Simulation analysis of drying of ternary polymeric solution coatings

Progress in Organic Coatings 78 (2015) 155–167

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Progress in Organic Coatings journal homepage: www.elsevier.com/locate/porgcoat

Simulation analysis of drying of ternary polymeric solution coatings Raj Kumar Arya a,∗ , Chitresh Kumar Bhargava b a b

Department of Chemical Engineering, Thapar University Patiala, Bhadson Road, Patiala 147004, Punjab, India Department of Chemical Engineering, Jaypee University of Engineering & Technology, Guna, A.B. Road, Raghogarh, Guna 473226, MP, India

a r t i c l e

i n f o

Article history: Received 6 March 2014 Received in revised form 19 August 2014 Accepted 22 September 2014 Available online 12 October 2014 Keywords: Multicomponent diffusion Thin film Free-volume theory Skinning

a b s t r a c t A simulation study on multicomponent polymeric solvent casting process was carried out using known different models and their relative performance of the model assessed in terms of residual parameters and solvent concentration profiles in the coating film. While all the models were found to predict nearly identical concentration profile for highly volatile solvents, significant variation in the profiles was observed in case of the less volatile solvents. Estimated values of the preexponential factors (D01 and D02 ) exhibited marginal differences implying perhaps a material balance deviation of the predictive models. The effect of air flow rate used for the drying process has been realized through its influence on the skinning behavior, low flow rate being the preferred while for maintaining the appropriate level of skinning. © 2014 Elsevier B.V. All rights reserved.

1. Introduction To produce the polymer coating, the casting of continuous layer of polymer solution is done on the substrate. These coatings are used in synthetic fiber, photographic films, magnetic media, etc. Photographic films, adhesives, image media, and magnetic media are different kinds of films which are made using thin film coating and drying technology. Drying is the last and quality controlling step in the polymeric coating. Drying conditions: air flow, oven temperature, and residence time, are responsible for final structure and properties of the coatings. Due to poor drying conditions, sometimes improper drying takes place, hence low quality, internal gradients, phase separations, colloidal transformations lead to wrong microstructure, inappropriate non-uniformities and stress related defects [1]. Simultaneous heat and mass transfer operations occur during the removal of volatile excess solvents from the coating [2]. The rate controlling step for drying shifts from external mass transfer to internal mass transfer within the coating, as the solvents evaporate. Diffusion and diffusion induced convection are responsible to move the solvents from the coating to the surface. Diffusion coefficient drops dramatically as the solvent concentration falls in case of polymeric coating [1]. The concentration of the solvent at the exposed surface drops during the course of drying. Hence, the drying rate falls steadily. This period is called falling rate

∗ Corresponding author. Tel.: +91 175 2393444; fax: +91 175 2393005. E-mail addresses: [email protected], [email protected] (R.K. Arya), [email protected] (C.K. Bhargava). http://dx.doi.org/10.1016/j.porgcoat.2014.09.011 0300-9440/© 2014 Elsevier B.V. All rights reserved.

period. The diffusion coefficient of solvent in the polymeric coating is the strong function of the solvent concentration, temperature and molecular size. In literature, several studies are reported for binary polymer–solvent coatings. The results of binary diffusion models compare well with experimental weight loss data [3–5]. Recently, the results of the binary model have been shown to compare well with depth profile measurements using confocal laser Raman spectroscopy [6]. All the multicomponent diffusion models [2,7–9] have been developed using Bearman’s friction factor theory by making some assumptions. Therefore, a comparative study of these models is required to find the suitable model for the prediction of drying behavior in multicomponent coatings. Only few studies [2,10] have been reported in literature in this field. Schabel et al. [11] have modified the Flory–Huggins theory and found good agreement with experimental and model predicted diffusion data in case of poly (vinyl acetate)–methanol–toluene system. They have not shown the comparison among the different multicomponent diffusion models. Recently, Arya and Vinjamur [12] have tested these diffusion models again the measured concentration profiles during drying. They have measured concentration profiles in ternary polymer–solvent–solvent systems using confocal Raman spectroscope. They have compared measured profiles with model predicted values and found that none of multicomponent diffusion model is able to predict complete concentration profiles for the less volatile solvent. However, predictions of generalized model are much better than the Alsoy and Duda [2] and Zielinski and Hanley [8] models. Therefore, a comprehensive simulation study is needed

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R.K. Arya, C.K. Bhargava / Progress in Organic Coatings 78 (2015) 155–167

ω1 =

c1 , c1 + c2 + (1 − c1 Vˆ 1 − c2 Vˆ 2 )polymer

ωi : mass fraction of species i, ci : concentration of species i, Vˆ i : specific volume of species i, polymer : density of polymer. Mass fraction of solvent 2 ω2 =

c2 c1 + c2 + (1 − c1 Vˆ 1 − c2 Vˆ 2 )polymer

Mass fraction of polymer 3 ω3 = 1 − ω1 − ω2 Fig. 1. Schematic of a drying coating. hg : heat transfer coefficients at the base side, hG : heat transfer coefficients at the surface of the coating, pGib : partial pressure of solvent i in the bulk gas, L: thickness of coating, cm, t: time, s, z: thickness of the coatings at time t, TG and Tg : gas/air temperature at top and bottom sides respectively.

to study various multicomponent free volume diffusion models and their limitations. In this paper, a simulation study in ternary coating of one polymer and two solvents using all the available free volume theory models have been reported.

Fig. 1 shows the schematic of a drying coating, which has been cast on impermeable substrate. As soon as the solvent reaches the surface, it evaporates into the air on the top side of the coating. Then the mass of solvents decreases with time and coating–gas interface moves closer to the substrate opposite to diffusion. There is no mass transfer from the substrate side and hence the fluxes will be zero at the substrate. 2.1. Mass transport Both the solvents are diffusing within the coating from substrate side to the coating side. At anytime total mass transfer of any diffusing species is the sum of the mass transfer due to its own concentration gradient and mass transfer due to concentration gradient of second solvent. The reference velocity is chosen to be volume average velocity because it is shown to be equal to zero if there is no change in volume on mixing [13]. Mass balance for solvent 1



∂c1 D11 ∂z



∂ + ∂z



∂c2 D12 ∂z

 (1)

