Patent Publication Number: US-2023149969-A1

Title: Slot Die Coating Using Concave Die Lip Over Deformable Back-Up Roll

Description:
BACKGROUND 
     There are two typical types of die coating configurations for applying coating material on a moving web: i) coating against a back-up roll or ii) coating on a tensioned web in a free span. Slot dies have been widely used as coating devices for applying coatings on a web.  FIG.  1   ′ illustrates a slot die  2 ′ for disposing a liquid material  7 ′ on a free span  3 ′ of a flexible web to form a coating  9 ′. 
     SUMMARY 
     There is a desire to improve coating uniformity when applying a coating on a moving web via a slot die. For example, in the process shown in  FIG.  1   ′, the free span  3 ′ of a baggy web may not maintain uniform tension across the width of the slot die  2 ′, leading to variations in coat weight/thickness across the baggy web. The present disclosure provides methods and apparatuses of applying a uniform coating on a web via a slot die over a deformable back-up roll. 
     Briefly, in one aspect, the disclosure describes a method of applying a coating onto a web. The method includes providing a back-up roll having a deformable inner layer with a surface thereof covered by a deformable outer layer; providing a slot die having one or more die lips extending along a cross direction, the die lips including a coating surface positioned proximate to the back-up roll; disposing a flexible web between the back-up roll and the die lips; and dispensing a liquid coating material from the slot die onto the flexible web. The flexible web at a contacting area is impressed into the back-up roll with an engagement depth D. At least a portion of the coating surface is substantially concave and facing to the back-up roll. 
     In another aspect, this disclosure describes a coating apparatus including a back-up roll having a deformable inner layer with a surface thereof covered by a deformable outer layer; a slot die having one or more die lips extending along a cross direction, the die lips including at least one coating surface that is substantially concave and positioned proximate to the back-up roll; and a flexible web disposed between the back-up roll and the slot die. The flexible web at a contacting area is impressed into the back-up roll with an engagement depth D, and the slot die is configured to dispense a liquid coating material onto the web. In some cases, the inner layer of the back-up roll is softer than its outer layer. At least a portion of the coating surface is substantially concave and facing to the back-up roll. 
     Various unexpected results and advantages are obtained in exemplary embodiments of the disclosure. One such advantage of exemplary embodiments of the present disclosure is that a substantially uniform coating can be formed on a moving web via a slot die including at least one concave coating surface over a deformable back-up roll. This can be achieved by engaging the concave coating surface with a deformable back-up roll having a deformable outer layer and a deformable inner layer, where a flexible web and the deformable outer layer at a contacting area can be impressed into the deformable inner layer with a certain engagement depth. The embodiments described herein can significantly expand the coating window when coating a moving web while significantly mitigating undesired coating effects. For example, coating on a free-span of a baggy web may result in variations in coat weight across the web, while coating against a rigid back-up roll, may create issues related to back-up roll nonuniformity and have limitations in coating thin layers of liquid on a web. 
     Various aspects and advantages of exemplary embodiments of the disclosure have been summarized. The above Summary is not intended to describe each illustrated embodiment or every implementation of the present certain exemplary embodiments of the present disclosure. The Drawings and the Detailed Description that follow more particularly exemplify certain preferred embodiments using the principles disclosed herein. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The disclosure may be more completely understood in consideration of the following detailed description of various embodiments of the disclosure in connection with the accompanying drawings, in which: 
         FIG.  1   ′ illustrates a perspective view of a slot die coating on a free-span web (prior art). 
         FIG.  1    is a perspective view of a coating apparatus applying coating on a moving web, according to one embodiment. 
         FIG.  2 A  is an enlarged portion view of  FIG.  1   , according to one embodiment. 
         FIG.  2 A ′ is an enlarged portion view of a coating apparatus including a rigid back-up roll. 
         FIG.  2 B  is an enlarged portion view of  FIG.  1   , according to another embodiment. 
         FIG.  2 C  is a perspective view of the web of  FIG.  2 A . 
         FIG.  3 A  is a schematic diagram of a back-up roll engaged with a test roller for mechanical compression testing. 
         FIG.  3 B  is a schematic diagram of a back-up roll engaged with a test plate for mechanical compression testing. 
         FIG.  4    illustrates force versus engagement curves for the mechanical compression testing in  FIGS.  3 A-B . 
         FIG.  5    illustrates plots of slope factor S versus engagement depth D for various back-up rolls. 
         FIG.  6 A  is a schematic diagram of a die lip positioned proximate to a back-up roll, according to one embodiment. 
         FIG.  6 B  is a schematic diagram of a die lip positioned proximate to a back-up roll, according to one embodiment. 
         FIG.  6 C  is a schematic diagram of a comparative die lip positioned proximate to a back-up roll. 
