Abstract

A distinct development of an exact analytical solution for power-law fluids during the spin-coating process is presented for temporal and spatial thickness evolution, after steady state conditions are attained. This solution leads to the definition of a characteristic time, related to the memory of the initial thickness profile. Previously obtained experimental data, for several rotation speeds and carboxymetilcellulose concentrations in water, are quantitatively analyzed through the evaluation of their characteristic times and compared with theoretical predictions, thus allowing better understanding of thickness profile evolution and of process reproducibility.

© 2014 Optical Society of America

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  1. P. H. Walker and J. G. Thompson, “Proceedings of the twenty-fifth annual meeting,” Am. Soc. Test. Mater. 22, 463–485 (1922).
  2. A. G. Emslie, F. T. Bonner, and L. G. Peck, “Flow of a viscous liquid on a rotating disk,” J. Appl. Phys. 29, 858–862 (1958).
    [CrossRef]
  3. B. D. Washo, “Rheology and modeling of the spin coating process,” IBM J. Res Devel. 21, 190–198 (1977).
    [CrossRef]
  4. A. Acrivos, M. J. Shah, and E. E. Petersen, “On the flow of a non-Newtonian liquid on a rotating disk,” J. Appl. Phys. 31, 963–968 (1960).
    [CrossRef]
  5. S. A. Jenekhe and S. B. Schuldt, “Coating of non-Newtonian fluids on a flat rotating disk,” Ind. Eng. Chem. Fundam. 23, 432–436 (1984).
    [CrossRef]
  6. J. P. F. Charpin, M. Lombe, and T. G. Myers, “Spin coating on non-Newtonian fluids with a moving front,” Phys. Rev. E 76, 0163121 (2007).
    [CrossRef]
  7. P. Temple-Boyer, L. Mazenq, J. B. Doucet, V. Conédéra, B. Torbiéro, and J. Launay, “Theoretical studies of the spin coating process for the deposition of polymer-based Maxwellian liquids,” Microelectron. Eng. 87, 163–166 (2010).
    [CrossRef]
  8. A. F. Michels, T. Menegotto, and F. Horowitz, “Optically monitored dip coating as a contactless viscometry method for liquid films,” Appl. Opt. 44, 912–915 (2005).
    [CrossRef]
  9. A. F. Michels, T. Menegotto, F. Horowitz, M. B. Susin, and H. P. Grieneisen, “Double optical monitoring of dip coating with a time-varying refractive index,” Appl. Opt. 45, 1491–1494 (2006).
    [CrossRef]
  10. A. F. Michels, P. L. G. Jardim, and F. Horowitz, “Laser monitoring of non-Newtonian liquids during dip coating,” AIChe J. 55, 3052–3055 (2009).
    [CrossRef]
  11. F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Real-time optical monitoring of spin coating,” J. Phys. III France 3, 2059–2063 (1993).
  12. F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Optical monitoring of the sol-to gel transition in spinning silica films,” Proc. SPIE 2288, 67–70 (1994).
  13. F. Horowitz, E. J. C. Dawnay, M. A. Fardad, M. Green, and E. M. Yeatman, “Towards better control of sol-gel film processing for optical device applications,” J. Nonlinear Opt. Phys. Mater. 6, 1–18 (1997).
    [CrossRef]
  14. F. Horowitz, A. F. Michels, and E. M. Yeatman, “Optical viscometry of spinning sol coatings,” J. Sol-Gel Sci. Technol. 13, 707–712 (1998).
    [CrossRef]
  15. P. L. G. Jardim, A. F. Michels, and F. Horowitz, “Optical interference monitoring for power-law fluids during spin coating,” Opt. Express 20, 3166–3175 (2012).
    [CrossRef]

2012 (1)

2010 (1)

P. Temple-Boyer, L. Mazenq, J. B. Doucet, V. Conédéra, B. Torbiéro, and J. Launay, “Theoretical studies of the spin coating process for the deposition of polymer-based Maxwellian liquids,” Microelectron. Eng. 87, 163–166 (2010).
[CrossRef]

2009 (1)

A. F. Michels, P. L. G. Jardim, and F. Horowitz, “Laser monitoring of non-Newtonian liquids during dip coating,” AIChe J. 55, 3052–3055 (2009).
[CrossRef]

