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Langmuir 2008, 24, 582-589
Modeling the Drying of Ink-Jet-Printed Structures and Experimental Verification D. B. van Dam Philips Research, High Tech Campus 4 (WAG01), 5656 AE EindhoVen, The Netherlands
J. G. M. Kuerten* Technische UniVersiteit EindhoVen, Department of Mechanical Engineering, Section Process Technology, EindhoVen, The Netherlands ReceiVed June 22, 2007. In Final Form: September 9, 2007 This article presents a numerical model that was developed for the drying of ink-jet-printed polymer solutions after filling the pixels in a polymer LED display. The model extends earlier work presented in the literature while still maintaining a practical approach in limiting the number of input parameters needed. Despite some rigorous assumptions, the model is in fair agreement with experimental data from a pre-pilot ink-jet printing line. Comparison inside a single pixel is shown, as well as a general trend in which the amount of polymer that is transported out of the central part of the pixel decreases with the rate of viscosity increase as a function of polymer concentration. Moreover, the effect of a varying solute diffusion coefficient is studied.
1. Introduction Interest in the drying of ink-jet-printed fluids appears in many different fields.1-3 In the more traditional graphical applications, the visual appearance of the ink depends on the drying process, and the limitation of the total drying time is important. In new applications of ink-jet printing, for example, for depositing electro(optical) materials in the display and electronics industry,4 the functionality of the layer is of crucial importance. This functionality often depends strongly on the layer thickness distribution of the deposited structure. This layer thickness distribution in turn is determined during the drying of the layer, when a significant redistribution of material can take place. In addition, in multilayer devices the thickness of layers deposited on top of a printed layer depends on the geometry of the printed structures. The research that is described in this article was performed in the framework of an activity to manufacture polymer LED displays.5 For this purpose, it is important to understand the formation of layers of light-emitting polymers with uniform thickness in predefined wells, which were formed by hydrophobic barriers. For this application, we aimed to find a relatively simple theoretical framework that enabled us to better guide the development of inks and of the printing process. The drying of ink-jet-printed droplets that form thin films of solute is a process that can show various hydrodynamic phenomena, such as Marangoni flow and hydrodynamic instability.6 However, even in a relatively simple form that still has * Corresponding author. E-mail:
[email protected]. Phone: +31 40 247 2362. Fax: +31 40 247 5399. (1) Dufva, M. Biomol. Eng. 2005, 22, 173-184. (2) Hjelt, K. T.; Van den Doel, L. R.; Lubking, W.; Vellekoop, M. J. Sens. Actuators 2000, 85, 384-389. (3) Rieger, B.; Van den Doel, L. R.; Van Vliet, L. J. Phys. ReV. E 2003, 68, 036312. (4) Bale, M.; Carter, J. C.; Creighton, C. J.; Gregory, H. J.; Lyon, P. H.; Ng, P.; Webb, L.; Wehrum, A. SID J. 2006, 14, 453-459. (5) Fleuster, M.; Klein, M.; Van Roosmalen, P.; De Wit, A.; Schwab, H. SID Symp. Digest Tech. Pap. 2004, 35, 1276. (6) Cuk, T; Trojan, S. M.; Hong, C. M.; Wagner, S. Appl. Phys. Lett. 2000, 77, 2063-2065.
practical significance, the description of the drying process already poses significant difficulties. In the case considered here, the main processes involved are the convective and diffusive transport of solute, coupled with the evolution of the free surface, the evaporation of the solvent, and the viscosity increase in the solution (Figure 1). In previous work, the occurrence of flow and particle transport in thin fluid films is recognized, which occurs as a result of local evaporation that does not match the energetically preferred fluid surface evolution.7,8 Significant work on the drying of droplets has been performed by Deegan,9-11 quantifying the surfacetension-driven mechanism of convective solute transport initiated by the evaporation of droplets with radii on the order of 1 mm. However, for smaller particles and droplets than Deegan studied, diffusion established by advectively induced concentration gradients can become important. Also, the increase in viscosity as the concentration of the solute increases can become important. In this study, we present a practical and approximate approach toward modeling the drying of ink-jet-printed structures containing polymers or colloidal particles in which we take these processes into account and compare model predictions with experimental results.
