# The Effect of Conductive Heat Transfer on the Morphology Formation in Polymer Solutions Undergoing Thermally Induced Phase Separation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Development

_{B}is the Boltzmann’s constant, N

_{1}and N

_{2}are the degrees of polymerization of the solvent and the polymer, respectively. In polymer solutions, the degree of polymerization of the solvent N

_{1}is 1 and the degree of polymerization of the polymer N

_{2}, was set to 100 in this study. The polar interactions and hydrogen bonding were considered in calculating Flory’s interaction parameter. The correlation that was utilized to determine the interaction parameter is the one used by Nistor et al. [47] in the simple thermodynamics of a binary mixture defined as [6,47]:

_{CHOL}is the molar volume of cyclohexanol and $\delta $ is the Hildebrand solubility parameter. It is beneficial to define the interaction of the polymer and the solvent through the solubility parameter. Hansen’s contribution theory, which is a useful explanation of polarity, designates the solubility parameter to depend on polar interactions (${\delta}_{\mathrm{p}}$), hydrogen bonding (${\delta}_{\mathrm{h}}$) and the dispersive term (${\delta}_{\mathrm{d}}$) in terms of [65]:

**j**is the interdiffusional flux determined as a function of chemical potential gradient:

_{1}and M

_{2}are the self-mobility of the solvent and polymer, respectively.

_{i}< 200 and is expressed as [71,72,73]:

_{1}is the volume fraction of the solvent, and c

_{2 =}1 − c

_{1}is the volume fraction of the polymer. If frictional coefficients of the solvent and the polymer are assumed to be equal and independent of concentration and temperature (${\xi}_{1}={\xi}_{2}=\xi $), and no interaction exists between polymer and solvent, the mobility is defined through the following equation as [12,15,72]:

_{p}* is the dimensionless specific heat. The two-dimensional Cahn-Hilliard equation and the Fourier heat transfer equation were derived using the scaling relations. Two dimensionless numbers were introduced in this study. The first one was $\mathsf{\Lambda}$ which was the square of the medium length over the product of the square of the radius of gyration and the degree of polymerization. This dimensionless number was defined as the dimensionless characteristic length. This parameter denotes the length of the sample with respect to the size of the polymer chain. The second dimensionless parameter $\lambda $ was a combination of $\mathsf{\Lambda}$, heat diffusivity $\alpha $, and the diffusion coefficient $D$. This dimensionless parameter can be explained as the combination of Lewis number Le which describes the rate of temperature spread through the material to the diffusion coefficient, and the $\mathsf{\Lambda}$ parameter. Hence, it provides information on the rate of heat diffusion with respect to polymer diffusion. The rate of heat conduction inside the solution influences the morphology formation as well. If heat diffusion is faster than the diffusion of mass, as is the case for high viscosity polymer solutions, heat will diffuse faster and reach the homogenous temperature in the entire sample before phase separation is completed.

_{2}= 10, and N

_{2}= 100 are presented in Figure 1 which are plotted using the Flory-Huggins theory and are in accordance with the experimentally and theoretically obtained phase diagram [6,28,40,47,76,77,78]. N

_{2}is the degree of polymerization of the polymer and N

_{1}is the degree of polymerization of the solvent, which is N

_{1}= 1. The simulation results provided in this study are based on N

_{2}= 100.

_{0}and the phase separated compositions of c

_{1}and c

_{2}as [28,80,81,82,83]:

_{q}and the initial solution concentration was demonstrated as c*. The finite element method was applied to obtain the set of time-dependent ordinary differential equations (ODEs) that were solved by the Newton-Raphson method. The Galerkin finite element method was utilized to solve the governing equations using the Hermitian basis function to discretize space [15,63]. A mesh of 80 × 80 was considered to discretize the square domain. As there were 8 values per node, the size of the Jacobian matrix was 52,448 × 52,448. The Forward-Backward Euler method was employed for the time integration. The convergence criterion was set to be the difference between the two consecutive solutions being less than 10

^{−6}. The Fortran programming language was utilized to simulate the process, and each simulation was performed for almost two weeks using the high-performance computers in the graduate computer lab of the Chemical Engineering Department, Toronto Metropolitan University and Digital Research Alliance of Canada.

