WO2011156648A1 - Method for producing films - Google Patents

Abstract

The present invention provides a process for producing thin solid films of a substance by drawing a thin liquid film of the substance from a bath, stabilizing the film in a liquid state by inducing a surface tension gradient along the length of the film, and subsequent cooling or curing to form a solid film. The present process permits contact- free thin films to be produced more rapidly than conventional processes, and allows precise control of the final film thickness.

Description

METHOD FOR PRODUCING FILMS

Reference to Related Applications

[0001] The present application claims the benefit of the earlier filing date of U.S. Patent Application No. 61/352,984, filed on June 9, 2010, the contents of which is incorporated by reference herein in its entirety.

Government Support

[0002] This invention was made with U.S. Government support under DMR-0213805 awarded by the National Science Foundation. The U.S. Government has certain rights in the invention.

Background of the Invention

[0003] Thin films have numerous useful applications. For example, thin inorganic films, also called ribbon crystals, can be used in a variety of electronic devices, including

semiconductor wafers and solar cells. Conventional processes for forming ribbon crystals include dendritic web growth, growth from a capillary shaping dye, growth with edge supports or "strings," and growth on a substrate.

[0004] Thin silicon sheet material or ribbon is particularly important in making solar cells. Conventionally, continuous silicon ribbon growth is carried out by passing two high temperature material filaments or strings vertically through a crucible that contains a layer of molten silicon. The strings serve to stabilize the edges of the growing ribbon and the molten silicon freezes into a solid ribbon just above the molten layer. The molten layer that forms between the strings and the growing ribbon is defined by the meniscus of the molten silicon.

[0005] However, control of the geometry of the resultant ribbon often is problematic. Thin films formed from a molten bath according to conventional string processes often evidence uncontrolled irregularities in width and/or thickness, which may reduce the usefulness or effectiveness of the films. For example, non-uniform thin silicon films cause variability in the relative spacing of the finished crystals on a solar cell, thus reducing the total power produced by the cells. Moreover, conventional ribbon processes tend to be extremely slow, typically producing silicon ribbon at the rate of about 2 cm/minute or less. Summary of the Invention

[0006] It is an objective of the present disclosure to provide an improved process whereby high quality thin films of silicon and other substances can be rapidly and efficiently produced.

[0007] The present disclosure provides a process for producing a thin solid film of a substance by drawing a thin liquid film of the substance from a bath under controlled conditions, imparting a desired surface tension gradient to the thin liquid film, and subsequent cooling or curing of the liquid film to form a solid film. The present process permits films to be produced more rapidly than conventional ribbon processes, and without the need for a substrate. Moreover, the present method allows precise control of the final film thickness.

[0008] The present method generally comprises providing a bath of the substance in a liquid state, and drawing a liquid sheet or film of the substance from the bath while inducing a surface tension gradient along the length of the emerging liquid sheet or film, thereby stabilizing the film in a liquid state. The surface tension gradient may be induced by applying a physical field, such as a temperature gradient, an electrical gradient, or a gradient in the concentration of a surfactant. The gradient is controlled, thereby allowing the physical properties of the film surfaces to be precisely modulated. The liquid film is induced to solidify once the desired properties are achieved, thereby forming a solid sheet having the desired properties (e.g., thickness, uniformity). The thickness of the film can be controlled by regulating the parameters of the applied gradient, and the speed at which the liquid film is pulled from the bath. In certain embodiments, the speed at which the liquid film is pulled can also affect the applied surface tension gradient. The method may further include curing or actively cooling the material in a controlled manner to form the solid film.

[0009] In one aspect, the method includes contacting a bath comprising the material to be formed into a thin sheet or film in the liquid or molten state, with a device capable of pulling or drawing a thin film of the liquid from the bath using the surface tension of the liquid to form the film. In certain embodiments, the direction of pulling can be substantially perpendicular to the surface of the liquid. However, the direction of pulling is not so limited. In one embodiment, the pulling device comprises two or more filaments or strings (referred to hereinafter as "strings"), which are positioned substantially parallel to one another. The strings can be drawn through the bath in a manner which allows the liquid material to form a film spanning the distance between the strings. As the film rises from the bath, a temperature gradient or other physical field can be applied in order to induce, impart and/or control a desired surface tension gradient at the liquid film surfaces (e.g., the surface tension increases with the distance from the bath), thereby stabilizing the liquid film such that it attains the desired properties. For example, for a material where the surface tension decreases with increasing temperature and a surface tension that increases away from the melt is desired, a desired surface tension gradient can be induced by locally heating the film at the meniscus or in the melt near the meniscus. Alternatively or in conjunction with the local heating, the desired surface tension gradient can be induced by locally cooling the film near the solidification front. Suitable techniques include localized heating elements, lasers, and the like. Once the desired properties are achieved, the film is allowed or induced to solidify, e.g., by cooling, UV curing, and the like.

[0010] The present method allows thin films to be produced more rapidly than

conventional processes, and without the need for a substrate. Using the present method, highly uniform thin films can be produced at speeds much greater than conventional ribbon methods, e.g., on the order of centimeters per second compared to centimeters per minute in

conventional processes. Films produced in accordance with the present method can have very smooth interfaces. Films produced in accordance with the present method can also be substantially free of substances that are not initially in the liquid bath from which the film was pulled. The present method allows precise control of the final film thickness through modulation of pulling speed and an externally applied temperature gradient or other physical phenomenon that induces a surface tension gradient in the emerging liquid film.

[0011] In certain embodiments, the present disclosure includes a process for producing a film. The process can include: providing a bath of a substance in a liquid state; pulling a liquid film of the substance from the bath; applying a predetermined a surface tension gradient along the length of the film that allows the liquid film to be pulled at a desired pulling speed; and solidifying the liquid film.

Brief Description of the Figures

[0012] FIG. 1 shows a method to produce thin films by applying a desired surface tension gradient in accordance with certain embodiments;

[0013] FIG. 2 is a schematic diagram illustrating a thin film of fluid being pulled from a bath in accordance with certain embodiments; [0014] FIG. 3 shows a method to calculate the temperature gradient that should be applied to obtain a desired surface tension gradient as a function of film thickness and pull speed in accordance with certain embodiments;

[0015] FIG. 4 is a schematic diagram illustrating a thin film of fluid being withdrawn from a bath of the same fluid in the presence of a temperature gradient, where the fluid solidifies as it cools some distance away from the bath in accordance with certain embodiments;

[0016] FIG. 5 is a graph showing the non-dimensional curvature H" of a silicon sheet as it reaches the bath versus non-dimensional length scale D over which the temperature varies along the j-direction (non-dimensionalized by the length of the dynamic meniscus £ which itself is set by parameters of the problem including the thermocapillary stress parameter Γ). D = 1 corresponds to a temperature drop of about 440 K over a length of about 1 mm for the example of a silicon sheet of 300 microns thickness withdrawn from the melt at 10 cm/s in accordance with certain embodiments;

[0017] FIG. 6 shows the result from simulation of the Stokes equations in COMSOL that does not include temperature dependence for the viscosity, nor does it include inertial effects, but it does include gradients of surface tension. This numerical simulation demonstrates the ability of surface tension gradients to thicken a film that may otherwise drain rapidly under the action of gravity. The left panel shows no surface tension gradient. The right panel shows a surface tension gradient where the surface tension changes from γ₀ at the bath (y = 0), to 1.05 γο at twice the capillary length £ _c above the bath (y = 200); and

[0018] FIGS. 7A and 7B shows the temperature distribution from the solidification front (T_coid=1687K for silicon) to the hottest temperature (T_hot). The different curves show the influence of the heat transfer from the film to the ambient. The smaller the heat transfer coefficient, the smaller will be the effective temperature gradient if we keep all other parameter settings the same. FIG. 7A and 7B show results for two different pulling speeds. The larger the pulling speed (above 1 cm/s), the larger will be the temperature difference required to produce a steady film of a given thickness.

Detailed Description

[0019] As used herein, a "film" refers to a material having a thickness that is much smaller than the width and/or the length. The terms "film," "sheet," and "ribbons" will be used interchangeably throughout the application. [0020] As used herein, a "liquid film" refers to a film that is above the melting point of the material. A "liquid film" also refers to a film that is at a temperature above the glass transition temperature of the material.

[0021] FIG. 1 shows a method of producing a film in accordance with certain

embodiments. In 101, a bath containing a liquid or molten material is provided. Techniques and apparatus for forming baths of liquid or molten material, crucibles and furnaces for high temperature melts, and devices and techniques for pulling the film from a bath (including strings) that are conventionally employed can also be utilized in the present disclosure. For example, U.S. Patent Nos. 4,322,263; 4,689, 109; 7,507,291 B2; and 7,407,550 B2, which are incorporated by reference herein in their entireties, describe certain apparatuses that can be utilized.

[0022] In 103, a liquid film is pulled from the bath containing the liquid or molten material. For example, parallel high-temperature filaments can be drawn through a bath of the film- forming material in a liquid or molten state, under conditions sufficient to induce the material to adhere to the filaments, forming a thin film that is suspended between the filaments as they emerge from the bath. In certain embodiments, the filaments can be positioned so that the film emerges from the bath in a substantially vertical direction. In certain embodiments, an existing solid film or horizontal wire filament mounted between the two withdrawal filaments may be placed in contact with the melt to form a meniscus.

[0023] In certain embodiments, the liquid film can be a free standing film wherein the sides of the film are not supported by any solid substrate, in contrast to films obtained by solution casting on a substrate where one side of the film is supported by the substrate. Rather, except for the contact with the filaments (or other devices) to initiate the pulling of the film, the film can be contact-free with other support structures.

