Building Model for Simulating the Effect of Main Factors on the PVDF Membrane Formation Process

The conservation equations are described using mathematical descriptions, and then the equations are solved discretely. The location of the moving interface (volume fraction of the polymer) was obtained using the interface tracking method ^13–16 and the results were output.

Assumptions of the model

Basic control equations

Control equations are mathematical descriptions of conservation laws (mass, momentum and energy) (all expressed in the form of scatter).

• (1)  
  Conservation of mass equation:

The fluid mass conservation equation is given in the form of a fluid volume fraction continuity equation:

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+\displaystyle \frac{\partial \left(\rho u\right)}{\partial x}+\displaystyle \frac{\partial \left(\rho v\right)}{\partial y}+\displaystyle \frac{\partial \left(\rho w\right)}{\partial z}=0\end{eqnarray} \tag{ 1 }$$

where ρ is density, ∇ is coordinate free spatial differential operator, Nabla operator, u, v, w are velocity components in x, y, z axis directions, respectively, i, j, k are unit vectors in x, y, z directions in the coordinate system respectively.

$$\begin{eqnarray}&&{\rm{\nabla }}=i\displaystyle \frac{\partial }{\partial x}+j\displaystyle \frac{\partial }{\partial y}+k\displaystyle \frac{\partial }{\partial z}\end{eqnarray} \tag{ 2 }$$

The density of the mixed system is defined as:

$$\begin{eqnarray}&&\rho ={\varphi }_{p}{\rho }_{p}+{\varphi }_{s}{\rho }_{s}+{\varphi }_{ns}{\rho }_{ns}\end{eqnarray} \tag{ 3 }$$

where $\varphi$ is volume fraction of each phase, subscripts "p", "s", "ns" denote polymer phase, solvent phase, and non-solvent phase, respectively.

The volume fraction of each phase is given by the following equation:

$$\begin{eqnarray}&&{\varphi }_{p}+{\varphi }_{s}+{\varphi }_{ns}=1\end{eqnarray} \tag{ 4 }$$

If the density is constant when the fluid motion of the fluid micro-element, its material derivative is 0:

$$\begin{eqnarray}&&\displaystyle \frac{D\rho }{Dt}=\displaystyle \frac{\partial \rho }{\partial t}+{\rm{\nabla }}\cdot \left(\rho \vec{U}\right)=0\end{eqnarray} \tag{ 5 }$$

Substituting 5 into 1 there is:

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot \vec{U}=0\end{eqnarray} \tag{ 6 }$$

• (2)  
  Volume fraction continuity equation:

The volume fraction continuity equations for the polymer phase, solvent phase, and non-solvent phase are given.

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial t}{\varphi }_{p}{\rho }_{p}+{\rm{\nabla }}\cdot \left({\varphi }_{p}{\rho }_{p}{U}_{p}\right)=0\end{eqnarray} \tag{ 7 }$$

• (3)  
  Conservation of momentum equation:

The conservation of momentum equation whose textual expression is: the rate of increase of fluid momentum in a micro-element.

$$\begin{eqnarray}&&\displaystyle \frac{\partial \left(\rho {\rm{u}}\right)}{\partial t}+{\rm{\nabla }}\cdot \left(\rho u\vec{u}\right)=-\displaystyle \frac{\partial P}{\partial x}+\displaystyle \frac{\partial {\tau }_{xx}}{\partial x}+\displaystyle \frac{\partial {\tau }_{yx}}{\partial y}+\displaystyle \frac{\partial {\tau }_{zx}}{\partial z}+{F}_{x}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial \left(\rho v\right)}{\partial t}+{\rm{\nabla }}\cdot \left(\rho v\vec{u}\right)=-\displaystyle \frac{\partial P}{\partial y}+\displaystyle \frac{\partial {\tau }_{xy}}{\partial x}+\displaystyle \frac{\partial {\tau }_{yy}}{\partial y}+\displaystyle \frac{\partial {\tau }_{zy}}{\partial z}+{F}_{y}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial \left(\rho {\rm{w}}\right)}{\partial t}+{\rm{\nabla }}\cdot \left(\rho w\vec{u}\right)=-\displaystyle \frac{\partial P}{\partial z}+\displaystyle \frac{\partial {\tau }_{xz}}{\partial x}+\displaystyle \frac{\partial {\tau }_{yz}}{\partial y}+\displaystyle \frac{\partial {\tau }_{zz}}{\partial z}+{F}_{z}\end{eqnarray} \tag{ 10 }$$