## 3. Results and Discussion

_{q3}= 0.97, T*

_{q2}= 0.99 and T*

_{q1}= 1.0 from all four sides of the sample are provided in Figure 2, Figure 3, Figure 4 and Figure 5 for an off-critical quench with a dimensional initial concentration of c* = 0.85 and the influence of the quench depth on morphology formation was evaluated at different times during phase separation. The initial dimensionless solution temperature was set to T* = 1.03. The dimensionless concentration profiles are presented on the left side, and the corresponding temperature profiles are presented in the right column. Most of the experimental studies and industrial processes conducted for polymeric materials’ formation using the TIPS process, apply quench to one or two sides of the medium. The results provided in this section go one step further and provide a qualitative analysis of the morphology formation when all four sides of the sample are quenched to the same temperature. The dimensionless concentration c* as a function of x* and y* at time t

^{*}= 1.03 × 10

^{−6}and the corresponding dimensionless temperature T* profile are presented in Figure 2. Phase separation did not initiate at this time yet and the infinitesimal thermal concentration fluctuations that were specified in the initial condition of the Cahn-Hilliard equation can be observed in the concentration profile for all quench depths. However, the temperature profiles are different at different quench depths, which indicates that increasing quench depth leads to higher driving force for heat transfer. The transient temperature evolves faster than the phase separation upon applying quench in the high-viscosity polymer solution.

^{−5}, as shown in Figure 3, the morphology started to evolve for the deep quench to T*

_{q3}= 0.97 from all four sides of the sample and advanced inside the medium gradually. However, no phase separation was accomplished for the shallow quench to T*

_{q1}= 1.0 yet. Due to the low interfacial tension between the two phases at this quench depth, the difference in the equilibrium concentrations between the two phases was not significant, which led to a lower phase separation rate. The dimensionless concentration profiles for the quench depths of T*

_{q2}= 0.99, and T*

_{q3}= 0.97 showed phase separation initiation from all four sides, which is more evident in the deep quench to T*

_{q3}

^{=}0.97. Anisotropic morphology would result due to the difference in the quench rate in the boundaries and inside the sample. Although there was a variation in the droplet size between shallow and deep quenches, due to the polar interactions and the hydrogen bonding effects of the solvent, the morphology difference between the shallow and deep quenches was not substantial, as verified by Nistor et al. [6]. The major difference between the shallow and deep quenches was the time it took for the entire system to phase separate.

^{−4}as shown in Figure 4. As is evident, the temperature reached almost its homogenous state while the phase separation was still progressing. However, the inhomogeneity in the dimensionless temperature profiles was due to the enthalpy of demixing effect, which will be discussed further. The number of the droplets increased as the quench depth increased, or the quench temperature decreased. On the other hand, the size of the droplets decreased as the quench depth increased even though the change was not considerable. The dimensionless temperature profile presented in Figure 4a shows that the temperature has almost become homogenous in the entire sample with a very low amount of heat released. Whereas the temperature profiles presented in Figure 4b,c show more heat being released during the phase separation process. The four sides of the medium reached the intermediate stage of phase separation at t* = 2.03 × 10

^{−4}while the interior parts were still in the initial stage due to the high viscosity of the solution that prevented convection driven heat transfer. Heat did not propagate fast enough to induce phase separation homogenously, and interior parts of the sample were still at higher temperatures than the quench sides, which led to slower phase separation rates. This led to gradation in the pore size due to the difference in the quench rate between various parts of the sample, and hence anisotropic morphology resulted. The heat that was released during phase separation performed as a shallow quench effect and increased the quench temperature, which led to phase separation being accomplished at a higher temperature than the initial quench temperature. Although this increase in temperature is not significant in polymer solutions and blends, it is verified in this study that it cannot be neglected. The comparison between the temperature profiles from the shallowest to the deepest quench revealed that increasing the quench depth led to higher rates of heat transfer and, consequently, higher rates of phase separation by increasing the driving force for heat transfer and phase separation due to the difference in the compositions between the two phases.

^{−4}. In the case of deep quench to T*

_{q3}= 0.97, more phase separation was accomplished, and phase separation reached its final stage on the four sides where the droplets started to coarsen by merging while the interior parts were still in the intermediate stage of phase separation. A comparison between the deep and shallow quenches reveals that there is a significant difference between the shallow quench to T*

_{q1}= 1.0 and the deep quench to T*

_{q3}= 0.97 which led to more phase separation being performed in the deep quench with smaller droplet formation. As the quench depth increased, the driving force for heat transfer increased.

