Temperature calculation in metal substrates during two-photon polymerization

: The two temperature model (TTM) as well as the heat conduction equation (HCE) are used to calculate the temperature at surfaces of metal substrates during two-photon polymerization (TPP). Using TTM, the change in reﬂectivity of the material during one laser pulse is simulated for diﬀerent initial temperatures. In addition, experiments with tempered metal substrates are performed to examine how the change of the material properties due to diﬀerent temperatures inﬂuence the polymerization process.

during interference lithography. Due to their experiments and calculations concerning substrate temperature and the dynamics of the radicals in the polymer, they found out, that changes in the radical concentration, thermal accumulation and the diffusion of oxygen from the surrounding strongly influence the shape of polymerized areas. When regarding TPP on metal surfaces, three areas have to be considered: The heating of the substrate due to absorption of laser power, the shape of a stationary wave occuring due to the superposition of incoming and reflected light and the polymerization process, that is influenced by the temperature on the metal substrate and the intensity distribution of the stationary wave. In the context of pulsed laser light interacting with metal surfaces, it must be distinguished between the two temperature model, that describes the electron and ion temperature as well as changes in reflectivity and the complex refractive index and the heat conduction equation. The HCE calculates the metal temperature in dependence on material properties like the heat conduction or heat capacity in a more macroscopic manner.
For a better control of TPP on metal surfaces, a model for describing the heating of metal substrates during TPP is presented within this paper. The heat conduction equation (HCE) as well as the two temperature model (TTM) are used to calculate the temperature in the metal. In addition, experiments with tempered metal substrates are performed to examine the influence of the substrate temperature on the polymerization process.

Calculation of the temperature with the heat conduction equation
In TPP, a pulsed laser is focused through liquid photosensitive polymer on the surface of a glass or metal substrate. For the latter, dip-in lithography is a possible method, where the substrate and photopolymer is used in an upside-down arrangement [16]. Due to the spatial and temporal concentration of energy, the polymer is exposed within a volumetric pixel, called voxel, in the focus. By moving the laser in lateral direction through the polymer and the substrate in axial direction, arbitrary geometries can be fabricated. To simplify the calculation of the temperature, only the writing of single lines in x-direction is regarded. The following thermic energy equation describes the temperature field in a metal substrate caused by a focused laser [17]: Where ρ is the density of the metal, c p is the specific heat capacity, T the temperature, ì v the velocity, λ the thermal conductivity and Q L corresponds to a heat source with the unit power (Watt) per volume (cubic meter). The first term of Eq. 1 represents the change of temperature with time. Convection is given by the second term and accords to the motion of the substrate, respectively the laser in this case. Diffusion and heat input from the laser are described by the two terms on the right side, while the time average of the laser power is used.

Discretization of the HCE
For a numerical solution of the HCE via a linear system of equations, it has to be discretized with the finite volume method (FVM). To do so, the regarded simulation area is divided into reference units in a Cartesian coordinate system. Fig. 2 illustrates the arrangement of the control volumes as a right-handed coordinate system [18]: While the x-axis represents the west-east direction and the y-axis accordingly the south-north direction, the z-axis is assigned to a down-up link. Starting from the center point C in the middle of the regarded reference unit P, the spatial steps δx w , δx e , δx s , δx n , δx d and δx u represent the distances to the centers of the adjacent reference units in the corresponding directions. The edge lengths ∆x, ∆y and ∆z define the size of a control volume [18]. In a first step, Eq. 1 has to be integrated over a reference cell, resulting in Eq. 2 To consider the change of temperature T within t + ∆t, following expression is used for the time independent part: For a better clarity, the following expressions are used to outline the discretized HCE [19], where P e,i is the Peclet number P e and the index i is used to show which unit is regarded. Since the Peclet number corresponds to the ratio of convection to diffusion and convection is only regarded in x-direction, the index i can become w or e.
The discretised HCE follows as where the volumetric source term is described by the laser power per cubic meter With I 0 = 2P L πr 2 L T ob j , whereP L is the time average of the laser power, r L represents the radius of the laser focus, α the attenuation index and T ob j corresponds to the transmission of the microscope objective. In Eq. 7, R is the reflectivity of the regarded surface and the middle term ∆A ∆V · e −αz − e −α(z+∆z) considers the attenuation of the laser intensity along the height of a unit cell ∆z with volume ∆V as well as upper and bottom surface ∆A perpendicular to the z-direction. The index P in T n p signals, that reference unit P is regarded at time step n, respectively in T n+1 p at time step n + 1.

