Received 26 April 2016; Revised 6 June 2016; Accepted 6 June 2016
DOI: xxx/xxxx
1 RESEARCH ARTICLE
2 Molecular dynamics modeling of a partially saturated clay-water
3 system at finite temperature
4 Xiaoyu Song* | Miaochun Wang
Engineering School of Sustainable Infrastructure and Environment, University Summary of Florida, Gainesville, FL 32611, USA The mechanical and hydraulic properties of unsaturated clay under non-isothermal Correspondence conditions have practical implications in geotechnical engineering applications such *Xiaoyu Song. Email: xysong@ufl.edu as geothermal energy harvest, landfill cover design, and nuclear waste disposal facil- Present Address ities. The water menisci among clay particles impact the mechanical and hydraulic 365 Weil Hall, Gainesville, FL 32611, USA. properties of unsaturated clay. Molecular dynamics (MD) modeling has been proven to be an effective method in investigating clay structures and their hydromechanical behavior at the atomic scale. In this study, we examine the impact of temperature increase on the capillary force and capillary pressure of the partially saturated clay- water system through high-performance computing. The water meniscus formed between two parallel clay particles is studied via a full-scale MD modeling at dif- ferent elevated temperatures. The numerical results have shown that the temperature
5 increase impacts the capillary force, capillary pressure, and contact angle at the atomic scale. The capillary force on the clay particle obtained from MD simula- tions is also compared against the results from the macroscopic theory. The full-scale MD simulation of the partially saturated clay-water system can not only provide a fundamental understanding of the impact of temperature on the interface physics of such system at the atomic scale and but also has practical implication in formulating physics-based multiscale models for unsaturated soils by providing interface physical properties of such materials directly through high-performance computing.
KEYWORDS: Molecular dynamics, temperature, water meniscus, unsaturated clay, capillary force, high-performance computing
6 1 INTRODUCTION
7 Multiphase porous media such as unsaturated clay and sand play an essential role in geotechnical and geoenvironmental engineer-
8 ing applications, for instance, unsaturated soil slope failures and the design of landfill cover systems. However, our knowledge
9 of the underlying fundamental mechanism governing failure or instability of multiphase porous media under the environmental 1,2,3,4,5 10 loads (e.g., humidity and temperature) is limited . Computational experiments or modeling play an increasingly important
11 role in obtaining a faithful understanding of the underlying failure mechanism as well as the complicated coupled processes of 6,7 12 the solid deformation and fluids flow in multiphase porous media . For example, recent mesoscale mixed finite element mod-
13 eling of unsaturated soils taking into account the material heterogeneities (e.g., density and the degree of saturation) for the first 2 Xiaoyu Song ETAL
14 time has demonstrated that both the spatial variations of porosity and degree of saturation have a first-order triggering role in 8,9,10,11,12,13 15 strain localization failures of unsaturated soils under the isothermal conditions . Another example is that both concur-
16 rent and hierarchical multi-scale computational modeling methodologies have been developed to study the instability of single 14,2 17 phase materials (e.g., metal or granular materials) . However, it is still a challenging job to formulate high-fidelity multi-scale
18 computational frameworks for modeling multiphase porous media, such as unsaturated soils. This deficiency is rooted in our
19 limited knowledge of the complex interfacial physics (e.g., the interfaces among soil particles, water and air for unsaturated
20 soils) of multiphase porous media at multiple length scales (i.e., from the nano to the particle or pore scale to the continuum 3 21 scale) and the linkages between these various length scales (i.e., bridge scales) . For unsaturated soils, water menisci among
22 clay particles impact the mechanical and hydraulic properties of such materials. For instance, the water bridge between two clay
23 particles is under tension, and this tensile capillary force generally increases the tensile strength of the clay particles by pulling 1,15 24 them together . The hydromechanical performance of unsaturated soils under non-isothermal conditions has practical impli-
25 cations in geotechnical engineering such as geothermal energy harvest and storage, landfill cover system design, and nuclear
26 waste disposal facilities.
27 Unsaturated soils are multiphase porous media comprised of three phases of matter: the solid, pore water and pore air. In
28 dealing with unsaturated soils, one requires not only the principles of mechanics and hydraulics but also the knowledge of
29 fundamental interfacial physics. Here, the interfacial physics refers to the thermodynamic principles describing equilibrium
30 among all three phases, the transition of matter from one phase to another, and the adsorption or desorption of one phase of matter 1 31 onto or from an adjacent phase of different matter . For example, at the particle level, the behavior of clay is controlled by the 16,17 32 van der Waals, double-layer due to electrostatic interactions, and capillary interactions . The problems involving unsaturated
33 soils are multiscale and multiphysics by nature. The multiscale modeling techniques are useful tools to model the mechanical and
34 hydraulic properties and failures of unsaturated soils. However, there are still four challenges in multiscale modeling of materials
35 in general. These four challenges are: (1) understanding the physical models at different scales; (2) understanding how models
36 of different complexity are related to each other; (3) understanding how models of different complexity can be coupled together
37 smoothly, without creating artifacts; and (4) understanding how to formulate models at intermediate levels of complexity, or
38 "mesoscale models", to tackle the real scientific and engineering problems . These challenges apply to model unsaturated soils 18,19,20,21 39 as a multiscale multiphysics process . To formulate a high-fidelity physics-based multiscale computational framework
40 for unsaturated soils, we first need to obtain the fundamental physical information at the interfacial scale of such materials.
