Evolutionary algorithm optimizes screen design for solar cell metallization

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Introduction
Flatbed screen printing is the process of choice for the metallization of Si-solar cells with over 95 % market share because of it's reliable and low cost production capabilities [1].The metallization step is a crucial part of the entire Silicon solar cell production chain because front-side printing pastes contain a high mass share of silver, making them one of the biggest cost contributors in the mass production PV market [1].The goal of screen printed metallization is to apply fine Ag-electrodes onto the wafer surface with an optimal trade-off between maximizing grid conductivity and minimizing the shaded area on the cell while keeping Ag-laydown as low as possible [2,3].In previous research, we have discussed how screen printing of Ag-pastes has become a tremendous success story by decreasing the printed electrode width from approximately 120 μm in 2007, reported by Mette [2] towards only 20 μm reported in 2020 [4].This was achieved by continuous efforts of the industry and several research groups to further advance the understanding of the relationship between flow characteristics of highly filled pastes and screen architectures [5][6][7][8][9][10][11][12][13].However, industrial screen printing lacks behind the scientific results by at least 10− 15 μm because printing speeds are much higher, resulting in additional spreading of Ag-electrodes due to the nature of the highly non-Newtonian flow characteristics of commonly used metal pastes [3].Furthermore, those scientific studies use specially designed R & D screens optimized for printability while lacking sufficient stability during mass production [4,14].The international roadmap for photovoltaic (ITRPV) predicts that industrial screen printing at competitive throughput rates will catch up with recent scientific demonstrations of Ag-electrode width below 20 μm by 2030 [1].This will directly lead to one of the fundamental challenges in rating screen designs: quantifying their actual objective.A screen for fine-line printing should be designed in such a way that a highly engineered printing paste is able to penetrate the opening structure in a precise and reproducible manner.At the same time, the manufacturing cost of fine-line screens needs to be as low as possible while maintaining or even increasing its life cycle during production.The required magnitude of precision in today's screen printed metallization industry has become so high that individual wires within the screen opening structure have become the non-avoidable bottleneck for obtaining the described objective.Fig. 1 shows a SEM image of a regular screen with its specification parameters, the channel width w n , the screen angle φ, the wire-to-wire distance d 0 and the wire diameter d [15].Further, a tiled side profile of a screen printed Ag-electrode on a silicon solar cell at a width of w f =20 μm is shown.It becomes clear that the opening structure of the corresponding screen leaves its imprint on the electrode geometry, limiting further optimization of the trade-off between sufficient grid conductivity and Ag consumption because the minimal cross-sectional area of the electrode directly correlates with the lateral electrode resistance.Proposed ideas for minimizing the number of wires within a screen by the introduction of so-called knotless screens showed promising results in respect to this problem.However, screen stability and subsequently the life cycle are not competitive to this date [16,17].Another way to change the opening structure and the corresponding number of exposed wires is the configuration of the underlying mesh itself.These meshes are usually woven into carpets at widths around 1-1.2 m.Production speed per weaving machine is usually mechanically limited, making a scaling of mesh production challenging.Therefore, the industry around mesh production is often committing to specific specifications of the mesh month or even years in advance of market entry.In our previous research, we have introduced a screen simulation that is able to predict the exact opening structure of any arbitrary screen specification [4,15].This simulation enables us to determine the shape, size and frequency of occurrence of any individual small opening within the entire screen opening structure.Further, we have utilized that data by deriving global parameters to quantify the overall quality of the screen opening regarding the described objective [9].
In this study, we present an evolutionary algorithm which utilizes this screen simulation tool in order to grade screen architectures for the purpose of selection and reproduction.An evolutionary algorithm is a stochastic approach to solve various search and optimization problems [18].It is designed after biological evolution which is driven by natural selection [19].In general, these algorithms have fixed number of individuals competing in an artificial exercise (e.g.simulation) for the right of passing on its characteristics (genes) to the next population.Over the course of many generations, exactly those characteristics will dominate which are able to complete this exercise most effectively.We use this approach to find promising screen candidates that satisfy a decent trade-off between sufficient screen stability while maintaining the required printability at an expected ever decreasing opening width w n .Further, we run a computational fluid dynamics (CFD) simulation in order to predict the required thickness of the emulsion layer (emulsion over mesh EOM) for all candidates presented by the evolutionary algorithm.Combining both simulations, screen architectures for opening widths down to only w n =6 μm are proposed, presenting the industry around screen production an indication for future designs well into the 2030s.

