**2. Microcellular foam injection molding theories**

According to above chapters, all the advantages and disadvantages are all caused by the SCF injected into the polymer melt. Before introduction microcellular foam injection molding theories, supercritical fluid is firstly discussed.

### **2.1 Supercritical fluid**

Supercritical fluid is any substance at certain temperature and pressure above its critical point, where distinct liquid and gas phases do not exist. It can effuse through solids like gas, and dissolve materials like liquid. In addition, close to the critical point, small changes in pressure or temperature result in large changes in density, and allowing many properties of supercritical fluid to be "fine-tuned". Supercritical fluids are suitable as a substitute for organic solvents in a range of industrial and laboratory processes. Carbon dioxide and nitrogen are the most commonly used supercritical fluids for microcellular foam injection molding. Figure 2-1 shows the Carbon dioxide pressure-temperature phase diagram.

Microcellular Foam Injection Molding Process 183

Table 2-2 shows that carbon dioxide generally has much greater solubility in molten polymers than nitrogen. It indicates that more carbon dioxide can be added to the polymer melt in microcellular foam processing than nitrogen. The result of higher blowing agent concentration in the polymer melt means more density reduction. Table 2-2 shows CO2 and N2 maximum

> Polymer Carbon dioxide (%) Nitrogen (%) PE 14 3 PP 11 4 PS 11 2 PMMA 13 1

However, because of the similar diffusion rates of nitrogen and carbon dioxide in polymers melt, as shown in the Table 2-3, nitrogen lends to generate smaller cells at the same concentration in polymer melt than carbon dioxide. And the driving force of nitrogen to devolve from the polymer-SCF single phase solution is greater than carbon dioxide. Thus more nucleation sites can be formed in the polymer-nitrogen mixer. Because the diffusion rates are similar, all nucleation sites grow at the same rate whatever nitrogen or carbon

Polymer Carbon Dioxide(cm2/s) Nitrogen(cm2/s) PS 1.3×10-5 1.5×10-5 PE 2.6×10-6 8.8×10-7 HDPE 2.4×10-5 2.5×10-5 LDPE 1.1×10-4 1.5×10-4 PTFE 7.0×10-6 8.3×10-6 PVC 3.8×10-5 4.3×10-5

The nucleation theory was established by Gibbs in early 20th century. Colton [31] proposed the classic nucleation theory, which should be classified into three types: the homogeneous

The main concern of classical nucleation theory is a thermodynamic description of initial stage of nucleation from embryo to nucleus with a little larger size than the critical one.

Homogeneous nucleation occurs in single phase solution system that has no impurity. During the pressure unloading process, every gas molecules will be a nucleation point. So theoretically the largest nucleation density and the smallest bubble size in the final parts will be obtained by homogeneous nucleation. However, due to the purity system, it need more energy to overcome the "energy barrier" to create stable and effective nucleus. Thus there

solubility in different polymer melt at 200℃temperature and 27.6MPa pressure[1].

Table 2-2. Estimated Maximum Gas Solubility at 200℃/27.6MPa[1].

dioxide is the blown agent. Thus nitrogen has smaller cell sizes.

Table 2-3. Estimated Diffusion Coefficient at 200°C [1].

nucleation, heterogeneous nucleation and cavity nucleation.

should be more super saturation in the polymer-SCF system.

**2.2.1 Theories of nucleation processing** 

**2.2 Nucleation theory** 

Fig. 2-1. Carbon dioxide pressure-temperature phase diagram [30].

In Figure 2-1, the boiling separates the gas and liquid region and ends in the critical point, where the liquid and gas phases disappear to become a single supercritical phase.

In general terms, supercritical fluids have properties between those of gas and liquid. In the Table 2-1, the critical properties are shown for some components, which are commonly used as supercritical fluids.



