Molecular modeling and electron transport in polyethylene

Molecular Modeling and Electron Transportin Polyethylene

Yang Wang, Kai Wu1State Key Lab. of Electrical Insulation and Power Equipment,

Xi'an Jiaotong University, Xi'an, 710049, China

David Cubero2, 1

2Departmento de Física Aplicada I,Escuela Politécnica Superior,

Universidad de Sevilla, Seville, 41011, Spain

and Nick Quirke3, 1

3Department of Chemistry, Imperial College, London, SW7 2AZ, UK

ABSTRACTPolyethylene is commonly used as an insulator for AC powercables. However it is known to undergo chemical and physicalchange which can lead to dielectric breakdown. Despite almosteighty years of experimental characterization of itselectrical properties, very little is known about the detailsof the electrical behaviour of this material at the molecularlevel. An understanding of the mechanisms of charge trappingand transport could help in the development of materials withbetter insulating properties required for the next generationof high voltage AC and DC cables. Molecular simulationtechniques provide a unique tool with which to studydielectric processes at the atomic and electronic level. Herewe summarise simulation methodologies which have been used tostudy the properties of PE at the molecular level, elucidatingthe role of morphology in the trapping of excess electrons. Wefind that polyethylene has localised states due toconformational trapping extending below the mobility edge(above which the electrons are delocalised), at -0.1±0.1eVwith respect to the vacuum level. These trap states withlocalisation lengths between 0.3 and 1.2nm have energies aslow as -0.4±0.1eV in the amorphous and interfacial regions ofpolyethylene with more positive values in lamella structures.Crystalline regions have a mobility edge at +0.46 ±0.1eV, sowe would expect transport by electrons excited above the

Manuscript received on 10 September 2013, in final form 15 February 2014, accepted 16 March 2014.

mobility edge to delocalised states to be predominantlythrough amorphous regions if they percolate the sample.

Index Terms — Molecular simulations, polyethylene,methodologies, electron trapping, electron transport,morphology.

1 INTRODUCTIONPOLYETHYLENE low electric conductivity,

low dielectric loss, high dielectricstrength and outstanding mechanicalproperties [1] make it a common choice asan insulator for AC power cables.However, under certain conditions,polyethylene may suffer from chemical andelectrical aging or degradation afterlong-standing operation under voltage,which may finally lead to dielectricbreakdown [2]. A major contribution to aging and

breakdown is thought to be made by trappedcharges, called the ‘space charge’ [2].The accumulation of space charge distortsthe local electric field, facilitatingphenomena such as treeing as well asgenerating chemical and physical defectsthrough the local release of trappingenergies (of the order of an eV),eventually leading to the degradation ofthe insulation, although the exactmechanisms operating are still not fullyunderstood. The formation andtransportation mechanism of space chargehas been the subject of extensiveexperimentation [2]. For example, bymonitoring the threshold electric fieldfor charge accumulation it has been foundthat charge injection is strongly affectedby the material of the electrode [3-5].Space charge formation and transportphenomena under DC stress can be observedusing the Pulsed Electro-Acoustic (PEA)method [6-9]. This technique has been usedto show that the interfaces either betweenmultilayers of PE or between PE and theelectrode, is prone to space chargeaccumulation, and thus might be consideredto be weak points with respect to

insulating properties [10-12].Despite a vast literature concerned with

the experimental characterization of itselectrical properties, very little isknown about the details of the electricalbehavior of this material, especially atthe molecular level. An understanding ofthe mechanisms of charge trapping andtransport in these insulators and how suchprocesses are affected by local physicalstructures and chemical impurities couldhelp in the development of materials withbetter insulating properties required forthe next generation of high voltage AC andDC cables. In what follows we summarizemolecular simulation methodologies whichhave been used to study the properties ofPE at the molecular level, elucidating therole of morphology as well as chemicalimperfections and additives on thetrapping of excess electrons.The paper is organized as follows:

Section II presents simulationmethodologies which have been found usefulin understanding PE, including: MolecularDynamics (MD) for modeling morphology,Density Functional Theory (DFT) forcalculating electronic properties usingthe generated morphologies, the Lanczosmethod which can be used to predict thedensity of states of excess electrons ina given PE structure, and the SurfaceHopping method which predicts the quantumtransitions between such states. SectionIII reviews current work focusing on therelationship between the electronic statesand PE morphologies. The findings arediscussed and some conclusions presentedin Section IV.

