A reaction-diffusion formulation to simulate EVA polymer degradation in environmental and accelerated ageing conditions

Solar Energy Materials & Solar Cells 164 (2017) 93–106

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A reaction-diffusion formulation to simulate EVA polymer degradation in environmental and accelerated ageing conditions

MARK

⁎

M. Gagliardi , P. Lenarda, M. Paggi IMT School for Advanced Studies Lucca, Piazza San Francesco 19, 55100 Lucca, Italy

A R T I C L E I N F O

A BS T RAC T

Keywords: Degradation Durability Reaction-diffusion system Poly(ethylene-co-vinyl acetate) Photovoltaics Finite element method

Among polymers used as encapsulant in photovoltaic (PV) modules, poly(ethylene-co-vinyl acetate), or EVA, is the most widely used, for its low cost and acceptable performances. When exposed to weather conditions, EVA undergoes degradation that affects overall PV performances. Durability prediction of EVA, and thus of the module, is a hot topic in PV process industry. To date, the literature lacks of long-term predictive computational models to study EVA aging. To fill this gap, a computational framework, based on the finite element method, is proposed to simulate chemical reactions and diffusion processes occurring in EVA. The developed computational framework is valid in either case of environmental or accelerated aging. The proposed framework enables the identification of a correspondence between induced degradation in accelerated tests and actual exposure in weathering conditions. The developed tool is useful for the prediction of the spatio-temporal evolution of the chemical species in EVA, affecting its optical properties. The obtained predictions, related to degradation kinetics and discoloration, show a very good correlation with experimental data taken from the literature, confirming the validity of the proposed formulation and computational approach. The framework has the potential to provide quantitative comparisons of degradation resulting from any environmental condition to that gained from accelerated aging tests, also providing a guideline to design new testing protocols tailored for specific climatic zones.

1. Introduction Poly(ethylene-co-vinyl acetate) (EVA) is one of the most widely used materials for photovoltaic encapsulants, due to its chemicophysical characteristics and low cost. Although raw materials used in module manufacture are generally stabilized with chemical additives, EVA copolymers suffer of thermo-photo-oxidative degradation, upon the prolonged exposure of photovoltaic (PV) installations to UV light, environmental agents and the high working temperature range. During degradation, EVA polymer chains composing the encapsulant layer lose atoms and small molecules, such as protons and acetic acid, with consequent changes at the macromolecular level. These macromolecular alterations causing deterioration of optical properties (so-called yellowing and browning), corrosion of electric connections and formation of snail trails. Moreover, its permeability to moisture and the presence of acidic degradation products induce a progressive oxidation of the grid line over the solar cell surface, and a reduction of mechanical adhesion and sealing [1–3]. Computational models represent nowadays a powerful tool to study the effects of climatic parameters in several fields, such as hydrology [4], irrigation [5], and

⁎

agriculture [6–8]. For EVA-related applications, the literature reports a valid overview of chemical reactions occurring in EVA degradation, and some interesting experimental data [9–12]. Concerning the analysis of PV cells, both exposed to weathering conditions or in climatic chambers, existing computational models only study water absorption and diffusion phenomena [1,13]. To date, comprehensive models to simulate the fate of chemical species, as well as the chemico-physical and mechanical properties changes after degradation, and undergoing different climatic loads, are lacking. On the other hand, such kind of study, indeed, is relevant for the PV community to assess (i) the impact of degradation of PV installations in climatic zones out of European countries and (ii) the correspondence between accelerated aging tests and real environmental conditions. Both aspects are currently considered open concerns for the Technical Committee of the Task 13 on Performance and Reliability of Photovoltaic Systems of the International Energy Agency, of which one author is member. Previous computational methods for the simulation of environmental degradation of PV modules available in the literature have been mostly focusing on the prediction of moisture diffusion, in some cases taking also into account the effect of temperature (see [1,3], among

Corresponding author. E-mail addresses: [email protected] (M. Gagliardi), [email protected] (P. Lenarda), [email protected] (M. Paggi).

http://dx.doi.org/10.1016/j.solmat.2017.02.014 Received 20 October 2016; Received in revised form 19 December 2016; Accepted 9 February 2017 0927-0248/ © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

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M. Gagliardi et al. Ea

others). To gain a complete picture of degradation phenomena in EVA, the literature on chemical degradation has reported the occurrence of a much more complex system of chemical reactions influencing polymer aging. Moreover, some chemical species are subjected to diffusion, so that a spatial variability of concentrations takes place. As we know from preliminary results [3], moisture diffusion tends to accelerate in the presence of cracks in Silicon solar cells, creating channels enhancing moisture percolation between the two layers of the encapsulating EVA. To effectively simulate these problems, semi-analytic solutions for mono-dimensional geometries are not fully adequate. There is therefore a urgent need for a mathematical framework and an efficient computational tool to solve the system of reaction-diffusion partial differential equations over the 2D and 3D domains of the PV module, considering the actual geometry. Moreover, the proposed method should allow the possibility to account for any realistic initial or boundary condition for temperature and moisture, to simulate accelerated aging tests and quantitatively compare degradation predictions with the outcome of environmental degradation in any climate zone. These topics are of paramount importance for the reliability and durability of PV systems. The accurate prediction of degradation phenomena has also an impact on the economical sustainability of Silicon PV, allowing for the accurate estimation of maintenance costs of PV systems depending on their location of installation. To give a significant contribution to this research field, we propose, for the first time, a computational tool to predict the evolution over time of chemical reactions and diffusion of small molecules within EVA encapsulant. The proposed framework will be developed to study any generic weathering condition. In addition to the study of chemical phenomena, we report some interesting results describing the decay of optical properties in EVA upon degradation. The final aim of the study is to provide an adaptable tool, working for different climatic conditions, to gain quantitative and predictive results on EVA thermo-photooxidation, and a relationship to identify the correlation between accelerated tests and environmental degradation.

k = A0 e− RT .

d[P] d[P*] = = k[P]. dt dt

(4)

where Δ is the Laplacian differential operator, D is the diffusion coefficient that, in the most general case, might depend upon the concentration of S and temperature [18,19], i.e.: ⎛ Ed ⎞ − γ [S ]⎟ ⎜− RT ⎠.

