# The Influence of Physical Properties on the Membrane Morphology Formation during the Nonisothermal Thermally Induced Phase Separation Process

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Development

_{2}= 100 is plotted in Figure 1 using the Flory–Huggins free energy density adopted from Kurata [42], which is consistent with the experimental phase diagrams for the polystyrene–cyclohexanol polymer solution available in the literature [5,19,21,23,43,44,45]. Since the viscosity of the polymer is highly dependent on molecular weight and temperature, the exact value is not provided at the temperature range studied. However, the viscosity of polystyrene in cyclohexanol is equal to 56.45 cP at 25 °C [46].

**n**is the outward unit normal and is expressed in Cartesian coordinates as follows:

_{0}and phase-separated compositions of c

_{1}and c

_{2}, the lever rule and Flory–Huggins theory are used to determine the difference in the mixing enthalpy between the original system and the demixed system as follows [43,44,53,54]:

_{q}, and the initial solution temperature was denominated as T*. T*

_{q}

_{1}and T*

_{q}

_{2}denote the two different quench temperatures that are applied to the two opposite sides of the membrane.

^{−6}, according to the established convergence criterion. The process simulation was carried out using the Fortran programming language.

## 3. Results and Discussion

_{q}= 0.97 are provided in Figure 2. The initial concentration and initial and quench temperatures are selected from our previous studies so that the change in concentration and temperature due to a change in the physical properties of the system can be used to provide a comparison with regard to the influence of the physical properties on phase separation and the morphology formation [8,9]. In addition, due to the number of graphs in each section, only the results of the simulations for the model developed based on the composition- and temperature-dependent physical properties are provided in this paper to avoid misperception. In order to evaluate the structure formation during the phase separation in the first step, a quench was applied simultaneously from all four sides of the membrane.

^{−6}, the initial thermal concentration fluctuations still exist in the one phase region of the phase diagram, and the corresponding temperature profile represents a very short time after the quench is applied. The reason for providing the very early times is to provide the range of change in solution density, heat conductivity, and heat capacity in the initial solution at a high temperature and right after the quench is applied, which will be discussed in the following paragraphs.

^{−5}. The droplets start to form at the sides where the quench is applied, while in the interior parts, no major phase separation can be detected. The yellow region represents the polymer-rich phase, and the blue region represents the solvent-rich droplets. The corresponding temperature profile in Figure 2b shows that there is a small amount of heat generated as a result of the enthalpy of demixing. Comparing these results with the dimensionless concentration and temperature patterns for the deep quench in our previous study reveals that phase separation has progressed more with smaller droplets being formed from the sides of the sample. In addition, the temperature profile exhibits more progress inside the sample, which also leads to more heat generation during this stage. This is due to the fact that considering the physical properties leads to an increase in heat diffusivity during phase separation, which is subject to the change in density, heat capacity, and thermal conductivity. Higher heat diffusivity means a higher heat transfer rate, which in turn leads to more phase separation. This phenomenon will be further elaborated.

^{−4}, as shown in Figure 2c, phase separation has progressed in the interior parts of the membrane, with droplets being formed in the entire sample. The droplets that are formed are smaller in size compared to our previous study, where a constant heat diffusivity was used. In addition, the difference in the size of droplets between the interior parts and the boundaries is more vivid. The inhomogeneity in the corresponding temperature profile accounts for the heat that is released during phase separation as a result of demixing. The amount of heat generation is not substantial in this stage. The variation in droplet size is clearly visible in Figure 2c, as the droplets located at the boundaries begin to increase in size, resulting in an anisotropic morphology.

^{−4}demonstrate that on the four sides of the membrane where the quench was applied, the phase separation has entered its final stage and the droplets that were previously formed have merged to reduce the interfacial area due to coarsening. In the interior parts of the sample, the phase separation is still in the intermediate stage. The increase in temperature, as can be seen in the temperature profile in Figure 2d, illustrates the increase in heat generated during phase separation. Terminating the phase separation at any stage would result in a membrane with an anisotropic morphology and desired pore size due to the difference in the size of droplets owing to different rates of heat transfer and phase separation at various parts of the membrane.

