Abstract

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.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

A. Gutermuth, J. Maassen, E. Harnisch, D. Kuhlen, A. Sauer-Budge, C. Skaazik-Voogt, and K. Engelmann, “Descemet’s membrane biomimetic micro-topography differentiates human mesenchymal stem cells into corneal endothelial-like cells,” Cornea 38, 110–119 (2018).

2017 (2)

E. Stankevicius, E. Daugnoraite, A. Selskis, S. Juodkazis, and G. Raciukaitis, “Photo-polymerization differences by using nanosecond and picosecond laser pulses,” Optics Express 25(5), 4813–48139 (2017).
[Crossref] [PubMed]

E. Harnisch and R. Schmitt, “Two-photon polymerization as a structuring technology in production: Future or fiction?” Proc. of SPIE 10115, 101150Q1 (2017).

2015 (1)

E. Harnisch, M. Russew, J. Klein, N. König, H. Crailsheim, and R. Schmitt, “Optimization of hybrid polymer materials for 2PP and fabrication of individually designed hybrid microoptical elements thereof,” Optical Materials Express 5(2), 456–461 (2015).
[Crossref]

2014 (1)

S. Rekstyte, T. Jonavicius, and M. Malinauskas, “Direct laser writing of microstructures on optically opaque and reflective surfaces,” Optics and Lasers in Engineering 53, 90–97 (2014).
[Crossref]

2013 (3)

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D polymer micro/nanostructures on metallic surfaces,” Applied Surface Science 270, 382–387 (2013).
[Crossref]

J. Fischer and M. Wegener, “Three-dimensional optical laser lithography beyond the diffraction limit,” Laser Photonics Rev. 7(1), 22–44 (2013).
[Crossref]

A. Zukauskas, M. Malinauskas, A. Kadys, G. Gervinskas, G. Seniutinas, S. Kandasamy, and S. Juodkazis, “Black silison: substrate for laser 3D micro/nano-polymerization,” Optics Letters 31(9), 1307–1309 (2013).

2012 (3)

T. Bückmann and et al.., “Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography,” Optical Materials Express 24(20), 2710–2714 (2012).

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D micro/nanostructures on opaque surfaces,” Proc. SPIE 8431, 843123 (2012).
[Crossref]

M. Roehrig, M. Thiel, M. Worgull, and H. Hoelscher, “3D Direct Laser Writing of Nano-and Microstructured Hierarchical Gecko-Mimicking Surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

2011 (2)

F. Klein, B. Richter, T. Striebel, C. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two component polymer scaffolds for controlled three-dimensional cell culture,” Advanced Materials 23(11), 1341–1345 (2011).
[Crossref] [PubMed]

Z. Zhong, M.H. Leong, and X.D. Liu, “The wear rates and performance of three mold insert materials,” Materials & Design 32(2), 643–648 (2011).
[Crossref]

2010 (1)

E. Majchrzak and Jolanta Poteralska, “Two-temperature microscale heat transfer model. Part II: Determination of lattice parameters,” Scientific Research of the Institute of Mathematics and Computer Science 9(1), 109–119 (2010).

2009 (1)

I. A. Abdallah, “Maxwell-Cattaneo Heat Convection and Thermal Stresses Responses of a Semi-infinite Medium due to High Speed Laser Heating,” Progress in Physics 3(12), 12–17 (2009).

2008 (1)

N. Uppal and P.S. Shiakolas, “Modeling of temperature-dependent diffusion and polymerization kinetics and their effects on two-photon polymerization dynamics,” Journal of Micro/Nanolithography, MEMS, and MOEMS 7(4), 043002 (2008).
[Crossref]

2007 (1)

B. Jia and M. Gu, “Two-Photon Polymerization for Three-Dimensional Photonic Devices in Polymers and Nanocomposites,” Aust. J. Chem 60(7), 484–495 (2007).
[Crossref]

2006 (2)

C. Brecher, F. Klocke, and M. Winterschladen, “Ultraschallunterstuetztes Hartdrehen für die Fertigung von gehaerteten Praezisionsstahlbauteilen,” Wt Werkstattstechnik online 6, 396–401 (2006).

C. Rheinhardt, S. Passinger, B.N. Chichkoc, C. Marquart, I.P. Radko, and S.I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons,” Optics Express 21(6), 6901–6909 (2006).

2005 (2)

J.K. Chen, W.P. Latham, and J.E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” Journal of Laser Applications 17(1), 63–68 (2005).
[Crossref]

L. Jiang and H.-L. Tsai, “Improved Two-Temperature Model and Its Application in Ultrashort Laser Heating of Metal Films,” Journal of Heat Transfer 127(10), 1167–1173 (2005).
[Crossref]

2004 (1)

J. Schneider, H. Iwanek, and K.H. Zum Gahr, “Charakterisierung des Verschleißverhaltens von Formeinsatz-Werkstoffen für das Mikro-Pulverspritzgießen,” Materialwissenschaft und Werkstofftechnik 35(10–11), 729–735 (2004).
[Crossref]

2002 (1)

I. Wang, M. Bouriau, and P.L. Baldeck, “Three-dimensional microfabrication by two-photon-initiated polymerization with a low-cost microlaser,” Optics Letters 27(15), 1348–1350 (2002).
[Crossref]

1998 (1)

A.D. Rakic and et al.., “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Applied Optics 37(22), 5271–5283 (1998).
[Crossref]

1997 (1)

1974 (1)

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel’man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39(2), 375–377 (1974).

1952 (1)

A.M. Turing, “The Chemical Basis of Morphogenesis,” Philosophical Transactions of the Royal Society of London B: Biological Sciences 237(641), 37–72 (1952).
[Crossref]

1912 (1)

P. Debye, “Zur theorie der spezifischen wärmen,” Annalen der Physik 344(14), 789–839 (1912).
[Crossref]

Abdallah, I. A.