Mass balance for solvent 2

∂c2 ∂ = ∂t ∂z

 D21

∂c1 ∂z

ω3 =

 +

∂ ∂z

 D22

∂c2 ∂z

 (2)

ci is the concentration of solvent i (i = 1,2), t is the time, z is the thickness of the coatings at anytime, D11 and D22 are main diffusion coefficients that characterize transport due to solvents own concentration gradient, D12 and D21 are cross diffusion coefficients that characterize transport due to other solvents concentration gradient. The concentration of polymer, balancing component, can be obtained by equating sum of mass fraction to one. Mass fraction of solvent 1

c3 c1 + c2 + (1 − c1 Vˆ 1 − c2 Vˆ 2 )polymer

Now equating mass fractions of polymer, 1−

c1 c1 + c2 + (1 − c1 Vˆ 1 − c2 Vˆ 2 )polymer c2

−

2. Model equations

∂c1 ∂ = ∂t ∂z

But mass fraction of polymer can be calculated using,

=

c1 + c2 + (1 − c1 Vˆ 1 − c2 Vˆ 2 )polymer c3 c1 + c2 + (1 − c1 Vˆ 1 − c2 Vˆ 2 )polymer

⇒ c3 = (1 − c1 Vˆ 1 − c2 Vˆ 2 )polymer

(3)

Several theories for predicting main-term and cross-term diffusion coefficients have appeared in the literature. The theories begin with Bearman’s statistical mechanical theory [14] that relates gradient of chemical potential of a species to frictional motion between the species and others of the system.

∂i  cj ij (i − j ) = ∂z n

(4)

j=1

(∂)/(∂z) is chemical potential gradient; ci , concentration of component i; Mi , molecular weight of component i;  ij is friction coefficient between component i and j; i and j are the mean velocities of component i and j respectively. According to Bearman, self diffusion coefficients are also related to friction is given by Di =

n

RT

(c /Mi )ij J=1 i

(5)

Di is self diffusion coefficient of species i, R is universal gas constant and T is absolute temperature. Friction factors  ij cannot be measured directly. Different assumptions on them led to different theories for diffusion in multicomponent mixtures. Zielinski and Hanley [8] related chemical potential gradient to mass average velocity and frictional force experienced by the molecule. They related the mass flux with respect to volume average velocity to the mass flux with respect to the mass average velocity. Mass flux with respect to mass average velocity is related to frictional force experienced by a molecule. They assumed ratios of friction factors equal to the ratios of molecular weight. Mass average velocity is chosen as the reference velocity in their work. Model equations for their case are given in Table 1. Dabral [9] modeled diffusive flux to the volume average velocity. He assumed friction between the solvents is equal ( 12 =  21 ).

Model

D11

D12

D21

Zielinski and Hanley [8]

D1 c1 (1  −∂ lnc1aVˆ 1+ 1 + c1 Vˆ 3 )

D1 c1 (1  −∂ lnc1aVˆ 1+ 1 + c1 Vˆ 3 )

D2 c2 (1  −∂ lnc2aVˆ 2+ 2 + c2 Vˆ 3 )

D2 c2 (1  −∂ lnc2aVˆ 2+ 2 + c2 Vˆ 3 )

ˆ3 − D2 c 1 c2 (V  ∂ ln a2 Vˆ 2 )

ˆ3 − D2 c 1 c2 (V  ∂ ln a2 Vˆ 2 )

ˆ3 − D1 c 1 c2 (V  ∂ ln a1 Vˆ 1 )

ˆ3 − D1 c 1 c2 (V  ∂ ln a1 Vˆ 1 )

D1 c1 (1  −∂ ln a  1 − c1 Vˆ 1 )

D1 c1 (1  −∂ ln a  1 − c1 Vˆ 1 )

D2 c2 (1  −∂ ln a  2 − c2 Vˆ 2 )

D2 c2 (1  −∂ ln a  2 − c2 Vˆ 2 )

D2 c1 c2 Vˆ 2

D2 c1 c2 Vˆ 2

D1 c1 c2 Vˆ 1

D1 c1 c2 Vˆ 1

∂c1

∂c2

∂c1

Dabral [9]

 ∂ ln a 

Alsoy and Duda [2]

Case 1 Case 2 Case 3 Case 4

D1

∂c1

c1 c2

1

 ∂ ln a 

∂c2

 ∂ ln a 

∂c1

2

∂c2

 ∂ ln a 

c2 c1

1

∂ ln c2

D2

 ∂ ln a 

∂c2

1

∂c1

 ∂ ln a 

1

 ∂ ln a  D2 ∂ ln c 2  ∂ ln a2 

2

∂ ln c1

2

0

0

0 D1 c1 (1  −∂ ln a  1 − c1 Vˆ 1 )

0 D2 c2 (1  −∂ ln a  2 − c2 Vˆ 2 )

D2 D2 c2 (1  −∂ ln a  2 − c2 Vˆ 2 )

D2 c1 c2 Vˆ 2

D2 c1 c2 Vˆ 2

D1 c1 c2 Vˆ 1

D1 c1 c2 Vˆ 1

∂ ln c1

 ∂ ln a 

 ∂ ln a c1 c2 Vˆ 2 D1 ∂c 1  ∂ ln1a

c1 c3 Vˆ 3 D1

1

∂c1

∂ ln a − D2 ∂c 2 + 1 ∂ ln a3 − D3

 ∂ ln a 

∂c2

2

∂c1

∂c1

 ∂ ln a c1 c2 Vˆ 2 D1 ∂c 1  ∂ ln2a

c1 c3 Vˆ 3 D1

1

∂c2

∂ ln a − D2 ∂c 2 + 2 ∂ ln a3 − D3

 ∂ ln a 

∂c1

2

∂c2

D2

∂c2

D1 D1 c1 (1  −∂ ln a  1 − c1 Vˆ 1 ) ∂c1

Price and Romdhane [7]

D1

∂c2

∂c1

∂c2

2

 ∂ ln a  D1 ∂ ln c 1  ∂ ln a1 

∂c1

∂c2

∂c1

D22

∂c2

 ∂ ln a c2 c1 Vˆ 1 D2 ∂c 2  ∂ ln1a

c2 c3 Vˆ 3 D2

2

∂c1

∂ ln a − D1 ∂c 1 + 1 ∂ ln a3 − D3

 ∂ ln a 

∂c2

1

∂c1

∂ ln c2

∂c1

1

∂c2

 ∂ ln a c2 c1 Vˆ 1 D2 ∂c 2  ∂ ln2a

c2 c3 Vˆ 3 D2

2

∂c2

∂ ln a1 ∂c2 ∂ ln a − D3 ∂c 3 2

− D1

+

R.K. Arya, C.K. Bhargava / Progress in Organic Coatings 78 (2015) 155–167

Table 1 Various multicomponent diffusion models.