^{−5}, which corresponds to the dimensionless concentration and temperature profiles and patterns represented in Figure 2b, the droplets of the solvent that are initially formed are more thermally conductive than the polymer solution and the polymer-rich phase. Hence, the thermal conductivity of the solvent-rich droplets increases, and the thermal conductivity inside the membrane also increases due to the increase in the mean-free path of the molecules as a result of the decrease in temperature, which is in accordance with the expected pattern in amorphous polymers.

^{−5}, which corresponds to the dimensionless concentration and temperature profiles depicted in Figure 2c, the thermal conductivity of the solvent-rich droplets continues to increase while the thermal conductivity of the polymer-rich phase decreases. This is due to the fact that the heat that is generated leads to a decrease in the thermal conductivity of the polymer solution. While on the sides of the membrane, the thermal conductivity keeps increasing as a result of the formation of thermally conductive droplets, which merge and increase the conductivity, and are also in contact with the cooling membrane during phase separation. The heat generation due to the enthalpy of demixing influences the thermal conductivity of the polymer solution in a similar way to the density.

^{−5}which is presented in Figure 3(IIb). The heat capacity of the solvent-rich droplets that are formed during phase separation is lower than the heat capacity of the initial polymer solution. This is also true for the polymer-rich membrane that ultimately forms. The same phenomenon occurs when there is a heat release due to the enthalpy of demixing during phase separation between the times t* = 2.03 × 10

^{−4}and t* = 8.03 × 10

^{−4}as depicted in the concentration and temperature profiles in Figure 2c,d and the corresponding patterns in Figure 3(IIc,d). As the temperature increases, the heat capacity also increases. At the very late stage of phase separation, the two phases are completely separated, and the heat capacity has decreased in the solvent-rich droplets, which have now merged and formed bigger droplets, as well as in the polymer-rich phase.