_{q3}= 0.97, the direction of anisotropy would change and the smaller droplets that were formed in the boundaries would merge due to coarsening, which would increase the size of the droplets in comparison with the interior parts. This was previously verified by Lee et al. [1,2] as a result of applying a linear temperature gradient in the sample.

_{q1}= 1.0, T*

_{q2}= 0.99, and T

^{*}

_{q3}= 0.97 at different stages during phase separation. As presented in Figure 6 at t* = 1.03 × 10

^{−6}, the initial thermal concentration fluctuations existed in the two-phase region of the phase diagram where phase separation had not yet initiated, while the temperature profiles are different due to the difference in the heat transfer rates between the shallow and deep quenches. As the quench depth increased, the rate of heat transfer in the system also increased, which is reflected through the temperature profiles presented in Figure 6, Figure 7, Figure 8 and Figure 9. As shown in Figure 7, at t* = 2.03 × 10

^{−5}for the shallow quench to T*

_{q1}= 1.0, no phase separation was accomplished while heat transfer progressed considerably inside the sample and reached its homogenous state. On the other hand, as can be seen in Figure 7b,c, phase separation was initiated but was in the very early stage of phase separation with the temperature almost reaching its homogenous state and the entire sample. In the deep quench to T*

_{q3}= 0.97, phase separation started and progressed inside the sample at t* = 2.03 × 10

^{−5}.

_{q3}= 0.97. As time proceeded to t* = 3.01 × 10

^{−4}as demonstrated in Figure 8, phase separation progressed in the entire sample and an interconnected morphology became apparent. As shown in the concentration profile in Figure 8c, phase separation progressed considerably compared to the shallow quench presented in Figure 8a,b. The corresponding temperature profile of each stage of phase separation reveals that even though the increase in the temperature due to the enthalpy of demixing is not substantial, its influence on the morphology formation during the thermally induced phase separation process cannot be neglected. At time equal to t* = 3.01 × 10

^{−4}, phase separation in the deep quench to T*

_{q3}= 0.97 reached the late stage of phase separation and the medium started to coarsen from all four sides. The corresponding temperature profile also presented the highest increase in temperature. On the other hand, as presented in Figure 8a,b, phase separation was in the intermediate stage, with more phase separation being accomplished in the four boundaries where the quench was applied.

^{−4}. The entire medium reached the late stage of spinodal decomposition and coarsened in the deep quench to T*

_{q3}= 0.97 as presented in Figure 9a. Figure 9b presents the dimensionless concentration and temperature profiles of the quench temperature to T*

_{q2}= 0.99. Compared to Figure 9a, more phase separation was carried out in the entire medium, and more heat was released due to phase separation. All the results provided for the case of critical quench, along with the results provided in Figure 2, Figure 3, Figure 4 and Figure 5, verify that deeper quench depths led to more phase separation with smaller droplet formation and a greater number of droplets. Besides, applying quench from four sides of the medium caused a gradient in the cooling rate between the sample boundaries and the interior parts, which led to anisotropic morphology formation.

_{q3}= 0.97 as provided in Figure 10 and Figure 11, respectively. The first column represents the spatial concentration profile, the second column illustrates the phase-separated pattern, and the third column presents the corresponding spatial temperature profile. The white area presents the polymer-rich region where c* < 0.88 and the black area presents the solvent-rich region where c* > 0.88. The droplet-type morphology was formed due to the off-critical quench and the droplets on the left side of the medium where quench was applied merged and coarsened faster with time than the other sides of the medium that were insulated. The same initial composition, boundary conditions, and the quench rate were considered for both cases presented in Figure 10 and Figure 11.

^{−5}. As it is shown, phase separation started in both cases from the left side of the sample where the quench was applied. However, the amount of the phase separation that was accomplished without considering the enthalpy of demixing was more than the case when the enthalpy of demixing was employed. This confirms the importance of considering the enthalpy of demixing during phase separation through spinodal decomposition because the heat increased the quench temperature to some extent and led to phase separation being accomplished at a higher temperature. Neglecting this parameter would cause loss of cost and energy during the industrial processes of porous polymeric materials’ formation.

^{−4}. Phase separation in the quench side progressed significantly, with smaller droplets forming in the quench side and larger droplets in the insulated sides. Considering Figure 10c and 11c confirms that the amount of the phase separation when the enthalpy of demixing was neglected was higher comparing to the case of considering it as presented in Figure 11c.