41 Fundamental physical information at the interface among different phases of unsaturated soils is the capillary force on the
42 soil particles imposed by water menisci. Capillary effects play a significant role in the mechanical and hydraulic properties of 22,1 43 unsaturated soils, due to intergranular forces causing an increase of cohesion and mechanical strength of such materials . For
44 instance, when a dry soil becomes partially saturated, capillary stresses develop between soil particles, resulting in shrinkage
45 in some soils like clay. The fundamental physical parameters related to capillary effects are the capillary force, the capillary
46 pressure, the contact angle, and the water surface tension. Here, the contact angle is the angle formed by the tangents of the solid-
47 water interface and the water-air interface. Both theoretical analysis and experimental data show that temperature influences the 23,24 48 capillary stress because the surface tension of water and the contact angle are dependent on temperature . Clays and related
49 phases present a particularly daunting set of challenges for the experimentalist and computational chemist. For this reason,
50 molecular computer simulations have become extremely helpful in providing an atomic-scale perspective on the structure and 25 17 51 behavior of clay minerals . While the Young-Laplace equation can be applied to compute the capillary force acting on clay
52 particles, it requires the knowledge of the water meniscus such as the contact angle between the solid-water-air interface and 26 53 water meniscus curvature, which is somewhat difficult to obtain experimentally . MD modeling is a powerful computational 27,28,29,30 54 method for studying physical and chemical phenomena occurring at the atomic scale . MD has been proved to be a 31,32,33,34,35 55 useful method for studying geomaterials under fully or partially saturated conditions .
56 In MD simulations, a force field is needed to define the physical interaction between atoms. In the geomechanics community 36 25 57 two force fields for clays, CHARMm and CLAYFF , have been applied to study the physical and mechanical properties of
58 clays. In the CHARMm force field, the total potential energy is expressed as a sum of the bonded energy and the nonbonded 37,31 32,38 59 energy as shown in Figure 1 . The CHARMm force field for clays has been used to study a clay-water-air system .
60 CLAYFF force field for clay minerals is a versatile force field built around the flexible version of the simple point charge (SPC) 39 61 water model to represent the water, hydroxyl, and oxygen-oxygen interactions. The geotechnical community has used the 33 40,38 41 62 MD technique to model the interactions of clay particles with ions , and a clay-water-air system . Ebrahimi et al. used
63 molecular dynamics with the CLAYFF force field to simulate the isothermal-isobaric water adsorption of interlayer Wyoming
64 Na-montmorillonite. In this work, the nanoscale elastic properties of the clay-interlayer water systems were calculated from the Xiaoyu Song ETAL 3
42 65 potential energy of the model system at 0 K as a function of water uptake. Carrier et al. investigated the elastic properties of
66 swelling clay particles at finite temperature upon hydration via MD and found that the elastic properties of montmorillonite at 43 67 finite temperature cannot be extrapolated from computation at 0 K. Hantal et al. first studied the fracture properties of clays 44 45 68 by means of a reactive molecular force field - ReaxFF . Teich-McGoldrick et al. adopted CLAYFF force field to determine
69 the pressure and temperature dependence of the elastic properties of muscovite. MD has been used to study the clay-water-air 32,38 70 system . However, these studies have focused on the unsaturated clay at the ambient temperature.
71 In this study, for the first time we investigate the impact of temperature variation on the fundamental physical properties
72 (e.g., capillary force) of the unsaturated clay-water system at the atomic scale via a full-scale MD modeling. In particular, the
73 meniscus formed between two parallel pyrophyllite clay particles and pure water at different temperatures is modeled through
74 full-scale MD simulations and high-performance computing. The temperature range is from 300 K (i.e., ambient temperature)
75 to 360 K, and the water-vapor interface is not considered in this study. We investigate the impact of temperature increase on
76 the capillary force, the capillary pressure, and the contact angle of the water meniscus between two parallel clay particles.
77 Furthermore, the capillary forces between the clay particles obtained directly from MD simulations at elevated temperatures are
78 compared against the results from the Young-Laplace equation, which requires the knowledge of the contact angle and capillary
79 pressure obtained directly from the same MD simulation. The MD simulations of unsaturated clay-water systems at elevated
80 temperatures have significant implication in physics-based multiscale (e.g., the atomic scale to the continuum scale) modeling
81 of thermal unsaturated soils by providing physical properties of unsaturated fine soils at the interfacial scale directly through
82 high-performance computing.
83 It is worth noting that an MD simulation consists of four distinct stages, namely, the build-up of an initial configuration of 30 84 the systems, the equilibration phase, the production phase, and the simulation analysis . We present these four stages for the
85 present study in the remaining part of the article. Section 2 presents the initial configuration of the system. Section 3 introduces
86 the simulations from the initial stage to the equilibrium phase. Section 4 provides the concerned physical properties calculated
87 during the production phase, the analysis of the simulated results and comparison with the results based on the macroscopic
88 theory, followed by the conclusion in Section 5.
89 2 MODEL SET-UP
90 2.1 Molecular dynamics 29 91 MD simulation is a technique for computing the equilibrium and transport properties of a classical many-body system . MD 30 92 computes the ‘real’ dynamics of the system, from which time averages of properties can be calculated . MD is a ‘deterministic’
93 method, which means that the state of the system at any future time can be predicted from its current state. In MD, the positions,
94 velocities, and accelerations of the atoms in a molecular system are computed by numerically solving the equations of Newton’s
95 second law of motion F i = miai, where F i is the force acting on atom i, mi is the mass of atom i, and ai is the acceleration 96 of atom i. The external force can be directly applied to selected atoms. The energy of the molecular system is expressed using
97 suitable empirical potential energy functions or force fields. Successful application of any computational molecular modeling
98 technique requires the use of interatomic potentials (force fields) that effectively and accurately account for the interactions
99 of all atoms in the model system. Figure 1 shows the consistent force field description of the potential energy function of a 46 100 molecule in which the terms shown have been used with little alteration since 1970 . There are two common ways to specify
101 the position of the atoms and molecules in the system to a modeling program, namely the Cartesian (x, y, z) coordinates and 30 102 internal coordinates in which the position of each atom is described relative to other atoms in the system . In this study, we
103 adopt the Cartesian coordinate system to build both the unit cell of clay and water as well as the system model.
104 2.2 Unit cell
105 In this study, the clay mineral chosen is pyrophyllite, which has a basic structure of the smectite-type clay mineral. Pyrophyllite 26 106 is the precursor to other smectite-type minerals exhibiting swelling . For instance, the mineral structure of pyrophyllite can lead
107 to other minerals in the smectite group through appropriate isomorphous substitutions. The chemical formula of pyrophyllite
108 is Al2Si4O10(OH)2. Figure 2 (a) shows the molecular structure of a unit cell of pyrophyllite. The dimension of a unit cell in −10 47,37 109 the x − y − z Cartesian coordinate system is 5.28Å × 9.14Å × 6.56Å (where 1Å = 10 m) . To perform a full-scale MD
110 simulation, we need to provide the initial atomic positions and a set of suitable parameters for the potential functions, known as 4 Xiaoyu Song ETAL
46 FIGURE 1 The consistent force field description of the potential engergy function of a molecule . Here Kb, b0, K, 0, K , , r0, and qi are the force field parameters obtained from experimental results, ab initio calculations, etc. b, , , r are the bond length, bond angle, dihedral angle, the distance between two non-bonded atoms, respectively, and qi is the partial charge of each atom.
