# A Modified Equation for Thickness of the Film Fabricated by Spin Coating

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- (a)
- The viscous PDMS (Polydimethylsiloxane) coated on the glass (Sylagard 184, Dowcoaning) is using the spin coating material.
- (b)
- The substrate of size $2\times 2\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cm}}^{2}$ is used to measure the film thickness at the center of the substrate, and the substrate of size $3\times 3\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cm}}^{2}$ is used to measure the overall thickness distribution of the PDMS film.
- (c)
- We fix the viscosity, density of coating material, initial thickness at $4000\phantom{\rule{3.33333pt}{0ex}}\mathrm{cP}$, $965\phantom{\rule{3.33333pt}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ and $0.105\phantom{\rule{3.33333pt}{0ex}}\mathrm{cm}$. Then, the rotation time is fixed at 300 s and the experiment is performed in 500 RPM units from 500 to 6000 RPM.
- (d)
- The spin coating is performed by Spin coater ACE-200 (DongAh Trade Corp, Seoul, South Korea).
- (e)
- Finally, we measure all samples thickness and thickness distribution to step measurement by surface profiler DektakXT (Bruker, Karlsruhe, Germany).
- (f)
- Thickness measurement is performed by measuring the thickness when the stylus of the DektakXT passed through the coated film from the uncoated section of the substrate.
- (g)
- We focus on a coating thickness range of 4 to $20$ $\mathsf{\mu}$m using experimental limits of $\omega =1000,2000$ and 3000 and $t=300,450$ and 600 s. In these experimental conditions, the equation for thickness of the film fabricated by spin coating is given by the formula:$$h\left(\omega \right)=\frac{1050}{\sqrt{1+0.00116671{\omega}^{2}}}\left(\mathsf{\mu}\mathrm{m}\right),$$

**Remark**

**1.**

## 2. A Modified Equation for Thickness of the Film Fabricated by Spin Coating via the Curve Estimation

#### 2.1. Fixed Time at 300 s

**Step 1.**We use the Curve Estimation of the regression analysis from the SPSS to make a curve estimate for the MT value of Table 1 above. There are 11 models available in the curve estimation menu. Among these, we select five models with the possibility of being suitable for MT data. They are Logarithmic, Inverse, Quadratic, Cubic and Power models. The results of the analysis are as follows (Figure 1).

**Step 2.**Let ${E}_{300}$ denote the difference of the TT value and the MT value. That is to say, let ${E}_{300}$ = TT −MT. Then, we obtain Table 3 below.

**Step 3.**We now establish the following hypothesis to test the function ${h}_{log}$ and the consistency of MT value. Let ${\mu}_{MT}$ and ${\mu}_{h}$ be population means of MT values and the estimated function ${h}_{log}$, respectively. Then, we formulate the following research hypothesis.

#### 2.2. Fixed RPM at 1000

#### 2.3. Other Cases

#### 2.4. Summary of Section 2

## 3. A Modified Equation for Thickness of the Film Fabricated by Spin Coating via the Polyhedron Approximation

#### 3.1. Polyhedron Approximation

#### 3.2. Verification of the Polyhedron Approximation

#### 3.3. Summary of Section 3

## 4. Application: Target Verification

## 5. Conclusions

#### 5.1. Importance of Results and Formulas in This Paper

#### 5.2. Another Approach

#### 5.3. Expected Results

#### 5.4. Capture of the Thickness Calculator by the Excel Program

**Remark**

**2.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

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RPM | TT ($\mathsf{\mu}$m) | MT ($\mathsf{\mu}$m) |
---|---|---|

500 | $61.38$ | $52.99$ |

1000 | $30.73$ | $25.09$ |

1500 | $20.49$ | $15.36$ |

2000 | $15.37$ | $11.16$ |

2500 | $12.30$ | $8.53$ |

3000 | $10.25$ | $6.60$ |

3500 | $8.78$ | $5.30$ |

4000 | $7.68$ | $4.26$ |

4500 | $6.83$ | $3.92$ |

5000 | $6.15$ | $3.54$ |

5500 | $5.59$ | $2.80$ |

6000 | $5.12$ | $2.72$ |

RPM | TT | MT | ${\mathit{h}}_{\mathit{inverse}}$ | ${\mathit{h}}_{\mathit{power}}$ |
---|---|---|---|---|