2. Modeling Framework Modeling should provide a description of various processes, which are the evaporation of solvent, the rheological evolution of the ink, the advective flow field within the droplet, the diffusive transport of solute, and the surface tension forces acting on the fluid. In the literature, several useful models were developed. The ones that we use as references are those of Deegan9-11 and Fischer12 because from the existing models these come closest (7) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3183-3190. (8) Parisse, F.; Allain, C. J. Phys. II 1996, 6, 1111-1120. (9) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature 1997, 389, 827-829. (10) Deegan, R. D. Phys. ReV. E 2000, 61, 475-485. (11) Deegan, R. D. Phys. ReV. E 2000, 62, 756-765. (12) Fischer, B. J. Langmuir 2002, 18, 60-67.
10.1021/la701862a CCC: $40.75 © 2008 American Chemical Society Published on Web 12/11/2007
Modeling the Drying of Ink-Jet-Printed Structures
Langmuir, Vol. 24, No. 2, 2008 583
Figure 1. Schematic overview of processes incorporated in our modeling.
to our experimental situation, although still showing room for improvement. In this article, we consider both viscous and inviscid models. 2.1. Assumptions Common to Models. We assume that the fluid is well mixed in the vertical direction. This implies that the concentration of solute does not depend on the vertical coordinate. This assumption can be made when diffusive transport in the vertical direction is dominant over advective transport in the radial direction, which is typically the case when
VH ,1 D
M)
() 2
L 12η 2 H
( )( )
∂σ c ∂cs s H2 L ) Ds 12ηDs Rc
() 2
( ( ))
Ve ) Ve,av(1 - λ) 1 -
r R0
2 -λ
(3)
where Ve is the evaporation rate, λ ) (π - 2θ)/(2π - 2θ), and θ is the contact angle between fluid and substrate. Ve,av is the average evaporation rate per unit of solid-fluid area. We assume that the evaporation velocity is not influenced by the possible gelation of the polymer during the drying process. 2.2. Governing Equations. The fluid flow in the droplet is governed by the Navier-Stokes equation and the continuity equation
F
(1)
where V is a characteristic lateral flow velocity (that is larger than the average evaporation velocity), H is a characteristic fluid layer thickness, and D is the solute’s diffusion coefficient. This assumption enables us to integrate the velocity over the z coordinate using a concentration that depends on lateral coordinates x and y only. (See Figure 1 for the coordinates that were used.) We assume as well that surface tension gradients can be neglected. For inks with one solvent, this assumption is reasonable for our materials because we measured no influence of solute concentration on surface tension. However, in inks consisting of two or more solvents having different evaporation velocities and surface tension, surface tension gradients can in general be expected. The solutal Marangoni number M describes the proportion of the time scale for the diffusive transport of one solvent in the other solvent and the time scale for convective solvent transport as a function of a surface tension gradient and can be expressed as
∂σ c ∂cs s Rc
Marangoni effects is on the verge of being significant, it is worthwhile to consider it as a source of error when comparing experiments with model results. Furthermore, we assume that the contact line remains pinned, which is the case for many practical situations. This condition is discussed by Deegan.10 By observation of the drying process, we see that this assumption also holds for our materials. In addition, we need to specify the evaporation velocity of the fluid. For a sessile droplet with radial symmetry and radius R of the solid-fluid interface, the evaporation velocity can be written as11
∂u + Fu‚∇u ) -∇p + η∇2u - giz ∂t
(4)
∇‚u ) 0
(5)
where u is the velocity vector, F is the fluid mass density, η is the dynamic viscosity, g is the acceleration of gravity and iz is the unit vector in the upward vertical direction. To determine the relative importance of the different terms in the governing equations, we take a typical length scale L for the horizontal size of the droplet. We use H as a typical length scale for the height of the droplet and the evaporation velocity Ve,av as the typical fluid velocity in the vertical direction. From eq 5, it follows that the typical velocity in the horizontal direction is equal to V ) Ve,avL/H. The appropriate scale for pressure is given by σ/Rc. Next, we decompose the fluid velocity into a horizontal and a vertical component according to u ) u | + wiz. The horizontal and vertical components of the scaled Navier-Stokes equation then read, respectively,