_{q2}= 0.99 are demonstrated. When the initial temperature of the solution was higher, as observed in Figure 12a it took more time for heat transfer to reach its homogenous state, which led to phase separation being accomplished at a higher temperature. Even if the quench was applied fast, the high viscosity of the solution prevented the temperature homogeneousness instantly. The droplet size was another factor that was influenced by the solution temperature. The high solution temperature led to longer quench time and, as a result, larger droplet formation as presented in the dimensionless concentration profiles in Figure 12. Comparing the dimensionless concentration profiles in Figure 12a,b, reveals that more phase separation was accomplished at the same time scale. Despite the fact that more heat was also released in the case presented in Figure 12b, the initial solution temperature was more predominant in increasing the phase separation temperature, which led to less phase separation. This effect was significant at the initial stages of phase separation when the phase separation was still in the early stage.

^{−4}, the amount of the heat generated in the initial solution temperature of T

^{*}= 1.03 was larger than that in the case with the initial solution temperature of T* = 1.05 as presented in Figure 13b. After this time, when the amount of the enthalpy of demixing was competing with the initial solution temperature, the difference between the concentration profiles for the two cases of different solution temperatures could be neglected. Hence, it is of great importance to choose the most suitable initial solution temperature for phase separation based on the system properties and characteristics. If the enthalpy of demixing is high in a system, the initial solution temperature should be designated based on the competing effects between the heat generated in the system and the initial solution temperature as it directly influences the phase separating morphology.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic phase diagram of the polystyrene-cyclohexanol polymer solution with two degrees of polymerization of N

_{2}= 10 and N

_{2}= 100.

**Figure 2.**Spatial concentration (

**left**column) and temperature (

**right**column) profiles for the off-critical quench to c* = 0.85 at t* = 1.03 × 10

^{−6}for three different quench depths of (

**a**) T*

_{q1}= 1.0, (

**b**) T*

_{q2}= 0.99 (

**c**) T*

_{q3}= 0.97.

**Figure 3.**Spatial concentration (

**left**column) and temperature (

**right**column) profiles for the off-critical quench to c* = 0.85 at t* = 3.03 × 10

^{−5}for three different quench depths of (

**a**) T*

_{q1}= 1.0, (

**b**) T*

_{q2}= 0.99 (

**c**) T*

_{q3}= 0.97.

**Figure 4.**Spatial concentration (

**left**column) and temperature (

**right**column) profiles for the off-critical quench to c* = 0.85 at t* = 2.03 × 10

^{−4}for three different quench depths of (

**a**) T*

_{q1}= 1.0, (

**b**) T*

_{q2}= 0.99 (

**c**) T*

_{q3}= 0.97.

**Figure 5.**Spatial concentration (

**left**column) and temperature (

**right**column) profiles for the off-critical quench to c* = 0.85 at t* = 8.03 × 10

^{−4}for three different quench depths of (

**a**) T*

_{q1}= 1.0, (

**b**) T*

_{q2}= 0.99 (

**c**) T*

_{q3}= 0.97.

**Figure 6.**Spatial concentration (

**left**column) and temperature (

**right**column) profiles for the off-critical quench to c* = 0.909 at t* = 1.03 × 10

^{−6}for three different quench depths of (

**a**) T*

_{q1}= 1.0, (

**b**) T*

_{q2}= 0.99 (

**c**) T*

_{q3}= 0.97.

**Figure 7.**Spatial concentration (

**left**column) and temperature (

**right**column) profiles for the off-critical quench to c* = 0.909 at t* = 2.03 × 10

^{−5}for three different quench depths of (

**a**) T*

_{q1}= 1.0, (

**b**) T*

_{q2}= 0.99 (

**c**) T*

_{q3}= 0.97.

**Figure 8.**Spatial concentration (

**left**column) and temperature (

**right**column) profiles for the off-critical quench to c* = 0.909 at t* = 3.01 × 10

^{−4}for three different quench depths of (

**a**) T*

_{q1}= 1.0, (

**b**) T*

_{q2}= 0.99 (

**c**) T*

_{q3}= 0.97.

**Figure 9.**Spatial concentration (

**left**column) and temperature (

**right**column) profiles for the off-critical quench to c* = 0.909 at t* = 8.01 × 10

^{−4}for three different quench depths of (

**a**) T*

_{q1}= 1.0, (

**b**) T*

_{q2}= 0.99 (

**c**) T*

_{q3}= 0.97.

