Modeling Heater Electrodes on Sheet Metal for Arbitrary Temperature Distributions

We present a method for the design of heater electrodes on substrates with high thermal conductivity such as sheet metal. The substrate is covered with a thin polymer insulation layer on both faces, which are, in turn, carrying screen printed, electrically conductive heater electrodes and another protective polymer overlayer. The temperature inside a predetermined optimization area is required to be as uniform as possible, which is desirable for various high-precision heating applications. We alternatively also aim to design a heater structure to create non-uniform temperature distributions such as a temperature dip, a peak or a gradient. These temperature distributions are required for droplet operations like splitting, merging, and moving droplets, subsequently realizing a newly proposed “lab on metal” approach in microfluidics. For this purpose, an optimization algorithm yields the required distances between linear heater electrodes. The resulting electrode distribution was numerically obtained for different optimization areas. Finally we describe the fabrication of a test device and show infrared measurements of the FEM modeled temperature distributions on the experimental realized heaters. We describe the design of heater electrodes, capable of triggering droplet motion on a “Lab-On-Metal”, as an alternative technology in Lab-On-a-Chip applications.


I. INTRODUCTION
S PECIFICALLY shaped temperature distributions on sur- faces can be utilized for many applications, e.g., hot plates for laboratory use or microfluidic lab-on-a-chip designs [1].Peltier elements to pre-heat liquids in a biological system are also very common [2].Fastly responding and uniformly tempered heating systems for Polymerase Chain Reaction (PCR) are investigated in [3], [4].Fluid flows can be manipulated by fluidic close-open paraffin micro-valves, where the whole heater system can be screen printed and the microfluidic parts are imprinted [5].In microfluidic regimes, mixing can be performed by natural convection or micro-mixing without the need of pumps or valves [6].Fluid manipulation by temperature dependent surface tension (droplet movement, splitting etc.) is investigated in [7], [8].
In general, the surface tension force along a liquid-gas interface acts in the direction of the higher surface tension.By providing a temperature gradient T across a droplet's surface one can create regions of locally higher and lower surface tensions γ (T ).This induces surface stresses and Marangoni flows, which can be utilized to manipulate droplets on top of a heated substrate to move them into the direction of the higher surface tension region [9], [8].Usually the surface tension decreases with rising temperature, so a droplet, for which we can maintain a temperature gradient along its liquid-gas interface, will move towards regions of lower temperature [1].
The present work describes the design and optimization of the heater part of what we call "Lab-On-Metal" device, which is highly parallelizable due to the potentially large area of the metallic substrate.We will show that virtually every desired temperature distribution is achievable by varying the distances d i between screen-printed conductive heater lines (electrodes) of width w EL on top of metallic substrate, which is insulated from the electrodes by a thin polymer coating.
The required temperature distributions are hard to achieve with conventional heaters such as a simple hotplate.Our approach of creating a particularly desired temperature pattern makes use of multiple heater lines, screen printed on top of  an insulated metallic substrate enabling a fast temperature response.In this work we focus on the design of four specific temperature distributions, namely a temperature dip, a temperature peak, a gradient and a distribution which provides a uniform temperature across a large area.By positioning a liquid droplet on top of the heater structure, e.g., a temperature gradient can be generated across the droplet, thus leading to the said gradient in surface tension (see section III).