Theory on screen design & simulation approach
The architecture of a screen opening is shown in Fig. 1.The structure is defined by the properties of the underlying mesh, thus the mesh count MC and wire diameter d, and the two screen parameters opening width w n and the screen angle φ.Furthermore, the EOM (emulsion over mesh) is defined as the thickness of the corresponding channel below the mesh.Based on these parameters, the so-called relative opening area OA % is calculated by Eq. 2.1, describing the average ratio between opened and total area [20].
In order to calculate OA % , the wire-to-wire distance d 0 has to be calculated by using Eq.2.2 [20].
Furthermore, the maximal possible screen tension γ screen_max in Eq. 2.3 is critical for a high screen life cycle during production [20]. (2. 3) The ultimate tensile strength σ uts_wire_max describes the maximal tensile strength before a particular wire will break while being stretched.Common wires in industrial application are made out of stainless steel with an ultimate tensile strength σ uts_wire_max ≈ 800 N/mm 2 [21].More, recently tungsten alloy wires are used with a ultimate tensile strength in the range of 1200 N/mm 2 [4,14,22].
In previous work, we have introduced and analyzed the dimensionless parameter screen utility index SUI presented in Eq. 2.4 [4,9].
This parameter quantifies the relationship between the printability of a screen and its durability.Each screen opening is the sum of individual openings separated by the underlying mesh and emulsion edges.The average size of these individual openings A Ind.Opening is fundamentally limiting the paste transfer and should be maximized.In previous research, we have discussed how increasing the average size of a blocking wire element d⋅d 0 will further limit the paste transfer by increasing the separation between openings [9].However, by decreasing the wire-to-wire distance d 0, the average lifetime of the screen is increased.Following this, the screen utility index SUI is calculated by calculating the ratio of the two, multiplied by a correction term that accounts for the angle dependency of the number of mesh units within the screen opening.To this date there is no analytical solution for calculating A Ind.Opening .Therefore, we present in Fig. 2 an overview how a virtual clone of arbitrary screen openings is created.This clone grants access to a comprehensive data set, including the size of each opening.The mathematical model behind this algorithm and its experimental verification is well described in previous work [15,17].

The evolutionary algorithm
Fig. 3 presents the flow chart of our evolutionary algorithm.First, a random starting population of n pop = 250 different screen configurations (further called "screen individuals") is created.The genes of each screen Fig. 1.On the top, a SEM image of regular screen opening is shown, highlighting the 2-dimensional geometric parameters wire-to-wire distance d 0 , wire diameter d, screen angle φ and opening width w n [15].On the bottom, a printed Ag-electrode on a silicon solar cell is shown, demonstrating how single mesh wires cause significant local deviation of the electrode height, thus limiting grid conductivity.
individual are randomly set to a value within the global boundaries for each parameter.This is done for all screen individuals of the initial population.In the next step, the corresponding fitness function is calculated for all individuals.The population is then ranked based on fitness from high to low values.Afterwards, the probability to be selected as a winner p w (i) within that given generation is calculated for all individuals by Eq. 3.1.
Following this, a full trial is run to determine all winners based on the winning probability p w (i).Each winner then selects randomly (with even probability) another winner as a partner for reproduction.In the next step, a number of n w pairs of winners are created which engage in a crossover sequence followed by a probabilistic mutation sequence.The crossover is done by randomly choosing each gene from one of the  parents for all genes in the genome G.During that procedure, the initial ranking of the two parents are not considered.Afterwards, a mutation sequence is added to prevent limited gene diversity and subsequent convergence in local maxima of the fitness function.Each gene will mutate with the probability p mut which is defined as a global parameter for the algorithm.Once a gene has been selected for mutation, the value for that gene will be randomly changed to a new value with a maximal relative deviation a mut towards its initial value.Further, the global limits for each parameter are always met.This reproduction procedure is now repeated for all winners, creating a list of offspring individuals.All offspring's are sorted by their calculated fitness in order to be ranked from low fitness to high fitness.Following, the offspring with the lowest fitness value first replaces a random individual from the loosing group.This is repeated for the entire list of offspring's in the described order.This procedure will naturally create a scheme where offspring's with lower fitness have a greater chance of immediately being replaced by better offspring's.How strong this effect turns out depends on the trial run for winner determination based on Eq. 3.1.Finally, the algorithm is repeated for n gen generations in order to create a global shift of the average genome G towards the absolute optimum of the parameter space.