500 | $61.375$ | $52.991$ | $52.550$ | $56.876$ |

1000 | $30.727$ | $24.541$ | $25.077$ | $24.428$ |

1500 | $20.490$ | $15.357$ | $15.920$ | $14.900$ |

2000 | $15.368$ | $11.280$ | $11.341$ | $10.492$ |

2500 | $12.295$ | $8.528$ | $8.594$ | $7.992$ |

3000 | $10.246$ | $6.530$ | $6.763$ | $6.399$ |

3500 | $8.783$ | $5.301$ | $5.454$ | $5.303$ |

4000 | $7.685$ | $4.264$ | $4.473$ | $4.506$ |

4500 | $6.831$ | $3.920$ | $3.710$ | $3.903$ |

5000 | $6.148$ | $3.536$ | $3.100$ | $3.433$ |

5500 | $5.589$ | $2.801$ | $2.600$ | $3.056$ |

6000 | $5.123$ | $2.718$ | $2.184$ | $2.748$ |

${R}^{2}$ | 0.999 | 0.997 |

RPM | TT | MT | ${\mathit{E}}_{300}$ |
---|---|---|---|

500 | $61.375$ | $52.991$ | $8.384$ |

1000 | $30.727$ | $24.541$ | $6.186$ |

1500 | $20.490$ | $15.357$ | $5.133$ |

2000 | $15.368$ | $11.280$ | $4.089$ |

2500 | $12.295$ | $8.528$ | $3.767$ |

3000 | $10.246$ | $6.530$ | $3.716$ |

3500 | $8.783$ | $5.301$ | $3.481$ |

4000 | $7.685$ | $4.264$ | $3.421$ |

4500 | $6.831$ | $3.920$ | $2.911$ |

5000 | $6.148$ | $3.536$ | $2.612$ |

5500 | $5.589$ | $2.801$ | $2.788$ |

6000 | $5.123$ | $2.718$ | $2.406$ |

RPM | TT | MT | ${\mathit{h}}_{\mathit{log}}$ | ${\mathit{h}}_{\mathit{inv}}$ |
---|---|---|---|---|

500 | $61.375$ | $52.991$ | $53.538$ | $52.534$ |

1000 | $30.727$ | $24.541$ | $24.462$ | $25.100$ |

1500 | $20.490$ | $15.357$ | $15.138$ | $15.935$ |

2000 | $15.368$ | $11.280$ | $10.665$ | $11.349$ |

2500 | $12.295$ | $8.528$ | $8.095$ | $8.598$ |

3000 | $10.246$ | $6.530$ | $6.456$ | $6.763$ |

3500 | $8.783$ | $5.301$ | $5.340$ | $5.452$ |

4000 | $7.685$ | $4.264$ | $4.543$ | $4.469$ |

4500 | $6.831$ | $3.920$ | $3.955$ | $3.705$ |

5000 | $6.148$ | $3.536$ | $3.510$ | $3.093$ |

5500 | $5.589$ | $2.801$ | $3.165$ | $2.593$ |

6000 | $5.123$ | $2.718$ | $2.895$ | $2.176$ |

${R}^{2}$ | 0.964 | 0.953 |

RPM | MT-${\mathit{h}}_{\mathit{inverse}}$ | MT-${\mathit{h}}_{\mathit{power}}$ | MT-${\mathit{h}}_{\mathit{log}}$ | MT-${\mathit{h}}_{\mathit{inv}}$ |
---|---|---|---|---|