II. STATE OF THE ART
Many applications in industry and science require exact temperature control.Combining the microfluidic tasks described above, a large scale alternative to the common lab-on-a-chip approach is thinkable by manipulating all involved fluids only by specifically impressed temperature patterns.This section will show some examples of already published works thus indicating the feasibility of the approach.Besides the quoted original source, all examples in this section can also be found in a review article by Miralles et al. [1] Fig. 1 shows a droplet squeezed between two glass plates where the bottom plate is covered by four electrical line heaters arranged in square where each side can be heated independently, using the associated heaters 1 to 4. The whole device is embedded in a PMMA casing [9].Fig. 2 shows the resulting movement of a droplet between the glass and cover plates [9].The setup is shown in Fig. 1.Powering up the heaters individually induces a temperature dependent surface tension gradient across the surface of the sample fluid.This leads to a movement of the liquid droplet in  direction of the surface tension gradient.In general, the value of the surface tension gets smaller with rising temperature.As an example, by activating two heaters the droplet will move towards the upper right corner (see t = 0 s to t = 20 s).
Fig. 3 shows the same concept of creating a certain temperature pattern on top of a substrate with a more complex electrode heater structure [8].The heater lines are distributed across the whole surface of the substrate.Like in the example from Fig. 1, it is possible to move a droplet in two dimensions.
Finally, a much simpler, but also easier to fabricate, electrode geometry is shown in Fig. 4. It shows electrodes of different width w E to create a linear temperature gradient along one axis of the device [10].A liquid droplet placed on top of a hydrophobic plate, heated with the resistor pattern shown below, will obviously only move in one dimension.
The present work represents a specific approach regarding the modeling of heater electrode positions and shapes on top of a suitable substrate.In our previous work we showed the modeling and fabrication of a heater stack with screen printed silver electrodes [11].There, we calculated the spatial distribution of the electrodes on top of the heater stack to enforce the formation of an uniform temperature distribution across predefined areas of the substrate.In the present work we will expand the modeling to create arbitrary temperature distributions like a temperature dip, a (non-gaussian) temperature peak, and a temperature gradient.Using the results, we fabricated the designed heater stacks for all four temperature distributions.We will show infrared measurements of the resulting temperature distributions.We will also take a look at suitable sample liquids by calculating the temperature dependent surface tension change of the respective liquid for a given temperature distribution.

III. BASIC THEORY AND PRINCIPLE
As shown in the last section, specific heater electrode distributions can yield particular temperature distributions, e.g., a temperature gradient across a heated surface.Imagine putting a liquid droplet (e.g.water) on top of such a surface.A local change of temperature emerges along the droplets surface.This leads to a local change of the surface tension and therefore the contact angle equilibrium of said droplet.This behaviour is given by Young's equation mentioned first in [12] and derived more in detail in [13].Fig. 5 shows the definition of the surface tensions γ and the corresponding contact angle θ of a liquid droplet for both, a wetting (θ< 90 • ) and a non-wetting (θ> 90 • ) droplet.The variables in Fig. 5 and Eq. 1 below are γ LG , γ SG and γ SL representing the surface tensions between the corresponding phases (L: liquid, S: solid, G: gaseous).The contact angle θ represents the inner angle between the solid surface and the tangent to the droplet's surface).The line where all three phases come in contact is called triple contact line.Young's equation can be derived by geometrically constructing the sum of all three surface tension force vectors (shown in red).Both cases lead to the same equation shown below (Eq.1).
The unit of γ is J/m 2 or N/m.The surface tensions γ LG , γ SG , and γ SL represent forces that are exerted tangentially to the surface of the respective phase.In microfluidics, a more commonly used unit is mN/m (e.g., Water-Air: γ LG ∼ 72.8 mN/m).Young's equation reads This formula indicates that a change in surface tension also leads to a change in the apparent contact angle θ between a liquid droplet and its solid substrate.An asymmetric change in θ (different contact angles at different positions at the systems triple contact line) leads to different line forces along the base of the droplet, thus resulting in a net line force [14].It has to be mentioned that Young's equation neglects effects like contact angle hysteresis, multi component substrates or the surface roughness of the substrate.
Fig. 6 shows a schematic cross section of the proposed heater stack and a rendering of the fabricated sample with a width W TOT = 20 cm and a length of the heater lines L HEAT = 40 cm.The length along the red cut-line, where the algorithm optimizes for e.g., a uniform temperature distribution, is called optimization length L OPT .Depending on the distance between the electrode heaters, one can achieve customizable temperature gradients, regions of uniform temperature and other kinds of temperature distributions [15].The optimization area is given by ( Regarding the design process associated with the targeted temperature distributions, we utilize optimization algorithms provided by the commercial FEM package COMSOL TM Multiphysics.These algorithms vary the distances between multiple electrode heater lines for achieving a certain desired temperature distribution in a predetermined area A O PT on top of the metallic substrate.For the uniform distributions, the number of distributable heater lines of width w EL = 1 mm has been held constant at n EL = 24 in the present study. The deviation from the chosen target temperature T g has been characterized in terms of the quadratic deviations from the actual temperature T (x) for each position x along the bottom insulation layer, where the microfluidic channels for our "lab on metal" applications are to be imprinted when using this platform for a microfluidic device in future works.E.g., the formula for the quadratic deviation f (T, x) for a uniform target temperature distribution corresponding to a constant temperature T g (60 • C or 333.15K in our example below) along the whole length of L OPT , reads The formulas for the other three distributions utilize the similar quadratic form but consist of piecewise combined terms f i (T i , x) defined for different areas on the substrate.

