Cavitation of Water Confined in Hydrophobic Nanoporous Materials

1,2Antonio Tinti*; 2Alberto Giacomello; 2Carlo Massimo Casciola;

1Max-Planck-Institut für Intelligente Systeme, Stuttgart, Germany; 2Sapienza Università di Roma, Rome, Italy

Abstract

Hydrophobic nanoporous materials immersed in water are emerging as promising means to store ordissipate energy [1-3]. In such applications, surface energy is accumulated when the water pressure isincreased causing liquid intrusion inside the pores and can be subsequently released by decreasingpressure and triggering cavitation inside the pores. Given the extreme confinement, which for zeolitesand metal organic frameworks can be below the nanometer, the phenomena of liquid intrusion andcavitation are expected to significantly deviate from the classical macroscopic laws: cavitation canhappen at pressures larger than tens of MPa. In this contribution, we study via molecular dynamics the nucleation of a vapor cavity insidenanometer-sized hydrophobic pores and the opposite process of liquid intrusion. By employing ad-vanced rare-event simulation techniques in order to tackle the long timescales typical of vapor nucle -ation, we obtain molecular-level insights into nanoconfined cavitation (and liquid intrusion) avoidingsimulation artifacts [4]. The simulation campaign reveals deviations from the macroscopic Kelvin-Laplace law for liquid intrusion in a capillary and a significant increase of the cavitation rate as com-pared to the predictions of the classical nucleation theory. Furthermore, the behavior of nanoporousmaterials as molecular springs or as vibration dampers is critically discussed and related to theirphysical characteristics.

Keywords: Cavitation, Nanoscale, Molecular Dynamics, Molecular Springs, Confinement

Introduction

Cavitation in extreme confinement is responsible for the innovative energetic application of heterogeneous lyopho-bic systems (HLS), in which a nanoporous material is paired with a non-wetting liquid: liquid-vapor transitionwithin the pores determines the accumulation of surface energy, owing to the immense surface area density of por-ous materials. During the intrusion phase, the liquid is forced by large liquid pressures inside the pores accumulatingsurface energy; during the extrusion phase, nanoconfined cavitation is triggered by the decreased pressure. Depend-ing on their characteristics, HLSs are able either to efficiently dissipate or to store mechanical energy. Energy dissi -pation is related to the difference in the cavitation (extrusion) and intrusion pressures. If this difference is minimal, itis possible to exploit such systems as efficient energy storage devices. A number of such systems were tested experi-mentally in previous works [1-3,5]. The behavior of these systems, associated with the intrusion and extrusion of water inside the nanopores, chal-lenges the present understanding based on macroscopic theories and classical nucleation theory (CNT) because ofthe peculiar behavior of water in extreme confinement. In this contribution we show how advanced molecular dy-namics techniques can be used to investigate these non-classical effects, which are often difficult to probe directlyvia experiments, owing to the nanometric size and fast timescales that characterize the phenomenon. A state-of-the-art rare-event technique is used to observe, without artifacts, the thermally-activated, microscopic mechanism of wa-ter intrusion and cavitation in the pores. In addition, in silico intrusion-extrusion experiments are presented, showinghow, as the size of the nanopore decreases, the intrusion-extrusion hysteresis is suppressed, making these materialsideal candidates for mechanical energy storage. Our simulations were able to measure the existence of non-classicaleffects, i.e., phenomena that are not accounted for in classical nucleation theory (CNT). In particular the cavitationfree-energy barrier is reduced as compared to CNT, accounting for an accelerated cavitation process, and the pres-sure of liquid intrusion is increased due to nanoscale confinement. A dependence of the intrusion/extrusion hystere-sis on the pore size is also exposed by the present simulations. The reported effects are able to provide a microscopicinterpretation of several experimental results on nanoporous materials.

