The Colloid Chemistry of Silica - American Chemical Society

21 Evaporation and Surface Tension Effects in Dip Coating Downloaded by NORTH CAROLINA STATE UNIV on May 3, 2015 | http://pubs.acs.org Publication Date: May 5, 1994 | doi: 10.1021/ba-1994-0234.ch021

A l a n J. H u r d Ceramic Processing Science, Department, Sandia National Laboratories, Albuquerque, NM 87185-5800

Evaporation sets an important time scale for the formation of structure in sol-gel films during dip coating, and surface tension is the dominant driving force influencing that structure. The action and interplay of these two phenomena were evaluated by experiments with pure and binary solvents. From the optically measured thickness of the steady-state film profile, accelerated evaporation near the drying line that sets stringent constraints on the time available for network formation was found. In binary solvents, there is evidence for strong flows driven by surface tension gradients; this flow gives rise to capillary instabilities. Aided by these flows, differential evaporation leads to regions rich in the nonvolatile component near the drying line.

D

IP C O A T I N G is T H E DEPOSITION O F A S O L I D F I L M on a substrate by

immersion in a sol or solution, withdrawal, and drying. The simplicity of dip coating, a cousin of painting, belies the fact that films of very high quality can be applied. Indeed, optical-quality films of controlled index and thickness are readily obtainable with simple, inexpensive apparati. Complex shapes can be coated in one step; this simplicity is not always possible with evaporative or sputtering techniques. For bulky objects, dip coating is far easier to scale up than vacuum techniques. Finally, the admirable purity of solution chemistry, such as the popular sol-gel route, can be exploited. According to a review by Schroeder (J), the technology of spin coating or dip coating inorganic sols to make stable films was pioneered in 0065-2393/94/0234-0433$08.00/0 © 1994 American Chemical Society

In The Colloid Chemistry of Silica; Bergna, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

Downloaded by NORTH CAROLINA STATE UNIV on May 3, 2015 | http://pubs.acs.org Publication Date: May 5, 1994 | doi: 10.1021/ba-1994-0234.ch021

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Germany and became widely known after W o r l d War Π. The physics involved in spin and dip coating has been reviewed (2) and remains under intense study (3, 4). The flexibility of sol-gel chemistry is demonstrated by the wide range of oxides and mixed oxides that have been used to form coatings (5, 6). The term "controlled index" means that the refractive index can be made smaller than that of the bulk precursor by controlling the microstruc ture via the porosity. W h e n silica is deposited, for example, the film index η can be varied over a wide range (7) from η = 1.1 to 1.5. This process control makes sol-gel coatings interesting for many optical, electronic, and sensor applications, but the evolution of the microstructure during film formation is not well understood, in spite of efforts to survey the variables (8, 9). This chapter reviews the important factors determining the microstructure of dip-coated films and explores at length two of them, evaporation and surface tension. Although easy to accomplish, the process of dip coating is complex because it proceeds through overlapping stages: W h e n the substrate is withdrawn slowly from a sol, a film of liquid, several micrometers thick at the bottom, becomes hydrodynamically entrained on the surface. If the solvent wets the substrate, the film thins through gravitational draining, capillary-driven flows, and evaporation. W h e n the recession speed of the drying film (relative to the substrate) matches the withdrawal rate, steadystate conditions prevail, and the entrained film terminates in a welldefined drying line that is stationary with respect to the reservoir surface. As described later, the presence of this edge in the evaporating film leads to dramatic effects. Meanwhile, the precursors in the entrained liquid experience a rapidly concentrating environment; they tend to gel or jam through chemical or physical interactions. Most likely, a transient chemical or physical gel network occurs fleetingly in the thin liquid film under these conditions, and it is my view that the porosity of the deposited film is a remnant of this network. Often the reservoir sol is unstable with respect to aggregation and gelation. However, the process of film formation forces reactions at a much accelerated rate: Although the bulk reservoir might require several hours or days to gel, the transit time from entrainment to drying line is of the order 10 s. Here is the first competition of time scales for film deposition. Clearly, network formation through the usual diffusion-limited and reac tion-limited schemes can be frustrated by the accelerating effects of evaporation. In this context, the evaporation can be viewed as a strong force field coupling to the suspended particles—analogous to a centrifugal or electrophoretic field—and forcing them to crowd together. (The centrifugal acceleration causing an equivalent rate of crowding is as much as 10 g.) W h e n the crowding is rapid, particles do not have time to find low-energy configurations, so porous microstructures result. Thus, sol-gel 6