IV. FINITE ELEMENT MODELING
The temperature distribution for a specific electrode distribution on top of the middle insulation layer, i.e. the insulation layer on top of the metal substrate, has been calculated with the help of the FEM (Finite Element Method) solver COMSOL TM Multiphysics.Some basic concepts of the FE method are explained in more detail in [16].Each electrode distribution yields a temperature distribution due to joule heating, originating from the current in the electrodes.The average of the quadratic deviation f (T, x) along the optimization length L OPT is minimized.Fig. 7 shows a cross section of the heater stacks'FEM geometry in the region of two neighboring electrodes with width w E = 1 mm and height h E = 5μm.The electrodes' material characteristics are modeled according the parameters provided for the used polymer-based silver screen print ink.The same parameters have been used in the experimental part of this work (see section VIII).The hybrid mesh used for the FEM calculation consists of a structured quadrilinear mesh in the region of the substrate and an unstructured triangular mesh in smaller regions such as the insulation layers and the electrodes.The following COMSOL TM packages have been used: "Optimization", "Joule Heating" and "Electric Currents".The triangular mesh has been refined by the longest edge algorithm discussed in [17] and [18].
The distance between two electrodes centers d i is varied.At the same time the heating power is varied by varying the heating voltage U H .Both variables are optimized such that the integrated temperature error E INT , defined by the line integral of the quadratic error along the optimization length L OPT is minimal: The numerical values for d i have been calculated by a Downhill Simplex optimization algorithm (error threshold E INT (T, x) ≤ 5e-4 mK 2 ), followed by a random Monte Carlo shifting of a small distance d i ∓ δd i around the Downhill Simplex solution, to ensure not to be trapped in a local optimum of the electrode positions and to lower the error even more.The optimization parameters were the heating voltage U H and the electrode center to electrode center distance d i between each neighboring heater element.Fig. 8 shows a scheme of the heater stack and how heat can be transported through the system.The dominant effects are the thermal conduction inside the material, emission of heat from the surface, and transport of heat due to thermal convection away from the surface.1.The insulation layer where the stack is positioned for the measurements with a very low thermal conductivity λ 0 (in our experiments: Polystyrene).The heat transfer coefficient α 1 is practically zero.2. The top insulation layer with thermal conductivity λ 1 .The electrode heater lines (red) are embedded in this layer.3. The middle insulation layer with a low thermal conductivity λ 1 .4. The metallic substrate with a high thermal conductivity λ 2 .5. The bottom insulation layer with a low thermal conductivity λ 1 .The bottom insulation has an emissivity ε.
It is in direct contact with the surrounding air, where α 2 = 3.95 W/(m 2 * K).
V. BOUNDARY CONDITIONS Boundary conditions (BCs) for both the thermal FEM simulations and the optimization of the electrode positions are required.The ambient temperature for the FEM simulations was set to T 0 = 293.15K.The exemplary target temperature for the uniform temperature distribution was T g = 333.13K.The target temperatures for the peak, dip and gradient distribution depend on the position on the substrate.The target temperature has been measured at the bottom surface of the bottom insulation layer (see Fig. 7).
The following boundary conditions have been used to calculate the convective and radiant heat transfer during heating the heater electrode lines.The FEM model uses the heat transfer coefficient α for air perpendicular on a metal plate, modified by air movement due to convective heat as suggested in [19]- [21] and [22].The emissivity ε of the polymer surface has been measured with a Flir TM E4 infrared camera, utilizing a test sample (ε = 0.9).The heat transfer coefficient at the transition 'bottom insulation -air' (5) and at the edges of For asymmetric distributions like the temperature gradient in Fig. 11, the electrodes have to be able to shift in a non-constrained manner along the whole substrate width W TOT .The drawback is a higher requirement on computing power.The number of distributable electrodes n EL depends on the desired temperature pattern.For the uniform distributions in Fig. 10 this number is n EL = 24.Note again that the uniform distribution features a center electrode which is fixed.

