A model for wet-casting polymeric membranes incorporating nonequilibrium interfacial dynamics, vitrification and convection

Journal of Membrane Science 354 (2010) 74–85

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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci

A model for wet-casting polymeric membranes incorporating nonequilibrium interfacial dynamics, vitrification and convection Hanyong Lee a , William B. Krantz b , Sun-Tak Hwang c,∗ a b c

R&D Center, Samsung Engineering Co. Ltd., Suwon, Gyeonggi-do 443-823, Republic of Korea Singapore Membrane Technology Center, Nanyang Technological University, Singapore 639798, Singapore Department of Chemical and Materials Engineering, University of Cincinnati, Cincinnati, OH 45221-0012, USA

a r t i c l e

i n f o

Article history: Received 16 March 2009 Received in revised form 23 February 2010 Accepted 26 February 2010 Available online 6 March 2010 Keywords: Polymeric membranes Wet casting Phase inversion Modeling Nonequilibrium interface Vitrification Convection Macrovoids Asymmetric membranes

a b s t r a c t A new model is developed for wet-casting polymeric membranes that address how the concentrations at the interface between the casting solution and nonsolvent bath adjust from initial nonequilibrium to equilibrium values on the binodal. Properly describing the evolution of the interface concentrations enables this new model to predict vitrification, which has been observed experimentally but not predicted heretofore. This new model also incorporates densification-induced convection that arises owing to density changes associated with the concentration gradients and contributes to the mass-transfer fluxes. The predictions for the cellulose acetate, acetone, and water system indicate that densificationinduced convection can increase the mass-transfer flux by nearly two orders-of-magnitude shortly after initiating wet-casting. This increased mass-transfer flux can have a marked effect on the properties of the functional layer of asymmetric membranes that is formed early in the casting process. The predictions for initial casting-solution thicknesses of 75 and 125 ␮m are markedly different. When densificationinduced convection is included, the 125 ␮m film is predicted to enter well into the metastable region, thereby allowing supersaturation that promotes macrovoid defects. Hence, this new model provides an explanation for the effect of casting-solution thickness on the occurrence of macrovoids. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The wet-casting process is a phase-inversion method for polymeric membrane formation. In the wet-casting process a homogeneous polymer solution (casting solution) is prepared with polymer and solvent (and sometimes nonsolvent) and cast into a thin film or hollow fiber geometry. The casting solution then is immersed into a precipitation bath of nonsolvent. Due to the concentration differences, solvent is transferred from the casting solution into the bath, while nonsolvent is transferred from the bath into the casting solution. As a result, the local composition at any point in the casting solution changes with time. The homogeneous casting solution eventually becomes thermodynamically metastable or unstable, which leads to phase separation and formation of the solid polymeric matrix of the membrane. The structure of the membrane is strongly dependent on the local composition in the casting solution. However, measuring this local composition as a function of time and position is very difficult due to the very

∗ Corresponding author. Tel.: +1 513 556 2791; fax: +1 513 556 2569. E-mail addresses: [email protected] (H. Lee), [email protected] (W.B. Krantz), [email protected], [email protected] (S.-T. Hwang). 0376-7388/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2010.02.066

rapid change within the thin film of casting solution [1]. Therefore, a mathematical model is useful for providing insight into the influence of various parameters that determine the structure and performance of a membrane. Prior modeling studies of the wetcasting process have made limiting and in some cases tenuous assumptions that are addressed in this study. The first attempt to model the wet-casting process was made by Cohen et al. [2]. Their description of the mass-transfer process was based on a steady-state diffusion model. Several researchers [3,4] have pointed out that this assumption is not reasonable. McHugh and Yilmaz [4] improved on this model by using an unsteady-state pseudo-binary mass-transfer formalism whereby any diffusion of the polymer is ignored. Hence, their governing equations are simplified by assuming zero polymer flux in the casting solution and negligible mass-transfer dynamics in the nonsolvent bath. Moreover, the pseudo-binary formalism implies no convective contribution to the mass-transfer flux within the casting solution. A convective contribution to the mass-transfer flux can arise during membrane casting owing to local density changes associated with the concentration gradients created by the mass transfer. In principle, this densification can cause either free convection that arises owing to buoyancy effects or can cause bulk flow in the absence of any gravitational body forces because of the interplay

H. Lee et al. / Journal of Membrane Science 354 (2010) 74–85

between the density and velocity dictated by the continuity equation. Free convection is not possible in conventional wet-casting owing to the very large solution viscosity and thin liquid film. However, convection can arise during wet-casting owing to the solvent removal and nonsolvent addition that causes the polymer molecules to assume a more compact configuration. The terminology densification-induced convection used henceforth in this paper will refer to this bulk flow effect that can arise solely due to density changes in the absence of any buoyancy effects. Reuvers et al. [5,6] further improved on the previous models by coupling the bath dynamics with the diffusion in the casting solution. They utilized a phenomenological approach based on nonequilibrium thermodynamics to describe the coupling effect of diffusion in both the casting solution and nonsolvent bath. Their detailed and appropriate description of the polymeric system was a significant improvement that has made their model a standard for many successive studies. However, they assume an infinitely thick casting solution and ignore any densification-induced convection, although they mention a calculation methodology for incorporating a finite thickness. The assumed continuity of mass fluxes at the interface between the casting solution and nonsolvent bath in their model greatly reduces the computational effort, but is valid only when the thickness of the casting solution does not change owing to densification-induced convection. They claim that the measured onset time for phase separation shows close agreement with their model predictions. However, since their model assumes that the interfacial composition of the casting solution is instantaneously located on the binodal, it cannot accurately predict the onset time for phase separation. This assumption implies that the phase separation should start nearly immediately rather than after the relatively long time (several seconds) determined by their experiments. Tsay and McHugh [7] improved on the model of Reuvers et al. [5,6] by allowing for a variable interfacial composition with time. Moreover, they formulated the ternary conservation of species equations in terms of mass fluxes relative to the volume-average velocity. If one assumes Flory–Huggins theory, the volume-average velocity is identically zero. Hence, the approach of Tsay and McHugh avoids having to explicitly account for any densification-induced contribution to the mass-transfer flux since this is implicitly incorporated into the diffusive fluxes in their describing equations. However, this approach is valid only for an equation-of-state for which the volume-of-mixing is identically zero. Indeed, it might be expedient to invoke Flory–Huggins theory to describe the solution thermodynamics while using an empirical equation for the density that does not imply that the volume-ofmixing is identically zero. Moreover, the approach of Tsay and McHugh does not permit assessing the importance of the densification contribution to the mass-transfer flux since it is implicitly incorporated into the diffusive flux terms. This assessment would require comparing the model predictions with and without the densification contribution to the mass-transfer flux. The only way to suppress the contribution of densification to the mass-transfer flux in the approach of Tsay and McHugh would be to assume a constant mass density; however, this would compromise the ability of the model to predict changes in the casting-solution thickness. The manner in which the composition at the interface between the casting solution and nonsolvent bath changes from its initial value to an equilibrium concentration on the binodal is also not considered in the model of Tsay and McHugh. Kesting [8] reports that a significant polymer accumulation can occur at the interface of the casting solution that can lead to skin formation via vitrification. Predicting the latter requires an accurate description of the interfacial composition from the inception of the casting process. Radovanovic et al. [9,10] showed that the description of diffusion in the bath made by Reuvers et al. [5,6] is in error and proposed