500 | $0.442$ | $-3.885$ | $-0.557$ | $0.457$ |

1000 | $-0.537$ | $0.113$ | $0.079$ | $-0.560$ |

1500 | $-0.563$ | $0.457$ | $0.219$ | $-0.578$ |

2000 | $-0.061$ | $0.788$ | $0.615$ | $-0.069$ |

2500 | $-0.066$ | $0.536$ | $0.434$ | $-0.069$ |

3000 | $-0.233$ | $0.131$ | $0.073$ | $-0.233$ |

3500 | $-0.153$ | $-0.001$ | $-0.039$ | $-0.151$ |

4000 | $-0.210$ | $-0.243$ | $-0.280$ | $-0.206$ |

4500 | $0.210$ | $0.017$ | $-0.035$ | $0.215$ |

5000 | $0.437$ | $0.104$ | $0.027$ | $0.443$ |

5500 | $0.201$ | $-0.255$ | $-0.364$ | $0.208$ |

6000 | $0.534$ | $-0.031$ | $-0.178$ | $0.542$ |

SSE | $1.490$ | $16.375$ | $1.181$ | $1.565$ |

TIME | TT | MT | ${\mathit{h}}_{\mathit{w}=1000}$ |
---|---|---|---|

100 | $53.176$ | $43.269$ | $43.311$ |

200 | $37.625$ | $29.576$ | $29.625$ |

300 | $30.727$ | $24.541$ | $23.818$ |

400 | $26.613$ | $19.710$ | $20.478$ |

500 | $23.905$ | $18.192$ | $18.270$ |

600 | $21.731$ | $16.591$ | $16.687$ |

700 | $20.121$ | $15.948$ | $15.490$ |

800 | $18.822$ | $14.431$ | $14.551$ |

RPM | MT${}_{\mathit{t}=450}$ | ${\mathit{h}}_{\mathit{t}=450}$ | MT${}_{\mathit{t}=600}$ | ${\mathit{h}}_{\mathit{t}=600}$ |
---|---|---|---|---|

500 | $38.969$ | $39.510$ | $36.454$ | $36.664$ |

1000 | $20.242$ | $18.806$ | $16.591$ | $16.418$ |

1500 | $11.639$ | $11.894$ | $10.177$ | $10.029$ |

2000 | $8.309$ | $8.436$ | $7.133$ | $7.012$ |

2500 | $6.134$ | $6.361$ | $4.996$ | $5.309$ |

3000 | $5.216$ | $4.978$ | $4.493$ | $4.243$ |

3500 | $4.199$ | $3.990$ | $3.580$ | $3.532$ |

4000 | $2.989$ | $3.248$ | $3.002$ | $3.037$ |

4500 | $2.199$ | $2.672$ | $2.478$ | $2.681$ |

TIME | MT${}_{\mathit{\omega}=2000}$ | ${\mathit{h}}_{\mathit{\omega}=2000}$ | MT${}_{\mathit{\omega}=3000}$ | ${\mathit{h}}_{\mathit{\omega}=3000}$ |
---|---|---|---|---|

100 | $19.908$ | $20.015$ | $12.224$ | $12.340$ |

200 | $13.986$ | $13.893$ | $8.493$ | $8.257$ |

300 | $11.280$ | $10.996$ | $6.530$ | $6.606$ |

400 | $9.240$ | $9.206$ | $5.662$ | $5.695$ |

500 | $7.977$ | $7.978$ | $5.265$ | $5.118$ |

600 | $7.133$ | $7.052$ | $4.493$ | $4.719$ |

700 | $5.984$ | $6.325$ | $4.472$ | $4.430$ |

800 | $5.703$ | $5.735$ | $4.235$ | $4.210$ |

Sub-Area | Domain |
---|---|

${D}_{1,2,3}$ | $\left\{(t,\omega )\right|300\le t<450,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1000\le \omega <4000-\frac{20}{3}t\}$ |

${D}_{2,3,5}$ | $\left\{(t,\omega )\right|300\le t<450,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}4000-\frac{20}{3}t\le \omega <2000\}$ |

${D}_{2,4,5}$ | $\left\{(t,\omega )\right|450\le t<600,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1000\le \omega <5000-\frac{20}{3}t\}$ |

${D}_{3,5,6}$ | $\left\{(t,\omega )\right|300\le t<450,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}2000\le \omega <5000-\frac{20}{3}t\}$ |

${D}_{4,5,7}$ | $\left\{(t,\omega )\right|450\le t<600,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}5000-\frac{20}{3}t\le \omega <2000\}$ |

${D}_{5,6,8}$ | $\left\{(t,\omega )\right|300\le t<450,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}5000-\frac{20}{3}t\le \omega <3000\}$ |

${D}_{5,7,8}$ | $\left\{(t,\omega )\right|450\le t<600,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}2000\le \omega <6000-\frac{20}{3}t\}$ |