Objective of the evolutionary algorithm
In section 2, we have described the geometric description of a typical screen.In order to use this definition as bases for screen optimization by an evolutionary algorithm, a string of genes is defined.Eq. 3.2 describes the genome G of a screen individual as a string of 7 genes.Table 1 summarizes the corresponding connection to real world parameters including the global range for each parameter.For this study, it is necessary to set limitations on the maximal range of certain genes in order to keep the algorithm's objective to solve real world problems instead of moving immediately into theoretical constructions.In future studies these extreme cases could be explored separately.As shown in Table 1, especially the mesh count MC and wire diameter d are restricted to values which are suitable for metal based wire materials.When the ultimate tensile strength of the corresponding wire material σ uts_wire_mat is not fixed, the algorithm may choose to derive architectures based on material compositions or configurations which are probably decades away from industrial application such as carbon nanotubes (see our previous study for more details [9]).In order to verify the algorithm, three different verification runs are executed with the fitness function presented in Eq. 3.3.
This simple fitness function has the objective of maximizing the opening rate as presented in Eq. 2.1.For the first run V1, only the mesh count MC and the wire diameter d are non-fixed genes, leading to the trivial solution of minimizing both variables.In the second verification run V2.1, the underlying mesh is fixed at a MC = 480 1/inch and a wire diameter d = 11 μm.Here, only the screen angle φ and the position of the screen opening on the mesh κ are non-fixed genes.The objective remains the same, leading to the mathematical solution of a knotless zero degree (φ = 0 • ) screen for which Eq. 3.4 applies.
The position of the screen opening in respect to the mesh κ must be placed exactly between two parallel wires.Further, Eq. 3.4 implies that the wire-to-wire distance d 0 is greater than the opening width w n .Otherwise, the screen configuration at φ = 0 • for which the knotless requirement remains true becomes mathematically impossible.Depending on the possible magnitude for gene mutation a mut , the algorithm may not be able to fully approach the absolute minimum value for the screen angle φ = 0 • because the set minimum within that simulation is the absolute physical/mathematical minimum itself.Therefore, another verification run V2.2 is executed.In this run the corresponding range for the screen angle gene is set between 0 -φ min (V2.1),allowing the algorithm to explore the regime close to the mathematical boundaries of the geometric problem.
Finally, in run A-C the objective of the algorithm is defined by the presented fitness function in Eq. 3.5.All screen individuals are competing for survival by maximizing their fitness F. The first exponential of Eq. 3.5 minimizes the difference between the screen utility index SUI of a particular screen individual and the nominal screen utility index SUI n = 2.The second exponential sets the screen tension to its nominal value γ n = 20 N/cm in order to guarantee sufficient snap-off mechanics.By defining the fitness function F o as a product of two exponentials, the evolutionary drift is very steep.Any deviation from the optimal parameters γ n and SUI n is punished by rapidly decreasing fitness value.

