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 A model for screen utility to predict the future of printed solar cell metallization Results and discussion . Introduction of the screen utility index . In order to generalize the screen opening pattern, a relationship between the screen manufacturing parameters (e.g. screen opening width w n , mesh count MC, wire diameter d and the screen angle φ) and the resulting opening pattern needs to be derived. Figure? 2 presents a way to deconstruct the screen opening channel, resulting in a single dimensionless parameter which describes the general utility of a screen. The screen utility index SUI is constructed in three parts: Figure 2 Definition of the screen utility index SUI, describing the relationship between the area of individual openings and the area of individual mesh bridges weighted by the amount of mesh units per screen opening channel. Full size image 1. The average area of an individual opening defines how much fluid is able to transfer through the screen opening channel w n at given pressure. An analytical solution for the dependency of this area A Ind.Opening on all described screen manufacturing parameters is unknown and therefore, requires the presented simulation approach. The average size of all openings is directly linked to the printability. 2. In order to quantify the impact of the underlying mesh on the screen performance, the area of a single mesh bridge A bridge is defined by Eq.?( 4 ). Using a fine mesh (high mesh count MC or small wire to wire distance d 0 ) will increase the screen lifetime due to an increased wire intersection coverage 15 . Furthermore, the screen tension is increased, improving the screen snap-off mechanics to minimize spreading effects 15 , 18 . On the other hand, the wire diameter d should be minimized because the paste transfer is strongly limited by a blocking cylindrical object. Whitney et al. analyzed the force–velocity relationship of a rigid cylinder moving through a highly non Newtonian fluid, showing that commonly used metal pastes with a flow index n? ? ?1 require forces more than one order of magnitude higher than for Newtonian flows with equal velocities and geometric conditions 24 , 25 . This scenario applies directly to the screen printing process during screen snap-off and indirectly to the flooding phase 3 . Combining both statements for the wire-to- wire distance d 0 and the wire diameter d, will lead to the conclusion to minimize the area of a mesh bridge A bridge . $${\text{A}}_{{{\text{bridge}}}} = {\text{d}}_{0} \cdot {\text{d}}{.}$$ (4) 3. The dependency of the amount of contributing mesh units within a screen opening channel on the screen angle φ is presented in Eq.?( 5 ). Due to this relationship, the ratio between the average area of individual openings A Ind.Opening and the area of single mesh bridges A bridge must be multiplied by the corresponding factor 1/cos(φ) in order to account for the decreased angle dependent number of mesh units per screen opening channel at nonzero angles. Each mesh bridge contributes to the expected stability of the emulsion edge during printing because it acts as a micro foundation. $$\frac{{{\text{n}}_{{{\text{mesh \;units}} \varphi }} }}{{{\text{n}}_{{{\text{mesh\; units}} \varphi = 0^\circ }} }} = \frac{1}{\cos \varphi }.$$ (5) Finally, the definition of the dimensionless screen utility index SUI can be given in Eq.?( 6 ) by combining the presented three statements. Any screen configuration, defined by its 2-dimensional geometric parameters has one specific screen utility index SUI. However, there is an infinite amount of theoretical screen configurations which result in the exact same value of the SUI. $${\text{SUI}} = \frac{{\overline{{{\text{A}}_{{{\text{Ind.Opening}}}} }} }}{{\cos \varphi \cdot {\text{d}} \cdot {\text{d}}_{0} }}.$$ (6) Following this statement gives rise to a classical optimization problem. What value for the SUI is good enough to ensure printability in respect to a fixed reference fluid or paste? In order to answer this question, we must analyze the special case where SUI?=?1 applies. In that case, the average size of an individual opening is equal to the area of a blocking wire bridge, weighted by the amount of mesh units across a screen opening channel. This relationship puts the impact of the mesh into context to the resulting screen opening channel, finding a balance between a fine mesh, optimized for high screen tension as well as the screen life time, and the task of providing a sufficient paste transfer at the desired screen opening width w n . In the regime where SUI??1 applies, the underlying mesh is fine enough to create a sufficient screen opening pattern and therefore not limiting the paste transfer more than the angled screen opening channel width w n would have done anyway. It must be noted that the special case of SUI?