2 MOLECULAR METHODOLOGIESIn many branches of science and

engineering, molecular simulation

techniques (including MD, DFT, etc.) arean integral part of modern researchprograms. They can provide a theoreticalbasis upon which to interpret experimentalresults as well as a tool for discovery intheir own right. For example, computersimulations can explore the properties of‘perfect’ materials free from chemical andmorphological contamination [13]. Once therelationship between the simulated andexperimental systems is established, thesimulation can be used to explore theeffects of systematically changing bothexternal (pressure, temperature, density,etc.) and internal (composition,morphology, defect structure, etc.)variables. In this section, we provide abrief introduction to simulationtechniques which have been found to beuseful for polymeric materials.

2.1 MOLECULAR DYNAMICS METHODS Molecular dynamics (MD) [13, 14] is

widely used to simulate the trajectoriesof molecules or atoms in a classical many-body system by solving Newton’s equationsof motion. The thermodynamic, structuraland transport properties of the system canthen be obtained by taking time averagesover the trajectories. This method is widely used for almost

any molecular system at moderate or hightemperatures (such as at normalconditions, 300 K), so that quantumeffects in the dynamics of atoms ormolecules can be safely neglected. The interactions between the molecules

are described using a force-field, thatis, a set of potential energy functions.These functions are usually empirical,consisting of intra-molecular termsrepresenting the energy associated withfor example atom-atom bonds, andorientation (valence and dihedral angles)and intermolecular terms representing thevan der Waals or dispersion interactions,Coulomb interactions between charges,directional bonds such as H-bonds and,where necessary, three or many-bodyterms.

Given the force-field and a set ofinitial conditions, the trajectory of thewhole system is usually integrated as afunction of time using a simple time-stepalgorithm such as the velocity Verletmethod [14, 15]. Clearly, it is importantto choose a time-step small enough sothat the generated trajectories arerealistic. Typical time-steps for apolymeric system are of the order of afew femtoseconds and averages are takenover thousands to hundreds of thousands ofconfigurations (time-steps) of the system. The standard MD simulation generates a

micro-canonical ensemble, with constantnumber of particles (N), volume (V) andenergy (E). The instantaneous temperatureT is obtained from the total kineticenergy as given by the equipartitiontheorem.By introducing to the simulation a

thermostat to fix the system temperature(popular choices are those given byBerenson, Andersen or Nose-Hoover) or abarostat to fix the pressure, MDsimulations can be also carried out in acanonical ensemble (NVT) or an isothermal-isobaric ensemble (NPT) [14, 15]. When only equilibrium properties are

required, Monte Carlo (MC) simulation [14]provides an alternative methodology. MCdoes not generate realistic trajectoriesbut samples configuration space usingMetropolis sampling to generate a set ofconfigurations consistent with thestatistical mechanical probabilitydistribution for the given ensemble (i.e.isothermal-isobaric corresponding toconstant number of atoms N, pressure P,and temperature T or grand canonicalcorresponding to constant chemicalpotential , volume V, and temperatureT.). The fictitious trajectories aregenerated by randomly changing themicroscopic degrees of freedom (atomcoordinates, etc). The name of the methodcomes from the famous Monte Carlo casino,due to the extensive use of randomnumbers. Thermal equilibrium is usually

guaranteed by implementing an ergodicsampling that also complies with detailedbalance. Thus far we have only considered

classical methods where the electronicdegrees of freedom are incorporated intothe potential function describing theinter-atomic interaction. For explicitelectronic properties, such as thedensity of electron states, or excesselectron dynamics, a quantum mechanicalsimulation method is required.

2.2 DENSITY FUNCTIONAL THEORYDensity Functional Theory (DFT) (see

book eg. reference [16]) is a verypowerful and successful method forinvestigating the electronic structure ofa many-body system (i.e. a multi-electronsystem). In many cases the results of DFTcalculations for solid-state systemsagree very well with experimental data.For covalent, metallic and ionic bondsDFT will usually be within 2-3% for thegeometry (bond lengths, cell parameters),and 0.2 eV for the bonding energies (GGA,see below) [17]. The starting point of the theory is the