#### **2.1.1 Nitrogen vs carbon dioxide**

Both nitrogen and carbon dioxide are widely used in microcellular foam processing. However, the choice of blowing agent affects the final parts bubble morphology. Therefore, the choice should be made depending on what microcellular foam bubble morphology is desired rather than on ease of use or blowing agent costs.

In Figure 2-1, the boiling separates the gas and liquid region and ends in the critical point,

In general terms, supercritical fluids have properties between those of gas and liquid. In the Table 2-1, the critical properties are shown for some components, which are commonly used

Solvent Molecular weight Critical temperature Critical pressure Critical density

Both nitrogen and carbon dioxide are widely used in microcellular foam processing. However, the choice of blowing agent affects the final parts bubble morphology. Therefore, the choice should be made depending on what microcellular foam bubble morphology is

CO2 44.01 304.1 7.38 (72.8) 0.469 N2 28 126.2 3.4 (33.6) -- H2O 18.015 647.096 22.064 (217.755) 0.322 CH4 16.04 190.4 4.60 (45.5) 0.162 C2H6 30.07 305.3 4.87 (48.1) 0.203 C3H8 44.09 369.8 4.25 (41.9) 0.217 C2H4 28.05 282.4 5.04 (49.7) 0.215 C3H6 42.08 364.9 4.60 (45.4) 0.232 CH3OH 32.04 512.6 8.09 (79.8) 0.272 C2H5OH 46.07 513.9 6.14 (60.6) 0.276 C3H6O 58.08 508.1 4.70 (46.4) 0.278

g/mol K MPa (atm) g/cm3

where the liquid and gas phases disappear to become a single supercritical phase.

Fig. 2-1. Carbon dioxide pressure-temperature phase diagram [30].

Table 2-1.Critical properties of various solvents [30].

desired rather than on ease of use or blowing agent costs.

**2.1.1 Nitrogen vs carbon dioxide** 

as supercritical fluids.

Table 2-2 shows that carbon dioxide generally has much greater solubility in molten polymers than nitrogen. It indicates that more carbon dioxide can be added to the polymer melt in microcellular foam processing than nitrogen. The result of higher blowing agent concentration in the polymer melt means more density reduction. Table 2-2 shows CO2 and N2 maximum solubility in different polymer melt at 200℃temperature and 27.6MPa pressure[1].


Table 2-2. Estimated Maximum Gas Solubility at 200℃/27.6MPa[1].

However, because of the similar diffusion rates of nitrogen and carbon dioxide in polymers melt, as shown in the Table 2-3, nitrogen lends to generate smaller cells at the same concentration in polymer melt than carbon dioxide. And the driving force of nitrogen to devolve from the polymer-SCF single phase solution is greater than carbon dioxide. Thus more nucleation sites can be formed in the polymer-nitrogen mixer. Because the diffusion rates are similar, all nucleation sites grow at the same rate whatever nitrogen or carbon dioxide is the blown agent. Thus nitrogen has smaller cell sizes.


Table 2-3. Estimated Diffusion Coefficient at 200°C [1].

#### **2.2 Nucleation theory**

#### **2.2.1 Theories of nucleation processing**

The nucleation theory was established by Gibbs in early 20th century. Colton [31] proposed the classic nucleation theory, which should be classified into three types: the homogeneous nucleation, heterogeneous nucleation and cavity nucleation.

The main concern of classical nucleation theory is a thermodynamic description of initial stage of nucleation from embryo to nucleus with a little larger size than the critical one.

Homogeneous nucleation occurs in single phase solution system that has no impurity. During the pressure unloading process, every gas molecules will be a nucleation point. So theoretically the largest nucleation density and the smallest bubble size in the final parts will be obtained by homogeneous nucleation. However, due to the purity system, it need more energy to overcome the "energy barrier" to create stable and effective nucleus. Thus there should be more super saturation in the polymer-SCF system.