${D}_{7,8,9}$ | $\left\{(t,\omega )\right|450\le t<600,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}6000-\frac{20}{3}t\le \omega <3000\}$ |

${D}_{4,7,10}$ | $\left\{\right(t,\omega \left)\right|600\le t<\infty ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}1000\le \omega <2000\}$ |

${D}_{7,9,11}$ | $\left\{\right(t,\omega \left)\right|600\le t<\infty ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}2000\le \omega <3000\}$ |

${D}_{9,11,12}$ | $\left\{\right(t,\omega \left)\right|600\le t<\infty ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}3000\le \omega <\infty \}$ |

${D}_{8,9,12}$ | $\left\{\right(t,\omega \left)\right|450\le t<600,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}3000\le \omega <\infty \}$ |

${D}_{8,9,13}$ | $\left\{\right(t,\omega \left)\right|300\le t<450,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}3000\le \omega <\infty \}$ |

Plane | Equation |
---|---|

${\Pi}_{1,2,3}$ | $t+0.391529\omega +29.4118h-1401.53=0$ |

${\Pi}_{2,3,5}$ | $t+0.677312\omega +64.2123h-2349.91=0$ |

${\Pi}_{2,4,5}$ | $t+0.635932\omega +60.2894h-2238.84=0$ |

${\Pi}_{3,5,6}$ | $t+0.295985\omega +64.2123h-1547.26=0$ |

${\Pi}_{4,5,7}$ | $t+0.977515\omega +102.669h-3276.9=0$ |

${\Pi}_{5,6,8}$ | $t+0.36767\omega +110.947h-2127.51=0$ |

${\Pi}_{5,7,8}$ | $t+0.340246\omega +102.669h-2002.36=0$ |

${\Pi}_{7,8,9}$ | $t+0.548209\omega +214.9h-3207.38=0$ |

${\Pi}_{4,7,10}$ | $t+1.46929\omega +154.321h-4623.61=0$ |

${\Pi}_{7,9,11}$ | $t+1.89665\omega +743.49h-9620.82=0$ |

${\Pi}_{9,11,12}$ | $t+1.07435\omega +743.494h-7153.9=0$ |

${\Pi}_{8,9,12}$ | $t+0.31053\omega +214.9h-2494.34=0$ |

${\Pi}_{6,8,13}$ | $t+0.214127\omega +110.947h-1666.86=0$ |

${\mathit{b}}_{\mathit{i}}$ | Coordinates | ${\mathit{Curve}}_{\mathit{appro}}$ | ${\mathbf{\Pi}}_{\mathit{i},\mathit{j},\mathit{k}}$ | ${\mathit{h}}_{\mathbf{\Pi}}\left({\mathit{b}}_{\mathit{i}}\right)$ |
---|---|---|---|---|

