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Existence of a Hexatic Phase in Porous Media VOLUME 89, NUMBER 7 PHYSICAL REVIEW LETTERS 12AUGUST 2002 Existence of a Hexatic Phase in Porous Media Ravi Radhakrishnan,1 Keith E. Gubbins,2 and Malgorzata Sliwinska-Bartkowiak3 1Massachusetts Institute of Technology, 77 Massachusetts Avenue, 66-021, Cambridge, Massachusetts 02139 2North Carolina State University, 113 Riddick Labs, Raleigh, North Carolina 27695 3Institute of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland (Received 7 August 2000; revised manuscript received 17 April 2002; published 25 July 2002) Molecular simulations for simple fluids in narrow slit-shaped carbon pores exhibit crystal-hexatic and hexatic-liquid transitions that are consistent with Kosterlitz-Thouless-Halperin-Nelson-Young theory. The temperature range over which the hexatic phase is stable is dramatically widened under confinement. Remarkably, the transitions, which are continuous for a single adsorbed layer, become weakly first order when the pore can accommodate two molecular layers. Nonlinear dielectric effect measurements for CCl4 and aniline in activated carbon fibers (pore width 1.4 nm) show divergence at these transitions, confirming the hexatic phase. DOI: 10.1103/PhysRevLett.89.076101 PACS numbers: 68.35.Rh Two-dimensional systems have a special significance for ment of thermal equilibrium because of large particle sizes phase transitions in which continuous symmetry is broken compared to molecular dimensions. (such as freezing transitions). The Mermin-Wagner theo- Activated carbon fibers (ACF) possess microcrystallites rem states that true long-range order cannot exist in such made up of graphene sheets that tend to align in similar systems [1]. Nelson and Halperin [2] proposed the directions, with slit-shaped voids between the microcrys- KTHNY (Kosterlitz-Thouless-Halperin-Nelson-Young) tals. The spontaneous ordering of the molecules adsorbed mechanism for melting of a crystal in two dimensions, in these voids into distinct two-dimensional molecular which involves two transitions of the Kosterlitz-Thouless layers makes the adsorbed phase a quasi-two-dimensional (KT) kind [3]: The crystal to hexatic transition occurs system, with none of the complications mentioned above. through the unbinding of dislocation pairs, and the hexatic Here, we investigate melting in a quasi-two-dimensional to liquid transition involves the unbinding of disclination system consisting of simple, near-spherical molecules ad- pairs. The KTHNY theory predicts that for the hexatic sorbed in slit-shaped pores, using both molecular simula- phase the correlation function associated with the bond tion and experiment. The results are compared with the orientational order parameter decays algebraically with predictions of KTHNY theory. exponent 0 <6 < 1=4 (quasi-long-range orientational In the simulations, carbon tetrachloride is chosen as the order), while there is only short-range translational order. adsorbate to make contact with recent experimental studies Early simulation studies on small, strictly two- of this system. The fluid-wall potential was modeled to be dimensional systems failed to provide compelling evidence of the form of the ‘‘10-4-3’’ Steele Potential, with parame- to support the KTHNY melting scenario (for a review, see ters chosen to represent the strongly attractive graphite Strandburg [4]). However, it was demonstrated by Bagchi pore [11]. The fluid-fluid Lennard-Jones (LJ) parameters et al. [5], using a systematic scaling analysis on large sys- were chosen to reproduce the bulk freezing temperature at tem sizes, and subsequently by Jaster [6], that the equili- 1 atm pressure [12]. Two pore widths, H 1ff fw brium properties of a two-dimensional system of disks with 0:911 nm and H 2ff fw 1:41 nm, where H rep- repulsive interactions are indeed consistent with the resents the shortest distance between the planes passing KTHNY theory of melting, suggesting the earlier studies through the carbon nuclei on the surface of the opposing were plagued by serious system size effects. pore walls, were chosen, so that the adsorbed phase had The hexatic phase was first observed experimentally in one and two molecular layers of CCl4, respectively. The an electron diffraction experiment on a quasi-two- calculations for the larger pore width also enable a direct dimensional system of a thin film of a liquid-crystalline comparison with experimental measurements for CCl4 material (see [7] and references therein). However, the confined in porous activated carbon fiber ACF A-10, of intrinsic presence of a herringbone symmetry and its cou- mean pore width H 1:4nm. The extent of the rectilinear pling to the hexatic symmetry in multilayer liquid-crystal- simulation cell was 180ff 180ff (93 nm 93 nm) line films