Supplementary material for Universal stretched exponential relaxation in nanoconfined water

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Supplementary material for Universal stretched exponential relaxation in nanoconfined water Adarsh Shekhar, 1 Rajiv K. Kalia, 1 Aiichiro Nakano, 1 Priya Vashishta, 1 Camilla K.

Materials Science, University of Southern California, Los Angeles, CA 90089-0242, USA 2 Department of Physics, University of Oslo, 0316 Oslo, Norway I.

1 Supplementary material for Universal stretched exponential relaxation in nanoconfined water Adarsh Shekhar, 1 Rajiv K. Kalia, 1 Aiichiro Nakano, 1 Priya Vashishta, 1 Camilla K. Alm, 2 and Anders Malthe-Sørenssen 2 1 Collaboratory for Advanced Computing and Simulations, Department of Physics & Astronomy, Department of Computer Science, Department of Chemical Engineering & Materials Science, University of Southern California, Los Angeles, CA , USA 2 Department of Physics, University of Oslo, 0316 Oslo, Norway I. PHASE DIAGRAM OF WATER Figure S1 shows the phase diagram of bulk water. The yellow region in the temperaturepressure phase diagram is the No Man s Land (NML). This metastable region has eluded experimental investigation because it is bypassed by crystal formation. Experimentally, rapid cooling of supercooled water leads to the formation of low-density amorphous (LDA) and highdensity amorphous (HDA) ice. In atomistic simulations, the cooling rates are high enough to reach NML from room temperature. It has been suggested that there are two types of water in NML: low-density liquid (LDL) and high-density liquid (LDL), which result from melting of LDA and HDA ice. The continuous black line in NML indicates a theoretical scenario, suggesting a first-order transition between LDL and HDL terminating at a critical point. FIG. S1. Phase diagram of bulk water showing No Man s Land. 1

2 II. SIMULATION METHODS A. Silica and water force fields The MD force fields for silica and water consist of 2-body and 3-body terms, which are combined with an environment-dependent potential that changes dynamically with changes in local atomic configurations. The 2-body terms account for steric repulsion, charge-charge, charge-dipole, and dipole-dipole interactions. The 3-body terms incorporate bond-bending and bond-stretching covalent interactions. The force field for water allows dissociation of water molecules. The functional form of the interatomic potential for both silica and water is, (2) (3)!! E = V ( r ) + V ( r, r ). (1) tot i< j k i< j< k ik (2) The two-body potential V depends on the distance r between atoms i and j and has four terms: steric repulsion, screened Coulomb, charge-dipole and van der Waals interactions; V (2) (r) = H + Z i Z j e r r r η 1s D r 2r 4 e r r 4 s w r 6, (2) where Z i is the charge on the i th atom, and r 1s and r 4s are screening lengths in Coulomb and (3) charge-dipole interactions, respectively. The three-body term V k has the form, V (3) jik ( r, r $ ξ ξ ' (cosθ ik ) = B k exp & + k cosθ 0 ) 2 % r r 0 r ik r ) ( 0 ( 1+ C k (cosθ k cosθ 0 ) 2 r,r ik r 0 ), (3) where θ k is the angle between r and r ik. The force-field parameters Z i, H, D, w, h, B k, C k, q k depend only on atom types, i.e., Si, O and H. These parameters were optimized to obtain good agreement with experimental measurements or quantum mechanical calculations of certain structural, mechanical, thermodynamic and dynamical properties, e.g., short-range order in silica and water (Si-O and O-H bond lengths and O-Si-O, Si-O-Si, H-O-H bond angles), melting temperature and elastic moduli of amorphous silica, diffusion in water, and silanol density at the silica-water interface. B. Force-field validation Silica: The MD results for structural properties are in correspondence with neutron scattering data for static structure factor and T(r) (= r 2 g(r), where g(r) is the radial distribution function). 1-3 The calculated values of Young s modulus (66.9 GPa), bulk modulus (39.2 GPa), and Poisson s ratio (0.22) of amorphous silica are in good agreement with experiments. 4 The fracture toughness (1 MPa m 1/2 ) of amorphous silica, which was not a part of the force field training set, is also within the range of experimental values ( MPa m 1/2 ). 4 Recently, we performed nanoindentation simulations with the MD approach and found the hardness of amorphous silica (10.6 GPa) 5 to be close to the experimental value (10 GPa). 6 Water: Our results for O-H bond length (0.97 Å) and H-O-H bond angle (104 ) agree well with X-ray diffraction experiments. 7 The MD results for silanol concentration at the silica-water interface are in the range of nm -2, which is in good agreement with experimental results. 8 The calculated value of the hydrolysis reaction energy between water and silica also falls within the range of values obtained by ab initio calculations ( ev), 9 and our results for the lowest energy structure of the gas phase orthosilicic acid are also in good agreement with ab initio calculations. 10 We calculated the heat of immersion of a hydrated silica slab in water and the value (0.18 J/m -2 ) is in good agreement with the experimental result. 11 2

