Drying of Glassy Polymer Films - ACS Publications - American

Apr 20, 2001 - ... Hadj Romdhane,* Peter E. Price, Jr., Craig A. Miller, Peter T. Benson, and ... 3M Center, Building 518-1-01, St. Paul, Minnesota 55...
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Ind. Eng. Chem. Res. 2001, 40, 3065-3075

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Drying of Glassy Polymer Films Ilyess Hadj Romdhane,* Peter E. Price, Jr., Craig A. Miller, Peter T. Benson, and Sharon Wang 3M Engineering Systems Technology Center, 3M Center, Building 518-1-01, St. Paul, Minnesota 55144-1000

During the drying of many polymer films, temperature and compositional changes induce a rubbery-glassy transition at the film surface. The transition front propagates toward the substrate as drying proceeds, eventually rendering the entire coating glassy. The glass transition induces structural changes that have a direct effect on the drying rate. We examine this process by measuring gravimetric and thermal drying profiles for poly(methyl methacrylate) films cast from a toluene-based formulation. The experimental conditions were chosen so that the films would pass through a glass transition. We seek an understanding of the observed behavior by comparing the experimental data to solvent concentration and temperature profiles computed with a mathematical model based on the work of Vrentas and Vrentas.1 The computational results show that the critical factors for capturing the drying behavior of this rubbery-glassy system are (i) the nonideal volumetric behavior exhibited by glassy polymer-solvent systems and (ii) the type of friction-based theory used to relate mutual- and self-diffusion coefficients. The model offers a quantitatively accurate alternative to the many viscoelastic-diffusion-based models that have appeared in the literature. Introduction Glassy polymers have abundant applications in the coating industry, including uses as primer, protective, and barrier coatings. They can be applied from waterbased, latex, and solvent-based formulations. Examples of some common amorphous glassy polymers include polymethyl methacrylate (PMMA), polystyrene (PS), and polyvinyl acetate (PVAc). The room-temperature glassy state of these polymers provides their functional properties, but they must be coated in the rubbery state. They become rubbery when they are heated above their glass-transition temperatures and/or plasticized with solvents or other additives. In many applications, glassy films are initially coated from solvent solutions and then dried in convective ovens. The choice of drying conditions (air temperature, air flow rate, and solvent humidity) can therefore lead to rubbery-glassy transitions as solvent is removed. A cartoon depicting the coating structure evolution at different stages of a typical drying process is shown in Figure 1. The coating is initially rubbery because the polymer is uniformly dissolved in sufficient solvent (∼20% solids). If the processing temperatures are sufficiently low, then during the initial stages of drying, as concentration gradients develop within the film, a rubbery-glassy transition will begin at the surface of the coating, forming a glassy skin. As drying proceeds, part of the coating will be in the rubbery state, and part of it will be in the glassy state. When sufficient solvent has been removed, the whole coating turns glassy. In some cases, operating conditions can be selected to maintain the coating in the rubbery state during the drying process. An obvious scenario is to select oven temperatures so that the coating remains above the glass-transition temperature of the pure polymer. Be* Author to whom correspondence should be addressed. Tel.: (651) 733-6163. Fax: (651) 736-3122. E-mail: [email protected].

cause of the high glass-transition temperature of some polymers, however, such conditions might actually boil the solvent and blister the coating. Another scenario is to humidify the atmosphere above the coating with solvent vapor. High solvent humidity can keep the coating surface sufficiently plasticized that it remains rubbery. In practical applications, however, the solvent humidity would eventually need to be reduced, as in a multiple-zone oven, to reach target residual specifications. During this latter stage, the coating could still undergo a rubbery-glassy transition and develop structural gradients across the film thickness. These process limitations suggest that any attempt to optimize drying of glassy polymer films requires fundamental understanding of rubbery-glassy transitions and their effects on the drying rate and coating structure. In our laboratory, we are often asked to optimize drying conditions for glassy coatings. We have therefore sought to develop a model capable of predicting the drying behavior and coating structure evolution of solvent-based coatings that pass through glass transitions. Several models exist in the literature to describe the drying behavior of rubbery polymers,2-4 viscoelastic polymer coatings,5 and semicrystalline glassy polymers.6 The drying model proposed in this paper differs from these models for the following reason: In rubbery coatings, it is reasonable to assume no volume change on mixing, and the modeling analysis is greatly simplified by using the volume-average velocity. In glassy polymer solutions, however, addition or removal of solvent leads to structural rearrangements in the polymer matrix as it gradually moves toward a denser equilibrium liquid configuration during sorption or toward an unrelaxed glassy state during desorption/ drying.1,7,8 These structural changes result in nonideal volumetric behavior, i.e., volume change on mixing. The proposed model includes these nonideal volumetric effects by allowing for a concentration dependence of the partial specific volumes and diffusion-induced convection. It also takes into account the effects of plasticiza-

10.1021/ie001110h CCC: $20.00 © 2001 American Chemical Society Published on Web 04/12/2001

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Figure 1. Coating structure evolution during the drying process.

nied by structural rearrangements that result in nonideal volumetric behavior. Because the partial specific volumes become concentration-dependent in the glassy coating, the species continuity equation for the solvent should be written as1

[ ]

∂F1 ∂F1 ∂(F1υq) ∂ + ) D ∂t ∂x ∂x ∂x

(1)

where the velocity field is represented by

∂υq D )- 3 ∂x F V ˆ

2

2

Figure 2. Schematic representation of the model system.