VI. SIMULATION RESULTS
Similar to Fig. 10, Fig. 11 shows the resulting temperature distribution for a temperature dip, peak and a gradient.The resulting temperature distribution (orange line) to minimize the difference between the target temperature T g and the target function f (T, x) (blue line) is again shown in the top row.The corresponding quadratic error (black line) for each position x along the optimization length is shown in the center row.The absolute Error E I NT is shown in the figure insets.The location of the electrodes on top of the primer is shown again as red lines in the bottom row.The number of distributable electrodes n EL depends on the desired temperature pattern.
The FEM results clearly show that an optimization of the electrode positions for every desired temperature distribution is possible.All four shown temperature profiles in Fig. 10 and Fig. 11 are capable to achieve an uniform temperature across the substrate, creating a temperature dip and peak, or gradually moving a droplet on top of the substrate.This can be done by providing a linear temperature gradient.
The quality of the achievable solutions strongly depends on the material characteristics like thermal conductivity and heat capacity of the substrate and insulation layers.Designing the electrode distribution for a temperature dip is, e.g., much more feasible on a substrate with a high thermal conductivity like the used sheet metal.A low thermal conductivity would not allow for a large enough temperature difference to obtain a surface tension gradient, large enough to move a droplet.This difficulty can be seen in Table I where the achievable T is listed in the fourth column.
The temperature dependent change in surface tension for different sample fluids will be shown later in section IX.A temperature gradient leads to a pulling force on the surface molecules towards regions with higher surface tension, i.e., regions with lower temperature.This induces the already mentioned Marangoni convection on the droplet surface and therefore a mass flow within the droplet.
The designed temperature patterns can be utilized to move droplets over a surface or through microfluidic channels.The uniform temperature distributions in Fig. 10 serve as test simulations.The dip, peak and gradient temperature patterns shown in Fig. 11 are harder to achieve because of their specific shape.Especially the gradient can be used to move a droplet along an appropriate heated surface.
The desired temperature distribution can be achieved with sufficient accuracy within a certain error range.
For the evaluation of the simulation results we first define a dimensionless number t S O L which represents the normalized computation time of the optimization algorithm for the different temperature distributions.The respective starting point for the calculation of t S O L is the solving time for the uniform distribution optimization (L OPT = 4 cm, t S O L = 1).All other computation times are normalized to this time (meaning e.g.: t S O L = 3.96 means that the temperature gradient problem takes 3.96 times longer to solve than the uniform distribution with L OPT = 4 cm, see table I, last line, third column).
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.As already mentioned, the time t SOL for a converged solution is lowest for the uniform distribution (t SOL = 1) (as the solution is symmetric around the center electrode), followed by the peak (t SOL = 1.3) and dip distribution (t SOL = 1.44).Note the corresponding boundary conditions for the electrode shifting in Fig. 9.The most challenging distribution is the gradient (t SOL = 3.96) as the distribution is not symmetric and the electrodes have to move freely over the entire domain W T OT .
Besides other results like the integrated total error E INT and the already mentioned normalized solution time t SOL for an optimization, Table I shows the achieved temperature change and the corresponding surface tension change leading to a variation of the contact angle and thus a potential movement of a droplet resting on the heater stack.From the first to the last column the following data is listed: Integrated error, normalized solution time, the achieved temperature difference between the minimal and maximal temperature and the corresponding total change in surface tension of water (−0.2057mN/mK, γ 0 = 72.8mN/m) according to section IX.
The last column shows the decrease in surface tension over the temperature range T for water as sample liquid.The achievable temperature difference T between the lowest and the highest temperature at the bottom insulation layer depends strongly on the shape of the distribution.The resulting change in surface tension γ has been calculated for water, using a linear surface tension model (see section IX).The corresponding percental decrease in surface tension over the achievable temperature range T is given in the last column of Table I.Summarized, a significant temperature dependent change in surface tension γ (T) seems to be possible with the proposed setup.