CFD model description and governing equations
In the presented CFD model, a flow of a commercial Ag-paste around an individual mesh wire is simulated.The goal of this model is to predict the necessary emulsion thickness for reestablishing a homogenous flow below the mesh.Commonly used Ag-pastes for solar cell metallization show non-Newtonian flow characteristics [23][24][25].They contain a high mass share of spherical Ag-particles with sizes around 1-3 μm [26].A liquid phase is added, containing binders, thixotropic agents and solvents to control the rheological behavior.These pastes show a strong shear thinning behavior after exhibiting a significant yield stress.Their shear rate dependent viscosity is well described by the power law model S. Tepner et al. [27,28].For this model, a creeping flow is used which is commonly described as a flow where the corresponding fluid velocities are relatively slow, the dimensions of the flow containing boundaries are small and fluid viscosities are high.Therefore, resulting Reynolds numbers are very small in creeping flow (Re < 1) which leads to neglecting the inertial term in the Navier-Stokes equation.Using this flow, the so-called stokes equation for conservation of momentum (eq.3.6) and the continuity equation (eq 3.7) for conservation of mass have to be solved in space and time.However, in this study, a 2-dimensional time-independent model is used, reducing the Stokes equation to the form presented in Eq. 3.6.Owing to the small scale of interest throughout this study, the equations do not consider gravity.Further, this model for fluid flow requires some crucial assumptions.First, the fluid is assumed to be a continuum, neglecting particle interactions entirely.It must be noted that this assumption is especially challenging for the presented scale of interested because metal pastes for solar cell metallization usually contain very high mass shares of Ag-particles with sizes in the range of 1− 3 μm.However, the presented study will still give valuable insights on optimization of screen architectures for PERC metallization.Changing the paste particle system may require further consideration of the model.
In terms of flow behavior, a power law model has been chosen.Its general expression for the viscosity function is given in Eq. 3.8.The flow behavior index n dictates the fundamental relationship between viscosity and shear rate.For n < 1, a fluid shows shear thinning behavior whereas for n > 1, the fluid is shear thickening.For the special case when n = 1 applies, the power low model simplifies to a Newtonian flow.The flow consistency index K gives scale to the magnitude of the viscosity, describing the viscosity when the fluid is experiencing a shear rate of γ = 1 s − 1 .When applying the power law fluid to a CFD model, one has to consider the calculation of the viscosity at very low shear rates.Following the power law expression, the viscosity approaches infinity for vanishing shear rates.This behavior is first physically inaccurate for most fluids and further creates challenges for the numerical solver in terms of convergence.Therefore, a minimum shear rate γmin = 0.1 s − 1 is defined and used wherever the model fulfills γ ≤ γmin .
Fig. 4 presents the schematic 2-dimensional (x,z) model in which a circular wire is placed within the flow field of a power law fluid with an homogenous constant velocity of v(x,z) = v 0 .For the power law modeling parameters, a flow consistency index K = 1250 Pas n , a flow behavior index of n = 0.2 and a density of ρ =5400 kg/m 3 are chosen (based on experimental data for common metal pastes from our previous research [3]).The blocking wire element has a diameter of d = {5, 10, 15, 20, 25} with a no slip boundary condition at the walls.Following this, the fluid velocity relative to the wall at the fluid-wall intersection is zero at all times.At a specific distance to the bottom point of the wire, the velocity component in z-direction across the x-axis is obtained.This distance is specified as the emulsion over mesh EOM parameter and varied between EOM = {6, 9, 12, 15, 18}.

Verification of the evolutionary algorithm
In order to verify the evolutionary algorithm, a set of simple fitness functions are evaluated.According to Eq. 2.1 the opening rate OA % depends on the wire-to-wire distance d 0 (subsequently on the mesh count MC) and the wire diameter d.In the verification run V1, the minimal mesh count is set to MC min (V1) = 100 1/inch (therefore: d 0_max (V1) =254 μm) and the wire diameter to d min (V1) = 1 μm.If we put these parameters in Eq. 2.1, we calculate the maximal opening rate OA % according to Eq. 4.1.
Fig. 5 on the left presents the evolution of the mesh count MC and wire diameter d for the verification run V1.The trend reveals quick convergences to d 0_max and d min and subsequently fitness values above F > 0.98 are reached.Taking a closer look at both genes, shows how the mesh count MC is strongly converging towards the minimum of MC min (V1) = 100 whereas the wire diameter d stays around d ≈ 2 μm.At this point, no clear explanation for this is known.Even though Eq. 4.1 indicates slower convergence for the wire diameter d compared to the wire-to-wire distance d 0 , the presented data shows that enough gene variation for d is present for over 400 generations at which the mesh count MC has already been converged to its set minimum.
In Fig. 5, results for the verification run V2.1 (n gen < 500) and V2.2 (n gen > 500) are shown in sequence.The goal was to optimize the opening rate OA % by adjustment of the screen angle φ and the screen opening alignment on the mesh κ.The mesh was kept constant to a mesh count MC = 480 1/inch and wire diameter d = 11 μm.In previous studies, we have discussed how the opening rate is not depending on the screen angle with the exception of specific knotless configurations.The strongest exception regarding the opening rate OA % occurs at a screen angle φ = 0 • when a knotless alignment on the screen is created.This requires that both channels defining emulsion edges are exactly placed between two parallel wires as discussed in section 3. Looking at the data in Fig. 5 shows how the mutation rate prevents convergence to the mathematical minimum at φ = 0 • .After a very quick progression towards small screen angles within the first ten generations, the algorithm reaches a local optimum at approximately φ ≈ 1 • .Here, the average magnitude of gene mutation prevents a further reduction, because the chance of mutating the angle towards such small values depends on the maximal possible range.In V2.2, the maximal range is set to 0 • < φ < 1 • , allowing the algorithm to focus on the remaining deviation of the screen angle gene to the mathematical optimum.Furthermore, the alignment of the screen opening on the underlying mesh κ converges to the smallest possible value that obeys Eq. mathematical maximum of the opening rate OA % in terms of choosing the correct mesh configuration as well as the optimal screen opening position on that mesh.