[0024] As shown in FIG. 2, pulling the film produces a meniscus 203 over the bath 201 which tapers off to the final thickness and eventually solidifies at the solidification front 205. In certain embodiments, the fluid solidifies as it cools some distance away from the bath.

Alternatively in the case of some polymer solutions, curing, e.g. , by UV light, could be used to solidify the thin film produced by the withdrawal process. The newly formed solid film can pull on the fluid phase, to keep the motion continuous.

[0025] Returning to FIG. 1, in 105, a desired surface tension gradient along the pulling direction is applied to the film surface 207. In 107, the liquid film is allowed to solidify at the solidification front 205. The desired surface tension gradient is applied to stabilize the film and provide a number of distinct advantages over the conventional art.

[0026] The present disclosure represents a substantial improvement over conventional ribbon processes in which thin films are produced by drawing strings through a liquid or molten bath of a film-forming material. Conventional string ribbon film processes rely upon heat transfer from the meniscus such that the film solidifies as it emerges from the bath, i.e., at the end of the meniscus. Generally, in a conventional technique, the bath 201 containing the molten material (Tbat_h) is kept at a temperature just above the melting temperature (T_m), such as T_m + a few degrees Celsius. Surface tension gradients were not actively established on the liquid film surface. Accordingly, the conventional techniques have suffered due to slow pulling speeds and/or uncontrolled film thickness variations. For example, conventional techniques for pulling silicon films were limited to centimeters per minutes. Faster pulling rates led to undesirable and uncontrolled film thickness variations.

[0027] In contrast, the present disclosure utilizes the fluid mechanical properties and/or physico-chemical properties of the molten material to adjust the surface tension and other fluid properties of the film to obtain a desired result. In particular, it has been discovered that inducing a surface tension gradient in the emerging liquid film stabilizes the liquid film, thereby permitting the use of faster pulling speeds and more precise control of film thickness. Accordingly, the process of pulling molten silicon out of a bath and letting it solidify can be sped up substantially. For example, whereas conventional techniques were pulled at a speed of about 2 cm/min (0.033 cm/s) or less, speeds on the order and in excess of several centimeters per second and even tens of centimeters per second (cm/s) can be achieved using the present method. For example, pulling speeds of about 0.5 cm/s, 1 cm/s, 2 cm/s, 5 cm s, 10 cm/s, 15 cm/s, 20 cm/s, and even about 50 cm/s or more may be possible.

[0028] In certain embodiments, inducing a surface tension gradient in the emerging liquid film can be utilized to achieve even slower pulling speeds and more precise control of film thickness than conventional techniques, if so desired. For example, because surface tension gradient can allow for fine control of the thickness including modulations of the thickness, it may be desirable to pull faster and slower as a function of time to produce corrugated films or films having varying thicknesses. The present methods provides significantly greater flexibility in pulling speeds, and is not limited by a small processing window of a limited range of pulling speeds of conventional techniques. [0029] The present method can be applied to any film- forming material, including crystalline materials, semiconductor materials, amorphous materials, glass, polymers or metals, that has the physical properties (e.g., surface tension or viscosity) necessary to form a solid film upon cooling, e.g. , by crystallization. The present method is particularly useful for making thin, uniform sheets or ribbons of silicon. The contact- free nature of the approach minimizes contamination.

Surface Tension Gradient

[0030] As noted above, the present disclosure provides a surface tension gradient in a controlled manner in the liquid film as it emerges from the bath to stabilize the film in a liquid state so that its properties can be manipulated and controlled prior to solidification. In some embodiments, one or more external gradients are applied to give rise to a gradient in surface tension, otherwise known as a Marangoni stress, as the liquid film emerges from the bath. Marangoni stresses may be generated by application of any number of different external fields or gradients, including a temperature gradient, gradients in surface concentration of surfactants, gradients in electric fields, pulling speeds, gradients induced by compositional variations in mixtures, and the like. The present method can utilize any source of Marangoni stress, so long as the applied stress is sufficiently large to stabilize the film until the time of solidification. Both the surface tension gradient and the rate of solidification are controlled, e.g., by active heating/cooling, to ensure that the solid film attains the desired properties.

[0031] There are numerous ways that a desired surface tension gradient can be imparted to the liquid film that emerges from the bath. In one embodiment, a surface tension gradient can be induced in a liquid film as it emerges from the bath by controlling the surrounding temperature around the liquid film. For example, the meniscus can be heated to induce a temperature gradient in the emerging film, which in turn can result in the formation of the surface tension gradient in the film. Depending on the material to be pulled and the desired surface tension gradient, heat can be applied in any suitable manner. For example, for a material for which surface tension increases with decreasing temperature and the desired surface tension gradient is to increase the surface tension as the temperature decreases, heat can be applied to the meniscus region that is closer to the bath of molten material so that the surface tension increases with the distance from the bath, thereby stabilizing the film in a liquid state. Alternatively, for a material for which surface tension increases with increasing temperature and the desired surface tension gradient is to increase the surface tension in the direction toward the solidification front, heat can be applied near the solidification front. Other embodiments will be readily apparent to one of ordinary skill in the art. The heat can be applied by any suitable techniques, such as radiative heat transfer techniques; local heating of the meniscus or the melt near the meniscus by laser (pulsed or continuous); placing metal objects close inside or outside of the melt which can be heated or cooled as needed; or using convective methods, such as directing hot gas toward or parallel to the pulling direction. Other active heating methods not explicitly disclosed herein but apparent to one of ordinary skill in the art may also be utilized.

[0032] In another embodiment, desired surface tension gradients can be obtained by actively cooling the liquid film near the meniscus and/or the solidification front, depending on the properties of the material to be pulled. For example, heat can be removed by contacting the solidified film near the solidification front with a natural or forced convective cooling fluid, such as a gas or a liquid. In other embodiments, heat can be removed by enhanced radiative heat transfer techniques (e.g., positioning a very cold surface near the solidification front). In other embodiments, cooling can be achieved using thermoelectric elements. In yet other embodiments, cooling can be achieved by evaporative cooling (e.g., spraying a fine mist of water or other liquid on the film). Other active cooling methods not explicitly disclosed herein but apparent to one of ordinary skill in the art may also be utilized. It should be noted that the active cooling described herein is distinguished from removal of latent heat of crystallization that is carried out in both the conventional techniques and the present methods. The active cooling described herein encompasses additional active cooling techniques not employed in conventional techniques.

[0033] In another embodiment, an induced surface tension gradient can be modified by controlling the speed at which the film is pulled from the bath. For example, assuming that all other settings (heat removal rate, temperature of bath, applied local heating power by laser or other device etc.) remain constant, increasing the pulling speed can lower the temperature gradient and alter the film thickness.

[0034] In certain embodiments, surfactant gradient concentration may be utilized to impart or induce a desired surface tension gradient. Use of a gradient in surfactant concentration at the surface may be particularly useful in low temperature materials, such as where

solidification occurs through a separate mechanism, such as polymerization, chemical reactions, cross-linking, UV curing, and the like. For instance, surfactant that is compatible (e.g., soluble, miscible, etc.) with the material of interest can be incorporated in the bath, where properties such as concentration, pH, activity, solubility, can alter the surface tension gradients. For example, the pulling action itself may induce a difference in the concentration of surfactants between the solidification front and the meniscus portion so that the surface tension is higher near the solidification front.

[0035] In certain embodiments, mixture component concentration gradients may be utilized to impart or induce a desired surface tension gradient. Use of a gradients of concentration of components in a mixture may be particularly useful in material combinations where one component evaporates at a much faster rate than other components in the mixture. For example, in a polymer/solvent mixture where greater amounts of the solvent leads to lower surface tension, the solvent may evaporate more quickly as a function of thickness so that the surface tension is higher near the solidification front than near the thick portion of the meniscus and the bath.

[0036] In certain embodiments, electric fields may be utilized to impart or induce a desired surface tension gradient. Use of an electric field may be particularly useful in materials that respond to electric fields, such as conducting polymers, piezoelectric materials, certain surfactants, and the like. For instance, metal plates can be placed near the meniscus or the solidification front, an electric field (e.g., dc or ac as appropriate) can be applied between them, which in turn can change the surface tension near the applied electric field. For example, the electric field can be set up to reduce the surface tension with its application and provided near the meniscus region and the bath.

[0037] In certain embodiments, one or more of the techniques described above to impart a desired surface tension gradient can be used in conjunction with each other.

[0038] Any desired surface tension gradient can be applied. For example, a surface tension gradient can be applied to the emerging film wherein the surface tension increases as the distance from the bath increases. In certain embodiments, the surface tension gradient can be set up as a function of time where a suitable field (e.g., heat) applied at the meniscus is periodically turned on and off to decrease or increase the local surface tension, respectively. In other embodiments, a varying surface tension profile can be established where the surface tension is maximum near the midpoint (or any other location) between the bath and the solidification front. In another embodiment, an undulating surface tension profile can be established. Many different surface tension gradients or profiles can be established as desired.

[0039] As a result of the applied surface tension gradient, the rate at which the film can be withdrawn from the bath can be substantially increased, and the thickness of the film can be precisely controlled. The particular profile and value of the surface tension gradient that is needed to achieve a particular pull speed and thickness will depend on the properties of the material to be pulled. However, generally for most materials, providing a larger surface tension gradient may provide the ability to pull at faster speeds and achieve larger thicknesses. The present method can also be particularly useful for making thin sheets or ribbons of silicon (or another molten material) where the thickness undulates either along the length of the ribbon, the width of the ribbon, or both. These undulations can be tuned by tuning the applied gradient locally and/or temporally. For instance, in the case a temperature gradient is imposed using technology shaped laser beam for instance, modulating the intensity in time can produce undulation of the film along the length of the film (i.e., x-direction in FIG. 2). Likewise, modulating the intensity of the laser in space can produce undulations of the film along the width of the film (i.e. z-direction in FIG. 2). Consequently, the two modulations imposed together can produce complex patterns.