C. Preparation of nanoporous silica Molecular dynamics simulation approach has been used to prepare low-density silica glass by applying negative pressure. 12 The flowchart in Fig.

2 g/cc) as the bulk amorphous silica system under ambient conditions. The system consisted of 65,856 atoms in a MD cell of dimensions 99.7 99.7 99.7 Å 3.

3 C. Preparation of nanoporous silica Molecular dynamics simulation approach has been used to prepare low-density silica glass by applying negative pressure. 12 The flowchart in Fig. S2 shows how we prepared the nanoporous silica system. The starting configuration of the system was a β-cristobalite lattice, which has the same mass density (ρ = 2.2 g/cc) as the bulk amorphous silica system under ambient conditions. The system consisted of 65,856 atoms in a MD cell of dimensions Å 3. Periodic boundary conditions were imposed in the x, y and z directions. The lattice was heated to obtain a well-equilibrated molten system at a temperature T = 4,500 K. The dimensions of the MD cell were increased by Å 3 over 5 ps, and the expanded system was again equilibrated at T = 4,500 K for 10 ps. Repeating this cycle of expansion and relaxation, we obtained several molten systems at 4,500 K. The molten systems with mass densities below 1.0 g/cc had nanometer-scale cavities. These low-density liquids were quenched to 10 K to obtain nanoporous silica at various mass densities between 1.0 and 0.1 g/cc. We chose to study the behavior of water in the nanoporous system at a mass density ρ = 0.5 g/cc. Under-coordinated silicon and oxygen atoms in this nanoporous system were passivated with OH and +H groups, respectively. The energy of this system was minimized with the steepest-descent method. The nanoporous system was divided into Å 3 voxels and water molecule were placed in empty voxels. The resulting system consisted of 404,379 atoms including 105,752 water molecules. We again used the steepest-descent method to minimize the energy of the nanoporous silica system embedded with water. Subsequently, the system was heated to 300 K over a time period of 250 ps and thermalized for 500 ps. FIG. S2. The flowchart shows how nanoporous silica systems were prepared in MD simulations. 3

Next, we calculated the moment-of-inertia tensor: (r iα R Cα )(r iβ R Cβ ) i T αβ =, R G = tr(t ) N (4) where α and β denote Cartesian coordinates, r iα is the position of the center of mass of the i

