# A geometrical multi-scale numerical method for coupled hygro-thermo-mechanical problems in photovoltaic laminates

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## Abstract

A comprehensive computational framework based on the finite element method for the simulation of coupled hygro-thermo-mechanical problems in photovoltaic laminates is herein proposed. While the thermo-mechanical problem takes place in the three-dimensional space of the laminate, moisture diffusion occurs in a two-dimensional domain represented by the polymeric layers and by the vertical channel cracks in the solar cells. Therefore, a geometrical multi-scale solution strategy is pursued by solving the partial differential equations governing heat transfer and thermo-elasticity in the three-dimensional space, and the partial differential equation for moisture diffusion in the two dimensional domains. By exploiting a staggered scheme, the thermo-mechanical problem is solved first via a fully implicit solution scheme in space and time, with a specific treatment of the polymeric layers as zero-thickness interfaces whose constitutive response is governed by a novel thermo-visco-elastic cohesive zone model based on fractional calculus. Temperature and relative displacements along the domains where moisture diffusion takes place are then projected to the finite element model of diffusion, coupled with the thermo-mechanical problem by the temperature and crack opening dependent diffusion coefficient. The application of the proposed method to photovoltaic modules pinpoints two important physical aspects: (i) moisture diffusion in *humidity freeze* tests with a temperature dependent diffusivity is a much slower process than in the case of a constant diffusion coefficient; (ii) channel cracks through Silicon solar cells significantly enhance moisture diffusion and electric degradation, as confirmed by experimental tests.

## Keywords

Thermo-visco-elasticity Moisture diffusion Cohesive zone model Geometrical multiscale model Photovoltaics## 1 Introduction and motivations

Photovoltaic modules are composite structures obtained by laminating layers of various materials. Some have the role to guarantee protection from the environment by a suitable sealing of the device, others have specific electric features to produce energy. The durability of these devices is a serious concern due to fracture events promoted by the mismatch between the thermo-mechanical properties of the material constituents, often amplified by the severe working conditions they are exposed to. Moreover, their modelling is changelling and it requires a multi-physics framework to gain an accurate picture of their overall working conditions, performance and degradation [1].

*damp heat*tests in [2] prescribed by the international qualification standards [4], where PV modules were exposed to a very aggressive environment at constant \(85\,^{\circ }\)C temperature and \(85\%\) of air humidity. In particular, it has been shown a progressive increase of dimmer areas in time in the electroluminescence (EL) images starting from the edges of the solar cells towards their center (see Fig. 1a). Correspondingly, the current-voltage of the PV module degrades, with a significant power-loss (see Fig. 1b).

The EVA polymer displays a strong thermo-visco-elastic constitutive response, as experimentally reported in [5, 6], with a variation of the elastic modulus of up to three orders of magnitude depending on temperature. Generalized Maxwell rheological models used so far generally provide exponential type relations for the relaxation modulus and, in order to approximate the experimentally observed power-law trend, a huge number of elements (and thus of model parameters) has to be taken into account. To significantly simplify the task of parameters identification, modelling the visco-elastic behaviour via fractional derivatives has been proved to be very effective [8, 9, 10]. For rheologically complex polymers as EVA, whose microstructure changes with temperature, the fractional calculus formulation in [11] allows the use of only two temperature dependent parameters for its complete description.

Another complexity regards the moisture diffusion properties of EVA, strongly temperature dependent as experimentally reported in [12], which implies coupling between moisture and temperature fields. In return, moisture is degrading the cohesive energy of the EVA encapsulant, promoting decohesion of the backsheet or separation of the glass cover from the solar cells [13, 14]. This mathematically corresponds to coupling between the mechanical and the diffusive fields, with a further acceleration of moisture diffusion and degradation as reported in [15], which is a feedback coupling from diffusion to mechanics.

*damp heat*test, its validity in the case of a cyclic variation of temperature from \(-40\) up to \(85\,^{\circ }\)C as in the

*humidity freeze*test is highly questionable.

To shed light into the above issues, and provide a comprehensive physical modelling and computational framework for the study of these phenomena, a geometrical multiscale approach is herein proposed by following the seminal work in [16] for biophysical systems. Starting from the evidence that moisture diffusion takes place in a physical domain with a lower dimension with respect to that of the thermo-mechanical and heat conduction problems, two different finite element models are used in parallel. The coupled thermo-mechanical and heat conduction problems are solved in the three-dimensional setting (or in the two-dimensional one in the case of a cross-section of the PV module). As a further simplification, the EVA encapsulant layers are modelled as zero-thickness interfaces, whose thermo-visco-elastic constitutive response is taken into account by a novel thermo-viscoelastic cohesive zone model. As compared to other cohesive zone model formulations available in the literature [17, 18], the present formulation is based on fractional calculus and it is able to simulate rheologically complex materials.

