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Probing Memory Effects in Confined Fluids via Diffusion Measurements Sergei Naumov,† Rustem Valiullin,*,† Peter A. Monson,‡ and Jo¨rg Ka¨rger† Abteilung Grenzflächenphysik, Fakulta¨t fu¨r Physik and Geowissenschaften, UniVersita¨t Leipzig, D-04103 Leipzig, Germany, and Department of Chemical Engineering, UniVersity of Massachusetts, Amherst, Massachusetts 01003 ReceiVed April 30, 2008. ReVised Manuscript ReceiVed May 28, 2008 Confinement of fluids in porous materials is widely exploited in a variety of technologies, including chemical conversion by heterogeneous catalysis and adsorption separations. Important fundamental phenomena associated with many-molecule interactions occur in such systems, including a remarkably long “memory” of the past when the actual amount of molecules in the pores dramatically depends on the history of how the external conditions have been changed. We demonstrate that the intrinsic diffusivity as measured by NMR serves as an excellent probe of the history-dependent states of the confined fluid. A remarkable feature of our results are differences in diffusivity between out-of-equilibrium states with the same density within the hysteresis loop. This reflects different spatial distributions of the confined fluid that accompany the arrested equilibration of the system in this region.
A critical component of the processing of raw materials into value-added products involves conversion by heterogeneous catalysis and chemical separation by molecular sieving and selective adsorption in porous materials. In all cases, the use of nanoporous catalysts and adsorbents ensures a particularly intimate interaction between the guest molecules and the internal surfaces of the host system, which gives rise to the desired processes.1,2 Recent developments in chemistry and materials science are making it possible to tailor the chemistry and nanostructure of porous materials to optimize their performance in specific catalytic and separations applications, with new applications ranging from small-scale electronics, to sensors, to diagnostic devices.3–6 Simultaneously, with the advent of structural and chemically well-defined nanoporous materials, important fundamental questions concerning the nature of fluid (liquid or gas) systems under confinement are becoming accessible to direct experimental exploration.7–14 It is such a problem that is the focus of this report. The particular problem with which we are concerned is the nature and existence of equilibrium states of confined fluids, especially where there is a tendency for the * To whom correspondence should be addressed. E-mail: valiullin@ uni-leipzig.de. † Universita¨t Leipzig. ‡ University of Massachusetts.
(1) Davis, M. E. Nature 2002, 417, 813. (2) Rolison, D. R. Science 2003, 299, 1698. (3) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature 1992, 359, 710. (4) Barton, T. J.; Bull, L. M.; Klemperer, W. G.; Loy, D. A.; McEnaney, B.; Misono, M.; Monson, P. A.; Pez, G.; Scherer, G. W.; Vartuli, J. C.; Yaghi, O. M. Chem. Mater. 1999, 11, 2633. (5) Taguchi, A.; Schu¨th, F. Microporous Mesoporous Mater. 2005, 77, 1. (6) Barthelemy, P.; Ghulinyan, M.; Gaburro, Z.; Toninelli, C.; Pavesi, L.; Wiersma, D. S. Nat. Photonics 2007, 1, 172. (7) Stapf, S. Nat. Phys. 2006, 2, 731. (8) Granick, S. Science 1991, 253, 1374. (9) Heuberger, M.; Zach, M.; Spencer, N. D. Science 2001, 292, 905. (10) Wallacher, D.; Kunzner, N.; Kovalev, D.; Knorr, N.; Knorr, K. Phys. ReV. Lett. 2004, 92, 195704. (11) Coasne, B.; Hung, F. R.; Pellenq, R. J.-M.; Siperstein, F. R.; Gubbins, K. E. Langmuir 2006, 22, 194. (12) Toda, R.; Hieda, M.; Matsushita, T.; Wada, N.; Taniguchi, J.; Ikegami, H.; Inagaki, S.; Fukushima, Y. Phys. ReV. Lett. 2007, 99, 255301. (13) Thommes, M.; Smarsly, B.; Groenewolt, M.; Ravikovitch, P. I.; Neimark, A. V. Langmuir 2006, 22, 756. (14) Zurner, A.; Kirstein, J.; Doblinger, M.; Brauchle, C.; Bein, T. Nature 2007, 450, 705.