where P is pressure on the fluid microelement, ρ is density of the mixed fluid, τ is viscous stress, F is volumetric force on the microelement.

• (4)  
  Energy equation:

The conservation of energy equation whose textual expression is: the rate of increase of thermodynamic energy in the micro-element is equal to the work done by the force acting on the micro-element is. ¹⁷

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho E}{\partial t}+{\rm{\nabla }}\cdot \left[U\left(\rho E+p\right)\right]={\rm{\nabla }}\cdot \left({k}_{eff}{\rm{\nabla }}T\right)+{S}_{h}\,\end{eqnarray} \tag{ 11 }$$

The energy is similarly shared by all phases, as follows: where E denotes energy; T denotes temperature; both of which are mass-averaged based variables.

$$\begin{eqnarray}&&E=\displaystyle \frac{{\varphi }_{p}{\rho }_{p}{E}_{p}+{\varphi }_{s}{\rho }_{s}{E}_{s}+{\varphi }_{ns}{\rho }_{ns}{E}_{ns}}{{\varphi }_{p}{\rho }_{p}+{\varphi }_{s}{\rho }_{s}+{\varphi }_{ns}{\rho }_{ns}}\end{eqnarray} \tag{ 12 }$$

where Ei for each phase is a function of the energy of each phase.

The materials used in this experiment are shown in Table I.

The main experimental apparatus used in this experiment is shown in Table II.

The specific experimental steps are as follows:

• (1)  
  The PVDF will be dried in the oven at 60 °C for 24 h to remove the water.
• (2)  
  Measure an appropriate amount of solvent in a conical flask, accurately weigh a quantitative amount of PVDF mixed in the solvent, and carry out ultrasonic dispersion for 30 min.
• (3)  
  Stirring at a constant temperature in an oil bath at 60 °C until PVDF is completely dissolved, and then standing in a water bath at 60 °C for 12 h to defoam. ¹⁸
• (4)  
  Finally, the membrane solution was cast on a glass plate with a membrane maker and quickly immersed into a pure water solution at room temperature of about 20 °C for phase separation. Subsequently, the prepared membrane was kept in pure water.
• (5)  
  After the preparation of the modified membrane is completed, the water is changed every 8 h for the first three days; the water is changed every 24 h for the next four days, at the end of which the membrane is considered to have completed phase separation and is kept in pure water for use.

For adding additives to the system, accurately weigh a certain amount of additives mixed in the solvent and follow the same steps as above.

In this paper, the mass fraction is used to determine the amount of solute and solvent in the experiment, while the volume fraction is used to express during the simulation. The transformation relationship is as follows:

Let the total mass of the experiment be M, the solute (PVDF) mass fraction be ω_p , and the density be ρ_p . solvent mass fraction of ω_s and density of ρ_s . The mass fraction of non-solvent (H₂O) is ω_ns and the density is ρ_ns ; the mass fraction of additive is ω_α and the density is ρ_α .

Then the volume fraction of solute (PVDF) is:

$$\begin{eqnarray}&&{\varphi }_{p}=\displaystyle \frac{M{\omega }_{p}/{\rho }_{p}}{M{\omega }_{p}/{\rho }_{p}+M{\omega }_{s}/{\rho }_{s}+M{\omega }_{ns}/{\rho }_{ns}}\end{eqnarray} \tag{ 13 }$$

Similarly, the volume fraction of solvent, non-solvent (H₂O) and additives can be found. Among them:

$$\begin{eqnarray}&&{\varphi }_{p}+{\varphi }_{s}+{\varphi }_{ns}=1\end{eqnarray} \tag{ 14 }$$