I. A. Abdallah, “Maxwell-Cattaneo Heat Convection and Thermal Stresses Responses of a Semi-infinite Medium due to High Speed Laser Heating,” Progress in Physics 3(12), 12–17 (2009).

Abeln, T.

T. Abeln, Grundlagen der Verfahrenstechnik des reaktiven Laserpraezisionsabtragens von Stahl (Herbert Utz Verlag Wissenschaft Muenchen, 2002).

Anisimov, S. I.

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel’man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39(2), 375–377 (1974).

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Festkoerperphysik (Oldenburg Wissenschaftsverlag GmbH, 2013).

Baehren, M.

M. Baehren, H. Hartmann, G. Hein, A. Loeffler, K. Manok, J. Polzin, and U. Tober, Photonik–Branchenreport 2013 (SPECTARIS, VDMA, ZVEI, BMBF, 2013).

Baldeck, P.L.

I. Wang, M. Bouriau, and P.L. Baldeck, “Three-dimensional microfabrication by two-photon-initiated polymerization with a low-cost microlaser,” Optics Letters 27(15), 1348–1350 (2002).
[Crossref]

Bastmeyer, M.

F. Klein, B. Richter, T. Striebel, C. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two component polymer scaffolds for controlled three-dimensional cell culture,” Advanced Materials 23(11), 1341–1345 (2011).
[Crossref] [PubMed]

Beraun, J.E.

J.K. Chen, W.P. Latham, and J.E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” Journal of Laser Applications 17(1), 63–68 (2005).
[Crossref]

Binder, H.H.

H.H. Binder, Lexikon der chemischen Elemente: das Periodensystem in Fakten, Zahlen u. Daten; Mit einem Geleitw. von I. Barbur (Hirzel Verl, 1999).

Bouriau, M.

I. Wang, M. Bouriau, and P.L. Baldeck, “Three-dimensional microfabrication by two-photon-initiated polymerization with a low-cost microlaser,” Optics Letters 27(15), 1348–1350 (2002).
[Crossref]

Bozhevolnyi, S.I.

C. Rheinhardt, S. Passinger, B.N. Chichkoc, C. Marquart, I.P. Radko, and S.I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons,” Optics Express 21(6), 6901–6909 (2006).

Brecher, C.

C. Brecher, F. Klocke, and M. Winterschladen, “Ultraschallunterstuetztes Hartdrehen für die Fertigung von gehaerteten Praezisionsstahlbauteilen,” Wt Werkstattstechnik online 6, 396–401 (2006).

Bückmann, T.

T. Bückmann and et al.., “Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography,” Optical Materials Express 24(20), 2710–2714 (2012).

Chen, J.K.

J.K. Chen, W.P. Latham, and J.E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” Journal of Laser Applications 17(1), 63–68 (2005).
[Crossref]

Chichkoc, B.N.

C. Rheinhardt, S. Passinger, B.N. Chichkoc, C. Marquart, I.P. Radko, and S.I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons,” Optics Express 21(6), 6901–6909 (2006).

Crailsheim, H.

E. Harnisch, M. Russew, J. Klein, N. König, H. Crailsheim, and R. Schmitt, “Optimization of hybrid polymer materials for 2PP and fabrication of individually designed hybrid microoptical elements thereof,” Optical Materials Express 5(2), 456–461 (2015).
[Crossref]

Date, A.

A. Date, Introduction to computational fluid dynamics (Cambridge University Press, 2005).
[Crossref]

Daugnoraite, E.

E. Stankevicius, E. Daugnoraite, A. Selskis, S. Juodkazis, and G. Raciukaitis, “Photo-polymerization differences by using nanosecond and picosecond laser pulses,” Optics Express 25(5), 4813–48139 (2017).
[Crossref] [PubMed]

Debye, P.

P. Debye, “Zur theorie der spezifischen wärmen,” Annalen der Physik 344(14), 789–839 (1912).
[Crossref]

Engelmann, K.

A. Gutermuth, J. Maassen, E. Harnisch, D. Kuhlen, A. Sauer-Budge, C. Skaazik-Voogt, and K. Engelmann, “Descemet’s membrane biomimetic micro-topography differentiates human mesenchymal stem cells into corneal endothelial-like cells,” Cornea 38, 110–119 (2018).

Fischer, J.

J. Fischer and M. Wegener, “Three-dimensional optical laser lithography beyond the diffraction limit,” Laser Photonics Rev. 7(1), 22–44 (2013).
[Crossref]

Franz, C.

F. Klein, B. Richter, T. Striebel, C. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two component polymer scaffolds for controlled three-dimensional cell culture,” Advanced Materials 23(11), 1341–1345 (2011).
[Crossref] [PubMed]

Gervinskas, G.

A. Zukauskas, M. Malinauskas, A. Kadys, G. Gervinskas, G. Seniutinas, S. Kandasamy, and S. Juodkazis, “Black silison: substrate for laser 3D micro/nano-polymerization,” Optics Letters 31(9), 1307–1309 (2013).

Gordienko, Y.

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D polymer micro/nanostructures on metallic surfaces,” Applied Surface Science 270, 382–387 (2013).
[Crossref]

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D micro/nanostructures on opaque surfaces,” Proc. SPIE 8431, 843123 (2012).
[Crossref]

Gu, M.

B. Jia and M. Gu, “Two-Photon Polymerization for Three-Dimensional Photonic Devices in Polymers and Nanocomposites,” Aust. J. Chem 60(7), 484–495 (2007).
[Crossref]

Gutermuth, A.