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films can differ greatly in structure from bulk xerogels or aerogels (10) i f desired. Just before he died, Ralph K. Her (personal communication, 1985) was working on a device for electrophoretically depositing particles with a controlled degree of order. In addition to the direct current field for deposition, he imposed an alternating current component. At low frequen cies, the ac component tended to unsnarl packing defects, whereas at high frequencies it apparently created a dipolar interaction between particles. Although the evaporation creates varying physicochemical states of the entrained sol with height from the reservoir, it is easy to show that in most situations the sol is essentially homogeneous across its thickness. [If it were not homogeneous, the particles might collect near the air interface of the entrained film as a sort of skin that would impede evaporation ( I I , 12).] In most situations the particles' transport by diffusion is fast enough to keep the concentration constant through the thickness: For a liquid film of thickness Ar « 1 μπι and a diffusivity Do of 1 0 cm /s (appropriate for a 10-Â moiety), it takes only a time At ~ Ar /D = 10~ s to relax thickness concentration gradients that might build up. For most positions on the film, this time scale can be considered short compared to other processes. Only very near the drying line itself would it be possible for concentration gradients normal to the substrate to "lock i n " . Some evidence suggests that such inhomogeneities exist (Fabes, B., personal communication and poster presentation at the Spring Meeting of the Materials Research Society, San Francisco, C A , 1990) through the thickness, but there does not appear to be a deep enough data base to conjecture about their origin. - 6

2

2

2

Q

Gravitational draining creates hydrodynamic shear throughout the entrained film. Unlike particle concentration, the shear rate is not the same throughout the thickness. In fact it must be zero at the air interface (because the gas cannot exert a shear force on the film, assuming, for the moment, no surface tension gradients exist) and nonzero at the substrate, which provides the force of lifting from the reservoir. Thus the shear rate must be maximal at the substrate: If ζ is the distance normal to the substrate and h is the thickness of the entrained liquid film, then the velocity (u) satisfying the Navier-Stokes equation d u/dz = 0 is parabolic: 2

/

h(x) ζ

_

2

J\

Here λ is a characteristic length of order 10 μτη given by λ = pg/rjuo, where ρ is liquid density, g is gravitational acceleration, η is shear viscosity, χ is height and m is substrate withdrawal velocity. The shear rate 7 is due solely to gravitational back flow in the absence of surface-driven forces: 2


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du

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(i

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ν


Thus, the shear is greatest near the reservoir, where the film is thickest, and next to the substrate wall ζ is 0. It is interesting to consider whether shear-induced particle encounters are slow compared to difiusional transport. The relevant parameter is the Peclet number

P e

_ "