     
    
    
     In the drawings, like reference numerals indicate like elements. While the above-identified drawings, which may not be drawn to scale, sets forth various embodiments of the present disclosure, other embodiments are also contemplated, as noted in the Detailed Description. In all cases, this disclosure describes the presently disclosed disclosure by way of representation of exemplary embodiments and not by express limitations. It should be understood that numerous other modifications and embodiments can be devised by those skilled in the art, which fall within the scope and spirit of this disclosure. 
     DETAILED DESCRIPTION 
     For the following Glossary of defined terms, these definitions shall be applied for the entire application, unless a different definition is provided in the claims or elsewhere in the specification. 
     Glossary 
     Certain terms are used throughout the description and the claims that, while for the most part are well known, may require some explanation. It should be understood that: 
     In this application, the terms “compressible” or “incompressible” refers to a material property, i.e., compressibility, of an object (e.g., an elastomer outer layer) which is a measure of the relative volume change of the material in response to a pressure. For example, the term “substantially incompressible” refers to a material having a Poisson&#39;s ratio greater than about 0.45. 
     The term “elastically deformable” means a deformed object (e.g., an inner layer of synthetic foam) being capable of substantially 100% (e.g., 99% or more, 99.5% or more, or 99.9% or more) recovering to its original, undeformed state. 
     The term “baggy web” refers to a web that shows non-planarity or distortions, at least in a portion of the surface of the web, when positioned on a flat surface. The web bagginess, which may be caused by differential tensions across the width of the web during the web manufacturing, can result in cross-web direction (CD) length variation. U.S. Pat. No. 6,178,657 describes a method and apparatus to measure the internal web length differences in the CD of sheet materials. In this application, the CD length variation of a baggy web can be equivalent to or smaller than, for example, 10,000 ppm (equivalent to 1% strain), or 1,000 ppm (equivalent to 0.1% strain). 
     The term “slot die” or “slot die coating” refers to a system or a method of dispensing a liquid coating material from a die body thereof to a web. The die coating described herein is a pre-metered coating process in which the amount of liquid applied to the web per unit area is substantially predetermined by a fluid metering device upstream, such as, for example, a precision gear pump. Typical slot die coating methods and systems are described in, e.g., Ian D. Gates, Slot Coating Flows: Feasibility, Quality, PhD Thesis, 1999, University of Minnesota. 
     The terms “liquid,” “liquid material,” or “liquid coating material” refers to any materials flowable at coating operation conditions described herein. 
     In this application, the terms “polymer” or “polymers” includes homopolymers and copolymers, as well as homopolymers or copolymers that may be formed in a miscible blend, e.g., by coextrusion or by reaction, including, e.g., transesterification. The term “copolymer” includes random, block and star (e.g. dendritic) copolymers. 
     In this application, by using terms of orientation such as “atop”, “on”, “over,” “covering”, “uppermost”, “underlying” and the like for the location of various elements in the disclosed coated articles, we refer to the relative position of an element with respect to a horizontally-disposed, upwardly-facing substrate (e.g., web). However, unless otherwise indicated, it is not intended that the substrate (e.g., web) or articles should have any particular orientation in space during or after manufacture. 
     In this application, by using the term “overcoated” to describe the position of a layer with respect to a substrate (e.g., web) or other element of an article of the present disclosure, we refer to the layer as being atop the substrate (e.g., web) or other element, but not necessarily contiguous to either the substrate (e.g., web) or the other element. 
     In this application, the term “machine direction” refers to the direction in which the web travels. Similarly, the term “cross-web direction” refers to the direction perpendicular to the machine direction (i.e., substantially perpendicular to the direction of travel for the web), and in the plane of the top surface of the web. 
     In this application, the terms “about” or “approximately” with reference to a numerical value or a shape means+/−five percent of the numerical value or property or characteristic, but expressly includes the exact numerical value. For example, a viscosity of “about” 1 Pa-sec refers to a viscosity from 0.95 to 1.05 Pa-sec, but also expressly includes a viscosity of exactly 1 Pa-sec. Similarly, a perimeter that is “substantially square” is intended to describe a geometric shape having four lateral edges in which each lateral edge has a length which is from 95% to 105% of the length of any other lateral edge, but which also includes a geometric shape in which each lateral edge has exactly the same length. 
     In this application, the term “substantially” with reference to a property or characteristic means that the property or characteristic is exhibited to a greater extent than the opposite of that property or characteristic is exhibited. For example, a substrate (e.g., web) that is “substantially” transparent refers to a substrate (e.g., web) that transmits more radiation (e.g. visible light) than it fails to transmit (e.g. absorbs and reflects). Thus, a substrate (e.g., web) that transmits more than 50% of the visible light incident upon its surface is substantially transparent, but a substrate (e.g., web) that transmits 50% or less of the visible light incident upon its surface is not substantially transparent. 