D = D0e⎝

(5)

In Eq. (5), the model parameters to be identified from experiments are: the diffusion coefficient for the limit case of a vanishing concentration, D0, the activation energy of diffusion, Ed, and a constant, γ. When the degradation process causes chain scission phenomena (case (ii)), the number of molecules in the volume of interest varies during time. In this case, the most general reaction scheme is: k

P⟶P′ + P″.

(6)

This reaction involves three chemical species and the rates of formation of P′ and P″ are equivalent to the rate of consumption of P since the stoichiometric coefficients are the same, i.e.:

−

d[P] d[P′] d[P″] = = = k[P]. dt dt dt

(7)

As regards case (iii), in the specific case of EVA copolymers used as encapsulant in photovoltaic modules, raw materials are cured resins with a degree of cross-linking significantly higher than the one generated during the degradation process. Hence, in this study the generated amount of cross-linking will be neglected, which is a reasonable assumption, according to [20]. 3. Degradation phenomena in EVA copolymers and their mathematical description 3.1. Degradation phenomena Poly(ethylene-co-vinyl acetate) copolymers (EVA) used as encapsulant in photovoltaic modules are macromolecular materials composed of ethylene (ET) and 28–33% in weight of vinyl acetate (VAc). During the lamination process, the viscous EVA resin is thermally cured through a cross-linking reaction that significantly changes its chemico-physical properties [21]. Even if the degradation process of EVA is not yet fully understood in weathering conditions, the literature reports a large number of experimental studies attempting at explaining the involved chemical reactions [9–11]. Accelerated degradation tests, such as thermogravimetry [22] or tests within climate chambers with controlled temperature and humidity [23], are useful methods to investigate the degradation phenomena, helping in formulating a general scheme for the overall chemical reactions involved in the process. However, acceler-

(1)

where P is the native polymer molecule, P* is the degraded product, and S denotes the small molecules possibly produced in the reaction. Denoting with k the rate of reaction, the rates of concentration change for the two chemical species P and P* and are defined as:

−

−1

∂[S] −Δ(D[S ]) = k[P], ∂t

The degradation of polymer materials takes place according to three different pathways: (i) without significant changes in molecular weight, (ii) by chain scission, or (iii) by cross-linking [14]. Degradation phenomena occurring in polymers are due to chemical reactions described through first-order kinetic equations [15]. In the sequel, we consider that each chemical specie is distinguishable within a class, such as the class of radicals, unsaturated or carbonyl compounds. The chemical species are denoted with capital letters (e.g. R•, U or Cb) and the molar concentration of their characteristic bonds (mol m−3) are written with capital letters in square brackets ([R•], [U] or [Cb]). When the molecular weight does not significantly vary during time (case (i)), transformations include only the loss of atoms, radicals or small molecules that diffuse across the polymer layer. In the most general scenario, the transformation of the macromolecular chain of the polymer can be written as: k

−1

where R = 8.314 [J K mol ] is the ideal gas constant, A0 [s ] is a preexponential factor, Ea [J mol−1] is the activation energy of the corresponding chemical reaction. The parameters A0 and Ea are usually determined from experiments. In this work we assume that kinetic constants involved in EVA degradation were dependent only on temperature. This assumption is justified by the high working temperatures of the module (up to 70 °C), coupled with the significantly low concentration of chromophores in the polymer [16], and the low dose of UV-B light that can trigger the EVA degradation [17]. When chemical reactions take place together with the formation of small molecules, also diffusive phenomena arise. In this instance, the concentration profile of the small generated molecules can be described by a reaction-diffusion model. Referring to the reaction reported in Eq. (1), the reaction-diffusion model for the specie S results:

2. Description of a generic reaction-diffusion process

P⟶P* + S ,

(3) −1

(2)

In Eq. (2), concentrations are functions of the extent of reaction, and therefore implicitly of time, while the rate k is a function of temperature according to the Arrhenius law: 94

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Fig. 1. Chemical reactions involving the ethylene and the vinyl acetate moieties in EVA macromolecules.

slowly and it migrates towards non-crystalline regions, where can react with other radical species present in the polymer. This initiation step can be schematized as follow:

ated degradation tests do not aim at reproducing the realistic environmental conditions, and hence the extrapolation of the obtained results to understand EVA behavior in weathering conditions should be attempted with caution. Furthermore, the analysis of the overall degradation of EVA, without distinguish between ethylene and vinyl acetate moieties, as seen e.g. in [24,12], adds some difficulties for a thorough comprehension of the phenomenon. As an example, it was reported in [25] that unsaturations (double bonds between C atoms) formed under UV exposure in the presence of oxygen can be only due to VAc deacetilation and the low ratio between VAc and ET does not allow the formation of conjugated dienes. On the other hand, the evidence of polyene formation was demonstrated in [26] via indirect methods. Considering that photo-oxidative degradation occurs on both monomeric moieties, we herein propose the comprehensive degradation scheme, reported in Fig. 1, along with the nomenclature of chemical species, summarized in Table 1. It is worth highlighting that previous experimental studies on this topic did not consider degradation reactions separately, but evaluated the global reaction kinetics for the overall degradation process. Each repeating unit of ET and VAc undergoes a degradation processes based on different chemical mechanisms, triggered by the temperature and the presence of oxygen. The linear portions of EVA between two cross-linking sites undergo H abstraction (Reactions (1) and (2)), oxidation (Reaction (3)), loss of acetate units (Reactions (4) and (5)) and chain cleavage (from the output of Reactions (3) and (4)) with a mathematical description in terms of molar concentration. ET units are relatively chemically inert, because of their small dipole moment associated with the C–H bonds. Copolymerisation with polar monomers, like vinyl acetate, increases their chemical reactivity. C–H and C–C bonds typical of ET do not absorb the UV radiation by themselves, thus the initiation of photo-oxidation in ET requires the presence of chromophores, such as carbonyl groups, additives and stabilizers [27]. The initiation step in ET degradation is represented by the generation of radical species [28] (Reaction 1 in Fig. 1). Radicals are formed after the loss of atomic hydrogen, leaving an unpaired radical in a C atom of the backbone or in a terminal position after the chain cleavage. The latter situation is not common in weathering degradation but can occur in molten polymers or under extreme shear stresses. After the cleavage of the C–H bond, the formed H • radical diffuses very

k1

ET ⟶R• + H •.