75

a modified version of the model. Their resulting diffusion model and experimental results showed that the structure of a polysulfone membrane strongly depends on the onset time of demixing. Cheng et al. [11] further modified the model of Reuvers et al. [5,6] by allowing for a finite casting-solution thickness. Although an advanced numerical method is used in their calculations, they do not allow for any densification contribution to the mass-transfer fluxes. Fernandes et al. [12] presented a simple mathematical model using the Fickian diffusion equation to describe the wet-casting process. In spite of utilizing an advanced computational method, their model is overly simplified since it neglects the coupling effects of diffusion and assumes an ideal solution. This brief review of prior wet-casting studies indicates that no model has properly allowed for the transition of the concentration at the casting solution interface between its initial nonequilibrium to an equilibrium composition on the binodal. This will result in an inaccurate description of the properties of the functional layer of the membrane since these are determined shortly after the initiation of the casting process. In particular, it will compromise the ability of a model to describe vitrification that can occur very rapidly after the initiation of casting. This review also indicates that only the model of Tsay and McHugh [7] has accounted for the effect of densification on the mass-transfer flux. However, their approach is restricted to the equation-of-state implied by modified Flory–Huggins theory since it formulates the mass fluxes relative to the volume-average velocity. Moreover, it does not permit assessing the relative contribution of densification to the mass flux. Clearly an improved model is needed for the wet-casting process that properly accounts for the transition from the initial nonequilibrium to a local equilibrium condition at the interface between the casting solution and nonsolvent bath and provides a general framework for incorporating densification-induced convection. Recently, Lee et al. [13] developed a rigorous method for obtaining an explicit equation for the mass-average velocity that arises owing to densification and thereby contributes to the total masstransfer fluxes. They applied their method to develop a coupled heat- and mass-transfer model for the dry-casting process for membrane formation. Their improved model was shown to agree more closely with the real-time data of Shojaie et al. [14] for the instantaneous overall casting solution mass and surface temperature and for the time required for the onset of phase separation for the dry-casting of the cellulose acetate, acetone, water system. This argues strongly for applying their method for incorporating densification-induced convection into an improved wet-casting model. However, this is not a trivial extension of the work of Lee et al. [13] for the dry-casting process since the wet-casting process presents two major complications. The first is that the mass-transfer dynamics in the nonsolvent bath are more complex than those encountered in the dry-casting process. The second is that an appropriate methodology must be developed to account for the change in the interfacial concentrations from their initial nonequilibrium values to a composition located on the binodal. This paper then is focused on the development of an improved wetcasting model that results in the following novel contributions: (1) a description of the mass-transfer dynamics in the nonsolvent phase justified by scaling analysis; (2) a proper description of the transition of the composition at the casting solution interface from nonequilibrium to local thermodynamic equilibrium; (3) a generalized framework for determining the mass-average velocity arising from densification; (4) the prediction of vitrification at the upper membrane surface in agreement with prior observations; and (5) an explanation for the effect of initial casting-solution thickness on the propensity to form macrovoid pores. This paper is organized as follows. First, the model equations incorporating densification-induced convection are described. The

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H. Lee et al. / Journal of Membrane Science 354 (2010) 74–85

Fig. 1. Schematic of the wet-casting process for membrane formation.

model then is used to describe the wet-casting process for the cellulose acetate, acetone, water system for two different initial casting-solution thicknesses. The results are discussed in the order in which the events occur during the wet-casting process: namely, from the initial casting-solution composition to the binodal; the change in the composition on the binodal to the vitrification boundary; and, the change in the bulk composition after interfacial vitrification. The importance of the densification-induced convection to the mass transfer is then assessed in terms of the magnitude of the modified Peclet number. Finally, the conclusions emanating from this study are summarized.

2. Model development 2.1. System of interest Fig. 1 shows a schematic of the assumed one-dimensional wet-casting process. This process involves casting a homogeneous solution of polymer, solvent, and possibly some nonsolvent as a thin planar film having a uniform initial thickness L0 and subsequently immersing it into a nonsolvent precipitation bath to cause phase separation and ultimately membrane formation. At time t = 0 the components in both the casting solution and nonsolvent bath begin to diffuse corresponding to their respective driving forces. The mass transfer is assumed to be one-dimensional, which is a good approximation for commercial wet-casting conditions. The solvent loss along with the nonsolvent gain in the casting solution lead to a change in the local composition and instantaneous casting-solution thickness, L(t). The nonsolvent bath also starts absorbing solvent from the casting solution. The negligible solubility of the polymer in the nonsolvent permits neglecting its mass transfer into the bath. However, the polymer can undergo mass transfer in the casting solution even if its diffusion coefficient is very small owing to densification-induced convection. To calculate the interfacial fluxes, one needs to know the local concentrations on both sides of the interface between the casting solution and nonsolvent bath. A reasonable assumption is that the transferring components are in local thermodynamic equilibrium at this interface. If the polymer cannot diffuse into the bath, the latter will be at most a binary solution of nonsolvent and solvent. Under this condition it is necessary to demand that the chemical potential of only one of the three components in the casting solution be equal at the interface between the casting solution and nonsolvent bath.