A. Gutermuth, J. Maassen, E. Harnisch, D. Kuhlen, A. Sauer-Budge, C. Skaazik-Voogt, and K. Engelmann, “Descemet’s membrane biomimetic micro-topography differentiates human mesenchymal stem cells into corneal endothelial-like cells,” Cornea 38, 110–119 (2018).

Harnisch, E.

A. Gutermuth, J. Maassen, E. Harnisch, D. Kuhlen, A. Sauer-Budge, C. Skaazik-Voogt, and K. Engelmann, “Descemet’s membrane biomimetic micro-topography differentiates human mesenchymal stem cells into corneal endothelial-like cells,” Cornea 38, 110–119 (2018).

E. Harnisch and R. Schmitt, “Two-photon polymerization as a structuring technology in production: Future or fiction?” Proc. of SPIE 10115, 101150Q1 (2017).

E. Harnisch, M. Russew, J. Klein, N. König, H. Crailsheim, and R. Schmitt, “Optimization of hybrid polymer materials for 2PP and fabrication of individually designed hybrid microoptical elements thereof,” Optical Materials Express 5(2), 456–461 (2015).
[Crossref]

Hartmann, H.

M. Baehren, H. Hartmann, G. Hein, A. Loeffler, K. Manok, J. Polzin, and U. Tober, Photonik–Branchenreport 2013 (SPECTARIS, VDMA, ZVEI, BMBF, 2013).

Hein, G.

M. Baehren, H. Hartmann, G. Hein, A. Loeffler, K. Manok, J. Polzin, and U. Tober, Photonik–Branchenreport 2013 (SPECTARIS, VDMA, ZVEI, BMBF, 2013).

Hoelscher, H.

M. Roehrig, M. Thiel, M. Worgull, and H. Hoelscher, “3D Direct Laser Writing of Nano-and Microstructured Hierarchical Gecko-Mimicking Surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

Iwanek, H.

J. Schneider, H. Iwanek, and K.H. Zum Gahr, “Charakterisierung des Verschleißverhaltens von Formeinsatz-Werkstoffen für das Mikro-Pulverspritzgießen,” Materialwissenschaft und Werkstofftechnik 35(10–11), 729–735 (2004).
[Crossref]

Jia, B.

B. Jia and M. Gu, “Two-Photon Polymerization for Three-Dimensional Photonic Devices in Polymers and Nanocomposites,” Aust. J. Chem 60(7), 484–495 (2007).
[Crossref]

Jiang, L.

L. Jiang and H.-L. Tsai, “Improved Two-Temperature Model and Its Application in Ultrashort Laser Heating of Metal Films,” Journal of Heat Transfer 127(10), 1167–1173 (2005).
[Crossref]

Jonavicius, T.

S. Rekstyte, T. Jonavicius, and M. Malinauskas, “Direct laser writing of microstructures on optically opaque and reflective surfaces,” Optics and Lasers in Engineering 53, 90–97 (2014).
[Crossref]

Juodkazis, S.

E. Stankevicius, E. Daugnoraite, A. Selskis, S. Juodkazis, and G. Raciukaitis, “Photo-polymerization differences by using nanosecond and picosecond laser pulses,” Optics Express 25(5), 4813–48139 (2017).
[Crossref] [PubMed]

A. Zukauskas, M. Malinauskas, A. Kadys, G. Gervinskas, G. Seniutinas, S. Kandasamy, and S. Juodkazis, “Black silison: substrate for laser 3D micro/nano-polymerization,” Optics Letters 31(9), 1307–1309 (2013).

Kadys, A.

A. Zukauskas, M. Malinauskas, A. Kadys, G. Gervinskas, G. Seniutinas, S. Kandasamy, and S. Juodkazis, “Black silison: substrate for laser 3D micro/nano-polymerization,” Optics Letters 31(9), 1307–1309 (2013).

Kandasamy, S.

A. Zukauskas, M. Malinauskas, A. Kadys, G. Gervinskas, G. Seniutinas, S. Kandasamy, and S. Juodkazis, “Black silison: substrate for laser 3D micro/nano-polymerization,” Optics Letters 31(9), 1307–1309 (2013).

Kapeliovich, B. L.

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel’man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39(2), 375–377 (1974).

Kawata, S.

Klein, F.

F. Klein, B. Richter, T. Striebel, C. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two component polymer scaffolds for controlled three-dimensional cell culture,” Advanced Materials 23(11), 1341–1345 (2011).
[Crossref] [PubMed]

Klein, J.

E. Harnisch, M. Russew, J. Klein, N. König, H. Crailsheim, and R. Schmitt, “Optimization of hybrid polymer materials for 2PP and fabrication of individually designed hybrid microoptical elements thereof,” Optical Materials Express 5(2), 456–461 (2015).
[Crossref]

Klocke, F.

C. Brecher, F. Klocke, and M. Winterschladen, “Ultraschallunterstuetztes Hartdrehen für die Fertigung von gehaerteten Praezisionsstahlbauteilen,” Wt Werkstattstechnik online 6, 396–401 (2006).

König, N.

E. Harnisch, M. Russew, J. Klein, N. König, H. Crailsheim, and R. Schmitt, “Optimization of hybrid polymer materials for 2PP and fabrication of individually designed hybrid microoptical elements thereof,” Optical Materials Express 5(2), 456–461 (2015).
[Crossref]

Kuhlen, D.

A. Gutermuth, J. Maassen, E. Harnisch, D. Kuhlen, A. Sauer-Budge, C. Skaazik-Voogt, and K. Engelmann, “Descemet’s membrane biomimetic micro-topography differentiates human mesenchymal stem cells into corneal endothelial-like cells,” Cornea 38, 110–119 (2018).

Latham, W.P.

J.K. Chen, W.P. Latham, and J.E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” Journal of Laser Applications 17(1), 63–68 (2005).
[Crossref]

Leong, M.H.