48 31,47 111 a force-field . The initial coordinates of the atoms in the unit cell of pyrophyllite are obtained from the literature . In this 49 112 study, the TIP3P water model is adopted for water where the water molecule is treated as rigid . The atomic structure of the water molecular used in the full-scale MD simulation is shown in Figure 2 (b).
oxygen silicon hydrogen
aluminum
oxygen hydrogen (a) (b)
FIGURE 2 (a) Atomic structure of a unit cell of pyrophyllite which consists of 24 atoms, (b) Atomic structure of a water molecular.
113
114 2.3 Clay-water system
115 In this part, we present the specific procedure to prepare the unsaturated clay-water molecular model. The Cartesian coordinates
116 are used to specify all atoms present in the initial configuration of the system. It is noted that the initial construction of the
117 molecular model often determine the success of molecular dynamics simulations.
118 2.3.1 Clay particles
119 In the x − y − z Cartesian coordinate system the size of a clay particle used in the analysis is 26.40Å × 457.00Å × 6.56Å. The
120 layer consists of 5 unit cells in the x direction, 50 unit cells in the y direction and 1 unit cell in the z direction. The total number Xiaoyu Song ETAL 5
121 of atoms in a clay particle layer is 20,000. In this study, we assume that the capillary force on the clay particles generated by the
122 water meniscus is not large enough to produce the substantial deformation of the clay particles. Moreover, the study is focused
123 on the impact of the increase in temperature on the overall capillary force. Therefore, we assume two clay particles are fixed at
124 their initial positions over the process of simulations. Figure 3 shows two clay particles prepared for the MD simulation. Both
125 clay particles in Figure 3 have a unit thickness in the z direction (e.g., one single layer). It is worth noting that in nature several 26 126 such layers are stacked up in the z direction at some basal spacing .
(a)
(b)
FIGURE 3 (a) Two parallel clay particles; (b) Zoom-in view.
127 2.3.2 Water preparation
128 For simulations of systems at equilibrium, it is suggested to choose an initial configuration that is close to the state which it is 30 38 129 desired to simulate . For this reason, we adopt a two-step procedure to prepare a water box which is placed in between clay
130 plates in the initial state. Figure 4 presents the schematic of the procedure to produce the desired water body at the desired temperature. In the first step, as shown in Figure 4 (a), an independent water box of size 26.40Å × 360.00Å × 180.00Å is
(a) (b) (c)
FIGURE 4 Schematic of the procedure to produce the desired water body at a desired temperature: (a) Pour water molecule into a simulation box; (b) Run simulation to equilibrate the water body in the isobaric-isothermal ensemble (NPT); (c) Cut a desired water body from the equilibrated water in (b).
131 29 132 equilibrated in the isobaric-isothermal ensemble (constant number, pressure, and temperature or NPT) with zero pressure and −15 133 three different temperatures 318 K, 338 K and 358 K, respectively. A time step of 1 fs (1 fs = 10 second) is adopted for
134 the simulations through high-performance computing with 64 central processing units (CPUs) on HiPerGator supercomputer. 6 Xiaoyu Song ETAL
135 The temperature and pressure of the water system are controlled via a thermostat and a barostat respectively in LAMMPS
136 (Large-scale Atomic/Molecular Massively Parallel Simulator), a classical molecular dynamics code with a focus on materials 50 137 modeling . Figure 5 (a) and (b) shows the variation of temperature and potential energy of the system with the simulation
138 time and the variation of average water pressure with the simulation time at the temperature of 318 K. The results in Figure 5
139 (a) demonstrate the potential energy of the water system takes longer to reach the equilibrium state than the temperature of the
140 system does. Figure 5 (a) and (b) shows that the pressure oscillates around the zero pressure when both the temperature and
141 potential energy of the system have reached a dynamic equilibrium.
400 -4.3 600 Temperature 390 Potential energy -4.4 500
380 -4.5 400
370 -4.6 300 Kcal/mole) 5 360 -4.7 200
350 -4.8 100
340 -4.9 0 Temperature (Kelvin) 330 -5 Pressure (atmospheres) -100 Potential Energy (10 320 -5.1 -200
310 -5.2 -300 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (ns) Time (ns) (a) (b)
FIGURE 5 (a) Variation of water temperature and potential energy with respect to the simulation time; (b) Variation of water pressure with respect to the simulation time at a target temperature (e..g., 318 K).
142 In the second step, we cut a rectangular shaped water body of 26.40Å × 180.00Å × 60.00Å with respect to the x − y − z
143 coordinates from the equilibrated water box through the text-based Tcl/Tk interface on VMD (Visual Molecular Dynamics), a 51 144 molecular visualization and analysis program . The obtained water body is used to build the unsaturated clay-water model in
145 the initial state.
146 2.3.3 Clay-water model
147 This study focuses on the water meniscus between two parallel clay particles which are held at their initial positions. The water
148 between clay particles is free to move. Figure 6 is a schematic of the unsaturated clay-water system used in the MD simulation.
149 In Figure 6 two clay particles are placed parallel to each other keeping a selected spacing (lz) between them. The space between 150 the two particles is assumed constant for all simulations. A distance of 1.5 Å is kept in between the water body and the clay 38 151 layer to minimize the instability at the beginning of the MD simulation . Considering the rectangular water box is 60 Å in the
152 z direction, the distance between the clay particles are 63 Å for all simulations reported in the succeeding section. It is noted
153 that this value which is greater than the well-hydrated interlayer spacing of 15.23 Å is selected to avoid any possible interlayer 52 154 interactions . The rectangular water box prepared in the step of water preparation is placed between the clay particles in Figure
155 6 . The cell size of the model in Figure 6 is 26.40Å × 457.00Å × 76.12Å in the x − y − z Cartesian coordinate system. The
156 periodic cell size (see more on the periodic boundary conditions in the succeeding section) shall be at least four cut-off distances
157 (van der Waals) larger than the model size in the y and z directions to avoid any influences from the atoms in the neighboring 25,38 158 cells. A cutoff distance larger than 8 Å is generally considered as sufficient distance for truncating the van der Waals force .