We
(
)
(
∂u/| ∂2u/| H2 L + u*‚∇u/| ) -∇p* + Ca *2 + 2 ∇2| u/| ∂t* Rc ∂z L
)
(6a)
and
(2)
where η is the viscosity of the solution, σ is the surface tension at the fluid-air interface, Ds is the diffusion coefficient of one solvent in the other solvent, cs is the concentration of one solvent in the ink, L is a typical size of the droplet in the lateral direction (along the x, y, or r coordinate), and Rc is a typical value for the radius of curvature of the fluid-air interface. At the beginning of drying, the value of M typically is O(1) for drying lightemitting polymer solutions (using typical values for ∂σ/∂cs cs ) 1 × 10-3 N m-1, η ) 1 × 10-2 Pa s, Ds ) 10-9 m2 s-1, Rc ) 10-3 m, and H ) 1 × 10-5 m). However, as drying proceeds the value of H2/Rc will decrease quickly, thus justifying our neglectance of Marangoni effects. Because the influence of solute
We
(
)
∂w* ∂p* L3 + u*‚∇w* ) + ∂t* ∂z* H2R c Ca
(
)
∂2w* H2 2 + 2 ∇| w* - Bd (6b) ∂z*2 L
where ∇| denotes the horizontal component of the gradient operator and an asterisk is used for a scaled variable. Moreover, Ca is the capillary number that is defined as
Ca )
ηV L2 σ H2
(7a)
We is the Weber number that is defined as
We )
FV2L σ
(7b)
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584 Langmuir, Vol. 24, No. 2, 2008
Figure 2. Viscosity vs concentration for polymer ink P1.
and Bd is the Bond number that is defined as
Bd )
FgL2 σ
(7c)
For a typical case at the start of drying, V ) 1 × 10-6 m s-1, L ) 1 × 10-4 m, H ) 1 × 10-5 m, Rc ) 1 × 10-3 m, σ ) 30 × 10-3 N m-1, F ) 1 × 103 kg m-3, Ve,av ) 10-7 m s-1, and η ) 10 mPa s. Hence, We ) 3 × 10-12, Ca ) 3 × 10-5, and Bd ) 3 × 10-3. During the evaporation process, the viscosity can increase by 4 orders of magnitude. Hence, the capillary number also increases by this factor. It can be concluded that for a typical case the small Bond number and Weber number justify the scaling of the pressure term. Moreover, it can be seen that for these typical magnitudes of the terms in the Navier-Stokes equation the convective terms on the left-hand side, the gravity term, the horizontal derivatives in the viscous terms in eq 6a, and the total viscous term in eq 6b can always be disregarded. Therefore, the governing equations reduce to
∂p )0 ∂z
(8a)
and
∇ |p )
R c ∂ 2 u| Ca 2 L ∂z
(8b)
where for convenience the asterisks have been omitted. These equations are similar to those derived by Fischer12 for the axisymmetric case, and another lubrication-based model has been used by Routh and Russel.13 The difference is that we assume that the viscosity depends on the solute concentration and is in this way a function of the horizontal coordinates and time. To show that explicitly in eq 8b, we define
Ca0 )
η0V L2 σ H2
(9)
where η0 is the viscosity at the initial solute concentration, so eq 8b can be rewritten as
∇ |p )
Rc η ∂2u| Ca0 2 L η0 ∂z
(Figure 3a). However, even at the edge of the droplet, of which the interface is given by z ) h(x, y), |∇|h| is less than 1, so the lubrication approximation remains, to good approximation, valid. In the lubrication approximation used here, the pressure at the interface between the droplet and surrounding air is given by
p)-
RcH 2 ∇| h L2
(11)
The constant on the right-hand side is on the order of unity for typical droplets considered here. Because according to eq 8a the pressure is constant in the vertical direction, the horizontal velocity follows from eq 10 if appropriate boundary conditions are specified. At the bottom of the drop, the no-slip condition u| ) 0 should hold, whereas at the interface the tangential stress balance leads to ∂u|/∂z ) 0. The solution of eq 10 is
u| )
η 0L 1 2 z - hz ∇|p ηRcCa0 2
(
)
(12)
and with eq 11 we find
(10)
For the typical parameter values mentioned above, the lubrication approximation is certainly accurate near the center of the droplet. Near the edge of the droplet, the estimates for the convective terms in the Navier-Stokes equation made above become more questionable because the curvature of the droplet interface and the substrate might become important, especially for the substrate structure considered in this article (13) Routh, A. F.; Russel, W. B. AIChE J. 1998, 44, 2088-2098.
