**5. Mass transfer across the interface between polymer solution and nonsolvent bath**

Polymer solutions are important for a variety of purposes, especially, for manufacturing fibers and membranes. Generally, solidification of polymer in shape of interest takes place by solvent evaporation from nonsolvent diffusion into the polymer solution. During the process, polymer solution is undergoing a change in the concentration of components. A schematic representation of mass transfer through the interface between polymer solution and nonsolvent bath is depicted in Fig. 9. During the quench period, solvent-nonsolvent exchange is doing with time and eventually polymer precipitation takes place. The ratio of solvent-nonsolvent exchange, predicting mass transfer paths on the thermodynamic phase diagram, is an important factor to control the ultimate structure of product (Karimi and Kish, 2009).

Fig. 9. Schematic presentation of mass transfer through the interface between polymer solution and nonsolvent bath.

Such a mass transfer process requires a consideration based on the thermodynamics of irreversible processes which indicates that the fundamental driving forces for diffusion through the interface of multiphase system are the gradients of the chemical potential of each of the system components. Here, considering polymer solution and nonsolvent bath as two-phase system, the amount of molecules passing through the interface could be described by Fick's law if the intrinsic mobility ( ) of each component could be known. The flux ( *J* ) due to pure diffusion is:

$$J = -(\frac{\mathbf{M}\,\phi}{N})(\frac{\partial\mu}{\partial\mathbf{x}}) \tag{23}$$

where *N* is Avogadro's number and is chemical potential (partial molar free energy). Replacing *RT a* ln in to the Equation 23 then,

$$J = -RT\frac{\mathbf{M}\phi}{N}(\frac{\partial \ln a}{\partial \mathbf{x}}) \tag{24}$$

Based on the water cup method, referring to ASTM E96-00 standard, the vapor passing through the polymer film measured as function of time. That is, an open cup containing water sealed with the specimen membrane, and the assembly is placed in a test chamber at the certain temperature, with a constant relative humidity of 50%, and the water lost

Polymer solutions are important for a variety of purposes, especially, for manufacturing fibers and membranes. Generally, solidification of polymer in shape of interest takes place by solvent evaporation from nonsolvent diffusion into the polymer solution. During the process, polymer solution is undergoing a change in the concentration of components. A schematic representation of mass transfer through the interface between polymer solution and nonsolvent bath is depicted in Fig. 9. During the quench period, solvent-nonsolvent exchange is doing with time and eventually polymer precipitation takes place. The ratio of solvent-nonsolvent exchange, predicting mass transfer paths on the thermodynamic phase diagram, is an important factor to control the ultimate structure of product (Karimi and

Fig. 9. Schematic presentation of mass transfer through the interface between polymer

<sup>1</sup> *J* <sup>2</sup> *J*

Such a mass transfer process requires a consideration based on the thermodynamics of irreversible processes which indicates that the fundamental driving forces for diffusion through the interface of multiphase system are the gradients of the chemical potential of each of the system components. Here, considering polymer solution and nonsolvent bath as two-phase system, the amount of molecules passing through the interface could be described by Fick's law if the intrinsic mobility ( ) of each component could be known.

Support

Nonsolvent bath

*J* ( )( ) *N x* 

ln ( ) *<sup>a</sup> J RT N x* 

*RT a* ln in to the Equation 23 then,

(23)

Diffusion layer

is chemical potential (partial molar free energy).

(24)

**5. Mass transfer across the interface between polymer solution and** 

isrecorded after certain period of time

**nonsolvent bath** 

Kish, 2009).

Replacing

solution and nonsolvent bath.

Cast polymer solution

The flux ( *J* ) due to pure diffusion is:

where *N* is Avogadro's number and

where *a* is the activity, *R* the gas constant, and *T* is the absolute temperature. Mobility coefficient some times are called self-diffusion coefficient ( *D* ), presenting the motion and diffusion of molecules without presence of concentration gradient and/or any driving force for mass transfer, given by / *D RT N* . Then

$$J = -D^\* \phi(\frac{\partial \ln a}{\partial \mathbf{x}}) \tag{25}$$

Since it is difficult to measure the self-diffusion coefficient, a thermodynamic diffusion ( *DT* ) is introduce to use in theories regarding the concentration and temperature dependence of diffusion, as given by

$$D\_T = D(\frac{\partial \ln c\_1}{\partial \ln a\_1}) \tag{26}$$

According to Equation 25 to determine the mass transfer of solvent and or nonsolvent across the interface, it should be given their chemical potentials. Flory-Huggins (FH) model is wellestablished to use for describing the free energy of the polymer solution, as given in Equation 26. It should be noted that the FH model can be extended for multi-component system if more than one mobile component exists for mass transfer; more details for Gibbs free energy of multi-components system are documented in literatures (Karimi, 2005, Boom, 1994)