Note that prior studies have ignored the process whereby the concentration of the interface changes from its initial nonequilibrium composition to an equilibrium value on the binodal. As the mass transfer proceeds, the local compositions in the casting solution and in the nonsolvent bath change. Eventually, the instantaneous composition at the interface between the casting solution and nonsolvent bath will reach the binodal. After this, the local equilibrium concentration at the interface can be calculated using the Flory–Huggins lattice model until vitrification or phase separation occurs. Although the model developed here is applicable to any ternary wet-casting system, the transport and thermodynamic properties will be particular to the chosen polymer, solvent, and nonsolvent system. The cellulose acetate, acetone, and water system is chosen here because the required transport and thermodynamic properties are available in the open literature [14,15]. 2.2. Thermodynamic model A thermodynamic model is needed to determine, activity coefficients, chemical potentials, and the ternary phase diagram. The lattice model used in classical Flory–Huggins theory assumes that the molecular volumes of the solvent and nonsolvent as well as that of the polymer repeat unit are equal, thereby implying that the excess volume-of-mixing is zero. A less restrictive assumption for the solution thermodynamics invokes modified Flory–Huggins theory that merely requires that the molecular volumes of the solvent, nonsolvent, and polymer repeat unit be equal to their pure component values, thereby implying that the volume-of-mixing rather than the excess volume-of-mixing be zero. In order to incorporate densification-induced convection, it is necessary to introduce an equation-of-state for the mass density of the casting solution. Here we will use an equation-of-state that is consistent with modified Flory–Huggins theory for which the total volume per mole of the polymer solution V is given by V = V1◦ x1 + V2◦ x2 + V3◦ x3

(1)

Vi◦

where is the pure component molar volume and xi is the mole fraction of the nonsolvent, solvent, or polymer denoted by the subscripts 1, 2, and 3, respectively. Eq. (1) can be expressed in terms of the mass concentration i or mass fraction ωi as follows: ¯ ¯ ¯ ¯ ¯ ¯ 1 ω2 M ω3 M M2 2 M M3 3 M ω1 M M1 1 M + + + ◦ + ◦ = = ◦ 1 M1  2 M2  3 M3  1◦ 2◦ 3◦ c

(2)

H. Lee et al. / Journal of Membrane Science 354 (2010) 74–85

where Mi and i◦ are the molecular weight and pure component mass density of component i, respectively, c is the molar density, ¯ is the average molecular weight. Eq. (2) can be rearranged and M into the form of the equation-of-state for the mass density used in this paper: 1 ω1 ω2 ω3 = ◦ + ◦ + ◦ = ω1  1 2 3



1 1 − ◦ 1◦ 3





+ ω2

1 1 − ◦ 2◦ 3



+

1 (3) 3◦

2.3. Densification-induced convection Let us first consider the implications of the equation-of-state given by Eq. (3) for densification-induced convection. This can be assessed by combining Eq. (3) with the overall mass and speciesbalance equations for one-dimensional mass transfer in a ternary system given by the following: ∂ ∂ ¯ = − (u) ∂z ∂t

(4)

∂n1 ∂1 =− ∂t ∂z

(5)

∂n2 ∂2 =− ∂t ∂z

(6)

where z and t are the spatial and temporal coordinates, respectively, u¯ is the mass-average velocity, and ni is the mass flux of component i with respect to a stationary reference frame. If Eqs. (3)–(6) are combined and the result integrated while applying the impenetrable boundary condition at z = 0, one obtains the following:



i

i

i◦

ui = 0

2.4. Mass-transport model for casting solution





i ui

i

V¯ i Mi



=

   i i

Mi /V¯ i

ui

(8)

where V¯ i is the partial molar volume of component i. The assumption in modified Flory–Huggins theory that the molar volume of each component in the mixture is equal to its pure component molar volume implies that Mi /V¯ i = i◦ . Hence, Eq. (7) can be identified with the volume-average velocity, which for the special case of modified Flory–Huggins theory is identically zero. This was recognized by Tsay and McHugh [7] who formulated their describing equations in terms of the mass fluxes relative to the volumeaverage velocity and thereby did not need to explicitly include any densification-induced convection terms in their model. The mass-average velocity in the z-direction is defined by the following: u¯ ≡

volume-average velocity formulation is not applicable for equations describing the concentration dependence of the density other than that given by Eq. (3); and (3) formulating the mass fluxes relative to the volume-average velocity introduces less familiar forms of the species-conservation equations; the latter are most often given in standard references in terms of the mass fluxes relative to the mass-average velocity. In particular, one would like to assess the magnitude of the densification-induced contribution to the mass fluxes. Moreover, one would like to have a general formalism for the describing equations that is not restricted to the special assumptions embodied in Eq. (3). Indeed, one might well assume Flory–Huggins theory to describe the solution thermodynamics but invoke an empirical equation that more accurately describes the solution density than that given by Eq. (3). Hence, it is advantageous to develop the describing equations in terms of the mass-average rather than the volume-average velocity. One reason that prior investigators have not explicitly incorporated densification-induced convection is because it results in having to solve coupled parabolic (species balance) and hyperbolic (overall mass conservation) differential equations. There is no numerical algorithm for solving these coupled equations. However, a judicious manipulation of the describing equations permitted Lee et al. [13] to obtain an explicit analytical equation for the massaverage velocity thereby obviating the need to numerically solve the coupled set of parabolic and hyperbolic differential equations. Lee et al. [13] used their approach to assess the importance of densification-induced convection effects in the describing equations for membrane formation via the dry-casting process. Here we will adapt the formalism of Lee et al. [13] for the wet-casting process.

