ) (cp. Eq. 1 with βp = βAr,ads , neglecting 15



the minor change in f ). The definition αAr,ads =

dβAr,ads dpL

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thus relates the measured effective

modulus to αAr,ads and the factor C:

αAr,ads = C

∂β ∂pL

(7)

Recently, the authors of Ref. 34 suggested that for the isothermal modulus the slope αT for nanoconfined liquids equals that for bulk liquids, provided that the confinement is not T T solvophobic, αAr,ads = αAr,bulk (a conclusion drawn from Monte Carlo simulations for argon

in silica mesopores). They also proposed to use this equality to estimate the elastic moduli of an unknown porous medium from ultrasonic data. In order to check this hypothesis, we calculate the proportionality factor in Eq. 1 or 7 ∂β as C = αAr,bulk / ∂p using the measured slope ∂β/∂pL [see Fig.2(a), transforming the x-axis L

from p/p0 to pL ] as well as a linear fit to the adiabatic bulk values of αAr,bulk from Ref. 58 [see the turquois line in Fig. 4, αAr,bulk = 3.4257 + 0.1145/K · T ]. The values for the modulus of the adsorbed argon calculated under this assumption, βAr,ads = C (β − β0 ) = ∂β αAr,bulk (β − β0 ) / ∂p , are shown in Fig. 6 (at saturation, that is for p = p0 ). Obviously, L

this method yields values for the adsorbate’s modulus that are far too low, most values are even below those for the modulus of bulk argon. This contradicts theoretical analyses (via DFT and MC simulations) that predict an enhancement of the adsorbate’s modulus as a result of the solvation pressure. 29,30,63 If we used the weaker pressure dependence from Ref. 57 for the bulk values of αAr,bulk (red lines in Fig. 4), the calculated moduli βAr,ads would be further reduced. Summarizing, we conclude that the pressure dependence of the adiabatic longitudinal modulus of nanoconfined argon is not equal to that of bulk argon, αAr,ads 6= αAr,bulk . Finally, let us check whether our effective medium analysis is self-consistent. For this purpose we compare the values for βAr,ads at p = p0 obtained using Eq. 1 to what we expect [Eq. 5 with αAr,ads = 15 ± 2 (see Fig. 3) as well as ∆psat S = 12 ± 4 MPa. The latter value covers both the data in Fig. 5 as well as a value obtained via density functional theory 16


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29 (∆psat An error in the determination of the pore size would S ≈ 16 MPa for rP = 3.8 nm).

increase the uncertainty (∆psat S ∝ 1/rP ), which would lead to an extension of the shaded area in Fig. 6 to higher values]. Obviously the data is not contradictory (see Fig. 6).

4

Conclusions and Outlook

With this ultrasonic study we have shown that the pressure dependence of the adiabatic longitudinal modulus of adsorbed argon in nanoporous Vycor glass is of a similar magnitude but about 20% stronger than that of bulk argon. The analysis of our measurements has shown that the dependence of the adiabatic longitudinal modulus on the Laplace pressure can be readily approximated by a linear Tait-Murnaghan equation. For the whole temperature range (76 − 86.9 K), αAr,ads =

dβAr,ads dpL

is nearly constant and has an average value of αAr,ads ≈ 15.3.

In contrast to bulk argon, the temperature variation (between 76 and 86.9 K) causes no noticeable change of αAr,ads . Likewise, there is no significant impact of temperature on the normal pore pressure at saturation, ∆psat S , which suggests that also the sorption-induced deformation is not significantly influenced by temperature (for 76 − 86.9 K). Previously, simulation for smooth spherical silica pores revealed no significant influence of nanoconfinement on the pressure dependence of the adsorbate’s isothermal modulus. 34 There are several possibilities to explain this discrepancy. If nanoconfinement would affect only adiabatic but not isothermal moduli, also the quantities relating both would be altered (i. e., T the heat capacity ratio, cp /cv = βAr,ads /βAr,ads ). On the other hand, the difference might be

related to the confinement studied: for the isothermal simulations these were smooth surfaces of pure silica, for our adiabatic measurements Vycor glass exhibiting rough pore walls with silanol-OH-groups. 35 Their presence can alter the interaction with the adsorbed argon and thus its modulus βAr,ads . In addition, the surface tension γSL and thus the normal pressure ∆pS may change, which in turn affects βAr,ads . In fact, previous simulations have shown that the interaction between adsorbate and pore surface significantly changes the pressure

17



dependence. 34 Consequently, the observed enhancement of the pressure dependence might be related to this interaction strength. Summarizing, the actual pressure dependence of the adiabatic modulus of a nanoconfined liquid is influenced by various parameters. At least for the system we have studied (Vycor glass), the method proposed in Ref. 34 does not give us a direct access to the elastic properties of the adsorbed argon. The experimental observation of the possibility to describe the pressure dependence of adsorbed argon in nanopores with a modified Tait-Murnaghan equation may help to describe the elastic behavior of natural materials (including porous rocks). Before a generalization of our results to other adsorbates (e.g., alkanes) and pore systems additional measurements and simulations are required. Thus, for an assessment of the impact of pore size on the adsorbate’s modulus and its pressure dependence, experimental data for different pores sizes are strongly needed. Previously, theoretical studies for smooth pores indicated a significant impact of the pore size on the longitudinal modulus. 29,30,63 Also the impact of a broader pore size distribution, which is typical for natural materials, remains to be studied. In addition, a variation of the adsorbate (and thereby of the interaction strength with the pore surface) seems a promising direction for further research. Such measurements will certainly contribute to a thorough understanding of the factors influencing elastic behavior in nanopores and the pressure dependence.

Supporting Information longitudinal modulus of bulk argon at the saturation vapor pressure pressure dependence of the longitudinal modulus of bulk argon

Notes and References (1) Huber, P. Soft Matter in Hard Confinement: Phase Transition Thermodynamics, Structure, Texture, Diffusion and Flow in Nanoporous Media. J. Phys.: Condens. Matter

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Liquid Argon in Nanopores: The Impact of Confinement on the

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