# Computer Simulation of Anisotropic Polymeric Materials Using Polymerization-Induced Phase Separation under Combined Temperature and Concentration Gradients

^{*}

## Abstract

**:**

_{ave}> was larger in this region. While smaller droplets formed later in the lower concentration/temperature regions, at the higher concentration/temperature side, the droplets went through phase separation longer, allowing them to reach the late stage of the phase separation where particles coarsened. In the intermediate stage of phase separation, <d

_{ave}> was found proportional to ${t}^{*\alpha}$, where $\alpha $ was in the range between $\frac{1}{3}$ and $\frac{1}{2}$ for the cases studied and was consistent with published results.

## 1. Introduction

## 2. Model Development

**j**= 0

**n**is the unit normal to the surface. For a two dimensional square domain, the natural boundary conditions become:

^{−6}. A mesh of 100 × 100 is used in the computer simulation. The computing code was written in C

^{++}and executed on the Compute Canada, Sharcnet Consortium, Graham Resource (128G/32Core); each run took 10−18 h.

## 3. Results and Discussion

_{1}[43], and degree of polymerization [4]. The first part includes the results showing phase-separated structure and morphology development for the eight cases listed in Table 1. The second part shows the size analysis obtained for the Cases 1, 4, and 5 using ImageJ software [44].

#### 3.1. Phase-Separated Structure and Morphology Development

#### 3.2. Size and Morphology Analysis

_{ave}> of the droplets. The method of exclusion [45] was utilized to assess the size development of the three cases mentioned above. The domain was divided into five equal intervals; sections 1–5 from left to right. The particles located on the boundary of two sections were considered in the right region. The particles located on the boundary of the system were assumed as half droplets. The results are presented on Figure 10.

^{th}section, the section facing a higher concentration/temperature earlier while they grew later from section 4 to section 1. Growth in the lower sections occurred later due to a higher process induction time and slower elevation of phase diagram at the lower concentration/temperature. Therefore, the particles of different sections entered the unstable region and went through the early, intermediate, and late stages of the phase separation at different times.

^{−4}), sections 4 and 5 were in the intermediate stage of phase separation while the others were still in the early stage of phase separation (section 3) or in the stable one phase region (sections 1 and 2). Therefore, the average diameters in different sections were different, and the morphology was anisotropic. At the later time (${t}^{*}$ > 2.19 × 10

^{−4}), however, the droplets in all of the sections were in the intermediate stage of phase separation, so the growth rates and the average sizes got close together. In addition, when the time proceeded, the quench depth was continuously increasing, and the samples were getting far from the critical point; as such, at the higher sections, which undergo the phase separation longer, the growth rate of the particles decreased because they had reached the late stage of phase separation. This let the droplets at the lower sections grow and their diameters get close to those at the higher sections.

## 4. Conclusions

^{*α}where α was found in a range between $\frac{1}{3}$ and $\frac{1}{2}$ for the cases studied, which was consistent with previous work.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic phase diagram of a binary mixture undergoing the polymerization-induced phase separation (PIPS) phenomenon. The symmetric phase diagram represents the initial state when the degrees of polymerization of both of the components are one. The phase diagram shifts upward to higher concentrations and becomes asymmetric with increasing polymer molecular weight through polymerization. Eventually, the sample is thrust into the unstable region and starts to phase separate. The blue box represents the domain in which all of the eight cases presented in this paper are located.

**Figure 2.**Spatial concentration profiles (first column) and phase-separated patterns (second column) for Case 1 at the following times: (

**a**) ${t}^{*}$ = 2.14 × 10

^{−4}, (

**b**) ${t}^{*}$ = 2.18 × 10

^{−4}, and (

**c**) ${t}^{*}$ = 2.24 × 10

^{−4}.

**Figure 3.**Spatial concentration profiles (first column) and phase-separated patterns (second column) for Case 2 at the following times: (

**a**) ${t}^{*}$ = 9.8 × 10

^{−5}, (

**b**) ${t}^{*}$ = 9.9 × 10

^{−5}, (

**c**) ${t}^{*}$ = 1.0 × 10

^{−4}, and (

**d**) ${t}^{*}$ = 1.01 × 10

^{−4}.

**Figure 4.**Spatial concentration profiles (first column) and phase-separated patterns (second column) for Case 3 at the following times: (

**a**) ${t}^{*}$ = 1.01 × 10

^{−4}, (

**b**) ${t}^{*}$ = 1.02 × 10

^{−4}, (

**c**) ${t}^{*}$ = 1.07 × 10

^{−4}.

**Figure 5.**Spatial concentration profiles (first column) and phase-separated patterns (second column) for Case 4 at the following times: (

**a**) ${t}^{*}$ = 4.312 × 10

^{−4}, (

**b**) ${t}^{*}$ = 4.32 × 10

^{−4}, (

**c**) ${t}^{*}$ = 4.34 × 10

^{−4}, and (

**d**) ${t}^{*}$ = 4.37 × 10

^{−4}.