159 In this study, a cut-off distance of 10 Å is used. In the x direction, the continuous feature of both clay particles and water are
160 maintained by keeping the width of the clay particle exactly the same as the size of the simulation box. The size of the simulation
161 box as sketched in Figure 6 are 26.40Å × 497.00Å × 116.12Å in the x − y − z Cartesian coordinate system. The number of
162 nitrogen and oxygen molecules occupied in the free space of the simulated model is extremely small. Therefore, air molecules
163 are not included in the unsaturated clay-water molecular model. Xiaoyu Song ETAL 7
simulation box
clay
water
clay z y x FIGURE 6 Schematic of the model set up for MD simulations.
164 3 SIMULATIONS
All MD simulations of the partially saturated clay-water model are performed through LAMMPS molecular dynamics simula- tor 50. All simulations are run in parallel on 256 CPUs of a supercomputer. The CHARMm force field 36 is adopted to model the
clay particles. In the CHARMm force field, the total potential energy Utot is expressed as a sum of the bonded energy Ub and the non-bonded energy Unb (refer to Figure 1 ). The bonded energy consists of three parts, namely, the stretching (bonds), the bending (angles), and the torsion (dihedral), as follows. É B 2 É A 2 É D Ub = k (r − r0) + k ( − 0) + k [1 + cos(n + )] , (1) bond angle dihedral B A D where k , k , and k are the force constants, r is the distance between the bonded atoms, r0 is its reference value, and are the rotational and torsional angles, respectively, 0 is the reference value of , n is the multiplicity, and is the phase angle of the dihedral cosine function. It is assumed that both the molecule of the clay and water are rigid, and hence the bonded energy equations are irrelevant. The non-bonded energy consists of the van der Waals part and long-range electrostatic part. In this
study, we adopt the following expression for the non-bonded energy, Unb. L0 112 0 16M É É ij ij É É qiqj U = 4" − + , (2) nb ij r r r van der Waals i≠j ij ij eletrostatic i≠j ij
165 where rij is the distance between atoms i and j, "ij and ij are constants governing the van der Waals energy, and qi and qj 166 are the electrostatic charges on atom i and j respectively. It is noted that the van de Waals part in Equation (2) is the standard 28 12 167 Lennard-Jones 6-12 potential . The term 1∕rij dominating at short interatomic distance models the repulsion between atoms, 168 whose physical origin is related to the Pauli principle - when the electronic clouds surrounding the atoms starts to overlap, the 6 169 energy of the system increase abruptly. The term 1∕rij dominating at large interatomic distance represents the attractive part 17 170 whose physical origin is van der Waals dispersion forces - dipole-dipole interactions in turn due to fluctuating dipoles . Figure
171 7 (a) shows the schematic of the standard Lennard-Jones 6-12 potential adopted for the unsaturated clay-water system. The
172 standard Lennard-Jones potential has an attractive tail at large rij , and it reaches a minimum around rij = 1.122ij , as shown in 173 Figure 7 (a). The parameters ij and ij are determined by fitting the physical properties of the material. Table 1 shows Van 37,31 174 der Waals parameters for clay and water used in all MD simulations. As usually adopted in molecuar dynamics modeling, √ 175 the parameters between two dissimilar atoms i and j are computed as, "ij = "i"j and ij = i + j ∕2, where (i, i) 176 are parameters associated with atom i, and (j , j ) are parameters associated with atom j. The values of the van der Waals 37 31 177 parameters ("ij and ij ) and the electrostatic charges (qi) are obtained from the work by Teppen et al. and Katti et al. . Figure 178 7 (b) plots the Lennard-Jones potentials for three pairs of dissimilar atoms in the clay-water molecular model.
179 For all simulations, the periodic boundary conditions are prescribed in all directions of the simulation box. Refer to Allen and 28 29 180 Tildesley and Frenkel and Smit for more discussions on periodic boundary conditions in MD simulations. The temperature 29 −15 181 of the system is controlled via a thermostat . A time step of 0.25 fs (1 fs =10 second) was used in all MD simulations. The 8 Xiaoyu Song ETAL
TABLE 1 Van der Waals Parameters for Clay and Water 37,31
Atom i(Kcal/mol⋅Å) i (Å) qi (e) Al 0.150 6.30 1.68 Si 0.001 7.40 1.40
H(clay) 0.0001 2.40 0.40 O(interior-1) 6.0 2.80 -0.96 O(interior-2) 6.0 2.80 -0.91 O(surface) 1.0 3.0 -0.70 H(water) 0.046 0.44 0.417 O(water) 0.152 3.53 -0.834
Total Energy OsiOsi 10 Repulsive Energy 10 OsiOw Attractive Energy OssOw
5 5
0 0 Energy E (Kcal/mol) Energy E (Kcal/mol)
-5 -5
2 2.5 3 3.5 4 4.5 5 2 2.5 3 3.5 4 4.5 5 Interatomic Seperation r (Angstrom) Interatomic Seperation r (Angstrom) (a) (b)
FIGURE 7 (a) Schematic of the Lennard-Jones 6-12 potential used for the unsaturated clay-water system; (b) Schematic of the Lennard-Jones 6-12 potential for three pairs of dissimilar atoms (Note: Osi = Oxygen in the interior layer of the soil molecular structure, Oss = Oxygen in the surface of layer of the soil molecular structure, and Ow = Oxygen in the water molecular).
