This work focuses on using computational techniques to comprehend the reasons that impact ion transport in gelled and nanoparticle embedded polymer electrolyte (PE) membranes. The final goal is to engineer new polymeric materials to replace traditional liquid electrolytes for rechargeable lithium batteries (RLBs). The additional goal is to verify these computational techniques against experiments. The verification is presented by directly comparing computationally calculated lithium ion conductivities with experiment for advanced PEs. After all experiments are finished, other may, with this verification evaluate the reliability of the models and methods used to set up collaborations with experimentalists in the field, and propose and verify new PEs with optimal lithium ion conductivities and mechanical properties.
Rechargeable Lithium Battery (RLB)
The need for energy is one of the most important issues and challenges facing our country and the world today. Improving electrochemical energy technologies, for example batteries, will be a vital part of the solution to our energy challenges. The huge economic and environmental benefits will be provided through these new technologies. Also, their use can indirectly reduce the dependence on imported fuels. In recent decades, rechargeable lithium batteries, especially the second generation, have been increasingly used in consumer electronics and military equipment, and have the potential for wide utilization in electric and hybrid vehicles. The most often compromised electrolytes for RLBs are liquids. However, the leakage of liquid electrolytes is a major safety consideration, caused by the highly reactive elements with lithium salts and metals. Furthermore, at high temperatures and in overcharging circumstances, traditional carbonate electrolytes react with the electrodes, creating gases that cause the batteries to break, resulting in fire or explosion. Due to the react ion with electrolytes, some of lithium becomes passivated and isolated from the bulk anode forever as finely divided lithium. This phenomenon is usual to all secondary generation lithium cells and is somewhat independent of the cathode .
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To eliminate these considerations, a common way is using a solid polymer electrolyte (PE) in place of the liquid electrolytes since it is virtually free of leakage. These composite solid polymer electrolytes offers other potential advantages, such as low cost design, flexibility in sizes and shapes, good electrochemical stability, low flammability and toxicity with the ability to form good interfacial contact with electrodes, and so on . The most investigated PE is poly(ethylene oxide) (PEO) with the addition of lithium salts of the imide anion, ClO4¯, [N(CF3SO3)2]¯(TFSI¯), and BF4¯ etc., to polymer to allow the conduction of lithium ions, in which the segmental motion of the PEO chains assists ion motion along the oxygen atoms. The ionic conductivity of PEs, however, at room temperature is on the order of 10-4~10-7 S/cm, while 10-3 S/cm is a good fit for an electrolyte to be commercially viable . A reasonable speculation for this non-viability is that the nature of PEs are crystalline near room temperature, hindering their efficiency and significantly reducing their practicality . In order to enhance the conductivity of PEs, two popular methods have been pursued : the introduction of plasticizers to form plasticized polymers (PPs), and utilization of nanoporous membranes.
Gelled/Plasticized Polymer Electrolytes (PPs)
The addition of plasticizers, such as cyclic carbonates, has been shown to increase the conductivity to practical levels . Consider the beneficial properties of solid-state PEs and the high conductivities of liquid electrolytes, plasticized polymers are a compromise proposes. While PPs provide conductivities close to those of liquid electrolytes, they have two main weaknesses. Since a large liquid component is added in polymer, the mechanical properties have been weakened. Also the separation of the liquid fraction from the polymer, indicated as syneresis, is a problem . One common way to improve the mechanical properties of gel PEs is using cross-linking the polymers in the gel .
The addition of nanoporous membranes (e.g. TiO2, Al2O3, and SiO2 etc.) to PEs has the benefit of increasing the solid-polymer interfacial area over spherical nanoparticles . Evidence reviews that stronger PEO oxygen-nanoparticle and anion-nanoparticle interactions aids faster lithium transport through the electrolyte . Based on this reasoning, further strengthening of these interactions, and increasing the overall volume fraction of PEO in touch with these interactions are enhance the conductivity and lithium transference number.