(8)

•

•

The H radical formed can diffuse in the EVA layer while R is formed in the polymer backbone and therefore it does not diffuse. After the formation of a radical on a C atom in the polymer backbone and the presence of a free H •, degradation can further propagate via two different and separate mechanisms. The first one is due to the high reactivity of the radical site in the backbone and to the presence of H • that can abstract another H • close to the unpaired C radical, leaving an unsaturated bond in the polymer backbone [29] and producing H2 molecules after the recombination of two H •, accordingly to the Norrish Type I reactions scheme [30] (see Reaction 2 in Fig. 1). The second propagation mechanism occurs in the presence of oxygen absorbed from the external environment, causing the formation of carbonyl groups in the backbone and free water molecules [31] (Reaction 3 in Fig. 1). Schematization of Reactions (2) and (3) is: k2

R•⟶U (R2) + H •,

(9a)

k3

R• + O2⟶Cb(R3) + H2O, (R2)

(9b)

Cb(R3)

Products U and denote the double-bonds and the carbonyl groups formed after Reaction 2 and 3 respectively. The small molecules H • and H2O formed by Reactions (2) and (3) diffuse in the EVA while unsaturated and carbonyl groups are formed in the backbone molecule and do not diffuse. The VAc fraction shows two different degradation mechanisms, mainly due to the loss of acetate units [32]. Also in this case, reactions follow a radical mechanism that can lead to the formation of acetaldehyde, through a Norrish type I mechanism (Reaction 4 in Fig. 1), or acetic acid, based on a Norrish type II mechanism [33] (see Reaction 5 in Fig. 1). These reactions are schematized as: k4

VAc⟶Cb(R 4) + CH3CHO, k5

VAc⟶U (R5) + CH3COOH , where Cb(R 4)

(R5)

(10a) (10b)

and U denote carbonyl groups and unsaturations formed within the backbone after Reaction 4 and 5, while CH3CHO and CH3COOH are small produced molecules that can diffuse. 95

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Table 1 Chemical species involved in photo-oxidative processes, nomenclature used and their definition.

Cb⟶Cb• + Rt•,

(11a)

k7

Cb⟶Cbt + Ut . Chemical structure

Nomenclature

Definition

ET

Ethylene units

VAc

Vinyl acetate units

U

Unsaturated C=C bond in the main chain (vinylene groups)

Ut

Unsaturated C=C bond in terminal position (vinyl groups)

R•

Radical with one unpaired electron in the main chain

(11b)

causing a reduction of the molecular weight of the EVA polymer. Considering that the native EVA is cross-linked, products of Reactions (6) and (7) are supposed to be large molecules and thus they do not diffuse. Finally, termination reactions occur when two free radicals react between them. In the case of EVA, radicals are formed in the backbone polymer and are not able to diffuse, reducing the possibility to provide this kind of reactions. Thus, in this work, the termination step is neglected. 3.2. Mathematical description of the reaction kinetics and diffusion of species The unknown variables of interest of the system are the starting species, [ET ] and [VAc], the intermediate product [R•], the species responsible for the browning process, [U ], [Cb], [Cb•], [Cbt ], and [Ut ], and two small molecules absorbed from the external environment and participating to chemical reactions, [O2] and [H2O]. In summary, reactions involved are: k1

ET ⟶R• + H •,

(12a)

• k2

R ⟶U + H •,

(12b)

k3

Rt•

R• + O2⟶Cb + H2O,

Radical with one unpaired electron on a terminal C atom

(12c)

k4

VAc⟶Cb + CH3CHO,

(12d)

k5

VAc⟶U + CH3COOH , Cb

Carbonyl bond C=O in the main chain

(12e)

k6

Cb⟶Cb• + Rt•,

(12f)

k7

Cb⟶Cbt + Ut . Cbt

• Cbr

(12g)

Each reaction can be described by the following reaction-diffusion system:

Carbonyl bond C=O close to a terminal C atom due to back-biting isomerization

Carbonyl bond C=O with radical on a terminal C atom

d[ET ] = − k1[ET ], dt

(13a)

d[VAc] = − (k 3 + k4 )[VAc], dt

(13b)

•

The overall concentration of carbonyl groups formed by Reactions (3) and (4) is given by their sum, as well as the overall concentration of unsaturations, which is the sum of products of Reactions (2) and (5). Carbonyl groups formed from ET and VAc can further degrade giving chain scission (Fig. 2):

d[R ] = k1[ET ]−k 2[R•]−k 3[R•][O2], dt

(13c)

d[U] = k 2[R•] + k5[VAc], dt

(13d)

d[Cb] = k 3[R•][O2] + k4[VAc] dt

−(k6 + k 7)[Cb],

(13e)

d[C•b] = k6[Cb], dt

(13f)

d[Cbt ] = k 7[Cb], dt

(13g)

∂[O2] − D7Δ[O2] = − k 3[R•][O2], ∂t

(13h)

∂[H2O] − D8Δ[H2O] = k 3[R•][O2]. ∂t

(13i)

In these equations, reaction rates ki (i=1,…,7) are temperaturedependent and described by the Arrhenius equations:

Fig. 2. Chain scission reactions on carbonyl groups formed on EVA main chains.