causes the phase behavior to be more compli- in the xy plane so that correlations up to 90ff were cated [8]. Quasi-two-dimensional systems consisting of captured in the simulations (z is normal to the pore walls). micron-sized colloidal particles confined between parallel Periodic boundary conditions were used in the x and the y glass plates do not have this complication [9,10], but in dimensions. We expect the approximation of a structure- some [9] cases are faced with the question of the attain- less graphite wall to be a good one here, since the diameter 076101-1 0031-9007=02=89(7)=076101(4)$20.00 2002 The American Physical Society 076101-1 VOLUME 89, NUMBER 7 PHYSICAL REVIEW LETTERS 12AUGUST 2002 1 of the LJ molecule (0.514 nm) is much larger than the C-C 10 σ bond length in graphite (0.14 nm). We use the Landau- 10 40 σ Ginzburg formalism [12] to calculate the free energy sur- 0 60 σ face as a function of the hexatic bond-orientational order 10 180 σ parameter, 6;j in layer j, as defined in Ref. [12]. GL γ Grand canonical Monte Carlo simulations were used to -1 10 study the freezing behavior of LJ CCl4 in our model graph- ite pore. Our simulation cell contained up to 64 000 mol- -2 ecules. The state conditions were such that the confined 10 phase was in equilibrium with bulk LJ CCl at 1 atm 300.0 320.0 340.0 4 T / K pressure. The simulations were started from a well equili- brated confined liquid phase at T 400 K, and in succes- FIG. 1. Ginzburg parameter, GL, for four different system sive simulation runs the temperature was reduced. Equili- sizes: 10ff 10ff H, 40ff 40ff H, 60ff bration was for a minimum of eleven billion steps. The 60ff H, and 180ff 180ff H. The simulations are two-dimensional, in-plane positional and orientational cor- for a pore width, H 1:4nm, consisting of two adsorbed layers relation functions [gj r and G6;j r of layer j], were of CCl4. Finite system size calculations are self-consistent in the monitored to keep track of the nature of the confined phase. regime GL 1. The positional pair correlation function is the familiar radial distribution function. The bond orientational pair the scaling properties of the orientational correlation func- correlation function is G h 0 i, where 6;j 6;j 6;j tion, we plot logI LB=I L as a function of logLB=L in ~ xe^ ye^ . x y Fig. 2, where L was chosen to be 180ff and three differ- Our results for the two confined molecular layers of ent values, LB 60ff, 90ff, and 135ff, were used. CCl at three different temperatures were as follows [13]: 4 The integral I LB is evaluated from G6;j for the 180ff The high temperature phase at T 360 K was character- system, by integrating over a subvolume of dimension ized by an isotropic g r and exponentially decaying LB LB H instead of the total volume. A slope of ÿ2 G6;j r, and, hence, was a liquid. At T 330 K, the posi- corresponds to having a finite correlation length (liquid), a tional pair correlation function was isotropic and the ori- slope of ÿ1=4 corresponds to an algebraic decay of the entational correlation function decayed algebraically, a orientational correlation function with exponent ÿ1=4 signature of the hexatic phase. At T 290 K, the confined (hexatic), and a zero slope corresponds to true long-range phase was a two- dimensional hexagonal crystal, with order [5]. In Fig. 2, the lines correspond to the scaling quasi-long-range positional order and long-range orienta- predicted by the KTHNY theory [5] and the symbols are tional order [13]. These features were verified using system our simulation results at three different temperatures; size scaling analysis as discussed below. Since simulation agreement between the simulations and theory is excellent. results are always for a finite system size, we have estab- Therefore, the correlations in the one-layer system and the lished the self-consistency of our calculations by comput- two-layer system are both consistent with the KTHNY ing the Ginzburg parameter [14], GL, for the different predictions. phases as a function of temperature and system size: 2 h 6;ji I L 3.5 ÿ 1; (1) GL 2 2 crystal, bilayer h 6;ji h 6;ji 3 liquid, bilayer 2 hexatic, bilayer where L H V, the volume of the system, and I L is crystal, monolayer given by 2.5 R liquid, monolayer 2 hexatic, monolayer V Rd~G6;j ) / I(L) ] I L : (2) B V d~ 1.5 The Ginzburg parameter is calculated by numerically in- 1 tegrating Eq. (2). If GL 1, the simulations are self- consistent, i.e., (i) fluctuations do not destroy the ordered log [ I(L 0.5 phase observed in the simulations, and (ii) the scaling 0 behavior is not mean-field-like. It is clear from Fig. 1 -1.5 -1 -0.5 0 that, for system sizes greater than 60ff 60ff H, log [ L /L ] and for the temperature range of our simulations, our B results are self-consistent.
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