2007 (1)

J. P. F. Charpin, M. Lombe, and T. G. Myers, “Spin coating on non-Newtonian fluids with a moving front,” Phys. Rev. E 76, 0163121 (2007).
[CrossRef]

2006 (1)

2005 (1)

1998 (1)

F. Horowitz, A. F. Michels, and E. M. Yeatman, “Optical viscometry of spinning sol coatings,” J. Sol-Gel Sci. Technol. 13, 707–712 (1998).
[CrossRef]

1997 (1)

F. Horowitz, E. J. C. Dawnay, M. A. Fardad, M. Green, and E. M. Yeatman, “Towards better control of sol-gel film processing for optical device applications,” J. Nonlinear Opt. Phys. Mater. 6, 1–18 (1997).
[CrossRef]

1994 (1)

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Optical monitoring of the sol-to gel transition in spinning silica films,” Proc. SPIE 2288, 67–70 (1994).

1993 (1)

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Real-time optical monitoring of spin coating,” J. Phys. III France 3, 2059–2063 (1993).

1984 (1)

S. A. Jenekhe and S. B. Schuldt, “Coating of non-Newtonian fluids on a flat rotating disk,” Ind. Eng. Chem. Fundam. 23, 432–436 (1984).
[CrossRef]

1977 (1)

B. D. Washo, “Rheology and modeling of the spin coating process,” IBM J. Res Devel. 21, 190–198 (1977).
[CrossRef]

1960 (1)

A. Acrivos, M. J. Shah, and E. E. Petersen, “On the flow of a non-Newtonian liquid on a rotating disk,” J. Appl. Phys. 31, 963–968 (1960).
[CrossRef]

1958 (1)

A. G. Emslie, F. T. Bonner, and L. G. Peck, “Flow of a viscous liquid on a rotating disk,” J. Appl. Phys. 29, 858–862 (1958).
[CrossRef]

1922 (1)

P. H. Walker and J. G. Thompson, “Proceedings of the twenty-fifth annual meeting,” Am. Soc. Test. Mater. 22, 463–485 (1922).

Acrivos, A.

A. Acrivos, M. J. Shah, and E. E. Petersen, “On the flow of a non-Newtonian liquid on a rotating disk,” J. Appl. Phys. 31, 963–968 (1960).
[CrossRef]

Bonner, F. T.

A. G. Emslie, F. T. Bonner, and L. G. Peck, “Flow of a viscous liquid on a rotating disk,” J. Appl. Phys. 29, 858–862 (1958).
[CrossRef]

Charpin, J. P. F.

J. P. F. Charpin, M. Lombe, and T. G. Myers, “Spin coating on non-Newtonian fluids with a moving front,” Phys. Rev. E 76, 0163121 (2007).
[CrossRef]

Conédéra, V.

P. Temple-Boyer, L. Mazenq, J. B. Doucet, V. Conédéra, B. Torbiéro, and J. Launay, “Theoretical studies of the spin coating process for the deposition of polymer-based Maxwellian liquids,” Microelectron. Eng. 87, 163–166 (2010).
[CrossRef]

Dawnay, E.

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Optical monitoring of the sol-to gel transition in spinning silica films,” Proc. SPIE 2288, 67–70 (1994).

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Real-time optical monitoring of spin coating,” J. Phys. III France 3, 2059–2063 (1993).

Dawnay, E. J. C.

F. Horowitz, E. J. C. Dawnay, M. A. Fardad, M. Green, and E. M. Yeatman, “Towards better control of sol-gel film processing for optical device applications,” J. Nonlinear Opt. Phys. Mater. 6, 1–18 (1997).
[CrossRef]

Doucet, J. B.

P. Temple-Boyer, L. Mazenq, J. B. Doucet, V. Conédéra, B. Torbiéro, and J. Launay, “Theoretical studies of the spin coating process for the deposition of polymer-based Maxwellian liquids,” Microelectron. Eng. 87, 163–166 (2010).
[CrossRef]

Emslie, A. G.

A. G. Emslie, F. T. Bonner, and L. G. Peck, “Flow of a viscous liquid on a rotating disk,” J. Appl. Phys. 29, 858–862 (1958).
[CrossRef]

Fardad, A.