Simulation of the temperature during TPP
Since the background of this simulation is to control TPP on metal surfaces for structuring molding tools, corresponding materials were chosen for the temperature calculation. Three established metals for the fabrication of molding tools are steel 1.2083 [20], brass CuZn37 [21] and aluminium RSA 905 [22], while the according parameters are given in table 1. The absorption coefficient for steel was calculated with the formula α = 4·π ·k λ from [23] (with k being the amount of the imaginary part of the complex refractive index), while the complex refractive index was derived from [24] for stainless steel, since no data for steel 1.2083 was found.
Evaluation was performed by measuring the highest temperature in the middle of the Gaussian temperature distribution. The results in Fig. 3 show, that the temperature increases linearly with the laser power from 10 mW to 50 mW and nearly no influence is given by the variation of the laser scanning speed between 100 µm/s and 1000 µm/s. Due to its high heat conduction and reflectivity, the lowest temperatures are reached on brass, since a smaller amount of energy is coupled into the material and the heat quickly dissipates. Due to the same reasons, a small heat conduction and reflectivity, the highest temperatures are found in steel. Initial temperature for the calculation was 298 K. For aluminium and brass, no influence of the writing speed on the resulting temperature was observed. Thus, only regarding the temperature in the substrate, brass seems to be most appropriate for structuring with TPP, since it heats up least. Heat capacity c p [ J kg·K ] 460 [25] 897 [26] 377 [28] Heat conduction λ [ W m·K ] 16 [25] 115 [27] 120 [28] Density ρ [ kg m 3 ] 7800 [25] 2950 [27] 8440 [28] Extinction  All calculations were performed with a laser wavelength of 780 nm, a pulse duration of 100 fs and a repetition rate of 80 MHz. A special metal thickness was not considered within the simulation, since the platelets used in the experiment have a thickness of 6 mm, that is comparable to infinity when considering the voxel size of 360 nm and the penetration depth. Scanning speeds of 100 µm/s and 1000 µm/s were chosen due to the aim of a high throughput during structure fabrication.
For steel, also calculations with a varying initial temperature were performed at 50 mW and a scanning speed of 100 µm/s. Corresponding to the experiments in chapter 4, the temperature was varied from 5 • C to 80 • C (283K and 358 K). Results show, that the final temperature increase linearly in dependence on the initial temperature, and that the change in the initial temperature is simply added to the results shown in Fig. 3 for 50 mW and 100 µm/s, see Fig. 4.

Calculation of the temperature with the two temperature model
Regarding material processing with laser pulses in the femtosecond range, there is a thermic non-equilibrium between the electrons and the ions. Consequently, the whole system is described by the two coupled differential equations 8 and 9, one for the electrons and one for the ions. Due to scattering processes, an equilibrium between the two systems can be established. If the time between two scattering processes is in the order of the duration of one laser pulse, the TTM has to be applied [31]. C e (T e ) is the heat capacity of the electrons, C i (T i ) the one for the ions and G(T e , T i ) corresponds to the coupling factor due to scattering. The first terms of both equations represent the change of the internal energy U, caused by diffusion flows, heat exchanges between electron and ions and the absorbed laser power. q is the heat flow due to heat conductivity and Q the absorbed laser power, which is only considered in the equation for the electrons. To take into account a delay of the heat conduction, the heat conduction model according to Cattaneo was used [32]. The temperature dependence of the heat capacity of the electrons is derived from literature and given by equation 10. Here, u e is the internal energy of the electrons referred to the volume and is defined by the Fermi distribution f F , the density of the energy levels g e and the corresponding energy level E. An overview of the Fermi temperatures, energies and electron densities for calculating C e is given in [33] for a choice of metals. Heat conduction of the electrons and its temperature dependences are determined, based on Drude's model.
Since the heat capacity of the ions mainly determines the heat capacity of the metal, it will be regarded in the following. Based on the phonon formalism and using the Debye model, Eq. 11 is given for the heat capacity of the ions, where the left one is the general form and the right one corresponds to a limit value observation for high temperatures, causing C i (T i ) to become constant [34]. In case of high temperatures (e.g. the Debye temperature), the approach of Dulong-Petit has to be used: C i (T i ) = 3n i k B with the density of energy levels n i . This expression will be used in this model.
with the ion density N i /V, the Debye temperature T D , the absolute temperature T l and the Boltzmann constant k B . ϑ is given by the expression ϑ = hω D /k B T i , where h is the Planck constant and ω D the Debye frequency. Consequently, the heat capacity of the ions corresponds to the derivation of the internal energy of the phonons with respect to the temperature [35]. In comparison to the heat conduction of the electrons, the heat conduction of the ions is very small and can be neglected.
The maximum temperature is also influenced by the electron-ion coupling factor and its dependence on the temperature [36]. Due to the proportional dependence on the electron-as well as on the ion temperature, the coupling factor increases during absorption of laser power, causing a higher heat flow from the electron to the ion system. Thus, the increase of the electron temperature is damped and that of the ions is accelerated, resulting in the following Eq. 12.
This expression becomes constant for T e = T i =room temperature, m e is the mass of an electron.
To describe the source term of the laser Q(x,t) with the absorption coefficient α, the pulse duration t P , J L the exposure dose and R the substrate's reflectivity is given [37]. The reflectivity R is calculated from the complex refractive index, which is derived from the complex dielectric conduction.