^{−3}, phase separation in the deep quench side has almost reached its final stage, and the droplets that were previously produced start to merge, while in the shallow quench side, phase separation is still in the intermediate stage. Heat is generated inside the membrane as a result of enthalpy of demixing. Heat generation increases as time passes and the heat moves to the deep quench side and induces a shallow quench effect. This phenomenon has been widely investigated in our previous paper [8,9]. However, comparing the results provided here reveals that, at the same time period during phase separation, the heat transfer rate is high owing to higher heat diffusivity. This leads to smaller droplet formation and weakens the influence of the heat of demixing during phase separation. The reason for choosing temperature and concentration dependence of density, heat conductivity, and heat capacity instead of heat diffusivity alone is to capture the variations of each physical property during phase separation and the influence of the enthalpy of demixing on each physical property.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Chan, P.K.; Rey, A.D. Computational analysis of spinodal decomposition dynamics in polymer solutions. Macromol. Theory Simul.
**1995**, 4, 873–899. [Google Scholar] [CrossRef] - Cahn, J.W. Free Energy of a Nonuniform System. II. Thermodynamic Basis. J. Chem. Phys.
**1959**, 30, 1121–1124. [Google Scholar] [CrossRef] - Alt, H.W.; Pawlow, I. A mathematical model of dynamics of non-isothermal phase separation. Phys. D Nonlinear Phenom.
**1992**, 59, 389–416. [Google Scholar] [CrossRef] - Van De Witte, P.; Dijkstra, P.J.; Van Den Berg, J.W.A.; Feijen, J. Phase separation processes in polymer solutions in relation to membrane formation. J. Membr. Sci.
**1996**, 117, 1–31. [Google Scholar] [CrossRef] - Tsai, F.J.; Torkelson, J.M. The roles of phase separation mechanism and coarsening in the formation of poly(methyl methacrylate) asymmetric membranes. Macromolecules
**1990**, 23, 775–784. [Google Scholar] [CrossRef] - Romay, M.; Diban, N.; Urtiaga, A. Thermodynamic Modeling and Validation of the Temperature Influence in Ternary Phase Polymer Systems. Polymers
**2021**, 13, 678. [Google Scholar] [CrossRef] [PubMed] - Xing, W.; Plawsky, J.; Woodcock, C.; Yu, X.; Ullmann, A.; Brauner, N.; Peles, Y. Liquid-liquid phase separation heat transfer in advanced micro structure. Int. J. Heat Mass Transf.
**2018**, 127, 989–1000. [Google Scholar] [CrossRef] - Ranjbarrad, S.; Chan, P.K. The Effect of Conductive Heat Transfer on the Morphology Formation in Polymer Solutions Undergoing Thermally Induced Phase Separation. Polymers
**2022**, 14, 4345. [Google Scholar] [CrossRef] - Ranjbarrad, S.; Chan, P.K. Morphology formation during the nonisothermal thermally-induced phase separation (TIPS) process. Can. J. Chem. Eng.
**2023**. advance online. [Google Scholar] [CrossRef] - Vandeweerdt, P.; Arnauts, J.; De Cooman, R.; Koningsveld, R. Calorimetric analysis of liquid-liquid phase separation. J. Chem. Phys.
**1994**, 101, 10420–10426. [Google Scholar] [CrossRef] - Ullmann, A.; Poesio, P.; Brauner, N. Enhancing heat transfer rates by inducing liquid-liquid phase separation: Applications and modeling. Interfacial Phenom. Heat Transf.
**2015**, 3, 41–67. [Google Scholar] [CrossRef] - Ullmann, A.; Gat, S.; Brauner, N. Flow Phenomena and Heat Transfer Augmentation during Phase Separation of Partially Miscible Solvent Systems. AIP Conf. Proc.
**2010**, 1207, 81–91. [Google Scholar] [CrossRef] - Molin, D.; Mauri, R. Enhanced heat transport during phase separation of liquid binary mixtures. Phys. Fluids
**2007**, 19, 074102. [Google Scholar] [CrossRef] - Matsuyama, H.; Berghmans, S.; Batarseh, M.T.; Lloyd, D.R. Formation of Anisotropic and Asymmetric Membranes via Thermally-Induced Phase Separation. ACS Symp. Ser.
**1999**, 744, 23–41. [Google Scholar] [CrossRef] - Lee, K.