     In this application, the singular forms “a”, “an”, and “the” include plural referents unless the content clearly dictates otherwise. Thus, for example, reference to fine fibers containing “a compound” includes a mixture of two or more compounds. As used in this specification and the appended embodiments, the term “or” is generally employed in its sense including “and/or” unless the content clearly dictates otherwise. 
     As used in this application, the recitation of numerical ranges by endpoints includes all numbers subsumed within that range (e.g. 1 to 5 includes 1, 1.5, 2, 2.75, 3, 3.8, 4, and 5). 
     Unless otherwise indicated, all numbers expressing quantities or ingredients, measurement of properties and so forth used in the specification and embodiments are to be understood as being modified in all instances by the term “about.” Accordingly, unless indicated to the contrary, the numerical parameters set forth in the specification and more particularly the Listing of Exemplary Embodiments and the claims can vary depending upon the desired properties sought to be obtained by those skilled in the art utilizing the teachings of the present disclosure. At the very least, and not as an attempt to limit the application of the doctrine of equivalents to the scope of the claimed embodiments, each numerical parameter should at least be construed in light of the number of reported significant digits and by applying ordinary rounding techniques. 
     Exemplary embodiments of the present disclosure may take on various modifications and alterations without departing from the spirit and scope of the present disclosure. Accordingly, it is to be understood that the embodiments of the present disclosure are not to be limited to the following described exemplary embodiments, but are to be controlled by the limitations set forth in the claims and any equivalents thereof. 
     Methods and apparatuses are described herein for slot die coating on a moving substrate. In a coating process described herein, a flexible web is disposed between a back-up roll and a slot die. The back-up roll has a deformable inner layer with a surface thereof covered by a deformable outer layer. The inner layer may be softer than the outer layer. The flexible web can be a baggy web that wraps around the back-up roll. 
     The slot die and the back-up roll can be pressed against each other with an engagement depth D. A liquid coating material can be dispensed from the slot die onto the flexible web to form a liquid layer (coating bead) between a coating surface of the slot die and the flexible web. The coating bead described herein refers to a volume of liquid contained between a substrate and a coating die. It is to be understood that the pressure in the coating bead can further impress the flexible web and the deformable outer layer into the deformable inner layer. 
     In some embodiments, the engagement depth D can be adjusted to enable a uniform coating on a web. In some embodiments, a positioning mechanism can be provided to control the distance between the slot die and the back-up roll so as to adjust the engagement depth D. This positioning mechanism may adjust the engagement depth D by moving the slot die relative to a fixed backup roll, moving the backup roll relative to a fixed slot die, or by moving both the backup roll and the slot die simultaneously. The pressure in the coating liquid may be altered as the engagement depth D is changed. In some embodiments, the engagement width W may correspond to the width of the coating surface of the slot die, which may be substantially a constant during coating. 
     The uniformity of a liquid coating may be impacted by a combination of many sources of imperfections and may result in variations in the appearance and amount of the coating that adheres to a substrate. The present disclosure addresses some issues that might impact the coating uniformity. In some embodiments, the average amount of the applied liquid coating can be metered by a solution handling system, which can be proportioned to the speed and width of the flexible web that is to be coated. The thickness variation in the cross-web direction of the applied liquid coating can be controlled by the performance of the die cavity, which shapes flow from a feed pipe into a sheet that emerges from a die slot. The thickness uniformity in the cross-web direction is referred to as the coating profile. In some embodiments, the thickness uniformity in the down-web direction can be controlled by solution handling (e.g., to control a down-web coating thickness variation due to the variation in the flowrate delivered by a pumping system) and web handling (e.g., to control a down-web coating thickness variation due to variation in the speed of a substrate). In the present disclosure, the coating profile can be controlled such that both the cross-web coating thickness and the down web coating thickness are substantially uniform over time. 
     In some cases, factors other than the performance of the solution handling, the die cavity and/or the web handling, may also affect the uniformity of the coating profile. For example, nonuniformities in the coating bead may create visible localized defects in the applied coating such as those brought on by entrainment of air between the coating and the web, break-up of the continuous coating bead into rivulets or repeating cross web bands, and surface roughness in the coated surface. These discontinuities and nonuniformities in the coating are generally referred to as coating defects. 
     In addition, coating defects may be produced by imperfections in a back-up roll. For example, when a back-up roll is used, it may deviate significantly from an ideal cylinder, as indicated by a total indicated runout (TIR). The requirement for a low TIR (e.g., less than 1 micrometer) back up roll can significantly increase the cost and complexity of a slot die coating system with a back-up roll. When a liquid coating is applied in free-span such as shown in  FIG.  1   ′, baggy lanes in the substrate may also lead to coating defects as the web bag can lead to an imbalance between the pressure produced by the web and that produced by the coating bead. 
