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Intrusion and extrusion of water in hydrophobic nanopores

  1. Carlo Massimo Casciolaa
  1. aDipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, 00184 Rome, Italy;
  2. bCIC Energigune, Mi?ano 01510, Spain
  1. Edited by Christoph Dellago, University of Vienna, Vienna, and accepted by Editorial Board Member John D. Weeks October 13, 2017 (received for review August 22, 2017)

  1. Fig. 2.

    Most probable path for the extrusion process computed via the string method in CVs (see also Movie S1). For visualization purposes, the Gibbs dividing surface corresponding to the isosurface <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>ρ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mi>ρ</mml:mi><mml:mi>l</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:math>ρ=ρl/2 is shown. Under the assumption that the process is quasistatic, the most probable intrusion process is obtained by inverting the direction of the arrows.

  2. Fig. 3.

    (Upper) Distance between consecutive density configurations along the discretized path for the atomistic string (black) and for the CNT path (red). The CNT results are obtained via Surface Evolver (65) calculations similar to those in refs. 4 and 5 (Figs. S2 and S3). Note that the final volume is different in the two cases due to excluded volume effects in the atomistic simulations. (Lower) The TS configurations in the (Left) atomistic and (Right) CNT cases.

  3. Fig. 4.

    Free-energy profiles at <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>T</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mn>300</mml:mn></mml:mrow></mml:math>T=?300 K and <mml:math><mml:mrow><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mpadded width="+1.7pt"><mml:mi>P</mml:mi></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>ΔP=?0 obtained via the atomistic string (blue symbols), via modified CNT (dashed orange line), and via modified CNT with terms proportional to the triple line and using an effective contact angle (green line). The two regions related to cavitation of the critical bubble (I) and to the sliding of two symmetric menisci (II) are indicated. “Modified CNT” free-energy profiles are obtained relaxing its third assumption, i.e., using the path calculated from the string isosurfaces shown in Fig. 2. Nanoscale effects are further added by considering an effective “line tension” (i.e., a term proportional to the triple line) <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>τ</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mrow><mml:mn>1.1</mml:mn><mml:mo>?</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>?</mml:mo><mml:mn>11</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:mrow></mml:math>τ=?1.1?10?11 N and an effective contact angle <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:msub><mml:mi>θ</mml:mi><mml:mi>eff</mml:mi></mml:msub></mml:mpadded><mml:mo>=</mml:mo><mml:msup><mml:mn>129</mml:mn><mml:mo>°</mml:mo></mml:msup></mml:mrow></mml:math>θeff=129° in Eq. 2. Additional comparisons with standard CNT and variations thereof are shown in Fig. S2.