Figure 3. (a) Photoluminescence microscope view of a pixel filled with polymer P1 and (b) schematic view of substrate geometry (not to scale).
u| )
η0L 1 2 z - hz ∇|∇2| h ηRcCa0 2
(
)
(13)
In a similar way as in Fischer,12 the continuity equation can be used to find an evolution equation for the height of the droplet
∂h ) -∇|(h 〈u|〉) - Ve ∂t
(14)
where the angular brackets denote a height average and Ve is the evaporation velocity. The substitution of eq 13 finally yields
Modeling the Drying of Ink-Jet-Printed Structures
(
Langmuir, Vol. 24, No. 2, 2008 585
)
η0L ∂h h3 ∇| ∇|∇2| h - Ve ) ∂t 3RcCa0 η
(15)
as the equation that describes the evolution of the droplet height. This equation can be solved if the viscosity is known as a function of the horizontal coordinates. This follows from the known relation between viscosity and solute concentration and the convectiondiffusion equation for the solute concentration:
∂ (ch) + ∇|(c〈u|〉h) ) ∇|(Dh∇|c) ∂t
(16)
2.3. Inviscid Approximation. For some materials, we found that the viscosity remains of the same order of magnitude during the drying process until a certain critical concentration ccr is approached, at which the value of the capillary number using the local viscosity sharply increases to a value that is large compared to unity.14 In such a case, the use of the equations derived in the previous section makes less sense because during almost the whole drying process the pressure term in eq 6a is completely dominant. In such a case, we use the same approach as introduced by Deegan.9 Because the pressure inside the droplet is constant, the shape of the droplet follows from the requirement that its curvature is constant. This leads to a unique shape if the volume of the droplet and the shape of the contact line are known. In this work, we assume that the contact line is pinned and the volume of the droplet follows from the known evaporation velocity profile integrated over the interface. To solve eq 16 for the solute concentration, the height-averaged velocity has to be known. Equation 14 still holds but is not sufficient to determine the fluid velocity if the droplet is not axisymmetric. An additional equation is found by averaging eq 12 over the droplet height. This results in15
∇|
η〈u|〉 h2
)0
(17)
The convection-diffusion equation (eq 16) for the solute concentration is used to calculate the time evolution of the solute concentration, but the concentration and local solute transport are limited to the critical concentration ccr. When we assume a radial geometry of the droplet, the condition of constant curvature reduces to the condition that a spherical cap is always maintained. The thickness of the fluid layer is then given by
h)
x(
)
2 R20 + hr)0 2 hr)0
2
- r2 -
(
)
2 R2 - hr)0 2hr)0
(18)
We can also infer that the problem is described by three dimensionless numbers and a parameter describing the evaporation rate profile
NH )
H0 R0
ND )
H0D Ve,avR20
Nc )
c0 ccr
λ0
(19)
where H0, R0, and λ0 are the central layer thickness, droplet radius, and λ at the start of drying. (14) van Dam, D. B. In Mechanics of the 21st Century: Proceedings of the 21st International Congress of Theoretical and Applied Mechanics; Gutkowski, W., Kowalewski, T. A., Eds.; Springer: Dordrecht, The Netherlands, 2005. (15) Popov, Y. O.; Witten, T. A. Phys. ReV. E 2003, 68, 036306.