Film-Forming Material

[0040] The present disclosure may be applicable to a wide range of materials, including but not limited to metals, semiconductors, polymers, and the like. In certain embodiments, the material has a melting temperature or a glass transition temperature that is above room temperature. In certain embodiments, the material has a decomposition temperature that is above the melting temperature of the material.

[0041] Some exemplary materials that can be formed into a film as described herein include silicon, silver, aluminum, gold, cobalt, copper, iron, nickel, germanium, and the like.

[0042] In certain embodiments, the film-forming material has the property that surface tension increases as temperature decreases. For example, silicon, silver, aluminum, gold, cobalt, copper, iron, nickel, germanium, and the like have material properties where the surface tension increases as temperature decreases.

[0043] It should be noted that the present disclosure is applicable to a wide range of different materials. Particularly, as will become evident through the fluid dynamic equations below and in the Appendices, the present method is independent of the viscosity of the material to be pulled. The material can have high viscosity (e.g., glass) or low viscosity (e.g., liquid metals), but the techniques described herein may be applicable to any of these materials without the need to adjust for the viscosity as was needed in conventional techniques. Fluid Dynamic Equations

[0044] Detailed fluid dynamic equations have been derived, as described below in

Appendix I.

[0045] The fluid dynamic model shown in Appendix I is derived for the case of ideal heat transfer (St=∞) and without taking into consideration inertia or gravity. Additionally taking inertia and gravity into considerations, the fluid dynamic model described in Appendix II can be derived.

[0046] Summarizing the key concepts discussed in Appendices I and II, the derived equations:

ho∞ H"(-∞) £ _c Γ, where Γ= γ' AT/y_s is the thermo capillary stress parameter not accounting for inertia, or,

h₀ oc H"(-∞) I _c Γ (l+WeA/2)^"1, accounting for inertia, where We is the Weber number and £ _c is the capillary length,

can be utilized to calculate the desired surface tension profile that can be imparted to the film surface to achieve a certain film thickness. To account for the thinning effect due to gravity drainage, the dimensionless curvature H"(-∞) of the dynamic meniscus should be multiplied by (1-1.16/?) where ?is the gravity number. Likewise, to account for the effect of non-ideal heat transfer effect that smoothes out the effective temperature gradient along the film surfaces, H"(-∞) should be multiplied by tanh(0.335 St+0.885), where St is the Stanton number.

Provided ?is not too large (typically β« 1), and St is not too small (typically St »1), the two corrections can be combined.

[0047] As shown, the final film thickness will depend on a number of variables, including (i) the pulling speed; (ii) the material properties of the film material, in particular the surface tension and viscosity, and their respective dependencies on temperature, as well as the density, the heat capacity and the heat conductivity; and (iii) the applied temperature gradient along the length of the film; and (iv) the rate of heat transfer to the ambient.

[0048] Taking the case of achieving a desired surface tension gradient by applying a temperature gradient, the needed temperature gradient (ΔΤ/ £ ) needed to achieve a certain film thickness (h₀) can be calculated as shown in FIG. 3. [0049] In 301, the properties of the liquid to be pulled are obtained. For example, density p, viscosity μ, surface tension γ, temperature dependence of the surface tension άγΙάΤ, heat capacity c_p, and heat conductivity κ can be obtained experimentally or from various references.

[0050] In 303, the range of control parameters, such as film thickness ho, pulling speed uo, and heat transfer coefficient a can be fixed.

[0051] In 305, model parameters, including capillary length , Weber number We,

temperature difference

, thermocapillary stress parameter length of the dynamic

meniscus , and temperature gradient

can be calculated for each set of [h₀, u₀] assuming a perfect heat transfer (i.e., α→∞).

[0052] In 307, corrections due to gravity drainage, including gravity number β, temperature difference ΔT⁽²⁾, thermocapillary stress parameter I⁽²⁾, length of the dynamic meniscus £^{2) , and temperature gradient

, can be calculated.

[0053] In 309, correction due to finite heat transfer coefficient a, including Stanton number St, temperature difference ΔT⁽³⁾, thermocapillary stress parameter I⁽³⁾, length of the dynamic meniscus , and temperature gradient can be calculated.

[0054] Although calculations to determine the needed temperature gradient to achieve a desired surface tension gradient Δγ is described in FIG. 3, similar calculations can be carried out if other means to provide the desired surface tension gradient is to be employed.

EXAMPLES

[0055] The present method is particularly suited for producing thin silicon films shown in FIG. 4. The following example demonstrates how the present method is utilized to produce a thin (300 micron thick) silicon film. The melting temperature of silicon is about 1687°K, and its surface tension γ is about 721 mN/m which decreases with temperature at about dy/dT = - 0.06 mN/m per degree K, although values of up to -0.7 mN/m per degree K have been reported. The viscosity of silicon μ at these temperatures is about 0.5 mPa.s, the heat capacity c_p is 910 J/kg/K and the density p is around 2600 kg/m³, such that the capillary length £ _c =(y/pg) ^1/2 , with g the gravitational acceleration, is about 5.3 mm.

[0056] Applying these parameters, the thickness of a silicon sheet produced in accordance with the present method scales with the non-dimensional capillary number Ca, defined as Ca = μ U I γ, as well as with the strength of the applied surface stresses as measured via the non- dimensional Marangoni number Ma= γ' AT/( η₅ U), and expressed in terms of the non- dimensional curvature H"(-∞) of the sheet as it connects to the bath:

ho∞ H"(-∞) £ c Γ, where Γ= Ca Ma, not accounting for inertia, or, ho∞ H"(-∞) £ c r(l + WeH2)^~' , accounting for inertia, where We = p £ _c U²//_s, where ho is the thickness of the ribbon. FIG. 5 shows, in the case of ideal heat transfer conditions and negligible gravity (valid for very thin films), how H"(-∞) depends on the non- dimensional length scale D (scaled by the length of the dynamic meniscus £ of the film) over which the temperature gradient is prescribed. For example, a sheet having a thickness of 300 microns can be made according to the present invention at a withdrawal velocity of about 10 cm/s as long as Γ> 0.031 , which is realized by dropping the surface tension by 16 mN/m over the length of the dynamic meniscus £ of the film. In the case of ideal heat transfer, depending on the precise value of dy/dT and accounting for gravity drainage, this requires lowering the temperature of the surface of the film by anywhere between about 40-440°K over a distance of about 1 mm, corresponding to D smaller than or equal to 1.

[0057] Whatever the cooling method, the cooling efficiency can be associated with a heat transfer coefficient a (W/m²/K). The applied temperature gradient should be adapted depending on the value of the heat transfer coefficient, as shown in Tables 1-3.

[0058] The present invention is illustrated by Tables 1-3 showing three sets of data for three different values of the heat transfer coefficient a. For each set, two pulling speeds uo are considered. The tables show the corresponding surface tension change that needs to be obtained by any means, such as a temperature gradient, electrical field gradient, concentration gradient, or combinations thereof to obtain the listed film thickness. For the case of an applied temperature gradient, the tables give the temperature gradient ΔΤ/ £ necessary to obtain a given film thickness in a controlled manner. Values in the table have been obtained for D = 1 , i.e. the length over which the temperature gradient is prescribed fits the length of the dynamic meniscus. The effects of drainage by gravity and heat transfer from the film to the ambient have been taken into account . The numbers in tables 1-3 were calculated for silicon using dy/dT = - 0.06 mN/m/K for the variation of surface tension with temperature, although the tables also show the necessary temperature gradients in the case where dy/dT is ten times larger, as it is found in at least one reference. The procedure to obtain the numbers in Tables 1-3 are as outlined in FIG. 3, whiched below with greater details

Fix the properties of the working fluid (here for silicon): density p = 2583 kg/m³, surface tension γ= 0.721 N/m, temperature dependence of the surface tension άγ ΙάΤ = 0.00006 N/m/K, heat capacity c_p = 910 J/Kg/K, heat conductivity κ= 22.1 W/K/m; using g = 9.81 m²/s for the gravity acceleration

Fix the range for control parameters: film thickness ho =

{ 10;20;50; 100;200;300} μιη, pulling speed = { 1 ; 10} cm/s, heat transfer coefficient a = { 1 ; 10;100} W/cm²/K Calculate the model parameters for each set [h₀,uo], assuming first perfect heat transfer, i.e. a ∞ :

b. Weber number:

c. Temperature difference:

d. Thermocapillary stress parameter

Length of the dynamic meniscus

f. Temperature gradient:

Calculate the correction due to gravity drainage: a. Gravity number:

b. Temperature difference (numerical-based correlation)

valid for β« \ . c. Get Γ⁽²⁾ ,£⁽²⁾ andVT⁽²⁾ as in steps 3d-f Calculate the correction due to finite heat transfer coefficient a. Stanton number:

b. Temperature difference (numerical-based correlation):

[0060] Results in step 5c are reported in Table 1-3 along with the variation of surface tension in case it is produced by another mean than a temperature gradient. Note

that such that if is ten times larger,

can be ten times smaller, as also

reported in Table 1-3.