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Optical monitoring of the sol-to gel transition in spinning silica films,” Proc. SPIE 2288, 67–70 (1994).

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Real-time optical monitoring of spin coating,” J. Phys. III France 3, 2059–2063 (1993).

Fardad, M. A.

F. Horowitz, E. J. C. Dawnay, M. A. Fardad, M. Green, and E. M. Yeatman, “Towards better control of sol-gel film processing for optical device applications,” J. Nonlinear Opt. Phys. Mater. 6, 1–18 (1997).
[CrossRef]

Green, M.

F. Horowitz, E. J. C. Dawnay, M. A. Fardad, M. Green, and E. M. Yeatman, “Towards better control of sol-gel film processing for optical device applications,” J. Nonlinear Opt. Phys. Mater. 6, 1–18 (1997).
[CrossRef]

Grieneisen, H. P.

Horowitz, F.

P. L. G. Jardim, A. F. Michels, and F. Horowitz, “Optical interference monitoring for power-law fluids during spin coating,” Opt. Express 20, 3166–3175 (2012).
[CrossRef]

A. F. Michels, P. L. G. Jardim, and F. Horowitz, “Laser monitoring of non-Newtonian liquids during dip coating,” AIChe J. 55, 3052–3055 (2009).
[CrossRef]

A. F. Michels, T. Menegotto, F. Horowitz, M. B. Susin, and H. P. Grieneisen, “Double optical monitoring of dip coating with a time-varying refractive index,” Appl. Opt. 45, 1491–1494 (2006).
[CrossRef]

A. F. Michels, T. Menegotto, and F. Horowitz, “Optically monitored dip coating as a contactless viscometry method for liquid films,” Appl. Opt. 44, 912–915 (2005).
[CrossRef]

F. Horowitz, A. F. Michels, and E. M. Yeatman, “Optical viscometry of spinning sol coatings,” J. Sol-Gel Sci. Technol. 13, 707–712 (1998).
[CrossRef]

F. Horowitz, E. J. C. Dawnay, M. A. Fardad, M. Green, and E. M. Yeatman, “Towards better control of sol-gel film processing for optical device applications,” J. Nonlinear Opt. Phys. Mater. 6, 1–18 (1997).
[CrossRef]

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Optical monitoring of the sol-to gel transition in spinning silica films,” Proc. SPIE 2288, 67–70 (1994).

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Real-time optical monitoring of spin coating,” J. Phys. III France 3, 2059–2063 (1993).

Jardim, P. L. G.

P. L. G. Jardim, A. F. Michels, and F. Horowitz, “Optical interference monitoring for power-law fluids during spin coating,” Opt. Express 20, 3166–3175 (2012).
[CrossRef]

A. F. Michels, P. L. G. Jardim, and F. Horowitz, “Laser monitoring of non-Newtonian liquids during dip coating,” AIChe J. 55, 3052–3055 (2009).
[CrossRef]

Jenekhe, S. A.

S. A. Jenekhe and S. B. Schuldt, “Coating of non-Newtonian fluids on a flat rotating disk,” Ind. Eng. Chem. Fundam. 23, 432–436 (1984).
[CrossRef]

Launay, J.

P. Temple-Boyer, L. Mazenq, J. B. Doucet, V. Conédéra, B. Torbiéro, and J. Launay, “Theoretical studies of the spin coating process for the deposition of polymer-based Maxwellian liquids,” Microelectron. Eng. 87, 163–166 (2010).
[CrossRef]

Lombe, M.

J. P. F. Charpin, M. Lombe, and T. G. Myers, “Spin coating on non-Newtonian fluids with a moving front,” Phys. Rev. E 76, 0163121 (2007).
[CrossRef]

Mazenq, L.

P. Temple-Boyer, L. Mazenq, J. B. Doucet, V. Conédéra, B. Torbiéro, and J. Launay, “Theoretical studies of the spin coating process for the deposition of polymer-based Maxwellian liquids,” Microelectron. Eng. 87, 163–166 (2010).
[CrossRef]

Menegotto, T.

Michels, A. F.

Myers, T. G.