182 procedure known as the "Ewald sum" which is implemented on LAMMPS is used to calculate the long-range electrostatic forces
183 accurately. The MD simulations of the unsaturated clay-water system are conducted in the canonical ensemble (constant number 28 184 of molecules, volume, and temperature or NVT) at different elevated temperatures (e.g., 318 K, 338 K, and 358 K). It is a good
185 practice to ensure that the initial configuration does not contain any high-energy interactions as these may cause instabilities in
186 the simulation. Such ‘hot spots’ can often be eradicated by performing energy minimization to the MD simulation itself. For
187 this purpose, in this study, we apply a Langevin thermostat to the water body, which models an interaction with a background 53,54 188 implicit solvent .
189 3.1 Monitoring the equilibrium state
190 The purpose of the equilibration is to enable the system to evolve from the initial configuration to reach equilibrium. Equilibra-
191 tion should continue until the values of a set of monitored properties become stable. The properties usually monitored include 30 192 thermodynamic quantities such as the system energy, temperature, and pressure . In this study, the simulation process is moni-
193 tored by the time-averaged energy curves and time-averaged system temperature curves with the simulation time, as well as the
194 capillary forces on clay particles obtained directly from the MD modeling. Figure 8 (a) illustrates the time-averaged system
195 energy curves with time for MD simulations of the unsaturated clay-water system at three different temperatures. Figure 8 (b)
196 plots the time-averaged system potential energy curves with time for the same simulations at three different temperatures. As
197 shown in Figure 8 , both the system temperature and potential energy jump at the beginning of all MD simulations, and then Xiaoyu Song ETAL 9
198 gradually decay. In general, when the variation of time-averaged energy and temperature has ceased, it indicates that the system
199 has reached a steady state. However, comparing Figure 8 (a) and (b) shows that it may take longer time for the potential energy
200 to reach a steady state than the system temperature does. Furthermore, the absolute value of the capillary force on the clay par- 38 201 ticle and suction pressure of water in a two-dimensional clay-water system may take longer to reach a steady state (refer to the
202 succeeding section for more discussions). Therefore, for all simulations in this study both the capillary force on clay particles
203 and negative pore water pressure (suction) are also tracked during the simulation process to determine whether the clay-water
204 system reaches a steady state. Figure 9 plots snapshots of configurations of the partially saturated clay-water system at four
205 different simulation stages from the initial to the final time step.
440 -1.286x106 T=318 K T=318 K T=338 K 6 T=338 K 420 T=358 K -1.288x10 T=358 K -1.29x106 400 -1.292x106 380 -1.294x106 360 -1.296x106 340 Temperature (Kelvin) -1.298x106 Potential Energy (Kcal/mole) 320 -1.3x106
300 -1.302x106 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (ns) Time (ns) (a) (b)
FIGURE 8 (a) Time variation of temperature of the system; (b) Time variation of potential energy of the system at three temperatures, 318 K, 338 K and 358 K, respectively.
206 4 RESULTS AND DISCUSSION
207 MD simulations enable predictions of the thermodynamic properties of systems for which there is no experimental data, or for
208 which experimental data is difficult (if not impossible) to obtain. MD simulations can also provide structural information about 30 209 the conformational changes in molecules and the distributions of molecules in the system . In this section, we will report the
210 numerical results directly obtained from the full-scale MD simulation at different elevated temperatures. In particular, we study
(a) (c)
(b) (d)
FIGURE 9 Snapshots of the configuration of the partially saturated clay-water system at four different simulation stages: (a) initial, (b) 0.033 ns, (c) 0.066 ns, and (d) 0.1 ns. 10 Xiaoyu Song ETAL
211 the impact of the system temperature on the capillary force induced by the water meniscus and the negative pore water pressure
212 in the water meniscus. We also examine the variation of the contact angle at the solid-water-air interface and the approximate
213 meniscus curvature with the temperature increase. Then, we compare capillary force obtained from MD simulations with the 55 214 results obtained from the macroscopic theory, i.e., Young-Laplace equation , at different elevated temperatures.
215 4.1 Capillary force The assessment of the interparticle forces that arise in capillary liquid bridges and their evolution are of primary importance because of their contribution to the solid deformation and fluid flow in unsaturated soils 1,56,57. Capillary force is a tensile force imposed on the clay particles by the water meniscus between clay particles. The capillary force consists of two parts which are associated with the water surface tension at the soil-water-air interface and negative pore water pressure, respectively. The capillary force plays a fundamental role in modeling unsaturated soils at both microscale (pore scale) ad continuum scale 1,15,58,4. Macroscopically, the capillary force tends to pull the soil grains toward one another, similar in effect to overburden stress or a surcharge load 1. From MD modeling, we can directly measure/output the capillary forces on the clay particles when the system reaches its equilibrium state. In MD, the pressure tensor (or stress tensor) can be defined from the varial 59,60 as H I 1 É 1 É ̄pab = − pnapnb∕mn + rnafnb , (3) V n 2 n where a and b are components of the pressure tensor, n is the particle index, p is the momentum of particle n, m is its mass, r is
its position, and f is the force. Then the forces, F x, F y, and F z on the top or bottom clay particle in the x − y − z directions respectively can be computed as
F x = ̄pxx × ly × lz, (4)
F y = ̄pyy × lx × lz, (5)
F z = ̄pzz × lx × ly, (6)
216 where ̄pxx, ̄pyy, and ̄pzz are the averaged pressure of each atom in the top or bottom clay particle along the x − y − z-directions, 217 respectively, lx, ly and lz are the length of the clay particle in the x − y − z directions, respectively. It is worth noting that F z can 50 218 be generated directly from the MD simulations on LAMMPS . Figure 10 (a) and (b) plots the variation of capillary force on
219 the top and bottom clay particles with the simulation time at three different temperatures. The results in Figure 10 (a) and (b)
220 show that the magnitude of the capillary forces on the top and bottom clay particles oscillates with different amplitudes for three
221 different temperatures at the steady state or the equilibrium state. To show the average values of capillary forces in the z direction
222 on both clay particles, Figure 10 (c) and (d) plots the time average of the capillary force on the top and bottom clay particles
223 with the simulation time at the three different temperatures. The average values of the capillary force on the clay particles in
224 the z direction at each temperature are summarized in Table 2 . The magnitude of the capillary force on the top and the bottom
225 clay particles in the z direction are almost identical at each temperature. Furthermore, comparing Figures 10 and 8 (a) shows
226 that the capillary force takes more time to obtain a dynamical equilibrium state than the system temperature does. The results
227 of the average capillary forces in Table 2 demonstrate the system temperature increase can decrease the overall capillary force
228 imposed on the clay particle by the water meniscus. The observed phenomena can be due to the fact that a temperature increase 1 229 will decrease the absolute water pressure and the surface tension of water in the meniscus . Both are major contributors of the
230 capillary stress in the z direction. Thus the capillary stress in the z direction decreases with the temperature increase.