[0061] It should be noted that the results shown in Tables 1-3 are meant to be indicative of the general trend. The numbers are obtained from the models described above, assuming that the material properties reported in the literature are accurate, and taking several simplifications and assumptions (e.g., calculated for the optimum value of D=l). The actual values reported in the Tables may change but one of ordinary skill in the art will be able to utilized the values reported herein as starting points for further optimization. Table 1 : α = 1 W/cm7K

Table 2: a = 10 W/cm K

[0062] FIG. 6 shows the thickening effect of surface tension gradients (i.e., Marangoni stresses) as simulated in COMSOL with an imposed surface tension gradient but without inclusion of a temperature-dependent viscosity. As shown in FIG. 6, induced Marangoni stresses lead to a thickening of the film, without altering the dependence on the withdrawal speed. This thickening is significant, since it helps to stabilize the thin film as it cools. In FIG. 6, the horizontal and vertical axes are measured in terms of wire diameter w; the wire is the object which initially pulls fluid out of the bath, and it is at the top of the figure in both panels. In these calculations, w/i _c = 0.01 , so for a typical value of I _c = 2 mm, w = 20 μιη. Also, the capillary number Ca = μ\]/γ= 0.01 in this simulation. For a liquid metal with typical viscosity μ = 10^"2 kg m^'V¹ (10 times more viscous than water at standard temperature and pressure) and γ = 0.5 N/m (10 times more surface tension than water at standard temperature and pressure), the pulling speed is about U= 50 cm/s.

[0063] In the present method, the temperature gradient, or other gradient, is configured to allow the film to solidify in a controlled manner. The actual parameters will depend on the properties of the material being used, and the thickness of the finished film. These parameters can be determined empirically based on well-established principles, as demonstrated above for silicon. For example, where a temperature gradient is used, detailed analysis of the heat transfer between a thin sheet of fluid and a cooler ambient environment demonstrates that as long as the film is thin enough, the temperature is practically uniform across the film (with a small parabolic correction) but will vary with the coordinate y as defined in FIG. 6; and if the ambient temperature T_a is kept lower than the bath temperature 7_¾, then the temperature in the film will decrease exponentially with distance away from the bath. The strength of the exponential decay will depend on the specific parameters of the process, and in many cases may be well-approximated as a linear decrease in temperature with distance from the bath.

[0064] FIG. 5 shows the non-dimensional curvature H" of a silicon sheet as it reaches the bath versus non-dimensional length D over which the required temperature drop is

accomplished. As explained before, H" plays a large role in determining film thickness. D is in

1/2

units of £ = /z₀/(2*(rr , which is about 1 mm for the example of 300 microns at 10 cm/s. The calculation includes the effect of inertia: the solid line corresponds to pulling speeds u₀ smaller or equal to 1 cm s while the dashed line corresponds to a pulling speed of about 20 cm/s. As shown, the effect of inertia (or speed effect) decreases the curvature of the dynamic meniscus, and hence the film thickness, keeping all other parameters that remains constant. Moreover, as shown, as long as D is smaller or equal to unity, the results are independent of this parameter, which in turn means that as long as the length over which the surface tension gradient is applied is smaller or equal to the length of the dynamic meniscus, the results depend largely on the temperature difference ΔΤ. [0065] FIGS. 7 A and 7B shows the dimensional temperature distribution from the solidification front (T_coi_d=1687K for silicon) to the hottest temperature (T_hot), see Appendix II for details. The different curves show the influence of the heat transfer from the film to the ambient. The smaller the heat transfer coefficient, the smaller will be the effective temperature gradient if we keep all the other parameter settings the same. FIGS. 7A and 7B show results for two different pulling speeds. The larger the pulling speed (above lcm/s), the largest will be the temperature difference requested to produce a steady film.

[0066] Other embodiments are within the scope of the claims which follow below.

APPENDIX I

We consider a liquid film withdrawn with speed n from a bath of temperature ¾ (Fig. 1). Symmetry is assumed about the x-axis. The prescribed far-field ambient temperature is denoted Taix). At steady state, a film of thickness h(x) eventually reaches a constant value h at a distance sufficiently far above the bath. We assume the fihn then solidifies. The density p and the viscosity t are taken to be constant because they do not change significantly over a modest temperature interval near the solidification temperatures of typical materials that interest us. We denote θίχ) as the cross-sectional average temperature of the film and assume a linear decrease of the surface tension y with temperature from the solidification temperature ¾, γ( θ ) = y_s— yt( 0— T_s)_t where y_s = y{T_s) and }'∑^■ = jdy/dF|_s as is the case for most liquid metals ·^'.

Following the region decomposition applied in the context of a foam lamella⁰, determination of the shape of the film requires solving the thin-film equation in an intermediate region of length t that connects the static meniscus at the bath with the flat film region near the solidification front. The static meniscus (dashed line in Fig. 1) is identical to a meniscus that would be attached to a perfectly wetting substrate with curvature h" = y/2 /(.<., where the prime denotes the .y-derrvative. and t_c =

the capillary length⁹. Because of the downward capillary suctio induced by the positive curvature of the free surface, no purely viscous film can be stably pulled out of a liquid bath⁵ unless sufficiently large shear stress is present at the free surface, which is ensured here by the Marangoni stress induced by the gradients of surface temperature on both sides of the film. According to Breward and Howell⁸ this "shear flow" regime corresponds to a distinguished limit where capillary and Marangoni effects provide the dominant balance in the intermediate region and where the exteiisional viscous stress is always negligible. This limit is applicable for

FIG. 1. Sketch of the pulling film problem. The dashed line indicates a static meniscus (not to scale). with e = ¾ <^■' and AT = ¾— T_s. The corresponding axial stress equation, neglecting inertia and gravity effects (see justifications hereafter), has the form

^ΑΑ"' - 2γτβ' = 0 . (2)

Terms in (2) account, respectively, for the capillary stress induced by the curvature of the interface and for the therraocapillary stresses at the two interfaces. Finally, in the case where the rate of heat transfer across the free surfaces is much larger than the rate of heat advected by the flow, the local temperature of the film takes the temperature of the ambient (the reader is referred to Scheicl et ctl ⁰ for details on this 'prescribed temperature limit'), s ch that the energy ecpiatioii reduces to θ(χ) fis T_a{x) . (3)

We also assume that heat released during solidification can be neglected because it occurs after the geometry of the film is no longer changing^{1 1}. In the present problem, we assume the ambient

temperature to be prescribed such that it varies, by the temperature difference AT over a distance d. For the sake of subsequent analytical deveiopment, we chose a continuous and integrahle function of the form

We note thai the choice of a function other than an hyperbolic tangent (such as an error function for instance) leads to the same conclusions as those presented in this letter. Because the problem is invariant by translation, and according to the sketch in Fig. 1 , we have also shifted in ce the temperature variation by the quantity d such that T_a{0) s¾ T_s. This is valid so long as AT/T_S 1.

We next nondiuiensionalize using the scalings

Defining the length scale ί of the intermediate region as

γτ Γ

with Γ (6)

2 yT ¾

leads to the parameferless equation

HH"' = Θ^' . (7)

Since the thickness is assumed to reach a constant near the solidification front the boundary conditions at A^" = 0 are H = 1 and H^! = H" = Θ = 0. Integrating (7) and combining wit ( 3) and (4) in dmiensionless form, yields

Λ

2HH" - H = I - tank .a I · I (8)

D winch straightforwardly relates the dimensionless length of the temperature (or interfacial stress) vaiiation D to the shape of the film H(X). Furthermore, this latter must match the curvature of the static meniscus as the film thickens, i.e. Hi.,,— v^'2 ^f " as X→— °°. Therefore, the matching condition that allows determination of h§, using (6), yields

where Hi'„, remains to be determined.

Ill the limit D 0 and for A^" < 0, the term tanh™—1 in C^'S so that an analytical solution can be found: H = 1 +X^,t/2. Replacing the corresponding curvature H__„ = 1 in the matching condition (9) yields the asymptotic determination of the film thickness:

¾ο = 2 ^/2 Γ as D→0 . (10.)

X

FIG. 2. Solutions to (8) showing H for various D and H" for £> = 1. The dotted line also corresponds to

H = l +X²/2.

0.0Ϊ 0.1 1 10 100

D = d/i

FIG. 3. Matching curvature Ht_M versus the length D over which the temperature difference is applied.

This prediction for the typical film thickness is the main result of this work showing that it only depends on the amplitude of the surface tensio variation, as measured by the parameter Γ. The film is thus pulled out of the intermediate region with a thickness that does not depend on the pulling speed. Increasing the speed will thus increase the rate of liquid passing through this 'virtual 'dot', whose the width is adjusted by the amplitude of the therniocapillary stresses. As the pulling speed increases, however, eventually inertia! effects wili enter the axial stress balance (2), and we comment on this influence below.

Equation (8) can be solved numerically for any value ofZ>: typical solutions are given in Fig. 2. We observe that even though the numerical solution for D = Ϊ is far from the analytical D→ 0 solution (contrary to the D— 0.1 solution), the curvature H'¹ still tends to a constant approaching unity as X→— ©©.

We next report in Fig. 3 the constant value H"^ for a wide range of D. It appears that Hi' ^ is well approximated by unit}-' for D ¾ 1 , which extends the range of validity of the asymptotic TABLE I. Properties given at the solidification temperature T_: for various elements'¹; C_c = > &/iPJf) ^aftd AT = 100 .

^■3 I. Egry and J. Brilio, J. Cheni. Eng. Data 54, 2347 (200:9). result (10). However, for D > 1, numerical solutions in Fig. 3 should be used instead for H"_«, which gives the film thickness by (9). The results show that the film thickness ¾ decreases as D increases, i.e. as the imposed temperature gradient decreases.