Scanning electron microscopy was used to characterize the surface morphology as well as the cross-sectional structure of the modified membranes, which can help this paper to analyze the various properties of the membranes from the microscopic morphology. The membrane samples to be tested were soaked in 30% glycerol for 24 h to protect the pore structure of the membranes, then the membranes were rinsed with anhydrous ethanol, and then dried naturally at room temperature and sealed in self-sealing bags for use. ¹⁹ The process was noted to record the numbering in time. The membranes were subjected to liquid nitrogen brittle fracture treatment before observing the morphological structure of the cross-section, and the microscopic morphological features of the upper and lower surfaces and cross-sections of the membranes were observed by field emission scanning electron microscopy.

The water flux is a basic parameter of the membrane, which is a direct reflection of the membrane filtration carrying capacity. In this paper, the water flux is measured in order to compare and analyze with the prediction of membrane pores in the simulation results graph, and then validate the model from the side.

This experiment is a pure water flux test of the membrane on a high pressure plate membrane tester. ²⁰ The pilot machine was rinsed with pure water before starting the test. After flushing, the membrane is pre-pressurized at 0.25 MPa for 30 min, and then the pressure is adjusted to 0.2 MPa to collect 1000 ml of filtrate and record the time, water temperature, room temperature and other conditions after the machine runs stably.

$$\begin{eqnarray}&&J=\displaystyle \frac{V}{At}\end{eqnarray} \tag{ 15 }$$

where J is pure water flux (L·m⁻²·h⁻¹), V is volume of filtered water (L), A is effective area of membrane (m⁻²), t is filtration time (h).

In this experiment, the dry and wet weight method is used to determine the porosity of the membrane. The membranes with completed phase separation were removed from pure water, dried on the surface of the membranes and weighed in wet state m_w , and then the membranes were put into the oven at 60 °C to be completely dried and weighed in dry state m_d . Three samples were tested with the same ratio of membrane, and the average value of ε was calculated as the test result according to Eq. 16.

$$\begin{eqnarray}&&\varepsilon =\displaystyle \frac{\left({m}_{w}-{m}_{d}\right)/{\rho }_{w}}{\left({m}_{w}-{m}_{d}\right)/{\rho }_{w}+{m}_{d}/{\rho }_{p}}\times 100 \% \end{eqnarray} \tag{ 16 }$$

where m_w is membrane wet state weight (g), m_d is membrane dry state weight (g), ρ_w is density of water at 20°C (0.998 g cm⁻³),ρ_P is PVDF density(1.78 g cm⁻³).

From the control equation of the model, we know that the magnitude of the pressure P is related to the momentum equation in the control equation, so the magnitude of its value is closely related to the model. Starting from the momentum equation of the model, the interfacial force source term on the right-hand side of the momentum equation corresponds to the incremental pressure due to the interfacial tension added at the interface, and the static pressure P affects the velocity field shared by all phases, which in turn affects the polymer diffusion. The specific setting values are shown in Table III. The pressure value P was taken as the only variable value, and the effect of cross-migration rate was ignored in the setting of the system because of the possible complex relationship between system pressure and cross-migration rate. The initial conditions of the two layers are shown in Table IV.

Table IV. Two-layer initial conditions.

Layering	$\varphi$ p	$\varphi$ s	$\varphi$ ns
Polymer solution layer	0.2	0.75	0.05
Coagulation bath	0.01	0.01	0.98

The simulation time of t = 7.0 × 10⁻⁵s was selected for comparison and analysis, as shown in Fig. 2. From the overall view of the figure, the membrane pores are all slowly changing from partially dispersed points to a continuous state, and the interfaces between the polymer solution layer and the solidification bath connection are all relatively flat, but the top layer of the polymer solution all show small fractures. With the increase of system pressure value, the graphs A0 ∼ A1 show that the thickness of inter-pore wall thickness decreases and the pore diameter becomes larger after membrane formation and Figs. A1 to A4 are the opposite of Figs. A0 to A1. Figures A4 to A5 are again similar to Figs. A0 to A1, which indicates that the effect of system pressure P value on membrane formation morphology possesses a certain peak, but its specific value needs to be further explored through experiments.

