A practical strategy to identify appropriate application patterns for adhesively bonded joints

This paper presents a simple experimental validation of a method for finding optimal application patterns in adhesive bonding, as previously introduced by Flaig et al. (2023). The study uses two examples to demonstrate that the patterns found in theory effectively fill the target gaps after pressing in practice. The results confirm that the identified application patterns fill the target gaps almost perfectly, ensuring excellent joint integrity after the pressing process. In particular, the applicability of the star pattern to fill square gaps is experimentally validated. In addition, the method of creating a rectangle by pressing a cut star connected by a straight beam is also effective. While the theoretical results in Flaig et al. (2023) were already promising, this research now provides experimental evidence of their practical applicability.


INTRODUCTION
To adhesively bond two substrates, whether in a scientific, industrial or domestic setting, users are usually confronted with the question of how the adhesive should be applied to the substrates so that the gap is completely filled after pressing.Ideally, corners should be completely filled, no air should be trapped and not too much adhesive runs out.This is important because incomplete filling may reduce the strength of the joint [1,2] and adhesive leakage is a waste of valuable raw materials and causes unnecessary cleaning.
Finding this optimal initial application pattern for adhesive joints is a challenging task.This is because the adhesive naturally spreads radially during the pressing process, making it difficult to fill non-circular geometries [3].As a result, completely filling rectangular gap geometries appears hard to realize.In order to find optimal application patterns, an approach was implemented in the literature [4] where the simulation is run backwards in time.This method was used to calculate, step by step an initial pattern with an associated gap height to finally reach the targeted pattern.By reversing time, the system simulates, under certain physical restrictions, a scenario in which the surfaces to be joined were separated by lifting the upper surface.This mathematical transformation makes it analogous to a lifting Hele-Shaw cell process [5].For this task, the Partially Filled Gap Model (PFGM) [3,[6][7][8][9] was used.It very efficiently models fluid flow in narrow gaps.In the literature [4] modifications have been made to the PFGM to allow the implementation of a time reversal F I G U R E 1 Schematic representation of a forward and a reverse simulation [4].
simulation.The conceptual framework is illustrated in Figure 1.As a further step, the aim of the current paper is to provide experimental evidence that the patterns found in theory can be reproduced in practice.
The reverse simulation published in the literature [4]  As a result, a four-branched star-like shape was identified as an effective initial pattern for achieving a square configuration by parallel compressing.Similarly, an elongated star shape emerged as the result for rectangles.For shapes that can be constructed from multiple rectangles, the use of compound star shapes, elongated star shapes or variations thereof as initial application patterns gave favourable results.A modified star shape can also be used for desired oblique target patterns or curved shapes.It has been shown that the final fluid distribution is strongly affected by the squeeze ratio , which is defined as (ratio of final gap height ℎ  to initial gap height ℎ  ).In the examples studied, it was found that the final distribution is not affected by viscosity, scaling of the system or absolute domain size and therefore volume.These variables only affect the force required to compress.These results were also found in another study by Kaufmann et al. [10] in which an optimal application pattern was found iteratively.There, an application pattern was compressed, overflowed fluid was traced back and deleted in the application pattern.A selection of the shapes studied in Flaig et al. [4] are shown in Table 1.In order to generalise the quantification of the deviation between the actual pattern and the target pattern, a relative volume deviation  with the following equation has been introduced.To calculate , it is used that in the PFGM the area is divided into cells labelled with indices  and  (subscripts: tar means target; act means actual; max is the maximum volume of the cell) [4]:

MATERIALS AND METHODS
This paper presents a simple experimental proof of the simulated shapes illustrated in Table 1.For this purpose, two different types of experiments are carried out.In the first experiment, a star that has been identified as the optimal application TA B L E 1 Selection of results for the reverse simulation, the simplified form and the forward simulation of the simplified form (actual pattern).
Note: The squeeze ratio is the ratio of the final gap height to the initial gap height.
pattern is applied and compressed.The aim is to prove that the compression results in the expected final distribution.In the second experiment, the complete process as described above is reproduced experimentally to prove that this method of inverse simulation is not only possible in theory but also in practice.For the samples studied, as mentioned in the introduction, viscosity was found to have no effect on the flow behaviour considered.So a skin care cream (Cien skin care cream Soft, Mann & Schröder GmbH) is used as a substitute for the adhesive.It meets the requirements that it is relatively easy to clean and obtain, it can be applied stably and its white colour provides an easily recognisable contrast to the base.The joining surfaces are two panes of glass.The upper pane is commercially available float glass.In Experiment 2 this pane has a size of 60 × 100 mm, in Experiment 1 it is large enough so that the liquid doesn't flow to the edge.The lower pane is acid-etched float glass, painted black to improve contrast with the white liquid for photography.Recent studies have shown that the coatings should have no effect on the flow result [11,12].

Experiment 1 -Compressing a star-shape
The star found in the literature [4] with a squeeze ratio of  = 1 2 is chosen (pattern top row of Table 1) because here no fluid in the star's tips sticks to the mould.This is applied with a thickness of ℎ = 0.8 mm so that the square obtained has a thickness of ℎ = 0.4 mm after pressing.A 3D-printed stencil is used to apply the star-shaped application pattern to the bottom pane of glass.The geometric dimensions and an image of the stencil are shown in Figure 2. The stencil is filled with liquid and then smoothly skimmed with a spatula.After application, the stencil is removed.A squeeze ratio of  = 1 2 was chosen because the star's tips are blunter than with a smaller value of  and therefore not as much material gets stuck in the star's tips.Now the top joint surface is applied and the pattern is pressed to a final gap height of ℎ = 0.4 mm for which a 0.4 mm spacer is used to ensure the correct final gap height.The resulting pattern is then visually compared, and the deviation is quantified using the formula 2.

Experiment 2 -Experimental reproduction of the inverse simulation
As a simulation running inverse in time is very similar to pulling apart two plates with fluid in between them, this method is used to reproduce this step of the simulation experimentally.The above-mentioned steps of the inverse simulation serve as a guideline for the experiment.The experiment contains the following five consecutive steps: a) Heavily overfill the gap and press the panes until the desired final gap height is reached b) Clean up the overflown material c) Slowly pull the mating surfaces apart in parallel until they separate (manually) d) Compressing the surfaces e) Evaluate the result In this experiment the influence of the lift-off velocity is also investigated.In the literature [4] it was found that the lift-off velocity also has an influence on the fluid flow.For this reason, the experiment is carried out with a higher lift-off velocity.All other parameters, such as the gap height in the compressed state, remain the same.

Experiment 1 -Compressing a star-shape
Figure 3A shows an applied star as determined by the literature [4] as a nominal perfect application pattern for a squeeze ratio of  = 1 2 to obtain a square.In Figure 3b this star has been squeezed between parallel plates from ℎ  = 0.8 mm to ℎ  = 0.4 mm.It can be seen that the final distribution is very similar to a square.At the same time it can be seen that there are air inclusions, especially in the outer regions.Looking at the progression from pattern 3A to pattern 3B, it is clear that there are two causes of the air inclusions.The air inclusions in the centre of the surface are caused by small air bubbles in the fluid.These bubbles become visible when the fluid is compressed.The inclusions at the edges are the result of the offset edge of the application pattern caused by the stencil.The pink framed area in 3b is the target distribution.For the comparison of the target and the actual distributions, air bubbles are neglected.Using Formula 2, a value for the volume deviation of  = 2.12 % is determined.The theoretically determined value of the volume deviation for this pattern in the literature [4] is  = 1.35 %.The experimental value is slightly worse at  = 2.12 %.This is due to imperfect application, manual pressing which is not perfectly parallel, and production related variations in the thickness of the spacers which determines the final gap height.Overall, however, both values are in the same order of magnitude and both deviations are very small.This experiment proves that the star found in the literature [4] is indeed an appropriate application pattern to create a square.It also proves, that the method introduced in [4] of finding the optimal application pattern is effective.However, the inclusion of air bubbles shows that, within such a simple experiment, it is not trivial to apply such a pattern without defects, even when a stencil is available.For practical purposes, further research can be done in this area, for example, more accurate ways of filling the stencil perfectly, or coatings for the stencil so that the fluid does not stick to the stencil when it is removed.