4 III. RESULTS A. Structural characterization of nanopores in silica Nanopores were characterized by their size, shape and anisotropy. First, we tessellated the system with Å 3 voxels to identify nanopores. Next, we calculated the moment-of-inertia tensor: (r iα R Cα )(r iβ R Cβ ) i T αβ =, R G = tr(t ) N (4) where α and β denote Cartesian coordinates, r iα is the position of the center of mass of the i th voxel in a pore, R Cα is the position of the pore center, and N is the total number of voxels in the pore. The trace of the moment-of-inertia tensor gives the radius of gyration of a nanopore R G. The anisotropy and shape parameters, Δ and S, were obtained by diagonalizing the T matrix in Eq. (4): Δ = i=1 (λ i λ) 2 (tr(t )) 2, S = 27 3 i=1 (λ i λ) (tr(t )) 3 (5) where {λ i } are the eigenvalues of T and λ = (λ 1 + λ 2 + λ 3 ) / 3. The value of S ranges between -1/4 and 2. Oblate ellipsoidal pores have negative and prolate ellipsoidal pores have positive values of S. Figure S3(a) shows a wide distribution of nanopore anisotropies in the silica system at ρ = 0.5 g/cc. The shape-parameter distribution in Fig. S3(b) shows that approximately 20% of the nanopores in the system are spherical, 20% are oblate ellipsoids, and the rest are prolate ellipsoids. FIG. S3. (a) Normalized distribution, P(D), of anisotropy parameter D, and (b) normalized distribution, P(S), of shape parameter S for nanopores in the amorphous silica system at r = 0.5 g/cc. Movie: The movie shows the structure of nanoporous silica at a density of 0.5 g/cc: 4

B. Density distribution in nanoconfined water Figure S4 shows mass-density distributions, P(ρ), for room temperature and supercooled water at T = 100, 200 and 300 K.

5 B. Density distribution in nanoconfined water Figure S4 shows mass-density distributions, P(ρ), for room temperature and supercooled water at T = 100, 200 and 300 K. The mass densities, ρ, were calculated with the Voronoi tessellation of the entire MD system using the Voro++ software ( The Voronoi tessellation was done with the MD data for the positions of Si and O atoms in silica and O atoms in water molecules. The positions of H atoms were not included in the construction of Voronoi cells. The mass densities at the positions of O atoms in water molecules were determined from the volumes of Voronoi cells around those atoms. The mass-density distributions were calculated for many MD configurations at regular intervals and averaged over a time period of 2 ns. FIG. S4. Mass-density distributions of supercooled water and room-temperature water in nanopores of the amorphous silica network at r = 0.5 g/cc. C. Cage dynamics of water molecules Dynamic heterogeneities in nanoconfined water are apparent in the mean square displacements (MSDs) of water molecules. Figure S5 shows the MSD as a function of time for a tagged water molecule in the interior region of the largest nanopore in silica. The MSD exhibits plateaus and intervening diffusive regimes. At a given temperature the durations of plateau and diffusive regimes vary from molecule to molecule, although on average the plateaus become longer upon cooling. Analyzing the dynamics of a tagged water molecule and its neighbors, we find that the plateau region arises from the trapping of the tagged molecule in a cage formed by its nearest-neighbor water molecules. When some of the nearest neighbors begin to diffuse, the cage breaks allowing the tagged particle to enter the diffusive regime. 5

6 FIG. S5. Mean square displacement as a function of time for a water molecule in the interior region of the largest nanopore in the nanoporous silica system at a mass density r = 0.5 g/cc and temperature T = 100 K. For the first 50 ps, the molecule is trapped in a cage formed by its nearest-neighbor water molecules. The cage was determined by Voronoi tessellation and is shown in the left inset. The cage breaks and the molecule diffuses for the next 50 ps. This is followed by the trapping of the water molecule in another cage of nearest-neighbor water molecules. The second cage is shown in the right inset. Cage trapping and breaking are observed at all temperatures. The molecules are trapped longer in their cages at lower temperatures. D. Identification of water molecules inside nanopores We classified water molecules in the nanoporous silica matrix into three categories. Water molecules within 5.5 Å from silica were marked as Surface molecules (blue region in Fig. S6) and then there was a 5.5 Å-thick Buffer region for water molecules. The remaining molecules in a pore were labeled Interior molecules (red region in Fig. S6). Cage correlation functions were calculated for water molecules in the Interior regions of nanopores. 6

!!! 1 N (6) Fs (k, t) = cos k [ ri (t) ri (0)], N i=1! where N is the number of atoms and ri (t) is the position of the i-th atom at time t.