Born-Oppenheimer adiabatic approximation,in which the nuclear and electronicdegrees of freedom are separated. Theelectrons are considered to adjust almostinstantaneously to a change in nuclearcoordinates which can be considered fixedwhilst the system is solved for theelectronic states.DFT is based on the Hohenberg–Kohn (HK)

theorems [16, 18]. The first one statesthat “The ground-state energy from theSchrödinger equation is a uniquefunctional of the electron density”,which implies that the ground-stateelectron density n(r) uniquely determinesall properties, including the energy andwave function, of the ground state. Thesecond HK theorem defines an importantproperty of the functional: “The electrondensity that minimizes the energy of theoverall functional is the true electrondensity corresponding to the full

solution of the Schrödinger equation”.Therefore, by minimizing the functionalof the electron density, the electronicstructure of the system can be thencalculated.The Schrödinger equation of the many

electron system can be rewritten as a setof single-particle equations for anauxiliary non-interacting system, theKohn-Sham (KS) equation [19],

The terms on the left-hand side of thisequation are, in order, the kineticenergy; the Coulomb interactions betweenthe electron and the nuclei; the Hartreeterm describing the electron-electronCoulomb repulsion; and the exchange-correlation potential which includes allthe many-particle interactions. Thislatter term can be defined as afunctional derivative of the exchange-correlation energy as,

VXC (r )=∂EXC (r )∂n (r )

(2 )

where n(r) refers to the electrondensity. If the exchange-correlationenergy is known as a functional of thedensity, we have a closed set of self-consistent equations yielding a solutionto the electronic structure problem.Since the Hartree term Vh(r) and Vxc(R)

depend on n(r), which depends on the KSwave-functions ψi(r) which in turn dependon the total effective potential V(r) +Vh(r) + Vxc(R), the problem of solving theKohn–Sham equation has to be done in aself-consistent way. Starting with aninitial guess for n(r), the correspondingeffective potential can be calculated andthe Kohn-Sham equations solved for ψi(r).From these a new density is calculatedand the process starts again. Thisprocedure is then repeated untilconvergence is reached.The exact form of the exchange-

correlation energy functional is only

known for the free electron gas. Thus, inpractice, Exc is calculated usingapproximations, such as local densityapproximation (LDA) [19], or generalizedgradient approximations (GGA) [17], orB3LYP [20], see also [21]. Finding theright functional can be considered theequivalent problem to finding thepotential function in a classicalsimulation.The Kohn-Sham wave functions are also

sometimes identified with electronicstates, and the KS energies are taken asestimates of single-particle energies,including that of an excess electron or ahole in the system. While this is not arigorous procedure, this identificationprovides a simple method to obtainestimates in many systems that would bevery difficult to obtain otherwise. Notehowever that the results of thisuncontrolled approximation are notexpected to be as accurate as thestandard results (total energy, totalelectronic density, plus the atomicpositions obtained when used forequilibration or dynamics) of DFT. Within the framework of the Born-Oppenheimer adiabatic approximation, anab initio Molecular Dynamics techniquecan be obtained when the forces exertedby the electrons on the nuclei arecomputed using DFT at every time-step.The nuclei are then moved using Newton’slaws as in the standard MD method. One ofsuch approaches is the so-called Car-Parrinello method [22]. As an artefact tospeed up the calculations, the Car-Parrinello method explicitly introducesthe electronic degrees of freedom asfictitious variables. The resultingfictitious dynamics keeps the electronson the electronic ground state, thusavoiding an explicit electronicminimization at each time-step. Fromamong all ab-initio Molecular Dynamicstechniques, the Car-Parrinello method isperhaps the most widely used.Finally, let us note that DFT

calculations do not provide fully abinitio solutions of the full Schrödingerequation because the exact functional isnot known, however, approximatefunctionals can be tuned to experimentaldata if sufficient information isavailable for the system of interest.Among the shortcomings of DFT is itsinability to account accurately for theweak van der Waals attractions that existbetween atoms and molecules (see [23] fora summary of current process). Inaddition, being strictly a ground statetheory standard DFT cannot predictexcited states (however Time-dependentdensity functional theory [24] canpredict the properties of excited statesof a multi-electron system) and is notcapable of predicting accurate band gapsfor semiconducting and insulatingmaterials. However many variations andelaborations of density functional theorycodes exist which address these problems(see for example [25, 26] for thesuccessful prediction of band gaps), andin general the method is extremelypowerful.In the following, we conclude this

section by discussing two methods whichallow quantum calculations of a singleelectron in the system. Thesecalculations are appropriate for thestudy of excess electrons in thedielectric, when the number of them isnot too large, so that we can treat theexcess electrons as independentparticles. These methods are moreamenable when compared to DFT, requiringmuch less computer power and without manyof the typical difficulties of the DFTtechniques.