tion on volumetric, solubility, and diffusion behavior.7-11 As will be demonstrated in this paper, the inclusion of nonideal volumetric behavior is an important component of an accurate model. There have also been a number of recent publications on the relationship between self- and mutual-diffusion coefficients in rubbery systems.12-14 Investigation of these results with our model shows that the choice of diffusion model also has a significant impact on the accuracy of the model predictions. Model Formulation The model system, shown in Figure 2, consists of a coated substrate in a continuous or batch convective dryer. The operating parameters on the top side of the coating are the upper air temperature, TG; the upper heat transfer coefficient, hG; and the solvent partial G . The operating parameters on pressure in the bulk, p1b the bottom side of the impermeable substrate are the lower air temperature, Tg, and the lower heat transfer coefficient, hg. The coating is composed of a solvent (component 1) and a polymer (component 2). At some point during the drying process, the coating might contain both rubbery and glassy regions. Under such conditions, the diffusion Deborah number can change from very low to very high values across the coating thickness. It seems reasonable to envision a viscoelastic, non-Fickian diffusion process at the rubber-glass interface. In our model formulation, however, we ignore viscoelastic effects because only a small part of the sample will have a Deborah number near unity. Thus, in our model, solvent transport through the coating is governed by Fickian diffusion processes: elastic in the glassy part of the coating, where the temperature is below the glass-transition temperature of the polymer-solvent system, Tgm, and viscous in the rubbery part of the coating, where the temperature is above Tgm. As solvent is removed from the coating, there is a rubbery-glassy transition. This transition is accompa-

( ) ∂F1 ∂x

2

∂2 V ˆ ∂ω12

(2)

Equations 1 and 2 also apply in the rubbery region of the coating. In the rubbery region, however, we assume that there is no volume change of mixing, so the local volume-average velocity gradient is zero. In the cases considered here, the rubbery region always extends upward from the base of the coating, where the volumeaverage velocity is, by definition, zero. Thus, in the rubbery region, the diffusion-induced convection term in the species continuity equation disappears. A jump mass balance at the coating-air interface, x ) X(t), gives the following boundary condition

F1υq - D

∂F1 dX G G - F1 ) kG 1 [p1i - p1b] ∂x dt

(3)

where the time dependence of the moving boundary position is given by

dX G ) υq - kG ˆ 1[p1i - pG 1 V 1b] dt

(4)

At the coating-substrate interface, there is no solvent flux, and the volume-average velocity vanishes

∂F1 ) 0 and υq(0,t) ) 0 ∂x

(5)

The initial solvent concentration in the coating is assumed to be uniform

F1(x,0) ) F10

(6)

The drying of coated webs is generally a nonisothermal process. In this work, we treat the experimental coating temperature data as input to the drying model, assuming negligible temperature gradients within the coating. This procedure circumvents the need to determine upper and lower heat transfer coefficients and requires only solution of the mass transfer problem. Equations 1-6 are converted to a set of ordinary differential equations by using Galerkin’s method with finite-element basis functions. The resultant systems of

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Figure 3. Concentration dependence of the specific volume for PMMA-toluene at 25 °C. The dashed line represents the rubbery state of the system.

equations are integrated in time using the differentialalgebraic system solver DASSL.4 Evaluation of Material Properties Computation of drying profiles requires evaluation of the volumetric, sorption, and diffusion behavior of the polymer-solvent system being modeled. Because these properties depend on the local state of the system (rubbery or glassy), the model first evaluates the glasstransition temperature at any point in the coating according to the linear approximation15

Tgm ) Tg2 - Aω1

(7)

where the value of the coefficient A depends on the nature of the solvent used to depress the glass-transition temperature of the polymer. The state of the coating is determined by whether the coating temperature is above or below Tgm, and the properties in the rubbery or glassy domains can be evaluated as described below. To help illustrate the concept of structural changes in the polymer matrix upon solvent removal, we show the behavior of PMMA-toluene polymer-solvent system at 25 °C throughout this section. The values of the parameters obtained for this system are detailed in the Results and Discussion section. Volumetric Behavior. Based on the work of Vrentas and Vrentas,7,8 the specific volume of the coating is calculated according to the following expressions

ˆ °1 + ω2 V ˆ °2[1 + k1(R2g - R2)(T - Tgm)] V ˆ ) ω1V T < Tgm (8) V ) ω1V ˆ °1 + ω2 V ˆ °2

concentrations below ω1 ) ω1E, the polymer solution is glassy, and its specific volume (solid line) is higher than that of the equilibrium liquid (dashed line). At the pure polymer limit, ω1 ) 0, the specific volume of the polymer is greater than its equilibrium value by about 3%. This means that solvent removal leads to a gradual expansion of the polymer matrix as it moves from its equilibrium liquid configuration (ω1 g ω1E) toward an unrelaxed glassy state. This structural change causes departures from volume additivity, and the partial specific volumes of the solvent and polymer become concentration-dependent. In the model, the following relations are used to calculate the partial specific volumes of the solvent and polymer:

V ˆ 1 ) (1 - ω1)

where k1 is a parameter that takes into account the coating preparation history. In this work, we measured the density of some of our dried samples and determined that a value of k1 ) 1 best describes the volumetric behavior of our glassy coatings. Because the densities of glassy polymers are often reported at their glass-transition temperatures, the following equation is used to determine the relaxed, or equilibrium liquid volume of the polymer at temperatures below Tg2:

(9)

Figure 3 shows the concentration dependence of the specific volume of PMMA-toluene at 25 °C. For this system, the solvent weight fraction at which Tgm becomes equal to 25 °C (ω1 ) ω1E) is 0.189. At solvent

∂V ˆ +V ˆ ∂ω1

(10)

ˆ 1 + F2V ˆ2 ) 1 F1 V

(11)

VLE Behavior. One of the assumptions imposed in the model is that interfacial equilibrium is established instantly at the coating surface. To describe the vaporliquid equilibrium at the coating-air surface, we use the Flory-Huggins theory above Tgm and an extension of this theory to temperatures below Tgm suggested by Vrentas and Vrentas7,16 G p1i ) φ1 exp[φ2 + χφ22]eF p°1

(12)

where

F)

T g Tgm

V ˆ °2(T) ) V ˆ °2(Tg2)[1 + R2(T - Tg2)]

Figure 4. Sorption isotherm for PMMA-toluene at 25 °C. The dashed line represents the rubbery state of the system.