96

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⎛ −E ⎞ ki(T ) = ki0 exp⎜ i ⎟ , ⎝ RT ⎠

i = 1, …, 7.

of the matter and it is a function of the thickness, affecting the intensity of the transmitted light. This phenomenon is described by the LambertBeer equation [37]:

(14)

The set of differential equations (13) forms a reaction-diffusion system defined in the space-time domain R × [0, t f ], where R is the EVA layer of lateral sizes B × L and thickness s. It consists of 7 non diffusive ordinary differential equations and 2 diffusive partial differential equations. Diffusion coefficients are temperature-dependent and can be described as:

Dj (T ) =

⎛ −E d ⎞ ⎜ j ⎟ 0 ⎜⎝ RT ⎟⎠ Dj e ,

j = 6, 7.

dΦ(z, λ ) = − ϵi(z, λ )[i ]Φ(z ), dz

where z is the depth of light penetration from the exposed surface, Φ is the light intensity and ϵ is the molar attenuation coefficient of the specie i and [i ] is the molar concentration of specie i across the thickness. Denoting with s the total thickness of the layer, the integration of Eq. (20) over the depth z leads to:

(15)

TR(λ ) =

Dirichlet boundary conditions for the reaction-diffusion system involving the diffusing species on the boundary ∂RD × [0, t f ] are prescribed by the values of equilibrium concentrations of species absorbed from the external environment, [O2]eq and [H2O]eq , and result:

[O2]eq = constant,

(16a)

lH O. [H2O]eq = S (T )·P 2

(16b)

5. Numerical scheme for the simulation of environmental degradation In case of environmental degradation, ambient temperature Tamb(t ) and relative humidity %RH (t ) can be provided from real annual climate data periodically repeated to obtain profiles over 20 years, to be used as boundary conditions for the degradation simulation. These environmental data are usually provided by environmental agencies, such as Agenzia Regionale per la Protezione Ambientale (ARPA) in Italy, for different locations and with a time resolution of 1 h. The module temperature can finally be computed from the ambient temperature as [39]:

(17) and write the previous system of differential equations together with boundary and initial conditions in vectorial form:

∂C(t ) − div(D(C(t ))∇C(t )) = F(C(t )), ∂t

in R × [0, t f ]

(18)

where D is the diagonal diffusion matrix having the coefficient Di on the diagonal. The vector F depends on the concentration vector C and is related to the right-hand side term in Eqs. (13d)–(13i). System (18) is completed with the following boundary conditions:

× [0, t f ],

(22)

Each chemical group has a specific molar extinction coefficient at a determined value of λ. In the case proposed in this work, the optical parameters used are summarized in Table 2 [38]. Assuming that the concentration of chromophores [U ] and [Cb] are homogeneous across the EVA thickness, values of absorbance calculated with Eq. (22) can be directly correlated with the discoloration of the polymer film.

C(t ) = (C1(t ), …, C7(t ))T = ([R•], [U ], [Cb], [Cb•], [Cbt ], [O2], [H2O])T

on

(21)

A(λ ) = − log10(TR(λ ))ϵi[i ]s.

humidity %RH . The temperature-dependent function S (T ) = S0e(Es /(RT )) denotes the solubility. In the sequel, we denote initial concentrations of chemical species as: [ET ]t=0 = [ET ]0 , [VAc]t=0 = [VAc]0 , [R•]t =0 = [R•]0 , [U ]t=0 = [U ]0 , [Cb•]t =0 = [Cb•]0 , [Cbt ]t =0 = [Cbt ]0 , and [Ut ]t =0 = [Ut ]0 . For the sake of simplicity, we introduce the concentration vector:

∂RDC

s ΦT − ϵ (z, λ )[i]dz = e ∫0 i , ΦI

where ΦT and ΦI are the intensity of the transmitted and incident light, respectively. The ratio between ΦT and ΦI is called transmittance TR and it is related to the absorbance A through the following relationship:

The first value is only dependent on the considered material (see Section 7.1). The last relation is the Henry's law relating the water concentration at the boundary of the EVA layer in contact with air and lH O , depending on the relative the partial pressure of the water vapor P 2

C(t ) = G*(t ),

(20)

T = Tamb +

(NOCT − 20) × Irr , 800

(23)

where T is the working temperature of the module (°C), Tamb is the measured external temperature (°C), NOCT = 50 °C and Irr is the solar irradiance (W m −1). For practical considerations, the temperature of the EVA layer was set equal to the working temperature of the module, since the ambient temperature is slowly varying with time. However, there are some evidences that temperature is not homogeneous across the module thickness [40], and therefore a refined analysis could take into account the high order effects. Multiplying the differential system (18) by a vectorial test function S, and integrating over the domain, using the divergence theorem, the weak form of the reaction-diffusion system reads: for all t ∈ [0, t f ], find the concentration vector C(t ) such that it satisfies:

(19)

where G*(t ) is the boundary condition vector imposed on ∂RDC , the Dirichlet part of the domain, specified by Eq. (16). 4. Relation between chemical species and overall degradation of the optical performance As previously explained, the degradation of EVA mainly involves the change of chemical bonds of polymer backbones and the formation of small organic molecules, like water and acetic acid. Ref. [24] reports that yellowing and browning of EVA are directly related to the presence of chromophores formed during degradation. When severe discoloration appears, this phenomenon causes a decrease of transparency of the encapsulant, contributing to the loss of electric efficiency of the modules [10,34]. Optical properties of EVA significantly change upon degradation, due to the formation of unsaturations and carbonyl bonds absorbing the sunlight in the middle field, and causing the discoloration of the modules and the loss of efficiency [35,36]. The presence of chromophores increases the attenuation coefficient of the encapulant, also called opacity index, partially blocking the transmitted light to the solar cell. At each light wavelength λ, the attenuation coefficient is a property

∫R ∂tC(t )·S dV + ∫R D(C(t ))∇C(t ): ∇S dV = ∫R F(C(t ))·S dV ,

∀ S. (24)

For the application of the mentioned method, the domain R occupied Table 2 Optical parameters for chromophores formed during EVA degradation.

Unsaturations Carbonyl bonds

97

Characteristic absorption wavelength λ (cm−1)

Molar extinction coefficient ϵ (L mol−1 cm−1)

909 1718

100 350

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M. Gagliardi et al.

by the portion of EVA material is discretized into a finite number of elements R(e), so that:

R ≈ ⋃R(e).