2.3 LANCZOS METHOD The Lanczos algorithm is a numericalmethod for diagonalising (i.e. findingthe eigenstates and eigenfunctions) for agiven matrix A. Here the matrix ofinterest is the Hamitonian He describingthe interaction of an excess electronwith PE through a pseudopotential

V(x,y,z), which is an input to thecalculation (for details see [27] andreferences therein). The Schrodingerequation is then solved for the groundand excited states of the excess electron(a single electron) for givenconfigurations of atoms, thesecorresponding to different morphologies.The Lanczos method is particularly usefulfor the purpose of solving theSchrodinger equation for a singleelectron on a three dimensional grid ofpoints in real space (x,y,z), because theHamiltonian is sparse, having manynegligible terms. It is based on thegeneration of a set (or a set of blocks)of basis functions which are designed toallow the efficient diagonalisation ofHe, so that the ground and low lyingexcited states of the excess electron arefound. This calculation directly providesthe information required to estimate thedensity of excess electron states of oursystem. This information can then be usedto calculate transport properties such asexcess electron mobilities due toextended states.We briefly describe here the Block

Lanczos method proposed by Webster,Rossky and Friesner [28, 29]. We startfrom a trial set of n basis functions{ψn

1} comprising the vectors ψi (forexample sine or cosine functions ofposition). The operator W = exp(−βHe ),where β is a positive constant, isapplied to this set to get a second setof eigenfunctions {ψn

2}. In each step theGram-Schmidt method [30] is used toguarantee that this new set is orthogonalto the first set and the basis functionsin a given set are kept orthogonalbetween themselves. The first set {ψn

1} isthe first ‘block’ of our space, (calledthe Krylov space), with which wediagonalise W. The set {ψn

2} is the secondblock, and so on. By repeated applicationof the operator W, we can obtain NB

blocks, which constitute the space withinwhich the operator W is diagonalised. Whas the same eigenvalues and

eigenfunctions as He and is used becauseit increases the convergence of thesolution towards the ground and lowerexcited states of the system, which areour primary interest. The results dependon the parameter β, which is varied untilthe solutions converge.

2.4 SURFACE HOPPING The Lanczos method provides the excesselectron properties for a single atomicconfiguration. The surface hoppingtechnique (see reference [31]) is amethod to simulate the dynamics of aquantum particle (here to be identifiedwith an excess electron) in interactionwith a larger classical system (such asPE), which allows us to go beyond theBorn-Oppenheimer adiabatic approximation.We describe here the algorithm proposedby Tully [32, 33], which accountsapproximately for the quantum transitionsbetween energy levels (surfaces) and theway these transitions (hopping) affectthe dynamics of atoms in the system,which are treated classically. The term‘surface’ here refers to the singleelectron energy eigenstates of thesystem.Let r=(x,y,z) refer to the coordinates

of the quantum particle and R(t)={Ri(t)}those of the classical molecules, thenthe total Hamiltonian describing theelectronic motion, He(r,R), will be atime-dependent operator. The wavefunction of our quantum particle ψ(r,t)can be solved using the time-dependentSchrödinger equation,

It is helpful to use a set of adiabaticbasis functions Фn(r, R), which can becomputed using the Lanczos algorithmdescribed in the previous Section, ψ (r,t )=∑

nan (t)Фn (r,R (t) )

Where the an(t) are expansioncoefficients. Substituting (4) into (3),multiplying from the left by Фm and

integrating over r givesdamdt =∑

ndmnan

where

By rewriting Eq. (5) in the equivalentdensity matrix notation, we define ρnm =am

* an , so that the Schrödinger equationcan be written as

The diagonal elements ρnn are theelectronic state population and the off-diagonal elements ρnm define thecoherence. Each term in the sum can beinterpreted as the number of transitionsfrom the state n to the state m per unittime. Therefore, during a small time stepδ we can express this quantity as theprobability that the electron is in staten multiplied to the probability of thetransition to other state:

Once a transition to other state has beendecided by using Monte Carlo transitionprobability [14], we must change thekinetic energy of the classical subsystemin order to conserve energy. This isusually done by scaling the velocities vj

of the classical particles in thedirection of the corresponding non-adiabatic coupling vector Dj

mn(t).In this implementation of the surface

hopping method, the classical particlesare moved at fixed intervals according toMD. Whether there is a quantum transitionis decided during the last time step(i.e. if t is the current time step, the

transition is decided in the intervalfrom t − ∆ to t.). If the hop is acceptedwe scale the atom’s velocities at t. Thequantum force over the atom j can bederived from the Helman-Feyman theorem as

This implementation of surface hoppingmethod makes it possible to study thenon-adiabatic dynamics of a small quantumsystem in contact with a larger classicalsystem in an approximate way. As such,it goes beyond the scope of DFT or theCar-Parrinello method, since theselatter techniques assume that the wholesystem always remains in the ground-state. 3 MOLECULAR MODELLING OF POLYETHYLENE

3.1 PE MORPHOLOGIES AND ITSCONFORMATIONAL DISORDERS

Early simulation studies on bulk PEused n-alkane short chains to mimic longchain PE. Simulations of PE chains, eachwith 50 CH2 units, below and above theglass transition temperature, wasperformed to study PE physical andconformational properties such as chainconformation, glass transition or free-volume distribution [34-37]. Thetransition between rotational isomers inPE chains of 100 CH2 units was analyzedin [38]. All-trans PE chains of 60 CH2

units was studied [39] around the meltingpoint. However, none of those simulationsobserved the chain folding phenomenonwidely seen in experiments, and believedto be one of the basic motifs of the PEbulk sample.Quantum effects on an assumed

orthorhombic phase of crystalline PE werestudied in reference [40] by means ofpath-integral Monte Carlo (PIMC)simulations. The low temperature range,from 25K to 300K, was explored in the NPTensemble by looking at PE chains of twolengths, C12 and C24 (with a total numberof atoms of 432 and 864). Structural

parameters such as the lattice constants,bond lengths, and bond angles and theirdependence with temperature wereanalyzed, showing their reduction as thetemperature is decreased. By performing MD simulations of PE

chains of various lengths, using aunited-atom model with torsional barriersof 2, 3 and 6 kcal/mol, with theremaining parameters being taken directlyfrom the Dreiding forcefield, the chainfolding phenomenon was observed [41, 42].They found that during simulations with atorsional barrier of 2 kcal/mol, PE chainfolding occurs when the chains have morethan 150 CH2 units, long enough that theaverage inter-molecular van der Waalsenergy is low enough to compensate theincrease in energy owing to torsion angleand bond angle deformation in the folds.The reported behaviour was in goodagreement with the experimentalobservations [43].More recently, multi-phase polyethylene

morphologies including crystalline,amorphous, lamellae and interfacialregions have been studied using MD [44,45]. As discussed in reference [2], apolyethylene sample may containamorphous, lamellae and crystallineregions of various sizes depending on thematerial processing details. In reference[44], model crystalline and lamellaeregions were created by geometryoptimization using the all-atom COMPASSforcefield in Materials Studio 5.5 [46].For the lamellae regions, gauche andanti-gauche defects were introduced intoten all-trans PE chains with 552 CH2

units in order to enable them to foldback and forth upon themselves, resultingin a lamellar thickness of around 5.0 nm.The structures were then imported intoLAMMPS simulation package [47], andequilibrated using a united-atom forcefield [48] optimized for long chain n-alkanes. The amorphous and interfacialregions were prepared by melting thecorresponding lamellae blocks either inNPT or NμT ensembles with a time-varying

thermostat in LAMMPS. The resultingmorphologies were in good agreement withexperiment.As observed experimentally [49],

polyethylene contains a significantnumber of nanometre sized voids. In orderto explore their influence on excesselectrons, nanometre sized voids wereproduced in the simulated amorphous phaseby expanding a test particle [44]. Bycalculating the Gibbs free energy tocreate different radius of sphericalvoids in amorphous PE, a surface tensionof 36 mN/M was obtained, in goodagreement with the experimental value of35.7 mN/M at 20 oC for linear HDPE [50].