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From the control equation of the model, it is obtained that the magnitude of the temperature T is related to the momentum equation in the control equation, so the magnitude of its value is closely related to the model. The specific setting values are shown in Table V. The temperature value T was taken as the only variable value, and the effect of cross-migration rate was ignored in the setting of the system because of the possible complex relationship between system pressure and cross-migration rate. The initial conditions of the two layers are shown in Table IV.

The simulation time of t = 8.0 × 10⁻⁵s was selected for comparative analysis, as shown in Fig. 3. From the overall view of the figure, the membrane pores are all slowly changing from dispersed points to a continuous state, and the interface between the polymer solution layer and the solidification bath connection are all relatively flat, but the top layer of the polymer solution is all slightly fractured. With the increase of the system temperature, among them, Figs. B0 ∼ B3 show that the thickness of the inter-pore wall thickness after membrane formation decreases in turn, and the pore diameter after membrane formation increases in turn, and the trend in Figs. B3 ∼ B5 is similar to that of B0 ∼ B3, but the trend pattern of polymer volume fraction in Figs. B2 ∼ B3 is opposite to that of B0 ∼ B3, which indicates that the influence of the system temperature T value on the morphology of membrane formation has a certain peak, but its specific value needs to be further explored through experiments.

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The initial concentration (volume fraction) of the polymer was fixed and the initial volume fraction of solvent and non-solvent was varied to study the effect of the initial concentration of solvent on the membrane morphology. The volume fraction of polymer in the fixed polymer solution was 5%, and the percentage of DMAc and H₂O were varied sequentially, and the specific ratios between them are shown in Table VI.

Table VI. Initial composition in polymer solutions.

Series	$\varphi$ p	$\varphi$ s	$\varphi$ ns
C1	0.05	0.05	0.90
C2	0.05	0.10	0.85
C3	0.05	0.15	0.80
C4	0.05	0.20	0.75
C5	0.05	0.25	0.70
C6	0.05	0.30	0.75

The simulation time of t = 5.5 × 10⁻⁴s was selected for comparison and analysis, as shown in Fig. 4. It can be seen from the figure that the simulated variation diagrams of the solvent systems with different concentrations follow roughly the same course, and when the volume fraction of the solvent increases, the morphology does not actually change significantly, despite the different final equilibrium concentrations. At the same time, as the solvent system in the polymer solution increases, a small change occurs in the solidification bath that is no longer the volume fraction of the lowest polymer in the time step, i.e., completely blue. It indicates that the polymer in the polymer solution layer is diffusing into the solidification bath. The volume fraction of PVDF in the polymer layer in C4 is smaller compared to the other series. With the increase of solvent volume fraction in the system polymer solution layer, the volume fraction of polymer in the solidification baths of Figs. C1 ∼ C6 increased sequentially with a slight trend, where Figs. C1 ∼ C2 illustrated that the thickness of inter-pore wall thickness increased sequentially after membrane formation, while the pore diameter decreased sequentially after membrane formation, the trend in Figs. C2 ∼ C4 was opposite to that of C1 ∼ C2, and Figs. C5 ∼ C6 were similar to that of Figs. C2 ∼ T4, but Figs. C4 ∼ C5 were similar to that of C1 ∼ C2, indicating that the initial component of solvent in the system polymer solution had some influence on the morphology of membrane formation.

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The initial concentration (volume fraction) of the polymer was fixed and the initial volume fraction of solvent and non-solvent was varied to study the effect of the initial concentration of the solvent on the membrane morphology. The volume fraction of polymer in the fixed polymer solution was 25%, and the percentage of DMAc and H₂O was varied sequentially, and the specific ratios between them are shown in Table VII.

Table VII. Initial composition in polymer solution.