  4. Fig. 5.

    (A) Hysteresis cycle for a thought intrusion/extrusion experiment on the <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>D</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mn>2.6</mml:mn></mml:mrow></mml:math>D=?2.6-nm pore; the pressure is varied between the intrusion value <mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:msubsup><mml:mi>P</mml:mi><mml:mi>int</mml:mi><mml:mi>sp</mml:mi></mml:msubsup></mml:mrow></mml:math>ΔPintsp and the extrusion one, <mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:msubsup><mml:mi>P</mml:mi><mml:mi>ext</mml:mi><mml:mi>sp</mml:mi></mml:msubsup></mml:mrow></mml:math>ΔPextsp (black lines). The plot is constructed by adding a term <mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mpadded width="+1.7pt"><mml:mi>P</mml:mi></mml:mpadded><mml:msub><mml:mi>V</mml:mi><mml:mi>v</mml:mi></mml:msub></mml:mrow></mml:math>ΔPVv to the atomistic free-energy profile at <mml:math><mml:mrow><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mpadded width="+1.7pt"><mml:mi>P</mml:mi></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>ΔP=?0 in Fig. 4 and by computing the spinodal pressures; the compressibility of the liquid and of the pore walls is not taken into account, resulting in a rectangular cycle (only the values of the pressure at the plateaus are actually computed; rounded angles help visualization). On the <mml:math><mml:mi>x</mml:mi></mml:math>x axis, the bubble volume <mml:math><mml:msub><mml:mi>V</mml:mi><mml:mi>v</mml:mi></mml:msub></mml:math>Vv is normalized with the maximum value such that it ranges from <mml:math><mml:mn>0</mml:mn></mml:math>0 (fully wet pore) to <mml:math><mml:mn>1</mml:mn></mml:math>1 (vapor bubble occupying the nanopore). The pressures at which intrusion (top plateau) and extrusion (lower plateau) happen for fixed experimental times <mml:math><mml:mi>t</mml:mi></mml:math>t are computed inverting Eq. 1 to obtain the intrusion and extrusion barriers <mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:msup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>?</mml:mo></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>P</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>ΔΩ?(ΔP) corresponding to the prescribed <mml:math><mml:mi>t</mml:mi></mml:math>t (colored lines). (B) Energy <mml:math><mml:msub><mml:mi>E</mml:mi><mml:mi>d</mml:mi></mml:msub></mml:math>Ed dissipated per cycle as a function of the frequency, from refs. 6, 38, 39, and 45 and from A. Experimental data, which are available over a limited range of frequencies, were originally given in different units, which required estimates of the porosity (38, 39) or interpolation of <mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:msub><mml:mi>P</mml:mi><mml:mi>int</mml:mi></mml:msub></mml:mrow></mml:math>ΔPint and <mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:msub><mml:mi>P</mml:mi><mml:mi>ext</mml:mi></mml:msub></mml:mrow></mml:math>ΔPext (45). We remark that experiments are performed on different materials, temperatures, and, for some of them (38), <mml:math><mml:msub><mml:mi>E</mml:mi><mml:mi>d</mml:mi></mml:msub></mml:math>Ed refers to the entire device.

  5. Fig. 6.

    (A) Intrusion/extrusion cycle for a nanopore with <mml:math><mml:mrow><mml:mpadded width="+1.7pt"><mml:mi>D</mml:mi></mml:mpadded><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mn>1.2</mml:mn></mml:mrow></mml:math>D=?1.2 nm showing reduced hysteresis. The black and red curves are computed from an in silico experiment in which the pressure is changed in steps of <mml:math><mml:mn>12.5</mml:mn></mml:math>12.5 MPa and allowed to stabilize for at least <mml:math><mml:mn>3</mml:mn></mml:math>3 ns; the average value of vapor filling is obtained by discarding the initial steps in which pressure is varied. The dashed green line is an illustration of the same experiment as predicted by classical theories: The intrusion pressure calculated via Eq. 3 yields <mml:math><mml:mrow><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mpadded width="+1.7pt"><mml:msubsup><mml:mi>P</mml:mi><mml:mi>int</mml:mi><mml:mi>KL</mml:mi></mml:msubsup></mml:mpadded></mml:mrow><mml:mo>=</mml:mo><mml:mo>?</mml:mo><mml:mn>111</mml:mn></mml:mrow></mml:math>ΔPintKL=?111 MPa. The classical extrusion spinodal pressure <mml:math><mml:mrow><mml:mo>→</mml:mo><mml:mrow><mml:mo>?</mml:mo><mml:mi mathvariant="normal">∞</mml:mi></mml:mrow></mml:mrow></mml:math>→?∞ because the related macroscopic barrier never vanishes: The “classical” intrusion/extrusion cycle is actually not closed for finite pressures. (B) Intrusion and extrusion cycle for MCM-41 from ref. 4. The intrusion and extrusion pressures for the largest and smallest pore diameters are indicated as <mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:msubsup><mml:mi>P</mml:mi><mml:mrow><mml:mi>int</mml:mi><mml:mo>/</mml:mo><mml:mi>ext</mml:mi></mml:mrow><mml:mi>min</mml:mi></mml:msubsup></mml:mrow></mml:math>ΔPint/extmin and <mml:math><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:msubsup><mml:mi>P</mml:mi><mml:mrow><mml:mi>int</mml:mi><mml:mo>/</mml:mo><mml:mi>ext</mml:mi></mml:mrow><mml:mi>max</mml:mi></mml:msubsup></mml:mrow></mml:math>ΔPint/extmax, respectively, showing a decrease in the hysteresis with the size of the nanopores.

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