Discretization of the TTM
To solve the differential equations given above, the finite difference method (FDM) is applied. By replacing the derivations with difference quotients, numerical solutions can be found. Using equation 8 and discretising it for the temperature in the next time-and spatial step T n+1 e k , results in the electron balance equation: The discretised ion balance equation results as follows:

Simulation of the temperature
As a time step width, 0.02 fs were chosen, the spatial step width was set to 1 nm. To determine the complex refractive index, respectively the reflectivity of the metal surface, the Drude-Lorentz oscillator model was applied [38], leading to a reflectivity of 87.6 % for aluminium [39] (in comparison to table 1, the reflectivity is over 10 % higher, this difference results from the fact, that the value in table 1 was determined with a reflectivity measurement on a surface with a roughness and the 87.6 % result from a simulation). In the following the TTM is applied to aluminium to determine the change in reflectivity and the complex refractive index during the pulse duration. The fluence of the laser source is set to J L = 873.01 J m 2 . Heat radiation at the substrate surface is given by Planck's law [40]: with the intensity of the heat radiation I H R , the emissivity and the Boltzmann constant k B . This causes electron temperatures of 15,000 K (see Fig. 5(a)), leading to an ion temperature of 774 K. Consequently, the change in the refractive index results in 5 % and that in the reflectivity 0.7 % (see Fig. 5(b)). The ion temperature of 774 K is smaller than the melting point at 933 K, so no material will be melt. Considering Wien's displacement law, electron temperatures of  7430 K cause the material to emit radiation at 390 nm, which is the exposing wavelength in TPP. Thus, during the decomposition of the imbalance between electrons and ions, polymerization processes on the substrate could be influenced. With increasing temperatures, the spectrum is shifted to smaller wavelengths, so the radiation takes influence until the electron temperature has fallen beyond 7430 K. For all calculations, a wavelength of 780 nm, a pulse duration of 100 fs and a repetition rate of 80 MHz as well as an initial temperature of 298K was used.
With regard to the experimental part, where the substrate temperature is changed during TPP, the reflectivity was calculated with the TTM for different initial temperatures. These temperatures were chosen to be between 278 K and 1240 K. On the one hand, the temperature variation in the experimental part was performed between 278 K and 358 K (compare Fig. 10), on the other hand, the maximum temperatures calculated on steel with the HCE for 10 mW to 50 mW was between 487 K and 1240 K (compare 3). The reflectivity due to this temperature variation decreased by 0.025, see Fig. 6(a).
To also consider the influence of the laser beam interacting with the metal surface, when the focus is in a distance of 20 µm from the substrate surface (as it is the case for the upper planes of the cubes written in chapter 4), the laser fluence on the metal surface was calculated in dependence on the focus distance from the surface, see Fig. 6(b). At a distance of 20 µm, the fluence has fallen by a factor of 10 4 in comparison to the the situation, when the laser is focused on the surface. Comparing the electron temperatures, a decrease from 15,000 K to 340 K occured. The ion temperature fell from 772 K to 300.9 K, meaning that the temperature change due to interaction of laser and metal does not affect polymerization at higher distances from the substrate.