157

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R.K. Arya, C.K. Bhargava / Progress in Organic Coatings 78 (2015) 155–167

In high molecular weight polymer and high concentrations of polymer, the friction factor between the polymer and solvents ( 13 and  23 ) are much greater than friction factor between solvent pair ( 12 ,  21 ) and themselves ( 11 ,  22 ). Model equations are given in Table 1. Alsoy and Duda [2] presented models for diffusion coefficients for four cases. In one case, the ratio of friction factors was assumed to be constant; in another case, the cross-term coefficients were set to zero; in yet another case, the cross-term coefficients were set to zero and the main-term coefficients were set equal to self-diffusion coefficients; in still another case, the friction factors were set to zero. Detailed derivation of the diffusion coefficients is available in Alsoy [15]. Table 1 shows the diffusion coefficients for all the four cases. Equations of case 4 are same as those of Dabral [9]. Price and Romdhane [7] presented a generalized theory, again based on Bearman’s friction theory, which unifies the above models. They defined ratio of friction coefficients as ij ik

˛j V˜ j ˛j Vˆ j Mj = , ˛k V˜ k ˛k Vˆ k Mk

=

where ˛ is a constant. Using this ratio, they derived equations for diffusion coefficients which are given below.





D11 = 1 − c1 Vˆ 1 1 − −



1−

˛2 ˛3



c2 Vˆ 2 c1 D2





D12 = 1 − c1 Vˆ 1 1 − −



1−

˛2 ˛3







1−



˛1 ˛3







−

1−

˛1 ˛3

˛2 ˛3

c1 Vˆ 1 c2 D1

D22 = 1 − c2 Vˆ 2 1 −



˛1 ˛3

c2 Vˆ 2 c1 D2

D21 = 1 − c2 Vˆ 2 1 − −

˛1 ˛3

˛2 ˛3

c1 Vˆ 1 c2 D1



c1 D1

∂ ln a1 ∂c1

∂ ln a2 ∂c1

c1 D1

(6)

∂ ln a1 ∂c2

∂ ln a2 ∂c2

c2 D2

(7)

c2 D2

(8)

∂ ln a2 ∂c2

∂ ln a1 ∂c2

(9)

In generalized model ratio of self diffusion coefficients were set jj , jk

equal

to

ratio

of

friction

⎛ N

D

factors, Dk = j

˛j Vˆ j Mj ˛k Vˆ k Mk

=

ij ik

=

j= / i, and i, k = 1, . . ., N − 1

The generalized diffusion equations predicted by them are given in Table 1. The generalized theory requires self-diffusion coefficient of the polymer – a shortcoming of the theory because few experimental data are available for this coefficient. Activity of the solvents for the ternary polymer–solvent–solvent system can be calculated using Flory–Huggins theory. All other previous models [2,8,9] are some special cases of generalized model of Price and Romdhane [7]. By setting different values to ˛i , the theories can be recovered. For Dabral [9] model, ˛i = 0, i = / N For Zielinski and Hanley [8] model, ˛i = 1/Vˆ i , i = 1, . . .N, For Alsoy and Duda [2] model, ˛i = 1, i = 1, . . . N Zielinski and Alsoy [16] checked the consistency of multicomponent diffusion models using Onsager reciprocal relations

⎞

ω Vˆ ∗ (iN /jN ) j=1 j j

Di = D0i exp ⎝−

∂ ln a2 ∂c1

∂ ln a1 ∂c1



(ORR). They showed that Zielinski and Hanley [8] model satisfies the Onsager relation for low molecular weight species but fails for higher molecular weight species. Alsoy and Duda [2] models was also unable to satisfy the Onsager relations. In case 1 of Alsoy and Duda models, ratio of friction factors was assumed constant and equal to the partial molar volumes of components which cannot hold throughout the concentration range. In case 4, friction coefficients between the diffusing components was assumed to be zero which gives ratio of friction factor between the solvent and solute equal to the ratio of their self diffusion coefficient. According to Bearman’s theory, friction factor between the components is inversely proportional to the self diffusion coefficient. Therefore, this model satisfied the Onsager relations. In case of Zielinski and Hanley [8], ORR lead to D1 /D2 = M2 /M1 and in case of Alsoy and Duda [2] it lead to D1 /D2 = (Vˆ 2 /Vˆ 1 )(M2 /M1 ). Therefore these relationships cannot be true universally. Conclusion of ORR are just mathematical and do not resemble the real behavior of the material. Price and Romdhane [7] checked the consistency of multicomponent diffusion models using Onsager Reciprocal Relations. They suggested to substitute values of ˛i , for Dabral [9] model, / N, for Zielinski and Hanley [8] model, ˛i = 1/Vˆ i , i = ˛i = 0, i = 1, . . .N, and for Alsoy and Duda [2] model, ˛i = 1, i = 1, . . . N. In generalized model ratio of self diffusion coefficients were set equal to ratio of friction factors, Dk /Dj = (˛j Vˆ j Mj )/(˛k Vˆ k Mk ) = ij /ik = / i and i, k = 1, . . . N − 1, gives the exactly same results as jj /jk , j = that of Zielinski and Alsoy [16] Onsager analysis. Zielinski and Alsoy [16] Onsager analysis get the results which is same as the assumptions made in model development. Therefore, all the models satisfy the Onsager reciprocal relations. D1 and D2 appearing in Table 1 are the self diffusion coefficients, which can be calculated using Vrentas and Duda [17], Vrentas and Duda [18] free volume theory

Vˆ FH /

⎠

(10)

i3 = (critical molar volume of a jumping unit of component i/ critical molar volume of the jumping unit of the polymer) = [(Vˆ i∗ Mji )/(Vˆ 3∗ Mj3 )] (Vrentas et al. [19]), and the hole free volume is given by: K11 Vˆ FH = ω1 (K 21 + T − Tg1 )   +

K13 K12 ω2 (K 22 + T − Tg2 ) + ω3 (K 23 + T − Tg3 )  

(11)