Z. Zhong, M.H. Leong, and X.D. Liu, “The wear rates and performance of three mold insert materials,” Materials & Design 32(2), 643–648 (2011).
[Crossref]

Liu, X.D.

Z. Zhong, M.H. Leong, and X.D. Liu, “The wear rates and performance of three mold insert materials,” Materials & Design 32(2), 643–648 (2011).
[Crossref]

Loeffler, A.

M. Baehren, H. Hartmann, G. Hein, A. Loeffler, K. Manok, J. Polzin, and U. Tober, Photonik–Branchenreport 2013 (SPECTARIS, VDMA, ZVEI, BMBF, 2013).

Maassen, J.

A. Gutermuth, J. Maassen, E. Harnisch, D. Kuhlen, A. Sauer-Budge, C. Skaazik-Voogt, and K. Engelmann, “Descemet’s membrane biomimetic micro-topography differentiates human mesenchymal stem cells into corneal endothelial-like cells,” Cornea 38, 110–119 (2018).

Majchrzak, E.

E. Majchrzak and Jolanta Poteralska, “Two-temperature microscale heat transfer model. Part II: Determination of lattice parameters,” Scientific Research of the Institute of Mathematics and Computer Science 9(1), 109–119 (2010).

Malalasekera, W.

H.K. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics – The finite volume method (Longman Scientific & Technical, 1995).

Malinauskas, M.

S. Rekstyte, T. Jonavicius, and M. Malinauskas, “Direct laser writing of microstructures on optically opaque and reflective surfaces,” Optics and Lasers in Engineering 53, 90–97 (2014).
[Crossref]

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D polymer micro/nanostructures on metallic surfaces,” Applied Surface Science 270, 382–387 (2013).
[Crossref]

A. Zukauskas, M. Malinauskas, A. Kadys, G. Gervinskas, G. Seniutinas, S. Kandasamy, and S. Juodkazis, “Black silison: substrate for laser 3D micro/nano-polymerization,” Optics Letters 31(9), 1307–1309 (2013).

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D micro/nanostructures on opaque surfaces,” Proc. SPIE 8431, 843123 (2012).
[Crossref]

Manok, K.

M. Baehren, H. Hartmann, G. Hein, A. Loeffler, K. Manok, J. Polzin, and U. Tober, Photonik–Branchenreport 2013 (SPECTARIS, VDMA, ZVEI, BMBF, 2013).

Marquart, C.

C. Rheinhardt, S. Passinger, B.N. Chichkoc, C. Marquart, I.P. Radko, and S.I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons,” Optics Express 21(6), 6901–6909 (2006).

Maruo, S.

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Festkoerperphysik (Oldenburg Wissenschaftsverlag GmbH, 2013).

Mosca, G.

P.A. Tipler and G. Mosca, Physik fuer Wissenschaftler und Ingenieure (SpringerSpektrum, 2015).

Nakamura, O.

Passinger, S.

C. Rheinhardt, S. Passinger, B.N. Chichkoc, C. Marquart, I.P. Radko, and S.I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons,” Optics Express 21(6), 6901–6909 (2006).

Perel’man, T. L.

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel’man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39(2), 375–377 (1974).

Polzin, J.

M. Baehren, H. Hartmann, G. Hein, A. Loeffler, K. Manok, J. Polzin, and U. Tober, Photonik–Branchenreport 2013 (SPECTARIS, VDMA, ZVEI, BMBF, 2013).

Poprawe, R.

R. Poprawe, Lasertechnik fuer die Fertigung (Springer-VerlagBerlin Heidelberg, 2005).

Poteralska, Jolanta

E. Majchrzak and Jolanta Poteralska, “Two-temperature microscale heat transfer model. Part II: Determination of lattice parameters,” Scientific Research of the Institute of Mathematics and Computer Science 9(1), 109–119 (2010).

Purlys, V.

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D polymer micro/nanostructures on metallic surfaces,” Applied Surface Science 270, 382–387 (2013).
[Crossref]

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D micro/nanostructures on opaque surfaces,” Proc. SPIE 8431, 843123 (2012).
[Crossref]

Raciukaitis, G.

E. Stankevicius, E. Daugnoraite, A. Selskis, S. Juodkazis, and G. Raciukaitis, “Photo-polymerization differences by using nanosecond and picosecond laser pulses,” Optics Express 25(5), 4813–48139 (2017).
[Crossref] [PubMed]

Radko, I.P.

C. Rheinhardt, S. Passinger, B.N. Chichkoc, C. Marquart, I.P. Radko, and S.I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons,” Optics Express 21(6), 6901–6909 (2006).

Rakic, A.D.

A.D. Rakic and et al.., “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Applied Optics 37(22), 5271–5283 (1998).
[Crossref]

Rekstyte, S.

S. Rekstyte, T. Jonavicius, and M. Malinauskas, “Direct laser writing of microstructures on optically opaque and reflective surfaces,” Optics and Lasers in Engineering 53, 90–97 (2014).
[Crossref]

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D polymer micro/nanostructures on metallic surfaces,” Applied Surface Science 270, 382–387 (2013).
[Crossref]

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D micro/nanostructures on opaque surfaces,” Proc. SPIE 8431, 843123 (2012).
[Crossref]

Rheinhardt, C.

C. Rheinhardt, S. Passinger, B.N. Chichkoc, C. Marquart, I.P. Radko, and S.I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons,” Optics Express 21(6), 6901–6909 (2006).

Richter, B.

F. Klein, B. Richter, T. Striebel, C. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two component polymer scaffolds for controlled three-dimensional cell culture,” Advanced Materials 23(11), 1341–1345 (2011).
[Crossref] [PubMed]

Roehrig, M.

M. Roehrig, M. Thiel, M. Worgull, and H. Hoelscher, “3D Direct Laser Writing of Nano-and Microstructured Hierarchical Gecko-Mimicking Surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

Russew, M.