(7)

where ui is the velocity of species i in the z-direction with respect to a stationary reference frame. The volume-average velocity in the z-direction u* is defined by the following: u∗ ≡

77



i



ui

(9)

When ni is expressed in terms of ji , the mass flux of component i relative to the mass-average velocity, Eqs. (5) and (6) assume the following form:





 

∂ω1 ∂ω1 + u¯ ∂t ∂z ∂ω2 ∂ω2 + u¯ ∂t ∂z

=−

∂j1 ∂z

(10)

=−

∂j2 ∂z

(11)



The diffusive mass fluxes are obtained from statistical mechanics and Flory–Huggins theory and are given by the following [15]:



j1 = f1

 j2 = f2

∂ω1 ∂z ∂ω1 ∂z





+ g1



 + g2

∂ω2 ∂z ∂ω2 ∂z



(12)

 (13)

where f1 , f2 , g1 , and g2 are parameters related to the chemical potentials via the following:

i

When Eq. (9) is compared to Eq. (7) one sees that modified Flory–Huggins theory implies the mass-average velocity is zero only for the trivial case for which 1◦ = 2◦ = 3◦ = ; that is, when all components in the mixture have the same pure component mass density for which no densification occurs. It is advantageous to formulate the describing equations in terms of the mass fluxes relative to the mass-average rather than the volume-average velocity for several reasons: (1) the volume-average velocity formulation does not permit assessing the contribution of densification to the mass fluxes without making the unrealistic assumption of constant mass density; (2) the



f1 =

f2 =

g1 =

g2 =

 

1 DB − CA 1 DB − CA

 

  ∂ 1



  ∂ 1



  ∂ 1



  ∂ 1



∂2 C +B ∂ω1 ∂ω1

1 DB − CA 1 DB − CA

∂2 D +A ∂ω1 ∂ω1

(14)

∂2 C +B ∂ω2 ∂ω2 ∂2 D +A ∂ω2 ∂ω2

(15)

(16)

(17)

78

H. Lee et al. / Journal of Membrane Science 354 (2010) 74–85

where A, B, C, and D are defined as: A=

13 ω2 12 + ω1 M2 M3

12 13 B= − M2 M3 ω1 12 23 C= + ω2 M1 M3 D=

membrane. The integral mass balance for the nonsolvent is given by the following:

1 − ω  2

(18)

ω1

1 − ω  1

(20)

ω2

(21)

in which  ij is the binary friction coefficient for components i and j that is obtained from binary diffusion coefficient data as described in Shojaie et al. [15]. Note that although the polymer mass fraction does not appear explicitly in Eqs. (10)–(21) since ω3 = 1 − ω1 − ω2 the polymer properties enter explicitly through the friction coefficients  13 and  23 . The equation for the mass-average velocity is obtained by combining Eqs. (4), (10), and (11) as described by Lee et al. [13]: z

u¯ = 0

∂j 1  f () i dz ∂z 2

u¯ = −

 −

0

(22)

i=1

1 1 − ◦ 1◦ 3 1 1 − ◦ 2◦ 3



∂ω1 ∂ω2 f1 + g1 ∂z ∂z

 f2

∂ω1 ∂ω2 + g2 ∂z ∂z





L(t)

0



∂1 ∂t

∞



+

∂ω2 = 0 at z = 0 ∂z

∞

1B dz = 0

(26)

L(t)



dz + 1 



∂t

L(t)

z=L(t)

dL dt



dz − 1B 

z=L(t)

dL =0 dt

(27)

Simplification of the above using the species-balance equation for the nonsolvent given by Eq. (10) then gives: dL = dt



   B  1 −  n1 − nB1

1

(28) z=L(t)

where ni and nBi denote the total mass fluxes of component i with respect to a stationary reference frame in the casting solution and nonsolvent bath, respectively. An explicit equation for n1 = j1 + ω1 u¯ can be obtained by combining Eqs. (3), (10), and (23):



n1 =



1 − ω1

 −ω1

1 1 − ◦ 1◦ 3

1 1 − ◦ 2◦ 3

 

f1



∂ω1 ∂ω2 + g1 ∂z ∂z

∂ω1 ∂ω2 + g2 f2 ∂z ∂z



 (29)

By combining Eqs. (28) and (29) the following final form of the boundary condition for the nonsolvent is obtained:



(1 − 1B )

dL = dt





1 − ω1



(23) −ω1

1 1 − ◦ 1◦ 3

1 1 − ◦ 2◦ 3

 



f1

∂ω1 ∂ω2 + g1 ∂z ∂z

∂ω1 ∂ω2 f2 + g2 ∂z ∂z





−nB1 at z = L(t)

(24)

The boundary conditions at the underlying impermeable support surface are given by ∂ω1 = 0, ∂z





The describing equations consisting of Eqs. (10)–(13) require two initial conditions and four boundary conditions along with an auxiliary condition to determine L(t), the instantaneous location of the interface between the casting solution and the nonsolvent bath. The initial conditions correspond to the known composition of the casting solution prior to the inception of the wet-casting process denoted by ωi0 : ω2 = ω20 at t = 0



∂1B

2.5. Initial, boundary, and auxiliary conditions

ω1 = ω10 ,

d dt

where the superscript B denotes a property in the nonsolvent bath. The first and second terms in Eq. (26) represent the accumulation of mass in the casting solution and nonsolvent bath, respectively. Application of Leibnitz rule of differentiation to both terms yields:

N

where f() is a known function for a specified equation-of-state for the mass density. Eq. (22) is a completely general equation for determining the mass-average velocity that is applicable for any equation for the mass density. As such, it permits determining the contribution of densification-induced convection for conditions other than the restrictive case of the volume-average velocity being identically zero. For the special case of the mass density being described by Eq. (3), it is possible to integrate Eq. (22) analytically to obtain the following equation for the mass-average velocity:



L(t)

1 dz +

(19)

12 23 − M1 M3





d dt

(25)

The boundary conditions at the interface between the casting solution and nonsolvent bath involve a moving boundary. Hence, they are obtained via integral mass balances on the nonsolvent and solvent. Some prior studies [5,6] have assumed an infinitely deep casting solution to avoid having to consider a moving boundary problem. However, as a result they cannot account for any effects of the casting-solution thickness on the properties of the resulting

(30)

A similar development leads to the boundary condition for the solvent given by the following: dL (2 − 2B ) dt



= −ω2

 +

1 1 − ◦ 1◦ 3



1 − ω2



∂ω1 ∂ω2 f1 + g1 ∂z ∂z

1 1 − ◦ 2◦ 3

−nB2 at z = L(t)

  f2



∂ω1 ∂ω2 + g2 ∂z ∂z



(31)