D0i is the pre-exponential factor for component i; ωj mass fraction of component j; Vˆ ∗ is the specific critical hole free volume of comj

ponent j required for a jump; Vˆ FH is the average hole free volume per gram of mixture;  is an overlap factor which is introduced because the same free volume is available to more than one molecule; Mji is the molecular weight of a jumping unit of component i; ((K11 /), K21 − Tg1 ) and ((K12 /), K22 − Tg2 ) are free volume parameters for solvent 1 and solvent 2 respectively; (K13 /), K23 − Tg3 are free volume parameters for polymer. Mutual diffusion coefficients in Table 1 also requires activity for the ternary polymer–solvent 1–solvent 2 system and can be calculated using Flory–Huggins theory [20]. Activity coefficient of solvent 1

 ln a1 = ln 1 + ln 1 = ln 1 +

1 − 1 −

V¯ 1 2 V¯ 2



R.K. Arya, C.K. Bhargava / Progress in Organic Coatings 78 (2015) 155–167

−

V¯ 1 3 + 13 32 + 12 22 + 2 3 V¯ 3



13 + 12 −

V¯ 1

23 V¯ 2

 (12)

ln a2 = ln 2 + ln 2 = ln 2



+

1−

V¯ 2 1 − 2 V¯ 1

V¯ 2 2 + 12 2 + 1 3 V¯ 1

 −



V¯ 2 3 + 23 32 V¯ 3

V¯ 2 V¯ 2

12 + 23 −

13 V¯ 1 V¯ 1

 (13)

 +

13

V¯ 3 V¯ 3 1 − 2 ¯ V1 V¯ 2

V¯ 3 V¯ 3 1 + 23 2 V¯ 1 V¯ 2



( 1 + 2 ) − 12

V¯ 3 1 2 V¯ 1

(14)

(15)

V¯ i is the partial molar volume of solvent i; ıi is the solubility parameter of solvent i; ıj is the solubility parameter of polymer j; and volume fraction is given by i = ci Vˆ i , where ci is the concentration of species i; Vˆ i is the partial specific volume of species i. Various multicomponent diffusion models in Table 1, for ternary polymer–solvent–solvent systems, require value of self diffusion coefficient which is in general is being calculated using Vrentas and Duda free volume theory as given by Eq. (10). Recently some modifications have been done in earlier free volume theories and discussed in the following section. Ohashi et al. [22] developed a new shell like free volume of diffusion of solvent in polymer based on microscopic concept of molecular collisions. The model equation is given by D1 = D0 exp −

(Vˆ 1∗ M1 )/NA



(16)

S1 (Vf,sys /(Ssys ))

where Ssys : total surface area of all molecules contained in the system, cm2 g−1 , S1 : molecular surface area of solvent cm2 g−1 , Vf,sys /: total free volume contained in the mixed system, cm3 /g, Vˆ ∗ : critical 1

volume of species i, cm3 /g, NA : Avogadro’s number, M1 : molecular weight of solvent, g mol−1 . The component free volume is calculated using Vrentas and Duda [17,18] theory as following Vf,i 

=

K1i (K2i − Tgi + T ) 



=

(17)

n  Vf,i

ωi

˛v0

vf

 

(1 + ωN )

(19)

the length; ω is mass fraction of solvent. Jiang and Han [25] have tested the prediction capabilities of Vrentas and Duda [17], Vrentas and Duda [18] free volume theory and conjunction with Simha–Somcynsky hole model to calculate the hole free volume. They found very good agreement between experiment and model predictions in case of several binary polymer–solvent systems. The recently developed modified free volume theory models have been tested only for few binary systems. These newly developed models require their extension for ternary systems and their validation with experimental data. The various parameters required for these new models are not yet documented properly. Therefore, Vrentas and Duda [17,18] free volume diffusion model is being used to calculate the self diffusion coefficients in the present work. 2.2. Boundary conditions at the free surface At the surface, both the solvents are evaporating into the gas/air. The solvent rate of evaporation per unit area is a product of the difference in the partial pressure of the solvent at the surface of coating and in the bulk of the nearby gas and mass transfer coefficient, which is the combined action of convective and diffusion mass transfer. The rate of evaporation is equal to sum of diffusive flux and convective flux at the surface of the coating. Since, volume average velocity is zero for this case. Hence, only flux is the diffusive flux. Flux of solvent 1 at free surface [10]

 −D11 =

where (K1i /) and (K2i − Tgi ) are the free volume parameters of component i. The total system free volume is calculated as Vf,sys

(18)

(/∗ ) − (/1∗ )

N is the chain length; f is mean free volume in polymer melt; ˛v0 = vc is critical volume activating diffusion;  is void distribution along

V¯ i 2 (ı − ıj ) RT i





/1∗

D01 and  are the fitting and need to regressed from the pure solvent data. In general,  should be 0.5 to 1. * is the density in Sanchez–Lacombe Theory and inverse of it is same as specific hole free volume as in the case of Vrentas and Duda free volume theory. Sabbagh and Eu [24] developed a modified free volume theory of diffusion of polymeric liquids based on integral equation theory for site–site pair correlation function and generic van der Waals equation of state. The solvent self diffusion coefficient is given by D = D0 exp −

where is the Flory–Huggins binary interaction parameter can be determined from the Bristow and Watson [21] semi-empirical equation given below,

ij = 0.35 +





Activity coefficient of polymer 3

ln a3 = ln 3 + (1 − 3 ) −

Sanchez–Lacombe thermodynamic theory. They have developed following equation to calculate the self diffusion coefficient of solvent D1 = D01 exp −

Activity coefficient of solvent 2

159



i=1

Costa and Storti [23] obtained a new formalism through which self-diffusion coefficients can be evaluated from pure solvent diffusivity data and easily accessible thermodynamic data using only one adjustable parameter by using a lattice theory: the

∂c1 ∂c2 − D12 ∂z ∂z

(1 − c1 V¯ 1 )k1G (pG 1i

   

z=L(t)

− pG ) − c1 V¯ 2 k2G (pG 1b 2i

− pG ) 2b

(20)

− pG ) 1b

(21)

Flux of solvent 2 at free surface [10]

 −D22 =

∂c2 ∂c1 − D21 ∂z ∂z

(1 − c2 V¯ 2 )k2G (pG 2i

   

z=L(t)

− pG ) − c2 V¯ 1 k1G (pG 2b 1i

160

R.K. Arya, C.K. Bhargava / Progress in Organic Coatings 78 (2015) 155–167

2.3. Boundary conditions at the base

By arranging all the terms, following equation is derived for change in the coating temperature.

Since, base is impermeable. Hence, there is no mass transfer through the base to the gas.



dT =− dt

hG (T − T G ) +

N−1 i=1

G H ˆ vi (pG − pG ) + hg (T − T g ) kgi ii ib p

p Cˆ p X(t) + s Cˆ ps H (27)

Flux of solvent 1 at the base [10]



∂c1 ∂c2 −D11 − D12 ∂z ∂z

   

hG =0

(22)

z=0

Flux of solvent 2 at the base [10]



∂c2 ∂c1 −D22 − D21 ∂z ∂z

   



hg

and are the heat transfer coefficients at the surface of the ˆ vi is the enthalpy of evapcoating and the base side respectively; H oration of solvent i; s is the density of substrate; p is the density of the coating, Cˆ p is the specific heat; superscripts: p, polymer; s, substrate.