(

)

M1 ω22(cp,2 - cpg,2)A T -1 RT Tgm

F)0

T < Tgm (13) T g Tgm

and the volume fractions are defined as

φ1 )

ω1V ˆ °1 ω1V ˆ °1 + ω2V ˆ °2

(14)

φ2 )

ω2V ˆ °2 ω1V ˆ °1 + ω2V ˆ °2

(15)

where it is assumed that the lattice is formed by using the equilibrium liquid volume of the polymer. In eq 12, the pure-component vapor pressure is evaluated using the Antoine equation. Figure 4 shows the concentration dependence of the G /p°1) for PMMA-toluene at 25 °C. solvent activity (p1i

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eq 19 can be valid, Vrentas and Vrentas14 later proposed the expression

Case 2:

[

]

M1 V ˆ °1 D1 (1 - φ1)(1 - 2χφ1) + φ1 M2 V ˆ °20 D) (20) M V ˆ °1 D1 2 1 φ1 + (1 - φ1)(1 + 2φ1) M2 V ˆ °20 D2 φ1)1

Figure 5. Concentration dependence of the self-diffusion coefficient for PMMA-toluene at 25 °C. The dashed line represents the rubbery state of the system.

The dashed line represents the predicted activity if the system were in the rubbery state throughout the entire concentration interval. At solvent concentrations below ω1E, however, the system is in the glassy state, and the theory (solid line) predicts enhanced solubility or reduced activity because of structural changes in the polymer matrix. Good quantitative predictions were obtained by Vrentas and Vrentas in describing the sorption isotherms of polycarbonate-carbon dioxide7,8 and PMMA-water.17 Self-Diffusion. The solvent self-diffusion coefficient is evaluated on the basis of the free-volume theory of diffusion in polymer-solvent systems18,19

[ ] [

D1 ) D0 exp -

]

ω1V ˆ /1 + ω2ξV ˆ /2 E exp RT V ˆ FH/γ

(16)

where the hole-free volume available for transport depends on whether the system is glassy or rubbery, as described in the original work of Vrentas and Vrentas9

V ˆ FH/γ ) ω2V ˆ °2(Tg2)(1 - Aω1R2)(T - Tg2 + Aω1)(R2g R2)/γ2 + |V ˆ FH/γ|TgTgm T < Tgm V ˆ FH/γ ) ω1K11/γ1(K21 - Tg1 + T) + ω2K12/γ2(K22 Tg2 + T) T g Tgm (17) where γ2 is given by

γ2 )

V ˆ °2(Tg2)R2 K12/γ2

(18)

Figure 5 illustrates the concentration dependence of the self-diffusion coefficient of toluene in PMMA at 25 °C. The dashed line represents the predicted selfdiffusion coefficients if the system were in the rubbery state over the entire concentration interval. Below ω1E, however, there is extra hole-free volume trapped in the polymer matrix, and the availability of this free volume for transport enhances the solvent diffusivity, as indicated by the solid line in Figure 5. Mutual Diffusion. The literature contains a variety of friction-coefficient-based theories that link mutualand self-diffusion coefficients above Tgm. Vrentas and Duda18 suggested the first case for low solvent concentrations

Case 1:

D ) D1(1 - φ1)2(1 - 2χφ1)

(19)

Recognizing the limited concentration range over which

()

where the ratio (D1/D2) at φ1 ) 1 is evaluated using Hickey’s20 correlation. Recently, Alsoy and Duda13 and Zielinski and Hanley12 proposed new forms of the friction-based theory for multicomponent diffusion. Their models lead to the following expressions for binary polymer-solvent systems, respectively:

Case 3:

D ) D1(1 - φ1)(1-2χφ1)

Case 4:

ˆ °1 - V ˆ °2)](1 - φ1)(1 - 2χφ1) D ) D1[1- F1(V

(21)

(23) It is interesting to note here that, at φ1 ) 1, case 2 shows that D becomes equal to the self-diffusion coefficient of the polymer, whereas cases 1, 3, and 4, in general, do not. This failure does not interfere with the drying model calculations, however, because the initial solvent weight fraction in a coating solution is always less than 1. Evaluation of the binary mutual-diffusion coefficient below Tgm is carried out according to the linear approximation proposed by Vrentas and Vrentas11

[

]

ω1 D D )1+ (ω )ω1E) - 1 D1 ω1E D1 1

(24)

where D/D1(ω1)ω1E) is calculated based on the theory above Tgm and ω1E is the solvent weight fraction at which T ) Tgm. Figure 6 shows the concentration dependence of the mutual-diffusion coefficient above and below Tgm for PMMA-toluene at 25 °C. The numbers on each curve refer to the different cases documented in this section. In all cases, the onset of rubbery-glassy transition is characterized by a break at ω1E. At low solvent concentrations, all cases yield similar mutual-diffusion coefficient values. At higher solvent concentrations, however, significantly different values of D are obtained depending on the choice of friction-based theory. From a drying standpoint, we expect that such differences will result in different predictions of the drying rate or residual profiles, especially if drying becomes diffusionlimited early in the process. We would expect significant differences in the model predictions even if the coatings remained rubbery. Certainly, a definitive evaluation of the proposed diffusion theories should be based on a direct comparison between mutual- and self-diffusion data. Because of the scarcity of mutual-diffusion data at high solvent concentrations, however, it is difficult to discern which theory best links D to D1. We hope to shed some light on this link by comparing drying predictions made using the various diffusion models to gravimetric drying data. Experimental Details Drying Apparatus. Figure 7 shows a drawing of the batch convective drying apparatus used in this work.

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Figure 6. Concentration dependence of the mutual-diffusion coefficient for PMMA-toluene at 25 °C. The numbers on each curve refer to the mutual-diffusion theories, cases 1-4.

Figure 7. Batch convective drying apparatus.