The thermo-mechanical problem, which is much faster than moisture diffusion, is solved first via a fully implicit solution scheme in space and time, see Fig. 2. Temperature and relative displacements computed in the Gauss points along the encapsulant interfaces are then projected to the nodes of another finite element model specific for the solution of moisture diffusion. This second model is used to discretize the domain where moisture diffusion takes place. In particular, it is represented by the mid-surfaces of the encapsulant layers and of the channel cracks through Silicon, see Fig. 4.

This article is structured as follows. In Sect. 2, the variational framework for thermo-mechanics and heat conduction for the layers is presented, along with the interface model for the thermo-visco-elastic encapsulant, as well as for moisture diffusion. The weak forms of the partial differential equations are established in Sect. 3 and the finite element discretizations are presented in Sect. 4. Details on the proposed numerical solution scheme are provided in Sect. 5 and numerical applications to photovoltaics and comparison with experiments are collected in Sect. 6. Conclusions and an overview of future perspectives of research complete the study.

## 2 Variational framework

In this section, the variational framework describing moisture diffusion and the thermo-visco-elastic response of a laminate made of linear elastic homogeneous and isotropic layers separated by polymeric thermo-visco-elastic laminae is presented. In these laminates, moisture diffusion takes place in the polymeric layers, which progressively percolates from the free edges and from the permeable backsheet towards the centre of the solar cells. EVA layers used to protect Silicon solar cells are permeable to moisture, which is one of the major concerns for the degradation of the electrical output of the PV module over time. Due to moisture diffusion, the adhesive properties of EVA progressively degrade and the corresponding layers may experience a lack of cohesion leading to separation of the backsheet from the Silicon cells, or between the Silicon cells from the glass superstrate. In order to efficiently simulate cohesive degradation of EVA and delamination, we propose to treat the polymeric layers as zero-thickness internal interfaces with suitable traction-separation relations accounting for their thermo-visco-elastic properties.

Let the laminate occupy a volume \(\varOmega = \cup ^n_{m=1} R(m) \subset \mathbb {R}^3 \) in the reference undeformed configuration, where each layer is geometrically identified by \(R(m)=[0, a] \times [0, b] \times [z_{m-1}, z_m]\), with \(z_0=0\), \(z_m=z_{m-1}+h_m\), where \(a,b\gg h_m\) for \(m=1,\dots , n\). Moreover, let model a generic polymeric layer of thickness *h* between laminae \(R(p)=R_{-}\) and \(R(p+1)=R_{+}\) \((1\le p\le n)\) as a plane surface \(S(p)=(x_1,x_2,z_p) \ : 0\le x_1 \le a, 0\le x_2\le b\}\), see Fig. 4.

The position of each material point inside \(\varOmega \) is identified by the coordinate vector \(\mathbf {x}=(x_1,x_2,x_3)^{\mathbf {T}}\) in a three-dimensional cartesian orthonormal frame \(\{\mathbf {e}_1, \mathbf {e}_2 , \mathbf {e}_3 \}\). Let \(u_I(x_1,x_2,x_3,t)\) \((I=1, 2, 3)\) and \(\theta (x_1,x_2,x_3,t)=T(x_1,x_2,x_3,t)-T_0\) be the displacement field and temperature variation from a reference one, \(T_0\), inside the material during the time interval \(0\le t\le t_f\). The index *I* is used to denote the component of the displacement field along the corresponding coordinate.

### 2.1 Thermo-mechanical formulation of the layers

*R*(

*m*) obeys the equations of coupled linear isotropic thermo-elasticity (see e.g. [19, 20]). The Cauchy thermal stress tensor is defined for each layer

*m*as:

*m*-th material.

*R*(

*m*), and assume that it is related to the temperature variation \(\theta \) by the Fourier law:

*m*-th material. Hence, the heat transfer partial differential equation is given by:

*m*-th material.

### 2.2 Thermo-visco-elastic polymeric interfaces

*S*(

*p*) as:

*S*(

*p*) during time \([0, t_f]\).