3. Numerical analysis =0

(23)

z=0

2.4. Change in the coating thickness Since both the solvents are evaporating from the surface of the coating to the gas flowing parallel to the coating surface. Rate of change of mass per unit area per unit time will give the change in coating thickness [10] dL = −V¯ 1 k1G (pG − pG ) − V¯ 2 k2G (pG − pG ) 1i 1b 2i 2b dt

(24)

where L is the thickness of coating; k1G and k2G are the convective mass transfer coefficient of solvent 1 and solvent 2 respectively; V¯ 1 and V¯ 2 are the partial molar volume of solvent 1 and 2 respectively; and pG are the partial pressure of solvent 1 and 2 in bulk gas pG 1b 2b respectively; pG and pG are equilibrium partial pressure of solvent 1i 2i 1 and solvent 2 respectively, and can be calculated by vap

(25)

vap

(26)

p1i = P1 (T ) · 1 · 1 p2i = P2 (T ) · 2 · 2 vap

i is the volume fraction of species i; P1 is saturation vapor pressure of species i;  i is the activity constants for species i. 2.5. Energy transport The coating is heated by hot air blown on top and bottom. Because coating is very thin, the conductive resistance of the coating is negligible compared to convective resistance in the air. Hence, the coating temperature is assumed to be uniform throughout the thickness [2]. Detailed heat transport model of Price and Cairncross [26] showed a temperature variation of about 0.1 ◦ C. Temperature of coating and the substrate is assumed to be same. Radiation heat transfer is also neglected because temperatures are usually lower than 150 ◦ C. Heat supplied by the hot air from the top side per unit area: hG (TG − T) Heat supplied by the hot air from the bottom side per unit area: hg (Tg − T) Heat taken away from the coating by the solvents vapors per unit N−1 G ˆ vi (pG − pG ) area: k H i=1 gi ii ib

Eqs. (1) and (2) are partial differential equations, and Eqs. (24) and (27) are ordinary differential equations; they are coupled and non-linear. Together they model the mass and heat transport during the drying. Transport equations are coupled nonlinear partial differential equations. Galerkin’s finite element method converts them to ordinary partial differential equations (ODEs). Three point Gauss quadrature converts ordinary partial differential equation into system of ordinary differential equations. Arya [27] has given detailed finite element formulation for multicomponent diffusion in polymer–solvent–solvent system. ODEs were then integrated with time to determine concentrations as a function of time and distance and temperature. The coating thickness was divided into n elements, at all instants of time. Number of elements should be decided based on the error analysis between two different elements. If the cumulative absolute error associated between different elements is lower than or equal to 0.5% then lower number of elements are chosen to reduce the computation time and for same accuracy. Generally, the number of the elements is doubled for the elemental analysis, 10, 20, and 40 and so on. The elements were made non-uniform with their size rising gradually from the top to the bottom. The elements near the top were chosen to be small to capture the precipitous drop in concentration there. A benefit of using non-uniform elements is reduction in computation time. A function, ri = ((i − 1)/n)2 L where i varies from 1 to n + 1 stretched the elements from the top to the bottom of the coating. The size of element i can be obtained by ri+1 − ri . The exponent in the stretching function can be changed to raise or lower stretching. The set of ordinary differential equations generated that was integrated by a stiff solver, ode15s, of MATLAB. In this work 50 numbers of elements are chosen. A typical run on a 2.66 GHz computer with a memory of 506 MB takes about 20 s. Code was tested with the earlier published results of Alsoy and Duda [2]. 4. Results and discussion To test the accuracy of code, published results were compared to those from the code. For this study, polystyrene–toluene–tetrahydrofuran system has been chosen because for this system data is available for comparison. Alsoy and Duda [2] reported the simulation study for the same system. All the free volume parameters are given in Table 2 and experimental conditions and other parameters are given in Table 3. The generalized model requires the self diffusion coefficient which is taken as inverse of the molecular weight of the polymer as stated by Price and Romdhane [7]. The molecular weight of the poly (styrene) is taken as 230,000 for the current simulation study.

p

Heat accumulation within the coating per unit area: p Cˆ p X(t)dT Heat accumulation within the substrate per unit area: s Cˆ ps HdT

4.1. Validation of simulation code

Heat taken by the coating + substrate = Heat supplied by the gas − Heat taken by the solvents vapors.

A MATLAB code is written using Galerkin’s finite element numerical solution scheme discussion elsewhere [27]. This code

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Table 2 Free volume parameters (Alsoy and Duda [2]). Parameter

Unit

PS/toluene

PS/tetrahydrofuran

D0 K11 / K12 / K21 K22 Tg1 Tg2 Vˆ 1∗ Vˆ ∗

cm2 s−1 cm3 /g K cm3 /g K K K K K cm3 /g cm3 /g

4.82 × 10−4 0.000145 0.000582 −86.32 −327 0 0 0.917 0.85 0.58 0.354

14.4 × 10−4 0.00075 0.000582 10.45 −327 0 0 0.899 0.85 0.45 0.34

2



Fig. 3. Total residual solvent with time in poly (styrene)–toluene–tetrahydrofuran system.

Fig. 2. Change in temperature with time in poly (styrene)–toluene–tetrahydrofuran system.

the coating is lower than the heat consumed for evaporation. Fig. 3 shows the residual solvent as function of time. Most of the drying occurs within first 20 s. It is because of faster diffusion rate at higher concentration of the solvents. After 20 s, residual solvent in the coating is approximately 20%. The period during which temperature of the coating falls is called warm up period, then drying takes place at nearly constant rate during that period most of the drying takes place, after that drying rate falls with time and then drying stops. Fig. 4 shows that thickness of the coating shrinks very fast initially due to high rate of mass transfer. Shrinkage of coating thickness is only because of evaporation of the solvents from the surface of the coating that is controlled by the diffusion rate within the coating. Initially diffusion is faster because diffusive mass transfer is a strong function of the concentration, hence high rate in the beginning. 4.2. Concentration profiles

was tested against the earlier published data using some other solution techniques. Fig. 2 shows that temperature predicted by the code is in good agreement with published results. All the three cases of Alsoy and Duda [2] seem to produce same results. The temperature drop during the initial drying stages is due to the evaporative cooling. During these stages, the heat transferred to Table 3 Experimental parameters for poly (styrene)–toluene–tetrahydrofuran system (Alsoy and Duda [2]). Initial conditions Temperature Coating thickness Initial composition of solvent 1 Initial composition of solvent 2