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The understanding of the ion transport mechanism on the molecular level of these systems, and the effect of these behaviors on the entire conductivities of PEs would avail the design of PEs for RLBs greatly. Computational methods have a valuable role to play in providing significant insight into molecular level interactions and structures, and predicting essentially exact results for problems in statistical mechanics. Particularly, molecular dynamics (MD) simulations are well suited for providing a direct route from the microscopic details of a system (the molecular geometry, the masses of the atoms, the interaction between them, etc.). The results of macroscopic properties, such as ionic conductivity, transport coefficients, and so on, may also be directly compared with those of real experiments.
The conduction mechanism of lithium in poly(ethylene oxide) (PEO) and in carbonates has been investigated widely by computational methods. Because of this, there is a pretty good understanding of the mechanism of lithium ion transport in neat amorphous and crystalline PEO. Nevertheless, the role of plasticizers on lithium conduction in PEO in low enough concentrations to be relevant for RLBs has not been studied computationally, and requires the development of new computational methodologies.
Regarding the modeling of PEs with a single embedded nanoparticle, there have been a few studies focused on lithium ion transport on the molecular level . These works provide some interesting qualitative insight into lithium ion motion for these systems. However, the mechanism of the influence of plasticizers on PEO structure and lithium conduction on the molecular level is not understood completely. Also, research of interactions between PEO, lithium, and its counter-ions with spherical nanoparticles affecting lithium ion conduction is still incomplete and lacks comparison with other experiments.
Computer Simulation Techniques
Sampling from ensembles
Statistical mechanics provides a link for relating the microscopic properties (atomic and molecular positions R, velocities v etc.) of atom and molecules to the macroscopic properties (pressure p, internal energy E etc.) of materials. Consider a one-component macroscopic system, which is usually defined by a small set of parameters (e.g. the number of particles N, the temperature T, the energy E, the volume V and the pressure p etc.). Use donates for a point in phase space, and calculate the instantaneous value of some macroscopic property, as a function. The experimentally observable 'macroscopic' property can be performed by averaging over all possible states:
where is the probability density for state . In general, , such as , etc., is a function defined by the chosen fixed macroscopic parameters. For convenience sake, can be written as a 'weight' function, with a partition function (also called the sum over states) acting as the normalizing factor:
Common Statistical Ensembles
There are four popular statistical mechanical ensembles in common use: the microcanonical (constant-NVE) ensemble, the canonical (constant-NVT) ensemble, the isothermal-isobaric (constant-NTp) ensemble, and the grand canonical (constant-µVT) ensemble. In this work, three of them, except the grand canonical (constant-µVT) ensemble, are used and explained below.
The microcanonical ensemble, also referred to as the constant-NVE ensemble, is very useful for theoretical discussions. This ensemble is the collection of all states with a fixed number of particles (N), the volume (V), and the energy (E). It describes a completely isolated system, as it does not exchange energy or mass with the rest of the universe.
The probability density for the constant-NVE ensemble is proportional to , where is the Hamiltionian of the system and is a Kronecker delta, taking values of 0 or 1 when the set of states is discrete; when the states are continuous, is the Dirac delta function. Then the microcanonical partition function can be written:
For a quasi-classical system, the partition function can be expressed using a factor of ,
Here, h is Planck's constant,stands for integration over all 6N phase space coordinates for the three-dimensional system of N spherical particles.
The most commonly used ensemble in statistical thermodynamics is the canonical, or constant-NVT, ensemble. Each of the systems can share its energy with a large heat reservoir or heat bath, and each also requires keeping the number of particles (N), the volume (V), and the temperature (T) constant.
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The density for canonical ensemble is proportional to and the partition function yields:
The quasi-classical form for an atomic system is:
since the Hamiltonian can be described as a sum of kinetic and potential energy functions of the set of coordinates and momenta of each molecule. We have, the partition function can be turned into a product of kinetic (ideal gas) and potential (excess) part:
The quasi-classical form for an atomic system is:
where is the thermal de Broglie wavelength given by:
here, is a configuration integral, m is the molecular mass, is Boltzmann constant.