Series	$\varphi$ p	$\varphi$ s	$\varphi$ ns
D1	0.25	0.75	0.00
D2	0.25	0.73	0.02
D3	0.25	0.71	0.04
D4	0.25	0.69	0.06
D5	0.25	0.67	0.08
D6	0.25	0.65	0.10

A comparative analysis of the polymer solution simulation plots for different non-solvent systems is shown in Fig. 5. With the increase of non-solvent volume fraction in the system polymer solution layer, where Figs. D1 ∼ D2, D3 ∼ D4, D5 ∼ D6 illustrate that the thickness of the inter-pore wall thickness after membrane formation increases in order, while the pore diameter after membrane formation increases in order, and Figs. D2 ∼ D3, D4 ∼ D5 illustrate that the thickness of the inter-pore wall thickness after membrane formation decreases in order, while the pore diameter after membrane formation increases in order, indicating that the non-solvent initial component in the system polymer solution has a certain influence on the membrane formation morphology.

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In order to verify the validity of the model, experimental membranes were made for different polymer concentrations and used for comparison with the simulation result plots. To investigate the effect of polymer concentration on the morphological evolution of PVDF membranes, the H₂O/DMAc/PVDF system was selected for experimental membrane production and numbered as group A membranes. Simulations were also carried out using different combinations of initial concentrations in polymer solutions and these combinations of initial concentrations, i.e. two layers of initial conditions, are shown in Table VIII. The $\varphi$ _p and $\varphi$ _s in the polymer solution were changed, while keeping $\varphi$ _ns constant. The set values of each parameter for the simulated process system are shown in Table IX.

Table VIII. Table of composition of each component.

Number		E1	E2	E3
Polymer solution layer	$\varphi$ p	0.15	0.20	0.25
	$\varphi$ s	0.80	0.75	0.70
	$\varphi$ ns	0.05	0.05	0.05
Coagulation bath	$\varphi$ p = 0.01，$\varphi$ s = 0.01，$\varphi$ ns = 0.98

Figure 6 shows the performance of the membranes prepared by PVDF casting solutions with different initial polymer concentrations. The order of the porosity size is: E2 > E1 > E3, and the order of the pure water flux size is: E1 > E2 > E3. Combining the porosity with the pure water flux, the performance of the membranes made by the E3 system is the worst among them. As the polymer concentration increases, the membrane pore size increases first and then decreases, and the pure water flux gradually decreases.

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The evolution of the membrane morphology (volume fraction of polymer ($\varphi$ _p ) vs time) during the phase transition is shown in Fig. 7. Red indicates the maximum volume fraction of the polymer at each time step, blue indicates the minimum volume fraction of the polymer at each time step, and other colors fall in between.

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The simulation starts with two layers of initial conditions, with the polymer solution (polymer and solvent) at the bottom at 30% and the solidification bath (non-solvent) at the top. The polymer stays almost completely in the polymer solution region, as shown in Fig. 7. The solidification bath is always blue in color (tiny polymer volume fraction). The phase separation process starts from the top of the polymer solution layer and proceeds downward in the laminar structure. ²² The final morphology of the PVDF membrane is an asymmetric structure ²³ with a dense epidermal layer on top of the porous support layer, which is consistent with experimental observations.

With the increase of polymer volume fraction in the polymer solution layer, Figs. E1 ∼ E2 illustrate that the thickness of the inter-pore wall thickness after membrane formation increases sequentially, while the pore diameter after membrane formation decreases sequentially, while the trend in Figs. E2 ∼ E3 is opposite to E1 ∼ E2, indicating that there is a peak in the effect of polymer volume fraction in the polymer solution layer of the system on the morphology of membrane formation, from which it is indicated that the membrane performance formed by the E2 system is better.

Figure 8 shows SEM images of the membranes prepared with PVDF casting solutions with different initial concentrations of polymer, giving SEM images of the upper surface, lower surface and cross section. The lower surface appears almost smooth and more pores can be observed on the upper surface. The prepared PVDF membranes were all asymmetric, and the membrane thickness increased with increasing PVDF concentration, the macropore structure within the membrane was suppressed with increasing PVDF concentration, and the sponge-like structure size increased. PVDF At higher concentrations, the formed membrane has too much polymer at the interface with the solidification bath, forming a porous surface layer with lower porosity and a smaller water flux in the formed membrane. The validity of the model is verified by combining the experimental electron microscopy plots as well as the membrane properties, and the simulation result plots can be matched with the experiments.

