Electrical properties dependency with electric field: SCLC

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Space Charge Limited Current (SCLC)

In this model, the conduction current density is driven by the flow of mobile charge carriers under an applied electric field. The measured current is given by: j = ρµE − D∇(ρ) + ∂εE + ∂P (1.3.2) ∂t ∂t with ρ [C/m3] the charge density, µ [V.m2/s1] the charge carrier mobility, D [m2/s] the diffusion constant and ε [F/m] the permittivity. The first term corresponds to the charge transport under electric field, the second term to diffusion current due to a concentration gradient, the third term to the displacement current and the last term to the polarization current. SCLC aims at describing current at steady state meaning that displacement current and polarization current are not taken into account. Furthermore diffusion current is often considered negligible due to the high value of applied electric field.
At low electric field, space charge is low in the bulk of the dielectric and injected charges have a velocityv = µE. Conduction current density coming from this charge transport is given by the Ohm’s law: jconduction = ρv = ρµE (1.3.3).
When the electric field is increasing, the number of charges injected is also increasing so as space charge. Above an electric field threshold, space charge becomes high enough to induce electric field modification according to Poisson’s relation:
div(E) = ρ (1.3.4).

Breakdown physical process

Electrically and thermally stressed polymer insulation systems see their microstructure change over time, leading to short and/or long term degradations. The former is referred to as break-down and the latter as ageing phenomena (see Figure 1.12). The three processes responsible for short time degradation are electric breakdown, thermal breakdown and electro mechanical breakdown [24].

Electrical breakdown

In the avalanche breakdown mechanism, charge carriers, gaining enough kinetic energy from the electric field, ionize the polymer macromolecules by collisions. Additional carriers are created from the ionization, increasing the collisions probability. This accelerates the structural degra-dation and creates reaction products such as gases that may damage the solid if the solubility limit is exceeded [24]. The energy released by these charge carriers which is the onset of electrical aging, was observed from optical emission measurements [64]. Several regimes can be considered depending on the kinetic energy of the charge as shown in Figure 1.13 [64]: 1. E > Ecrit(2): Impact ionization regime with creation of electron-cations pairs (AB + e−hot → AB+ + 2e−).

Dakin, inverse power and Eyring models (1948)

These models were first used to treat thermal ageing or chemical reactions kinetics. They were then adapted for electrical ageing although none does take explicitly into account the effect of space charges on ageing. Dakin’s model proposes an electric ageing law based on the Arrhenius law [74]. Eyring’s thermodynamic ageing model is based on the equation that describes the kinetics of a chemical reaction versus temperature, adapted to take into account the electric field as well. These models do not take into account the complex structure of polymers and do not explain the physical origin of the insulation degradation. However, they are quite simple and still used for the design of insulation systems [75].

Crine model

In this model, ageing is described by polymer macromolecules deformation due to electrome-chanical forces [73]. A threshold electric field is considered for the triggering of macromolecules deformation in the amorphous phase when weak van der Waals bonds are broken, leading to nanocavity formation. Then electronic avalanche occurs after the formation of cavities. Charge carriers ionize the polymer macromolecules after crossing these cavities which enlarge the cavi-ties. The larger the cavities, the higher the kinetic energy of electrons breaking the intramolec-ular bonds. Strong charge injection occurs only after nanocavity formation, i.e. Crine’s model considers space charge as a consequence and not as a cause of ageing.

DMM (Dissaldo/Montanari/Mazanti) model

Dissado/Montanari/Mazzanti model considers space charges as the driving force for ageing [10, 71, 72]. In this model, formation rate of defects (called moiety by the authors) into insulation is described with a thermally activated degradation process between two states, i.e un-aged and aged states. An energy barrier represents the transfer between both states during the ageing process. This degradation is physically triggered by space charge accumulation. Space charge accumu-lation induces, by polaronic effect, a local storage of electromechanical energy that lowers the energy barrier between the two states. The transfer rate of defects between the two states increases due to space charge accumulation, electric field enhancement and temperature effect (thermal activation). In this model, breakdown is considered to occur when a fraction of moiety has reached the aged states. Space charge accumulation occurs in microscopic morphological de-fects such as voids located at interfaces between crystalline and amorphous regions. Mechanical stress created by accumulated space charges localized in this void is given by: σ = 1 αE2 (1.3.27). where α [N/V2] is the electrostriction coefficient. As no macromolecule is present in the cavity, highest stress applied on polymer is present at the interface of this cavity. Due to this stress, a progressive enlargement of this cavity occurs. Then hot electrons are formed and electronic avalanche starts followed by partial discharges and cavity erosion. The main limitation is that this model does not take into account the heterogeneous semi-crystalline structure of PE and the dynamics of macromolecules. Furthermore it considers only two states instead of a distribution of states.