### **5.1 Mass transfer paths**

Polymer membrane which is obtained by the so-called nonsolvent-induced phase separation (NIPS) (Fig. 9), has a structure determined by two distinct factors: (1) the phase separation phenomena (thermodynamics and kinetics) in the ternary system, and (2) the ratio and the rate of diffusive solvent-nonsolvent exchange during the immersion (Karimi, 2009, Wijmans, 1984). The exchange of the solvent and the nonsolvent across the interface initiates the phase separation of the polymer solution in two phases; one with a high polymer concentration (polymer-rich phase), and the other with a low polymer concentration (polymer-lean phase). The morphology of the membranes, the most favorable feature, is strongly related to the composition of the casting film prior and during the immersion precipitation. The compositional change during the phase separation has been frequently discussed theoretically (Tsay and McHugh, 1987), but the experimental results for the composition of the homogeneous polymer solution prior to precipitation of polymer are scarce (Zeman and Fraser, 1994, Lin, 2002). In particular, composition changes of all components prior to the demixing stage are necessary. In order to find out the change of composition during the phase inversion process it needs to determine the rate of solvent outflow ( <sup>1</sup>*J* ) and nonsolvent inflow ( <sup>2</sup>*J* ) through the diffusion layer. By this measurement the calculation of the changes in polymer content becomes possible.

As stated earlier, the flux for mobile components involved in the NIPS process can be described by Equation 27

$$J\_i = -(RT)^{-1} D\_i(\phi) \phi\_i(\frac{\partial \mu\_i}{\partial \mathbf{x}}) \tag{27}$$

Diffusion in Polymer Solids and Solutions 33

decreases along the route; Solidification of the system is possible as a result of phase transition, but loose heterogeneous structures result. In contrast, when 1 2 *d d*

polymer concentration increases and solidification results from the increase of polymer content as well as from possible phase transitions. *Region III,* yields the most dense and

Diffusion in polymer solution has been studied for decades using several techniques such as gravimetry (Hu, 1996), membrane permeation (Smith, 1988), fluorescence (Winsudel, 1996), and dynamic light scattering (Asten, 1996), Raman spectroscopy (Kim, 2000, Tsai and

Through the FTIR spectroscopy a spot near the interface of a thin layer of casting solution has been examined. There are two problems with this method. One is the difficulty of introducing the nonsolvent (especially water) into the liquid cell by a syringe due to the capillary action of water. The other is the saturation of coagulation bath with solvent due to limitation of circulation in the nonsolvent bath. It seems that the investigations of the phase demixing processes by such arrangements limits the information about the compositional

FTIR-ATR as a promising toll is recently used (Karimi and Kish, 2009) to measure the compositional path during the mass transfer of immersed polymer solution. A special arrangement to determine the concentration of components in the diffusion layers under quench condition prior to the phase separation and the concentration of all components in front of the coagulation boundary was introduced. This technique allows a simultaneous determination of solvent outflow and water inflow during the immersion time. Determination

To measure the concentration of each component, polymer solution was cast directly on surface of the flat crystal (typically ZnSe, a 45o ATR prism), similar arrangement as shown in Fig. 7. The flat crystal is equipped with a bottomless liquid cell. The penetrant is

3800 3600 3400 3200 3000 2800 2600 2400

Wavenumbers (cm-1)

0.2 0.4 0.6 0.8 1.0

A/At inf

0 30 60 90 120 150 0.0

Time (sec)

of the composition of all components becomes possible by using the calibration curves.

 / 

1 , polymer content

 / 1 ,

1 , i.e. the path parallel to S~NS line. When 1 2 *d d*

homogeneous structures as located outside the two-phase region.

Torkelso, 1990), and recently ATR-FTIR (Karimi, 2009, Lin, 2002).

**5.2 Measurement of mass transfer paths** 

change prior the phase demixing step.

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

Fig. 11. Intensity (H2O/AC/PMMA system).

Absorbance (a.u)

1 2 *d d* / 

where *<sup>i</sup> J* is the volume flux of component *i* . Here the self-diffusion coefficient ( *Di* ) is considered as concentration-dependence.