VII. CHARACTERIZATION
The electrode distributions resulting from the FEM simulations can be characterized in various ways.In this work we will take a closer look at the electrode distribution which leads to a uniform temperature distribution.The reason is the easily understandable and nice result which acts as an illustrative example.
The uniform distributions can be characterized by the individual distances d CE of all distributable electrodes to the fixed center electrode n 0 .The total number of distributable electrodes is n TOTAL = 25 (24 distributable electrodes and one fixed center electrode).
Fig. 12 shows the results for optimizing the electrode positions for three different optimization lengths L OPT .The algorithm is configured such that electrodes can be shifted outside the optimization area L OPT .
The top Fig. 12 shows the distances d CE between the center electrode n 0 and electrode n.With rising distance from the center electrode the slope gets lower.This behaviour is plotted  The bottom Fig. 12 shows the total integrated temperature error E INT vs. L OPT .As the desired area of constant temperature A OPT becomes larger with rising L OPT , the quadratic temperature error, and therefore the value for E INT , increases due to the constant number of distributable electrodes n EL = 24 (25 -1 center).

VIII. HEATER STACK PREPARATION AND EXPERIMENTS
As a first example we will explain the fabrication and measurements of the uniform heater with L OPT = 6 cm.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The heater electrodes have been printed on top of the middle insulation layer according to the resulting distributions from Fig. 10, center column and using the fabrication method we reported earlier in [23].
The heater electrode lines are realized by screen-printing a polymer-based silver screen print ink on top of sheet metal substrates with pre-coated organic primer coatings and cured at 140 • C for 240 s (electrode width w E = 1 mm and height h E = 5μm.).The screen is flooded with paste and the heater structures are printed through the mesh (120/34) with a semiautomatic screen print machine RokuPrintSD05.
Fig. 13 shows the first temperature measurement results for an uniform distribution at the bottom insulation layer.These results have also been reported in one of our previous works [11].The measurement in Fig. 13 is intended to illustrate how the following measurements work and for comparative reasons when focusing on the arbitrary temperature distributions later.
The upper part of the figure shows photos of the stack before covering it with the top insulation layer to show the electrode geometry.
The electrical contacts (see red points in Fig. 13) have been realized by soldering terminal clamps directly to the dry silver paste using induction soldering.The electrical contacts should be located in the center of the substrate at W TOT /2 to prevent an uneven distribution of the current density inside the heaters.This would lead to an unpredictable temperature distribution.
The pictures have been taken with a Sony Alpha 560 DSLR camera using a 100 mm f 1/2.8 macro lens.A very close look at the inset shows the resolution of the used screen printing mesh as one can see the terracing of the print.
The lower part of Fig. 13 shows an infrared measurement using a Flir TM E4 infrared camera.The temperature in the top left edge is the temperature at the crosshair (T = 60.3 • C).The temperatures at the heat map legend are the maximum (61 • C) and the minimum temperature (Background temperature of 26.3 • C) of the measurement area.The numerical temperature values have been crosschecked using a Pt100 temperature probe, visible on the left edge of the substrate.
The simulated temperature plateau of T g = 333.15K (60 • C) has been hit almost exactly.The results are sufficient, compared to the modeling.The uniform distribution for L OPT = 6 cm is clearly visible within the expectations from the simulation.