=?1 mainly applies for a homogeneous fluid which is either particle free or contains a particle size distribution where the majority of particles are small compared to the individual opening size. Therefore, we suggest a threshold for the ratio between the size of the majority of particles (e.g. particles with a diameter smaller than d 99% , assuming a normal distribution) and the individual average opened area in Eq.?( 7 ). $$\frac{1}{4}\pi {\text{d}}_{99}^{2} \ll \overline{{{\text{A}}_{\text{Ind.Opening}} }} .$$ (7) If the d 99% value of a highly filled suspension (e.g. metal pastes for solar cell metallization) becomes too big, certain small openings included in the nominator of Eq.?( 6 ) are not able to contribute anything to the overall paste transfer due to immediate clogging by individual particles or agglomerates. As soon as this effect cannot be neglected anymore, the threshold of the minimal required SUI value for a sufficient printability (SUI min ) becomes a function of the clogging probability itself. At this point, we are going to suggest an empirical value for SUI min for commonly used high temperature Ag-paste for PERC front-side metallization in the following experimental section. In order to further model the correlation between the SUI min and the clogging probability of individual opened areas within a screen opening channel, an experimental method to measure the clogging event during the screen printing process needs to be developed. At this point, the mechanics of the screen printing process do prevent an easily accessible method for direct measurements. Predictability of the simulation approach . In Fig.? 3 , we present the experimental verification of the simulation approach by comparing simulated values for the area of individual openings to measurements of the corresponding area by microscope images. As described in section “ Experimental verification of the simulation approach ”, two different commercial available screens with a screen angle φ?=?22.5° are used. The first screen has a mesh count MC?=?360?1/inch with a wire diameter d?=?16??m (w n ?=?40??m) and the second screen is made out of a mesh, using a MC?=?380?1/inch with a wire diameter d?=?14??m (w n ?=?30??m). The presented deviation between measured and simulated sizes for the exact same individual opening is not caused by the simulation approach itself, rather than resolution limitations of microscopy. Furthermore, manufacturing tolerances of w n , d, φ and further deviations due to the mesh calendaring are causes for the deviation. However, the overall predictability of the presented simulation approach for the area of individual openings is verified and shall be used to obtain values for the presented dimensionless parameter screen utility index SUI because the area of individual openings is the only parameter within Eq.?( 6 ) which must be simulated by this model. Figure 3 Experimental verification of the simulated area of individual openings. For this verification, 27 microscope images per screen of the screen opening channels were taken and analyzed regarding the area of individual openings. Two screens with different mesh types are chosen. The first mesh has a mesh count MC?=?380?1/inch and a wire diameter d?=?14??m (open symbols) with w n ?=?30??m and the second mesh has a mesh count MC?=?360 1/inch and a wire diameter d?=?16??m with w n ?=?40??m (closed symbols). The screen angle of both screens is φ?=?22.5°. The deviation of measurements is caused by manufacturing tolerances of $${\text{w}}_{{\text{n}}}$$ , MC, d and $$\varphi$$ , the mesh calendaring and an insufficient resolution of microscope images. Full size image Correlation of the SUI with screen printing experiments . In Fig.? 4 , we present experimental data from our previous publication 4 , demonstrating the impact of the SUI on screen printed metallization by increased lateral finger resistance R L . For this experiment, the paste sample, the emulsion height EOM, the rate of calendaring of the mesh and all printing parameters has been kept constant. In “ Results and discussion ”, we have discussed the theoretical meaning of SUI?=?1, highlighting the change when the mesh starts to contribute significantly to the limitation of further paste transfer. The presented data supports this critical point where SUI?=?1 applies and further shows that even approaching SUI?