2.4. Numerical Method. In both the viscous approach and the inviscid approximation, eq 16 is discretized with a finite volume method on a structured grid using second-order accurate central differences for the diffusion terms and a first-order accurate upwind method for the convective terms. Integration in time is performed by the second-order accurate implicit Crank-Nicolson method. Because the equation is linear in concentration and the discretization uses only nearest neighbors, the linear system of equations can be solved by a direct method. In the viscous approach, the evolution equation for the height (eq 15) is discretized in space with a second-order accurate finite volume method in space. For integration in time, a method based on the trapezoidal rule and especially suited for stiff systems is used. Because the equation is highly nonlinear, the time step has to be restricted to a small value. In the inviscid approximation, the equation that determines the shape of a surface with constant curvature and a given volume is discretized on a structured grid with second-order accurate central differences. In the lubrication approximation, the equation is linear and is solved by a direct method. If the droplet height is known, then the governing equations for the height-averaged horizontal fluid velocity are also linear and are solved in the same way. 3. Experimental Method and Materials 3.1. Materials Used. The light-emitting polymers were all obtained in solution, from various suppliers. A number of different kinds of polymers and solvents were used. The average molecular weight of all polymers was around 3 × 105 daltons. The solutions contained one, two (in most cases), or three organic solvents. The mix of polymer and solvents was tailored to have a viscosity of 10 mPa s, measured at a shear rate of about 102 s-1. These solvent-polymer combinations were evaluated during the ink development process when aiming for a flat polymer layer after drying while at the same time the various specifications for processability had to be met. A combination of two solvents with different evaporation rates and solubility for the polymer in general facilitates the achievement of a relatively flat polymer layer after drying.16 In this article, we present a direct comparison between measured polymer layer thicknesses in an experiment using polymer ink P1 and in our numerical simulations. Polymer ink P1 is based on m-xylene and decalin (1:1 cis/trans mixture) in a weight proportion of 40:60 and a green-light-emitting polymer with an average molecular weight of 3 × 105 daltons. The initial polymer concentration for this ink is 1.1 wt %. 3.2. Material Characterization. For different inks, differences in polymer, its molecular weight distribution, and solvents are all represented by differences in the measured shear viscosity as a function of polymer concentration. For the light-emitting polymer inks, the expected evolution of shear viscosity during evaporation was measured in a cone-plate rheometer. The samples at higher concentration than c0 were obtained in two ways. First, the polymer solute was partially evaporated in a thin film evaporator. Samples were taken at different concentrations, and in this way a curve of polymer concentration versus shear viscosity was obtained. Second, expected concentrations of solvents during evaporation were calculated. Then, polymer solutions with the relevant proportions of solvents were made, and the viscosity of these solutions was measured. Figure 2 shows the results for ink P1, for which the samples were obtained by the thin-film evaporation method. The viscosity was determined at a shear rate of about 0.1 s-1, which is representative of the conditions during drying. The diffusion coefficient of a number of polymers was measured with NMR, using the ink before evaporation took place. From eight different inks based on a range of polymers and solvents, the diffusion (16) Lyon, P. J.; Carter, J. C.; Bright, C. J.; Cacheiro, M. WO 02/069119 A1, 2002.
586 Langmuir, Vol. 24, No. 2, 2008 coefficients were found to be between 0.4 × 10-11 m2 s-1 and 1.3 × 10-11 m2 s-1. For an ink that resembles the P1 ink (containing the same polymer and based on a 70:30 mixture of m-xylene/decalin), the diffusion coefficient of the polymer was (4.7 ( 0.5) × 10-12 m2 s-1. The diffusion coefficient of m-xylene in this ink was measured to be (1.1 ( 0.1) × 10-9 m2 s-1. An accurate assessment of the evaporation rate during drying is rather complicated. First, the drying conditions during the printing process are strongly dependent on the local flow conditions. These conditions depend on such factors as the geometry of the experimental setup and the air flow setting of the fume hood in which the experiments are conducted. The printed pattern is important as well because the total evaporation rate of a small structure and its distribution over the surface of that structure are in general dependent on the size of the structure and on the presence of other fluid structures in the vicinity. In this study, we took the pragmatic approach to assessing the evaporation time from the total drying time of the printed plate, which was several minutes. Because the average layer thickness is approximately 10 µm, we estimate the average evaporation rate to be between 1 × 10-7 and 1 × 10-8 m s-1. Simulations