Fable I shows the physical properties of various metallic elements, which allows an assessment of the dimensionless parameter Γ for a fixed value of the temperature difference AT = 100 K. We see that Γ sw 10^-2 for all of the elements reported in Table I, except for silico which has a higher value. However, Zhou et al. report smaller values of y. = 0.72 N/m and γτ = 0.06mN/(mK) for silicon, winch gives Γ = 0.01. The film thickness is therefore of the order of 100 ,um for all of the elements in Fable I. Taking a practical example, to produce an iron film of 150 Jim thick would require a value of r = 0.01 , or a variation of surface tension by 20 m /in over a length smaller than or about ( = t_c γ 2Γ as found from Eq. (6) with the asymptotic result (10). This case requires lowering the temperature of the surface of the film by 50 K over a distance smaller than or about 0,8 mm (i.e. D ,¾ 1 ). If the temperature difference is to be imposed over a longer distance, the numerical results in Fig. 3 are to be used with (9). For instance, for D = 10. it is necessary to increase T^' by 60% to keep the same film thickness as that obtained for D 1 ; in the case of a 150 j .m thick iron foil, tiiis approach requires a temperature difference of S0 K over a distance of about 6 mm.

We now determine the conditions that allow neglect of gravity and inertia: both of them remain small as compared to capillary effects, so long as G = pgf.f/Ys ¾C 1 and We = «o/% 1 - respectively. Note the largest length scale of the system. t_c, has been taken to ensure a conservative evaluation of G and We. For typical molten materials, the first condition is always true since G ¾: i0^_1 for all of the elements in Table I. The second condition indicates that rnertial effects can be neglected for speeds IIQ vgi^'c- In the case of a 150 tim iron foil, we require - 22 cm s. For larger speeds, inertia will tend to thin the film as compared to the present theory. Detailed analysis including inertia is to be reported elsewhere.

As mentioned earlier, the present results are applicable in the 'shear' distinguished limit where thermocapillary stresses are large and exteusioual viscous stresses are negligible. Using (1) together with (6) and ( 10) gives a condition on the pulling speed, u_Q < (¾/ η ) or in the case of iron with η ¾' 1Q^_J Pa.s, ¾?<_, ¾ 1 in/*.

In this letter, we show that a liquid film can be pulled out of a bath by using thermocapillary stresses prescribed at the free surfaces. The resulting film thickness is proportional to the capillary length of the liquid t_c and to a parameter Γ that measures the amplitude of the surface tension change at the interface. If this change is imposed over a distance d that is larger than the characteristic length i— i ν^''2Γ of the system, the film thickness decreases with increasing d. otherwise it is independent f d. A significant feature of the present theory is that the film thickness is also independent of the pulling speed of the film, at least when inertia is neglected, which means that the flow rate can be changed without modifying the film thickness. REFERENCES

E. R Degamio, J. T. Black, and R. A. ohser, Materials and Processes in Manufacturing (Wiley, New York. 2003), 9th ed.

²D. Helmreich and E. Sirtl, J. Cryst, Growth 79. 562 (1986).

^•'A. G. Schoeneeker and K. I. Steinbach, Method and device for producing metal foils. Patent EP17433S5(B I) (2008).

⁴J. D. Zook and S. B. Schuldi J. Cryst. Growth 50, 51 (1980).

⁵E. A. van Nierop, B. Sclieid, and H. A. Stone, J. Fluid Meek 602, 119 (2008), Corrigendum: 630, 443 (2009).

bK. J. Mysels, K. Shinoda. and S. Frankel, Soap Films: Studies of Their Thinning (Pergamon, London, 1959).

⁷H. M. Lu mid Q. Jiang, J. Phys. Chem. B 109, 15463 (2005).

SC. J. Breward and P. D. Howell J. Fluid Mech. 458, 379 (2002_.).

⁹P.-G. de Gemies, F. Brackart-Wyart, and D. Quere, Gouttes, BuHes, Perles et Chides (Belin (Pans), 2005 ).

°B. Sclieid, S. Qniligotti, B. Iran, R. Gy, and H. A. Stone, J. Fluid Mech. 636. 155 (2009).

^! S. Smith and I). Stolle, Polym, Eng. Sc. 40, 1870 (2000).

²Z. Zhou, S. Mukherjee, and A.-K, Rhim, J. Cryst. Growth 257, 350 (2003).

APPENDIX II

II. PROBLEM FORMULATION

We consider a liquid film withdrawn with speed ?¾■ from a bath of temperature ¾ as shown in Fig. 1 , We imagine that the film is then cooled and solidifies at, some distance above the bath. Symmetry is assumed about the x-axis. The far-field ambient temperature,, denoted T_a(x), is assumed to vary along the film. The film of thickness h{x) eventually approaches, a constant value tj at a distance sufficiently far above the bath and before it solidifies, as sketched in Fig. 1 . The density p is taken to be constant and the viscosity ϊ) (Γ ) depends on the temperature T. We denote the surface temperature of the film as 2*(x) and consider a linear decrease of the surface tension

FIG. 1 : Sketch of the pulling film problem. The dashed line indicates a static meniscus (not to scale). with temperature, winch applies to most liquids including molten silicon, γ( ) = y_s -^■/( - ) , (1 ) where γ.₅ = J{ T_S ) , T_s is the solidification temperature, and γ = άγ/άΤ is a positive constant.

A. Conservation equation a d the stress balance

The mass conservation equation has the form h_; + (hu)_x = 0 , (2) where it is the cross-sectionally average velocity, and the subscripts / and x indicate the time and space derivatives, respectively. For a steady state. (2 ) reduces to

hi the framework of lubrication theory, and following Breward [7] . we ca write a master equation including all possible sources: of axial stresses in the liquid film (see e.g. [4, 8]):

4 ( η (T)Jm_x)_x - 2γ¾_χ + ^hb_xxx - pgk = hi ^ + m_x) , (A) where )) and T are the cross-sectionally averaged viscosity and temperature, respectively. The asymptotic expansion leading to (4) is given i appendix A. The first term on the left-hand sid in (4) represents the exteiisional viscous stress, where the factor 4 is the Trouton ratio. The second term accounts for themiocapitlary stresses at both interfaces. The third term accounts for the stress induced by the gradient of the film curvature. The fourth term represents stresses due to gravity. The right-hand side accounts for inertia! effects.

B. One-fHiiieasioiiitl temperature equation

As for the stress balance, we assume that the one-dimensional assumption applies to the temperature distribution, namely

T (x v, / )≡ Tix ) = Ί (x, t ) . (5 } where Tlx. ) is the cross-sec tionaliy average temperature. This approximation is a consequence of the thin-film asymptotic expansion, at least to first order, as outlined in Appendix B. The resulting energy equatio is

pcph (T( + ΰΤ_χ) =—2a(T— T_a) _r (6) where c_p is the specific heat capacity and ce is the heat transfer coefficient at the free surfaces. We can neglect energy released during solidification here because it occurs after the geometry is no longer changing [9],

Filially, the viscosity is presumed to decrease linearly with temperatiire,

where ¾ = and }^f = dn /dF is a positive constant. This approximation is valid for small temperature changes, i.e. ( T— T_s)/ - 1 .

C. on-dimeiisionailzattoH

We next nondimensionalize using the scalings

x h —

X— - . H =— . U =— r = ^

t ^' _ a^■

Θ = Ϊ _ τ_α - τ₅

A=T= . a, AT '

where is the characteristic length scale m the .Y-direction mid AT = Ti_>— T_s. As we are interested in steady pulling, we consider here stationary conditions, i.e. from (3).

OH = i . (8)

The stress balance (4) thus becomes

4ε² ( Y— ] + ΙεΜαθγ -—HH _T + GH - eRe^ = 0. ⁽9) H }_x 2Ca H- where ε = Q/( is the slendemess parameter, and the Marangoni. capillary, gravity a d Reynolds numbers are. respectively.

The dimensionless viscosity function has the form

Υ(θ) = 1 - μΘ , (10) where μ = ί]'ΔΓ/ϊ], measures the sensitivity of the viscosity to temperature changes.

Likewise, the steady temperature equation (6) becomes

<¾· St { H: > . (1 1)

The Stanton number. Si which measures the rate of heat transferee at both interfaces relative to the rate of heat advected by the main flow, is defined as

2a( S ^' ., , 2a

Si = ≡ with Si = :— :— . ( 12} pCp 'QfiQ e a pCpiYsffls)

The parameter Si' only depends on the fluid properties. Note that (6) is decoupled from the stress equation and can thus be solved independently provided the ambient temperature T_a{x) is specified.

D. Controlled ambient temperature

To close the system of equations, it remains to give die form of the ambient temperature. We assume here that it is prescribed and varies smoothly, in dimensionless form, from unity in the liquid bath to zero in the fiat film region where the liquid solidifies. We can choose for instance a hyperbolic tangent function of the form

where D is roughly the dimensionless distance over which the temperature difference Δ7^" is applied. We note that other functions such as an error function variations would lead to similar results as those which follow. Because the problem is invariant by translation, we have shifted the temperature variation by the quantity D such that Θ,,(0) s¾ 0, so that the coordinate x = 0 coincides approxiruatively with the solidification front.