To determine the mass transfer path for the polymer solution immersed into the nonsolvent bath the ratio of mass flaxes of nonsolvent to which of solvent should be plotted. Therefore we have

$$\frac{J\_1}{J\_2} = \sigma = \frac{\phi\_1}{\phi\_2} \cdot \frac{d\mu\_1}{d\mu\_2} \tag{28}$$

The assumption made here is that the ratio of 11 2 21 2 *D D* ( , )/ ( , ) is constant and unity. If the differentials of the chemicals potential are expressed as functions of the volume fractions, one finds

$$\frac{d\phi\_1}{d\phi\_2} = \frac{\phi\_1 \left(\left\|\left<\mu\_1/\left<\partial\phi\_2\right>\right\rangle\_{\Gamma,P,\phi\_1} - k\phi\_2 \left(\left<\partial\mu\_2/\left<\partial\phi\_2\right>\right)\_{\Gamma,P,\phi\_1}\right)\_{\Gamma,P,\phi\_1}\right.}{k\phi\_2 \left(\left<\partial\mu\_2/\left<\partial\phi\_1\right>\right)\_{\Gamma,P,\phi\_2} - \phi\_1 \left(\left<\partial\mu\_1/\left<\partial\phi\_1\right>\right)\_{\Gamma,P,\phi\_2}\right.\tag{29}$$

This first order differential equation yields the relation between 1 and <sup>2</sup> in the diffusion layer as a function of the ratio *J J* 1 2 / and one of the boundary conditions.

With the aid of the Flory-Huggins expressions for the chemical potentials together with Equation 29, Cohen and co-workers (Cohen, 1979) calculated, for the first time, the composition paths within the ternary phase diagram and discussed them in relation to the formation of membranes. Equation 29 has been derived using the steady-state condition.

When a solution of a polymer in a solvent is immersed in a bath of a nonsolvent, there are, depending on the preparation condition, various possible outcomes as the solvent release from polymer solution and is replaced to a greater or lesser extent by nonsolvent (i.e. *d d* 1 2 / ? ) (Karimi, 2009, Stropnik, 2000): *Region I,* as demonstrated in Fig. 10, the total polymer concentration decreases along the route. The change of composition in this region is a one-phase dilution of polymer solution without solidification. *Region II,* the routes intersect the binodal curve and enter the two-phase region. In this region there are two outcomes that depend on the location of the routes with respect to the route assigned by

Fig. 10. Mass transfer paths in a triangle phase diagram.

where *<sup>i</sup> J* is the volume flux of component *i* . Here the self-diffusion coefficient ( *Di* ) is

To determine the mass transfer path for the polymer solution immersed into the nonsolvent bath the ratio of mass flaxes of nonsolvent to which of solvent should be plotted. Therefore

> 1 11 2 22 . *J d J d*

the differentials of the chemicals potential are expressed as functions of the volume

 1 1

 

With the aid of the Flory-Huggins expressions for the chemical potentials together with Equation 29, Cohen and co-workers (Cohen, 1979) calculated, for the first time, the composition paths within the ternary phase diagram and discussed them in relation to the formation of membranes. Equation 29 has been derived using the steady-state condition. When a solution of a polymer in a solvent is immersed in a bath of a nonsolvent, there are, depending on the preparation condition, various possible outcomes as the solvent release from polymer solution and is replaced to a greater or lesser extent by nonsolvent (i.e.

 1 2 / ? ) (Karimi, 2009, Stropnik, 2000): *Region I,* as demonstrated in Fig. 10, the total polymer concentration decreases along the route. The change of composition in this region is a one-phase dilution of polymer solution without solidification. *Region II,* the routes intersect the binodal curve and enter the two-phase region. In this region there are two outcomes that depend on the location of the routes with respect to the route assigned by

112 2 22 <sup>1</sup> 2 2 21 1 11

*k d*

The assumption made here is that the ratio of 11 2 21 2 *D D* ( , )/ ( , )

*d k*

 

This first order differential equation yields the relation between

Fig. 10. Mass transfer paths in a triangle phase diagram.

(28)

1 and 

is constant and unity. If

<sup>2</sup> in the diffusion

 

(29)

2 2

, , , , , , , , *T P T P T P T P*

*J J* 1 2 / and one of the boundary conditions.

 

considered as concentration-dependence.

we have

*d d* 

fractions, one finds

layer as a function of the ratio

1 2 *d d* / 1 , i.e. the path parallel to S~NS line. When 1 2 *d d* / 1 , polymer content decreases along the route; Solidification of the system is possible as a result of phase transition, but loose heterogeneous structures result. In contrast, when 1 2 *d d* / 1 , polymer concentration increases and solidification results from the increase of polymer content as well as from possible phase transitions. *Region III,* yields the most dense and homogeneous structures as located outside the two-phase region.