=?1 will have consequences in terms of significant increase of the lateral finger resistance R L and thus reduced cell efficiency and non-optimal silver consumption. In order to understand this, we elaborate on the underlying optimization problem of solar cell metallization. The shading of the active cell area by the metallization grid is determined by the cell layout (e.g. number of busbars), the interconnection concept, the finger geometry (mainly the width) and the number of fingers. An increase in shading losses directly results in a significant reduction of the short circuit current density J sc and subsequently solar cell efficiency. As these shading losses of the grid should be minimized, one must also consider the series resistance contribution of the grid 26 . Here, the lateral finger resistance as well as the contact resistance at the metal–semiconductor contact plays a crucial role. The latter is mainly determined by paste formulation, configuration of the firing process as well as actual properties of the solar cell precursor. However, on the other hand, the lateral finger resistance for a given paste is predominantly determined by the geometry of the printed structure and therefore strongly correlates with printing results. The finger resistance increases whenever the cross-sectional area of a contact finger is locally reduced across its length due to insufficient printing. In our previous publication we calculated the maximal tolerable lateral finger resistance for different interconnection concepts when a maximal finger series resistance contribution of r s ?=?0.1 Ω?cm 2 is assumed 27 showing that there is hard limit for the maximal tolerable lateral finger resistance per given cell layout and interconnection concept. On top of that, the overall goal for the metallization process remains always to minimize silver consumption while meeting the described performance requirements. When we now come back to the screen utility index SUI, we must consider the correlation between the SUI and experimental data for the lateral finger resistance for each paste separately. As discussed in section “ Introduction of the Screen Utility Index ”, if one uses a highly filled suspension for which Eq.?( 7 ) does not remain true, the margin for minimal SUI shifts towards higher levels because the effective average area of individual openings is reduced due to clogging by single particles or agglomerates. Based on the presented data in Fig.? 4 , we suggest for high temperature Ag-pastes (HT-Ag), used for PERC front-side metallization, a minimal threshold of SUI min ?(HT-Ag)?=?1.25. Furthermore, for low temperature Ag-paste (e.g. metallization of HJT solar cells) we predict a suitable margin of SUI min ?(LT-Ag)?>?1.6 and for Al-paste for rear side metallization of bifacial PERC, we predict a necessary margin of SUI min ?(HT-Al)?>?1.9. However, to this date, there is no specific evidence for the last two predictions. Furthermore, we would like to point out that small deviations on screen configurations due to manufacturing tolerances might cause a deviation of the SUI, negatively (or positively) influencing the expected printing result further. In future studies, these deviations should be experimentally investigated in order to directly link manufacturing tolerances to a potential reduction in printability. Figure 4 Correlation of the average finger resistance on the SUI value. The predicted change of printability at a SUI?=?1 is supported by experimental data. For values where SU?I??1 applies, the mesh will not limit the paste transfer more than natural limitation of the screen opening channel w n . Different screens are plotted for 24??m, 21??m, 18??m and 15??m screen openings. The data is taken from our previous publication 4 . Values for the 380/0.014/22.5° screen are taken from 3 . Full size image In Fig.? 5 , we present accumulated data from successful screen printing experiments at Fraunhofer ISE (Freiburg, Germany) for Si-solar cell metallization over the last ten years, demonstrating the evolution of the SUI in a research environment. A variety of different mesh counts MC and wire diameters d has been used to ensure printing through an ever decreasing screen opening width w n . However, without realizing it at the time, the SUI has been reduced over the years, indicating that mesh manufactures where not able to keep up their development of finer meshes with the reduction rate of the screen opening width d w n / d t. Nevertheless, the absolute value for the SUI over the years was still suitable for mass production because not even the SUI?=?1.25 limit was passed. This offers a potential explanation why the evolution of published results for printed finger width over the last 15?years was achieved at an outstanding reduction rate of more than 7??m per year 28 . Further paste development was enough to drive this evolution as SUI values during that time span were far beyond SUI?