The boundary conditions given by Eqs. (30) and (31) are applied at the moving boundary denoted by L(t) whose instantaneous location is determined by an appropriate auxiliary condition. The latter is obtained via an integral mass balance whose development is analogous to that leading to Eqs. (30) and (31) except that the overall mass density rather than the species mass concentration is involved. The resulting equation is given by the following: u¯ − (nB1 + nB2 ) dL = at z = L(t) dt  − B

(32)

When Eqs. (3), (12), (13), and (23) are substituted into Eq. (32), the following form of the auxiliary condition for determining the

H. Lee et al. / Journal of Membrane Science 354 (2010) 74–85

instantaneous location of the moving boundary is obtained: ( − B )

dL = − dt

 

−

1 1 − ◦ 1◦ 3 1 1 − ◦ 2◦ 3



f1

 f2

∂ω1 ∂ω2 + g1 ∂z ∂z ∂ω1 ∂ω2 + g2 ∂z ∂z

tion and nonsolvent bath phases denoted by the superscripts I and II, respectively:



Ii = IIi



−(nB1 + nB2 ) at z = L(t)

(33)

The solution to Eq. (33) is coupled to that of Eqs. (10) and (11) and requires an initial condition that corresponds to the known casting-solution thickness at the inception of the wet-casting process denoted by L0 : L = L0 at t = 0

(34)

2.6. Mass-transport model for the nonsolvent bath Polymer transport into the nonsolvent bath will be assumed to be negligible. As Tsay and McHugh [7] have pointed out, this assumption implies a discontinuity in the polymer concentration at the interface between the casting solution and the nonsolvent bath. This assumption is justified for casting solutions consisting of cellulose acetate, acetone, and water because the polymer-lean side of the binodal for this system has a negligible polymer concentration. In the absence of any polymer, the mass-transfer flux of nonsolvent in the bath is described by Fick’s law for binary convective diffusion as follows: nB1 = (B )

2



1 1 − ◦ 1◦ 2



D12

∂ω1B ∂z

(35)

where D12 is the binary diffusion coefficient. In principle, Eq. (35) requires solving the unsteady-state binary species-balance equation in the nonsolvent bath. However, this presents numerical problems because the thickness of the nonsolvent bath is much greater than that of the casting solution (i.e., several centimeters versus a few hundred microns). Allowing for these vastly different length scales in a numerical solver greatly increases the computation time and also compromises the accuracy of the predictions. However, scaling analysis indicates that the mass transfer in the nonsolvent bath involves a solutal boundary layer whose thickness is described by penetration theory and is given by [16]: ı=

D12 t

79

(36)

The penetration theory approximation permits obtaining an analytical solution to the species-balance equation in the nonsolvent bath via the method of combination of variables. The concentration gradient at the interface required in Eq. (35) then is given by

    ∞  ω1Bo  2ω1Bo D12 n2 2 t  = − exp − (37) + ∂z  ı ı ı2 z=L(t)

∂ω1B 

n=1

where ω1Bo is the initial weight fraction of nonsolvent in the nonsolvent bath. A similar procedure is used to obtain the equation for the mass flux of solvent. The binary diffusion coefficient for water and acetone used in this simulation was determined from an experimental data fit to the following equation [17]: D12 = 0.0002ω24 − 0.0002ω23 + 0.0002ω22 − 6.0 × 10−5 ω2 + 10−5 (38)

2.7. Interfacial local equilibrium concentration The continuous chemical potential at the interface provides the following criterion for local equilibrium between the casting solu-

(39)

Handling the initial interfacial composition presents a problem in wet-casting that does not occur in modeling the dry-casting process. In the latter the initial casting solution can instantaneously establish appropriate equilibrium partial pressures in the contiguous gas phase at the interface without changing its composition. However, in wet-casting the initial casting solution cannot instantaneously establish equilibrium with the contiguous nonsolvent phase at the interface without changing its composition since all equilibrium concentrations of the casting solution with the nonsolvent phase are located on the binodal. The casting-solution composition cannot jump from a nonequilibrium to an equilibrium value on the binodal instantaneously as has been assumed in some models [5–7]. Hence, a defensible strategy is needed for describing the manner in which the casting-solution composition progresses from its initial value to a point on the binodal. If one adopts the reasonable assumption that negligible polymer is transferred into the nonsolvent bath, the latter then is at most a binary solution of nonsolvent and solvent. Hence, it is reasonable to assume that the progression from the initial casting-solution composition to that corresponding to some point on the binodal occurs under conditions of equal chemical potentials in the casting solution and nonsolvent bath for both distributing components. However, since the nonsolvent bath is a binary solution, it is sufficient to use only one of the chemical potential equalities. Here we choose the solvent chemical potential for which this equality is expressed as follows: x2B 2B = exp

   2

(40)

RT

where x2B and 2B are the equilibrium mole fraction and activity coefficient of the solvent in the nonsolvent bath, respectively, and 2 is the difference between the actual and reference chemical potential of the solvent that can be calculated using Flory–Huggins theory. However, the latter is not appropriate for determining the activity coefficient in the nonsolvent bath. Therefore, the activity coefficient of the solvent in the nonsolvent bath was determined using the NRTL (non-random two-liquids) model with the experimental parameters suggested in the literature [18–21] for water and acetone given by the following:

 ln(2B )

=

2 (x1B )

 1.065

0.566 x1B + 0.566x2B



2 +

 0.686 (x1B + 0.340x2B )

2

(41)

Eqs. (40) and (41) are used to determine the concentrations at the interface between the casting solution and nonsolvent bath only during the pseudo-equilibrium period when the casting solution is progressing from its initial composition to one on the binodal. Once the binodal is reached, true equilibrium exists between the two phases. Flory–Huggins theory then can be used to determine the interfacial equilibrium concentrations of both the casting solution and the nonsolvent bath. It is also possible that the interfacial composition of the casting solution may reach a point where vitrification occurs. These compositions for casting solutions consisting of cellulose acetate, acetone, and water were calculated as a function of temperature by Prakash [22]. In this simulation it is assumed that the thin vitrified layer maintains all the properties of the liquid state and that its composition remains fixed at the intersection of the vitrification boundary with the binodal as suggested by Reuvers et al. [5].