Microcellular Foam Injection Molding Process 185

Equation (2-5) shows that the frequency factor of the gas molecules joining a nucleus to make it stable varies with the surface area of the nucleus. Generally, ������������� can be

Knowing the surface energy of the system as a function of pressure and temperature, the

Equations 2-1, 2-2, 2-5, 2-6 form a complete set of the nucleation model for polymer-SCF

In order to calculate the total number of nuclei generated in the system at given saturation conditions. The rate of nucleation needs to be integrated over the time period of nucleation. Generally the gas pressure falls as a function of time. Thus the starting saturation pressure (*P*sat) and the pressure at which the polymer vitrifies (*P*g) define the time scale over which the rate of nucleation should be integrated. Therefore, the total number of nuclei,�������, can

��

����

Based on the above nucleation model, the main nucleation process parameters include saturation pressure, mixer temperature and SCF concentration. In this chapter, the effect of the three parameters and the interaction among them on the part cell morphology will be

The simulation experimental model is a thin box. The size is 15.5mm×14mm×13mm. the thickness varies from 0.35mm to 1.8mm. Figure 2-2 shows the cavity distribution, gate system and cooling channels. The characteristic point position is also selected near the gate. The PS/CO2 foam system is bulit and PS brade is Vestgran 620. The each level of three process parameters are shown in the Table 2-4. Besides the studied three parameters, the

> (A)Saturation pressure/ MPa 11 16 21 (B) Melt temperature/ � 220 240 260 (C) Gas concentration/ % 0.3 0.55 0.8

Factors Level1 Level2 Level3

�� � ��

�������������� (2-5)

∆P (2-6)

�� ��� �� � �

� (2-7)

�� � ������

critical size of the nuclei can be calculated at any conditions by Equation 2-6.

������ �� ������� � � �����

**2.2.3 Effect of nucleation process conditions on bubble morphology** 

�

**2.2.3.1 Simulation experimental model and Taguchi method** 

initial values of other process parameters are set in the Table 2-5.

Table 2-4. Level of process parameters.

Substituting Equation 2-4 into Equation 2-3:

Where *r*c is the radius of the critical nucleus.

regarded as a fitted parameter.

be calculated by Equation 2-7.

solution.

discussed.

Heterogeneous nucleation considers that there will be some impurity dispersed in the polymer-SCF mixer. Because there will be more interfacial energy at the impurity solid surface, the nucleation driving force at the impurity solid surface is bigger than the other places. It means that less free energy should be overcome for the nucleus generation. Compared with homogeneous nucleation, heterogeneous nucleation is easier to generate nuclei.

Cavity nucleation is that many nuclei are generated at the cavity places. The gas will be absorbed in the cavity by the nucleating agent or any other micro impurities. Polymer melt can't enter the split wedges at the roughness surface. However the gas will be trapped in these split wedges. During the nucleation process, the gas is tended to enter these cavities to form the nuclei. At the same time, these cavities can save the nucleation energy. And then the stable nucleus can be generated easily.

In this chapter, based on the classical homogeneous nucleation, the microcellular foam nucleation theory is introduced.

#### **2.2.2 Homogeneous nucleation**

#### **Classical homogeneous nucleation [19]**

The main concern of classical homogeneous nucleation theory has been a thermodynamic description of initial stage of nucleation from embryo to nucleus. When the thermodynamic equilibrium is broken and the change of free energy of mixer is more than the "energy barrier", the phase transition occurs and the nuclei are generated. When the nuclei are bigger than the critical one, the nuclei become stable and continue to grow up to bubbles. The rate of homogeneous nucleation can be described by the following Equation 2-1.

$$N\_{homo} = C\_0 f\_0 \exp\left(\frac{-\Delta G}{KT}\right) \tag{2-1}$$

where ܰ is the number of nuclei generated per cm3 per second. *C*0 is the concentration of the gas (number of molecules per cm3). *f*0 is the frequency factor of the gas molecules. *K* is the Boltzmann's constant. And *T* is absolute temperature. The term οܩ is the "energy barrier" for homogeneous nucleation. οܩ can be calculated by Equation 2-2:

$$
\Delta \mathbf{G} = \frac{16\pi\chi^3}{3\Delta P^2} \tag{2-2}
$$

where οܲ is magnitude of the quench pressure and γ is the surface energy of the bubble interface.