6πα? ~W

du dz

(for spheres of radius a, where k is the Boltzmann constant and Τ is the temperature), which is the ratio of these time scales. When P exceeds 1, it is well known that the structure of the dispersion is constantly forced by the shear to nonrandom states, typically ordered sheets or strings. For P « 1, diffusion randomizes the structure. Gravitationally driven shear is generally not strong enough to create "shear-induced ordering", even near the substrate at the base of the entrained film, where P is only 10~ . However, shear driven by surface tension gradients can be quite large, so shear-induced order might be exploited to affect film microstructure (13) in coatings derived from colloidal suspensions (14). Indeed, surface tension is arguably the dominant force in dip coating, at least at the point of entrainment and at the drying line where interfacial curvatures are significant. At the point of withdrawal from the reservoir, a meniscus forms to balance the pressure imposed by the curved surface against that of the gravitational " h e a d " . The relatively large volume of liquid pulled into the gravitational meniscus (radius of curvature about 1 mm) is indicative of the large surface tension of dip-coating liquids, and the effect (15) of surface tension is large enough to change the dependence of the entrained thickness h(0) (proportional to deposited mass) on withdraw al speed m from wo to m l . The deposited thickness need not scale in this way with withdrawal speed because coating porosity can vary with other factors (16, 17). Between the meniscus and the drying line, the radius of curvature is too large (—10 km!) for surface tension to have any effect, but, at the drying line, capillary pressures are again significant. Probably the most important surface tension effect is that of "capillary collapse" of the transient networks as they are invaded by the gas phase during the final phase of evaporation (18). This process is identical to the collapse of a sponge upon drying. The network can resist the invading menisci up to the point when their radii are small enough that the capillary stresses exceed the yield stress of the network (19). The network then compresses uniaxially until its modulus becomes high enough again to resist the capillary stress while the solvent is completely removed. e

e

e

1/2

2 s


4


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(Uniaxial compression is not the only possible mode of collapse. Large, stable pores could form, for example, by the lateral retraction of material.) By now the structure is denser and, possibly, completely different from that of the transient network. Nevertheless, the extent of capillary collapse can be controlled to some extent (7) by physical and chemical means to achieve a desired porosity. Surprisingly large capillary pressures are theoretically possible i n drying films. (In order of magnitude, there is little decrease i n surface tension at a rapidly evaporating interface, or even a boiling one.) Because the final pore size can be smaller than 1 nm, the pressure i n the liquid during the final stages of drying could exceed —100 atm (—10,000 kPa)! The negative sign indicates that the liquid is under tension. Although most liquids can support large tensions in small pores if no gas is present (20, 21) owing to suppression of nucleation, it is not clear how the liquid in open pores is similarly prevented or delayed from boiling away (22a). Nevertheless, strong evidence for suppressed vaporization is the hysteresis i n adsorption isothermus of microporous solids (20): L i q u i d is reluctant to leave microsize cavities once it has filled them, because, presumably, the pores are smaller than a critical vapor nucleus. Moreover, recent experiments in deflection of sol-gel-coated beams (22b) indicates drying stresses of enormous negative pressure (2000 atm, or 21,000 kPa) generated by the capillary pressure. The choice of solvent mixtures may be the route to controlling capillary forces. The main subject of the remainder of this chapter is the variety of effects that occur i n mixed solvent systems during dip coating. These effects arise when differential volatility gives rise to concentration gradients, thence to surface tension gradients, which have surprisingly large effects on flow. A few of these effects are discussed. First a description is given of the evaporation constraints, because these determine, to first order, the local composition of the film.

Experiments on Evaporating Thin Films Clean substrates of silicon were used to entrain liquid films of various compositions i n a dip-coating geometry, as i n Figure 1 with a = 0. Interference images were obtained in reflection with monochromatic light. The liquid index of refraction and the angle of incidence were known, so the film thickness profiles were obtained from the position of the interference fringes. The process of evaporation during dip coating was studied only recently (2, 22a), although the essential physics of evaporation has been well known for over a century. According to Fuchs (23), James Clerk Maxwell wrote an article on diffusion for the Encyclopedia Britannica i n which he considered the stationary evaporation of a spherical droplet in an



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Figure 1. Geometry of dip coating. The angle of the substrate a can be adjusted to vary the effects of gravity g. A stagnation point in the velocityfieldu occurs in the gravitational meniscus. The thickness h(x) is in the range Oto 10 μm, and the height χ is of order 1 cm. (Reproduced with permission from reference 22a. Copyright 1990.)