are done for both values of the evaporation rate. Because the pixel containing the printed fluid is surrounded by many other filled wells, we assume that the evaporation rate is independent of the position in the pixel. If the pixel shape had been axisymmetric, then this would have corresponded to λ ) 0 in eq 3. This approach also denies the multisolvent character of the ink, which leads to evaporation rates that vary in position and time. 3.3. Experimental Setup and Method. The drying data of the light-emitting polymers were obtained from samples that were inkjet printed on an experimental pre-pilot line, where inks and print heads were changed regularly. In this line, two types of multinozzle print heads were used as obtained from Spectra. These types were the Spectra Galaxy head (256 nozzles) and the Spectra SX head (128 nozzles), which use piezoelectric actuation of the fluid. Glass substrates with dimensions of 15 cm × 15 cm were used, and they contained photolithographically defined Novolak structures. The substrates were cleaned using O2 plasma, and subsequently CF4 plasma treatment was applied, which preferentially made the organic structures on the substrate hydrophobic. In this way, a contrast in surface energy was reached that enabled us to contain the printed fluid effectively within the organic structures (referred to as wells or pixels). During printing and drying, the substrates were resting on a metal chuck that was kept at 30 °C. In addition, the substrates experienced rather strong airflow in order to remove the harmful solvent vapor. After printing, an ∼60 nm thin aluminum layer is deposited by evaporation on top of the substrate. Afterward, the surface shape is measured using white-light interferometry. The empty pixels are used for reference to determine the absolute value of the layer thickness. Figure 3a shows an optical micrograph of the substrate, with some pixels filled with polymer. The pixels have a pseudoelliptical shape. Figure 3b shows a schematic cross section along the minor axis of the pixel. It shows the geometry and lateral length of the SiO layer (250 nm thickness) and the Novolak resist structure (top height of about 2.5 µm). These structures are similar along the major axis of the ellipse, where the lateral distance from the top of the resist structure to the center of the pixel is 92.5 µm. These are the designed dimensions, but the real distances can be slightly different. In the numerical model, the edge of the SiO layer is perpendicular to the substrate. In reality, however, the edge is somewhat tapered.
4. Results of Experiments and Simulations 4.1. Comparison of Various Models. For the specific pixel geometry, Figure 4 shows a comparison of the viscous and the inviscid model. In the viscous simulation, the following input parameters are used: the initial volume of the droplet is 82 pL, D ) 0 m2 s-1, Ve,av ) 10-7 m s-1, and the viscosity curve from Figure 2 is used. The evaporation rate is independent
Van Dam and Kuerten
Figure 4. Comparison between the inviscid (top) and viscous (bottom) models. The colors indicate the height of the polymer layer.
of position and constant in time. For the inviscid simulation, we have to determine a critical dimensionless concentration ccr/ c0 above which the transport of polymer effectively stops. To estimate ccr/c0, we consider the proportion of the pressure term and the viscous term in eq 6a. We can then estimate ccr/c0 by determining the viscosity at which Ca Rc/L is O(1). From geometrical considerations, we can deduce that Rc/L ) L/2H when H2 , L2 for a surface in the form of a spherical cap. Assuming that L/H ) 10, σ ) 0.03 N m-1, and V ) 10-6 m s-1, we find that for η ) O(102), Ca Rc/L is about O(1). From Figure 2, we can now determine the value of ccr/c0 to be 1.84. Significant differences can be observed in Figure 4. The inviscid model in general predicts less polymer in the center than does the viscous model. This example is representative of the comparison between viscous and inviscid simulations in general. Another striking difference is the sharp peak in polymer thickness along the straight edges of the pixel in the viscous simulation. This peak becomes less pronounced for higher values of the evaporation rate (Figure 5). Although in some experiments similar sharp peaks in polymer thickness have been observed, these peaks are partially due to incomplete physical modeling. During evaporation of the droplet, the solute concentration keeps growing near the edges of the droplet. Therefore, in the numerical model a maximum solute concentration, corresponding to close packing, is introduced, after which the convection of the solute is stopped. A decrease of this maximum solute concentration results in a reduction of the observed peak in polymer height but has a negligible influence on the polymer thickness away from the edge of the droplet. 4.2. Comparison with Experiment. Figure 3a gives typical examples of printed pixels using polymer ink P1. On average, the contact line of the ink is positioned at 30% of the total width of the resist structure. Figure 5 shows the comparison
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Langmuir, Vol. 24, No. 2, 2008 587
Figure 5. Comparison of measured and simulated layer thickness profiles, with varying evaporation rates and diffusion coefficients.