Compressed air is fed at a controlled flow rate to an online heater connected to a temperature controller. The controller uses a thermocouple to monitor the air temperature in the drying chamber upstream of the sample. The heated air flows through a 2-ft-long drying chamber. The flow chamber is constructed of rectangular aluminum tubing and has inner dimensions of 3.25 × 1.5 in. Approximately halfway down the length of the chamber, a 1/4-in. thick aluminum sheet (3 × 5.5 in.) rests on a top-loading Ohaus balance by means of a metal shaft that passes through a small hole in the bottom wall of the chamber. The aluminum sheet acts as a platform for the coating sample and is fixed at the midplane of the chamber. A hole in the platform allows flush mounting of a 6-cm-diameter steel sample pan. A thermocouple sensor monitors the coating temperature during a drying experiment. A removable glass window on the upper wall of the chamber is positioned right above the platform. This window allows easy introduction of and access to the sample, as well as visualization of the drying film. The Ohaus balance consists of a scale accurate to 0.01 g and is connected to a personal computer through an RS-232 interface. Experimental Procedure. In a typical experiment, the air supply is adjusted to the desired temperature and flow rate. The sample is introduced as follows: First, the glass window is removed, and the pre-tared sample pan and thermocouple sensor are placed on the platform. With the airflow momentarily switched off, approximately 5 g of the coating solution is quickly poured onto the pan. The thermocouple is then immersed in the coating, the glass window is repositioned, and the air flow is switched back on. During this sample

introduction, both sample weight and temperature are recorded in order to pinpoint the initial conditions of the coating. The information gathered in each drying experiment consists of time, air temperature, sample weight, and sample temperature. The air velocity in the drying chamber was occasionally measured with an anemometer placed at the end of the tunnel. Results and Discussion Experimental Results. General Observations. We measured drying data for a coating solution of PMMA (Aldrich Chemical; 18,223-0) in toluene (Aldrich Chemical; 27,037-7) at 19% solids. Drying experiments were conducted at air temperatures of 25 and 50 °C with air velocities ranging from 20 to 200 fpm. The results of each experimental run consist of sample weight and temperature data as a function of time. Figure 8 shows typical sample weight and temperature profiles obtained at 25 °C. The coating temperature profiles in this case are characterized by an initial period during which evaporative cooling exceeds the energy supply rate of the air stream. The time scale of this period varies inversely with the air flow rate. In all of our experiments, the drying becomes diffusion-limited early in the process, as indicated by the gradual decrease in the rate of weight loss from the samples. This phenomenon was expected in light of the thickness of our samples and the low solvent diffusivities at the operating temperatures. At 50 °C, evaporative cooling effects were less dramatic, but the trends in gravimetric data were analogous. Anomalous Behavior. By conducting drying experiments at different air flow rates, we had hoped to

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Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 Table 1. Model Parameters for PMMA-Toluene Fs (g/cm3) κs [cal/(s cm K)]

V ˆ 2°(Tg2) (cm3/g) κ2 [cal/(s cm K)] Tg2 (K) R2g - R2 (K-1) V ˆ 1° (cm3/g) κ1 [cal/(s cm K)] ∆Hvap,1 (cal/g) ba Figure 8. Experimental sample weight and temperature profiles for PMMA-toluene at 25 °C. The numbers on each curve represent the air velocity in feet per minute.

determine whether any anomalous behavior occurs during the drying of PMMA-toluene solutions. One particular anomaly that has long been a controversial issue is known as trapping skinning. Cairncross21 defines trapping skinning as occurring when more severe drying conditions (higher air flow or lower gas saturation) lead to higher final solvent residuals. Our drying data for samples with equivalent initial weights but different air flows showed no obvious signs of trapping skinning, at least within the accuracy of the scale (0.01 g) and the allowed drying time. Visual observations of our drying experiments did reveal, however, an anomaly that might be related to trapping skinning. We observed standing waves developing during the drying of PMMA-toluene samples. During the initial stages of drying, waves formed perpendicular to the air flow direction and along the downstream edge of the sample pan. The amplitude of these waves appeared to increase with increasing air flow. As drying proceeded, however, the waves relaxed and eventually disappeared or left only tiny ripples on the dried sample. In their investigation of solvent removal rates from PMMA-toluene solutions, Powers and Collier22 reported higher percentage solvent residuals at higher temperatures. The initial sample weights, available only in Powers’ thesis,23 show that much thicker samples (by 20-40%) were used in the higher-temperature experiments. Their percentage solvent residual data are, therefore, inconclusive and cannot be used to demonstrate trapping skinning. Powers and Collier did, however, report waves during solvent removal experiments from their PMMA-toluene solutions, similar to our observations. They attributed this phenomenon to flow instabilities caused by the large differences in viscosity and diffusivity between the “skinned-over” coating surface and the bulk of the coating. They also claimed that the presence of these instabilities would enhance solvent removal rates as the formation of standing waves yields larger surface areas. It is possible that the higher-amplitude waves observed at higher air flows increased the sample surface area, and therefore the drying rates, sufficiently to prevent trapping skinning from being detected. We discuss this issue further below, when we compare drying model predictions to experimental data. Model Parameters. The model parameters for PMMA-toluene are compiled in Table 1. The physical properties of the substrate and toluene were obtained from an unpublished 3M database. All properties of

Do (cm2/s) E (cal/mol) ξ V ˆ /1 (cm3/g) V ˆ /2 (cm3/g) χ a

Substrate Properties 7.848 csp [cal/(g K)] 0.114 H (cm)

0.11 2.1336 × 10-2

Polymer Properties 0.8695 cp,2 [cal/(g K)] 4.5 × 10-4 M2 (g/mol) 378.15 R2 (K-1) -3.71 × 10-4 cp,2 - cpg,2 [cal/(g K)] Solvent Properties 1.155 cp,1 [cal/(g K)] 3.475 × 10-4 M1 (g/mol) 98.89 aa 1344.8 ca

Free-Volume Parameters 3.51 × 10-4 K11/γ1 [cm3/(g K)] 0.0 K21 - Tg1 (K) 0.54 K12/γ2 [cm3/(g K)] 0.917 K22 - Tg2 (K) 0.788 A (K) 0.45

0.35 120 000 5.8 × 10-4 0.076 0.44 92.14 4.07383 -53.668

1.57 × 10-3 -90.5 2.884 × 10-4 -308 423

a, b, and c are coefficients used in the Antoine equation.4

Figure 9. Correlation of weight-fraction activity coefficient data for PMMA-toluene at 48.5 °C. The solid circles are experimental data,26 and the solid line represents the theoretical fit. The dashed vertical line represents the solvent weight fraction at which the glass transition occurs.