231 Figure 11 (a) and (b) plots the variation of capillary force in x and y directions (see Figure 6 ) on the top clay particle with
232 the simulation time at three different temperatures. Similarly, Figure 12 (a) and (b) plots the variation of capillary force in the x
233 and y directions (see Figure 6 ) on the bottom clay particle with the simulation time at three different temperatures. The results
234 in both Figures 11 and 12 show that the capillary force on the clay particles in the x and y directions are almost zero for the
235 symmetry of the clay-water model although they oscillate with different amplitudes around zero. The results about the average
236 capillary force on the clay particles in the x and y directions shown in Table 3 are almost zero due to the symmetry of the
237 clay-water molecular model.
238
239 Xiaoyu Song ETAL 11
500 -500 T=318 K T=318 K T=338 K T=338 K 400 T=358 K -400 T=358 K
300 -300
200 -200
100 -100
Force (Kcalmol-A) 0 Force (Kcalmol-A) 0
-100 100
-200 200 0.02 0.04 0.06 0.08 0.1 0.02 0.04 0.06 0.08 0.1 Time (ns) Time (ns) (a) (b)
100 100 T=318 K T=318 K T=338 K T=338 K T=358 K T=358 K
50 50
0 0 Force (Kcalmol-A) Force (Kcalmol-A) -50 -50
-100 -100 0.02 0.04 0.06 0.08 0.1 0.02 0.04 0.06 0.08 0.1 Time (ns) Time (ns) (c) (d)
FIGURE 10 Time variation of the capillary force in the z direction on (a) the top clay particle; (b) the bottom clay particle (see Figure 6 ); and average time variation of the capillary force in the z direction on (c) the top clay particle; and (d) the bottom clay particle at three temperatures, 318 K, 338 K and 358 K, respectively.
TABLE 2 Average capillary forces on the top and bottom clay particles and capillary pressure of water at three different temperatures (Note: 1 Atm = 101.1 kPa).
Average capillary force (F z) (Kcal/mol⋅Å) System temperature Top clay particle Bottom clay particle Average capillary pressure (Atm) T = 318 K -15.26 15.61 −201.63 T = 338 K -13.41 13.95 −192.15 T = 358 K -12.39 12.64 −187.49
240 4.2 Pore water pressure
241 The pressure of the pore water phase is negative in the unsaturated clay-water system if we assume the pore air pressure is zero.
242 The negative pore water pressure (matric suction or capillary pressure) in the water meniscus is a fundamental ingredient in
243 understanding the mechanical and hydraulic behavior of unsaturated porous media. For instance, the negative pore water pressure 61,62,63,64,65,66,67,68 244 is an essential component of the effective stress for unsaturated soils . However, it is not possible to directly 69 245 measure the capillary pressure between nanoparticles and water by experiments in general as discussed in Binks and Clint . 12 Xiaoyu Song ETAL
80 80 T=318 K T=318 K T=338 K T=338 K T=358 K T=358 K 60 60
40 40
20 20
0 0 Force (Kcalmol-A) Force (Kcalmol-A)
-20 -20
-40 -40 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time (ns) Time (ns) (a) (b)
FIGURE 11 Time variation of capillary force (a) in the x direction and (b) in the y direction on the top clay particle (see Figure 6 ) at three temperatures, 318 K, 338 K and 358 K, respectively.
80 80 T=318 K T=318 K T=338 K T=338 K T=358 K T=358 K 60 60
40 40
20 20
0 0 Force (Kcalmol-A) Force (Kcalmol-A)
-20 -20
-40 -40 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time (ns) Time (ns) (a) (b)
FIGURE 12 Time variation of capillary force (a) in the x direction and (b) in the y direction on the bottom clay particle (see Figure 6 ) at three temperatures, 318 K, 338 K and 358 K, respectively.
246 Figure 13 (a) plots the variation of the negative pore water pressure with the simulation time at three different temperatures,
247 318 K, 338 K, and 358 K, respectively. To show the average values of negative pore ater pressure in the z direction on both clay
248 particles, Figure 13 (b) plots the time-average of the negative pore water pressure with the simulation time at the three different
249 temperatures. The time-averaged suction pressure at the three different temperatures is listed in Table 2 . The magnitude of
250 capillary pressure at T = 318 K is 201.63 Atm, which is about 2 million kPa. With the increase of the system temperature, the
251 capillary pressure decreases. For instance, the magnitude of capillary pressure at T =358 K is 187.49 Atm, which 14.14 Atm
252 (or about 1414 kPa) smaller than that at T = 318 K. The results in Table 2 demonstrate that the temperature increase generally
253 decreases the matric suction in the water meniscus between clay particles. This observed phenomenon may be due to the water
254 expansion when the system temperature increases. It is also worth noting that this observation on the micro-scale (or nanoscale) 1 255 from MD modeling is consistent with the experimental data at the macroscale (or continuum scale) . Xiaoyu Song ETAL 13
TABLE 3 Average capillary forces in the x and y directions on the top and bottom clay particles at three different temperatures (Note: 1 Atm = 101.1 kPa).
Average capillary force (F x) (Kcal/mol⋅Å) Average capillary force (F y) (Kcal/mol⋅Å) System temperature Top clay particle Bottom clay particle Top clay particle Bottom clay particle T = 318 K -0.083 0.096 −0.059 0.093 T = 338 K -0.057 -0.052 −0.429 -0.019 T = 358 K 0.006 -0.099 −0.031 -0.463
150 50 T=318 K T=318 K 100 T=338 K T=338 K 0 T=358 K T=358 K 50 0 -50
-50 -100 -100 -150 -150 -200 -200 Pressure (Atm) Pressure (Atm) -250 -250 -300 -350 -300
-400 -350 0.02 0.04 0.06 0.08 0.1 0.12 0.02 0.04 0.06 0.08 0.1 0.12 Time (ns) Time (ns) (a) (b)
FIGURE 13 (a) Time variation of capillary pressure of the water meniscus at three temperatures, 318 K, 338 K and 358 K, respectively. (b) Average time variation of capillary pressure of the water meniscus at three temperatures, 318 K, 338 K and 358 K, respectively.