between experimental and simulated layers for ink P1. Four curves are calculated. First, there is the inviscid model using ccr/c0 ) 1.84. Second and third, there are results of the viscous model with two evaporation rates that are constant over the whole pixel. In these three calculations, the diffusion coefficient D equals zero. Fourth, there is the result of the viscous model using the lower evaporation rate of 10-8 m s-1 and a diffusion coefficient equal to 4.7 × 10-12 m2 s-1 that is independent of polymer concentration. To make a comparison between experiment and simulation, a value for the density of the printed layer Fdry after drying has to be assumed. We choose this value such that the simulated curve that includes diffusion and the experimental measurements in Figure 5 have the same layer thickness in the center of the pixel. This yields a value of Fdry equal to 1.05 × 103 kg m-3, which appears to be realistic. In calculating this value, we used a density of ink equal to 0.88 × 103 kg m-3. Going from the center of the pixel toward the edge along the minor axis, the SiO and resist structures are present at x/X0 ) 0.61 and 0.81. Along the major axis of the pixel, these are located at y/X0 ) 1.95 and 2.15. These positions give rise to numerical inaccuracies because the droplet height in eq 16 is discontinuous, which causes fluctuations in the calculated layer thickness. For values of x/X0 > 0.61 and y/X0 > 1.95, a good comparison between experiment and simulation is not possible. At these positions, the polymer rests on top of the SiO layer and/or resist structure. Because the thickness of these layers can vary somewhat and is significantly larger than the polymer layer thickness, the polymer layer thickness cannot be accurately inferred from the measured profile.
Figure 6. Comparison of simulated layer thickness profiles with varying diffusion coefficients.
A comparison between the experiment with ink P1 and the simulations shows that most experimental features are reproduced in the simulations, that is, the convex shape in the center and the slight increase at x/X0 ≈ 0.5 that appears to be related to the presence of diffusion. However, the increase in the measured layer thickness at 1.5 < y/X0 < 1.95 is not visible in the simulations. To study the effect of solute diffusion in more detail, a series of simulations with different diffusion constants have been performed, all with the lower evaporation rate of 10-8 m s-1. The results are shown in Figure 6. It can be seen that increasing diffusion counteracts the convection of solute toward the edges of the droplet and results in a less flat polymer layer in the center of the pixel. The highest solute concentration is no longer obtained near the edge but approximately halfway between center and edge, where it shows as a sudden increase in polymer thickness. Figure 7 shows a comparison between experiments and simulations for 24 different inks based on 16 different solvents and 5 different polymers. In this Figure, two parameters are plotted. On the y axis, we plot H0c0F/FdryHdry, where Hdry is the central layer thickness after drying and Fdry is the density of this dry layer. This parameter expresses the amount of polymer that has moved out of the center of the pixel. We estimate Fdry/F to be equal to 0.9 for all experiments, although this value is
Van Dam and Kuerten
588 Langmuir, Vol. 24, No. 2, 2008
Figure 7. Comparison of mass transport out of the center (represented by H0c0F/FdryHdry) between experiments and simulation models.
probably different for the various solvent-polymer combinations. H0 is calculated from the number of droplets deposited, where for all fluids printed with the Spectra SX head a volume of 6 droplets of 14 pL is used and for all droplets printed with the spectra Galaxy head a volume of 5 droplets of 17 pL is used. These droplet volumes represent average values for the inks considered. In the simulations, a volume of 85 pL is used. For the viscous simulation and the experiments, we plot on the x axis c1 Pa s/c0, which is the increase in concentration, with respect to the initial concentration, that is needed to reach a 100-fold increase in viscosity. For the inviscid simulations, the value on the x axis represents ccr/c0. Error bars of (30% relative error are plotted in the vertical direction, mainly representing the variation in Fdry/F and the random experimental error in the ink jet printing experiments. Error bars of (15% relative error are plotted in the horizontal direction, mainly representing the error in the measured viscosities. In the viscous simulation, we used an evaporation rate of Ve,av ) 10-7 m s-1. It turns out that the curve is relatively insensitive to variations in the evaporation rate. For a higher value of Ve,av, the curve tends to get flatter. We scale the shape of the viscosity curve in the visous simulation results in Figure 7 to the curve presented in Figure 2. We assumed the diffusion coefficient D to be zero. The error in the simulated curve as a result of the error in the input parameters is not indicated in Figure 7. The data points in Figure 7 show a relatively large error because of the errors in Fdry, droplet volume, and the determination of c1 Pa s, on top of the error caused by the variation in the print process in the experimental pre-pilot line. The simulation results are subject to errors in the input parameters and the neglected processes such as Marangoni flow, which may be significant in a few cases. Still, a coherent trend can be observed in the data and in simulations in which the amount of polymer that moves out of the center of the pixel increases with ccr/c0. Fukai et al.17 experimentally found only a small effect of variable initial polymer concentration on the resulting polymer film thickness in a case where the droplet is not pinned to the substrate. In that case, the wetting diameter of the droplet decreases with time.