which he considered the stationary evaporation of a spherical droplet in an infinite medium. (Maxwell was interested in wet bulb thermometry.) Not only did he realize that the rate of mass loss by the droplet is limited by vapor diffusion away from the surface, he correctly assumed that the vapor concentration at the surface of the drop is equal to its equilibrium saturation concentration (true when the vapor mean free path is small compared to the dimensions of the droplet). A l l that remains is to solve the steady-state diffusion equation for the vapor concentration c, V

2

c

=

0

(1)

on a sphere of radius a, with an additional boundary condition at infinity. The flux from the surface, which can be defined as a local evaporation rate (£), is governed through Fick's law by the vapor concentration gradients there: Ε

=

mass loss per unit area per unit time

=

ôc —D — | r = a

(2)

Although the diffusion constant D does not appear in the steady-state diffusion equation, it does appear in kinetic factors such as the time it takes for a droplet of a given initial mass to evaporate. The solution to equation 1


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is a concentration c that decreases as r from the droplet so that the total evaporation rate 4πα Ε from equation 2 is proportional to the product aD. Thus, it is not the surface area that controls the rate of mass loss, but the radius. Other geometries can readily be worked out. As a useful analogy, the concentration c in equation 1 can be viewed as the electrostatic potential around a conductor of potential co. The analog to the local evaporation rate is the electric field, evaluated at the surface of the conductor. B y this analogy a fresh set of intuitive ideas can be brought to bear on evaporation problems. For example, it is not surprising that the vapor density around an infinite cylindrical source drops logarithmically with radial distance, and that the evaporation rate varies inversely with the radius of the cylinder. Similarly, the vapor concentration above an infinite sea drops linearly with distance, whereas the evaporation rate is constant every where on the surface. _ 1


2

Dip-coating geometries are not perfect one-, two-, or three-dimension al structures, however. Usually a sharp boundary is involved, such as the edge formed by the drying line. When coating on flat substrates, the film can be considered a thin, finite sheet. A dip-coating film can be approximated by a semi-infinite sheet, mathematically formed from a wedge in the limit that the wedge angle approaches zero. The edge carries a field singularity (24) of the form r / . In fact, for an arbitrarily shaped finite sheet, the field singularity remains r~ l as long as the edge is locally straight on the scale of the sheet's thickness. This fact can be seen from the exact solution for the field above a thin disk, which can be readily shown to be encircled by a field singularity of the form r / , where r is the distance to the edge (25). -1

2

l

2

-1

2

Such evaporation singularities can be very easily observed because the thickness profile of a thin liquid film locally reflects the rate of solvent removal. For very thin films, especially near the drying line, back flow due to gravitational draining can be neglected. The mass flux h(x-rdx)m carried into a fluid element dx, as shown i n Figure 1, is balanced by the flux carried out by the substrate h(x)uo and the mass lost through evaporation E(x)dx; this observation leads to the continuity equation dh dx

E(x)

(3)

UQ

which can be integrated for Ε ~~ x~v to give h(x) ~

χ

(η

0

(5a)

x >

*2 , h =

χ

Χ 2 )

(0 < χ < χ ) 2


( 7 α )

(7b)

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CN

ο


(Ν

Φ

Figure 7. Thickness profile of 50:50 (vol) propanol-water film. The doublechin "phase separation" is due to differential volatilities and surface-driven flows. (Reproduced with permission from reference 13. Copyright 1991.)

Φ

Ε

3 Ô >

χι Figure 8. Schematic plot of volume fraction of water-rich phase φι for propanol-water film, based on Figure 7.