{Re}naI =

∫

(25)

e

{Ce}bJ

By introducing linear Lagrange 1 shape functions, the finite element approximation of each component CK of the concentration vector reads:

∑ Φa(ξ1, ξ2)CaK (t ), 1 ≤ K ≤ 7 a =1

∑∫ e

=

∑∫

R (e )

e

⎡ ∂{R }n ⎤ 1 e aI ⎢ ⎥= [Me]aIbJ + ⎣ ∂{Ce}bJ ⎦ Δt ⎛ +⎜ ⎝ −

(27)

[Ke]aIbJ =

∫R(e) ΦbΦadV ,

{Fe}aI =

∫R(e)

K

Solubility parameters: S 0, Es

(28c)

D {C} + [K ]{C} = {F} Dt

Initialize: {C}1 , tol, norm = 1 for n = 1, …, N time steps do

Compute temperature and partial pressure n

lH 2O; T n, P Kinetic constants, diffusivities and solubility ki(T n ) = ki0 exp( − (Ei /(RT n )));

(29)

{C}n +1 − {C}n + [K ]{C}n +1 = {F}n +1 . Δt

Dj (T n ) = Di0 exp( − (Ejd /(RT n ))); S (T n ) = S 0 exp(Es /(RT n ))); while (norm ≥ tol) do Update reaction vector and diffusion matrix n +1 F(nk+1 ) , D (k ) ;

Form the residual vector: {R}(nk+1 ) ; Solve the linearized reaction − diffusion system :

(30)

n +1 {C}(nk+1 ) ← {C}(k ) ;

The reaction term was treated in an explicit manner, considering {F} ({C}n ) instead of {F} ({C}n+1). The algorithm for the proposed time integration of the reaction-diffusion problem is detailed in Algorithm 1. Diffusion coefficients, expressed as:

D͠ j (c, T ) = Dj (T )

⎛ ⎞ 0 ⎜γ c ⎟ D͠ j e⎝ j ⎠,

norm ← ∥ {R}(nk+1 ) ∥ end Update concentration vector: {C}n +1 ← {C}n ;

j = 6, 7,

end

(31)

leads to a diffusion matrix [K ] which is not constant, but it is a nonlinear function of {C}. Hence, the Newton-Raphson iterative procedure with fully implicit approximation of the reaction vector is proposed to solve this problem. Hence, the residual vector {R} is defined as:

{R({C}n )} =

(35)

Kinetic and diffusion parameters: ki0, Ei, Dj0 , Ejd ;

FI (C)Φa dV ,

where [M ] and [K ] result from the finite element discretisation and assembly, {F} is the discretized reaction vector and {C} is the nodal concentration vector. Adopting an implicit Euler time integration scheme, at each time step tn the following algebraic set of equations was solved:

[M ]

∂F

∫R(e) ∂CI (Cen+1)ΦaΦbdV .

(28b)

where the Einstein summation notation has been adopted. Local matrices and vectors are then assembled over all elements R(e) as customary to obtain global matrices and vectors leading to the differential system:

[M ]

K

Algorithm 1. Numerical scheme for the solution of the reactiondiffusion system for environmental degradation.

∂Φ ∂Φ K

K

⎞ ∂DIL n +1 ∂Φl ∂Φa (C e ) Φb dV ⎟{Ce}lL ∂CJ ∂xK ∂xK ⎠

Given the solution of system (34), a better approximation for {C}n+1 n +1 n +1 is obtained as {C}(nk+1 +1) = {C}(k ) + {ΔC}(k ) . This procedure is repeated n +1 ∥ { R } ∥ is less than a prescribed tolerance. until the residual norm (k )

(28a)

∫R(e) DIJ (C) ∂x b ∂x a dV ,

∫R(e)

∂Φ ∂Φ

∫R(e) DIJ (Cen+1) ∂x b ∂x a dV

J

The expressions of matrices and vectors entering the discretized weak form (27) at the element level are:

[Me]aIbJ =

(34)

where the components of the tangent operator are:

∑∫

FI (C)ΦaSaI dV = 0

(33)

⎡ ∂{R} ⎤n +1 n +1 n +1 ⎢ ⎥ {ΔC}(k ) = − {R}(k ) , ⎣ ∂{C} ⎦(k )

(26)

∂Φ ∂Φ DIJ (C) a b CaI SbJ dV R (e ) ∂xL ∂xL

e

∫R(e) FI (Cen+1)Φa dV

Suppose that at the time step n is given, which is an approximation of {C}n+1, then the following linearized system is solved:

(e ) is a basis of local linear shape functions defined in where {Φa(ξ1, ξ2 )}aN=1 the natural reference system −1 ≤ ξ1, ξ2 ≤ + 1 and N (e) is the number of element nodes which is equal to 3 in the case of linear triangular Lagrange elements. After introducing this expression in the weak form, its discretized version is obtained:

∂C ΦaΦb aI SaI dV + R (e ) ∂t

−

{C}(nk+1 ) ,

N (e )

CK (x1, x 2 , t ) =

1 n [Me]aIbJ ({Ce}bJ − {Ce}bJ ) Δt ⎛ ⎞ ∂Φ ∂Φ +⎜ DIJ (Cen +1) b a dV ⎟ ∂xK ∂xK ⎝ R (e ) ⎠

[M ] ({C} − {C}n ) + [K ({C})]{C} − {F ({C})}, Δt

6. Numerical scheme for the simulation of accelerated degradation In the damp-heat test defined by the international standard IEC61215 [41], PV modules are subjected to a cyclic temperature from −40 °C up to 85 °C, according to the following ramps (see also the sketch in Fig. 3):

(32)

which, in components, at the element level, reads: 98

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Table 3 Model parameters. k0 (1/yr)

Ea (kJ/mol)

R1

3.17 × 107 [44]

64.8

R2

3.17 × 107 [44]

64.8

R3

2.08 × 104 [45]

104.7 [45]

R4

1.03 × 1013 [12]

107.5

R5

1.03 × 1013 [12]