III. DOMAIN DECOMPOSITION

In the stationary regime, the film-pulling problem represented in Fig, 1 can be decomposed into three regions, each of them corresponding to a specific force balance: (A) a capillary static meniscus near the bath where gravitational and capillary forces balance (third and fourth terms in (4»; (B) a fiat film region of constant film thickness ¾; (C) a transition region between the two others in which, in principle, all forces can be in balance. An analogous decomposition was used by Breward and Howell [10] to describe the drainage of a foam lamella. Next, we detail the specifics of each region:

(A) In the static meniscus region, the curvature can be calculated by balancing gravity and capillary forces. The curvature reaches constant value, Η„ = /2fi_c, with $i_c = _\Ζ% ίρ#) the capillary length (see [11] for instance), near the transition region (C). This is identical to the treatment of this region in the classical LLD problem [12, 13]. (B) As mentioned above, the geometry is assumed to no longer change beyond the transition region, and up to the solidification front. Relaxing this assumption would require also solving for the film thickness in this region, hi such a case, there will be a balance between at least the three first terms in (9), with a plug flow velocity profile (to leading order). The flow in this region would indeed be primarily extensional, with a decrease of the film thickness along the x -direction until the solidification front. According to [10] this extensional flow regime corresponds to the distinguished limit of the master equation (9) where Ma = 0[έ) and Ca = 0{e). Nevertheless, we can show that in the limit of large Marangoni stresses, the effect of extensional viscosity can be neglected, and the film thickness can effectively be assumed constant, as considered below.

(C) As poi ted out in [4] no free film can be stably pulled out of a liquid bath in the extensions! flow limit, hi order to pull a film, sufficiently large shear stress should be present at the free surface, as it is assumed to be the case here. The flow is therefore shear-driven in this region with a parabolic velocity profile across the film. Consequently, capillary and Marangoni effects primarily balance and the extensional viscous stress is negligible. Again, according to [10], this shear flow regime corresponds to the distinguished limit of the master equation (9) where Ma = 0{ε^~ι) and Ca = Oieⁱ ) , This limit thus requires the Marangoni number to be much larger tha unity.

Based on the above domain decomposition, the film thickness equation (9). with only the relevant terms, should be solved in the transition region (C) with the boundary conditions;

H 1 , H_x 0 χχ— 0 as X (14) i.e. approaching the flat, film region (B) above the bath. Furthermore, the film thickness profile must match with the curvature of the static meniscus (A) as the film thickens, namely ¾· = ν'2ί"- i ^' _ch(₃ ) as X→— «*. Therefore, the matching condition allows the determination of ¾:

where %[ -«] remains to be determined. Also, ε needs to be explicitly given, which depends on the scaling for iv. PARAMETRIC: STUDY

Based on the physical properties of silicon arid on the control parameters, we assess the values of the dimensionless numbers in Table I. Parameters G. Re and ε are not given in the table since they depend on ¾, which is an unknown of the problem. We still need to assess the slenderness parameter e . Because the length t should satisfy the dual condition .¾ < << £c, we first make an estimate of £' as the geometric mean P. = y'Ao^, such that ε = 0{ / o/t . }- For films of about lOOj m, which is typically the order of magnitude we are interested in, ε = Οί 10^-1 )· From Table I, we then get Ma > 0{€^{~ l} ) and Ca < £?(ε³), which matches the conditions for the "shear" distinguished limit for which the extensional viscous effects can be safely neglected. Moreover, we find in Table I that μ = 0(e)_f and the assumed dependence of the viscosity with temperature (10) ensures that Y remains of the order of unity. These estimates are trae at least for liquids with a viscosity' that does not diverge when approaching the solidification temperature, as is the case for sihcon[ 17] .

Now, as m our previous work [6], we construct the length scale in the A-direction for the transition region from the dominant balance between surface tension and Marangoiii effects. This step yields

TABLE I: Set of parameters for silicon. The physical properties are given at the melting temperature

/: = 1687 K [14],

which measures the relative change of surface tension with temperature. Multiplying

2Ca/e³ and using (16) leads to

f Hr \ - Hv where

c, =— = . 8 =—— -if and d = , ( 19) v I ^' 4P - ¾ 2Γ f_c

with f!¾ = -(-_cUQ/Y_S the Weber number. According to Table I. using 100 ML we get ζ = 0{ 10^~4) , β = Of 10^{_ i} ) and δ = O(10^_i ) such that viscous, g avity and inertia terms remain small as compared to capillary and Marangoni terms in ( 18), as assumed initially. Nevertheless, because We varies with the square of the pulling speed, for the parameters in Table I, a pulling speed of a few tenths of centimeters per second would make S— Oi i ). which thus requires including inertia in the calculation, increase the thickness would also make significant the influence of gravity. Both inertia and ¾ravitv effect are adresses below.

A. Role of inertia

Integrating (18) with ζ = β = 0, using H→ 1 , Ηχ ~* 0, Ηχχ→ 0 and Θ -→ 0 as X— «\ we obtain

This equation should be coupled to the heat equation (11), which can be simplified in the limit of large Stanton number. From (12) and Table I, St = 0( 10 ;. Scheid el al. [15] have shown that in the case of St 1, Θ ¾ <¾ such that Eq. (20), using (13 ), becomes

The case of St— 0{ l) will be treated in section V. Solving (21) numerically with the boundary conditions H(0) = 1 and Ηχ{ O i = 0, the solutions tend to a constant curvature as X→—∞, The value of Ηχχ{—<∞) versus D is reported in Fig. 2 with and without ineitial effects, i.e. for 6 = 1 and 0. Using the value of —∞) in ( 15) finally allows to determine the film thickness of a themiocapillary-assisted contact-free film. Now, we can observe in Fig. 2 that Ηχχ{— «*) tends to a constant as D tends to zero, and more specifically for Z> < 1. Contrary to the case with no inertia (t = 0) as treated in [6], we have not been able to determine an analytical solution for ¾(-«) in the limit of D— > 0. Nevertheless, we numerically obtain the relation

Hvx(-∞) ¾ 1 - - for D < 1 (22)

Combining (15) with (22), and using (19), leads to

100

FIG. 2: Matching curvature ¾¾·( ---∞>) versus the dimensioiiles length D over which the temperature varies along the JT-directiofi: solid line corresponds to no inertia, δ = 0: dashed line corresponds to finite inertia, δ = i.

This result shows that the thickness is proportional to the amplitude of the surface tension change along the interface, provided it occurs aioug a distance shorter or equal to Additionally, the thickness decreases as the Weber number or equivalent!}' as the pulling speed increases.

Talcing an example, for producing a silicon film of 300 i in thick at a pulling speed of 10cni s would require a value of Γ— 0.023. or a variation of surface tension by 1 m /m over at most the length f ss ( y^'2— We)t_c r of the transition region (i.e. D < 1). This requires lowering the temperature of the siuface of the fiim by about 2S0 K over a distance smaller or equal to 1 mm. We observe here that the estimate is very similar for a pulling speed of 10 cm/s than for 1 cm/s and this is hue as long as We 1. In the limit of We— 0, we recover our previous result reported in

[6] for which the thickness was assumed independent on the pulling speed.

B. Role of gravity

We now look at the influence of gravity by considering (18) with ζ = δ = 0. It appears that the solutions do not tend to a constant curvature as X—>—∞ Instead, it passes through a maximum before decreasing and eventually becoming negative. Following the work by de Ryck and Quere [16] relative to plate coating, we propose to match the static and the dynamic menisci using this maximum curvature, denoted by i¾y|_niilx hereafter. This choice for the matching curvature possibly overestimates the film thickness but has the advantage to recover the same curvature as the one obtained in absence of gravity for β→ 0. We plot in Fig. 3 and for various values of the

0.01 0.05 0.1 0.5 I 5 10

D

FIG. 3: Matching curvature ¾d_mas versus the dimensionfcss length D for various β solving (18) with ζ = δ = 0. The curve for β = 0 lias been matched with j¾y(-») as previously. The thick dotted line is the loci of the minimum of ¾r|_rass correspoudiug to the "drainage limit". gravity number β, t e maximum curvature Hod max versus the length D that controls the temperature gradient along the interface. Because by increasing the parameter D, the maximum curvature ¾d∞ax P^{as es} through a minimum before diverging, we only plotted the curves before that minimum whose the locus is represented by the thick dotted line in Fig. 3. Since the thickness should only decrease (and not increase) when reducing the strength of the temperature gradient, we actually conclude that the divergence of ¾τ|_ιη3Χ (not shown) is unphysical and is the signature that the Marangoni stresses become too weak to counterbalance the gravity drainage and maintain a stable film. The thick dotted line thus corresponds to the "drainage limit" beyond which no stable film can be stably formed. The fact that this "drainage limit" intersects with the curve obtained hi the absence of gravity is because the two curves are constructed with two different matching curvatures. Nevertheless, this intersection occurs for β < 0.01, i.e. for regimes where gravity should preferably be neglected a.s including it in the calculation will overestimate the thickness prediction.

To illustrate the limitation due to drainage, we plot in Fig. 4 the thickness profile for various β and at a fixed value of the parameter D = 2,5. The profiles for β = 0.1 and 0.3 shows a minimum as a consequence of the drainage. Indeed, conditions are such tha t they lie beyond the ''drainage iimi ' as depicted in Fig. 3. These conditions should therefore be ruled out as they would lead to

-5 -4 -3 -2 -ί 0

FIG. 4: Thickness profiles for D = 2.5 and various β . solving (i 8; with ζ = <5 = 0 uapliysicaUy large film thicknesses (especially tor β = 0,3).