303 K 0.00577 cm 0.321 cm3 /g 0.321 cm3 /g

Substrate parameters Heat capacity Density Base thickness

1.25 J/g K 1.37 cm3 /g 0.003556 cm

Coating parameters Heat Capacity Density of polymer Heat of evaporation of solvent 1 Heat of evaporation of solvent 2

2.12 J/g K 1.083 cm3 /g 360 J/g 435 J/g

Operating conditions Base side heat transfer coefficient, hg Coat side heat transfer coefficient, hG Bottom air supply temperature, Tg Top air supply temperature, TG Mass transfer coefficient of solvent 1 Mass transfer coefficient of solvent 2 Mole fraction of the solvent 1 in the air Mole fraction of the solvent 2 in the air

0.9228 × 10−3 W cm−2 K−1 10.944 × 10−4 W cm−2 K−1 333 K 333 K 1.85 × 10−9 s/cm 1.71 × 10−9 s/cm 0 0

Fig. 5a–c shows the concentration profiles of toluene, tetrahydrofuran, and poly (styrene) in poly (styrene)–toluene– tetrahydrofuran system for Alsoy and Duda [2] case 1. Initially poly (styrene) concentration is same through out the coating (see Fig. 5c). After 5 s, poly (styrene) concentration at the surface is lower than the concentration at the base. But the toluene and tetrahydrofuran (THF) concentration is higher compared to base (see Fig. 5a and b). Tetrahydrofuran concentration decreases very rapidly and in 20 s entire tetrahydrofuran is removed as shown in Fig. 5b. Volume

Fig. 4. Change in thickness of the (styrene)–toluene–tetrahydrofuran system.

coating

with

time

in

poly

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Fig. 5. (a) Toluene concentration profiles during the drying of poly (styrene)–toluene–tetrahydrofuran system for Alsoy and Duda case 1. (b) Toluene concentration profiles during the drying of poly (styrene)–toluene–tetrahydrofuran system for Alsoy and Duda case 1. (c) Poly (styrene) concentration profiles during the drying of poly (styrene)–toluene–tetrahydrofuran system for Alsoy and Duda case 1.

of coating solution decreases due to the evaporation of tetrahydrofuran. Concentration of low volatile toluene increases to maximum and begins to fall after 5 s as shown in Fig. 5a. In increase in toluene concentration is due to the high volatility and higher diffusivity of low molecular weight solvent tetrahydrofuran. Both the solvents add free volume to the polymer solution therefore increase in toluene concentration enhances the removal of tetrahydrofuran. Even though the overall mass of toluene decreases initially, its concentration in gram per unit volume goes through a maximum due to volume change of solution due the removal of tetrahydrofuran. Mass of the solvents drops as the drying takes places due to the evaporation of the solvents from the surface of the coating. Concentration of both the solvents is calculated using mass balance equations for solvent 1 (tetrahydrofuran) and solvent 2 (toluene). There is no equation to calculate the balancing component, poly (styrene), concentration separately. Concentration of balancing component, poly (styrene), was calculated by equating total volume fraction equal to one: polmer = 1 − solvent1 − solvent2 . Volume fraction of solvent in surface increases initially due to departure of high volatile tetrahydrofuran. Hence, the lower concentration of polymer in the surface as compared to bottom as shown in Fig. 5c. Initially evaporation rate is lower than the diffusion rate due evaporating cooling. Diffusion in polymeric solution is strong function of concentration and free volume available as compared to

temperature [17,18] however evaporation is strong function of temperature resulting higher concentrations of solvents at the surface in the beginning. In the later stages, drying is influenced by internal diffusion. In the beginning, tetrahydrofuran leaves the coating sooner than toluene because of higher volatility. This leads to an increase in the concentration of toluene; in fact, model predictions show that the increase is found at all locations in the coating. Later, when the concentration of tetrahydrofuran plateaus, the concentration of toluene falls everywhere in the coating. Fig. 5c shows that the concentration of poly (styrene) rises in the coating. This is because the solvents depart the coating and the polymer being highly non-volatile does not. As the coating temperature increases during the course of drying, the evaporation rate becomes higher than the diffusion rate within the coating, hence concentration becomes negligible at the surface compared to the layer beneath and that gives rise to develop steep concentration gradient at the surface as shown in Fig. 5a and c. Fig. 6a–c shows the comparative study for the different models. All the four models are predicting different profiles for poly (styrene), toluene, and tetrahydrofuran. Initially, all these models are predicting different profiles at the surface, and as drying stops virtually profiles are different at the base. Fig. 6a and c shows significant differences in concentration profiles predicted using different

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Fig. 6. (a) Toluene concentration profiles during the drying of poly (styrene)–toluene–tetrahydrofuran coating. (b) Tetrahydrofuran concentration profiles during the drying of poly (styrene)–toluene–tetrahydrofuran coating. (c) Poly (styrene) concentration profiles during the drying of poly (styrene)–toluene–tetrahydrofuran coating. Alsoy and Duda case 1 (–), Alsoy and Duda case 2 (– – –), Alsoy and Duda case 3 (· · · · · ·), Alsoy and Duda case 4 (– · –), Zielinski and Hanley (), and Generalized model ().

models for toluene and poly (styrene) respectively. However, all models are predicting nearly same profiles for high volatile solvent: tetrahydrofuran as shown in Fig. 6b. Zgadzai and Makcakov [28] have studied diffusion in poly (styrene)–ethylbenzene system using NMR field gradient spin echo technique. They have compared the experimental diffusion coefficient values with the free volume theory model predicted values. They have found the free volume model predictions are not in agreement with experimental values. They suggested that self diffusion activation energy cannot assumed negligible in less volatile solvent like poly (styrene)–ethylbenzene. They found that self diffusion activation energy decreases significantly in solvent weight fraction interval of 0.1–0.6. In the present work, free volume models are not able to predict concentration profiles in poly (styrene)–toluene system. However, free volume predictions in more volatile solvent systems are in good agreement. Unfortunately, self diffusion activation energy data are not available for polymer–solvent systems. Similar results have been observed by Arya and Vinjamur [12] in case of poly(styrene)–tetrahydrofuran–p-xylene and poly(methyl methacrylate)–tetrahydrofuran–ethylbenzene systems using confocal Raman spectroscopy. They have found that none of multicomponent diffusion models are able to capture complete drying behavior of polymer–solvent–solvent systems. Models predicted values are in good agreement with experimental depth profile data for more volatile solvent: tetrahydrofuran. However