107.5

ρc

Cn +1 − Cn + div(D(Cn +1)∇Cn +1) = F. Δt

0 ≤ t < t1* t1* ≤ t < t2* t2* ≤ t < t3*

∂T − κ ΔT = QT (t ), ∂t −3

Algorithm 2. Numerical scheme for the simulation of degradation in the accelerated aging tests.

t4* ≤ t < t5*

(36)

in R × [0, t f ] −1

Input: kinetic and diffusion parameters:

ki0, Ei, ΔHi, Dj0 , Ejd , κ , ρ , c ; Initialize: {C}1 , T 1tol, norm = 1 Given {C}n , T n for n = 1, …, N time steps do

Compute ki(T n ), QT (T n ); Solve the thermal problem :

(37) −1

−1

ρc∂tT n +1 − κ ∇2 T n +1 = QT ;

−1

where ρ [kg m ], κ [W m K ] and c [J kg K ] denote, respectively, the EVA density, the thermal conductivity and the specific heat capacity. QT is the internal heat generated by chemical reactions, given by:

QT , ET = ΔHR1k1(T )[ET ],

Update temperature: T n +1 ← T n; Update kinetic constants and diffusion coefficients: ki(T n +1), Dj (T n +1); while (norm ≥ tol) do

(38a) •

QT , R• = (ΔHR2k 2(T ) + ΔHR3k 3(T )[O2])[R ],

(38b)

QT , VAc = (ΔHR 4k4(T ) + ΔHR5k5(T ))[VAc],

(38c)

QT , Cb = (ΔHR6k6(T ) + ΔHR7k 7(T ))[Cb].

(38d)

Update reaction vector and diffusion matrix n +1 F(nk+1 ) , D (k ) ;

Form the residual vector: {R}(nk+1 ) ; Solve the linearized reaction − diffusion system : n +1 {C}(nk+1 ) ← {C}(k )

end Update the concentration vector:

Values of ΔHi for chemical reactions are reported in Table 5. The coefficients ki are function of those parameters according to Eq. (3). Boundary conditions are expressed in terms of temperature on the border of EVA, ∂R :

T (t ) = T *(t ),

in ∂R × [0, t f ],

(41)

A sketch of the operations is provided in the Algorithm 2.

t3* ≤ t < t4*

where T1* = 85 °C , T2* = − 40 °C , and t1* = 0.5 h , t2* = 1.5 h , t3* = 2.5 h , t4* = 3.5 h , t5* = 4.5 h . In this case, the temperature inside the module, and especially in the EVA layer, cannot be assumed homogeneous and equal to the temperature inside the climate chamber, since it is rapidly varying in time. Thus, the temperature becomes an additional variable of the problem to be solved and coupled with the other equations constituting the whole reaction-diffusion system:

ρc

(40)

2. Then update the kinetic constants ki(T n+1) and diffusion coefficients Dj (T n+1), with the new values of T n+1 and solve the reaction-diffusion problem:

Fig. 3. Values of T and ceq(T) during damp-heat test.

⎧ T* ⎪ 1 t, ⎪ t1* ⎪ * ⎪T1 , ⎪ T * − T2* (t3* − t ), T *(t ) = ⎨T2* − 1 t3* − t2* ⎪ ⎪ * ⎪T2 , ⎪ T2* (t5* − t ), ⎪ ⎩ t5* − t4*

T n +1 − T n − κ ∇2 T n +1 = QT ; Δt

{C}n +1 ← {C}n ; end

(39)

where T *(t ) is given by the ramps defined in Eq. (36). The characteristic rate of the heat conduction process is ruled by the ratio κ /(ρc ).. Typical values for EVA are ρ = 0.96 kg m −3, c = 1400 J kg−1 K−1 and κ = 0.34 W m −1 K−1. Considering experimental values of kinetic constants and diffusion coefficients reported in Table 3, κ /(ρc ) ≫ ki and Dj. Thus, heat conduction inside EVA was much faster than the reactiondiffusion processes, then it was possible to split the heat conduction dynamics, proposing the following scheme:

7. Numerical examples: validation and predictions 7.1. Model validation To validate the proposed numerical tool and identify the input data of the reaction-diffusion model, two case studies, reporting experimental data or benchmark solutions, were examined. The first test regards the time evolution of EVA absorbance due to aging under UV lamp at 50 °C and evaluated by Fourier Transform Infra Red Analysis at the wave number 909 cm−1 (characteristic of double bonds, denoted as A909 in the sequel) and at 1718 cm−1 (characteristic of carbonyl

1. Given the concentration vector Cn at time tn, compute QT and solve the thermal problem:

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Fig. 4. Finite element mesh used to discretize the EVA layer.

Fig. 6. Moisture concentration profiles in EVA after 1, 2, 4, 8 years. (a) results by [1]; (b) present model predictions.

while the following boundary conditions for the diffusing species are:

[O2] = 1.58 mol m −3, [H2O] = 0.036 mol m −3. The concentration of [O2] at the boundaries has been estimated by considering the partial pressure of oxygen in air (0.2 atm) and the solubility coefficient in EVA is S=0.0023 cm3(STP)cm−3 cm Hg−1 [43]. Temperature was considered constant and equal to 50 °C. The equation system reported in (12) contains the following primary concentration as unknowns [ET ], [VAc], [R•], [Cb], [U ] and [O2]. The model parameters entering the reaction-diffusion ordinary differential equations identified to match the experimental results are reported in Table 3, along with reference data reported in the literature, where available. The solution of the present model provides the concentrations [ET ], [VAc], [R•], [Cb], [U ], [O2] and [H2O] for each time step. The simulation was performed up to 300 h, with a time step of 0.01 h. Based on the value of these primary unknowns, the absorbance of EVA in the center of the sample has been computed and the finite element predictions are shown in Fig. 5, in comparison with experimental data taken from [42] for the same testing conditions. The experimental trend is correctly captured by the present model. The present approach was also validated for water transport inside EVA, considering the test problem in [1]. The finite element mesh and discretization were the same as in Fig. 4, but with a lateral size of 0.4 m. The temperature imposed at the boundary of the domain was equal to T=27 °C and kept constant over time. Similarly, a constant air relative humidity of %RH = 71% was imposed along the same boundary. This