PMMA were obtained from Brandrup and Immergut,24 except for R2g - R2 and cp,2 - cpg,2, which came from Vrentas and Vrentas.17,25 The χ parameter value of 0.45 was determined from a nonlinear regression of the weight-fraction activity coefficient (WFAC) data reported by Tait and Abushihada26 at 48.5 °C. Figure 9 shows the quality of the theoretical fit to the data. Table 1 also shows the free-volume parameters collected from different sources. The solvent free-volume parameters (V ˆ /1, K11/γ1, K21 - Tg1) were obtained from ˆ /2 from Zielinski and Duda,27 Vrentas and Vrentas,10 V and the parameters (Do, E, ξ, K12/γ2, K22 - Tg2) from Arnould and Laurence.28 Figure 10 shows that the compiled free-volume parameters give a good correlation of the diffusion data reported by Ju et al.29 at finite solvent concentrations and Arnould and Laurence at infinite dilution. Over the concentration range shown in Figure 10, the different mutual-diffusion models, cases 1-4, yield very similar results. Such behavior is expected, as the self- and mutual-diffusion coefficients should approach each other at low solvent concentrations. The A parameter was calculated based on the ω1E value of 0.189 reported by Wang et al.30 at 25 °C. The parameters in Table 1 can be used to describe the

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Figure 10. Comparison of experimental28,29(symbols) and theoretical (lines) mutual-diffusion coefficients for PMMA-toluene at different concentrations and temperatures.

Figure 11. Concentration dependence of the self-diffusion coefficient for PMMA-toluene at different temperatures.

concentration dependence of the specific volume, activity, and self/mutual diffusion of PMMA-toluene at different temperatures. The behavior of these properties at 25 °C was shown earlier in Figures 3-6. In addition, the concentration dependence of the self-diffusion coefficient, above and below ω1E, at different temperatures is shown in Figure 11. Model Predictions. For each set of experimental conditions, drying model predictions were made with each of the diffusion theories described above (cases 1-4). Drying profiles were also computed for each case assuming no volume change on mixing. When no volume change on mixing is assumed, the calculations do not include either diffusion-induced convection or the concentration dependence of the partial specific volumes. A total of eight drying profiles were thus computed for each experimental condition. In all calculations, the solvent partial pressure in the bulk was always set equal to zero. The mass transfer coefficient, kG 1 , was determined from the initial slope of the sample weight data so that all the simulations (cases 1-4) would match the initial experimental drying rate. In one case, however, we determined the heat and mass transfer coefficients independently, from pure solvent drying data obtained at the same process conditions (air temperature and flow rate). This case is included below to demonstrate the capability of the model to also predict the coating temperature. Tests of the Drying Model. Figure 12a shows experimental and predicted sample weight profiles for a PMMA-toluene coating dried at an air temperature of 25 °C and an air velocity of 20 fpm. The numbers on each curve refer to the mutual-diffusion theories, cases 1-4, described above. Figure 12b shows the counterpart

Figure 12. Comparison of experimental and predicted weight profiles for a PMMA-toluene coating dried at an air temperature of 25 °C and an air velocity of 20 fpm. The model predictions are shown (a) with and (b) without volume change on mixing. The numbers on each curve refer to the mutual-diffusion theories, cases 1-4. (c) Comparison of experimental and predicted weighttemperature profiles for a PMMA-toluene coating dried at an air temperature of 25 °C and an air velocity of 20 fpm [hg ) hG ) 4.0 -3 g/(s cm2 atm)]. × 10-4 cal/(s cm2 °C) and kG 1 ) 1.38 × 10

model predictions when no volume change on mixing is assumed. Several observations can be made from Figure 12a and b. First, the choice of friction-based diffusion model has a significant impact on the predicted drying profile. This behavior was expected, as the drying becomes diffusion-limited early in the process, and the diffusion coefficients resulting from cases 1-4 differ significantly (see Figure 6), especially at relatively high solvent concentrations. Second, the model predictions in Figure 12a show excellent agreement with the data when the friction-based theory of Zielinski and Hanley12 is used. If no volume change on mixing is assumed, none of the predictions match the sample weight, and all cases yield final residual solvent levels below the experimental value, as shown in Figure 12b.

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Figure 13. Comparison of experimental and predicted weight profiles for a PMMA-toluene coating dried at an air temperature of 50 °C and an air velocity of 100 fpm. The numbers on each curve refer to the mutual-diffusion theories, cases 1-4.

The validity of the model was further substantiated by also predicting the coating temperature profile using the heat/mass transfer coefficients obtained from independent drying data for pure toluene under the same conditions. This was accomplished by solving the energy balance equations4 coupled to the mass transfer problem. The thermal properties for the coating and substrate (steel pan) are also shown in Table 1. The results of these predictions, using Zielinski and Hanley’s12 diffusion model (case 4) with and without volume change on mixing, are shown in Figure 12c. The results again demonstrate the superiority of the model when structural rearrangements of the polymer matrix are taken into account. Similar conclusions were reached at different air velocities and temperatures. Figure 13, for example, shows comparisons between the model predictions and the experimental data recorded at an air temperature of 50 °C and an air velocity of 100 fpm. The diffusion model of Zielinski and Hanley12 yields the most accurate drying predictions. Additional simulations for this case suggest that the friction-based theory of Vrentas and Vrentas14 can yield quantitatively accurate predictions if (D1/D2) at φ1 ) 1 is set equal to 5500. According to Hickey’s20 correlation, however, this ratio should be less than 100 based on the molecular weight of PMMA used. We have also carried out predictions using the new scheme suggested by Vrentas and Vrentas11 to evaluate the hole-free volume of the polymer-solvent system. Their model predicts higher diffusion coefficients than eq 17. As a result, even when the friction-based theory of Zielinski and Hanley was used, the model predicted a slightly higher drying rate, yielding less satisfactory fits than those shown in Figures 12a and 13. Effects of Volume Change on Mixing. The model predictions and experimental data shown in Figure 12a-c suggest that the solvent removal rate is hindered by the nonideal volumetric behavior resulting from structural rearrangements in the glassy region of the coating. To understand this behavior, we analyzed the individual effects of the volume-average velocity, υq, and the concentration dependence of the partial specific volumes on the predicted drying rate of a PMMAtoluene sample at an air temperature of 25 °C and an air velocity of 20 fpm. According to eqs 2 and 5, the volume-average velocity is negative or zero throughout the coating. One could prematurely infer that this would hinder the solvent removal rate because the convective