**Figure 6.**Spatial concentration profiles (first column) and phase-separated patterns (second column) for Case 5 at the following times: (

**a**) ${t}^{*}$ = 5.93 × 10

^{−5}, (

**b**) ${t}^{*}$ = 6.0 × 10

^{−5}, and (

**c**) ${t}^{*}$ = 6.1 × 10

^{−5}.

**Figure 7.**Spatial concentration profiles (first column) and phase-separated patterns (second column) for Case 6 at the following times: (

**a**) ${t}^{*}$ = 1.095 × 10

^{−4}, (

**b**) ${t}^{*}$ = 1.1 × 10

^{−4}, (

**c**) ${t}^{*}$ = 1.11 × 10

^{−4}, and (

**d**) ${t}^{*}$ = 1.12 × 10

^{−4}.

**Figure 8.**Spatial concentration profiles (first column) and phase-separated patterns (second column) for Case 7 at the following times: (

**a**) ${t}^{*}$ = 5.66 × 10

^{−5}, (

**b**) ${t}^{*}$ = 5.7 × 10

^{−5}, (

**c**) ${t}^{*}$ = 5.8 × 10

^{−5}, and (

**d**) ${t}^{*}$ = 5.9 × 10

^{−5}.

**Figure 9.**Spatial concentration profiles (first column) and phase-separated patterns (second column) for Case 8 at the following times: (

**a**) ${t}^{*}$ = 5.62 × 10

^{−5}, (

**b**) ${t}^{*}$ = 5.65 × 10

^{−5}, (

**c**) ${t}^{*}$ = 5.67 × 10

^{−5}, and (

**d**) ${t}^{*}$ = 5.8 × 10

^{−5}.

**Figure 10.**The average equivalent diameter of droplets developed within the five sections of a sample. Figures in rows

**a**,

**b**, and

**c**represent Cases 1, 4, and 5, respectively. The left column is scaled according to t

^{*}–t

_{0}

^{*}, where t

_{0}

^{*}is the polymerization lag time; the right column is scaled based on the t

^{*}.

**Table 1.**The parameters are defined for the eight cases studied in the current work. For all cases, ${E}_{a}^{*}$ = 10 and $\psi $ = 1.

Case | Condition | T^{*} | c_{o}^{*} | D | A^{*} |
---|---|---|---|---|---|

Case 1 | • Δc_{o}*• Left to the critical region | 0.6 | 0.55–0.6 | 2 × 10^{5} | 5 × 10^{10} |

Case 2 | • Δc_{o}*• Across the critical region | 0.6 | 0.6–0.7 | 4 × 10^{5} | 10^{11} |

Case 3 | • Δc_{o}*• Right to the critical region | 0.6 | 0.7–0.75 | 4 × 10^{5} | 10^{11} |

Case 4 | • ΔT* • Left to the critical region | 0.54–0.55 | 0.55 | 4 × 10^{5} | 10^{11} |

Case 5 | • ΔT^{*}+ Δc_{o}^{*}• Same Direction • Left to the critical region | 0.595–0.6 | 0.55–0.6 | 4 × 10^{5} | 2 × 10^{11} |

Case 6 | • ΔT^{*}+ Δc_{o}^{*}• Opposite directions • Left to the critical region | 0.595–0.6 | 0.56–0.6 | 4 × 10^{5} | 10^{11} |

Case 7 | • ΔT^{*}+ Δc_{o}^{*}• Opposite directions • Left to the critical region | 0.595–0.6 | 0.54–0.6 | 4 × 10^{5} | 2 × 10^{11} |

Case 8 | • ΔT^{*}+ Δc_{o}^{*}• Opposite directions • Left to the critical region | 0.595–0.6 | 0.552–0.6 | 4 × 10^{5} | 2 × 10^{11} |

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**MDPI and ACS Style**

Ghaffari, S.; Chan, P.K.; Mehrvar, M.
Computer Simulation of Anisotropic Polymeric Materials Using Polymerization-Induced Phase Separation under Combined Temperature and Concentration Gradients. *Polymers* **2019**, *11*, 1076.
https://doi.org/10.3390/polym11061076

**AMA Style**

Ghaffari S, Chan PK, Mehrvar M.
Computer Simulation of Anisotropic Polymeric Materials Using Polymerization-Induced Phase Separation under Combined Temperature and Concentration Gradients. *Polymers*. 2019; 11(6):1076.
https://doi.org/10.3390/polym11061076

**Chicago/Turabian Style**

Ghaffari, Shima, Philip K. Chan, and Mehrab Mehrvar.
2019. "Computer Simulation of Anisotropic Polymeric Materials Using Polymerization-Induced Phase Separation under Combined Temperature and Concentration Gradients" *Polymers* 11, no. 6: 1076.
https://doi.org/10.3390/polym11061076