### **5.2 Measurement of mass transfer paths**

Diffusion in polymer solution has been studied for decades using several techniques such as gravimetry (Hu, 1996), membrane permeation (Smith, 1988), fluorescence (Winsudel, 1996), and dynamic light scattering (Asten, 1996), Raman spectroscopy (Kim, 2000, Tsai and Torkelso, 1990), and recently ATR-FTIR (Karimi, 2009, Lin, 2002).

Through the FTIR spectroscopy a spot near the interface of a thin layer of casting solution has been examined. There are two problems with this method. One is the difficulty of introducing the nonsolvent (especially water) into the liquid cell by a syringe due to the capillary action of water. The other is the saturation of coagulation bath with solvent due to limitation of circulation in the nonsolvent bath. It seems that the investigations of the phase demixing processes by such arrangements limits the information about the compositional change prior the phase demixing step.

FTIR-ATR as a promising toll is recently used (Karimi and Kish, 2009) to measure the compositional path during the mass transfer of immersed polymer solution. A special arrangement to determine the concentration of components in the diffusion layers under quench condition prior to the phase separation and the concentration of all components in front of the coagulation boundary was introduced. This technique allows a simultaneous determination of solvent outflow and water inflow during the immersion time. Determination of the composition of all components becomes possible by using the calibration curves.

To measure the concentration of each component, polymer solution was cast directly on surface of the flat crystal (typically ZnSe, a 45o ATR prism), similar arrangement as shown in Fig. 7. The flat crystal is equipped with a bottomless liquid cell. The penetrant is

Fig. 11. Intensity (H2O/AC/PMMA system).

Diffusion in Polymer Solids and Solutions 35

The authors believe that mass transfer during the process can describe the morphology

 Fig. 12. SEM micrographs of PMMA membrane formed from different solvents; (a) Acetone,

Fig. 13. Effect of membrane thickness on PAN membrane structure (dope, PAN/DMF: PAN

Diffusion is an important process in polymeric membranes and fibers and it is clear that mass transfer through the polymeric medium is doing by diffusion. Analyzing the diffusion

1. Various mechanisms are considered for diffusion, which it is determined by time scale

2. The rate of permeability can be controlled by loading impenetrable nano-fillers into the

3. Driving force for mass transfer across the interface of multiphase systems is chemical

As well described in this chapter, measuring the diffusion of small molecules in polymers using Fourier transform infrared-attenuated total reflection (FTIR-ATR) spectroscopy, is a promising technique which allows one to study liquid diffusion in thin polymer films in situ. This technique can be successfully employed for quantifying the compositional path during the mass transfer of immersed polymer solution, in which it is strongly involved to

which is basically formulated by Fick's laws, lead to the following conclusions

50 µm 50 µm

5 µm 5 µm 5 µm

20 wt %: casting temperature 25 oC: coagulant, water).

development if it can be possible to measure.

(b) N-dimethylformamide.

**6. Conclusion** 

structure

potential-base

the structure development.

of polymer chain mobility

transferred into the cell, and simultaneously the recording of the spectrum is started. Fig 11 shows a series of recording spectra at wave number ranging from 2400 to 3800 cm-1 for the water/acetone/PMMA system.

Having the FTIR-ATR spectrum measured and the system calibrated, the composition of polymer solution at the layer close to ATR prism could be accurately determined if the method described in part 4.1 could be properly chosen. In the case of polymer solution, generally, simple Beer's law deosn't result correctly. Other methods are normally evaluated with standard solutions to choose the proper one. Advantage of these methods is found to be satisfactory for solving complex analytical problems where the component peaks overlap. The principle component regression (PCR) is succefully used by Karimi (Karimi and Kish, 2009) to resolve the bands in overlapped regions for water/acetone/PMMA and water/DMF/PMMA systems. Since there exists overlapped characteristic peaks the spectra, simple Beer's law deos not predict reliable results.

### **5.3 Composition path and structure formation**

Undoubtedly, the rate of compositional change as well as the ratio of mass transfer of components across the interface of polymer solution are affecting factors controlling the ultimate structure, however, it should not be neglected that the thermodynamics is also the other affecting factor. Mass transfer paths can be derived from the model calculations defined in two different ways: The composition path can represent the composition range in the polymer solution between the support and the interface at a given time. The composition path can also be defined as the composition of a certain well defined element in the solution as a function of time. Some researchers have attempted to make relation between mass transfer and ultimate structure of polymer system. Many of them have referred to the rate of solvent and nonsolvent exchange, postulating instantaneous and delay demixing. This classification was clarified many observations about membrane morphologies. But there are several reports in the literatures that they didn't approve this postulation.