${b}_{1}$ | $(400,1000)$ | $20.4781$ | ${\Pi}_{1,2,3}$ | $20.7520$ |

${b}_{2}$ | $(300,1500)$ | $15.1329$ | ${\Pi}_{1,2,3}$ | $17.5020$ |

${b}_{3}$ | $(500,1000)$ | $18.2702$ | ${\Pi}_{2,4,5}$ | $18.3349$ |

${b}_{4}$ | $(300,2500)$ | $8.0908$ | ${\Pi}_{3,5,6}$ | $8.6660$ |

${b}_{5}$ | $(700,1000)$ | $15.4906$ | ${\Pi}_{4,7,10}$ | $15.9110$ |

${b}_{6}$ | $(500,2000)$ | $7.9778$ | ${\Pi}_{5,7,8}$ | $8.0531$ |

${b}_{7}$ | $(400,3000)$ | $5.6956$ | ${\Pi}_{6,8,13}$ | $5.7239$ |

${b}_{8}$ | $(700,2000)$ | $6.3254$ | ${\Pi}_{4,7,10}$ | $6.8300$ |

${b}_{9}$ | $(600,2500)$ | $5.3086$ | ${\Pi}_{7,9,11}$ | $5.6600$ |

${b}_{10}$ | $(500,3000)$ | $5.1175$ | ${\Pi}_{8,9,12}$ | $5.0570$ |

${b}_{11}$ | $(300,4500)$ | $3.9517$ | ${\Pi}_{6,8,13}$ | $3.7739$ |

${b}_{12}$ | $(700,3000)$ | $4.4294$ | ${\Pi}_{9,11,12}$ | $4.5120$ |

${b}_{13}$ | $(450,4500)$ | $2.6720$ | ${\Pi}_{6,8,13}$ | $2.4239$ |

${b}_{14}$ | $(600,4500)$ | $2.6812$ | ${\Pi}_{9,11,12}$ | $2.5420$ |

$\mathit{h}\left(\mathsf{\mu}\mathbf{m}\right)$ | $\mathit{\omega}$ | t | t | $\mathit{\omega}$ |
---|---|---|---|---|

4 | 1000 2000 3000 | - 1251 925 | 300 450 600 | 4453 3493 3150 |

5 | 1000 2000 3000 | - 956 526 | 300 450 600 | 3692 2990 2623 |

6 | 1000 2000 3000 | - 752 360 | 300 450 600 | 3181 2613 2265 |

7 | 1000 2000 3000 | - 606 269 | 300 450 600 | 2809 2321 2002 |

8 | 1000 2000 3000 | 6194 498 211 | 300 450 600 | 2522 2087 1799 |

9 | 1000 2000 3000 | 3007 415 171 | 300 450 600 | 2294 1896 1637 |

10 | 1000 2000 3000 | 2062 350 143 | 300 450 600 | 2106 1737 1503 |

15 | 1000 2000 3000 | 749 174 71 | 300 450 600 | 1511 1224 1078 |

20 | 1000 2000 3000 | 418 100 44 | 300 450 600 | 1187 945 846 |

Target Thickness $\left(\mathsf{\mu}\mathbf{m}\right)$ | Suitable RPM |
---|---|

11–20 | less than 1000 |

7–10 | 1000–3000 |

4–6 | more than 3000 |

$\mathit{h}\left(\mathsf{\mu}\mathbf{m}\right)$ | t | $\mathit{\omega}$ | ${\mathit{h}}_{\mathbf{\Pi}}\left(\mathsf{\mu}\mathbf{m}\right)$ | MT${}_{\mathit{t}=450}\left(\mathsf{\mu}\mathbf{m}\right)$ |
---|---|---|---|---|

4 | 300 450 600 | 4453 3493 3150 | $3.863$ $4.602$ $4.432$ | 3.777 |

5 | 300 450 600 | 3692 2990 2623 | $5.309$ $5.271$ $5.340$ | 4.956 |

6 | 300 450 600 | 3181 2613 2265 | $6.280$ $6.515$ $6.271$ | 6.058 |

7 | 300 450 600 | 2809 2321 2002 | $7.337$ $7.479$ $6.955$ | 6.390 |

8 | 300 450 600 | 2522 2087 1799 | $8.571$ $8.251$ $8.981$ | 7.481 |

9 | 300 450 600 | 2294 1896 1637 | $9.552$ $9.757$ $10.510$ | 8.587 |

10 | 300 450 600 | 2106 1737 1503 | $10.360$ $11.426$ $11.783$ | 9.723 |

15 | 300 450 600 | 1511 1224 1078 | $17.356$ $16.813$ $15.820$ | 15.163 |

20 | 300 450 600 | 1187 945 846 | $21.665$ − − | 21.872 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lee, U.G.; Kim, W.-B.; Han, D.H.; Chung, H.S.
A Modified Equation for Thickness of the Film Fabricated by Spin Coating. *Symmetry* **2019**, *11*, 1183.
https://doi.org/10.3390/sym11091183

**AMA Style**

Lee UG, Kim W-B, Han DH, Chung HS.
A Modified Equation for Thickness of the Film Fabricated by Spin Coating. *Symmetry*. 2019; 11(9):1183.
https://doi.org/10.3390/sym11091183

**Chicago/Turabian Style**

Lee, Un Gi, Woo-Byoung Kim, Do Hyung Han, and Hyun Soo Chung.
2019. "A Modified Equation for Thickness of the Film Fabricated by Spin Coating" *Symmetry* 11, no. 9: 1183.
https://doi.org/10.3390/sym11091183