fluid to undergo transitions between dense and dilute states, analogous to the liquid and vapor states of bulk fluids. For bulk fluids it is always possible to establish the state of equilibrium coexistence between vapor and liquid phase by placing samples of these phases in contact. On the other hand, for confined fluids one can only change the state of the system via changes in the pressure (or chemical potential) and/or temperature of the bulk phase. Two phases can only appear via the transformation of a single-phase system to a two-phase system by, for example, nucleation, and the phase change process is generally accompanied by very large free energy barriers.15,16 In practice, changes of state associated with large changes in density for confined fluids are typically accompanied by hysteresis.17,18 Such hysteresis is a well-known feature of adsorption/desorption isotherms for light gases, such as nitrogen, at cryogenic temperatures, and, depending on the pore structure of the adsorbent, may involve the existence of a very large number of completely reproducible but nonequilibrium states that are accessible via scanning experiments.17 Traditional adsorption measurements give us the relationship between the confined fluid density and the bulk chemical potential (via the bulk pressure) but do not tell us anything about the density distribution within the system. In this report, self-diffusion measurements by pulsed field gradient nuclear magnetic resonance (PFG NMR)19 are used to probe the state of a fluid (cyclohexane) confined within the internal pore space of a model porous material, namely Vycor porous glass. We provide direct experimental evidence that states within hysteresis loops that have the same average fluid density may have different average self-diffusivity, a direct reflection of differences in the density distributions between the states. The experimental setup used in our work consists of a flask with a liquid connected over a large reservoir with the sample tube placed inside of the NMR spectrometer coil. The volume of the reservoir exceeds that of the rest of the system, thus, the (15) Oxtoby, D. W. J. Phys.: Condens. Matter 1992, 4, 7627. (16) Sethna, J. P. Statistical Mechanics: Entropy, Order Parameters, and Complexity; Oxford University Press: New York, 2006. (17) Everett, D. H. Adsorption hysteresis. In The Solid-Gas Interface; Alison Flood, E., Ed.; Marcel Dekker, Inc. : New York, 1967; p 1055. (18) Evans, R. J. Phys.: Condens. Matter 1990, 2, 8989. (19) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Clarendon Press: Oxford, 1991.
10.1021/la801349y CCC: $40.75 2008 American Chemical Society Published on Web 06/07/2008
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sorption process does not change the number of molecules in the reservoir significantly. The liquid, reservoir, and the sample are separated by valves. A vacuum pump connected to the system allows one to reduce the vapor pressure in the reservoir and is utilized to activate the sample. Such a setup gives the possibility to prepare the adsorbate-adsorbent system under various conditions, adjusting the external pressure stepwise in the desired way. As a porous material, Vycor porous glass has been used.20,21 The amorphous monolithic particles possess a highly interconnected pore network (porosity of about 28% and internal surface area of about 200 m2/g) with a narrow pore size distribution and a mean pore diameter around 5 nm. NMR provides the possibility to probe the amount adsorbed as function of time as well as the self-diffusivities.22,23 The former has been measured by following the NMR free induction decay signal intensity, which is directly proportional to the number of nuclei in the sample. The selfdiffusivities have been measured by means of PFG NMR using a home-built NMR spectrometer.24 All measurements have been performed at 297 K. The primary data emerging from our experiments are plotted in Figures 1 and 2. Figures 1a and 2a show the conventional adsorption and desorption isotherms, namely, the amount θ of cyclohexane molecules in the pores normalized to the value at saturation pressure Ps as a function of the relative pressure, P/Ps, together with some sample adsorption and desorption scanning curves within the hysteresis loop. The data obtained on the complete adsorption branch (open circles, pressure increasing from zero to Ps) and desorption branch (filled circles, pressure decreasing from Ps to zero) clearly reveal a so-called hysteresis loop (for 0.4 < P/Ps < 0.75). In addition, Figure 1a also shows the results of measurements where pressure was reduced before the end of the hysteresis loop on adsorption, namely at 0.68 Ps (squares) and 0.65 Ps (triangles). These are examples of desorption “scanning curves”. Figure 2a displays data obtained when the desorption scanning curve (filled triangles) was not continued to pressures below the hysteresis range. The open triangles in Figure 2a reveal the situation when, after the pressure reaches 0.44 Ps, the scanning desorption curve (which started at 0.65 Ps) is reversed into an adsorption scanning curve. Similarly, the squares in Figure 2a show the situation when an incomplete adsorption scanning sequence (open squares) is followed by the desorption scanning sequence (filled squares). Our density versus pressure results are consistent with previous adsorption/desorption measurements.17 Recent theoretical analyses using mean field theory25,26 and Monte Carlo simulations27 provide an understanding of the observed behavior in terms of large numbers of metastable states associated with different ways of distributing the fluid density throughout the pore network. Scanning curves, under these circumstances, are loci connecting these metastable states. Within the hysteresis region for a given bulk pressure (chemical potential), there is an essentially infinite number of states with different densities or, for a given density, there is an essentially infinite number of states with different (20) Elmer, T. H. Porous and reconstructed glasses. In Engineered Materials Handbook; ASM International: Metals Park, OH, 1992; Vol. 4, p 427. (21) Levitz, P.; Ehret, G.; Sinha, S. K.; Drake, J. M. J. Chem. Phys. 1991, 95, 6151. (22) Valiullin, R.; Kortunov, P.; Ka¨rger, J.; Timoshenko, V. J. Chem. Phys. 2004, 120, 11804. (23) Valiullin, R.; Naumov, S.; Galvosas, P.; Ka¨rger, J.; Woo, H. J.; Porcheron, F.; Monson, P. A. Nature 2006, 443, 965. (24) Galvosas, P.; Stallmach, F.; Seiffert, G.; Karger, J.; Kaess, U.; Majer, G. J. Magn. Reson. 2001, 151, 260. (25) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Sarkisov, L.; Tarjus, G. Phys. ReV. Lett. 2001, 87, 055701. (26) Woo, H. J.; Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 7472. (27) Woo, H. J.; Monson, P. A. Phys. ReV. E 2003, 67, 041207.