The frequency factor of gas molecules in the Equation 2-1, *f*0 , can be expressed as:

$$f\_0 = Z\beta \tag{2.3}$$

where, Z, the Zeldovich factor, accounts for the fact that a large number of nuclei never grow, but rather dissolve. The rate at which the molecules are added to the critical nucleus, β, can be calculated as surface area of the critical nucleus times the rate of impingement of gas molecules per unit area. The calculation method can be expressed as Equation 2-4.

$$\beta = (4\pi r\_c^3) R\_{impingement} \tag{2.4}$$

Heterogeneous nucleation considers that there will be some impurity dispersed in the polymer-SCF mixer. Because there will be more interfacial energy at the impurity solid surface, the nucleation driving force at the impurity solid surface is bigger than the other places. It means that less free energy should be overcome for the nucleus generation. Compared with homogeneous nucleation, heterogeneous nucleation is easier to generate

Cavity nucleation is that many nuclei are generated at the cavity places. The gas will be absorbed in the cavity by the nucleating agent or any other micro impurities. Polymer melt can't enter the split wedges at the roughness surface. However the gas will be trapped in these split wedges. During the nucleation process, the gas is tended to enter these cavities to form the nuclei. At the same time, these cavities can save the nucleation energy. And then

In this chapter, based on the classical homogeneous nucleation, the microcellular foam

The main concern of classical homogeneous nucleation theory has been a thermodynamic description of initial stage of nucleation from embryo to nucleus. When the thermodynamic equilibrium is broken and the change of free energy of mixer is more than the "energy barrier", the phase transition occurs and the nuclei are generated. When the nuclei are bigger than the critical one, the nuclei become stable and continue to grow up to bubbles.

The rate of homogeneous nucleation can be described by the following Equation 2-1.

ܩ൬െο ݔ݂݁ܥ ൌ ܰ

barrier" for homogeneous nucleation. οܩ can be calculated by Equation 2-2:

ο

The frequency factor of gas molecules in the Equation 2-1, *f*0 , can be expressed as:

Ⱦ ൌ ሺͶݎߨ

where ܰ is the number of nuclei generated per cm3 per second. *C*0 is the concentration of the gas (number of molecules per cm3). *f*0 is the frequency factor of the gas molecules. *K* is the Boltzmann's constant. And *T* is absolute temperature. The term οܩ is the "energy

ൌ ͳߛߨ<sup>ଷ</sup>

where οܲ is magnitude of the quench pressure and γ is the surface energy of the bubble

where, Z, the Zeldovich factor, accounts for the fact that a large number of nuclei never grow, but rather dissolve. The rate at which the molecules are added to the critical nucleus, β, can be calculated as surface area of the critical nucleus times the rate of impingement of gas molecules per unit area. The calculation method can be expressed as Equation 2-4.

ܶܭ

(2-1) ൰

͵οܲ<sup>ଶ</sup> (2-2)

(2-3) ߚܼ ൌ ݂

ଷሻܴ௧ (2-4)

nuclei.

interface.

the stable nucleus can be generated easily.

nucleation theory is introduced.

**2.2.2 Homogeneous nucleation** 

**Classical homogeneous nucleation [19]**

Substituting Equation 2-4 into Equation 2-3:

$$f\_0 = Z(4\pi r\_c^3) R\_{lmplingenement} \tag{2-5}$$

Equation (2-5) shows that the frequency factor of the gas molecules joining a nucleus to make it stable varies with the surface area of the nucleus. Generally, ������������� can be regarded as a fitted parameter.

Knowing the surface energy of the system as a function of pressure and temperature, the critical size of the nuclei can be calculated at any conditions by Equation 2-6.

$$
\eta\_c = \frac{2\eta}{\Delta \mathbf{P}} \tag{2-6}
$$

Where *r*c is the radius of the critical nucleus.