Figure 14. Effect of diffusion-induced convection on solvent removal rate and film shrinkage for a PMMA-toluene coating dried at an air temperature of 25 °C and an air velocity of 20 fpm.

flux, F1υq, is opposite to the direction of the solvent diffusive flux. According to eq 4, however, the negative volume-average velocity should actually increase the rate at which the coating shrinks. It becomes apparent, therefore, that the effect of diffusion-induced convection on drying rate depends on its magnitude with respect to the diffusive flux and the rate of shrinkage. The ratio of convective to diffusive fluxes at x ) X(t) and the percent contribution of υq to the rate of shrinkage are shown in Figure 14. We find that diffusion-induced convection is negligible compared to the diffusive flux and only slightly affects the rate of shrinkage (∼5%), although this effect is only manifested during the last stages of drying. Simulations carried out by setting υq ) 0 yield drying profiles that are identical to those shown in Figure 12a except during the last stages of drying, where slightly higher residuals are obtained. Thus, diffusion-induced convection tends to enhance, not hinder, the drying rate. It becomes clear at this point that the overall effect of volume change on mixing cannot be explained solely in terms of the volume-average velocity. In eq 4, the rate of coating shrinkage is influenced not only by υq but also by the partial specific volume of the solvent. Because the rate of shrinkage is evaluated at x ) X(t), as soon as the coating surface transitions from a rubbery to a glassy state, the solvent partial specific volume at x ) X(t) is no longer equal to the pure-solvent specific volume. Once the surface becomes glassy, the solvent partial specific volume is computed from eq 10. The effect of this transition is illustrated in Figure 15, which shows the variation of the solvent partial specific volume at x ) X(t) with time. Because of structural rearrangements at the surface of the coating, the solvent partial specific volume decreases by about 10%. This decrease lowers the rate of coating shrinkage, also shown in Figure 15, which, in turn, increases the diffusion resistance in the coating and hinders solvent removal. This analysis suggests that the drying rate of the PMMA-toluene coating is significantly influenced by the concentration dependence of the solvent partial specific volume that results from the nonideal volumetric behavior of the glassy polymer solution. Although the analysis was shown only for PMMA-toluene at certain conditions, similar conclusions were drawn for different process conditions considered in this study as well. Coating Structure Evolution. In addition to the results described above, the model can be used to visualize the

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Figure 15. Model predictions of the variation of solvent partial specific volume at x ) X(t) and film thickness profiles for a PMMA-toluene coating dried at an air temperature of 25 °C and an air velocity of 20 fpm.

Figure 16. Coating structure evolution for a PMMA-toluene coating dried at (a) an air temperature of 25 °C and an air velocity of 20 fpm and (b) an air temperature of 50 °C and an air velocity of 100 fpm.

evolution of the coating structure during drying by superimposing process paths on a temperature-concentration phase diagram. Figure 16a shows the evolution of the PMMA-toluene coating structure when the coating is dried at air temperature of 25 °C and an air velocity of 20 fpm. The computations here include both mass and energy balances. In this diagram, the rubbery and glassy regions are separated by a line that represents the effective glass-transition temperature of the polymer-solvent system. The process paths start at a single concentration and temperature, labeled “t ) 0”. For the sake of clarity, only the process paths at the coating-substrate interface, at the free surface, and at

Figure 17. Model predictions of the effect of flow rate on weight and temperature profiles for a PMMA-toluene coating dried at an air temperature of 25 °C.

a point near (top 2%) the free surface are drawn. As the coating dries, concentration and small temperature gradients develop, and the process paths vary across the thickness of the coating. As expected, the free surface is the first to undergo a rubbery-glassy transition, indicated in the figure by the process path crossing the Tgm line. The rest of the coating remains rubbery until a much later time (4.5 h), when the layer near the free surface becomes glassy. Over the time allowed for drying, the model predicts that only the top 15% of the film becomes glassy. Figure 16b shows a similar diagram for the coating structure of PMMA-toluene when the coating is dried at air temperature of 50 °C and an air velocity of 100 fpm. At the higher temperature and air velocity, solvent is removed more quickly, and the rubbery-glassy transition front moves across the coating at a faster rate. At the end of the calculation, the top 20% of the film had become glassy. Trapping Skinning Phenomena. Trapping skinning occurs when more severe drying conditions (higher air flow, lower solvent humidity, higher temperature) lead to higher solvent residuals. In our experimental studies, the results (as in Figure 8) did not show any clear indication of trapping skinning. In our attempts to model each of the experiments, however, we found that the computational results do exhibit trapping skinning. The model predicted higher initial drying rates at higher air flows, as expected, but the final residual levels at higher air flows were between 0.01 and 0.04 g higher than the final predicted residuals at low air flows. Figure 17 shows the predicted sample weight and temperature profiles for two PMMA-toluene coatings with equivalent initial weights. The profiles were computed for an air temperature of 25 °C and air velocities of 20 and 100 fpm. After 9 h, the final predicted sample weights were 1.34 and 1.38 g at 20 and 100 fpm, respectively. Model predictions assuming no volume change of mixing do not show this trapping skinning behavior. The capability of the model to predict trapping skinning is indeed due to the incorporation of volume change on mixing in the model, through the concentration dependence of the partial specific volumes. Figure 18 shows the film thicknesses and solvent-specific volumes corresponding to the drying profiles in Figure 17. At higher air flow rates, the surface of the film becomes glassy more quickly. Once the surface transitions to the glassy state, the solvent partial specific volume decreases. At higher air flow rates, the earlier

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Figure 18. Model predictions of the effect of flow rate on film thickness and solvent partial specific volume [at x ) X(t)] profiles for a PMMA-toluene coating dried at an air temperature of 25 °C.