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445


assuming the no-flow profiles of equation 4 hold as adopted for equations 5a-5e. Not surprisingly, the abrupt disappearance of phase 2 at the false drying line xi leads to singular gradients. But the simple no-flow picture of equation 4 can no longer hold in view of equations 7a and 7b. At the liquid-vapor boundary, the viscous shear force must balance the force imposed by surface tension gradients, ηάη/άζ = da/dx (z = h). This boundary condition leads to a linear flow profile toward the drying line, U

=

1 η

da ζ ax

-j-

-

Uo

(8)

so the profiles hi and /12 have to be recalculated. Profiles for static menisci of binary solvents have been calculated (28). However, far from the singularity at the false drying line X2, equation 8 should describe the physics accurately enough so that the strength of the surface-driven flows can be appreciated. Figure 9 shows the thickness profile of a binary mixture of toluene and methanol during film formation. The flows are strong enough to distort greatly the foot profile, creating, in fact, a thickened " t o e " of toluene near the drying line. A crude estimation on the shear rate in the thin region is Δσ/Δχ « (10 dyne/cm)/(10 cm), hence du/dz ~ 10 s . For this sort of flow, P exceeds 1 for particles over 30 nm or so; thus shear-induced ordering would be expected. Temperature gradients can often be considered small in dip coating; this assumption is not generally true in thin-film evaporation problems (28). However, because the substrate moves relatively rapidly past the _1

4

_1

e

10 χ

15

20

(mm)

Figure 9. Thickness profile of50:50 (vol) methanol-toluenefilm.A steady-state "bubble" of toluene forms because of strong surface-driven flows. (Repro duced with permission from reference 13. Copyright 1991.)



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drying line, it constantly supplies energy for evaporation and renews the thermal field. A n interesting manifestation of surface flows in dip coating is a " r i b instability" often observed near the reservoir in binary solvent systems and in some pure solvents (ethanol). Figure 10 shows a set of wavy interference fringes, indicative of a 20-nm thickness undulation that has a wave vector that runs perpendicular to the withdrawal direction. Typically, this undulation is a standing wave, but under some conditions (not well defined at this time) the ribs have been observed to fluctuate in position considerably. A study of the instability in water and methanol reveals that its wavelength is inversely dependent on the withdrawal velocity m and that it is insensitive to solvent composition (Figure 11). A n increase in uo fattens the entrained film (18); this result would be expected to increase the wavelength, not decrease it. Furthermore, although the instability has been observed rarely in pure solvents, it is more definite in binary mixtures; hence it is probably aided by surface-driven flows, but, as the data show, the wavelength is relatively insensitive to the mixture. These observations suggest that the ribs are related to the " P l a teau-Rayleigh instability" of cylindrical surfaces of liquids (29), such as a jet of water, the cylindrical surface for dip coating being the concavity in the gravitational meniscus. First treated in the 19th century by Plateau and L o r d Rayleigh, this instability results from the fact that a cylindrical liquid surface can decrease its area by undulating longitudinally; from dynamic

Figure 10. Imaging ellipsometry of rib instability near the reservoir meniscus (dark region).


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instability wavelengths

Ο

Ο CN


Ε ο Ε q en c ω ο ω oo δ 6 Ο CD

Ο

χί-

0.00

0.50

1.00

withdrawal

speed

1.50

2.00

(mm/s)

Figure 11. Wavelength of rib instability as function of withdrawal speed for several concentrations of methanol-water.

considerations, one particular wavelength emerges as the fastest-growing unstable mode (although not necessarily the wavelength that fully devel ops beyond the linear-response regime). Unstable conditions can exist for concave cylindrical surfaces, but the complex dynamics determining the wavelength represent an unsolved problem. Because higher withdrawal speeds decrease the wavelength and, eventually, smooth out the ribs altogether, it probably takes some time for the instability to organize. The balance point for this competition of time scales is generally within the processing window of typical dip-coating operations.

Summary: Time Scales The limitation on structure formation in dip coating is best appreciated by considering the mean separation between two reactant molecules. By the conservation of nonvolatile mass, the concentration at χ is inversely related to the entrained film thickness (22), and because the mean separation is the inverse cube root of the concentration, A s

~

*1/6

Americao Chemical Society

Library

1155 ltth St. N.W. Washington, D.C. 20036 In The Colloid Chemistry of Silica; Bergna, H.; Advances in Chemistry; American Chemical Society: Washington, DC, 1994.

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The Colloid Chemistry of Silica - American Chemical Society

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