Equations 2-1, 2-2, 2-5, 2-6 form a complete set of the nucleation model for polymer-SCF solution.

In order to calculate the total number of nuclei generated in the system at given saturation conditions. The rate of nucleation needs to be integrated over the time period of nucleation. Generally the gas pressure falls as a function of time. Thus the starting saturation pressure (*P*sat) and the pressure at which the polymer vitrifies (*P*g) define the time scale over which the rate of nucleation should be integrated. Therefore, the total number of nuclei,�������, can be calculated by Equation 2-7.

$$N\_{total} = \int\_0^t N\_{homo} dt = \int\_{P\_{sat}}^{P\_g} N\_{homo} \frac{dP}{\left(dP \Big/\_{dt}\right)}\tag{2-7}$$

#### **2.2.3 Effect of nucleation process conditions on bubble morphology**

Based on the above nucleation model, the main nucleation process parameters include saturation pressure, mixer temperature and SCF concentration. In this chapter, the effect of the three parameters and the interaction among them on the part cell morphology will be discussed.

#### **2.2.3.1 Simulation experimental model and Taguchi method**

The simulation experimental model is a thin box. The size is 15.5mm×14mm×13mm. the thickness varies from 0.35mm to 1.8mm. Figure 2-2 shows the cavity distribution, gate system and cooling channels. The characteristic point position is also selected near the gate.

The PS/CO2 foam system is bulit and PS brade is Vestgran 620. The each level of three process parameters are shown in the Table 2-4. Besides the studied three parameters, the initial values of other process parameters are set in the Table 2-5.


Table 2-4. Level of process parameters.

Microcellular Foam Injection Molding Process 187

A(MPa) B(°C) x1x2 C(%) x1x3 x2x3 Cell Size(um)

Table 2-6. <sup>13</sup>

<sup>27</sup> *L* (3 ) Orthogonal table and experimental results.

(A), SCF concentration (C) and mixer temperature (B).

According to Table 2-6, the S/N is calculated and the effect trend of each factors on the S/N also are gotten. Figure 2-3 shows the details. According to Figure 2-3, the significance order from big to small of the effect of each process parameters on cell size is saturation pressure

Fig. 2-2. Experimental model and characteristic point position (a): CAE analysis model; (b): Characteristic point position.


Table 2-5. Othe rocess parameters list.

#### **2.2.3.2 Taguchi method**

Taguchi method is used as an experiment arrangement and parameters optimization method. Based on the setup of parameters and levels, the <sup>4</sup> <sup>9</sup> *L* (3 ) orthogonal array is selected to arrange the experiments. Table 2-6 shows the orthogonal array. The variable analysis is used to calculate the effect order of each process parameters on the cell size and obtain the process parameters optimization combination. At the same time, the experimental results are directly analyzed, that is to calculate the average value of cell size under the three levels of the each process parameter. Here, the cell size is considered that the smaller is better. Therefore it is a minimum value issue. The calculation formula is shown as Equation 2-8 [20]:

$$m = \frac{1}{n} \sum\_{i=1}^{n} y\_i \tag{2-8}$$

where *m* is the average value of process parameter under a certain level, *n* is the number of the level, *<sup>i</sup> y* is the result value of the process parameter under the level. Then the difference *R*diff of each process parameter can be calculated by the maximum average value subtracting the minimum average one. Based on the *R*diff value, the effect of process parameter on the cell size can be achieved.

#### **2.2.3.3 Results and discussion**

#### *Experiment result and signal-to-noise analysis*

The simulation experiments are arranged according to <sup>13</sup> <sup>27</sup> *L* (3 ) orthogonal table. At the same time, each experiment's cell size at characteristic point is obtained. The results are shown in the Table 2-6.

Characteristic point

(a) (b)

minimum value issue. The calculation formula is shown as Equation 2-8 [20]:

(b): Characteristic point position.