and more abrupt decrease in solvent partial specific volumes leads to a thicker film and, thus, higher residuals. We used the model to investigate other scenarios leading to trapping skinning. The model does predict the counterintuitive behaviors of faster drying at higher solvent humidities and faster drying at lower air temperatures. The latter case, however, only occurs when the effect of temperature on solvent partial specific volume outweighs its effect on solvent vapor pressure and diffusivity. In both cases, the mechanism is the same as described above: Earlier glass transitions at the coating surface lead to lower solvent partial specific volumes, thicker films, and higher residuals. The magnitude of the predicted trapping skinning behavior is on the same order as our experimental uncertainty. This suggests that several key points warrant further scrutiny in an effort to determine whether trapping skinning really occurs. From a theoretical standpoint, the expression for the solvent partial specific volume (eq 10) is probably the most critical element of the model. We need to carry out density measurements of the sample at different drying times to verify the evolution of the volumetric behavior. If the actual volumetric variations were slightly less than what the present model suggests, then trapping skinning behavior could be well below our detection resolution. From an experimental standpoint, we need to increase the resolution of our scale. Addition of a solvent humidifier would also be useful, as the model suggests that trapping skinning could be enhanced by varying both air flow and solvent humidity. We intend to pursue this investigation further. Conclusions We have implemented a drying model to predict solvent removal rates during drying of rubbery-glassy polymer films. When a polymer film passes through a glass transition as it dries, structural rearrangements of the polymer matrix result in nonideal volumetric behavior, i.e., volume change on mixing. The model accounts for these nonideal volumetric effects by including the concentration dependence of the partial specific volumes and diffusion-induced convection. The model also accounts for the effects of plasticization and the polymer glass transition on the volumetric, solubility, and diffusion properties of the polymer-solvent system. Based on comparisons between model predictions and gravimetric drying data for a PMMA-toluene coating

formulation, we found that the concentration dependence of the partial specific volumes is a critical component of an accurate model. We also found that the form of the friction-based theory used to relate mutualand self-diffusion coefficients plays an important role in capturing the drying behavior of this rubbery-glassy coating. Our results suggest that the Zielinski-Hanley friction-based theory gives the best drying predictions for the PMMA-toluene system. Additional polymersolvent systems and mutual-diffusion coefficient data at high solvent concentrations are required, however, to discern the optimal link between self- and mutualdiffusion coefficients. The model also predicts trapping skinning behavior. Unfortunately, the predicted behavior is on the same order as our experimental accuracy, and additional effort is required to prove or disprove the existence of this phenomenon. Finally, we believe that the model presented here is sufficiently accurate to use as a tool to optimize pilot- or production-scale drying processes of glassy coating formulations, as well as to visualize structure evolution in drying films. Acknowledgment Two of us, I.H.R. and P.E.P, are Penn State graduates. Our careers as research engineers have been profoundly influenced by Dr. J. Larry Duda, both directly, through his mentorship, and indirectly, through his leadership of the Department of Chemical Engineering at PSU. All of us have had the good fortune of working under Shuzo Fuchigami, who has supported and encouraged our interactions with Dr. Duda and his colleagues. 3M has profited greatly from these interactions. The foundations of this paper were laid by Drs. Duda and Vrentas and their students and collaborators. We have been continually amazed at the success of their theories in addressing our very real industrial problems. We hope this paper reflects the depth of our gratitude. Note Added after ASAP Posting This article was released ASAP on 4/12/01 with an error in eq 1. The correct version was posted on 4/20/01. Nomenclature A ) parameter defined by eq 7 a, b, c ) coefficients used in the Antoine equation cp,i ) specific heat capacity of solvent or equilibrium liquid polymer, cal/(g °C) cpg,2 ) specific heat capacity of glassy polymer, cal/(g °C) csp ) specific heat capacity of substrate, cal/(g °C) D ) polymer/solvent mutual-diffusion coefficient, cm2/s D0 ) constant preexponential factor, cm2/s Di ) self-diffusion coefficient of component i, cm2/s E ) activation energy, cal/mol H ) thickness of substrate, cm hg ) lower heat transfer coefficient, cal/(s cm2 °C) hG ) upper heat transfer coefficient, cal/(s cm2 °C) ∆H ˆ vap,1 ) latent heat of vaporization of solvent, cal/g k1 ) parameter defined by eq 8 2 kG 1 ) mass transfer coefficient of solvent, g/(s cm atm) K1i/γi ) free-volume parameter of component i, cm3/(g K) K2i ) free-volume parameter of component i, K Mi ) molecular weight of component i, g/mol p°1 ) pure-solvent vapor pressure, atm pG 1b ) equilibrium solvent partial pressure at the coatingair interface, atm

Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3075 G p1i ) solvent partial pressure in the bulk air, atm R ) ideal gas constant [e.g., 1.987 cal/(mol K)] t ) time, s T ) temperature of coating, °C Tg ) lower air temperature, °C TG ) upper air temperature, °C Tgi ) glass-transition temperature of component i, °C Tgm ) glass-transition temperature of coating, °C υq ) volume-average velocity, cm/s V ˆ ) specific volume of coating, cm3/g V ˆ °2(Tg2) ) specific volume of equilibrium liquid polymer at Tg2, cm3/g V ˆ FH/γ ) average hole-free volume per gram of coating, cm3/g V ˆ i ) partial specific volume of component i, cm3/g V ˆ i° ) specific volume of solvent or equilibrium liquid polymer, cm3/g V ˆ /i ) specific critical hole-free volume of component i, cm3/g x ) position in the direction normal to the coatingsubstrate interface, cm X ) thickness of the coating, cm

Greek Letters R2 ) thermal expansion coefficient of equilibrium liquid polymer, K-1 R2g ) thermal expansion coefficient of glassy polymer, K-1 χ ) Flory-Huggins interaction parameter φi ) volume fraction of component i κi ) thermal conductivity of component i, cal/(s cm K) κS ) thermal conductivity of substrate, cal/(s cm K) F ) total mass density of coating, g/cm3 FS ) density of substrate, g/cm3 F10 ) initial solvent concentration in coating, g/cm3 Fi ) mass density of component i, g/cm3 ω1E ) solvent mass fraction at which the coating becomes glassy at temperature T ωi ) solvent mass fraction of component i ξ ) ratio of solvent and polymer jumping units