J. P. F. Charpin, M. Lombe, and T. G. Myers, “Spin coating on non-Newtonian fluids with a moving front,” Phys. Rev. E 76, 0163121 (2007).
[CrossRef]

Peck, L. G.

A. G. Emslie, F. T. Bonner, and L. G. Peck, “Flow of a viscous liquid on a rotating disk,” J. Appl. Phys. 29, 858–862 (1958).
[CrossRef]

Petersen, E. E.

A. Acrivos, M. J. Shah, and E. E. Petersen, “On the flow of a non-Newtonian liquid on a rotating disk,” J. Appl. Phys. 31, 963–968 (1960).
[CrossRef]

Schuldt, S. B.

S. A. Jenekhe and S. B. Schuldt, “Coating of non-Newtonian fluids on a flat rotating disk,” Ind. Eng. Chem. Fundam. 23, 432–436 (1984).
[CrossRef]

Shah, M. J.

A. Acrivos, M. J. Shah, and E. E. Petersen, “On the flow of a non-Newtonian liquid on a rotating disk,” J. Appl. Phys. 31, 963–968 (1960).
[CrossRef]

Susin, M. B.

Temple-Boyer, P.

P. Temple-Boyer, L. Mazenq, J. B. Doucet, V. Conédéra, B. Torbiéro, and J. Launay, “Theoretical studies of the spin coating process for the deposition of polymer-based Maxwellian liquids,” Microelectron. Eng. 87, 163–166 (2010).
[CrossRef]

Thompson, J. G.

P. H. Walker and J. G. Thompson, “Proceedings of the twenty-fifth annual meeting,” Am. Soc. Test. Mater. 22, 463–485 (1922).

Torbiéro, B.

P. Temple-Boyer, L. Mazenq, J. B. Doucet, V. Conédéra, B. Torbiéro, and J. Launay, “Theoretical studies of the spin coating process for the deposition of polymer-based Maxwellian liquids,” Microelectron. Eng. 87, 163–166 (2010).
[CrossRef]

Walker, P. H.

P. H. Walker and J. G. Thompson, “Proceedings of the twenty-fifth annual meeting,” Am. Soc. Test. Mater. 22, 463–485 (1922).

Washo, B. D.

B. D. Washo, “Rheology and modeling of the spin coating process,” IBM J. Res Devel. 21, 190–198 (1977).
[CrossRef]

Yeatman, E.

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Optical monitoring of the sol-to gel transition in spinning silica films,” Proc. SPIE 2288, 67–70 (1994).

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Real-time optical monitoring of spin coating,” J. Phys. III France 3, 2059–2063 (1993).

Yeatman, E. M.

F. Horowitz, A. F. Michels, and E. M. Yeatman, “Optical viscometry of spinning sol coatings,” J. Sol-Gel Sci. Technol. 13, 707–712 (1998).
[CrossRef]

F. Horowitz, E. J. C. Dawnay, M. A. Fardad, M. Green, and E. M. Yeatman, “Towards better control of sol-gel film processing for optical device applications,” J. Nonlinear Opt. Phys. Mater. 6, 1–18 (1997).
[CrossRef]

AIChe J. (1)

A. F. Michels, P. L. G. Jardim, and F. Horowitz, “Laser monitoring of non-Newtonian liquids during dip coating,” AIChe J. 55, 3052–3055 (2009).
[CrossRef]

Am. Soc. Test. Mater. (1)

P. H. Walker and J. G. Thompson, “Proceedings of the twenty-fifth annual meeting,” Am. Soc. Test. Mater. 22, 463–485 (1922).

Appl. Opt. (2)

IBM J. Res Devel. (1)

B. D. Washo, “Rheology and modeling of the spin coating process,” IBM J. Res Devel. 21, 190–198 (1977).
[CrossRef]

Ind. Eng. Chem. Fundam. (1)

S. A. Jenekhe and S. B. Schuldt, “Coating of non-Newtonian fluids on a flat rotating disk,” Ind. Eng. Chem. Fundam. 23, 432–436 (1984).
[CrossRef]

J. Appl. Phys. (2)

A. Acrivos, M. J. Shah, and E. E. Petersen, “On the flow of a non-Newtonian liquid on a rotating disk,” J. Appl. Phys. 31, 963–968 (1960).
[CrossRef]