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3. Solution methodology The solution for the wet-casting model equations was obtained using a commercially available partial differential equation solver, D03PPF in the NAG (National Algorithms Group) Fortran library [23]. This numerical code requires applying the boundary conditions at fixed locations. Hence, the coordinate transformation z˜ ≡ z/L(t) was introduced to permit applying the boundary conditions at z˜ = 0 and z˜ = 1. Since the mass-transfer fluxes for the wet-casting process were expected to be much larger than those for the dry-casting process, it was assumed that at short contact times they would result in very steep concentration gradients confined to a thin region near the interface. Hence, in this study 100 of the 131 numerical calculation points were located within the top micron of the casting solution in order to increase the accuracy in this critical region. The model calculations with double precision were done on a UNIX system for which the machine precision is 10−300 . The goals of this simulation were multifold. First, we wanted to describe the initiation of the wet-casting process during which the interface between the casting solution and the nonsolvent bath are not in thermodynamic equilibrium. Second, we wanted to determine if vitrification could be predicted since this has been observed but not predicted by prior modeling studies [8]. Third, we wanted to assess the importance of incorporating densificationinduced convection explicitly. Fourth, we wanted to study the effect of casting-solution thickness since some prior modeling studies have claimed that it should have no significant effect on the functional layer of the resulting membrane [5,6]; moreover, we sought to explain its effect on the appearance of macrovoid pores. In view of these goals this simulation assumed an initial castingsolution composition having weight fractions of 0, 0.85, and 0.15 of water, acetone, and cellulose acetate, respectively, and initially pure water for the nonsolvent bath. Initial casting-solution thicknesses of 75 and 125 ␮m were chosen since the prior experimental studies of Paulson et al. [24] have shown that they lead to markedly different membranes particularly insofar as the appearance of macrovoids.

Fig. 2. Composition change of the interface between the casting solution and nonsolvent bath during the wet-casting of the water, acetone, and cellulose acetate system having initial mass fractions of 0, 0.15, and 0.85, respectively; for 0 < t ≤ 3.3 × 10−10 s the interface is adjusting from a nonequilibrium to an equilibrium concentration; for 3.3 × 10−10 ≤ t ≤ 2.0 × 10−9 s the interface is on or within the binodal until it vitrifies; initial casting-solution thicknesses of 75 and 125 ␮m give identical results owing to negligible penetration.

analyses [5–7]. Accounting for this adjustment in the interfacial composition properly is very important in establishing the correct starting point on the binodal for the subsequent vitrification or phase separation. Note also that the much faster mass transfer of the acetone relative to the water results in a large increase in the weight fraction of the cellulose acetate, which in turn implies significant densification and associated convective mass transfer. This increase in the polymer concentration also favors vitrification rather than phase inversion at the interface as will be discussed in the next section.

4. Presentation and discussion of results

4.2. Local equilibrium

4.1. Local pseudo-equilibrium

After the composition at the interface of the casting solution reaches the binodal, local thermodynamic equilibrium prevails between the two phases at this boundary. Hence, the compositions of both the casting solution and nonsolvent bath at the interface can be determined from the tie lines predicted by Flory–Huggins theory. The path for 3.3 × 10−10 ≤ t ≤ 2.0 × 10−9 s in Fig. 2 shows the change in the composition at the interface immediately after local equilibrium is achieved. The model predicts that the vitrification boundary is reached for both initial casting-solution thicknesses within 2.0 × 10−9 s. Again, composition changes occur in the casting solution only within 0.05 ␮m of the interface. These simulation results indicate that the interface vitrified well before any change in composition occurs in the bulk of the casting solution. Vitrification rather than phase separation occurs because the time scale is too short for nucleation and growth to occur. These predictions are supported by the polymer accumulation observed at the interface during the wet-casting process [8]. Prior models have not been able to predict vitrification at the interface owing to not accounting for the transition of the interfacial concentration from its initial nonequilibrium value to an equilibrium composition on the binodal. When this is properly accounted for as done in the present analysis, considerably higher polymer concentrations are predicted in the interfacial region between the casting solution and nonsolvent bath.

This part of the wet-casting process simulation involves describing the path by which the concentrations at the interface change from those of the initial casting solution to a composition on the binodal. Note that the instantaneous pseudo-equilibrium interfacial concentrations of the casting solution were obtained using the Flory–Huggins model and those of the nonsolvent bath using the NRTL model. The path for 0 < t ≤ 3.3 × 10−10 s in Fig. 2 shows the instantaneous composition of the upper interface during the initial part of the wet-casting process. Note that within 3.3 × 10−10 s the predicted composition of the upper interface reaches the binodal. The results for the two different initial thicknesses (75 and 125 ␮m) are identical as might be expected in view of the very short duration for this part of the wet-casting process. The composition changes for both initial thicknesses occurred only in top 0.05 ␮m of the casting solution. This is of the same order as the solutal boundary-layer thickness

ıs estimated from scaling analysis for

∼ which ıs = D23 t = (1 × 10−7 cm2 /s)(3.3 × 10−10 s) ∼ = 0.01 ␮m [16]. From these simulation results one can conclude that the interfacial concentration of the casting solution adjusts from its initial nonequilibrium to an equilibrium composition extremely rapidly; however, not instantaneously as has been assumed in prior

H. Lee et al. / Journal of Membrane Science 354 (2010) 74–85

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Fig. 3. Composition changes during the wet-casting of the water, acetone, and cellulose acetate system having initial mass fractions of 0, 0.15, and 0.85, respectively, for an initial casting-solution thickness of 75 ␮m; each curve shows the composition from the underlying impermeable substrate to the interface with the nonsolvent bath for successive times after immersion: (a) predictions incorporating densification-induced convection and (b) predictions in the absence of densification-induced convection.