model predicted values were always higher in case of less volatile solvents (p-xylene and ethylbenzene). It means the actual diffusion process is much faster than the model predicted one. Arya and Vinjamur [29] performed the sensitivity analysis of free volume theory parameters in polymer–solvent–solvent systems. They found that model predictions are highly sensitivity to thermodynamic parameters ( 13 and  23 ). Therefore these parameters need to be estimated using great care while performing regression analysis. Mutual diffusion coefficient in polymeric coating is calculated using Vrentas and Duda [17], Vrentas and Duda [18] free volume theory. Theory needs several free volume parameters out of which few parameters are not the pure component properties. These parameters should be estimated using regression analysis. Slight change in these parameters could give unrealistic results. In the present work all the free volume parameters were change to ±50%. Model predicted profile changes for each parameter change. However, change in preexponential factor D01 from 4.82 × 10−4 cm2 s−1 to 4.72 × 10−4 cm2 s−1 , gives unrealistic results. Poly (styrene) concentration near the surface comes out to be negative for Alsoy and Duda [2] case 4 that is not theoretically acceptable as shown in Fig. 7a. The concentrations of toluene were: −2.037 × 10−2 g cm−3 and − 5.004 × 10−3 g cm−3 and at 6 s and 10 s respectively. But for the Alsoy and Duda case 1, case 2, case 3, Zielinski and Hanley [8], and Price and Romdhane [7] solutions are still in theoretical agreement as shown in Fig. 7a–c however

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Fig. 7. (a) Toluene concentration profiles for the drying of poly (styrene)–toluene–tetrahydrofuran coating, D01 = 4.72 × 10−4 cm2 s−1 all other conditions are in Tables 2 and 3. (b) Tetrahydrofuran concentration profiles for the drying of poly (styrene)–toluene–tetrahydrofuran coating, D01 = 4.72 × 10−4 cm2 s−1 all other conditions are in Tables 2 and 3. (c) Poly (styrene) concentration profile for the drying of poly (styrene)–toluene–tetrahydrofuran coating, D01 = 4.72 × 10−4 cm2 s−1 , all other conditions are in Tables 2 and 3. Alsoy and Duda case 1 (–), Alsoy and Duda case 2 (– – –), Alsoy and Duda case 3 (· · · · · ·), Alsoy and Duda case 4 (– · –), Zielinski and Hanley (), and Generalized model ().

exact value of polymer self diffusion coefficient (D03 ) is not known. It is just taken as inverse of molecular weight of poly(styrene) (D03 = 1/230, 000 = 4.3478 × 10−6 cm2 s−1 ). This shows the uniqueness of the generalized model. The unrealistic results for Alsoy and Duda case 2 shows that one should take extreme care during the calculation of preexponential factor form the regression analysis of binary experiments. Fig. 7a shows the importance of preexponential factors (D01 , D02 , and D03 ) which are not the pure component parameters. They have to be estimated from binary drying data by regression analysis. There can be several regressed values but all of them cannot be used for the prediction in ternary coatings. Slight change in these values will lead to negative concentration profiles. Nauman and Savoca [30] show that computed concentration of balancing component in multicomponent diffusion calculation could be negative under some circumstances for Zielinski and Hanley [8] and Alsoy and Duda [2] models if values of self diffusion coefficient are kept constant. Price and Romdhane [7] have also discussed the similar problem. They said that these multicomponent diffusion models are derived by making assumptions that Dk /Dj = Mj /Mk . Violation of material balance constraint may occur whenever ratio of self diffusion coefficient is not equal the inverse ratio of molecular weight. There is no constraint of dependency of self diffusion coefficient on the concentration. Calculation of mutual diffusion coefficient is related to self diffusion coefficient and Flory–Huggins thermodynamics terms. Therefore assumption

of constant mutual diffusion coefficient could lead to material violation constraint. They have made several combinations of self diffusion coefficient and molecular weight and found that violation of material balance constraint may occur only when the ratio of self diffusion coefficient is not inversely proportional to the ratio of their molecular weights. Arya and Vinjamur [12] have not mentioned ratio of self diffusion coefficient and molecular weight while comparing experimental data with the model predicted values. They said that results are in good agreement up to 500 s of drying of ternary coating. In their work, most of highly volatile solvent (tetrahydrofuran) leaves the coating in first 500 s. After that coating behaves just like a binary coating and ratio of self diffusion coefficient will change significantly however ratio of molecular weight will remain constant since self diffusion coefficient is strong function of weight fractions of the solvents and polymer. It could be one of the reasons for differences in measured and experimental depth profile study performed by them. 4.3. Skinning phenomena in drying coating Temperature of the oven, composition of the coating, thickness of the coating, and air flow are only parameters that can be adjusted to dry a coating to meet required solvent specification without generating any defects such as blisters. Fig. 8 shows that at any given air flow or oven temperature, residual solvent plateaus off during

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Fig. 8. Residual solvent tetrahydrofuran system.

remaining

with

time

in

poly(styrene)–toluene–

later stages of drying. This is because of low drying rate due to low diffusion coefficients at low solvent concentrations. The residual solvent does not change for practical purposes, this is called figurative skinning. Another kind of skinning is literal skinning where a solid layer can be peeled off from the top surface of the coating. Yet another

165

kind is trapping skinning where drying under higher flow rates raises residual solvent [31]. It occurs because of increasing diffusional resistance to mass transfer in the polymer phase due to strong influence of the solvent concentration on the mutual binary diffusion coefficient [32]. Skinning may results because of high rate of evaporation at the surface compared to diffusion rate within the coating [1]. Vinjamur and Cairncross [31] studied and found that the solvent remains inside the coating due to a sharp decrease in the diffusion coefficient at the coating surface due to low solvent concentrations. In the production of polymer coatings, skinning is undesirable because it contributes to a lower drying rate, cracking at the surface, and bubble formation inside the coatings. Literal skinning occurs due to solidification of the top layer while the deeper coating is still liquid. Figurative skinning occurs when diffusion or internal resistance controls drying. Trapping skinning is an anomalous drying behavior where residual solvent increases with an increase in the drying gas flow rate and temperature. Fick’s law of diffusion, which states that the mass flux equals the product of the diffusion coefficient and concentration gradient; cannot predict trapping skinning. Glass transition, phase transformation, and reactions at the surface are some of the possible mechanisms for trapping skinning. Edwards [32] employed a stepwise diffusion coefficient and relaxation time in an isothermal model and showed that trapping skinning occurs at higher mass-transfer coefficients.