5. Discussion We presented a numerical model for the drying of fluids that was verified by comparison with ink-jet-printed polymer solutions. In this verification, a number of assumptions has been made. One of the main assumptions is the disregard of solute (17) Fukai, J.; Ishizuka, H.; Sakai, Y.; Kaneda, M.; Morita, M.; Takahara, A. Int. J. Heat Mass Trans. 2006, 49, 3561-3567.
Marangoni effects. In addition, the evaporation rate was taken to be constant with time. Both of these assumptions deny the multisolvent character of the inks. The evaporation velocity and its distribution over the surface of the fluid were unclear, which was another complication. We took a pragmatic approach here by defining the boundaries of the evaporation rate and noting that the numerical model was relatively insensitive to the evaporation rate for the relevant values of the other parameters. The results in Figure 5 show that a higher evaporation rate leads to a polymer layer that is thicker in the center and has a more convex shape. This can be understood by considering that the higher evaporation rate leads to higher fluid velocities and more viscous dissipation and, as a result, less solute transport out of the center of the pixel. This also explains why the sharp peak near the straight edges of the pixel becomes less pronounced for higher evaporation rates. If diffusion is taken into account, then the layer thickness in the center increases somewhat. This effect is comparable to the mechanism identified earlier in drops of colloidal silver suspension.14 In that paper,14 the radial inviscid model shows the transition from diffusion-dominated drying (ND . 1) to convection-dominated drying (ND , 1). (Please note that the experimental verification in ref 14 suffers somewhat from the insecurity about actual evaporation rates.) In the case of diffusion-dominated drying, no significant concentration gradients exist, and effectively no polymer transport occurs. It is instructive to calculate for this case the value of ND (based on the inviscid model) in eq 19. This yields ND ) 0.6 (using H0 ) 10-5m, D ) 4.7 × 10-12 m2 s-1, Ve,av ) 10-8 m s-1, and R0 ) 87.5 µm as inferred from the half distance between the contact lines along the major axis). This indicates that diffusion indeed has some effect, and this agrees fairly well with the differences we see between the simulations in Figures 5 and 6. The comparison of the experiment with polymer ink P1 and the simulations shows reasonable agreement and a small diffusion effect. However, along the major axis of the pixel the accumulation of polymer near the edge of the pixel appears to be underestimated in the simulation. It is unclear what causes this mismatch, especially because it is not present along the minor axis of the pixel. When comparing the model and experimental results presented in Figure 7, it is hard to draw a conclusion on a preferred drying model based on data points with 1.5 < c1 Pa s/c0 < 4. Some data points do not agree with either model, which may be explained by the rigorous assumptions made in the models and by the error in the input parameters in the simulation (e.g., the neglecting of diffusion in some cases). However, when looking at data points with 5 < c1 Pa s/c0 < 7, a clear preference for the viscous model can be inferred.
6. Conclusions We built a numerical model suitable for simulating the drying of ink-jet-printed structures. The model makes some rigorous assumptions, thus limiting the number of input parameters required. These assumptions were justified for comparison with our experimental results. Although the comparison was of limited extent and uncertainty exists about some of the input parameters in the model, fair agreement between experiments and simulation was obtained, thus showing the prospects of the model as a starting point for more detailed modeling approaches. In particular, our work indicates that the viscous model describes the experiments better than the inviscid model.
Modeling the Drying of Ink-Jet-Printed Structures
In our opinion, the current model presents a good compromise between complexity on the one hand and usability in an industrial environment on the other hand. The model prediction can be used to steer the basics of ink development. In addition, it can be extended relatively easily to include more material properties, if necessary, or to change the evaporation rate during the simulation to represent the multisolvent character of an ink.
Langmuir, Vol. 24, No. 2, 2008 589
Acknowledgment. We thank Roland van de Molengraaf for the NMR measurements, Martin Hack for the viscosity measurements, Harry Nulens for the measurement of the layer thickness, and Dirk Roos and Willem Huugen for performing the printing experiments. J.G.M.K. thanks Philips Research for their hospitality during his stay there. LA701862A