Reasons for a deviation can be a variance in the thickness of the electrode print or other involved layers like the primer coating (i.e. the coating of the substrate).The spatial dimensions of L HEAT in y-direction of the whole heater stack will also affect the correct formation of the desired temperature distribution.Generally spoken, the length of L HEAT should be sufficiently large compared to the total width of the stack W T OT .
The next step is the fabrication of the heater stacks with electrode distributions enabling the generation of arbitrary temperature distributions.Being able to create uniform temperatures is not enough for a heater system which is supposed to be capable to act as driving force for microfluidic droplet operations.Operations like splitting, moving and merging require more specialized electrode distributions, some of them (e.g., gradient) even require asymmetric distributions (see Fig. 11.third column) Fig. 14 shows the fabricated heater stacks, designed according to the data obtained from the simulation results shown in Fig. 10 and Fig. 11.The upper column of Fig. 14 shows photographs of the prints before adding the top insulation layer.Therefore, the silver electrode heater lines are visible.The lower part shows infrared measurements using a Flir TM E4 infrared camera.From the left to the right Fig. 14 shows: • A print which enables us to form a uniform temperature distribution along the predefined optimization length L OPT .• A print which allows us to generate a dip in temperature in the center of the heater stack.• A print which allows us to generate a non-Gaussian temperature peak in the center of the heater stack • A print where the temperature rises from a low temperature plateau to a high temperature plateau.In between the print is designed to show a linear temperature gradient.
The infrared measurements in Fig. 14 show that we are not only able to create an electrode distribution that leads to a uniform temperature distribution, but also virtually every  other desired distribution which is specified by an arbitrary target function.The small temperature deviations between measurement and simulation can be primarily explained by the influence of changing ambient temperature and air flow, leading to an altered heat transfer coefficient α.Besides the distribution of the electrodes, this coefficient has the largest influence on the temperature values in the simulations and is therefore a critical parameter to consider.It has also to be noted that the prints can suffer from small hotspots if the contacts are not placed, so that the current density is evenly distributed.

IX. CHOICE OF THE SAMPLE LIQUID
The question arises which sample liquid is suited best for actuation by a temperature dependent local manipulation of its surface tension.A liquid which shows a strong change in surface tension for a given temperature change is favorable.
Table II shows the surface tension values for various fluids at T 0 = 293.15K in the first column.The corresponding critical temperature T c is shown in column 2. All data has been gathered from [24] and [25].The value of γ 0 is used as measured calibration value in the surface tension model equations ( 5) to (7).The second column shows the critical temperature T C .(5) Several authors proposed a non-linear model which can be seen in comparison to the linear model in Fig. 16 [27].The blue line represents the linear model whereas the orange line represents the non-linear model.The slope of the blue line in Fig. 16 shows the decrease in surface tension with rising temperature for water.Water shows a strong dependence of -0.2057 mN/mK, therefore we will use it for future droplet moving experiments.
The surface tension data in Fig. 16 is plotted up to the boiling point of water.The numerical correction terms have been proposed by Patek et al. and Kalová and Mares [28], [29] The formula for the non-linear model reads γ = 241.322τ 1.26 1 − 0.0589τ 0.5 − 0.56917τ (8) The absolute error between both surface tension models for water in its liquid state is shown in Fig. 17.Due to the chosen reference points, the deviation is obviously minimal at T = T 0 .We conclude that the deviation between the two models is negligible for our purposes as the maximum deviation is 2.5 mN/m near the boiling point of water.