Prediction of future mesh choices
Fig. 6 shows the distribution of screen individuals within the mesh countwire diameter plane.Each figure shows a snapshot of the distribution at different generation's n gen = {1, 10, 100, 500}.Here, the evolutionary objective is set to a screen opening width of w n =9 μm.The competition for survival rapidly removes individuals far outside the constraint for sufficient screen tension and replaces them with better offspring.In all four examples, a population of n pop = 250 is shown, although the data suggests that many have very similar genes.After this initial concentration of individuals, a slower evolutionary drift occurs, maximizing the screen utility while maintaining sufficient screen tension.Even for advanced generations there are still offspring's created outside the hotspot.The pre-defined mutation rate prevents full convergences of the entire population because genes are randomly selected and changed each generation.
In Fig. 7, virtual clones of screen openings at different opening widths w n for generation n gen = 1 and n gen = 500 are presented for optimization run C.It becomes clear, that the algorithm chooses different mesh configurations for different objectives and further converges for all three optimization runs into a knotless structure at φ = 45 • .This result highlights the trade-off between stability and printability.The common knotless 0 • architecture is favorable in terms of printability at the expense of minimal wire intersection coverage at the emulsion edges.In previous research, we have studied this coverage parameter in more detail, demonstrating how an angled screen benefits from a structure with increased wire intersection coverage [17].Here, the algorithm chooses the other extreme with a screen angle of φ = 45 • because the screen utility index SUI heavily favors architectures with high numbers of mesh units interacting with the opening structure to increase stability.
Fig. 8 presents the evolution of the mesh count MC and wire diameter Fig. 5. Simulation results for the verification run V1 (left) and V2.1 and V2.2 (right).In V1, the algorithm searches through the mesh countwire diameter for the trivial solution presented in Eq. 2.1.In V2.1 and V2.2, the algorithm searches for the zero degree knotless optimum.Here, the simulation is divided into two runs because the average magnitude of gene mutation scales with the global range for that gene.the optimization objective.Most evolutionary runs benefited from sudden mutation and its spread across the population.On the bottom of Fig. 8, we summarize the final values for the mesh count MC with corresponding fitness values F. Throughout the last 15 years, various screen opening widths (e.g.w n =50 μm) have been applied to solar cell metallization.Fig. 8 further highlights common screen architectures for these opening widths, indicating a slight underestimation of mesh counts by the evolutionary algorithm compared to common values in the industry.In the past, typical meshes with mesh counts between 200 inch − 1 < MC < 300 inch − 1 and wire diameters between 20 μm < d < 30 μm have been very common and their manufacturing was efficient and cost-effective.The presented simulation results therefore indicate that mesh development was ahead of the minimum requirement for the majority of the last 15 years.It becomes clear that the algorithm's success is heavily dependent on the assigned objective.In optimization run C we set the goal to achieve a screen utility index of SUI min = 2 at a sufficient screen tension of γ screen_max = 20 N/cm.Results of that run indicate that for smaller opening width w n the algorithm fails to converge at decent fitness values because such screen architectures do not exist when tungsten alloy is chosen to be the set wire material.Once different woven techniques for different wire material properties are introduced to the gene pool, new directions for developments might emerge.Furthermore, the mesh configuration itself could become the focus of an evolutionary algorithm by removing constraints such as having the same number of wires at equal diameters in x-and y direction.The algorithm also could address arbitrary opening structures beyond straight lines used for solar cell metallization.In theory, screen utility could be quantified for arbitrary opening structures when certain symmetry rules are reassessed such as those for the screen angle [17,20].thickness EOM min because a sufficient height will reduce the deviation of the printed Ag-electrode height across its length.However, this does not give any information about the maximal possible emulsion thickness EOM max .The squeegee movement during the printing sequence induces a downward pressure impulse on the paste within a flooded mesh unit, resulting in acceleration towards the bottom of the screen opening channel.As soon as the pressure difference vanishes with continuing motion of the squeegee, the non-Newtonian flow characteristic of the paste will quickly lead to deceleration and subsequent reduction of its velocity until any motion stops entirely (adhesion forces to the walls are far greater than gravity).If the emulsion thickness EOM becomes too thick for the pressure induced paste flow during squeegee motion, the screen channel remains only partially flooded.In this case, the paste does not reach full contact with the substrate, preventing any adhesion to the substrate and subsequently any paste-screen separation.Finally, Fig. 9 on the top left shows the full dataset for the presented CFD simulation.The non-dimensional velocity component into z-direction v z /v 0 is given for all EOMd combinations within the pre-defined parameter range of this study.Further, we have defined an arbitrary requirement for a sufficient paste flow below the wire in Eq. 4.2.