-W.D.; Chan, P.K.; Feng, X. A Computational Study into Thermally Induced Phase Separation in Polymer Solutions under a Temperature Gradient. Macromol. Theory Simul.
**2002**, 11, 996–1005. [Google Scholar] [CrossRef] - Kukadiya, S.B.; Chan, P.K.; Mehrvar, M. The Ludwig-Soret Effect on the Thermally Induced Phase Separation Process in Polymer Solutions: A Computational Study. Macromol. Theory Simul.
**2009**, 18, 97–107. [Google Scholar] [CrossRef] - Huston, E.; Cahn, J.W.; Hilliard, J. Spinodal decomposition during continuous cooling. Acta Met.
**1966**, 14, 1053–1062. [Google Scholar] [CrossRef] - Krantz, W.B.; Ray, R.J.; Sani, R.L.; Gleason, K.J. Theoretical study of the transport processes occurring during the evaporation step in asymmetric membrane casting. J. Membr. Sci.
**1986**, 29, 11–36. [Google Scholar] [CrossRef] - Heijden, P.V.D. A DSC-Study on the Demixing of Binary Polymer Solutions. Doctoral Dissertation, Universiteit Twente, Twente, The Netherlands, 2001. [Google Scholar]
- Molin, D.; Mauri, R. Spinodal decomposition of binary mixtures with composition-dependent heat conductivities. Chem. Eng. Sci.
**2008**, 63, 2402–2407. [Google Scholar] [CrossRef] - Tsai, F.J.; Torkelson, J.M. Microporous poly(methyl methacrylate) membranes: Effect of a low-viscosity solvent on the formation mechanism. Macromolecules
**1990**, 23, 4983–4989. [Google Scholar] [CrossRef] - Di Fede, F.; Poesio, P.; Beretta, G. Heat transfer enhancement in a small pipe by spinodal decomposition of a low viscosity, liquid–liquid, strongly non-regular mixture. Int. J. Heat Mass Transf.
**2012**, 55, 897–906. [Google Scholar] [CrossRef] - Nistor, A.; Vonka, M.; Rygl, A.; Voclova, M.; Minichova, M.; Kosek, J. Polystyrene Microstructured Foams Formed by Thermally Induced Phase Separation from Cyclohexanol Solution. Macromol. React. Eng.
**2017**, 11, 1600007. [Google Scholar] [CrossRef] - Atkinson, P.M.; Lloyd, D.R. Anisotropic flat sheet membrane formation via TIPS: Thermal effects. J. Membr. Sci.
**2000**, 171, 1–18. [Google Scholar] [CrossRef] - Ariyapadi, M.V.; Nauman, E.B. Gradient energy parameters for polymer–polymer–solvent systems and their application to spinodal decomposition in true ternary systems. J. Polym. Sci. Part B Polym. Phys.
**1990**, 28, 2395–2409. [Google Scholar] [CrossRef] - Miranville, A.; Schimperna, G. Nonisothermal phase separation based on a microforce balance. Discret. Contin. Dyn. Syst.-Ser. B
**2005**, 5, 753–768. [Google Scholar] [CrossRef] - Chan, P.K.; Rey, A.D. A numerical method for the nonlinear Cahn-Hilliard equation with nonperiodic boundary conditions. Comput. Mater. Sci.
**1995**, 3, 377–392. [Google Scholar] [CrossRef] - Caneba Gerard, T.; Soong, D.S. Polymer membrane formation through the thermal-inversion process. 2. Mathematical modeling of membrane structure formation. Macromolecules
**1986**, 18, 2545, Erratum in Macromolecules**1986**, 19, 2671. [Google Scholar] [CrossRef] - Cahn, J.W.; Hilliard, J.E. Free Energy of a Nonuniform System. I. Interfacial Free Energy. J. Chem. Phys.
**1958**, 28, 258–267. [Google Scholar] [CrossRef] - Barton, B.F.; Graham, P.D.; McHugh, A.J. Dynamics of Spinodal Decomposition in Polymer Solutions near a Glass Transition. Macromolecules
**1998**, 31, 1672–1679. [Google Scholar] [CrossRef] - Manzanarez, H.; Mericq, J.; Guenoun, P.; Chikina, J.; Bouyer, D. Modeling phase inversion using Cahn-Hilliard equations—Influence of the mobility on the pattern formation dynamics. Chem. Eng. Sci.
**2017**, 173, 411–427. [Google Scholar] [CrossRef] - Wall, F.T. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, USA, 1953. [Google Scholar]
- Vonka, M.; Nistor, A.; Rygl, A.; Toulec, M.; Kosek, J. Morphology model for polymer foams formed by thermally induced phase separation. Chem. Eng. J.
**2016**, 284, 357–371. [Google Scholar] [CrossRef] - Matyjaszewski, K.; Moeller, M. Polymer Science: A Comprehensive Reference; Elsevier Science: Amsterdam, The Netherlands, 2021. [Google Scholar]
- Hansen, C.M. Hansen Solubility Parameters: A User’s Handbook; CRC Press: Boca Raton, FL, USA, 2007; ISBN 9780849372483. [Google Scholar]