Lewis electromechanical model

In this model, polymer degradation comes from electromechanical forces [25, 12]. High electric field stresses polymer macromolecules in the orthogonal direction to the applied electrical field in the amorphous phase. As the stress increases, macromolecules, that connect two adjacent crystalline lamellae, extend between the two lamellae and then pull up from one of the lamellae. These macromolecules fail successively until there is decohesion of the two lamellae and crack formation, as illustrated in Figure 1.15. The crack propagates by breaking all the adjacent polymeric chains to finally form voids. The failure of the material thus corresponds to cavity formations extending under the influence of an electric field. In this model, space charge effect is not considered as a cause of the ageing process.

Physical heterogeneity effect

In the case of HVDC insulation, the polymer resin commonly used is Low Density Polyethy-lene (LDPE). This polymer is obtained from polymerization of ethylene monomer at very high pressure (1000 bars) which results in a highly branched PE with a density ranging between 0.91 g/cm3 and 0.92 g/cm3 [54]. Contrarily to purely amorphous polymer such as poly(methyl methacrylate) (PMMA), LDPE has a semi-crystalline structure, making it morphologically het-erogeneous by nature. Furthermore crosslinking of polyethylene is performed to increase its mechanical resistance at elevated temperatures [76]. Chemical crosslinking of PE is based on the thermal decomposition of a peroxide molecule into radicals. For high voltage applications, dicumyl peroxide (DCP) is the most used peroxide [77]. DCP decomposition in two radicals reacts with PE chains by withdrawing an H+ and thus starting the crosslinking [76]. This reac-tion yields to the formation of chemical species that can contribute to conduction increase and trapping [78, 79, 42]. This last point is further discussed in section 1.5.3.1.
This semi-crystalline crosslinked structure induces physical heterogeneities in the polymer. This chapter first aims at defining the changes in heterogeneities that one may expect from electric field and thermal stresses. Secondly, consequences in the electrical properties are described from models and experiments performed in the literature.

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Behavior under temperature and electric field

After crystallization, thermal treatment of polymer at temperatures comprised between glass transition and melting temperature affects their semi-crystalline morphology. Annealing at a temperature below melting temperature results in a recrystallisation of thinnest crystal lamellae into thicker ones. This recrystallization is either due to partial melting of these lamellae or to the diffusion of crystalline defects out from the crystalline phase by an α-relaxation process [99] (see section 1.4.2.2). Indeed, Hestad et al. [100] observed a clear influence of thermal history on the morphology change of PE. Differential Scanning Calorimeter (DSC) measurements were performed on virgin XLPE and XLPE after annealing at several temperatures. Despite the authors measured same crystalline fractions from one sample to another, a change in the ther-mogram profile was observed and ascribed to recrystallisation. For PE, this effect is maximum for an annealing time of 30 min. No further structural change occurred at longer annealing times [99]. For PP annealed at 90➦C, a new melt peak appears close to the applied temperature of 90➦C [101]. The authors observed a dependency of the maximum melting temperature of this secondary peak to cooling rate [101]. The secondary peak is located at a lower temperature for a higher cooling rate [101]. The change in melting temperature may be responsible for the change in PP mechanical properties with a decrease of tensile strength after thermal treatment [101]. As insulation is submitted to thermal cycles during cable operation, this annealing effect is likely to occur.

Table of contents :