Fig. 9. (a) Toluene concentration profiles with time. (b) Tetrahydrofuran concentration profiles with time. (c) Poly (styrene) concentration profiles with time. For low heat transfer coefficient (8.36 × 10−4 W cm−2 K−1 ) and all other data are given in Tables 2 and 3.

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Fig. 10. (a) Toluene concentration profiles with time. For high heat transfer coefficient (1.5 × 10−2 W cm−2 K−1 ) and all other data are given in Tables 2 and 3. (b) Tetrahydrofuran concentration profiles with time. For high heat transfer coefficient (1.5 × 10−2 W cm−2 K−1 ) and all other data are given in Tables 2 and 3. (c) Poly (styrene) concentration profiles with time. For high heat transfer coefficient (1.5 × 10−2 W cm−2 K−1 ) and all other data are given in Tables 2 and 3.

Fig. 8 shows the residual solvent profiles at low and high values of heat transfer coefficients. All the operating conditions and the free volume parameters are given in Tables 2 and 3 except initial concentration of the toluene and tetrahydrofuran which is taken to 0.2643 g cm−3 respectively and mass transfer coefficients are calculated for the two different heat transfer coefficients using Chilton–Colburn analogy for heat and mass transfer. In skinning study, the heat transfer coefficients were set equal on both sides of the coatings. In the first case, they were set to the lowest possible value of heat transfer coefficient (8.36 × 10−4 W cm−2 K−1 ) and in another case they were set to the highest possible value of heat transfer coefficient (1.5 × 10−2 W cm−2 K−1 ). Residual solvent in a coating dried at high air flow (high heat transfer coefficient) is more compared to that at low heat transfer coefficient as shown in Fig. 8. When coating is dried at high air flow, step concentration gradient develops for low volatile solvent at the surface sooner than it does at low air flow (see Figs. 9 and 10). Fig. 9a–c shows the concentration profiles at lowest air flow condition. In low air flow condition, coating is dried at very slow rate and the steep concentration gradient are developed after very long drying time as shown in Fig. 9a and c. Hence, large amount of solvent is moved from the coating and less amount of solvent is remained in the coating. The drying rate plummets by several orders of magnitude as the steep gradient develops and therefore, rate of removal of solvents slow down significantly. However, steep gradients are

developed earlier when coating is dried at high air flows and drying shuts off early as shown in Fig. 10a–c. Hence, high amount of residual solvent remained in the coating. To get the skinning behavior of the thin film polymer coating, experiments should be performed using thermo gravimetric apparatus having high accuracy.

5. Conclusions Total residual solvent remaining and the concentration profiles are different for all multicomponent diffusion models. Free volume diffusion models need several free volume parameters for mutual diffusion coefficient calculation. Few of the parameters need to estimated using regression analysis of binary weight loss data like D01 and D02 . Slight change in preexponential parameters D01 and D02 will lead to negative concentration profiles. Hence, these parameters need to be estimated using great care. Self diffusion coefficients should be changed in such a way that ratio of self diffusion coefficient should always be equal to inverse ratio of molar volume otherwise material balance violation may occur in case of Alsoy and Duda [2] model and Zielinski and Hanley [8] model. At higher air flows, skinning phenomena may occur which will cause defects in the coating like phase separation, nonhomogenous composition and blister formation.

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Acknowledgements Authors are very much thankful to Prof. Madhu Vinjamur, Department of Chemical Engineering, IIT Bombay, India, for his support, guidance and thoughtful discussion. Authors are also thankful to Prof. N.N. Dutta, Department of Chemical Engineering, Jaypee University of Engineering & Technology, Guna, India, for his comments and suggestions. Authors are also thankful to Prof. Hari Mahalingam, Head of Chemical Engineering, Jaypee University of Engineering & Technology, Guna, India, for his continuous support and guidance. References [1] G. Wypych, Handbook of Solvents, ChemTec, Toronto, 2001, pp. 386–389. [2] S. Alsoy, J.L. Duda, Modeling of multicomponent drying of polymer films, AIChE J. 45 (4) (1999) 896–905. [3] R.A. Yapel, The Physical Model of Drying of Coated Films (M.S. thesis), University of Minnesota, Minneapolis, USA, 1988. [4] S. Alsoy, Predicting drying in multiple-zone ovens, Ind. Eng. Chem. Res. 40 (14) (2001) 2995–3001. [5] S. Alsoy, J.L. Duda, Drying of solvent coated polymer films, Drying Technol. 16 (1–2) (1998) 15–44. [6] W. Schabel, P. Scharfer, M. Mueller, I. Ludwig, M. Kind, Measurement and simulation of concentration profiles in the drying of binary polymer solutions, Chem. Ing. Tech. 75 (9) (2003) 1336–1344. [7] P.E. Price Jr., I.H. Romdhane, Multicomponent diffusion theory and its application to polymer–solvent systems, AIChE J. 49 (2) (2003) 309–322. [8] J.M. Zielinski, B.F. Hanley, Practical friction-based approach to modeling multicomponent diffusion, AIChE J. 45 (1) (1999) 1–12. [9] M. Dabral, Solidification of coatings: theory and modeling of drying, curing and microstructure growth (Ph.D. thesis), Univ. of Minnesota, Minneapolis, MN, USA, 1999. [10] M. Dabral, L.F. Francis, L.E. Scriven, Drying process paths of ternary polymer solution coating, AIChE J. 48 (1) (2002) 25–37. [11] W. Schabel, P. Scharfer, M. Kind, I. Mamaliga, Sorption and diffusion measurements in ternary polymer–solvent–solvent systems by means of a magnetic suspension balance–experimental methods and correlation with a modified Flory–Huggins and free-volume theory, Chem. Eng. Sci. 62 (2007) 2254– 2266. [12] R.K. Arya, M. Vinjamur, Measurement of concentration profiles using confocal Raman spectroscopy in multicomponent polymeric coatings-model validation, J. Appl. Polym. Sci. 126 (6) (2013) 3906–3918, http://dx.doi.org/10.1002/ app.38589. [13] R.A. Cairncross, Solidification Phenomena during Drying of Sol-to-Gel Coatings (Ph.D. thesis), University of Minnesota, Minneapolis, 1994.

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