Fig. 5. Experimental data (black dots) compared to results obtained by means of the developed FE model (black line) in simulated accelerated degradation, at T=50 °C [42].

bonds, referred to as A1718) [42]. To simulate this problem, a 2D square EVA layer of lateral size L = 65 × 10−3 m and thickness s = 4 × 10−3 m is considered. The domain R has been decomposed into 20 linear triangular Lagrangian finite elements Re per side, for a total of 800 finite elements (Fig. 4). Initial conditions of the problem are prescribed over the whole domain:

[ET ]0 = 0.64 mol m −3, [VAc]0 = 0.32 mol m −3, [R•]0 = [Cb]0 = [U ]0 = [O2]0 = [H2O] = 0 mol m −3, 100

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Fig. 7. Experimental values and related interpolations of: (a) ambient temperature Tamb , (b) relative humidity percent RH%. (c) Module temperature T and (d) equilibrium concentration of water at the exposed EVA edges over a period of 20 years.

Table 4 Fitting coefficients used in Eq. (45). i

ai

bi

ci

1 2

33.44 30.93

0.03 52.36

1.61 −1.49

Table 5 Heat of reaction for the chemical reactions.

ΔHR /kJ mol−1 R1 R2 R3 R4 R5

−30.0 [48] −288.0 [48] −247.0 [49] −1076.5 [50] −1076.5 [50]

value provided the water concentration at the boundary as:

lH O(T )S (T ), [H2O]eq (T ) = P 2

(42)

Fig. 8. Geometry of the mesh (not in scale) used to compare obtained results for different loading scenarios at point P.

lH O is the saturation pressure of water in the air and S(T) where P 2 denotes the solubility of water in the EVA. The former was computed as lH O = PHsatO%RH /100 , where PHsatO was determined according to the P 2 2 2 Antoine equation (Eq. (43)), using the coefficients A=4.654, B=1435.264, and C = − 64.848 [46]:

log10(PHsat2O ) = A −

101

B T+C

(43)

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Fig. 9. Concentration of ET and VAc vs. time in the point P inside EVA, for simulated environmental conditions.

lH O(27 °C) = 2.84 × 10−2 atm . The solubiwhich, for T=27 °C, leads to P 2 lity S(T) is provided by the Arrhenius Law:

S (T ) = S0e−Es /(RT ),

(44)

which, for T=27 °C, gives according to [47] S(27 °C) = 3.61 ×10−10 mol m −3 atm −1 . As initial condition on water concentration, [H2O]0 = 0 mol m −3 was considered inside the domain. The diffusion coefficient D H2O was equal to 2.0 × 10−3 m2/y . The system in Eq. (12) was solved for the primary unknown water concentration, [H2O], for each time and the results are plotted in Fig. 6(a) for the present 2D model. The trend was very similar to the 1D model prediction reported in [1] and shown in Fig. 6(b) for a visual comparison. 7.2. Environmental degradation vs. accelerated ageing In this section, the effect of environmental loading or accelerated testing conditions on optical degradation and water uptake of EVA is investigated. Regarding environmental data, air temperature, air relative humidity, and solar irradiance for the location of Piacenza (North of Italy) were taken from the database provided by Agenzia Regionale per la Protezione Ambientale (ARPA), with a time resolution of 1 h. Data, referring to the year 2010 characterized by a very cold winter, are shown in Fig. 7. From air temperature and irradiance, the module temperature was determined, according to Eq. (23). To provide a closed-form equation for the module temperature to be imposed at each time step to EVA, the experimental data are fitted according to the following expression:

Fig. 10. Concentration of U, Cb, and Cb• vs. time in P inside EVA, for simulated environmental conditions.

Tamb(t ) = a1sin(b1t + c1) + a 2sin(b2t + c2 ).

(45)

Fitting coefficients matching the sinusoidal time dependency and capture the extremal values of temperature are listed in Table 4. Regarding the relative humidity, which displayed very strong daily oscillations, two maximum values were considered as representative of the warm and cold seasons, as a worst case scenario for water uptake. Of course, any other continuous or discrete temperature and relative humidity profiles can be considered as input of the finite element 102

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Fig. 11. Water concentration in the point P inside EVA vs. time for simulated environmental conditions.

Fig. 13. Concentration of ET and VAc vs. cycle no. in the point P inside EVA, for accelerated aging test conditions.

In this case, temperature inside the EVA layer was treated as an additional unknown of the problem. Heat of reactions contributed to the heat conduction partial differential equation, reaction enthalpies have been taken from the literature and are listed in Table 5. Test geometry used to compare the effect of the two aging scenarios on EVA degradation is shown in Fig. 8 (dimensions not in scale). The point P inside EVA is selected to show the time dependency of the species concentration in that location. Periodic boundary conditions were imposed along the vertical edges, while zero flux boundary condition on the diffusive species was imposed on the interface between glass and EVA, and on the internal interfaces between Silicon and EVA. Finally, the backsheet has been treated as a zero thickness layer, contributing to the water flux evaluated as D H2O, BS × SH2O, BS × plH O / tBS , where D H2O, BS is the water diffusion coeffi2 cient of the backsheet, evaluated through the Arrhenius equation, with parameters Ed=39.1 kJ mol−1, D0,BS=1.89×10−2 m2 y−1, S0,BS =3.33×10−3 mol m−3 atm−1, Es,BS=40 kJ mol−1. This test geometry was proposed in [13] for similar purposes. Examining environmental conditions, the predicted consumption profiles of ET and VAc species vs. time is shown in Fig. 9. Results indicated an overall degraded fraction of the polymer of 0.22% after 20 years. The steps are due to temperature oscillations, influencing the kinetic rates of reactions ki. Concentration profiles of chemical species produced after degradation (U, Cb and Cb•) are shown vs. time in Fig. 10. The concentration of Cb• was much smaller than the other, being related to a secondary reaction type with a very high activation energy.