>?1.25, revealing that the screen was never the limiting factor when it comes to printability. In Fig.? 7 , we present the gap between the constant blue line and actual evolution of the SUI, giving a qualitative measure for this contribution of the paste development. Those improvements on the paste printability were able to compensate the (at the time) hidden reduction of the SUI. On the other hand the red line shows the theoretical evolution of the SUI when no mesh improvements since 2010 would have been achieved at all. The gap between this trend and the actual evolution gives a qualitative measure for the mesh development. Especially in recent years, the paste development has an increasing impact on the further reduction of the achieved finger width. Figure 5 History of screen printing experiments at Fraunhofer ISE using screens with the shown screen utility indices SUI. The progression towards smaller SUI values shows the natural evolution of the fine line screen printing process for metallization of Si-solar cells. The nominal screen opening width was continuously reduced over the years. The development of finer mesh patterns was not able to keep up with this trend, resulting in an average SUI reduction of approx. 0.05 points per year. In 2019, Fraunhofer ISE has challenged the screen printing process with an intense reduction of screen opening structures down to 15??m. Full size image Furthermore, we would like to highlight the fact that the overwhelming industry standard for the screen angle φ?=?22.5° was dominating even the research activities in a way that almost no data for different screen angles φ are available 29 , 30 , 31 , 32 , 33 . Only in recent years so 0° knotless screens and 30° angled screens have been investigated 4 , 34 . Figure? 5 further reveals that in recent years we have challenged the screen printing process to the point where usual screen architectures fail completely. In 2019, significant reductions of the screen opening width w n from initial 27??m towards a novel test pattern with screen openings ranging from 24??m to only 15??m have cut the resulting SUI almost in half. This result highlights how the mesh development is a critical step of overall screen development. Therefore, the rate in which screen manufactures decrease the screen opening width w n should not be done as quickly as possible as it requires a strong communication with mesh manufactures beforehand. Simulation of the SUI . Optimization of the mesh count MC and wire diameter d . Figure? 6 shows the SUI dependency on both mesh parameters for a screen opening width of w n ?=?20??m with a screen angle φ?=?22.5°. As discussed in section “ Introduction of the Screen Utility Index ”, the nominator of the SUI is depending on the wire to wire distance d 0 and the wire diameter d itself, resulting in the presented nonlinear relationship between the SUI and the mesh parameters. This result gives rise to a classical optimization problem because a mesh with a very low mesh count MC and a small wire diameter d will maximize the SUI, but at the same time minimizes the screen stability due to Eq.?( 3 ). In order to highlight this circumstance, we have added red curves for constant SUI values as well as curves for the maximal possible screen tension γ screen_max ?=?20?N/cm for stainless steel and tungsten alloy wires. The intersection of a constant SUI line (e.g. SUI min ?=?1.25) with the curve for the constant screen tension gives the minimal requirement for the mesh in terms of minimal mesh count MC and maximal wire diameter d at which the SUI?>?1.25 threshold is fulfilled. If an intersection point between the constant SUI line and the maximal possible screen tension curve for a given wire material does not exist, the desired configuration is physically impossible. In such a case, a new wire material with an increased ultimate tensile strength σ uts_wire_mat needs to be developed. This approach reveals the threshold at which a given wire material with an ultimate tensile strength σ uts_wire_mat is able to fulfill the requirements for a screen with a certain screen tension γ screen (as long as γ screen ??SUI min . Figure 6 The SUI is simulated for 68,600 different mesh configurations with a screen opening width of w n ?=?20??m at a screen angle of φ?=?22.5°. In red, constant SUI lines are presented, highlighting the minimal threshold at SUI?=?1. Further, the SUI?=?1.25 line is shown, indicating the minimal barrier for a sufficient printability of commonly used Ag-pastes. The black and grey lines indicate a constant screen tension for tungsten alloy and stainless steel wires at γ screen ?=?20?N/cm. The intersection point of the SUI?=?1.25 line with the screen tension function defines the optimal mesh choice. Full size image In Fig.? 7 , we present a suggestion for future wire materials by plotting Eq.?( 3 ) for a broad range of mesh counts MC and wire diameters d with a screen tension γ screen ?