*Corresponding Author, Antonio Tinti: atinti@is.mpg.de

Cavitation at the Nanoscale

Experimental studies of confined cavitation often result in the measurement of the nucleation rates at given experi -mental conditions [2]. The exponential dependence of the nucleation rates on the thermodynamic properties (pres-sure, surface tension, contact angles, etc.) makes these experiments an exceptional probe for elusive quantities suchas the line tension or the curvature dependence of the surface tension [6]. In these cases, a nucleation theory, provid -ing a theoretical expression for the rates, is needed in order to extract these quantities from the experimental mea -surements of the nucleation rates. Simple continuum theories based on the classical nucleation theory (CNT) rely on a number of approximations.These simplifications [4,7], which are often reasonable at a micrometric or mesoscale level, tend to break downwhen the system size falls below a few nanometers. The original CNT [8-10] can be extended to cope with heteroge-neous and confined cases [7,11], by studying the energetics of a cavitation bubble growth as a function of the bubblevolume. More sophisticated approaches, based on atomistic simulations can also be adopted. In particular, molecular dynam-ics (MD), coupled with state-of-the-art rare events methods such as the string in collective variables [12,13], is ableto provide an unbiased picture of nanoconfined cavitation including all genuine nanoscale phenomena, such as ther -mal fluctuations, packing and line effects, etc. A detailed comparison of MD and extended CNT techniques for cavi-tation in cylindrical confinement is presented in [14].

Fig. 1) Snapshot of the atomistic system during a string method calculation; a single nanopore, excavated from a hydrophobic Lennard-Jonescrystal, is surrounded by ca. 13000 TIP4P/2005 molecules. In order to drive the string method we adopt, as a set of collective variables, the

coarse-grained density field of water, computed on a grid of 4×4×10 cubic cells, whose sides measure ca. 7Å. The liquid pressure is kept con-stant, around liquid-vapor coexistence, using two solid walls acting as pistons (not shown in figure).

Simulation Methods

Atomistic simulations have been deployed in order to study cavitation inside a cylindrical nanopore with an un -precedented spatial and temporal resolution. These simulations combine molecular dynamics with the string methodin collective variables [12] in order to simulate the formation of a vapor bubble inside a cylindrical hydrophobicnanopore (confined cavitation) and compute the related free energy. The pore, having a diameter of 2.6 nm, is exca -vated from a crystalline material whose atoms interact with the oxygen atoms of the water molecules via a modifiedLennard-Jones potential. The strength of the attractive term is tuned in order to render the desired contact angle of119º on a flat surface, as measured in an independent simulation. The TIP4P/2005 water model [15] was adopted inthe simulations, as it reproduces the main features of water (and in particular the surface tension) at the temperatureof our interest (300 K). The string method uses a coarse grained density field as the collective variable. For furtherdetails on the simulations we refer to [16] and for the string method in collective variables to the original article[12]. A snapshot of the atomistic system is presented in Fig. 1. The main result of the rare event atomistic simulationof cavitation inside the model hydrophobic pore is the free-energy profile shown in Fig. 2, in which the volume ofthe vapor bubble is used as the progress variable for the cavitation process. This profile will be discussed in the nextsection. Another, simpler simulative protocol is used in order to study the intrusion and extrusion process of water ina hydrophobic pore, which, however, does not yield the detailed information on the free energy. By varying the pres-sure in discrete steps, and letting the system equilibrate for a sufficient time, it is possible to mimic in silico an intru-sion-extrusion experiment. This technique, similar to the one used in [17], has been adopted in order to measure theintrusion and extrusion pressures for a smaller hydrophobic pore with a diameter of 1.2 nm excavated from the samecrystalline material.

Fig. 2) Landau free energy profiles (T = 300 K, ∆P = 0) obtained via the atomistic string method (red symbols), via extended CNT (green line),and via extended CNT with the addition of terms proportional to the triple line and an effective contact angle (light blue line). We can distinguishbetween two regions, indicated in figure as I and II. The first one is associated with the cavitation of a critical bubble, whereas in the second re-

gion we observe the sliding of two symmetric liquid-vapor menisci. “Modified CNT” free-energy profiles are obtained by evaluating the free en-ergy functional Ω = ∆PVv + γlv(Alv + cos(θY) Asv) along the path returned by the atomistic string calculations. The green line shows how it is pos-sible to correct these predictions for nanoscale effects by adding an effective “line tension” (i.e., a term proportional to the triple line) τ = −1.1 ·

10−11 N along with the adoption of an effective contact angle θeff = 129◦ in the same free energy functional.