Fig. 12. Optical degradation vs. time for the point P inside EVA, for simulated environmental conditions.

algorithm, without any lack of generality. Considering Eq. (23) to convert ambient temperature to module temperature, and taking into account the temperature dependent solubility as in Eq. (44) and the lH O , the final input data extrapolated over a period of 20 expression for P 2 years are shown in Fig. 7. The sinusoidal variation of [H2O]eq in time lH O . was due to temperature oscillations, affecting S(T) and P 2

For accelerated aging, temperature and relative humidity were imposed by the ramp functions inside the climate chamber (Fig. 3).

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Fig. 15. Water concentration in the point P inside EVA vs. cycle no. for simulated accelerated conditions.

Fig. 14. Concentration of U, Cb, and Cb• vs. cycle no. in P inside EVA, for simulated accelerated conditions.

Fig. 16. Optical degradation vs. cycle no. for the point P inside EVA, for simulated accelerated conditions.

Moisture uptake vs. time is shown in Fig. 11 and presents an oscillating trend, tending towards a saturation after about 5 years. The corresponding optical degradation was finally quantified and reported in Fig. 12. It is remarkable to note that the amount of degradation was analogous to that measured in the case of accelerated tests under UV at 50 °C (Fig. 5). The same predictions regarding the accelerated aging test are now reported in the following figures, for a time frame of the simulations covering up to 400 cycles. As compared to environmental conditions, the consumption of ET and VAc (Fig. 13) displayed an analogous trend

with comparable consumption at 20 years after 400 cycles. Values of [U ], [Cb], and [Cb•] (Fig. 14) increased with the number of cycles. In particular, values of [U ] and [Cb] after 400 cycles doubled values predicted in the case of environmental aging. Regarding water uptake, the saturation value achieved in the environmental aging after about 5 years was reached notably earlier in accelerated aging, after about 20 cycles. Moisture uptake vs. time (Fig. 15) presented a sigmoid trend, as in environmental conditions.

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The corresponding optical degradation was finally quantified (Fig. 16). The degradation of the optical properties appears to be much higher than in the field.

risks of extrapolation. The same model can provide other additional results, such as the production of acetic acid due to EVA deacetylation, the corrosion of metallic fingers and the formation of snail trails, that are additional important phenomena affecting PV performances. Concerns related to these aspects will be studied in a future work, providing a relationship between the decay of electrical performances and corrosion phenomena. In the present study, transport of chemical substances and reactiondiffusion phenomena in EVA, due to temperature, moisture, oxygen and exposure to the environment, has been the main focus. Moreover, the problem of the interplay between the thermal stresses and EVA degradation is very interesting and challenging. Indeed, thermoelastic stresses play a role for the mechanical performances and the adhesive properties of EVA. The study of thermal stresses is also particularly complex, considering that working temperatures of solar cells exposed to the external environment straddle the glass transition temperature of EVA (around 45 °C). At present, further experimental evidences are requested to fully support the development of a quantitative thermoelastic degradation model for EVA due to thermal expansion, combined with the effect of swelling due to moisture. Indeed, with the increasing demand for new PV installations in emerging countries, a computational framework for the understanding of physico-chemical degradation phenomena in EVA for climate zones other than the European one will represent a useful tool for quality assessment and testing.

8. Conclusion and future perspectives The present study reports a comprehensive methodology to study degradation and diffusive phenomena in EVA encapsulants used in PV manufacturing. For the thorough study of the EVA layer, we have proposed a reaction-diffusion mathematical system to describe the complex scheme of coupled chemical reactions and diffusion of small molecules. Such mathematical system was solved with a procedure based on the finite element method. We have analyzed two different cases, environmental aging and accelerated testing conditions, and optimized the computational effort. The computational tool to treat a coupled set of ordinary and partial differential equations was implemented in the open source FreeFem++ software, and firstly applied to simulate moisture diffusion and optical degradation phenomena, comparing obtained results with data available from the literature. The application of the proposed method to these benchmark tests has permitted to identify all the model parameters. Afterwards, the tool has been applied to the prediction of degradation of chemico-physical properties of EVA occurring in environmental or accelerated conditions. In the former case, climate data from Piacenza, in the North of Italy, have been used as input to the model. In the latter, conditions typical of a damp-heat test have been considered. Model predictions allow quantitatively comparing the dynamics of aging in the two cited cases, providing useful hints on the equivalence and differences between the two tests. In particular, we have found a correspondence between the number of accelerated cycles required to achieve an environmental degradation after a 20 years-exposure. Moisture uptake was much faster in accelerated tests than in weathering conditions, while the consumption of the chemical species ET and VAc, and consequently the production of chromophore species, were comparable with those obtained in 20 years of environmental exposure and 150 damp-heat cycles. The reported computational model has shown the potential to provide quantitative data on degradation phenomena in EVA encapsulants, in different temperature and moisture conditions. On the other hand, the proposed computational model can be easily adapted to study not only different climatic conditions but also different standard tests or for the design of new testing protocols tailored for specific climate zones. The proposed numerical schemes have been designed so that any environmental or accelerated aging condition can be effectively simulated. To this aim, all the expressions for the temperaturedependent material parameters have been included in the model, based on the collection of data taken from a wide specialistic literature on the matter. Moreover, a specific numerical scheme has been proposed to simplify the computation cost and the complexity of the environmental aging problem, where EVA temperature and moisture are mostly dictated by the ambient conditions. This has highly simplified the problem dynamics as compared to the much faster accelerated aging conditions, where the transient regime has to be fully solved for the accurate prediction of the time evolution of temperature and moisture inside the EVA. To reduce uncertainties, the authors have decided to build up a mathematical formulation describing EVA degradation and aging starting from the accurate representation of the very fundamental chemical reactions and diffusion processes taking place inside EVA. This high level of accuracy in the model description is considered to be a good strategy to avoid significant uncertainties that might occur in using phenomenological models providing just an overall representation of material degradation. Moreover, since the temperaturedependency of chemical reactions is particularly relevant, having considered the explicit dependency between the reaction-diffusion parameters and temperature is a pro, which allows for the applicability of the method to any environmental condition without suffering from

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