- de Gennes, P.G. Dynamics of fluctuations and spinodal decomposition in polymer blends. J. Chem. Phys.
**1980**, 72, 4756–4763. [Google Scholar] [CrossRef] - Novick-Cohen, A.; Segel, L.A. Nonlinear aspects of the Cahn-Hilliard equation. Phys. Lett. A
**1984**, 10, 277–298. [Google Scholar] [CrossRef] - De Gennes, P.G. Reptation of a polymer chain in the presence of fixed obstacles. In P.G. Gennes’ Impact on Science, Soft Matter and Biophysics; World Scientific Publishing Co. Pte. Ltd.: Singapore, 2009; pp. 35–42. [Google Scholar] [CrossRef]
- De Gennes, P.G. Dynamics of Entangled Polymer Solutions. II. Inclusion of Hydrodynamic Interactions. Macromolecules
**1975**, 9, 594–598. [Google Scholar] [CrossRef] - Debye, P.; Jacobsen, R.T. Direct Visual Observation of Concentration Fluctuations in a Critical Mixture. J. Chem. Phys.
**1968**, 48, 203–206. [Google Scholar] [CrossRef] - Debye, P. Angular Dissymmetry of the Critical Opalescence in Liquid Mixtures. J. Chem. Phys.
**1959**, 31, 680–687. [Google Scholar] [CrossRef] - Kurata, M. Thermodynamics of Polymer Solutions; Hardwood Academic: New York, NY, USA, 1982. [Google Scholar]
- De Rudder, J.; Berghmans, H.; Arnauts, J. Phase behaviour and structure formation in the system syndiotactic polystyrene/cyclohexanol. Polymer
**1999**, 40, 5919–5928. [Google Scholar] [CrossRef] - Hikmet, R.; Callister, S.; Keller, A. Thermoreversible gelation of atactic polystyrene: Phase transformation and morphology. Polymer
**1988**, 29, 1378–1388. [Google Scholar] [CrossRef] - Guo, H.F.; Laxminarayan, A.; Caneba, G.T.; Solc, K. Morphological studies of late-stage spinodal decomposition in polystyrene–cyclohexanol system. J. Appl. Polym. Sci.
**1995**, 55, 753–759. [Google Scholar] [CrossRef] - Song, S.-W.; Torkelson, J.M. Coarsening effects on the formation of microporous membranes produced via thermally induced phase separation of polystyrene-cyclohexanol solutions. J. Membr. Sci.
**1995**, 98, 209–222. [Google Scholar] [CrossRef] - Cussler, E.L. Diffusion Mass Transfer in Fluid Systems; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Ebert, M.; Garbella, R.W.; Wendorff, J.H. Studies on the heat of demixing of poly (ethylene acrylate)-poly (vinylidene fluoride) blends. Die Makromol. Chem. Rapid Commun.
**1986**, 7, 65–70. [Google Scholar] [CrossRef] - Coker, A.K. Ludwig’s Applied Process Design for Chemical and Petrochemical Plants; Gulf Professional Publishing: Houston, TX, USA, 2014. [Google Scholar]
- van Krevelen, D.W.; te Nijenhuis, K. Properties of Polymers: Their Correlation with Chemical Structure; Their Numerical Estimation and Prediction from Additive Group Contributions; Elsevier: Amsterdam, The Netherlands, 2009. [Google Scholar]
- Maloney, J.O. Perry Chemical Engineers Handbook; The McGraw-Hill Companies, Inc.: New York, NY, USA, 2008. [Google Scholar]
- Mark, J.E. (Ed.) Physical Properties of Polymers Handbook; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar] [CrossRef]
- Aubert, J.; Clough, R. Low-density, microcellular polystyrene foams. Polymer
**1985**, 26, 2047–2054. [Google Scholar] [CrossRef] - Sakai, V.G.; Higgins, J.S.; Trusler, J.P.M. Cloud Curves of Polystyrene or Poly(methyl methacrylate) or Poly(styrene-co-methyl methacrylate) in Cyclohexanol Determined with a Thermo-Optical Apparatus. J. Chem. Eng. Data
**2006**, 51, 743–748. [Google Scholar] [CrossRef] - Choy, C.L. Thermal conductivity of polymers. Polymer
**1977**, 18, 984–1004. [Google Scholar] [CrossRef] - Muthaiah, R.; Garg, J. Temperature effects in the thermal conductivity of aligned amorphous polyethylene—A molecular dynamics study. J. Appl. Phys.
**2018**, 124, 105102. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of the phase diagram for polystyrene–cyclohexanol solution with N