Direct measuring the time dependence of concentration of system components (for instance 1 ,<sup>2</sup> , and <sup>3</sup> ) during the immersion precipitation process can clarify some obscure aspects of the structure formation. When the polymer solution comes to contact with nonsolvent bath, the solvent release may become higher than the nonsolvent penetration (i.e 2 1 *d d* / 1 ) that leads to an increase in polymer content of the dope and the solvent moves to mix with water in the water bath. The development of structure of this system will differ from which the solvent release is lower than the nonsolvent penetration (i.e 2 1 *d d* / 1 ). Different morphologies for PMMA membrane followed by different ratio of solvent and water exchange, *d d* 2 1 / **,** have been reported by Karimi (Karimi and Kish, 2009). Fig 12 shows such morphologies for PMMA membrane in which they developed from H2O/DMF/PMMA and H2O/AC/PMMA systems.

However the determination of mass transfer is very applicable to make clear some aspects of membrane morphologies, but there is a limitation regarding to the rate of data capturing. Some interesting morphologies were observed during the fast mass transfer in membraneforming system. For example, Azari and et al. (Azari, 2010) have reported a structure transition when the thickness of cast polymer solution is changed. Fig 13 shows different structures of poly(acrylo nitrile) membrane preparing by same system (H2O/DMF/PAN).

transferred into the cell, and simultaneously the recording of the spectrum is started. Fig 11 shows a series of recording spectra at wave number ranging from 2400 to 3800 cm-1 for the

Having the FTIR-ATR spectrum measured and the system calibrated, the composition of polymer solution at the layer close to ATR prism could be accurately determined if the method described in part 4.1 could be properly chosen. In the case of polymer solution, generally, simple Beer's law deosn't result correctly. Other methods are normally evaluated with standard solutions to choose the proper one. Advantage of these methods is found to be satisfactory for solving complex analytical problems where the component peaks overlap. The principle component regression (PCR) is succefully used by Karimi (Karimi and Kish, 2009) to resolve the bands in overlapped regions for water/acetone/PMMA and water/DMF/PMMA systems. Since there exists overlapped characteristic peaks the spectra,

Undoubtedly, the rate of compositional change as well as the ratio of mass transfer of components across the interface of polymer solution are affecting factors controlling the ultimate structure, however, it should not be neglected that the thermodynamics is also the other affecting factor. Mass transfer paths can be derived from the model calculations defined in two different ways: The composition path can represent the composition range in the polymer solution between the support and the interface at a given time. The composition path can also be defined as the composition of a certain well defined element in the solution as a function of time. Some researchers have attempted to make relation between mass transfer and ultimate structure of polymer system. Many of them have referred to the rate of solvent and nonsolvent exchange, postulating instantaneous and delay demixing. This classification was clarified many observations about membrane morphologies. But there are several reports in the literatures that they didn't approve this

Direct measuring the time dependence of concentration of system components (for instance

of the structure formation. When the polymer solution comes to contact with nonsolvent bath, the solvent release may become higher than the nonsolvent penetration (i.e

Different morphologies for PMMA membrane followed by different ratio of solvent and

shows such morphologies for PMMA membrane in which they developed from

However the determination of mass transfer is very applicable to make clear some aspects of membrane morphologies, but there is a limitation regarding to the rate of data capturing. Some interesting morphologies were observed during the fast mass transfer in membraneforming system. For example, Azari and et al. (Azari, 2010) have reported a structure transition when the thickness of cast polymer solution is changed. Fig 13 shows different structures of poly(acrylo nitrile) membrane preparing by same system (H2O/DMF/PAN).

 / 1 ) that leads to an increase in polymer content of the dope and the solvent moves to mix with water in the water bath. The development of structure of this system will differ from which the solvent release is lower than the nonsolvent penetration (i.e 2 1 *d d*

<sup>3</sup> ) during the immersion precipitation process can clarify some obscure aspects

**,** have been reported by Karimi (Karimi and Kish, 2009). Fig 12

 / 1 ).

water/acetone/PMMA system.

postulation.

2 1 *d d* 

water exchange, *d d*

2 1 / 

H2O/DMF/PMMA and H2O/AC/PMMA systems.

1 ,<sup>2</sup> , and 

simple Beer's law deos not predict reliable results.

**5.3 Composition path and structure formation** 

The authors believe that mass transfer during the process can describe the morphology development if it can be possible to measure.

Fig. 12. SEM micrographs of PMMA membrane formed from different solvents; (a) Acetone, (b) N-dimethylformamide.

Fig. 13. Effect of membrane thickness on PAN membrane structure (dope, PAN/DMF: PAN 20 wt %: casting temperature 25 oC: coagulant, water).