Determination of mesh limitation by CFD simulation
The flow velocity v |z− d|=EOM is defined as the velocity component of the flow in z-direction below the wire at a distance equals to the emulsion height EOM.If this velocity is at least 75 % as high as the reference flow v 0 at d = 0, the flow is assumed to be sufficient in terms of the described objective.

Conclusions
Screen printing for Silicon solar cell metallization requires advanced screen designs which enable reliable and fast fine-line printing of highly filled metal pastes.Further, strict requirements on screen life time during mass production as well as a constant pressure to decrease production cost make new developments challenging.For this reason, we presented an evolutionary algorithm for screen design optimization.This algorithm optimizes several mesh and screen parameters in order to improve the screen utility (trade-off between printability and screen stability) by using a genetic selection approach.Typical screen design parameters are defined as genes which then determine the architecture of a particular screen individual.During each generation, all individuals of a population compete for survival by evaluation of their fitness (screen utility).Additionally, we have performed a computational fluid dynamics (CFD) simulation to estimate the optimal emulsion thickness in relation to predicted wire diameters of such winning individuals.This is especially important because a stable and reproducible screen printing process requires sufficient paste flow around single metal wires during the screen flooding sequence.For the CFD simulation, the flow behavior of common metal pastes is described by applying a power law model in stationary flow condition.Flow velocities across the screen opening are obtained at particular distances to the wire which are equal to the emulsion thickness.In Table 2, we present a summary of all simulated solutions for mesh configurations and minimal emulsion height EOM min for three optimization objectives.This variation of different objectives includes an evaluation of stainless steel and tungsten alloy as the chosen wire material.When tungsten alloy is chosen as the wire material with an ultimate tensile strength of σ uts_tungsten_alloy ≈ 1100 N/mm 2 compared to stainless steel with σ uts_stainless_steel ≈ 800 N/mm 2 , the algorithm prefers to use a smaller wire diameter instead of further decreasing the mesh count, respectively.This suggestion is not obvious at first glance because maintaining a specific screen tension while maximizing the printability would lead us to use fewer wires with bigger diameters.However, this can be explained by looking at Eq. 2.3 which states that the screen tension is linearly dependent on the mesh count MC and quadratically dependent on the wire diameter, leading to a better compromise of screen utility when the mesh count is increased at the benefit of using smaller wires.For screen openings below 10 μm, the algorithm always predicts mesh count above MC > 1000 and wire diameters below d < 9 μm.These types of configurations are not available with state of the art mass production of mesh.However, there is an ongoing effort of the industry to further decrease wire diameters.To our knowledge it is a matter of years rather than decades before fine-line screens of such dimension become accessible.Now with the knowledge of optimal mesh choice, our CFD simulation predicts the minimum emulsion layer thickness in order to guarantee a sufficient paste flow around those wires.As discussed in section 4.3, we have set the requirement for a sufficient paste flow around such wires when the flow velocity in z-direction at a distance to the wire equal to the emulsion thickness is at least 75 % of the velocity above that wire.This approach yields us the minimum emulsion layer thickness in respect to the wire diameter as presented in Table 2.In future studies, the maximum emulsion layer thickness needs to be investigated because a sufficient paste transfer requires contact with the substrate during the printing sequence.Depending on process parameters such as squeegee speed and angle, all screen parameters discussed in this study as well as paste properties, the opening channel may only be partially filled during squeegee movement.For this reason, the correct choice for the emulsion layer thickness is especially important when a robust and reliable fineline screen printing is desired.