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Figure 1. The amount θ of cyclohexane in Vycor porous glass at T ) 279 K normalized to the value at saturation pressure Ps (a) and corresponding diffusivities (b) as a function of the relative pressure P/Ps. The diffusivities have been measured after sufficiently long equilibration times following a pressure step when no measurable change in the amount observed was observed. The open and filled circles show the data obtained on the complete adsorption and desorption branches by a pressure increase from zero to Ps and from Ps to zero, respectively. The filled triangles and squares give the results of desorption scanning experiments, where pressure was first increased from zero to 0.68 Ps (squares) and 0.65 Ps (triangles), and then the corresponding data were measured upon reducing pressure to zero. The lines are shown to guide the eye.
bulk pressures.27 The scanning curves in Figure 2a also exhibit two important features: (i) return point memory, i.e., upon incomplete adsorption-desorption or desorption-adsorption cycles, the system returns to its initial state, and (ii) lack of congruence reflected by the fact the shape of hysteresis subloops is not preserved upon changing the initial state. The first is a feature of many systems exhibiting hysteresis including magnets,28 suggesting that the system predominantly evolves under variation of the external conditions and further equilibration by thermalization is prohibited because of high free energy barriers. The second feature is a signature of structural disorder in the pore network,29 in contrast to independent pore models (or the Preisach model in magnetic systems). Figures 1b and 2b show the self-diffusivities corresponding to the states shown in Figures 1a and 2a, indicating that the major behavior in density is also seen in the diffusivities. PFG NMR allows us to explore the total probability distribution of molecular displacements (the “propagator”),30 via the spin-echo diffusion (28) Sethna, J. P.; Dahmen, K.; Kartha, S.; Krumhansl, J. A.; Roberts, B. W.; Shore, J. D. Phys. ReV. Lett. 1993, 70, 3347. (29) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Tarjus, G. J. Phys.: Condens. Matter 2002, 14, 9295. (30) Ka¨rger, J.; Heink, W. J. Magn. Reson. 1983, 51, 1.
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Figure 3. The diffusivities in Figures 1b and 2b plotted versus the amount adsorbed θ from Figures 1a and 2a demonstrating different diffusivities for states with the same density.
Figure 2. The same as in Figure 1, but the data are obtained using more complex adsorption and desorption cycles. The open and filled circles are the data on the complete adsorption and desorption from Figure 1. The filled squares show the data obtained for the desorption scan from 0.65 Ps to 0.44 Ps followed by an adsorption scan from 0.44 Ps to 0.65 Ps, respectively. The open and filled squares refer to the data obtained in an adsorption scan from 0.43 Ps to 0.59 Ps, followed by a desorption scan from 0.59 Ps to 0.43 Ps, respectively.
attenuation functions, which were found to exhibit an exponential dependence on the square of the applied magnetic field gradient strength, revealing clear evidence of a Gaussian propagation. In turn, this implies fast exchange between different regimes of molecular mobility (namely, in the capillary-condensed, adsorbed, and gaseous phases).31 This allows a straightforward explanation of the general trends in the diffusivities. We can identify three main contributions to the average self-diffusivity in the system. A slow diffusion contribution from fluid in close contact with the pore walls makes its largest fractional contribution to the average diffusivity at low pressures. A second fast diffusion contribution comes from the gas-like fluid at the center of the pores away from the pore walls, making its largest fractional contribution to the average diffusivity over an intermediate range of pressure. Finally a slow diffusion contribution from fluid in liquid-like states at the center of the pores makes its largest fractional contribution at higher pressure.22 In nice agreement with this view, the diffusivities on the adsorption branches notably exceed those on the desorption branches. This is a consequence of the fact that, at a given pressure, during desorption conditions a larger fraction of molecules occurs in the less-mobile liquid phase than during adsorption. The most important feature of our results emerges when the diffusivity data from Figures 1b and 2b are plotted versus the (31) Ka¨rger, J.; Pfeifer, H.; Heink, W. AdV. Magn. Reson. 1988, 12, 2.