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As shown in Fig. 2. As the system pressure increases, the polymer diffusion rate accelerates and the more blue fraction appears, i.e., the smaller the volume fraction of polymer in the grid in the time step. The schematic of each color can be seen in Fig. 1. And the blue part in the A5 chart is relatively reduced again. This indicates that the pressure value of the system can accelerate the phase separation rate and thus change the membrane formation structure, where the pressure value may slow down the phase separation rate relatively when it is too high. The obvious polymer lamellar structure layer can be seen in the A0 plot, the lamellar structure layer weakens in the A1 plot, and the lamellar structure layer gradually disappears in the A2 ∼ A5 plots, indicating that this lamellar structure is gradually broken with the increase of the pressure value. In the analysis of the simulated plot of the parameter of addition pressure compared with the previous parameters, the diffusion plot of the polymer tends to regularize the diffusion after its addition pressure.

The effect of system temperature on the membrane formation morphology can be seen in Fig. 3. As the temperature of the system T increases, the polymer diffusion rate accelerates and the degree of polymer dispersion gradually increases, and the blue part appears gradually, i.e., the smaller the volume fraction of polymer in the grid in the time step, and the schematic representation of each color can be seen in Fig. 1. The increase in temperature facilitates faster phase separation of the membrane, which in turn affects the membrane formation morphology.

With the increase of solvent in the polymer solution system, the polymer solution layer and the solidification bath connect the interface with a small fracture, but the undulation of its interface is small. This indicates that the top layer is destroyed during diffusion and that the concentration difference gradient is too large in diffusion and its diffusion drive breaks the energy setting at the interface. As the volume fraction of solvent in the polymer solution layer increases, the morphology changes from separated droplets to a bicontinuous mode. This also reduces the solvent gradient, which delays the onset of phase separation. The increase of the solvent system in the polymer solution leads to the increase of the interaction parameter between solvent and non-solvent, which leads to the increase of the mutual solubility zone in the phase diagram of the system, and then affects the membrane formation morphology.

From the overall view in Fig. 5, as the volume fraction of non-solvent in the polymer solution increases, the membrane pores gradually increase and the pore size also gradually increases, the morphology changes from separated droplets to a bicontinuous mode, and the interface between the polymer solution layer and the solidification bath connection is smoother. The non-solvent at low volume fractions moves downward in a laminar structure in the diffusion of the polymer, and breaks the laminar structure at higher volume fractions.

From the simulation results Fig. 7 as a whole, the membrane pores decrease but the pore size increases as the PVDF concentration increases. At the same time, the greater polymer volume fraction in the system is dispersed to a large extent, and the smaller polymer volume fraction is co-existed by droplets and continuums, implying the coexistence of finger-like pores and spongy pores in the system. At the same time, as the volume fraction of the polymer in the polymer solution layer increases, the effect on the bipartite line of the system gradually increases, affecting the interaction parameters between the components and, consequently, the membrane formation morphology. Meanwhile, this conclusion is fully verified in the experimental results in Fig. 8.

The initial moulding conditions of series C, D and E groups were described in the phase diagram by means of ternary phase diagrams, and then the effect of the concentration of each component on the formation process of PVDF membrane was compared and analyzed according to the graph of simulation results.

Figure 9 is a summary of the simulation results for each group of membranes according to the different variables, simulating the effect of different initial conditions in the polymer solution on the membrane formation morphology. The dots in the ternary diagram represent the initial conditions for the corresponding simulation results, where the green dots represent the different initial conditions for the simulated solvents in the Table IV series, the black dots represent the different initial conditions for the simulated non-solvents in the Table V series, and the red dots represent the simulation results with different initial polymer concentrations for comparison. The effect on the membrane formation process under different initial conditions is also well represented.

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