*S*(

*p*). The relations between those fields and \([[ u_I ]]\), \([[ \theta ]]\) are provided by the constitutive relations for the interface. By assuming the continuity of the traction vector components \(t_I\), the in-plane sliding and out-of-plane tearing components of the traction vector read:

In order to obtain a structural response of the interface equivalent to that of the real EVA layers, the modulus \(K_3\) of the zero-thickness interface is related to the actual stiffness of the layer in the direction \(\mathbf {n}\), i.e., it can be evaluated as the ratio between the EVA Young’s modulus, \(E_{\text {EVA}}\), and its thickness, \(h_{\text {EVA}}\), i.e., \(K_3=E_{\text {EVA}}/h_{\text {EVA}}\). Similarly, the shear response is matched by selecting \(K_1=K_2=E_{\text {EVA}}/\left[ 2(1+\nu _{\text {EVA}})\right] \).

*t*. Instead of using a Prony series representation, a fractional calculus approach [8, 9] is herein adopted to synthetically characterize those dependencies. This approach has been proved in [11] to be very effective for parameters identification. Accordingly, \(E_{\mathrm {EVA}}\) is defined as follows:

*h*(

*t*,

*T*) is a history dependent function of time and temperature used to model thermo-rheologically complex materials where the principle of time-temperature superposition does not apply. This is due to a modification of the internal microstructure of the polymer driven by a temperature change above a threshold. Hence,

*h*(

*t*,

*T*) is equal to the current time

*t*minus the time \(t_0\) corresponding to such a microstructure modification.

*S*(

*p*), which is a reasonable approximation for thin polymeric layers. Hence, \(q_1=q_2=0\) and \(q=q_3\) is given by

### 2.3 Moisture diffusion along polymeric interfaces

Durability tests of PV modules inside a climate chamber are characterized by temperature and moisture dependent on time according to specified ramps. In these composites, moisture diffusion takes place along the layers of the polymeric encapsulant, or along channel cracks in Silicon. Since their thickness is very small, it is possible to neglect moisture flux in the direction orthogonal to the EVA layers. Under these assumptions, moisture diffusion can be modelled as a diffusion process taking place over the mid-surface of a generic encapsulant layer, which corresponds to *S*(*p*).

Hence, the aim of the numerical method reduces to simulate and predict diffusion of water content \(c(x_1,x_2,t)\) along the encapsulant mid-surface *S*(*p*) for each point and time.

*D*is the encapsulant diffusivity.

*D*. Considering characteristic values for EVA, the ratio between the velocities of the two phenomena is:

*D*is assumed to be a linear increasing function of the normal gap \([[u_3 ]]\), for \([[ u_3 ]]\) larger than \(\delta _3^c\).

Due to the very different time scales of the diffusion processes, a staggered solution scheme is proposed, where the average temperature and the crack opening computed from the solution of the coupled thermo-mechanical problem are passed as input to the diffusion process by a suitable update of the value of *D*.

## 3 Weak forms

The partial differential equations governing the dynamic equilibrium of the body, Eq. (3), and heat conduction, Eq. (5), for each layer *R*(*m*) \((m=1, \dots , n)\) and the constitutive relations for the interfaces, Eqs. (8), (9) and (13) for each *S*(*p*), define an initial boundary value problem describing the debonding of a thermo-mechanical layered PV panel with thermo-visco-elastic polymeric interfaces.

*S*(

*p*) and by integrating the result on each domain

*R*(

*m*). After applying the divergence theorem as customary and by dropping the index

*m*to simplify notation, we obtain:

*S*(

*p*) and by integrating the result on each domain

*R*(

*m*). After some calculation and dropping the index

*m*to simplify notation, we get:

## 4 Finite element discretization

### 4.1 Discretized weak forms for the thermo-elastic and heat conduction problems

*R*is discretized into a finite number of bulk \(R_e\) and interface \(\tilde{S}_e\) elements so that:

The class of interface elements considered here consists of two surface elements coincident with the facets of the bulk elements used to discretize the continuum that are bricks or tetrahedra. For consistency between interfaces and bulk, the same order of interpolation is used. In the case of 2D simulations on cross-sections of a laminate, the present formulation still holds, provided that bulk elements are represented by quadrilateral or triangular plane strain finite elements and interface elements are given by two opposing lines. Again, the same interpolation order has to be used.

*N*(

*e*) is the number of element nodes, which is equal to 8 for a 3

*D*linear brick element, or 4 for a 2

*D*linear 4-node plane strain element.