Fig. 2 .
Fig. 2. Digital clone for screen architectures.In a) the schematic representation is shown.In b) a microscope image of a regular screen opening is shown.c) and d) show the virtual representation of the screen opening and the corresponding quantification of the opening rate as a function of the position.

Fig. 3 .
Fig. 3. Flow chart of the evolutionary algorithm applied for screen optimization.An initial population is randomly created and corresponding fitness values are calculated.Afterwards all individuals are ranked and winners are selected by running a Bernoulli experiment.In the next step, winners choose reproduction partners.The off-spring is created by crossover sequence, followed by mutation.Off-springs replace loosing individuals to form a new population.Tis procedure is repeated for G generations.

3 . 4 .Fig. 4 .
Fig. 4. On the top left, the basic CFD model is shown.A single wire with an adjustable wire diameter is placed at a certain adjustable distance to the measurement line.The flow field expands beyond that measurement line.

Fig. 6 .
Fig. 6.Evolution of 250 different screen individuals on the mesh countwire diameter plane.Different generation's n gen = {1, 10, 100, 500} are shown, indicating the convergences to local hotspots.Here, the screen opening width w n =9 μm is kept constant.

Fig. 7 .
Fig. 7. Virtual clones of screen openings at an opening width of w n ={9, 15, 35}μm for generation n gen = 1 and n gen = 500.Here, results of run C are presented.For all runs, the algorithm choose to use a specific mesh and knotless φ = 45 • screen design.

Fig. 8 .
Fig. 8. On the top, the evolution of the mesh count MC (left) and wire diameter d (right) for optimization run C is shown.Here, seven different screen openings w n are shown, indicating that convergence speed is dependent on the algorithm's objective.On the bottom, a full overview on different mesh counts and corresponding fitness values for all three optimization runs are shown.Further, common mesh options in industry are shown for the given wire materials, respectively.However, to this date, screen openings below 30 μm are barely used in mass production of PERC metallization.

Fig. 9 Fig. 9 .
Fig.9presents simulation results for the flow of a commercial Agpaste around a single wire.On the top left, the magnitude and direction of the flow profile around a wire with a diameter of d = 15 μm is shown.Here, velocities are presented as dimensionless ratio to an arbitrary reference flow for d = 0 μm which fulfills the creeping flow requirement of Re << 0. At a specific distance EOM, the velocity component in z-direction is obtained and displayed on the bottom of Fig.9for different wire diameter and EOM combinations.Comparing these results to a Newtonian fluid with the same viscosity as the presented power law fluid at a shear rate of γ = 1 s − 1 (blue line), indicates how the shear thinning nature of the Ag-paste improves the flow around the wire.The paste experiences an excessive shear rate when hitting the wire boundaries, resulting in a significant drop in viscosity.For very small wire diameters, the flow path around the wire boundaries allows the paste to reunite below the wire very quickly compared to Newtonian flow.Depending on the distance to the bottom of the wire, the velocity profile has only partially recovered its homogeneous nature.This result allows predicting another design requirement for the minimal emulsion

Table 1
Overview on different types of genes per screen individual.Each gene has a specific value range depending on the simulation run.Throughout this study, 3 verification runs and 3 optimization runs are performed.

Table 2
Overview on investigated simulation runs for different screen opening widths w n .The mesh count MC and wire diameter d are derived from the evolutionary algorithm whereas the minimal emulsion layer thickness EOM min is obtained by a CFD simulation.The * indicates results for which only very weak convergence was achieved.