X. CONCLUSION AND OUTLOOK
One major challenge of Lab-On-a-Chip applications is the restriction to small area substrates and therefore a restricted potential for high parallelization.Using large scale metallic substrates promises to be advantageous in that respect.Many microfluidic applications rely on heating and mixing fluids.
This work provides the heater part of a future "lab on metal" and shows exemplarily how to model the electrode positions for four different temperature distributions.This is done by a downhill simplex optimization of the electrode to electrode center distances.
The utilized metallic substrate is favorable in terms of a fast temperature response in heating and cooling due to its high thermal conductivity.We took a closer look at the electrode distribution for the uniform temperature distribution.There, the electrode distances d CE between the fixed center electrode n 0 to electrode n show a lowering slope with rising electrode number n.Samples for each of the four desired temperature distributions have been fabricated and infrared measurements have been performed on each sample.
Finally, the temperature dependent surface tension change for various liquids has been evaluated by utilizing two mathematical models, choosing water as ideal test liquid for future works.There we will deal with the integration of heater distributions for fast heating, movement, and mixing of fluids on top of a metallic substrate.Future work will also deal with imprinting microfluidic channels in the insulation layer to create a fully usable microfluidic device.
In summary, the present work addresses opportunities and methods, which will assist for the future design and fabrication of a fully functional "lab on metal" heater system as an alternative to the common fluid manipulation techniques in a lab-on-a-chip device.

Fig. 1 . 4 -
Fig. 1. 4-corner heater structure in PMMA casing.The droplet moves between the glass and cover plate because of the temperature dependent surface tension change induced by the four heaters [9].The figure is reprinted with permission of the authors.

Fig. 2 .
Fig. 2. Resulting movement of a water droplet in two dimensions by modification of the temperature dependent surface tension [9].The figure is reprinted with permission of the authors.

Fig. 3 .
Fig. 3.A more complex example of heater structures to move a droplet in two dimensions [8].The figure is reprinted with permission of the authors.

Fig. 4 .
Fig. 4. Linear temperature gradient realized by heaters of different dimensions in y direction [10].The figure is reprinted with permission of the authors.

Fig. 5 .
Fig. 5.The sum of the surface and interfacial energies determines the wetting of the liquid droplet on top of a surface.An useful manifestation of this surface tension equilibrium is called contact angle θ. a) shows the non-wetting case (e.g.: Water droplet on PTFE) and b) shows the wetting case (e.g.: Water droplet on glass).

Fig. 6 .Fig. 7 .
Fig.6.Layer structure and rendering of the electrode heater with indicated optimization area A OPT and red optimization line with length L OPT .

Fig. 9 .
Fig. 9. Possible heater electrode line distributions and corresponding boundary conditions for different states during the optimization process.BC b) and c) can be extended to the whole width W TOT or restricted to L OPT , depending on the application.theheater has been evaluated by comparing a test sample of the stack and a preceding FEM simulation.The value of α has been optimized until the measured temperature profile of the test sample matched an adjusted temperature profile of the FEM simulation.It amounts to α = 3.95 W/(m 2 K).The surface at the top insulation (layer 1 in Fig.7) is modeled by setting α = 0 W/(m 2 K).This insulation has been realized by placing this surface on polystyrol in the experiments.The boundary conditions (BCs) for the Downhill Simplex distribution optimization of the electrode lines with width w E L is shown in Fig.9.The upper part shows possible distributions of the heater electrodes during the optimization process from a) to c).The corresponding mathematical expressions for the electrode distances are shown in the lower part of Fig.9for a distributable number of electrodes n EL = 6.The position of this six electrodes can be varied.The position of the seventh center electrode is fixed.This can exclusively be used for symmetric electrode distributions.Starting with the initial distribution b), only half of the electrode distances d i have to be altered by the algorithm until the desired temperature distribution is achieved.The reason is the symmetric distribution around a fixed center electrode.BC a) ensures that the electrode lines never overlap.BC c) ensures that the electrodes do not move outside of L OPT .For asymmetric distributions like the temperature gradient in Fig.11, the electrodes have to be able to shift in a non-constrained manner along the whole substrate width W TOT .The drawback is a higher requirement on computing power.