*S*(

*e*) is the number of nodes of the interface element which is equal to 8 for a 3

*D*interface element compatible with bricks, or 4 for a 2

*D*interface element compatible with plane strain elements. The nodal displacement vector is:

Similarly, the operator \([\Delta _e]_{ab}\) applied to the nodal temperatures of the interface element leads to the temperature jumps between the \((+)\) and the \((-)\) interface flanks, i.e., \([\Delta _e]_{ab} \varTheta _a=[[\varTheta _a]]\) \((1\le a\le S_e,\, 1\le b\le 2S_e)\). Analogous expressions hold for the test functions \(\delta U_K , [[ \delta U_K ]] \), \(\delta \varTheta \), and \([[ \delta \varTheta ]]\).

*N*stands either for

*N*(

*e*) for a bulk element, or for 2

*S*(

*e*) for an interface element. Expressions for the matrix operators are detailed in Appendix.

*N*stands for either

*N*(

*e*) for the bulk elements, or 2

*S*(

*e*) for the interface elements, Eqs. (23) and (24) combine as:

*P*] has been used, see again Appendix for more details.

### 4.2 Discretized weak form of moisture diffusion

The discretization of the weak form for moisture diffusion is derived by introducing a finite element mesh of the mid-surface *S*(*p*) of the encapsulant layer. In principle, since a staggered geometrical multiscale solution scheme is adopted, the spacing of the mesh used to solve moisture diffusion can be different from that used for the discretization of the thermo-mechanical problem. In that case, a projection of the nodal temperatures from the discretized thermo-mechanical problem to the nodes of the mesh used to solve moisture diffusion has to be performed via a suitable interpolation scheme. In the sequel, without any loss of generality, we consider a finite element discretization for moisture diffusion coincident with the middle surface discretization of each interface element, i.e., \(S(p)\approx \bigcup _e S_e\).

*t*is

*B*] is time dependent because it contains the diffusivity

*D*which changes with time according to (16).

## 5 Proposed numerical solution scheme

The algorithm for the proposed time stepping method with a staggered scheme is detailed in Algorithm 1. The Newton-Raphson iteration is performed until machine precision is achieved, i.e., up to a tolerance in the norm of the residual vector \(\mathrm {tol}=1\times 10^{-15}\).

## 6 Application to photovoltaics

In this section we propose the simulation of the two tests prescribed by international standars [4], namely the *damp heat* test and the *humidity freeze* test. While the former allows the complete uncoupling between the solution of the thermo-mechanical problem from moisture diffusion and allows the derivation of a closed form solution useful for benchmarking, the latter requires the present fully-coupled solution scheme. Moreover, the role of a temperature-dependent diffusion coefficient and the role of cohesive cracks in Silicon are investigated, in comparison to experimental results.

### 6.1 Damp heat test

Let us consider a laminate of span \(L=125\) cm and made of a Glass-Glass structure separated by EVA as in Fig. 6. The thickness of each glass is 3 mm, while the thickness of the EVA is 0.5 mm. In this laminate, moisture is diffusing from the free edges towards the centre, since glass is not permeable to moisture.

As far as the initial and boundary conditions are concerned, let us consider the prescriptions by international standards [4] for the *damp heat* test, that is, a constant temperature of \(85\,^{\circ }\)C and an air relative humidity of \(85\%\). This relative humidity corresponds to a moisture content \(c^*=0.55\) g/cm\(^3\) imposed at the free edges of the laminate, i.e., for \(x_1=0\) and \(x_1=L\), where \(x_1\) is the distance from the left edge of the PV module.

### 6.2 Humidity freeze test

*damp heat*test. In particular, the cohesive properties of EVA have to be updated during the simulation, as well as its diffusivity. More specifically, as far as the Young’s modulus of EVA is concerned, the parameters \(\alpha (T)\) and

*a*(

*T*) are herein considered to be temperature-dependent as experimentally evaluated in [6] and interpreted via a fractional calculus model in [21], see the plot for \(\alpha (T)\) and

*E*(

*T*) in Fig. 9.

*D*(

*T*) for EVA as reported in [12] and shown in Fig. 10. Such trends can be fitted according to the Arrhenius type equation (16).

*G*is the area below the traction-separation curve, the following relation holds:

*G*(

*T*) experimental data can be converted in \(\delta _3^c(T)\) data based on the known temperature dependency of the Young’s modulus of EVA as in Fig. 8a, evaluated for the asymptotic condition of an infinite time. Based on these data, we obtain the correlations shown in Fig. 11 and used as input for the numerical simulations.