Experimental
TPP was performed with a Photonic Professional device in combination with a galvoscanner-based upgrade for higher writing speeds, both from Nanoscribe GmbH. The instrument was used in the so called Dip-in Laser Lithography (DiLL) mode, where the beam of a pulsed fibre laser with a wavelength of 780 nm is guided through a 63x microscope objective with a numerical aperture of 1.3. The objective directly immerses into the photosensitive polymer and focuses the laser on the bottom side of the substrate. Due to the spatial and temporal bundling of the laser power, the energy density within a volumetric pixel (voxel) in the focus is high enough to activate the photoinitiator in the polymer and thus start the polymerization reaction. By moving the laser beam through the polymer in lateral direction and the substrate in axial direction, arbitrary, up to three dimensional structures can be realized.
As photosensitive polymers, IP-Dip from Nanoscribe GmbH and OrmoComp from microresist technology GmbH (MRT) were used. The corresponding developers and adhesion promoters  The aim of the experimental part is to examine the influence of temperature on the TPP process. Cubes were written on steel substrates by varying the exposure dose via the laser power. The same experiment was performed for different substrate temperatures, while the substrate was tempered as follows: For cooling and heating, the Peltier element PF 127-14-25-S from Telemeter Electronic GmbH in combination with the power supply unit EA-PS 3150-04-B from EA Elektro-Automatik GmbH & Co. KG was used. Heat was dissipated from the Peltier element during cooling by a heat sink (LA ICK 17 x 17 F 12) with an integrated cooling fan. A thermal paste enhanced the thermal conductivity between the Peltier element and the steel substrate. To measure the temperature during experiments, a platinum resistance thermometer (PT100) was used in combination with a digital multimeter. The reflectivity was measured over a range of 250 to 1300 nm and here the values for 780 nm at 298 K are used. Fig. 7 depicts the set up for tempering the substrate during TPP. At the bottom side of the substrate, where the structures are written, the PT100 element is fixed beside the polymer area to control the surface temperature of the substrate. Temperature was set by the direct current value of the power supply. Scanning electron microscope (SEM) measurements of the structures were taken with a Zeiss Neon 40 EsB device from ZEISS AG.

Heating of metal substrates during TPP
To examine the influence of temperature on the TPP process, structures were written on steel substrates. Experiments with TPP were repeated for different temperatures from 5 • C to 80 • C .
As test structures, cubes were defined (see table 2), where the exposure dose was varied by the laser power (see table 2). The goal of this parameter variation was to write also underand overexposed structures to observe how the usable power dynamic range changes with the temperature. In addition, different heights were realized, to find out, if certain effects during tempering occur at specific axial distances from the substrate. Fig. 8 shows the complete structure at room temperature.
The substrates were cooled down, respectively heated from 5 • C to 80 • C, while the room temperature was measured to be 21,3 • C. Tests were carried out for the two photosensitive polymers IP-Dip and OrmoComp. In addition, the structuring beam was focussed before each  row, to exclude missing structures due to poor adhesion. Fig. 9 shows the classification of the different polymerization grades. Phase 1 and 2 correspond to underexposure, while in phase 2, the shape of the geometry is already visible in contrary to phase 1, where only a small part of the geometry is polymerized. In phase 3, the laser power value is ideal, so a structure with high shape accuracy is realized. Overexposure is defined by phase 4, were the shape accuracy is reduced by too much energy input and blistering occurs.

Results for IP-Dip
Fig . 10 shows the cubes written in IP-Dip for temperatures from 5 • C to 80 • C. It becomes clear, that the energy input is reduced with increasing temperature. While for 5 • C to 20 • C, phase 4, corresponding to overexposure, dominates, no overexposure is observed from 30 • C to 80 • C. Phase 3 (ideal laser power) is present up to 40 • C and from 50 • C to 80 • C only underexposure occurs.The respective phases 2 and 3 become smaller with higher temperatures, so from 60 • C to 80 • C all structures are underexposed (phase 1). The insets in Fig. 10(c) and (g) illustrate, how the surface quality changes with increasing temperature. For 20 • C the surface is even and the single writing lines are partly visible. At 60 • C, the surface texture has become rough and grainy. Fig. 10(a) to (c) demonstrate, that the energy input is higher in the lower planes of a cube, since blistering is more present here. This effect can be explained by the back reflection of a spheric wave at the metal surface. Due to divergence, the energy and thus the laser power is damped with increasing axial distance from the surface, so less energy is present in the upper planes of a cube. Fig. 10(f), right column, also emphasizes this assumption, because the cube with the smallest number of planes has a good shape accuracy and the cube with the highest number of planes is underexposed. This is the reason, why in Fig. 10(a) to (c) the cubes in the two right coloumns are marked as overexposed, although most of them look like they were exposed with the correct dose. Fig. 6(b) and the corresponding discussion about the influence of the laser fluence on the temperature confirm this observation: A higher distance of the focus from the  substrate surface causes the laser fluence on the surface to decrease exponentially, which makes the metals temperature not to change anymore due to the radiation.