E. Harnisch, M. Russew, J. Klein, N. König, H. Crailsheim, and R. Schmitt, “Optimization of hybrid polymer materials for 2PP and fabrication of individually designed hybrid microoptical elements thereof,” Optical Materials Express 5(2), 456–461 (2015).
[Crossref]

Sauer-Budge, A.

A. Gutermuth, J. Maassen, E. Harnisch, D. Kuhlen, A. Sauer-Budge, C. Skaazik-Voogt, and K. Engelmann, “Descemet’s membrane biomimetic micro-topography differentiates human mesenchymal stem cells into corneal endothelial-like cells,” Cornea 38, 110–119 (2018).

Schmitt, R.

E. Harnisch and R. Schmitt, “Two-photon polymerization as a structuring technology in production: Future or fiction?” Proc. of SPIE 10115, 101150Q1 (2017).

E. Harnisch, M. Russew, J. Klein, N. König, H. Crailsheim, and R. Schmitt, “Optimization of hybrid polymer materials for 2PP and fabrication of individually designed hybrid microoptical elements thereof,” Optical Materials Express 5(2), 456–461 (2015).
[Crossref]

Schneider, J.

J. Schneider, H. Iwanek, and K.H. Zum Gahr, “Charakterisierung des Verschleißverhaltens von Formeinsatz-Werkstoffen für das Mikro-Pulverspritzgießen,” Materialwissenschaft und Werkstofftechnik 35(10–11), 729–735 (2004).
[Crossref]

Selskis, A.

E. Stankevicius, E. Daugnoraite, A. Selskis, S. Juodkazis, and G. Raciukaitis, “Photo-polymerization differences by using nanosecond and picosecond laser pulses,” Optics Express 25(5), 4813–48139 (2017).
[Crossref] [PubMed]

Seniutinas, G.

A. Zukauskas, M. Malinauskas, A. Kadys, G. Gervinskas, G. Seniutinas, S. Kandasamy, and S. Juodkazis, “Black silison: substrate for laser 3D micro/nano-polymerization,” Optics Letters 31(9), 1307–1309 (2013).

Shiakolas, P.S.

N. Uppal and P.S. Shiakolas, “Modeling of temperature-dependent diffusion and polymerization kinetics and their effects on two-photon polymerization dynamics,” Journal of Micro/Nanolithography, MEMS, and MOEMS 7(4), 043002 (2008).
[Crossref]

Skaazik-Voogt, C.

A. Gutermuth, J. Maassen, E. Harnisch, D. Kuhlen, A. Sauer-Budge, C. Skaazik-Voogt, and K. Engelmann, “Descemet’s membrane biomimetic micro-topography differentiates human mesenchymal stem cells into corneal endothelial-like cells,” Cornea 38, 110–119 (2018).

Stankevicius, E.

E. Stankevicius, E. Daugnoraite, A. Selskis, S. Juodkazis, and G. Raciukaitis, “Photo-polymerization differences by using nanosecond and picosecond laser pulses,” Optics Express 25(5), 4813–48139 (2017).
[Crossref] [PubMed]

Stichel, T.

T. Stichel,“Die Herstellung von Scaffolds aus funktionellen Hybridpolymeren für die regenerative Medizin mittels Zwei-Photonen-Polymerisation," PhD thesis, Julius-Maximilians-UniversitaetWuerzburg (2016).

Striebel, T.

F. Klein, B. Richter, T. Striebel, C. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two component polymer scaffolds for controlled three-dimensional cell culture,” Advanced Materials 23(11), 1341–1345 (2011).
[Crossref] [PubMed]

Thiel, M.

M. Roehrig, M. Thiel, M. Worgull, and H. Hoelscher, “3D Direct Laser Writing of Nano-and Microstructured Hierarchical Gecko-Mimicking Surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

Tipler, P.A.

P.A. Tipler and G. Mosca, Physik fuer Wissenschaftler und Ingenieure (SpringerSpektrum, 2015).

Tober, U.

M. Baehren, H. Hartmann, G. Hein, A. Loeffler, K. Manok, J. Polzin, and U. Tober, Photonik–Branchenreport 2013 (SPECTARIS, VDMA, ZVEI, BMBF, 2013).

Tsai, H.-L.

L. Jiang and H.-L. Tsai, “Improved Two-Temperature Model and Its Application in Ultrashort Laser Heating of Metal Films,” Journal of Heat Transfer 127(10), 1167–1173 (2005).
[Crossref]

Turing, A.M.

A.M. Turing, “The Chemical Basis of Morphogenesis,” Philosophical Transactions of the Royal Society of London B: Biological Sciences 237(641), 37–72 (1952).
[Crossref]

Uppal, N.

N. Uppal and P.S. Shiakolas, “Modeling of temperature-dependent diffusion and polymerization kinetics and their effects on two-photon polymerization dynamics,” Journal of Micro/Nanolithography, MEMS, and MOEMS 7(4), 043002 (2008).
[Crossref]

Versteeg, H.K.

H.K. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics – The finite volume method (Longman Scientific & Technical, 1995).

von Freymann, G.

F. Klein, B. Richter, T. Striebel, C. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two component polymer scaffolds for controlled three-dimensional cell culture,” Advanced Materials 23(11), 1341–1345 (2011).
[Crossref] [PubMed]

Wang, I.

I. Wang, M. Bouriau, and P.L. Baldeck, “Three-dimensional microfabrication by two-photon-initiated polymerization with a low-cost microlaser,” Optics Letters 27(15), 1348–1350 (2002).
[Crossref]

Wegener, M.