Material parameters for the layers

| \(\alpha \) | \(\rho \) (Kg/\(m^3)\) | \(c_{\varepsilon }\) (W/mK) | | |
---|---|---|---|---|---|

Glass | 73 | \(8e-6\) | 2300 | 500 | 0.8 |

Si | 130 | \(2.49e-6\) | 2500 | 715 | 148 |

B.S. | 2.8 | \(5.04e-5\) | 1000 | 300 | 0.36 |

*humidity freeze*test is \(\Delta t=18\) s. At each time step, the Newton-Raphson iterative scheme used to solve the nonlinear thermo-mechanical block has a quadratic convergence and it requires a maximum of four iterations to satisfy the condition of a relative residual error norm less than the machine precision. For given variables computed from the solution of the thermo-mechanical block, the moisture diffusion block is linear and therefore it converges in one single iteration. In terms of CPU time, results are shown in Fig. 16, where the plot of the logarithm of CPU time for the solution of the nonlinear thermo-mechanical block, the moisture diffusion block, and the total time of the staggered scheme is shown for the first 90 time steps of the simulation of the

*humidity freeze*test, corresponding to the first temperature ramp from 0 up to \(85\,^\circ \)C. Simulations have been run on the server Proliant DL585R07 (128 GB Ram, 4 processors AMD Opteron 6282 SE 2.60 GHz, 16 cores). In the case of delamination, which has not been observed in the simulation, the update of the EVA fracture energy based on the moisture content would require a further solution of the thermo-mechanical problem.

This minimodule was then subjected to the *humidity freeze* test inside a climate chamber at Politecnico di Torino, Italy. Electroluminescence images taken regularly during the test (the reader is referred to [22] for more details about this nondestructive technique) show electrically damaged areas (black region) near the crack, see the EL image after 2400 h of testing in Fig. 18a. This electric degradation is presumably induced by chemical oxidation of the grid line deposited on the surface of the solar cell, enhanced by moisture. The presence of a crack appears to be relevant, since all the other solar cells show much lower degradation, mostly in form of dimmer areas at the edges of the solar cells due to moisture diffusion from the backsheet through the EVA interspace.

*humidity freeze*test. The presence of cracks enhances moisture diffusion in the central cell, with a contour plot of moisture concentration that correlates very well with the EL image of the electrically damaged areas (compare the red areas in Fig. 18b having \(c=c^*\) with the black areas in Fig. 18a, confirming that electric degradation can be significantly enhanced by the presence of cracks.

## 7 Conclusions and outlook

A comprehensive finite element computational framework for the simulation of coupled hygro-thermo-mechanical problems in photovoltaic laminates has been proposed. To achieve the computational efficiency required to simulate large scale commercial PV modules consisting of up to 60 solar cells, the EVA encapsulant layers have been modelled as zero-thickness interface elements whose traction-separation relations take into account the complex thermo-visco-elastic rheological response of the polymer as per experiments. Moreover, a staggered solution scheme has been proposed and implemented in FEAP [23] to solve the partial differential equations governing heat transfer and thermo-elasticity in the three-dimensional domain of the laminate, and then predict moisture diffusion in the two dimensional domains represented by the EVA layers and channel cracks in Silicon by considering a material diffusivity dependent on temperature and crack opening. Fractional calculus has been used to model the visco-elastic behaviour of the EVA material layer, while moisture diffusion has been assumed to be of classical type. The use of fractional derivatives in partial differential equations (PDEs) related to diffusion is also a viable modelling approach and it is usually related to a diffusion process taking place across a non-Euclidean domain, see e.g. [24]. However, due to a lack of specific experimental evidence of fractal diffusive domains, the classic PDE for diffusion has been considered in the present study.

The proposed methodology has been successfully applied to the simulation of the qualification tests required by the International Electrotechnical Commission [4], namely the *damp heat* test and the *humidity freeze* test. In the latter, due to a continuous variation of temperature during the test, we have shown that coupling between the thermo-mechanical field and moisture diffusion has to be taken into account to correctly predict the spatio-temporal evolution of moisture in the PV module. Moreover, percolation through channel cracks in Silicon solar cells has been found to significantly enhance moisture diffusion. This trend has been confirmed by experimental tests performed by the present authors and showing an increased electric degradation in cracked solar cells with respect to the intact ones.

## Notes

### Acknowledgments

The authors acknowledge the European Research Council for the support to the ERC Starting Grant “Multi-field and multi-scale Computational Approach to Design and Durability of PhotoVoltaic Modules” - CA2PVM, under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 306622. The authors would also like to acknowledge Dr. M. Köntges from the Institute for Solar Energy Research Hamelin for providing the cracked PV module, and the Ph.D. students I. Berardone and A. Infuso at Politecnico di Torino for the electroluminescence images taken during the accelerated aging test.

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