3.2 ELECTRONIC PROPERTIES OF PESurface trapping states of PE was

studied [51] using DFT in the gradient-corrected local density approximation(LDA), supplemented with empirical long-range tails in order to properly accountfor van der Waals forces. Two PE surfaceswere considered, representative of twoclasses of surfaces that differ in theorientation of the PE chains with respectto the planar surface i.e. with thechains oriented either parallel orperpendicular to the surface vector. Anegative electron-affinity was found,with values of -0.17eV and -0.10 eV forsurfaces with chains perpendicular andparallel to the surface, respectively.Negative electron-affinities are inagreement with experiment and othersimulation results (see below), though inthis DFT study the analysis is based onthe identification of KS eigenvalues(section 2) with single-particle energylevels. Dynamical DFT simulations have also

been performed [52] within the frameworkof the Car-Parrinello (CP) technique.Calculations were carried out with aninjected electron neutralized by abackground charge in a crystalline cellwith four PE chains (each of them with 7CH2 units), starting from an equilibriumconfiguration. A similar calculation was

also performed with an injected hole. Aself-trapped state for the injectedelectron was found, localizing near aninter-chain area which involved a pair oftrans-gauche defects. The hole wasobserved to remain delocalized throughoutthe simulation cell, being of an intra-chain nature. In reference [53], an electron-holepair, an exciton, was injected in asimilar crystalline cell, being observedto be long-lived, displaying no apparentdirect channel for non-radiactiverecombination. In contrast, very recentDFT calculations [54] have found a directchannel for the recombination of theexciton via the breaking of a C-H bond.The discrepancy with the results in [53]might be due to the exchange-correlationfunctional employed or the differentmethods chosen to improve thedeficiencies of standard DFT. However,reference [54] presents a convincingscenario in which the recombination of anelectron-hole pair under electricalstress could lead to degradation of PE.

Research focused on the electrontrapping density of states identifyingboth physical and chemical defects werealso carried out. To model the physicaldefects [55], MD simulations of amorphoustridecane (n-C13H28) were used as a modelof short sections of PE to generatelocalized conformational defects. Thetrap energy was defined as the differencebetween the electron affinity of the waxmolecule with and without conformationaldefects (i.e. not all trans) as Et=EA(C13H28)-EA(n-C13H28)all-trans. The electronaffinity was obtained from DFT asimplemented in the code DMol [56]. Asimilar methodology was used to estimatetrapping energies for chemical defectsand an approximate excess electrondensity of states obtained. Other workershave used similar methods [57-59]. Someresults for chemical defects are given intable 1, clearly there are some very deeptraps (>0.5 eV) but also some shallow

traps (<0.5eV) which overlap in energywith those caused by conformationaldefects.

Table 1. Trap depths of some chemical impuritiesand docomposition products in PE [57].

Molecule Trap depth(eV)

5-decanone(C10H20O)

-0.453

5-vinyl(C10H20)

-0.122

5-decanol(C10H21O)

-0.186

5-decanal(C10H20O)

-0.445

Cumylalchohol(C9H12O)

-0.28

Acetophenone(C8H8O)

-0.9

Alpha-methylstyreneC9H10

-1.53

Cumene(C9H12)

-0.04

A more detailed study of the physicaldefects in amorphous and crystalline PEand their effect on excess electrons wascarried out [27, 60] using Lanczos methodto compute the excess electronic statesof static configurations taken from MDsimulations (for ethane, methane,propane, crystalline PE, and amorphous PEusing a single chain with 300 CH2 units).The pseudo-potential, describing theinteraction of the excess electron withthe atoms, was adjusted to fit theexperimental data for the threshold ofconduction in fluid ethane and propane.It contained short-range repulsivepotentials, and an attractive part, whichaccounts for the polarization interactionbetween the excess electron and thedielectric, based on multi-centrepolarizabilities obtained by fully abinitio methods. The amorphous PE samplesshowed Anderson localization [61], with amobility edge, separating localized anddelocalized states. The electronicdensity of states (DoS) and the electronmobility at the amorphous phase wascalculated using the Kubo–Greenwoodequation [62]. Further work [63], using

non-adiabatic simulations of an excesselectron in amorphous PE, permitting thedeformation of the material due to thepresence of the electron, showed thespontaneous formation of localized smallpolaron states in which the electron isconfined. Despite allowing non-adiabatictransitions by using Tully’s surfacehopping algorithm [64], the simulationsshowed mainly adiabatic dynamics.Very recently, the analysis of