256 4.3 Average contact angle and approximate water meniscus curvature
257 One quantitative measure of the liquid-solid interaction is the contact angle, , formed by a liquid when placed against a solid
258 as shown in Figure 16 . The assumption is that the third phase involved is the equilibrium vapor of the liquid or some inertia 70 259 gas such as air . On rough solid surfaces, the three-phase interline may be quite complex. To avoid ambiguity, is defined
260 rigorously as the angle made between the normals to the solid surface (n1) and liquid surface (n2) (refer to Figure 16 ) at the point −1 261 of interest along the interline, i.e., = cos (n1 ⋅n2). The physical interline between the gas, liquid and solid phases is ultimately 262 made up of individual molecules, and at that scale, the contact angle may differ from the observed one macroscopically. At the
263 macroscopic scale, the contact angle is defined as the angle formed between the interface of water and solid and the interface 17 264 of water and air . Contact angle at the solid-water-air interface is a measure of the wettability a fluid on a solid surface. The
265 magnitude of contact angle characterizes the water adsorption capability of the solid. For instance, a zero contact angle means o 266 a perfect wettability (hydrophilic) and a 180 means a perfect non-wettability (hydrophobic).
267 At the molecular scale, it is rather difficult to measure the contact angle for nanodroplets on surfaces (e.g., a water meniscus
268 between clay particles) because there are significant fluctuations in the shape of the droplet or the water meniscus. Furthermore,
269 for tiny nanoclusters, the surface tension of water is a function of the curvature, and the change in the line tension with cur-
270 vature is also an important factor affecting the contact angle. In the analysis of MD simulations, contact angles are commonly
271 determined by using two-dimensional slices of the droplet and fitting its density profile to an empirical function, such as a cir- 71,72 38 272 cular section . For instance, Amarasinghe et al. graphically drew a fitting curve directly on the averaged positions of the
273 last few frames to determine the contact angle. For the study here, the tangent of the water-solid interface is a straight line par-
274 allel to the solid particle surface. To determine the contact angle, we need to accurately locate the topology of water meniscus
275 because the tangent line to the water-air interface depends on the shape of water meniscus. Figure 14 (a) and (c) plots the final 14 Xiaoyu Song ETAL
276 configuration of the meniscus from the front view and the back review, respectively, at T = 318 K. Figure 14 (b) shows the
277 zoom-in information of the water meniscus at the same temperature. The results in Figure 14 confirm that the water menis-
278 cus has the same configuration in the z direction as expected for the plane strain model. Thus we determine the contact angle
279 approximately referring to the front view of the average configuration over the last few frames. Figure 15 plots the snapshots
280 of the last configuration of the unsaturated clay-water system at three different temperatures.
(a) Front
(b)
(c) Rear
FIGURE 14 Snapshots of the last configuration of the unsaturated clay-water system at T=318 K: (a) Front view, (b) Zoom-in of the front view, and (c) Rear view.
281 For simplicity, here we graphically drew a fitting curve directly on the averaged positions of the last several hundred frames 38 282 to determine the contact angle as shown in Figure 15 . Table 4 summarizes the average contact angle for the simulations
283 at three temperatures and the corresponding approximate meniscus curvatures based on equation (9) in the succeeding section.
284 The results in Table 4 demonstrate that the contact generally decreases with increasing temperature which is consistent with 24 285 the macroscopic observation . This observation may be due to the decrease of the adhesion between the meniscus water and
286 the clay platelet and the surface tension of the meniscus water by the temperature increase.
TABLE 4 Summary of contact angle and approximate meniscus curvature
Temperature (Kelvin) Average contact angle (o) Approximate meniscus curvature (Å) T = 318 K 34 38.0 T = 338 K 30 36.4 T = 358 K 25 34.8 Xiaoyu Song ETAL 15
(a)
(b)
(c)
FIGURE 15 Snapshots of the last configuration of the unsaturated clay-water system at three different temperatures, (a) 318 K, (b) 338 K, and (c) 358 K, respectively.
287 4.4 Comparison with the macroscopic theory Given the contact angle and suction pressure at the nanoscale, we can determine the capillary force on the clay particle imposed by the water meniscus through the Young-Laplace equation 55. Figure 16 sketches the two components of the capillary force on the clay particle under plane strain condition. For a two-dimensional unsaturated clay-water system shown in Figure 16 ,
Clay θ θ z γ θ d w γw y pw x R Water Clay
FIGURE 16 Schematic of the decomposition of capillary force on clay particles under plane strain condition (Note: w is the water surface tension, is the contact angle at finite temperature, pw is water pressure in the water meniscus, and R is the approximate radius of the water meniscus assuming the water-air interface is a circle).
assuming that the water meniscus can be approximated by a circular arc, the Young-Laplace equation can be written as Γ p − p = w , (7) a w R
288 where pa is the air pressure outside the water meniscus, pw is the water pressure (capillary pressure) within the water meniscus, 289 Γw is the water surface tension, and R is the radius of the water meniscus curvature. The total force F z acting on the top clay 290 particle due to the water meniscus (capillary water) consists of two parts, i.e., the surface tension force and the tensile force
291 generated by the negative pore water pressure in the water meniscus, which can be expressed as follows,
F z = pwLxLy − 2ΓwLx sin , (8)
292 where Lx and Ly are the lengths of the rectangular interface area between the water meniscus and top clay particle. 16 Xiaoyu Song ETAL
From Equation (7), we can obtain the surface tension as a function of pore water and pore air pressures and the radius of the meniscus curvature. Based on the geometrical relationship shown in Figure 16 , the radius of the meniscus curvature can be expressed as d R = , (9) cos where d is one half of the gap between the two clay particles (i.e., 31.5 Å). Substituting Equation (7) and (9) into Equation (8) and assuming passive pore air pressure give the capillary force as a function of pore water pressure, the radius of meniscus curvature, and contact angle as follows,
F z = pwLx(Ly + 2d tan ). (10)
293 Thus, given the suction pressure pw and contact angle , we can determine the capillary force in the z direction on the clay 294 particle imposed by the water meniscus. Table 5 lists the average capillary force in the z direction on the clay particle obtained
295 from MD simulations and the macroscopic theory with and without considering the water surface tension on the boundary of
296 the water meniscus at three different temperatures. The results in Table 5 demonstrate that the capillary force from molecular
297 dynamics simulations is close to the results from the macroscopic theory at elevated temperatures. However, the latter needs
298 the capillary pressure and the approximate meniscus curvature obtained from molecular dynamics simulations. Furthermore,
299 the results in Table 5 also show that the negative pore water pressure contributes more than 80% of the capillary force for the
300 particular clay-water molecular model at elevated temperatures. It is noted the capillary force obtained directly from the MD
301 simulations does not distinguish between the contributions from the water surface tension surrounding the water meniscus and
302 the negative water pressure of the water meniscus.