=?20?N/cm. Furthermore, we are adding constant SUI lines which highlight the need for further developments of novel wire materials because commercial available wire materials like stainless steel and tungsten alloys already show very limited capabilities for further reduction of the screen opening width w n . For example, there exist different types of fiber glass with high ultimate tensile strengths which could be a suitable option for woven mesh wires 35 . Usually, the glass of such fibers is amorphous, providing a homogenous structure along and across the fiber, however the production of these fibers at diameters below 10??m is challenging, because even small scratches on the surface will dramatically influence the mechanical properties 36 . On the other hand, there exist fibers made out of carbon. They are widely used in the industry to produce strong and ultra-light components for a broad range of applications. Kumar et al. tested single carbon fibers with diameters down to 7??m, reporting ultimate tensile strengths of up to 3200?MPa 37 . Arshad et al. produced carbon fibers with an electrospinning approach, with diameters below 0.5??m and an ultimate tensile strength in the range of 4500?MPa 38 . Finally, if we further examine the thought experiment of using the finest possible “wire” with the maximal obtainable ultimate tensile strength, we will eventually arrive at Iijima and Ichihashi, who published the discovery of carbon nanotubes in 1993 39 . Takakura et al. measured for the first time the ultimate tensile strengths of an individual structure-defined, single-walled carbon nanotube with values for the ultimate tensile strength ranging from 20?GPa to over 50?GPa 40 . Furthermore, Zhang et al. was able to produce over 50?cm long carbon nanotubes by a floating chemical vapor deposition process in 2016 41 , making an industrial application for mesh production potentially a matter of years rather than multiple decades. The ultimate tensile strength of one these potential wire or fiber materials need to be higher than the minimum requirement for the desired SUI. For example, if a screen with a nominal screen opening channel of only w n ?=?5??m is manufactured, the underlying mesh cannot be made out of conventional wires. New technologies for the mass production of very thin and strong wires or fibers (e.g. carbon nanotubes) have to be developed in order to prevent an upcoming dead end of ultra-fine line metallization of Si-solar cells. Figure 7 The simulation of the ultimate tensile strength σ uts_wire of individual wires. The mesh parameters are varied between 100–1500?1/inch for the mesh count MC and 1–50??m for the wire diameter d. A fixed screen tension of 20?N/cm was used. Furthermore, constant lines for certain materials are shown and discussed (e.g. stainless steel, tungsten, glass fibers, carbon fibers, carbon nanotubes). Finally, constant SUI lines are given for different desired screen opening channels w n. Full size image Optimization of the screen angle φ and screen opening width w n . In Fig.? 8 on the right, a full simulation of the SUI for screen angles between 0°–45° and screen opening channel width w n between 5 and 40??m are shown, revealing a nonlinear relationship between the SUI and the screen angle φ. Red lines highlight constant SUI curves, including the SUI?=?1.25 margin discussed earlier. The common industry standard of φ?=?22.5° results in one of the worst configurations if a reduction of w n is desired. The screen manufacturer should switch to reducing or increasing the screen angle to avoid the regime where the SUI shows the strongest reduction with further reducing w n . However, increasing the screen angle will also increase the total area of mesh per screen required due to increased cutting losses during production. These cutting losses contribute significantly to the overall production costs and should be minimized. The industry produces meshes on weaving machines, creating a “mesh carpet” on a roll with a width of usually 1?m. Afterwards, single sheets of mesh are cut out of this mesh roll. Figure 8 On the left, the screen utility index is simulated for all possible screen angles—screen opening width w n combinations between 0°??SUI min , the chosen wire material does not offer a solution for the desired printed finger width w f and screen tension γ screen . In such a case, the designer has to research for new wire materials or reevaluate its initial technological and economic decision for the smallest available wire diameter. Figure 10 Design approach for the definition of the 2-dimensional screen architecture. In order to reach a desired printed finger width w f , a series of design choices has to be made. First, the screen opening width w n needs to be derived by rheological investigation of the paste at the desired printing speed and determination of the expected spreading offset. Furthermore, the margin for the SUI (e.g. SUI min ?>?1.25), the smallest available wire diameter for mesh production and the desired screen tension γ screen are defined. After calculating the mesh count MC by Eq.?( 3 ), the SUI min requirement is controlled. Depending on the result, the screen angle φ is chosen by minimizing the cutting losses, defined by Eq.?( 9 ). If no configuration is available which fulfills all requirements, new wire materials need to be developed. Full size image . From：