references [27, 60] has been extended tolarger systems and more representativeregions of PE, including lamellae andinterfaces between lamellae and amorphous[44, 45]. A clear correlation betweenlocal atomic density (calculated as thenumber of CH2 united atoms per unitvolume in a cube of side 0.5nm at a givenposition) and electron probability hasbeen found in all PE phases, showing thatthe electron is more likely to localizein lower density regions. Simple visual inspection of the

localized states showed that the electronis sitting at regions with a reducedlocal atomic density, as illustrated inFigure 1. In the interfaces betweenlamellae and amorphous phases, the lowestatomic density values are found either inthe amorphous regions or near theinterface [45]. Figure 2 shows a typicalground state localized in that area.In order to make this association more

quantitative, we have calculated thePearson correlation coefficient betweenthe probability density of the excesselectron and the local PE atomic density[45]. This dimensionless coefficient isusually defined as

where X and Y are the variables ofinterest, here to be identified with theelectronic probability density (thesquare of the wave function) in a givenelectronic state and the local atomic

density. The index i in (12) runs throughthe 3D grid points in the system. Thecoefficient r measures the strength ofthe linear association between bothvariables: a value of zero indicates nocorrelation, while nonzero values showcorrelation or anti-correlation. In theamorphous phase, starting with the groundstate, the Pearson correlation shows asignificant anti-correlation value ofabout -0.4 in regions of the materialwhere the electron has a probability of90% . The next first 10 excited states,still below the mobility edge, show acorrelation of about -0.2, while for thefirst 10 states above the mobility edgethe correlation is reduced to -0.1, whichis still significant, especially takinginto account that these latter states areextended.

Another way to look at this correlationis presented in reference [44]. First,the free volume or cavity numberdistribution in the system is computedafter identifying cavities with localminima of the atomic density. A radius isassigned to each cavity based on theradial distribution function from thecentre of the cavity. Then, eachlocalized state is associated with a

cavity, providing an average energy andlocalization length as a function of thecavity radius. The cavity numberdistribution can be then mapped toproduce a prediction of the density ofstates (DoS) based on the free volumedistribution. This prediction is shown[44] to accurately predict the observedDoS in the three PE bulk

phases studied- amorphous, lamellae andcrystalline, when a correction based onthe localization length is carried out.The correction is needed becauselocalized states with higher energyvalues are not restricted to singlecavities, instead extending throughadjacent ones.

4 DISCUSSION AND CONCLUSIONSAs we have seen molecular modeling

techniques have enabled some progress tobe made in understanding the fundamentalsof charge trapping and electron transportin polyethylene. Previous work hasidentified physical, (conformational)[52, 55] and chemical (impurities anddecomposition products) [57-59] electrontraps in models of polyethylene andcharacterized them using ab initiomethods. Some of this information wasused to create a preliminary distributionfunction representing the density oftrap states (DoS) as a function ofelectron energy and employed in a MonteCarlo simulation to predict the current-voltage [65] characteristics of modelpolyethylene, showing how once the DoS isknown the electrical properties can be

(a) Atomic Density

(b) Electronic Probability Density for GroundState

Figure 1. Slab sections showing the atomicdensity (upside) and the electronic probability

estimated. A more complete treatment ofconformational trapping using the fastFourier transform block Lanczosdiagonalization algorithm described insection 2 has resulted in densities ofstates for the different morphologiesfound in polyethylene [44, 45]. A simpletheory based on this work now links thephysical traps in polyethylene tonanovoids associated with regions of lowdensity in the material. Our current understanding of

conformational excess electron trappingbased on molecular modeling [44] haslocalized states extending below themobility edge (above which the electronsare delocalized), at -0.1±0.1eV withrespect to the vacuum level. These trapstates with localization lengths between0.3 and 1.2nm have energies as low as -0.4±0.1eV in the amorphous andinterfacial regions of polyethylene withmore positive values in lamellastructures. Note crystalline regions havea mobility edge at +0.46 ±0.1eV, so wewould expect transport by electronsexcited above the mobility edge todelocalized states to be predominantlythrough amorphous regions if theypercolate the sample.At low electron concentrations,