TABLE 5 Comparison of average capillary forces F z (Unit: Kcal/mol⋅Å) on the top and bottom clay particles and the capillary force from the MD simulations and the results from the macroscopic theory at elevated temperatures.
Capillary force F z from Young-Laplace equation
Temperature (Kelvin) Average magnitude of F z With surface tension Without surface tension T = 318 K 15.43 17.23 13.94 T = 338 K 13.68 15.97 13.29 T = 358 K 12.51 15.37 12.96
303 4.5 Discussions
304 The numerical results presented above have demonstrated that MD simulations can be utilized to obtain the capillary force and
305 suction of the water meniscus between two clay particles with a separation greater than the well-hydrated interlayer spacing
306 at different elevated temperatures. Thus the MD modeling here is not intended to probe the strong oscillation of the capillary 73 307 force at a smaller separation of clay particles, i.e., less than 50 Å (5 nm), as usually observed in MD simulations . Given the 55 308 contact angle and capillary pressure, the macroscopic theory can be applied to estimate the capillary force between the solid
309 and the pore water at elevated temperatures. However, the macroscopic theory cannot capture the oscillation of the capillary
310 force because the macroscopic theory (i.e., the Young-Laplace equation) only considers contributions from the surface tension
311 around the circumference of the meniscus and the pressure difference over the cross-section of the meniscus (e.g., Equation (8)).
312 Furthermore, the macroscopic theory does not take into account the contribution of the line tension between the solid-water-air 74,75,76 313 interface which can play an essential role in determining the capillary force in a three-dimensional case , which will be
314 addressed in a subsequent publication. It is also noted that tha disjoining pressure can influence the equilibrium interfacial profile 77 315 in the transition zone between a capillary meniscus and a thin film (e.g., the water in the meniscus) . The approximate contact
316 angle used in the macroscopic theory is determined by graphically drawing a fitting curve directly on the averaged positions of 78 317 the last few frames. It is worth noting that such an approach may provide inconsistent results , e.g., the potential larger error on
318 the measured contact angle using two-dimensional slices of molecular snapshots. Furthermore, the interface layer is diffuse and 79 319 spans a width of several molecular diameters. This fact may introduce another potential source of error . It is recommended to 80,81 320 adopt a more accurate method to characterize the variation of the contact angle with temperature through MD simulations. Xiaoyu Song ETAL 17
321 5 CONCLUSION
322 In this article, we have studied the capillary force on the clay particles by the water meniscus at elevated temperatures through
323 the full-scale molecular dynamics modeling. Since MD modeling is computationally expensive we run all simulations through
324 high-performance computing using 256 CPUs on a supercomputer to reduce the wall-clock time for the unsaturated clay-water
325 molecular model to reach a physical equilibrium. The study has been focused on the impact of temperature increase on the
326 capillary force imposed on the clay particle by the water meniscus between two parallel clay particles, which is different from 38,82 327 the MD simulations under the isothermal conditions in the literature . In the numerical model, the clay particle is modeled
328 by pyrophyllite, the pore water is modeled by a rigid water model, and the pore air is represented by empty space. The range
329 of temperature of the unsaturated clay molecular model is from 300 K (i.e., ambient temperature) to 360 K. Thus the water-
330 vapor interface is not taken into account in this study. The numerical results have demonstrated that the temperature increase
331 generally decreases the capillary force and the negative pore water pressure (suction) which is consistent with experimental
332 results. We also compared the numerical results against the results from the macroscopic theory (i.e., Young-Laplace equation)
333 by assuming that the meniscus curvature can be approximated by a circular curve. The capillary force on the clay particle at an
334 elevated temperature obtained by the macroscopic theory is consistent with the results from the MD simulation. However, the
335 macroscopic theory requires the knowledge of the capillary pressure, and the contact angle and the surface tension of water at
336 the target temperature although these parameters can be obtained from the MD simulation. Finally, it is worth noting that the
337 MD simulation of unsaturated clay at elevated temperatures in this article has not only provided a fundamental understanding of
338 the impact of temperature on the interfacial physics of the unsaturated clay at the atomic scale. It also has significant implication
339 in formulating physics-based multiscale models for thermal unsaturated soils by providing physical properties (e.g., capillary
340 pressure or capillary force) at the atomic scale (or nanoscale) through high-performance computing instead of laboratory tests 3 341 which are usually difficult or even impossible at the atomic scale .
342 ACKNOWLEDGEMENTS
343 Support for this work was provided by the Geotechnical Engineering and Materials Program of the US National Science Foun-
344 dation (NSF) under Contract number CMMI-1659932 to the University of Florida. The support of Dr. Richard Fragaszy is
345 gratefully acknowledged. Any opinions or positions expressed in this article are those of the authors only and do not reflect any
346 opinion of positions of the NSF. We also acknowledge the high-performance computing resource - HiPerGator supercomputer
347 provided by the Research Computing at the University of Florida.
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How to cite this article: Song X, and M. Wang (2018), Molecular dynamics modeling of a partially saturated clay-water system
506 at finite temperature, J Numer Anal Methods Geomech, 2018;00:1–20.