Having studied the influence of gravity shows that the distance D or alternatively the imposed temperature gradient should be large enough to sustain a free film and not to be overcome by the drainage. Based on Fig. 3 as well as on the estimate β = Oi l ), if the temperature gradient is imposed on a length equal to the length of the dynamic meniscus (i .e. D ¾^: 1 ), the thickness would be about 10% smaller than the thickness predicted by (23) due to the action of gravity. SOLVING THE TEMPERATURE EQUATION A. Influence of finite Stanton numbers

In this section, we relax the prescribed temperature assumption for infinite heat transfer coefficients, i.e. Θ = Θ,,, and solve (11 ) for finite Stanton number. The solution is

Stl) SiD

Θ{Χ] = Hypergeometric2F I L— Λ + π— (24)

4π 4π

where the integration constant has been set to zero to avoid the divergence of the solution as — f — «.*, ensuring that &(—<∞) = 1 , Figure 5 shows the dependence of the temperature distribution in the film with the Stanton number. We see that decreasing the Stanton number (i.e. decreasing the heat transfer coefficient), has the effect: of increa ing the effective length, say over which the temperature decreases in the ,r direction. It thus weakens the thernioeapiilary stresses at the free surfaces. Note also that the Stanton number decreases with increasing the pulling speed, thus

FIG. 5: Temperature distribution for various, values, of t e Stanton number St. with D = 1 reducing also the effective temperature gradient. Nevertheless, the important feature here is that solving the temperature equation for fimte St does not change qualitatively the results obtained in tire limit of St→ °». A parametric study could then be undertaken to replace D by a correlation on the form D_ta = fiSr. D ) in order to read Fig. 2 while accounting for finite St. This said, in the case of silicon, 5^'/ «s 10 for a pulling speed of 1 cin s and a heat transfer coefficient of 1

which gives a very small deviation from the limit of St→ ^«>. as shown in Fig. 5.

B. Role of heat diffusion

As a consequence of considering finite St, we should also investigate the influence of heat diffusion in the liquid along the flow direction, as to some extend both effects (heat transfer and heat diffusion) might become of the same order of magnitude. To that purpose, we rewrite the averaged temperature equation including a. diffusion term:

€>_r =—St (Θ - a,) + χΗθχχ , (25) where / —

A Pe

is the heat diffusion parameter whose expression results from the scaling. According to Table I, % = Oi l ), winch indicates that longitudinal heat diffusion can have a non-negligible influence, though essentially limited to small speeds. Since ( 25) now depends on the thickness H, this equation should be solved together with (18) and the corresponding boundary conditions. We used

0.01 0.05 0.10 0.50 1.00 5.00

X

FIG. 6: Matching curvature versus t e heat diffusion parameter χ for various values of the Stanton number St, The results are obtained numerically ζ— S = β— 0 and D = 1. the software COMSOL to compute solutions. The matching curvature versus the heat diffusion parameter χ is plotted in Fig. 6 for various Stanton numbers and a fixed value of D = I . Inertia and gravity effects have been neglected for the sake of clarity. As it could be expected, increasing heat diffusion decreases the effective temperature gradient and therefore reduces the matching curvature, hence tlie film thickness. This effect becomes negligible as St —r∞ but can be significant for St — Oi l ). It should be pointed out here that the effect of finite heat transfer coefficient as investigated in the previous section are included here.

As an example, for St = 10, the film thickness will be reduced due to heat diffusion/transfer by about 10% as compared to the one calculated with (23) for Si =∞>.

VI. TRANSVERSE MARA GONI INSTABILITY OE A UNIFORM FILM

In this section, we address the transverse stability of a film formed by thermocapillary-assisted pulling as described in the present paper. For the sake of simplicity, we consider the one- dimensional case, or equivalently the stability of a two dimensional film of uniform thickness hi,. Denoting z the transverse coordinate, t the time, w the thickness-averaged transverse velocity, and considering the unsteady case, the conservation equation takes the form k + (hw)_z = 0 , (26) while the balance equation is similar to (4) with no gravity:

4η {kw₂), - 2γΤ_ί∑ + ^hb^ - p {w_t = 0 . (27)

As there is no le gth scale in the transverse direction, we shall scale z by i. And as t e transverse component of the velocity can only exists through perturbations of the main flow, we shall scale w by a small quantity SUQ. The scaling for the new variables is therefore

z - w £

Z = - . W = . τ = 1 ,

The dimensionless transverse stress-balance equation becomes

2We H{W_v + WWz) = H zzz - < + Zai(HW_z)_z , (28) where We± = ε '8/2 = (h$ff_c)We. This equation needs to be complemented by an equation for the interfacial temperature (^■¾. For large heat transfer, the tempeiatiue profile across the film can be assumed parabolic, as outlined in appendix B 2, and (B 14) provides

As we consider the stability- of a. film of uuifomi thickness, the thickness-averaged temperature should also be uniform. Let us thus set Θ = 1 and <¾ = 0, such that the only parameter that accounts for the temperature constraint is the temperature difference, say AT , between the mean temperature of the film and the temperature of the ambient. The interfacial temperature then becomes

which only depends on the thickness H. A perturbation leading to a thickening (resp, thinning) of the film will induce a decrease (resp. increase) of the interfacial temperature, hence an increase (resp. decrease) of the surface tension. This is precisely the mechanism responsible for the long-wave thennocapillary (Marangoni) instability. Assuming normal mode perturbations of the uniform film, one can pose

H = l + a ^kz+GX , (31)

where a.b < I are small amplitudes, i = y'— 1 , k is the dimensionless wavenumber and σ is the dimensionless growth rate. Inserting these perturbations in (28-30) and linearizing with respect to a and b, leads to the dispersion relation σ = ( !lW _± ( | - k²} + 16 - 4C j . (33) where the cut-off wavensniber for which σ vanishes in a iiontrivial way, is · 5/>V ^'

1 he film is therefore unstable for a > 0, i.e. for 0 < k < A We can then write the growth rate and the wavenuniber of the most unstable mode:

From Table I. one finds that SCa/ /2We± «t. I - such that in the large surface tension limit (C — >^■ 0), we obtain the simplified forms

Dispersion curves are show in Fig. 7 where we observe that increasing the Biol number enhances the instability: it not only increases the growth rate of the instability but also the range of unstable wavenunibers. In dimensional form, one filiall gets

where fj_ = γ' The wavelength of the most amplified mode is

, _* 2ν 2 π

We plot in Fig. 8 the wavelength and growth rate of the most amplified modes. We observe that the

FIG. 7: Dispersion curve (37.» for various Biot numbers: Bi = 0.1 (dot-dashed line), Bi = 0.5 (dotted line), Bi = 1 (dashed line) and 5 = 2 (solid line). The black dots represent the most amplified modes corresponding to ½ and <¾■ wavelength increases with the film thickness and that the maximum amplification of perturbations will occur where the thickness is the thinnest, i.e. in the fiat film region. Therefore, for a molten silicon film of 300 μηι thick, a heat transfer coefficient = I W/cni'/K and a cross-stream temperature difference ΔΤ±_ = 1 , the wavelength of the Marangoni instability is 70 cm and the growth rate is about 0.01 s^_i . Furtiierinore, any perturbations of the wavelength smaller than 50 cm (i.e.

70/ v 2) will be stabilized by the system. It means that a silicon film winch is less than 50 cm wide will be unconditionally stable. Now the picture gets worse if the heat transfer coefficient and/or the temperature difference Δ2 _ is increased, as shown in Fig. 8.

VII. CONCLUSIONS

In this paper, we have studied the formation of a liquid film pulled out of a bath by using thermoeapillary stresses prescribed at the free surfaces. The resulting film thickness is proportional to the capillary length £_c of the liquid and to the strength Γ of the surface tension change at the interfaces. If this change is imposed over a distance d larger than the characteristic length / = \/2Γ of the system, the film thickness decreases with increasing d, otherwise it is independent of if. The resulting film thickness is also affected by inertia and decreases with increasing the pulling speed. Gravity drainage is also shown to be restrictive as it prevents to make stable film if 0.50

0.02

ΙχΗΓ⁵ 2.«:10-^ί 5Χ ·<Γ⁵ I χ ΗΓ* 2 :< ί<Γ⁴ 5χΙ<Γ* O.O i

«0

I ί0^"ί: 2χΚΓ^; χίΟ^"5 i SO^"4 2χΚΓ⁴ ίχΐΰ^" 9-301

FIG.8: Wavelength and growth rate of the most amplified mode, versus the thickness /¾ for various heat transfer characteristics; solid line - ΛΓ: = 1 and a = 1 W/cni² ; dashed line - & ± = 10 K and a = i W/ciir/K: dotted line - &Τ± = I K and a = 10 WcirrK. The other parameters are taken from Table I. a is typically larger than t. We also showed that t e larger the heat transfer coefficient is, the larger the thermocapillary stress at the interfaces, hence the larger is the resulting film thickness. Also, if the heat transfer coefficient is not large enough, longitudinal heat diffusion can become significant and smooth out the temperature gradient aloes the interlace. We finally address the stability in transverse direction and show that sufficiently thin films can experience a long-wave Marangoni instability for sufficiently large heat transfer coefficient and/or large temperature difference across the film. Appendix A: Stress balance in the shear-like description

In Eq. (4). we give the relevant stress balance by considering all possible contributions to axial stress, wit an implicit assumption of extensional flow. However, as indicated in the text and exposed by equations such and so. the final balance of stresses actually leads to a shear-like description of the flow; a necessary condition to allow for stable film formation [4], Here, we show that this seeming inconsistency is in fact consistent after all. since an axial balance that starts from the implicit assumption of shear flow ends up with the same stress balance.