J. Fischer and M. Wegener, “Three-dimensional optical laser lithography beyond the diffraction limit,” Laser Photonics Rev. 7(1), 22–44 (2013).
[Crossref]

F. Klein, B. Richter, T. Striebel, C. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two component polymer scaffolds for controlled three-dimensional cell culture,” Advanced Materials 23(11), 1341–1345 (2011).
[Crossref] [PubMed]

Winterschladen, M.

C. Brecher, F. Klocke, and M. Winterschladen, “Ultraschallunterstuetztes Hartdrehen für die Fertigung von gehaerteten Praezisionsstahlbauteilen,” Wt Werkstattstechnik online 6, 396–401 (2006).

Worgull, M.

M. Roehrig, M. Thiel, M. Worgull, and H. Hoelscher, “3D Direct Laser Writing of Nano-and Microstructured Hierarchical Gecko-Mimicking Surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

Zhong, Z.

Z. Zhong, M.H. Leong, and X.D. Liu, “The wear rates and performance of three mold insert materials,” Materials & Design 32(2), 643–648 (2011).
[Crossref]

Zukauskas, A.

A. Zukauskas, M. Malinauskas, A. Kadys, G. Gervinskas, G. Seniutinas, S. Kandasamy, and S. Juodkazis, “Black silison: substrate for laser 3D micro/nano-polymerization,” Optics Letters 31(9), 1307–1309 (2013).

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D polymer micro/nanostructures on metallic surfaces,” Applied Surface Science 270, 382–387 (2013).
[Crossref]

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D micro/nanostructures on opaque surfaces,” Proc. SPIE 8431, 843123 (2012).
[Crossref]

Zum Gahr, K.H.

J. Schneider, H. Iwanek, and K.H. Zum Gahr, “Charakterisierung des Verschleißverhaltens von Formeinsatz-Werkstoffen für das Mikro-Pulverspritzgießen,” Materialwissenschaft und Werkstofftechnik 35(10–11), 729–735 (2004).
[Crossref]

Advanced Materials (1)

F. Klein, B. Richter, T. Striebel, C. Franz, G. von Freymann, M. Wegener, and M. Bastmeyer, “Two component polymer scaffolds for controlled three-dimensional cell culture,” Advanced Materials 23(11), 1341–1345 (2011).
[Crossref] [PubMed]

Annalen der Physik (1)

P. Debye, “Zur theorie der spezifischen wärmen,” Annalen der Physik 344(14), 789–839 (1912).
[Crossref]

Applied Optics (1)

A.D. Rakic and et al.., “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Applied Optics 37(22), 5271–5283 (1998).
[Crossref]

Applied Surface Science (1)

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D polymer micro/nanostructures on metallic surfaces,” Applied Surface Science 270, 382–387 (2013).
[Crossref]

Aust. J. Chem (1)

B. Jia and M. Gu, “Two-Photon Polymerization for Three-Dimensional Photonic Devices in Polymers and Nanocomposites,” Aust. J. Chem 60(7), 484–495 (2007).
[Crossref]

Cornea (1)

A. Gutermuth, J. Maassen, E. Harnisch, D. Kuhlen, A. Sauer-Budge, C. Skaazik-Voogt, and K. Engelmann, “Descemet’s membrane biomimetic micro-topography differentiates human mesenchymal stem cells into corneal endothelial-like cells,” Cornea 38, 110–119 (2018).

Journal of Heat Transfer (1)

L. Jiang and H.-L. Tsai, “Improved Two-Temperature Model and Its Application in Ultrashort Laser Heating of Metal Films,” Journal of Heat Transfer 127(10), 1167–1173 (2005).
[Crossref]

Journal of Laser Applications (1)

J.K. Chen, W.P. Latham, and J.E. Beraun, “The role of electron–phonon coupling in ultrafast laser heating,” Journal of Laser Applications 17(1), 63–68 (2005).
[Crossref]

Journal of Micro/Nanolithography, MEMS, and MOEMS (1)

N. Uppal and P.S. Shiakolas, “Modeling of temperature-dependent diffusion and polymerization kinetics and their effects on two-photon polymerization dynamics,” Journal of Micro/Nanolithography, MEMS, and MOEMS 7(4), 043002 (2008).
[Crossref]

Laser Photonics Rev. (1)

J. Fischer and M. Wegener, “Three-dimensional optical laser lithography beyond the diffraction limit,” Laser Photonics Rev. 7(1), 22–44 (2013).
[Crossref]

Materials & Design (1)

Z. Zhong, M.H. Leong, and X.D. Liu, “The wear rates and performance of three mold insert materials,” Materials & Design 32(2), 643–648 (2011).
[Crossref]

Materialwissenschaft und Werkstofftechnik (1)

J. Schneider, H. Iwanek, and K.H. Zum Gahr, “Charakterisierung des Verschleißverhaltens von Formeinsatz-Werkstoffen für das Mikro-Pulverspritzgießen,” Materialwissenschaft und Werkstofftechnik 35(10–11), 729–735 (2004).
[Crossref]

Opt. Lett. (1)

Optical Materials Express (2)

E. Harnisch, M. Russew, J. Klein, N. König, H. Crailsheim, and R. Schmitt, “Optimization of hybrid polymer materials for 2PP and fabrication of individually designed hybrid microoptical elements thereof,” Optical Materials Express 5(2), 456–461 (2015).
[Crossref]

T. Bückmann and et al.., “Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography,” Optical Materials Express 24(20), 2710–2714 (2012).

Optics and Lasers in Engineering (1)

S. Rekstyte, T. Jonavicius, and M. Malinauskas, “Direct laser writing of microstructures on optically opaque and reflective surfaces,” Optics and Lasers in Engineering 53, 90–97 (2014).
[Crossref]

Optics Express (2)

C. Rheinhardt, S. Passinger, B.N. Chichkoc, C. Marquart, I.P. Radko, and S.I. Bozhevolnyi, “Laser-fabricated dielectric optical components for surface plasmon polaritons,” Optics Express 21(6), 6901–6909 (2006).