_{2}= 100.

**Figure 2.**The dimensionless concentration (column I and II) and temperature profiles and patterns (column III and IV) for an off-critical quench with c* = 0.85 and T* = 1.03 and T*

_{q}= 0.97 at the following times: (

**a**) t* = 1.03 × 10

^{−6}, (

**b**) t* = 3.03 × 10

^{−5}, (

**c**) t* = 2.03 × 10

^{−}

^{4}, (

**d**) t*= 8.03 × 10

^{−4}, and (

**e**) t* = 8.03 × 10

^{−2}.

**Figure 3.**The variations in density (column I), heat conductivity (column II), and heat capacity (column III) during phase separation at the following times: (

**a**) t* = 1.03 × 10

^{−6}, (

**b**) t* = 3.03 × 10

^{−5}, (

**c**) t* = 2.03 × 10

^{−4}, (

**d**) t* = 8.03 × 10

^{−4}, and (

**e**) t* = 8.03 × 10

^{−2}.

**Figure 4.**The dimensionless concentration (column I and II) and temperature profiles and patterns (column III and IV) for an off-critical quench with c* = 0.909 and T* = 1.05 and T*

_{q}= 0.95 at the following times (

**a**) t* = 7 × 10

^{−4}, (

**b**) t* = 1 × 10

^{−3}, (

**c**) t* = 6 × 10

^{−3}, and (

**d**) t* = 9 × 10

^{−3}.

**Figure 5.**The variations in density (column

**I**), heat conductivity (column

**II**), and heat capacity (column

**III**) during phase separation at the following times: (

**a**) t* = 7 × 10

^{−4}, (

**b**) t* = 1 × 10

^{−3}, (

**c**) t* = 6 × 10

^{−3}, and (

**d**) t* = 9 × 10

^{−3}.

**Figure 6.**The dimensionless concentration (column I and II) and temperature profiles and patterns (column III and IV) for an off-critical quench with c* = 0.909 and T* = 1.05 and T*

_{q}= 0.95 at the following times (

**a**) t* = 7 × 10

^{−4}, (

**b**) t* = 1 × 10

^{−3}, and (

**c**) t* = 9 × 10

^{−3}.

**Figure 7.**The variations in density (column I), heat conductivity (column II), and heat capacity (column III) during phase separation at the following times (

**a**) t* = 7 × 10

^{−4}, (

**b**) t* = 1 × 10

^{−3}, and (

**c**) t* = 9 × 10

^{−3}.

The Property | The Equation Utilized in the Model Development | Eq. | References |
---|---|---|---|

The total free energy of a system | $F=\int \left[f\left(c\right)+\kappa {\left(\nabla c\right)}^{2}\right]dV$ | (1) | [2,29,31] |

The free energy of the homogenous system | $f\left(c\right)=\frac{{k}_{B}T}{\nu}\left[\frac{c}{{N}_{1}}\mathrm{l}\mathrm{n}\mathrm{c}+\frac{1-c}{{N}_{2}}\mathrm{ln}\left(1-c\right)+\chi c(1-c)\right]$ | (2) | [32] |

Flory’s interaction parameter | $\chi =\frac{{V}_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{l}}}{RT}{({\delta}_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{l}}-{\delta}_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{y}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{e}})}^{2}+0.34$ | (3) | [23,33] |