In this section, we thus derive the stress balance equa tion in the context of lubrication approximation and of the shear-like distinguished limit for which Ma = 0{ε^{~ ι} ) and Ca = 0{e^s ). The continuity and the siieamwise Navier-Stokes equation in steady conditions have the forms c + e^v = 0 . (Al) p (iifXu + νν a ) ^■r)_xp + ^'¾ (d_xxu + f^y ) — pg , (A2) where the viscosity ¾ is assumed constant. Following the lubrication theory, the pressure is assumed to be uniform in a film cross-section and differs only from the ambient pressure by the Laplace pressure— y¾, (ft/2)_ such that d_xp = — γ3_χχχ{}ι /2 ) . This result would have been obtained by solving the cross-stream component of the Navier-Stokes equation, together with the normal stress balance, what we have by-passed here for the sake of clarity. The symmetry conditions apply at y = 0 (see Fig. 1 ),

d_y. = 0 , v = 0 , (A3) and the tangential stress condition applies at y = ft/2;

where n - V / I + {fkhV. The dimensionless variables are

Tlie diinensionless equations truncated at order t° except for the surface tension term, then become

V_Y = -½-, (A5)

Uyr = + e.Re (UU_X + VUr 1 - ²U_n- -— ½ (A6)

V = 0 at }^' 0. (A?)

( _r 0 at 7 = 0 , ί AS)

H

Γ.Α- = -eMrf- -*^*( 4H £/*- - Fr> at 7 =— . ί A9)

Jⁱ " ' 2

where the subscripts indicate the derivatives. Assuming€ = O(e') and AJ¾ = 0{e^~l) as indicated above, as well as Zte = 0( 1 ) and G = 0{ε), and expanding the variables in power of £ such as ϊ/ = ί/^ί° + _£ί/^ί1ϊ and F = F<⁰>+eF^, fAlO) the system of equations at leading order in ε has tlie form ] = - ° , (All) = ( 12) (⁰⁾ - 0 at 7 = 0. (A13)

L¾?⁰ = 0 at 7 = 0. f

H

-εΜαθ,ν at 7 \5\

Integrating (A 12) twice with respect to 7, using (A14) as well as the mass conservation

rH/2 . ,

i/^l°'d7 = 1 _t (A16) gives the leading-order streamwise velocity profile f^ = l ₊ ^¾_Tf^-¾ . (AIT) H 2Ca ' , 2 t 2 / ^'

Inserting (A 17) into (A15) yields the leading-order longitudinal stress balance

2εΜαθ_ίΧ - —HH^ = 0. ί Al 8)

To proeeed to next order in tlie e expansion, one first need to determme tlie leading order transverse velocity by integrating (All wit

The problem to be solved at first-order is e.Re [ IJ^{<3}Up + F¾ ⁰⁾ + G . (A20) Uy ' = 0 at = 0 . (A21 ) that can be integrated twice with respect to Γ, while forcing the mass conservation wit

The first- order correction to the velocity profile is thus obtained as

¹ = { ^ - ⁽ή { 24 - τ) ^{+ α ε } fA23)}

Substituting U = U^^0> + elfi ¹ ' into (A9) trimcaied at first order leads finally to

2εΜιΘ_; Y -—— ΗΗνγχ + GH - eRe^ = Q . ⁽A24) which is identical to (9) without the exten ional viscous term that is an ε -order term in the asymptotic expansion. Note also that the one-dimensional assumption for the temperature field implies that f¾ = Θ. This assumption is discussed in Appendix B.

Appendix B: One-dimensional temperature equation

In tiiis appendix, we demonstrate how the one-dimensional equation for the temperature field is derived and how it can be corrected to account for a temperature variation across the film induced by mterfaeial heat transfer. Neglecting the slreamwise temperature diffusion, the steady temperature equation has the form c_K (ttd_xT + vdyT) = KdypT , (Bl) where c_D is the specific heat capacity and κ is the thermal conductivity. The symmetric condition applies at v = 0,

d_vT = 0 , (B2) and Newton's cooling law applies at the interface v = ft/2,

Kdv T = - (T - ¾ fx) ! . (B3) where small slope, i.e. <¾ /; - 1, has been assumed. Using the same sealing as before, the dimen- sionless equations for the temperature become

0_TF = εΡβίυθχ+ν&γ) , (B4) θγ = - θ-¾) at ί^' =— , (BS) Θ_Γ = 0 at 7 = 0. (B6) where the Pee let and Biot numbers are

A- and Bi=^.

K K

Even though we consider 5^'/ = 2Bi/{ePe) > 1, the way the one-dimensdonal temperature equation is obtained will differ slightly depending on the magnitude of the heat advected by the flow and transferred to the surrounding, as discussed in the following.

1. Small transfer; Bi = Ofe) and Pe =0(l)

In the case of small heat transfer from the film to the surrounding, and expanding the temperature field like Θ = θ'°·' + εθ^ίί\ the system of equations (B4-B6) at leading order becomes θ¾ί = 0 , (B?)

H

Q> = 0 at Y =— . (B8) θί^0! = 0 at 7 = 0, (B9) the solution of which is simply a uniform temperature profile across the film &⁰⁾ = &{X). At first order, the system becomes

= -Βϊ(θ-Θ_α) at ]"=-, (Bll) = 0 at 7 = 0, (Bi2) which can be integrated across the thickness using (A 17) and (A ) to yield

§_X=-St (θ-€¾) , (B13) with St = IBi/iePe). This equation is identical to (11). 2. Large transfer: Bi = i and Pe = 0( 1 /e)

In the case of large heat transfer from the melt to the siirroundings, the temperature profile across the film cannot be uniform anymore at leading order. One can for instance assume a parabolic profile that satisfies the boundary conditions (B5) and (B6). and is written in terms of the average temperature Θ = (2/H ' ^" Θ ·^>:

Inserting (Bid), (A 17) and (A1 ) into (B4) yields

where the higher order terms in (A17,A19) have been dropped. We note that the only difference between (B15) and (B13) is the denominator of the r.h.s hat accounts for the parabolic temperature profile across tlie film. This equation has been used for instance i [ 15] to model the heat transfer in stretching glass sheets, where it has been shown that a non-uniform temperature profile across the film does not qualitatively change the film profile as c ompared to the case of a uniform temperature profile.

Appendix C: Coupling between gravity and inertia effects

When considering inertia! effects, the curvature calculated from (IS) with β = χ = 0 passes through a maximum H ' iis as X is decreased before reaching a constant value ¾_¾·(— o) as X—— <∞. In section IV A, we have investigated the sole influence of inertia using ¾^■(-«>) in the matching condition (15) while in section IV B where we investigated the influence of gravity only, we have used -¾Χ|ΒΜΚ as a matching curvature as the curvature never reaches a constant value in this case. Now that we want to analyze the influence of both inertia and gravity together, we should first assess the consequences of using -¾y|_max instead of_¾y(—<∞) as a matching condition. Fig. 9 shows both possible matching curvatures in function of the iiierta parameter 5. We see that as S increases, ¾IY|<_?1s>: decreases much less than ¾i—∞ ) . Consequently, using Horlmax instead of ¾(~ea) in the matching condition (15) will overestimate the film thickness he.

Now, we evaluate the effect of both inertia and gravity using Htirlmax as the curvature does not reaches a constant value as X→— =» Results are shown in Fig. 10.

0.0 0.2 0.4 0.6 0.8 i.O

(5

FIG. 9: Two different matching curvatures versus tiie inertia number δ with χ = β— 0 and = 1.

0.0 0.2 0.4 0.6 O.S 1.0

FIG. 10: Matching curvature -¾Yl_m__x for various gravity number β with χ = 0 and D = I ,

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[17] For liquids that experience a glass transition however, because of the exponential increase of die viscosity in the x direction due to cooling, there might be a location where becomes equal or larger than f^~2. This would correspond to an intermediate regime in which capillary. Marangoni and exteiisioiial viscous forces all balance.

Claims

CLAIMS What is claimed is:

1. A method for producing a film, comprising: a. providing a bath of a substance in a liquid state; b. pulling a liquid film of the substance from the bath; c. applying a predetermined surface tension gradient along the length of the film that allows the liquid film to be pulled at a desired pulling speed; and d. solidifying the liquid film.

2. The method of claim 1 wherein the surface tension gradient is applied by applying a temperature gradient, electrical gradient, surfactant concentration gradient, pulling speed, or combinations thereof.

3. The method of any one of the preceding claims, wherein the substance is silicon, glass or metal.

4. The method of any one of the preceding claims, wherein the substance is a polymer.

5. The method of claim 4, wherein said solidifying the liquid film includes curing the polymer.

6. The method of any one of the preceding claims, wherein said solidifying includes enhanced radiative heat transfer, evaporative cooling, thermoelectric elements, convective cooling, or combinations thereof.

7. The method of any one of the preceding claims, wherein the surface tension gradient is applied by applying a temperature gradient.

8. The method of claim 7, wherein the temperature gradient is applied by heating the liquid film.

9. The method of claim 7, wherein the surface tension gradient is applied by cooling the liquid film.

10. The method of any one of the preceding claims, wherein said pulling a liquid film forms a meniscus.

1 1. The method of any one of the preceding claims, wherein the desired pulling speed is greater than 1 cm/s.

12. The method of any one of the preceding claims, wherein the desired pulling speed is greater than 5 cm/s.

13. The method of any one of the preceding claims, wherein the desired pulling speed is greater than 10 cm s.

14. The method of any one of the preceding claims, wherein said predetermined surface tension gradient changes as a function of time.

15. The method of claim 14, wherein the thickness of the film changes as a function of the applied surface tension gradient.

16. The method of any one of the preceding claims, wherein the predetermined surface tension gradient changes along the width of the film.

17. The method of any one of the preceding claims, wherein the thickness of the film changes along the length of the film.