Discussion

Figure 2 shows a comparison of the free-energy profiles computed via the atomistic string method and via the ex-tended CNT [11,14], revealing the occurrence of a first non-classical effect: the atomistic cavitation free-energy bar-rier are reduced six-fold as compared to the CNT predictions. This is consistent with the experimental observationsby Charlaix and coworkers [2] on MCM-41 who reported extrusion at pressures as large as 10 MPa – thenanoporous material which inspired our model MD system. The low cavitation free-energy barrier computed byatomistic simulations indeed accounts for the observation of cavitation even at positive pressures within a timescalecompatible with the experimental one. A second important nanoscale effect is related to the intrusion pressures. Using a procedure explained in detail else -where [16], it is possible to add a pressure term, linear in the volume, to the atomistic free energy, in order to renderthe cavitation free-energy profile at different thermodynamical conditions (This term mimics the ∆PVv contributionin the classical expression Ω = ∆PVv + γlv (Alv + cosθY Asv ) for the free energy of the system). By approaching the conditions that result in the vanishing of the intrusion and extrusion barriers it is therefore possi -ble to obtain an estimate for the intrusion and extrusion pressures which determine the “spinodal” conditions. Forthe 2.6 nm pore, object of our string calculations these spinodal intrusion and extrusion pressures amount respec -tively to ∆Psp

int = 85 MPa and ∆Pspext = −22 MPa, defining a large hysteresis loop. The intrusion pressure is much

higher than the 54 MPa predicted by the classic Kelvin-Laplace equation, showing a significant increase due tonanoscale confinement. A third nanoscale effect which results from the atomistic simulations is related to the sup-pression of this large hysteresis loop when considering smaller pore radii. The in silico intrusion-extrusion experi-ments for the pore with 1.2 nm diameter, presented in Fig. 3 reveal a drastic decrease of the intrusion-extrusion hys-teresis and thus of the dissipation of mechanical energy associated with each intrusion-extrusion cycle. The ∼20MPa hysteresis observed in simulations is further reduced in real-world scenarios in which a macroscopic time is al-lowed for the event, which allow thermal fluctuations to activate both processes. This trend of hysteresis with thepore size is consistent with the experimental observations on zeolites and metal-organic frameworks available in theliterature [1,3,18-21]. For a more quantitative discussion of these nanoscale effects, along with an in-depth compari-son with individual experimental results, we refer the interested reader to [16].

Fig. 3) Intrusion/extrusion cycle for a D = 1.2 nm nanopore showing a reduced hysteresis. The yellow and red curves result from an in silico MDexperiment in which the pressure is changed in discrete steps of 12.5 MPa and the system is then allowed to stabilize for at least 3 ns; The initialsteps in which pressure is varied are discarded when computing the average value for the liquid filling of the nanopore. The blue dashed line rep-

resents the outcome of the same conceptual experiment according to the predictions of classical nucleation theories.

Conclusion

In this contribution we provide a microscopic picture of cavitation in extreme hydrophobic confinement –a scenariowhich is relevant for energy system based on nanoporous materials. These extreme conditions result in deviationsfrom the macroscopic theories, which are responsible for the technologically interesting phenomenology associatedwith intrusion and extrusion of a non-wetting liquid inside lyophobic nanoporous materials. These non-classical ef -fects include reduced cavitation free-energy barriers, allowing for the occurrence of spontaneous cavitation even atpositive pressures, deviations from the Kelvin-Laplace prediction for the intrusion pressure, and the suppression ofintrusion/extrusion hysteresis for smaller nanometer-sized pores. The simulative effort presented in this contributionreveals how such deviations from the macroscopic theory of cavitation due to extreme confinement are associatedwith controllable physical features allowing for the design of innovative devices of technological interest for the ef -ficient, inexpensive, and green storage or dissipation of mechanical energy.

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