4.3. Extended simulation after vitrification at the interface For the extended simulation after vitrification at the interface, the 130 calculation points were distributed equally throughout the thickness of the casting solution in order to account for concentration changes in the bulk. The interfacial composition was assumed to be fixed at the intersection point between the binodal and the vitrification boundary. Fig. 3a shows the predictions of the full model incorporating densification-induced convection for the composition at each point in the casting solution extending from its interface with the nonsolvent bath to the underlying impermeable substrate as a function of contact time for a casting solution with an initial thickness of 75 ␮m. After only 3.4 s all the compositions throughout the entire casting solution are predicted to be on or slightly within the binodal with no significant supersaturation predicted anywhere. This indicates the error that can be incurred in models that assume an infinitely deep casting solution. One of the advantages of incorporating densification-induced convection explicitly into a model is that it permits direct assessment of the importance of this contribution to the mass-transfer flux. This

Fig. 4. Thickness change during the wet-casting of the water, acetone, and cellulose acetate system having initial mass fractions of 0, 0.15, and 0.85, respectively, for an initial casting-solution thickness of 75 ␮m.

can be done by comparing the model predictions with the massaverage velocity u¯ = / 0 to the model predictions with u¯ = 0. This is not possible with the implicit approach for which any contributions of densification to the mass-transfer flux are incorporated into the diffusive fluxes relative to the volume-average velocity. Fig. 3b shows the comparable composition profiles predicted using the same describing equations while suppressing densificationinduced convection; that is, by setting u¯ = 0. A comparison of the composition profiles at the same time (e.g., 3 s) indicates that including densification-induced convection increases the amount of both the water and acetone in the casting solution. This indicates that the densification-induced convection aids mass transfer of water into the casting solution and hinders the mass transfer of acetone out of it. Fig. 4 shows the predictions of the full model incorporating densification-induced convection on the instantaneous castingsolution thickness for an initial thickness of 75 ␮m. Initially the thickness decreases markedly and then more slowly until it reaches approximately 64 ␮m. The marked initial decrease in thickness

Fig. 5. Mass-average velocity profiles during the wet-casting of the water, acetone, and cellulose acetate system having initial mass fractions of 0, 0.15, and 0.85, respectively, for an initial solution thickness of 75 ␮m. Each curve shows the velocity from the underlying impermeable substrate to the interface with the nonsolvent bath for successive times after immersion.

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concentration profile also showed a pronounced change when this ratio was approximately one. This is explained by the near exponential decrease in the diffusion coefficient of acetone in cellulose acetate at this ratio of solvent to polymer. Fig. 6 shows the corresponding modified Peclet number at the interface between the casting solution and nonsolvent bath as a function of time. The modified Peclet number is a measure of the ratio of the convective to the diffusive contributions to the mass-transfer flux and is defined as follows: PeM ≡

Fig. 6. Modified Peclet number as a function of time during the wet-casting of the water, acetone, and cellulose acetate system having initial mass fractions of 0, 0.15, and 0.85, respectively, for an initial solution thickness of 75 ␮m.

is caused by a combination of very large initial mass-transfer fluxes and significant densification occurring owing to the increase in polymer concentration at the interface. At longer times the mass-transfer fluxes asymptotically decrease to zero and the densification becomes less pronounced. If the densification-induced convection is ignored, the model predicts that the casting-solution thickness will not change. This occurs because in the absence of a convective contribution to the mass flux, the boundary condition at the upper interface in combination with binary diffusion in the coagulation bath imply that the mass fluxes of the two transferring components are equal in magnitude and opposite in direction. The corresponding mass-average velocity profiles as a function of normalized thickness at various times for an initial thickness of 75 ␮m are shown in Fig. 5. Note that the mass-average velocity is negative; this implies that the mass-flux of nonsolvent into the casting solution will be enhanced, whereas the loss of solvent will be retarded owing to the densification-induced convection. A dramatic change in the shape of the mass-average velocity profile occurs when the mass ratio of acetone to cellulose acetate is approximately one. In the dry-casting model results for the cellulose acetate, acetone, water system reported by Lee et al. [13], the

Lu¯ D23

(42)

¯ and D23 . in which instantaneous local values are used for L, u, The modified Peclet number results indicate that the initial densification-induced convective mass transfer is 65 times greater than the purely diffusive mass transfer. Therefore, neglecting densification-induced convection can result in serious inaccuracies in the predictions particularly early in the wet-casting process when the properties of the functional layer are determined. Fig. 7a shows the composition at each point in the casting solution extending from its interface with the nonsolvent bath to the underlying impermeable substrate as a function of contact time for a casting solution with an initial thickness of 125 ␮m. The composition in the top 7 ␮m crosses the binodal after 1 s, which implies that the top 7 ␮m region of the casting solution enters the metastable region nearly instantaneously, unlike the predictions for an initial thickness of 75 ␮m. Furthermore, it takes 7.5 s for the entire casting solution to cross the binodal, which is much longer than predicted for the 75 ␮m thickness. Therefore, even with identical initial casting-solution compositions, the effect of the initial thickness is significant. Having a substantial portion of the casting solution in the metastable region undoubtedly will result in a different membrane morphology. The differences between the model predictions in the absence of densification-induced convection (i.e., setting u¯ = 0) and those that include this effect (i.e, u¯ = / 0) are much more pronounced for the initial film thickness of 125 ␮m as shown in Fig. 7b. Whereas the model predictions incorporating densification-induced convection shown in Fig. 7a indicate that a substantial thickness of the casting solution is well into the metastable region of the binodal, Fig. 7b does not show significant supersaturation. Significant supersaturation promotes rapid phase transition and also is known to promote macrovoid formation [25,26]. Clearly, ignoring densification-induced convection

Fig. 7. Composition changes during the wet-casting of the water, acetone, and cellulose acetate system having initial mass fractions of 0, 0.15, and 0.85, respectively, for an initial casting-solution thickness of 125 ␮m; each curve shows the composition from the underlying impermeable substrate to the interface with the nonsolvent bath for successive times after immersion: (a) predictions incorporating densification-induced convection and (b) predictions in the absence of densification-induced convection.

H. Lee et al. / Journal of Membrane Science 354 (2010) 74–85

Fig. 8. Thickness change during the wet-casting of the water, acetone, and cellulose acetate system having initial mass fractions of 0, 0.15, and 0.85, respectively, for an initial solution thickness of 125 ␮m.

can result in unrealistic model predictions for important casting solution conditions such as the degree of supersaturation and the propensity for macrovoid formation. Fig. 8 shows the predictions of the full model incorporating densification-induced convection on the instantaneous thickness of the casting solution for an initial thickness of 125 ␮m. The thickness decreases to approximately 107 ␮m within the time it takes for the entire casting solution to enter the metastable region. Fig. 9 shows the corresponding mass-average velocity profiles for an initial thickness of 125 ␮m as a function of the normalized thickness at different times. The marked change in shape of the velocity profiles again corresponds to the point where the ratio of acetone to cellulose acetate is approximately one. The corresponding modified Peclet number as a function of time for an initial thickness of 125 ␮m is shown in Fig. 10. This also indicates that the densification-induced convective contribution to the mass-transfer flux is more than 65 times greater than the purely diffusive flux during the early part of the wet-casting process. One of the interesting consequences of incorporating densification-induced convection into the model is the possibility of having supersaturation through a substantial thickness of the casting solution near the interface as seen in Fig. 7a. However, a comparison of Figs. 3a and 7a indicates that this is possible only for thicker initial casting-solution thicknesses; that is, supersaturation is predicted for the 125 ␮m but not the

83

Fig. 10. Modified Peclet number as a function of time during the wet-casting of the water, acetone, and cellulose acetate system having initial mass fractions of 0, 0.15, and 0.85, respectively, for an initial solution thickness of 125 ␮m.