Fig. 10
shows the resulting uniform temperature distribution (orange line) to minimize the target function f (T, x) (blue line) and the corresponding error (black line) along L OPT .The absolute Error E INT is shown in the figure insets.For constant n EL , E INT rises as the optimization length L OPT rises.The locations of the electrodes on top of the middle insulation layer are shown as red lines in the bottom row.

Fig. 10 .
Fig. 10.Upper row: Target Temperature (blue line / target function) vs. simulated temperature along the corresponding optimization line.Middle row: Quadratic deviation from the target temperature at position x.The inset gives the integrated absolute error of the modeled distribution to the ideal distribution.Bottom row: Electrode distribution to reach the uniform temperature distribution seen in the upper row.For each figure, the x-axis indicates the x position on the metallic substrate.The number of distributable electrodes n EL = 24 (25 electrodes -one fixed center electrode).

Fig. 11 .
Fig. 11.Temperature distribution, quadratic error and electrode distribution for different temperature patterns (dip, peak and gradient).The annotation of the axes is similar to that in Fig. 10.The number of distributable electrodes n EL depends on the desired temperature distribution.

Fig. 12 .
Fig. 12. Top: Center Electrode to electrode n distances d CE starting from the center electrode 0 to electrode 1 for equally broad heating lines (w EL = 1 mm).Bottom: E INT for L OPT = 4, 6 and 8 cm.
for all three optimization lengths L OPT = 4, 6 and 8 cm.The distribution is symmetric around the center electrode n 0 .

Fig. 13 .
Fig. 13.Top: Detailed Photos of the screen printed electrodes on top of the large scale metal sheet before covering them with the insulation overlayer (L OPT = 6 cm).The insets show detailed views of the heater lines and the position of the electrical contacts.Bottom: Infrared measurement and temperature profile, captured with a Flir TM E4 camera, showing the temperature at the crosshair on the top left and the minimum and maximum temperature of the measurement area above and below the heat map legend.

Fig. 14 .
Fig. 14.Top: Detailed Photos of the screen printed electrodes on top of the large scale metal sheet before covering them with the insulation Bottom: Infrared measurements on all four fabricated heater stacks, captured with a Flir TM E4 camera.The uniform print is shown again for comparison reasons.The dip, peak and gradient print have been optimized to utilize the whole width of the substrate W TOT = 20 cm.Axis labels are similar to Fig. 13.

Fig. 15 .
Fig. 15.Temperature dependent surface tension for various liquids up to their critical temperature T C (Linear model).

Fig. 16 .
Fig.16.Linear and non-linear model for the temperature dependent surface tension γ T of water in its liquid temperature range.

Fig. 15
Fig.15shows a linear model of the decreasing surface tension with temperature for various liquids[26].The linear model which was used to calculate the temperature dependent

Fig. 17 .
Fig. 17.Absolute error between the linear and non-linear surface tension models from Fig. 16 in its liquid temperature range.surface tension γ (T) for all liquids shown in Fig. 15 is obtained as follows.T 0 and T C are the ambient temperature and the critical temperature of the fluid.The surface tension value γ 0 is the surface tension at ambient temperature T 0 = 293.15K.The formula reads γ = γ 0 (1 + β (T − T 0 )) (5)

TABLE I RESULTING
DATA FROM THE FEM SIMULATIONS AND OWN CALCULATIONS

TABLE II SURFACE
TENSION VALUES FOR VARIOUS FLUIDS