7 FIG. S6. Snapshot showing Interior (red), Buffer (green) and Surface (blue) water molecules in a nanopore. The silica network is shown here in magenta. E. Self-intermediate scattering function Neutron scattering experiments can probe the dynamics of water confined in nanoporous silica. We calculated the self-intermediate scattering function (SISF),!!!! 1 N (6) Fs (k, t) = cos k [ ri (t) ri (0)], N i=1! where N is the number of atoms and ri (t) is the position of the i-th atom at time t. Figure S7 shows the SISF for oxygen atoms for three values of wavenumbers, k = 0.78, 1.7 and 2.24 Å-1. The first wavenumber corresponds to the average pore radius, the second wavenumber to the peak in the structure factor in water, and the third to the average cage size. Figure S7 shows stretched exponential relaxation in all SISFs and the exponent β increases with a decrease in the value of k. The wavenumber dependence of β has been reported for water confined in vycor.13 ( ) 7

8 FIG. S7. Calculated self-intermediate scattering function at 100 K for wavenumbers k = 0.78, 1.7 and 2.24 Å -1. 8

9 REFERENCES 1 Johnson, P. A. V., Wright, A. C. & Sinclair, R. N. Neutron scattering from vitreous silica II. Twin-axis diffraction experiments. J Non-Cryst Solids 58, (1983). 2 Susman, S. et al. Intermediate-range order in permanently densified vitreous SiO 2 : A neutron-diffraction and molecular-dynamics study. Phys Rev B 43, (1991). 3 Wright, A. C. The comparison of molecular dynamics simulations with diffraction experiments. J Non-Cryst Solids 159, (1993). 4 Wang, Q., Saunders, G. A., Senin, H. B. & Lambson, E. F. Temperature dependences of the third-order elastic constants and acoustic mode vibrational anharmonicity of vitreous silica. J Non-Cryst Solids 143, (1992). 5 Nomura, K., Chen, Y. C., Kalia, R. K., Nakano, A. & Vashishta, P. Defect migration and recombination in nanoindentation of silica glass. Appl Phys Lett 99, (2011). 6 Miyake, K., Satomi, N. & Sasaki, S. Elastic modulus of polystyrene film from near surface to bulk measured by nanoindentation using atomic force microscopy. Appl Phys Lett 89, (2006). 7 Narten, A. H. & Levy, H. A. Liquid Water: Molecular correlation functions from X-ray diffraction. J Chem Phys 55, (1971). 8 Zhuravlev, L. T. The surface chemistry of amorphous silica. Zhuravlev model. Colloids Surf A 173, 1-38 (2000). 9 Bakos, T., Rashkeev, S. N. & Pantelides, S. T. Reactions and diffusion of water and oxygen molecules in amorphous SiO 2. Phys Rev Lett 88, (2002). 10 Sauer, J. Molecular structure of orthosilicic acid, silanol, and H 3 SiOH...AlH 3 complex: models of surface hydroxyls in silica and zeolites. J Phys Chem 91, (1987). 11 Hassanali, A. A. & Singer, S. J. Model for the water-amorphous silica interface: the undissociated surface. J Phys Chem B 111, (2007). 12 Kieffer, J. and Angell, C., A. Generation of fractal structures by negative pressure rupturing of SiO 2 glass. J. Non-Crystalline Solids 106, (1988). 13 Bellissent-Funel, M. C., Longeville, S., Zanotti, J. M. & Chen, S. H. Experimental observation of the a relaxation in supercooled water. Phys Rev Lett 85, (2000). 9

T=293 K is shown in Fig. 1. The figure shows that in. Crystalline fragments in glasses ' ' PHYSICAL REVIEW B VOLUME 45, NUMBER 13

T=293 K is shown in Fig. 1. The figure shows that in. Crystalline fragments in glasses ' ' PHYSICAL REVIEW B VOLUME 45, NUMBER 13 ), PHYSICAL REVIEW B VOLUME 5, NUMBER 13 1 APRIL 1992-I Crystalline fragments in glasses Giomal A. Antonio Departamento de Fisica, Universidade Fstadual Paulista Bauru, Caixa Postal 73, 1733 Bauru, So