density data from Figures 1a and 2a. Remarkably, the resulting representation (Figure 3) reveals states with the same average density that have different diffusivities. Understanding this behavior requires an assessment of how the fluid density distributions on adsorption and desorption (or in adsorption or desorption scans) may be different for the same overall average fluid density in the Vycor. We anticipate two mechanisms leading to the observed behavior. One of the key points is that, on desorption, the liquid-like regions may correspond to somewhat expanded (or stretched) liquid states with density as much as 10% lower than the average density when the Vycor is saturated with liquid at the pressure Ps. This estimation follows, for example, from Figure 1a where, upon desorption, starting from Ps until the onset of the evaporation transition, the amount adsorbed decreases by about 10%. Such a stretching, however, does not occur on adsorption. These expanded or stretched liquid states are due to spatial variations in the pore diameter that give rise to situations analogous to the so-called inkbottle pore geometry. This geometry has been the subject of several recent simulation and theoretical studies32–34 describing the mechanism of adsorption and desorption, and highlighting the differences between density distributions between adsorption and desorption. Regardless of whether the desorption occurs via cavitation or via pore blocking, the liquid in the wider pore region (bottle) is in a stretched state. This stretching effect means that, for one and the same overall concentration, during desorption the liquid phase occupies a notably larger part of the pore space than during adsorption. Although the diffusivity in the “stretched” liquid is somewhat higher than that of the dense liquid, it is still very much smaller than in the vapor. As a consequence, the larger fraction of “open” pores during adsorption leads, for a given total loading, to somewhat larger overall diffusivities than during desorption. The second mechanism, which may also contribute to the difference in the diffusivities on adsorption and desorption, is related to differences in the microscopic distribution of the liquid phase within the sample. Indeed, ultrasonic attenuation and light scattering studies of filling and draining of n-hexane in Vycor reveal a uniform filling process while draining is accompanied by long-range correlations in the liquid distribution.35 Our experiments, however, indicate complete homogeneity over the (32) Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 7600. (33) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2002, 18, 9830. (34) Libby, B.; Monson, P. A. Langmuir 2004, 20, 4289. (35) Page, J. H.; Liu, J.; Abeles, B.; Deckman, H. W.; Weitz, D. A. Phys. ReV. Lett. 1993, 71, 1216.
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total sample since the propagator results as an ideal Gaussian. This means that, over distances of more than about 1 µm as the typical molecular displacements traced in our experiments, we do have all over the sample completely identical filling properties. Otherwise, we should have observed a distribution of diffusivities. This rules out the idea that emptying of the porous space proceeds via gas invasion, i.e., beginning at the glass boundary and gradually percolating into the sample interior. However, interplay between pore-blocking and cavitation may effect that, during desorption, more extended regions of the liquid and gaseous phase are formed than during adsorption. We anticipate that the former case yields slower diffusivities due to tortuosity. The relevance and relative impacts of these two effects are accessible by computer modeling. This work is under progress. Besides this general tendency in the “history dependence” of the diffusivities for a given overall concentration, one clearly has to be aware of further influences related to the differences in the density distribution within the porous space. They ultimately correspond to the differences in the system’s history, e.g., in the way the external pressure has been varied to attain exactly the given state. We note that, as revealed by the constancy of the measured diffusivities over time periods from hours to days, these different states are preserved over essentially unlimited intervals of time. The diffusivities presented by Figure 3 indicate the existence of several different states at one and the same loading. Further states, however, may of course be reached by choosing further ways of system preparation. Within each of these states, the confined fluid molecules rapidly vary their
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positions as reflected by their diffusivities. Importantly, the molecular displacements recorded during typical PFG NMR observation times on the order of milliseconds exceed the typical pore size by more than 2 orders of magnitude. Thus, within the high-dimensional phase space formed by spatial and kinetic coordinates of all molecules, the molecular trajectories within these different states define subspaces strictly separated from each other.16 This separation occurs irrespective of the dramatic rate of molecular redistribution within the system. This means that this separation can only be eliminated by collective effects. The overall diffusivity has thus been identified as a sensitive probe of the given state of a confined liquid. Different states emerge from the different history of the system. They are associated with different out-of-equilibrium distributions of the fluid within the pore network, which, in turn, give rise to different overall mobilities of the system. Being separated by large barriers in the system free energy, these states are found to remain stable over very long intervals of time. We have created quite different series of states by appropriate variation of the pressure regime in the surrounding atmosphere, reflected by a corresponding spectrum of diffusivities. Large time-span observation of such diffusivities opens a novel route of experimentally exploring the time evolution of fluid states under confinement. Acknowledgment. This work has been supported by the German Science Foundation (DFG) and the NSF. LA801349Y