A. G. Emslie, F. T. Bonner, and L. G. Peck, “Flow of a viscous liquid on a rotating disk,” J. Appl. Phys. 29, 858–862 (1958).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

F. Horowitz, E. J. C. Dawnay, M. A. Fardad, M. Green, and E. M. Yeatman, “Towards better control of sol-gel film processing for optical device applications,” J. Nonlinear Opt. Phys. Mater. 6, 1–18 (1997).
[CrossRef]

J. Phys. III France (1)

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Real-time optical monitoring of spin coating,” J. Phys. III France 3, 2059–2063 (1993).

J. Sol-Gel Sci. Technol. (1)

F. Horowitz, A. F. Michels, and E. M. Yeatman, “Optical viscometry of spinning sol coatings,” J. Sol-Gel Sci. Technol. 13, 707–712 (1998).
[CrossRef]

Microelectron. Eng. (1)

P. Temple-Boyer, L. Mazenq, J. B. Doucet, V. Conédéra, B. Torbiéro, and J. Launay, “Theoretical studies of the spin coating process for the deposition of polymer-based Maxwellian liquids,” Microelectron. Eng. 87, 163–166 (2010).
[CrossRef]

Opt. Express (1)

Phys. Rev. E (1)

J. P. F. Charpin, M. Lombe, and T. G. Myers, “Spin coating on non-Newtonian fluids with a moving front,” Phys. Rev. E 76, 0163121 (2007).
[CrossRef]

Proc. SPIE (1)

F. Horowitz, E. Yeatman, E. Dawnay, and A. Fardad, “Optical monitoring of the sol-to gel transition in spinning silica films,” Proc. SPIE 2288, 67–70 (1994).

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Figures (2)

Fig. 1.
Fig. 1.

Reflectance profile for 0.1% wt., 0.2% wt., and 0.5% wt. CMC at a rotation speed of 5000 rpm. (Reflectance signal is inverted due to lock-in amplifier response polarity).

Fig. 2.
Fig. 2.

Physical thickness variation for 0.1% wt., 0.2% wt., and 0.5% wt. CMC at a rotation speed of 5000 rpm.

Tables (2)

Tables Icon

Table 1. Refraction Indices, n1, for CMC Weight Concentrations, Measured by an Abbe Refractometer

Tables Icon

Table 2. Rheological Constants (K), Powers (l), Characteristic Times (tc), Total Flow Times (tt), and Initial Thicknesses in the Steady State (ho) for 0.1% wt., 0.2% wt., and 0.5% wt. CMC at Several Rotation Speeds ω

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

τrzz=ρrω2,
vr(z=0)=0
τrz(z=h)=0
τrz=K|vrz|l1vrz.
μrzeff=K|vrz|l1,
vr(z)=(ρKω2r)1l(ll+1)(hl+1l(hz)l+1l).
q=0hvr(z)dz,
q=l2l+1(ρKω2r)1lh2l+1l.
h(r,t)t+1r(rq)r=0
ht+(l+1)(2l+1)h2l+1l(ρω2K)1lr1ll+(ρω2K)1lr1lhl+1lhr=0.
h(r,t)=R(r)T(t)
(l+1)(2l+1)Rl+1lr1ll+r1lR1ldRdrσ=0
1T2l+1ldTdt=σ(ρω2K)1l.
T(t)=[T0(l+1)l+l+1lσ(ρω2K)1lt]ll+1,
R(r)=Cr(l1)(l+1).
C=h0[r0(l1)(l+1)T0]1.
h(r,t)=h0(rr0)(l1)(l+1)[1+(l+1)lσT0(l+1)l(ρω2K)1lt]l(l+1),
σT0(l+1)l=h0(l+1)lr0(1l)l(3l+1)l(2l+1)(l+1).
h(r,t)=(rr0)(l1)(l+1)[h0(l+1)l+(3l+1)(2l+1)r0(1l)l(ρω2K)1lt]l(l+1).
ttc,
tc(2l+1)(3l+1)r0(l1)lh0(l+1)l(Kρω2)1l,
R=R(cosδ),
δ=4πλ0n1h,
Δhn1=mλ04,wherem=1,2,3,.

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