**Figure 10.**Spatial concentration (

**left**column), phase separated pattern (

**middle**column) and temperature (

**right**column) profiles for off-critical quench without considering enthalpy of demixing c* = 0.88 at (

**a**) t* = 1.03 × 10

^{−6}, (

**b**) t* = 2.03 × 10

^{−5}, (

**c**) t* = 8.03 × 10

^{−4}, and (

**d**) t* = 1.03 × 10

^{−3}.

**Figure 11.**Spatial concentration (

**left**column), phase separated pattern (

**middle**column) and temperature (right column) profiles for off-critical quench considering enthalpy of demixing c* = 0.88 (

**a**) t* = 1.03 × 10

^{−6}, (

**b**) t* = 2.03 × 10

^{−5}, (

**c**) t* = 8.03 × 10

^{−4}, and (

**d**) t* = 1.03 × 10

^{−3}.

**Figure 12.**The influence of initial solution temperature on phase separation through TIPS process with two different dimensionless initial solution temperatures of (

**a**) T* = 1.05 and (

**b**) T* = 1.03 to the same quench temperature of T*

_{q2}= 0.99 at time t* = 1.03 × 10

^{−5}.

**Figure 13.**The influence of initial solution temperature on phase separation through TIPS process with two different dimensionless initial solution temperatures of (

**a**) T* = 1.05 and (

**b**) T* = 1.03 to the same quench temperature of T*

_{q2}= 0.99 at time t* = 5.03 × 10

^{−4}.

**Table 1.**Conditions studied in this paper including boundary condition, quench depths, and initial average concentrations of the solvent.

Case | ${\mathit{c}}^{*}$ | ${\mathit{T}}^{*}$ | ${\mathit{T}}^{*}{}_{\mathbf{q}1}$ | ${\mathit{T}}^{*}{}_{\mathbf{q}2}$ | ${\mathit{T}}^{*}{}_{\mathbf{q}3}$ | Boundary Conditions |
---|---|---|---|---|---|---|

1 | 0.85 | 1.03 | 1.0 | 0.99 | 0.97 | ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{x}^{*}=0$ ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{x}^{*}=1$ ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{y}^{*}=0$ ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{y}^{*}=1$ |

2 | 0.909 | 1.03 | 1.0 | 0.99 | 0.97 | ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{x}^{*}=0$ ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{x}^{*}=1$ ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{y}^{*}=0$ ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{y}^{*}=1$ |

3 | 0.88 | 1.03 | ${\mathit{T}}^{\mathit{*}}{}_{\mathbf{q}}=0.97$ | ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{x}^{*}=0$ $\frac{\partial {T}^{*}}{\partial x}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{x}^{*}=1$ $\frac{\partial {T}^{*}}{\partial y}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{y}^{*}=0$ $\frac{\partial {T}^{*}}{\partial y}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{y}^{*}=1$ | ||

(a) With enthalpy of demixing | (b) Without enthalpy of demixing | |||||

4 | 0.89 | ${\mathit{T}}^{\mathit{*}}{}_{\mathbf{1}}$ | ${\mathit{T}}^{\mathit{*}}{}_{\mathbf{2}}$ | ${\mathit{T}}^{\mathit{*}}{}_{\mathbf{q}}$ | ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{x}^{*}=0$ ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{x}^{*}=1$ ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{y}^{*}=0$ ${T}^{*}={T}^{*}{}_{\mathrm{q}}$$\mathrm{at}{t}^{*}0$$\mathrm{and}{y}^{*}=1$ | |

1.03 | 1.05 | 0.99 |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/polym14204345

**AMA Style**

Ranjbarrad S, Chan PK.
The Effect of Conductive Heat Transfer on the Morphology Formation in Polymer Solutions Undergoing Thermally Induced Phase Separation. *Polymers*. 2022; 14(20):4345.
https://doi.org/10.3390/polym14204345

**Chicago/Turabian Style**

Ranjbarrad, Samira, and Philip K. Chan.
2022. "The Effect of Conductive Heat Transfer on the Morphology Formation in Polymer Solutions Undergoing Thermally Induced Phase Separation" *Polymers* 14, no. 20: 4345.
https://doi.org/10.3390/polym14204345