Table 2-5. Othe rocess parameters list.

Based on the setup of parameters and levels, the <sup>4</sup>

**2.2.3.2 Taguchi method** 

cell size can be achieved.

shown in the Table 2-6.

**2.2.3.3 Results and discussion** 

*Experiment result and signal-to-noise analysis* 

The simulation experiments are arranged according to <sup>13</sup>

Fig. 2-2. Experimental model and characteristic point position (a): CAE analysis model;

Process parameters Value Mold temperature/ Ԩ 50 Injection time/ s 0.6 Cooling time/ s 35 Open mold time/ s 5

Taguchi method is used as an experiment arrangement and parameters optimization method.

the experiments. Table 2-6 shows the orthogonal array. The variable analysis is used to calculate the effect order of each process parameters on the cell size and obtain the process parameters optimization combination. At the same time, the experimental results are directly analyzed, that is to calculate the average value of cell size under the three levels of the each process parameter. Here, the cell size is considered that the smaller is better. Therefore it is a

1

*i i m y n*

1 *<sup>n</sup>*

where *m* is the average value of process parameter under a certain level, *n* is the number of the level, *<sup>i</sup> y* is the result value of the process parameter under the level. Then the difference *R*diff of each process parameter can be calculated by the maximum average value subtracting the minimum average one. Based on the *R*diff value, the effect of process parameter on the

same time, each experiment's cell size at characteristic point is obtained. The results are

<sup>9</sup> *L* (3 ) orthogonal array is selected to arrange

(2-8)

<sup>27</sup> *L* (3 ) orthogonal table. At the


Table 2-6. <sup>13</sup> <sup>27</sup> *L* (3 ) Orthogonal table and experimental results.

According to Table 2-6, the S/N is calculated and the effect trend of each factors on the S/N also are gotten. Figure 2-3 shows the details. According to Figure 2-3, the significance order from big to small of the effect of each process parameters on cell size is saturation pressure (A), SCF concentration (C) and mixer temperature (B).

Microcellular Foam Injection Molding Process 189

three arrays. And every combination has three experimental results. The average value of

*A*3 12.53 9.47 34.86

According to Table 2-8, the smallest cell size is in the *A*3*C*2 array. Thus the optimization parameters combination is *A*3*B*3*C*2. And the experiment result is validated in the Figure 2-4.

Fig. 2-4. Cell size distribution based on the optimized process parameters combination.

distribution is reasonable. Thus the optimization parameters combination is suitable.

From the Figure 2-4, the cell radio at the characteristic point is 3 um. And the cell size on the part is between 5 um and 10 um. It means that the cell size in the part is acceptable and the

When the nucleation is completed, bubbles begin to grow up. Because the pressure of the mixer is higher than the pressure inside bubbles, SCF in the mixer diffuses into the bubbles and the bubbles grow up. Until the pressure inside the bubbles equals to the outside one or

Initially, the growth and collapse of gas bubbles in both viscous Newtonian and viscoelastic non-Newtonian fluids has been investigated to research on the effect of mass transfer, and the hydrodynamic interaction between the bubble and the liquid was neglected. Barlow et al. [21] are the first to study the phenomenon of diffusion-induced bubble growth in a viscous Newtonian fluid with both mass and momentum transfer. To predict the diffusion of the dissolved gas in the viscous liquid, they used a thin shell approximation. It is assumed that the gas concentration outside the shell always remained equal to the initial concentration. The simplified diffusion equation and an analytical solution were obtained to describe the

*C*1 *C*2 *C*3

each three experimental results is shown in the Table 2-8.

Table 2-8. *A*3×*C* combination table.

**2.3 Bubble growth process** 

**2.3.1 Classic bubble growth model** 

the melt is frozen, the bubbles will keep growing up.

initial stage of the growth at low Reynolds numbers.

Fig. 2-3. Effect of each factors on S/N ratio.