The isothermal-isobaric ensemble (constant-NTp ensemble) is an ensemble of systems in which the individual systems have N, T, and p fixed. The constraints would be on the total energy and total volume of the ensemble.
The density for the isothermal-isobaric ensemble is proportional to and the partition function is
The quasi-classical form for an atomic system is:
where is a basic unit of volume.
The configuration integral in this ensemble is:
General Simulation Methods
There are two predominant types of simulation methods that are employed to study and calculate thermodynamic properties of molecular systems: molecular dynamics (MD) and Monte Carlo (MC).
Molecular Dynamics (MD)
Molecular Dynamics (MD) is a very useful method of computer simulation of atom and molecule modeling based on statistical mechanics. MD simulation consists of the numerical, step-by-step, solution of classical equations of motion, which for a simple atomic system may be generated by integrating Newton's second law or the equation of motion, , where is the force exerted on a particle of mass and is its acceleration. The result is a trajectory that describes the positions, velocities and accelerations of the particles in the system as they vary with time.
The most widely used numerical integration scheme was that first used by Verlet in 1967 , which is derived by truncating the Taylor expansion of at :
The MD method is deterministic; once the positions and velocities of each atom are known, the state of the system can be predicted at any time in the future or the past. Moreover, the basic MD method is quite standard and can be used to study a large variety of systems, allowing for its wide use by experts and novices alike. The disadvantage of MD is that it is time consuming and computationally expensive. Since MD is an analytical solution to the equations of motion it cannot be determined for a complex molecular system. The simulations have to be split up into individual time steps ranging from about 0.5 ~ 5 fs (1 fs = 10-15 s).
Monte Carlo Methods (MC)
In contrast to molecular dynamics, most Monte Carlo methods do not follow any deterministic procedure, but follow a Markov Chain process. For a Markov Chain, any change in system configuration of a single step only depends on the previous step. In MC, the system moves between different states in a stochastic matter. A MC trajectory is generated by performing a random walk through configuration space.
Metropolis Monte Carlo
The Metropolis algorithm was the first method developed using Eqn (2.1)
Gibbs Ensemble Monte Carlo (GEMC)
Connectivity-altering Monte Carlo (CAMC)
Configurational-bias Monte Carlo (CBMC)
Self-Adapting Fixed-Endpoint Configurantional-Bias Monte Carlo (SAFE-CBMC)
Cross-linked Polymeric Gel (Prop_Poly page 11)
A force field (or forcefield) is a set of parameters and equations used in molecular mechanics simulations. Force field is used to calculate the potential energy of system of particles (typically but not necessarily atoms). Its functions and parameter sets are derived from both experimental work and high-level quantum mechanical calculations.
It is a mathematical function that describes how atoms/molecules move, stretch, vibrate, rotat and interact with each other. In the force field function, the presence of electros is ignored.
A general form for the total energy in a force field can be written as:
where indicates the potential energy, it is a function of the positions (r) of N particles (usually atoms).
The force field used for PEO in this research is the transferable potentials for phase equilibria-united atom (TraPPE-UA). It utilizes pseudo-atoms located at the center of carbon atoms for alkyl groups and treats all other atoms explicitly . This model uses Lennard-Jones (LJ) interactions potentials of the 12-6 form with electrostatic charges fixed. It has been found to do a good job of reproducing PEO densities over a fairly wide range of temperatures and pressures . PEO chains are considered semi-flexible with fixed bond lengths and flexible bond angles and dihedrals.
where and are the force constant, and , are the equilibrium bond length and angle, respectively. For all dihedral interactions except O-CH2-CH2-O, a cosine series form is used:
where is the dihedral angle, and are constants. For the pairwise nonbonded interaction energy between atoms and, separated by a distance of , the standard Lennard-Jones (LJ) 12-6 and Coulombic potentials are used:
where and are LJ diameter and well depth respectively, and is the Coulombic charge assigned to atom . For unlike interactions, the standard Lorentz-Berthelot combining rules are used:
A potential truncation of rcut with analytical tail corrections is used for all LJ interactions, and Ewald-summation is used to account for long-ranged Coulombic interactions.