Results for OrmoComp
For OrmoComp, the maximum energy input was reached at 30 • C, since it is the only setting, where overexposure occured. With increasing temperatures, less structures are realized at ideal laser power values and underexposure dominates. Regarding structures with different heights, overexposure in the lower planes in comparison to the upper planes cannot be observed. Fig.  11(a) middle column, (f) middle column and (i) second column from right show, that with additional planes, the footprint of a cube becomes larger. It can be assumed that laser power, which did not lead to polymerization is saved in the unexposed polymer, so polymerization occurs faster, when the area is exposed a second time (compare also [7]). Related to IP-Dip, the shrinking in the upper planes is more present. To find out, at which laser powers blistering occurs at higher temperatures, the experiments for For 80 • C, blistering occured that strong from 60 mW, that no polymerization process was possible anymore, since the substrate surface was not visible for the autofocus. At 10 • C and 30 • C, extremely strong blistering occurred, resulting in a Turing pattern over an area, much larger than the structured area (see Fig. 11(b), (d) and Fig. 13). This effect was only observed for OrmoComp, giving the hint to an underlying reaction-diffusion-problem [41]. As introduced by Nitin Uppal et al. [42] (and applied to Ormoceres in [43]), polymerization by TPP can be described by five coupled differential equations (DEQ), giving the change of the temperature and the concentrations of the monomer, the photoinitiator, the radical and the inhibitor. Beside the equation for the temperature, all other DEQ contain a term for diffusion. In addition, the five DEQ are coupled to show, that the regarded quantities influence each other. The diffusion terms are caused by concentration gradients of the regarded quantities. During TPP, polymerization is induced very locally, so large concentration gradients occur and diffusion becomes very important. Thus, we can say, that the reactions are diffusion controlled and TPP is based on a diffusion-controlled system.
Turing patterns are also known as diffusion-driven instabilities. In literature, the stability of reaction-diffusion systems is often described by two coupled, partial differential equations, each containing diffusion terms. The solution is given by an exponential function, according to plane waves, described by wavenumbers. While for ordinary differential equations, it is interesting to find the range, where the stationary state is stable, for the reaction-diffusion-equation, it is of high interest, in which parameter range the spatially homogenous state is not stable. When doing these stability analysis mathematically, one has to find the relation between the Eigenvalue of the  matrix resulting from the differential equations and the wavenumber describing the exponential solution, leading to a curve that describes the square of the wavenumber in dependence on the Eigenvalue. This curve gives a window in which the system is unstable. Unstable means, that a perturbation in the order of the inverse of the wavenumber will be amplified by the system and causes a pattern with a corresponding wavelength (inverse of wavenumber). If the perturbation is not within the window given by the curve, it is not amplified and will not cause a Turing pattern, respectively an instability [41].
As shown in the experiments (Fig. 11), the instability occured at 10 • C and 30 • C, but not at 20 • C. This can be explained by the fact, that there exists a specific parameter range, in which pattern formation can occur. If pattern formation occurs in this parameter range or not depends on small, statistical perturbations, respectively initial conditions. This is similar to bifurcation, where a small change of the initial condition leads to completely different results. Here, the  parameter range, where a Turing pattern is possible is assumed to be between 10 • C and 40 • C. If the initial conditions like the distribution of the concentration of the polymer's ingredients, vary a bit, no pattern formation occurs anymore. This is a very popular effect for strong nonlinearities like reaction-diffusion systems. For higher temperatures, the necessary conditions for Turing patterns are not fulfilled anymore due to underexposure.

Conclusion
Within this paper the temperature in a metal substrate during TPP was calculated with the HCE as well as with the TTM. As expected, both models show a clear increase of the temperature during exposure with the laser. For the TTM, also a small amount of UV radiation is observed. To get an idea of the influence of the substrate's temperature on the polymerization process, teststructures were written in IP-Dip and OrmoComp while varying the temperature. The results show, that with increasing temperatures less polymerization occurs.
In contrast to our results, an increase of polymerization with temperature was shown in [15] and [44]. An explanation, why less polymerization occurs with higher temperatures is given by Fig. 6. The calculation with the TTM in aluminium showed, that the reflectivity decreases with higher temperatures. Considering, that above experiments were performed on steel while consulting the calculation results for steel with the HCE (see Fig. 3), it becomes clear, that steel heats up much more than aluminium for different laser powers (800 K difference for steel, 70 K for aluminium) due to the strongly reduced heat conduction (table 1). Thus, it can be assumed, that heat accumulates much more in steel in comparison to aluminium, causing a stronger decrease of the reflectivity. Additionally, an increase of the refractive index is mentioned in the caption of Fig. 6, confirming, that the material properties change significantly with the temperature. These observations lead to the conclusion, that the temperature of the metal substrate mainly influences the polymerization by changing the reflectivity and that the combination of laser power and reflectivity (in dependence on temperature) determine the optimum exposure dose. Within our experiments, the reduced reflectivity caused an underexposure of the structures.