E. Stankevicius, E. Daugnoraite, A. Selskis, S. Juodkazis, and G. Raciukaitis, “Photo-polymerization differences by using nanosecond and picosecond laser pulses,” Optics Express 25(5), 4813–48139 (2017).
[Crossref] [PubMed]

Optics Letters (2)

A. Zukauskas, M. Malinauskas, A. Kadys, G. Gervinskas, G. Seniutinas, S. Kandasamy, and S. Juodkazis, “Black silison: substrate for laser 3D micro/nano-polymerization,” Optics Letters 31(9), 1307–1309 (2013).

I. Wang, M. Bouriau, and P.L. Baldeck, “Three-dimensional microfabrication by two-photon-initiated polymerization with a low-cost microlaser,” Optics Letters 27(15), 1348–1350 (2002).
[Crossref]

Philosophical Transactions of the Royal Society of London B: Biological Sciences (1)

A.M. Turing, “The Chemical Basis of Morphogenesis,” Philosophical Transactions of the Royal Society of London B: Biological Sciences 237(641), 37–72 (1952).
[Crossref]

Proc. of SPIE (1)

E. Harnisch and R. Schmitt, “Two-photon polymerization as a structuring technology in production: Future or fiction?” Proc. of SPIE 10115, 101150Q1 (2017).

Proc. SPIE (1)

S. Rekstyte, A. Zukauskas, V. Purlys, Y. Gordienko, and M. Malinauskas, “Direct laser writing of 3D micro/nanostructures on opaque surfaces,” Proc. SPIE 8431, 843123 (2012).
[Crossref]

Progress in Physics (1)

I. A. Abdallah, “Maxwell-Cattaneo Heat Convection and Thermal Stresses Responses of a Semi-infinite Medium due to High Speed Laser Heating,” Progress in Physics 3(12), 12–17 (2009).

Scientific Research of the Institute of Mathematics and Computer Science (1)

E. Majchrzak and Jolanta Poteralska, “Two-temperature microscale heat transfer model. Part II: Determination of lattice parameters,” Scientific Research of the Institute of Mathematics and Computer Science 9(1), 109–119 (2010).

Small (1)

M. Roehrig, M. Thiel, M. Worgull, and H. Hoelscher, “3D Direct Laser Writing of Nano-and Microstructured Hierarchical Gecko-Mimicking Surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

Sov. Phys. JETP (1)

S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel’man, “Electron emission from metal surfaces exposed to ultrashort laser pulses,” Sov. Phys. JETP 39(2), 375–377 (1974).

Wt Werkstattstechnik online (1)

C. Brecher, F. Klocke, and M. Winterschladen, “Ultraschallunterstuetztes Hartdrehen für die Fertigung von gehaerteten Praezisionsstahlbauteilen,” Wt Werkstattstechnik online 6, 396–401 (2006).

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N. W. Ashcroft and N. D. Mermin, Festkoerperphysik (Oldenburg Wissenschaftsverlag GmbH, 2013).

T. Abeln, Grundlagen der Verfahrenstechnik des reaktiven Laserpraezisionsabtragens von Stahl (Herbert Utz Verlag Wissenschaft Muenchen, 2002).

Filmetrics, “Refractive index database,"(Filmetrics, 2014), https://www.filmetrics.com/refractive-index-database/Stainless+Steel .

Stauber Werkzeugstahl und Edelstahl, “Datenblatt 1.2083," https://www.stauberstahl.com/werkstoffe/12083-werkstoff-datenblatt/ .

H.H. Binder, Lexikon der chemischen Elemente: das Periodensystem in Fakten, Zahlen u. Daten; Mit einem Geleitw. von I. Barbur (Hirzel Verl, 1999).

RSP Technology, “Alloy Overview: RSP Technology RSA-905 AE Aluminium Super Alloy," http://www.rsp-technology.com/datasheets-news.html .

Deutsches Kupferinstitut Copper Alliance, “Werkstoffe: Datenblaetter," https://www.kupferinstitut.de/de/persoenlicheberatung/downloads/downloads/werkstoffe/werkstoff-datenblaetter.html .

RefractiveIndex.Info, “Refractive index database," (RefractiveIndex.INFO website, 2008-2018), https://refractiveindex.info/?shelf=3d&book=metals&page=aluminium .

RefractiveIndex.Info, “Refractive index database," (RefractiveIndex.INFO website, 2008-2018), https://refractiveindex.info/?shelf=3d&book=metals&page=brass .

M. Baehren, H. Hartmann, G. Hein, A. Loeffler, K. Manok, J. Polzin, and U. Tober, Photonik–Branchenreport 2013 (SPECTARIS, VDMA, ZVEI, BMBF, 2013).

R. Poprawe, Lasertechnik fuer die Fertigung (Springer-VerlagBerlin Heidelberg, 2005).

A. Date, Introduction to computational fluid dynamics (Cambridge University Press, 2005).
[Crossref]

H.K. Versteeg and W. Malalasekera, An introduction to computational fluid dynamics – The finite volume method (Longman Scientific & Technical, 1995).

T. Stichel,“Die Herstellung von Scaffolds aus funktionellen Hybridpolymeren für die regenerative Medizin mittels Zwei-Photonen-Polymerisation," PhD thesis, Julius-Maximilians-UniversitaetWuerzburg (2016).

Mathworks, “Drude-Lorentz and Debye-Lorentz models for the dielectric constant of metals and water," (MathWorks website, 1994-2018), http://de.mathworks.com/matlabcentral/fileexchange/18040-drude-lorentz-and-debye-lorentz-models-for-the-dielectric-constant-of-metals-and-water .