Literature Cited (1) Vrentas, J. S.; Vrentas, C. M. Integral Sorption in Glassy Polymers. Chem. Eng. Sci. 1998, 53, 629. (2) Vrentas, J. S.; Vrentas, C. M. Drying of Solvent-Coated Polymer Films. J. Polym. Sci. B: Polym. Phys. 1994, 32, 187. (3) Cairncross, R. A.; Jeyadev, S.; Dunham, R. F.; Evans, K.; Francis, L. F.; Scriven, L. E. Modeling and Design of an Industrial Dryer with Convective and Radiant Heating. J. Appl. Polym. Sci. 1995, 58, 1279. (4) Price, P. E., Jr.; Wang, S.; Hadj Romdhane, I. Extracting Effective Diffusion Parameters From Drying Experiments. AIChE J. 1997, 43, 1925. (5) Cairncross, R. A.; Durning, C. A Model for Drying of Viscoelastic Polymer Coatings. AIChE J. 1996, 42, 2415. (6) Ngui, M. O.; Mallapragada, S. K. Quantitative Analysis of Crystallization and Skin Formation During Isothermal Solvent Removal From Semicrystalline Polymers. Polymer 1999, 40, 5393. (7) Vrentas, J. S.; Vrentas, C. M. Sorption in Glassy Polymers. Macromolecules 1991, 24, 2404. (8) Vrentas, J. S.; Vrentas, C. M. Hysteresis Effects for Sorption in Glassy Polymers. Macromolecules 1996, 29, 4391. (9) Vrentas, J. S.; Vrentas, C. M. Fickian Diffusion in Glassy Polymer-Solvent Systems. J. Polym. Sci. B: Polym. Phys. 1992, 30, 1005.

(10) Vrentas, J. S.; Vrentas, C. M. Solvent Self-Diffusion in Glassy Polymer-Solvent Systems. Macromolecules 1994, 27, 5570. (11) Vrentas, J. S.; Vrentas, C. M. Predictive Methods for SelfDiffusion and Mutual Diffusion Coefficients in Polymer-Solvent Systems. Eur. Polym. J. 1998, 34, 797. (12) Zielinski, J. M.; Hanley, B. F. Practical Friction-Based Approach to Modeling Multicomponent Diffusion. AIChE J. 1999, 45, 1. (13) Alsoy, S.; Duda, J. L. Modeling of Multicomponent Drying of Polymer Films. AIChE J. 1999, 45, 896. (14) Vrentas, J. S.; Vrentas, C. M. A New Equation Relating Self-Diffusion and Mutual Diffusion Coefficients in PolymerSolvent Systems. Macromolecules 1993, 26, 6129. (15) Ferry, J. D. Viscoelastic Properties of Polymers; Wiley: New York, 1970. (16) Vrentas, J. S.; Vrentas, C. M. Differential Sorption in Glassy Polymers. J. Appl. Polym. Sci. 1999, 71, 1431. (17) Vrentas, J. S.; Vrentas, C. M. Isotherm Shape for Penetrant Sorption in Glassy Polymers. J. Appl. Polym. Sci. 1992, 45, 1497. (18) Vrentas, J. S.; Duda, J. L. Diffusion in Polymer-Solvent Systems: I. Reexamination of the Free-Volume Theory. J. Polym. Sci. B: Polym. Phys. 1977a, 15, 403. (19) Vrentas, J. S.; Duda, J. L. Diffusion in Polymer-Solvent Systems: II. A Predictive Theory for the Dependence of Diffusion Coefficients on Temperature, Concentration, and Molecular Weight. J. Polym. Sci. B: Polym. Phys. 1977b, 15, 417. (20) Hickey, P. C. An Evaluation of Models For the Prediction of Diffusion Coefficients in Polymer-Solvent Systems. M.S. Thesis, The Pennsylvania State University, State College, PA, 1995. (21) Cairncross, R. A. Solidification Phenomena During Drying of Sol-To-Gel Coatings. Ph.D. Thesis, University of Minnesota, Minneapolis, MN, 1994. (22) Powers, G. W.; Collier, J. R. Experimental Modeling of Solvent-Casting Thin Polymer Films. Polym. Eng. Sci. 1990, 30, 118. (23) Powers, G. W. The Effect of Temperature on the Solvent Removal from Solution-Cast Thin Polymer Films. M.S. Thesis, Ohio University, Athens, OH, 1988. (24) Brandrup, J.; Immergut, E. H. Polymer Handbook; Wiley: New York, 1989. (25) Vrentas, J. S.; Vrentas, C. M. Predictions of Volumetric Behavior for Glassy Polymer-Penetrant Systems. J. Polym. Sci. B: Polym. Phys. 1990, 28, 241. (26) Tait, P. J. T.; Abushihada, A. M. Comparative Studies on the Use of Gas Chromatographic and Vapour Pressure Techniques for the Determination of the Interaction Parameter. Polymer 1977, 18, 810. (27) Zielinski, J. M.; Duda, J. L. Predicting Polymer/Solvent Diffusion Coefficients Using Free-Volume Theory. AIChE J. 1992, 38, 405. (28) Arnould, D.; Laurence, R. L. Size Effects on Solvent Diffusion in Polymers. Ind. Eng. Chem. Res. 1992, 31, 218. (29) Ju, S. T.; Liu, H. T.; Duda, J. L.; Vrentas, J. S. Solvent Diffusion in Amorphous Polymers. J. Appl. Polym. Sci. 1981, 26, 3735. (30) Wang, B.-G.; Yamaguchi, T.; Nakao, S.-I. Solvent Diffusion in Amorphous Glassy Polymers. J. Polym. Sci. B: Polym. Phys. 2000, 38, 846.

Received for review December 18, 2000 Revised manuscript received February 21, 2001 Accepted February 26, 2001 IE001110H