#### *ANOVA analysis*

In order to further analyze the effect of each factors and the interaction among these factors on the cell morphology, ANOVA analysis is calculated according to above S/N results and experiment values. The calculation results are shown in the Table 2-7.


Table 2-7. ANOVA analysis results.

According to Table 2-7, the conclusion of effect of saturation pressure (*A*), SCF concentration (*B*) and mixer temperature (*B*) on the cell morphology is same as the S/N results. However the interaction among the three factors is taken into account in the ANOVA analysis. Also according to Table 27, the significance order is: saturation pressure (*A*) possess 67.93%, the interaction between saturation pressure (*A*) and SCF concentration (*C*) posses 18.51%, SCF concentration (*C*) is 4.91%, Mixer temperature (*B*) is 2.89%. Compared with the S/N results, the interaction between saturation pressure (*A*) and SCF concentration (*C*) is also a very important factor to affect the cell morphology. According to F value, the effect of other factors on the cell morphology is less. So these factors belong to the error range.

Therefore, the optimization parameters combination is mainly determined by the factor *A* and *A*×*C.* Because the smaller cell size is better, the value of *A* and *B* should be the *A*3 and *B*3 in the optimization combination. Due to three levels of *C*, the *A*3×*C* combination has

A1 A2 A3 B1 B2 B3 C1 C2 C3

In order to further analyze the effect of each factors and the interaction among these factors on the cell morphology, ANOVA analysis is calculated according to above S/N results and

mean square

A 2 12620.1 6310.05 24.1638 67.93% \*\*\*

C 2 912.518 456.259 1.74721 4.91% ᇞ

A×C 4 6880.48 1720.12 6.58706 18.51% \*\*

According to Table 2-7, the conclusion of effect of saturation pressure (*A*), SCF concentration (*B*) and mixer temperature (*B*) on the cell morphology is same as the S/N results. However the interaction among the three factors is taken into account in the ANOVA analysis. Also according to Table 27, the significance order is: saturation pressure (*A*) possess 67.93%, the interaction between saturation pressure (*A*) and SCF concentration (*C*) posses 18.51%, SCF concentration (*C*) is 4.91%, Mixer temperature (*B*) is 2.89%. Compared with the S/N results, the interaction between saturation pressure (*A*) and SCF concentration (*C*) is also a very important factor to affect the cell morphology. According to F value, the effect of other

Therefore, the optimization parameters combination is mainly determined by the factor *A* and *A*×*C.* Because the smaller cell size is better, the value of *A* and *B* should be the *A*3 and *B*3 in the optimization combination. Due to three levels of *C*, the *A*3×*C* combination has

error F value Significance Significance

experiment values. The calculation results are shown in the Table 2-7.

B 2 536.500 268.250 1.02724 2.89%

A×B 4 1158.04 289.510 1.10865 3.11%

B×C 4 979.170 244.792 0.93741 2.64%

factors on the cell morphology is less. So these factors belong to the error range.

Sum of square of deviations

Error 8 2089.08 261.135

Fig. 2-3. Effect of each factors on S/N ratio.

‐40 -35 -30 -25 -20 -15 -10 -5 0

**S/N**

Degree of freedom

Sum 26 23086.8 Table 2-7. ANOVA analysis results.

*ANOVA analysis* 

three arrays. And every combination has three experimental results. The average value of each three experimental results is shown in the Table 2-8.


Table 2-8. *A*3×*C* combination table.

According to Table 2-8, the smallest cell size is in the *A*3*C*2 array. Thus the optimization parameters combination is *A*3*B*3*C*2. And the experiment result is validated in the Figure 2-4.

Fig. 2-4. Cell size distribution based on the optimized process parameters combination.

From the Figure 2-4, the cell radio at the characteristic point is 3 um. And the cell size on the part is between 5 um and 10 um. It means that the cell size in the part is acceptable and the distribution is reasonable. Thus the optimization parameters combination is suitable.