P.A. Tipler and G. Mosca, Physik fuer Wissenschaftler und Ingenieure (SpringerSpektrum, 2015).

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Figures (13)

Fig. 1
Fig. 1 (a) Pyramid written with TPP on a glass substrate, (b) Pyramid written with TPP on a metal substrate. Each pyramid has a height of 10 μ m, an edge length of 20 μ m and was written with 16 mW and 1000 μ m / s in IP-Dip.
Fig. 2
Fig. 2 Arrangement of control units for the discretization of the HCE.
Fig. 3
Fig. 3 Resulting maximum temperatures for (a) aluminium and brass and (b) steel, each for varying laser powers at 100 μ m / s and for steel also at 1000 μ m / s.
Fig. 4
Fig. 4 Resulting temperatures in steel for 50 mW, 100 μ m / s and varying initial temperatures from 10°C to 80°C.
Fig. 5
Fig. 5 (a) Electron- and ion temperature in dependence on the time, (b) Relative change of the refractive index and the extinction index in dependence on the time, the laser pulse and the reflectivity.
Fig. 6
Fig. 6 (a) Change of reflectivity in dependence on the initial temperature between 278 K and 1240 K. Within this range, the real part of the refractive index increased linearly from 2.636 to 3.129, (b) Laser fluence at the metal surface in dependence onthe focus distance from the surface.
Fig. 7
Fig. 7 Set up for temperature variation during TPP.
Fig. 8
Fig. 8 Test cubes for temperature variation at room temperature in IP-Dip, each written with 5000 μ m / s. The number of planes is reduced from up to down.
Fig. 9
Fig. 9 Definition of the different exposure phases by the example of the cube.
Fig. 10
Fig. 10 Cubes in IP-Dip with constant writing speed and decreasing number of planes (from up to down, see Fig. 8) for different temperatures. From left to right, the laser power increases in each picture corresponding to Fig. 8.
Fig. 11
Fig. 11 Cubes in OrmoComp with constant writing speed and decreasing number of planes (from up to down) for different temperatures. From left to right, the laser power increases from 10 mW to 30 mW in 5 mW steps from left to right.
Fig. 12
Fig. 12 SEM picture of cubes written at 80°C from 10 mW to 65 mW with a writing speed of 5000 μ m / s.
Fig. 13
Fig. 13 Structures in OrmoComp at 30°C. Strong blistering during the writing process caused a Turing pattern (c) and small spherical polymer clusters (d).

Tables (2)

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Table 1 Material properties of the metals used for the simulation.

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Table 2 Parameters of the teststcubes for temperature variation.

Equations (16)

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ρ c p [ T t + v T ] = λ 2 T + Q L ( x , y , z )
t t + Δ t d u s n w e ρ c p T t d x d y d z d t + t t + Δ t d u s n w e ρ c p v T d x d y d z d t = t t + Δ t d u s n w o ( λ 2 T + Q L ( x , y , z ) ) d x d y d z d t .
t t + Δ t T d t = Δ t T p n + 1
F = ρ c p v Δ x i Δ y i P e , i = ρ c p v T x T x ( λ T x ) A ( P e , i ) = 1 | P e , i 2 |
 
ρ c p Δ x Δ y Δ z Δ t T p n + Q ¯ Δ x Δ y Δ z = [   λ n Δ x Δ z δ y n λ s Δ x Δ z δ y s + [ λ e Δ x Δ y δ x e A ( P e , e ) + m a x ( 0 , F ) ] [ λ w Δ x Δ y δ x w A ( P e , w ) + m a x ( 0 , F ) ] + λ u Δ x Δ y δ z u λ d Δ x Δ y δ z d   ] T p n + 1 λ n Δ x Δ z δ y n T n n + 1 + λ s Δ x Δ z δ y s T s n + 1 [ λ e Δ x Δ y δ x e A ( P e , e ) + m a x ( 0 , F ) ] T e n + 1 + [ λ w Δ x Δ y δ x w A ( P e , w ) + m a x ( 0 , F ) ] T w n + 1 λ u Δ x Δ y δ z u T u n + 1 λ d Δ x Δ y δ z d T d n + 1 ,
Q L ( x , y , z ) = I 0 ( 1 R ) Δ A Δ V [ e α z e α ( z + Δ z ) ] e 2 ( x 2 + y 2 r L 2 ) .
U e ( t , z ) t = C e ( T e ) T e ( t , z ) ) t = z ( q ˙ e ) G ( T e , T i ) ( T e T i ) + Q ( t , z )
U i ( t , z ) t = C i ( T i ) T i ( t , z ) ) t = G ( T e , T i ) ( T e T i )
C e ( T e ) = u e T e = 0 f F ( E , μ , T e ) T e g e ( E ) E d E
C i ( T i ) = 9 N i V k B ( T l T D ) 3 0 T D T l ϑ e ϑ ( e ϑ 1 ) 2 d ϑ C i ( T i ) = 3 n i k B
G ( T e , T i ) = π 2 m e v l 2 n e 6 τ e ( T e , T i ) T e G ( T e = T i = T r o o m ) = π 2 m e v l 2 n e 6 B = c o n s t .
Q ( x , t ) = 0.94 α J L t P ( 1 R ) e ( 2.77 ( t t P 2 t P ) 2 α x )
T e k n + 1 = T e k n + Δ t C e k n ( Δ t 2 Δ z τ e k n [   ( q ˙ e k + 1 n q ˙ e k 1 n ) ( τ e k n Δ t 1 ) κ e k n T e k + 2 n 2 T e k n + T e k 2 n 2 Δ z   ] G ( T e k n T i k n ) + Q k n )
T i k n + 1 = T i k n + Δ t G ( T e k n T i k n ) C i k n
I H R = ϵ k B T 4

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