75 ␮m initial casting-solution thickness. Note also that supersaturation is not predicted even for the 125 ␮m initial thickness when densification-induced convection is neglected. Incorporating densification-induced convection is a necessary but not a sufficient condition for predicting supersaturation for the initial casting-solution composition in this study. Densification-induced convection is necessary to enhance the mass transfer of water into the casting solution, which favors crossing into the metastable region. However, it also retards the mass transfer of acetone out of the casting solution, which delays crossing into the two-phase envelope. Note that Figs. 5 and 9 indicate that at 3 s, both the 75 and 125 ␮m initial casting-solution thicknesses have nearly the same velocity profile. Hence, densification-induced convection alone does not explain why the 125 ␮m initial thickness shows a supersaturated region whereas the 75 ␮m does not. A thicker casting solution provides acetone at depth that can diffuse towards the interface, thereby also delaying crossing the binodal. The latter effect of casting-solution thickness can be seen by comparing Figs. 3a and 7a; after slightly more than 3 s the entire casting solution for the 75 ␮m initial casting-solution thickness has reached the binodal. In contrast, at 3 s the concentration of the lower region of the 125 ␮m thickness has experienced very little change in composition. In summary, thicker initial casting solutions can provide solvent from depth that delays crossing the binodal in the upper region of the casting solution; this delay favors creating supersaturation. However, including densification-induced convection is critical to predicting supersaturation conditions since it enhances the mass transfer of nonsolvent into the casting solution. 5. Conclusions

Fig. 9. Mass-average velocity profiles during the wet-casting of the water, acetone, and cellulose acetate system having initial mass fractions of 0, 0.15, and 0.85, respectively, for an initial solution thickness of 125 ␮m. Each curve shows the velocity from the underlying impermeable substrate to the interface with the nonsolvent bath for successive times after immersion.

An improved model for the wet-casting of polymeric membranes that provides a more realistic description of the mass transfer at the interface between the casting solution and the nonsolvent bath permitted describing the initiation of the wet-casting process during which the casting solution concentrations at the interface change from their initial nonequilibrium to an equilibrium composition on the binodal. This model is able to predict the occurrence of vitrification at the interface that has been observed in prior experimental studies [8]. This improved model also explicitly incorporates the contribution of densification-induced convection to the mass-transfer. The value of the initial modified Peclet number indicates that densification-induced convective mass transfer can be nearly two

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orders-of-magnitude larger than that due to diffusion near the interface between the casting solution and the nonsolvent bath. This implies that ignoring densification-induced convection can result in significant errors in model predictions for the interfacial region of the casting solution that strongly influences the properties of the functional layer of asymmetric membranes. Model predictions for the cellulose acetate, acetone, and water system show the formation of a relatively thick supersaturated region beneath the interface between the casting solution and nonsolvent bath for a thicker initial casting-solution thickness (125 ␮m) but not for a thinner one (75 ␮m). This is consistent with the results of prior experimental studies that show that the casting-solution thickness can strongly influence the resulting membrane morphology [24]. The occurrence of supersaturation for thicker initial casting-solution thicknesses is a consequence of both enhanced mass transfer of nonsolvent into the casting solution owing to densification-induced convection and the delay of crossing into the metastable region owing to diffusion of solvent from depth towards the interface. Prior experimental studies indicate that having a relatively thick region of the casting solution within the metastable region creates the supersaturation conditions conducive to macrovoid pore formation [25,26]. Hence, these simulation results offer an explanation as to why macrovoids are observed only for thicker initial casting-solution thicknesses [24].

ui V Vi◦ V¯ i xi xiB z z˜ ı iB i Ii IIi

i  ij  B i

Acknowledgment

iB

The authors gratefully acknowledge support of this research by the NASA Office of Biological and Physical Sciences Research via Grant No. NAG3-2451.

i◦ ωi ωi0

Nomenclature A B c C D Dij f1 f2 g1 g2 ji L(t) L0 Mi ¯ M n ni nBi PeM R T t u¯ u*

function defined by Eq. (18) function defined by Eq. (19) molar density function defined by Eq. (20) function defined by Eq. (21) diffusion coefficient for a binary mixture of components i and j function defined by Eq. (14) function defined by Eq. (15) function defined by Eq. (16) function defined by Eq. (17) mass flux of component i relative to the massaverage velocity instantaneous location of the interface between the casting solution and nonsolvent bath initial thickness of the casting solution molecular weight of component i average molecular weight integer mass flux of component i in the casting solution with respect to a fixed reference frame mass flux of component i in the nonsolvent bath with respect to a fixed reference frame modified Peclet number defined by Eq. (34) universal gas constant absolute temperature temporal coordinate mass-average velocity volume-average velocity

ωiB0

velocity of species i relative to a stationary reference frame total volume per mole pure component molar volume of component i partial molar volume of component i mole fraction of component i mole fraction of component i in the nonsolvent phase spatial coordinate z dimensionless spatial coordinate ≡ L(t) thickness of solutal boundary layer from penetration theory activity coefficient of component i in the nonsolvent phase chemical potential of component i chemical potential of component i in phase I chemical potential of component i in phase II difference between the actual and reference chemical potential of component i friction coefficient between components i and j mass density of the casting solution mass density of the nonsolvent bath mass concentration of component i in the casting solution mass concentration of component i in the nonsolvent phase mass density of pure component i mass fraction of component i mass fraction of component i in the initial casting solution mass fraction of component i in the initial nonsolvent bath

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