Experiment 2 -Experimental reproduction of the inverse simulation
Figure 4 shows the results of the second experiment.All of these images are unprocessed, in particular images 4c 1 and 4c 2 are the direct result of lifting the cover plate, no further smoothing has been performed.It can be seen that during the lifting a cut star is formed which is joined in the centre by a straight bar.In this star, the arms get thinner towards the tips.This is due to two effects: The width of the arms and the height of the arms decrease towards the outside.It can also be seen that after pressing, the target surface is perfectly filled and almost free of air bubbles.
For a further comparison, Figure 5 shows the results of the experiment in which the glass panes were pulled apart much faster.It can be seen that the shape is much more fissured than with the slow lift-off.
Comparing the pattern with the shape identified in the literature [4] and shown in the bottom row of Table 1, a great similarity can be noted.In both cases the star is cut and joined with a straight bar, the only difference is that in the literature [4] a constant filling height is required and the reduction of material towards the star tips is therein only achieved by the width of the arms.
The experimentally demonstrated influence of the lift-off velocity also confirms the simulation results of literature [4], where it was found that a faster lift-off velocity leads to stronger fissuring.Therefore, in order to find an optimal application pattern, a slow lift-off velocity should be aimed for.
Thus, this experiment proves that the method presented by Flaig et al. [4] is suitable for finding an optimal application pattern with a simple experiment.

CONCLUSION
In conclusion, it has been shown that the method presented by Flaig et al. [4] is suitable for finding perfect application patterns.Based on simple experiments, for two examples, it has been shown that the identified perfect patterns fill the target gap almost perfectly.These tests have confirmed that the star is a possible perfect application pattern when a gap in the form of a square has to be filled.The same holds true for a targeted rectangle.In addition, a practical application strategy was described to get a feel for what an application pattern for a specific bonded joint might actually look like.
As discussed in the literature [4], the applicability of these patterns is difficult in practice.Such a star shape is difficult to apply with a dispensing system, and applying it with a stencil is not trivial either.Further research is needed to find patterns that can be applied with industrial dispensing systems.An alternative research topic is to find coatings for the stencils that prevent the adhesive from sticking to them, thus making these application patterns applicable in industry.
This experimental evidence has led to a practical strategy for finding an individual optimal application pattern.The challenge in practice is that most joining surfaces are not square or rectangular and also the surfaces are not always smooth and parallel.With the method presented in experiment 2, an application pattern can be found for any shape of joint surface.Everything, what is needed, is the repetition of the presented steps with the own individual target pattern.The most important thing here is that the surfaces are pulled apart slowly.The resulting pattern can be scanned and used as an initial design for an optimal, individual application pattern.For the examples in the literature [4] and [10] it has been shown that in practice the viscosity has less influence on the distribution of the liquid.Therefore, any type of mass can be used for this procedure.Based on the experience of this work, it may be helpful to choose a liquid that is easy to handle.

A C K N O W L E D G M E N T S
We gratefully acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG), under Project ID: 445254897.Furthermore, we would like to express our gratitude for the financial support of the Publication Fund of the TU Braunschweig.
Open access funding enabled and organized by Projekt DEAL.

O R C I D
consists of the five consecutive steps: a) Define the target distribution b) Reverse simulation using the reformulated PFGM to run backwards in time c) Simplification of the pattern d) Forward simulation using the existing PFGM e) Comparison of the result of the forward simulation (actual pattern) with the defined target pattern

F I G U R E 2
Stencil for applying the star application pattern.

F I G U R E 3 2 .
Perfect application pattern according to the literature[4] to fill a square joint area after pressing.The pink dashed line marks the target pattern -Squeeze ratio  = 1

F I G U R E 4 F I G U R E 5
Experimental validation of Flaig et al.'s [4] method of finding a perfect application pattern for a rectangular joining surface.The pink dashed line marks the top edge of the transparent pane.Pattern created when the panes are pulled apart faster.