transport above the mobility edge will bedominated by the filling of the deepesttraps which are likely to be chemicaltraps ~ -1eV. As the deepest traps arefilled the excess electron mobility willincrease dramatically towards a valuethat corresponds to multiple trappingbetween more shallow conformational traps[65].Nonadiabatic simulations (Section 2) of

an excess electron in amorphous PE atroom temperature showed the spontaneousformation of localized small polaronstates in which the electron was confinedin a spherically shaped region with atypical localization length of 0.5nm. Theself-trapping energy was ~-0.06 ±0.03eV,with a lifetime on the time scale of afew tens of picoseconds. The smallness of

the self-trapping energy is consistentwith an adiabatic hopping mechanismassisted by phonons, as observed in thesimulations. The contribution to themobility due to hopping between theseself-trapped states may well be of sameorder of magnitude as the mobility due toexcited electrons above the mobilityedge.The model studies confirm the picture

of deep chemical and shallowconformational electron traps current in[2]. They also however provide the meansto go further and begin to investigatethe influence of trapped electrons, thespace charge, on the surroundingmaterial. For example there has been verylittle work on the fate of the energygiven up by trapped electrons inpolyethylene (but see [58] for trappedexcitons) which may alter the environmentleading to local damage and ageing. Thevery recent DFT results [54] suggestingPE degradation by exciton recombinationprovide an interesting scenario thatneeds to be confirmed. In this regard,though molecular modeling studies providea great insight in the understanding ofpolymeric dielectrics, these studies needto be assisted by experiments on verypure polymeric materials (includingpolyethylene) in which the effects ofchemical traps from additive chemicalsand/or radiation damage on electricalproperties can be separated from those ofphysical traps related to the variouspolymer morphologies present in realmaterials.

ACKNOWLEDGMENTThe authors wish to thank the funding

support from State Key Lab. of ElectricalInsulation and Power Equipment underProject EIPE13203.

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Yang Wang was born in Jilin, China,in 1984. He received the B.S.degree in applied physics fromXi’an Jiaotong University, Xi’an,China, in 2007. Now, he is a Ph.D.degree student in the Department ofElectrical Engineering in Xi’anJiaotong University. During his

Ph.D. study from 2010 to 2012, he was in UniversityCollege Dublin for a joint cultivating programfocusing his research on molecular simulation onelectronic states in polyethylene.

Kai Wu (M’00) was born in China in1969. He received the M.S. andPh.D. degrees in electricalengineering from Xi'an JiaotongUniversity, China in 1992 and 1998,respectively. He was a postdoctoralfellow from 1998 to 2000 and thenjoined the staff from 2000 to 2003

at Nagoya University, Japan. In 2003, he worked asa research associate at the University ofLeicester, UK. In 2004 and 2005, he was a visitingresearcher at the Central Research Institute ofElectric Power Industry, Japan. Since 2006, he hasbecome a professor of Xi’an Jiaotong University,China.

David Cubero was born in Spain in1971. He received his Ph.D. degreein physics from the University ofSeville in 1996. He was a post-doctorate researcher in ImperialCollege London, UK, from 2001 to2004. He is now an associateprofessor at the University ofSeville (Spain) specializing in the

area of nonequilibrium statistical physics,including transport theory and resonant phenomema.He has published about 30 papers in leadingjournals on a wide range of topics in StastisticalMechanics, including granular flows, Brownianmotors, relativistic systems, and electronictransport.

Nick Quirke was born in London,UK,in 1952. He received his B.S. andPh.D. degrees in physics from theUniversity of Leicester in 1973 and1977, respectively. He is now aProfessor of Chemical Physics at

Imperial College, London. His group conductstheoretical research in the general area ofnanomaterials with particular interest in theirinteraction with biological interfaces andpolymers. He is a Fellow of the Royal Society ofChemistry, and the Institute of Nanotechnology,Editor-in-Chief of the international Journals,Molecular Simulation, and the Journal ofExperimental Nanoscience. He received the 1998Royal Society of Chemistry Medal for Thermodynamicsand was Royal Society / Kan Tong Po Professor atthe University of Hong Kong for 2003/2004. He wasawarded the 2006 NSTI Fellow Award for OutstandingContributions in Advancing Nanotechnology and in2013 a Yangtze river fellowship at Xi’an JiaotongUniversity. His career has included leadershippositions in both academia and industry. He wasPrincipal Research Associate at BP. Most recentlyhe was Vice President and Principal of engineeringand science, University College Dublin.