Introduction
1 State of the Art 
1.1 Context
1.1.1 HVDC cable manufacturing
1.1.2 HVDC cable system testing
1.1.3 HVDC cable lifetime modeling
1.2 Electrical properties and related methods
1.2.1 Conductivity
1.2.2 Space Charge (SC)
1.3 Charge transport and aging modeling
1.3.1 Analytical model for charge transport
1.3.1.1 Space Charge Limited Current (SCLC)
1.3.1.2 Poole-Frenkel [1]
1.3.1.3 Variable Range Hopping (VRH)
1.3.2 Analytical model for charge injection
1.3.2.1 Fowler-Nordheim injection
1.3.2.2 Schottky injection [2]
1.3.2.3 Interface effect
1.3.3 Breakdown physical process
1.3.3.1 Electrical breakdown
1.3.3.2 Mechanical breakdown
1.3.3.3 Thermal runaway
1.3.4 Aging phenomenology and modeling
1.3.4.1 Dakin, inverse power and Eyring models (1948)
1.3.4.2 Crine model
1.3.4.3 DMM (Dissaldo/Montanari/Mazanti) model
1.3.4.4 Lewis electromechanical model
1.3.4.5 Summary
1.4 Physical heterogeneity effect
1.4.1 Description and formation
1.4.1.1 Description
1.4.1.2 Formation
1.4.2 Behavior under temperature and electric field
1.4.2.1 Annealing effect
1.4.2.2 Relaxation processes
1.4.2.3 Behavior under electric field
1.4.3 Effect on electrical properties: model and experiment
1.4.3.1 Charge transport model in specific heterogeneous semicrystalline polymeric structure
1.4.3.2 Effect on conductivity and space charge
1.5 Chemical heterogeneities
1.5.1 Charge transport model in polymeric structures with chemical residues .
1.5.2 Behavior under temperature and electric field
1.5.2.1 Behavior under temperature
1.5.2.2 Behavior under electric field
1.5.3 Peroxide decomposition products (PDP)
1.5.3.1 Formation
1.5.3.2 Diffusion properties
1.5.3.3 Effect on space charge and conductivity
1.5.4 Antioxidants
1.5.4.1 Formation
1.5.4.2 Effect on space charge and conductivity
1.5.5 Water content
1.5.5.1 Formation
1.5.5.2 Diffusion properties
1.5.5.3 Effect on space charge and conductivity
1.5.6 Oxidation
1.5.6.1 Formation
1.5.6.2 Effect on space charge and conductivity
2 Experimental approach 
2.1 Materials
2.1.1 PE-based material
2.1.2 PP-based material
2.1.3 PET-based material
2.2 Physical and chemical characterization
2.2.1 Morphological analysis
2.2.1.1 Differential scanning calorimetry (DSC) method
2.2.1.2 Analysis on material model s
2.2.1.3 Crystallinity variation with temperature
2.2.2 Chemical composition
2.2.2.1 Methods
2.2.2.2 Results
2.3 Dielectric spectroscopy measurement
2.3.1 Setup description
2.3.2 Morphological impact on permittivity
2.3.3 Glass transition and crystalline phase influence on conductivity
2.4 Current density measurement
2.4.1 Setup description
2.4.2 Test procedures
2.4.3 Crystallinity impact on conductivity
2.4.4 PDP influence on conduction
2.4.5 Interface effect
2.5 Space charge measurement
2.5.1 Setup description
2.5.1.1 Stimulus and measurement systems
2.5.1.2 Reduction of waves reflections
2.5.1.3 Signal resolution
2.5.1.4 Data processing
2.5.2 Test procedures
2.5.3 Glass transition influence on space charge
2.5.4 Crystallinity impact on space charge
2.5.5 PDP influence on space charge
3 Genetic model description 
3.1 Electrical properties output
3.1.1 Electric field
3.1.2 Charge density
3.1.3 Permittivity and temperature
3.2 Development of evolution laws
3.2.1 Charge injection
3.2.2 Charge transport
3.2.3 Charge extraction
3.2.4 Charge trapping and detrapping
3.3 Evolution of the system over time
3.3.1 System evolution from evolution laws
3.3.2 Current density calculation
3.3.3 Step time calculation
3.4 Simulation results
3.4.1 Leakage current and space charge measurements
3.4.2 Current density dependency with electric history: space charge effect
3.4.3 Electrical properties dependency with electric field: SCLC
4 Genetic evolution of semi-crystalline polymer 103
4.1 Heterogeneous semicrystalline structure simulation
4.1.1 Distribution of random spherulites in the model
4.1.2 Distribution of random lamellae related to spherulite
4.1.3 Algorithm for crystalline fraction calculation
4.1.4 Local permittivity calculation from local crystalline fraction
4.2 Evolution laws development
4.2.1 Charge injection and transport
4.2.2 Charge trapping
4.2.3 Microstructure modification with temperature: annealing
4.3 Criticity of model parameters
4.3.1 Parameters for charge transport and injection
4.3.2 Parameters for charge trapping
4.4 Comparison with experiments
4.4.1 Current density behavior
CONTENTS 9
4.4.2 Space charge profile
4.4.3 Dependency with crystallinity
4.4.4 Glass transition temperature
5 Genetic evolution of undegassed insulation system 133
5.1 PDP distribution simulation
5.1.1 Distribution in the polymer
5.1.2 Impact on local permittivity
5.1.3 Influence of degassing time
5.2 Evolution law development
5.2.1 Genetic behavior of ACP
5.2.1.1 Diffusion of ACP from concentration gradient
5.2.1.2 Impact on electrical properties: deep traps for electrons . . . . . 137
5.2.2 Genetic behavior of αCA
5.2.2.1 Diffusion of αCA from concentration gradient
5.2.2.2 Impact on electrical properties: ionic transport
5.2.3 Effect of macroscopic interfaces
5.3 Criticity of model parameters
5.3.1 Parameters for αCA genetic behavior
5.3.2 Parameters for ACP genetic behavior
5.4 Comparison with experiment
5.4.1 Impact of PDP in space charge distribution of XLPE
5.4.1.1 Effect of αCA
5.4.1.2 Effect of ACP
5.4.2 Impact of PDP in current density of XLPE
5.4.2.1 Degassing time
5.4.3 Interface effect
5.5 Summary
Conclusion 157
Bibliography 161

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