The solubility parameters of the solvent and the polymer | ${{\delta}_{\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{y}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{e}}}^{2}={{\delta}_{\mathrm{p}}}^{2}+{{\delta}_{\mathrm{h}}}^{2}+{{\delta}_{\mathrm{d}}}^{2}=22.68$ ${{\delta}_{\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{l}}}^{2}={{\delta}_{\mathrm{p}}}^{2}+{{\delta}_{\mathrm{h}}}^{2}+{{\delta}_{\mathrm{d}}}^{2}=22.401$ | (4) | [34,35] |

The nonlinear Cahn-Hilliard equation | $\frac{\partial c}{\partial t}=\nabla \xb7\left[M\nabla \left[\frac{\partial f}{\partial c}-2\kappa {\nabla}^{2}c\right]\right]$ | (5) | [36,37] |

The mobility | $M=\frac{\nu c\left(1-c\right)}{\xi}$ | (6) | [1,15,27,30] |

The gradient energy coefficient | $\kappa =\frac{RT\chi {l}^{2}}{6}$ | (7) | [36,38,39,40,41] |

**Table 2.**The equations utilized to designate the temperature- and composition-dependence of physical properties.

Property | Solvent (Cyclohexanol) | Polymer (Polystyrene) |
---|---|---|

Specific heat capacity (J/kg·K) | ${c}_{p,s}=-A+BT-\mathrm{C}{T}^{2}+D{T}^{3}$ ^{a}$A=470,B=19$ $C=47,\mathrm{D}=48$ | ${c}_{p,p}=A+BT$ ^{b}$A=1049.2,B=2.236$ |

Density (kg/m ^{3}) | ${\rho}_{s}=\frac{A}{{B}^{(1+{\left(1-\frac{T}{C)}\right)}^{D}}}$ ^{c}$A=82.43$ $B=0.26546$ $C=650$ $\mathrm{D}=0.2848$ | ${\rho}_{p}=A-BT-C{T}^{2}$ ^{d}$A=1.067$ $B=5.02\times {10}^{-4}$l $C=0.135\ast {10}^{-6}$ |

Heat conductivity (W/m·K) | ${k}_{s}=A-BT$^{e}$A=0.2092$ $B=2.5\ast {10}^{-4}$ | ${k}_{p}=k\left({T}_{g}\right)\ast \left(1.2-\frac{0.2T}{{T}_{g}}\right)$ ^{f,g} |

**Table 3.**The initial solvent volume fractions and the initial and quench temperature selected in this study.

Case | Initial Solvent Volume Fraction | Initial Solution Temperature | Quench Temperature |
---|---|---|---|

1 | c* = 0.85 | T* = 1.03 | T*_{q} = 0.97 |

2 | c* = 0.909 | T* = 1.05 | T*_{q}_{1} = 0.95T* _{q}_{2} = 0.95 |

3 | c* = 0.909 | T* = 1.05 | T*_{q}_{1} = 0.95T* _{q}_{2} = 0.99 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ranjbarrad, S.; Chan, P.K.
The Influence of Physical Properties on the Membrane Morphology Formation during the Nonisothermal Thermally Induced Phase Separation Process. *Polymers* **2023**, *15*, 3475.
https://doi.org/10.3390/polym15163475

**AMA Style**

Ranjbarrad S, Chan PK.
The Influence of Physical Properties on the Membrane Morphology Formation during the Nonisothermal Thermally Induced Phase Separation Process. *Polymers*. 2023; 15(16):3475.
https://doi.org/10.3390/polym15163475

**Chicago/Turabian Style**

Ranjbarrad, Samira, and Philip K. Chan.
2023. "The Influence of Physical Properties on the Membrane Morphology Formation during the Nonisothermal Thermally Induced Phase Separation Process" *Polymers* 15, no. 16: 3475.
https://doi.org/10.3390/polym15163475