     In some cases, producing coatings with a wet thickness of about 50 micrometers or less can be challenging when using a slot die. To produce such a thin coating against a rigid back-up roll such as shown in  FIG.  2 A ′, the slot die  2  may be positioned at a very close proximity to the rigid back-up roll  10 ′ (e.g., less than 100 micrometers), and therefore requires increasing precision not just in the positioning system, but also in the uniformity of the die and roll surfaces, and this quickly becomes impractical. A free-span coating does not have this challenge since there is no back-up roll, and so in theory one may be able to position the web arbitrarily closely to the slot die, and so produce arbitrarily thin coatings. In practice, a free-span coating may require increasingly accurate control of the tension and thickness in the substrate upon which the coating is applied, as these parameters can lead to changes in the pressure in the coating bead, and therefore variations in the local thickness of the coating. 
     Methods and apparatuses are described herein for slot die coating which can address the above described issues. In some embodiments of the present disclosure, a deformable back-up roll can be used, which allows the web to lay against the deformable back-up roll, diminishing the impact of web bag as compared to free-span coating, while also allowing the back-up roll to deflect under the fluid pressure in the coating bead, diminishing the impact of any surface nonuniformities in the back-up roll as compared to coating against a rigid back-up roll. 
     Some embodiments of the present disclosure can further address variation of coating thickness due to a splice. When changing from a first input roll of substrate to a second input roll of substrate, it is common to tape the trailing end of the first input roll of substrate to the leading edge of the second input roll of substrate, producing what is commonly referred to as a splice. In practice, this produces a significant thickness variation in the substrate at the location of the splice, due to the thickness of any tape used to hold the two substrates together, as well as due to any overlap between the two layers of substrate. When coating with a slot die against a rigid back-up roll, this may force the coating practitioner to temporarily increase the gap between the slot die and the rigid back-up roll so that the splice does not get stuck at the slot die, which typically may result in the web breaking and would therefore lead to a significant interruption in the coating operation. This may not typically be a problem when coating in free span, since there is no back-up roll to trap the splice. In the present disclosure, a rigid back-up roll is replaced with a soft, deformable back-up roll, and the splice can pass through the gap between the die and the back-up roll without tearing the splice due to deformation of the roll, resulting in a less significant interruption to the coating operation. 
     Various exemplary embodiments of the disclosure will now be described with particular reference to the Drawings. Referring now to  FIG.  1   , a perspective view of a coating apparatus  100  for applying a liquid coating on a moving web via a slot die over a back-up roll, according to some embodiments.  FIGS.  2 A and  2 B  illustrate an enlarged portion view of the coating apparatus  100  in  FIG.  1   , according to some embodiments. 
     The coating apparatus  100  includes a back-up roll  10  and a slot die  20 . The slot die  20  has a die lip  22  that engages with the back-up roll  10  to form a coating zone  120 . In the depicted embodiment, the die lip  22  includes an upstream lip  22   a  and a downstream lip  22   b  which provide an upstream coating surface at  22   a  and a downstream coating surface at  22   b , respectively. A flexible web  3  of indefinite length material is conveyed in a machine direction  5  into the coating zone  120 . It is to be understood that the web may not be limited to the specific wrap angles as it enters/exits the coating zone shown schematically in  FIG.  1   . Also, the vertical position of the slot die  20  compared to the back-up roll  10  may not be limited to what is depicted in  FIG.  1   . 
     The slot die  20  includes a die body  21  defining an internal manifold  24 . The die lip  22  of the slot die  20  has a die opening  25  in fluid communication with the internal manifold  24  via a slot channel  23 . The die lip  22  is positioned proximate to the back-up roll  10  and extends along a cross direction of the web  3 . The slot die  20  and the back-up roll  10  are pressed against each other with a footprint having an engagement depth D and a machine-direction width W as shown in  FIGS.  2 B-C . 
     A coating material  7  is provided to the internal manifold  24 , flows through the slot channel  23 , and is dispensed from the die opening  25 . The die lip  22  of the slot die  20  provides a coating surface (e.g., a surface of the upstream die lip portion  22   a , and/or a surface of the downstream die lip potion  22   b ) that is engaged with flexible web  3  wrapped around the back-up roll  10 . When the coating material  7  is dispensed from the slot die  20  onto the flexible web  3 , a liquid layer (coating bead)  92  is present between the coating surface of the slot die  20  and the flexible web  3 . 
     In the embodiment of  FIG.  2 A , the slot die  20  is pressed against the back-up roll  10  such that the coating surface of the slot die  20  at least partially surpasses the un-deformed surface  201  of the back-up roll  10 . 
     In the embodiment of  FIG.  2 B , the slot die  20  is pressed against the back-up roll  10  where the coating surface of the slot die  20  does not surpass the un-deformed surface  201  of the back-up roll  10  and there is a gap between the coating surface of the slot die  20  and the un-deformed surface  201  of the back-up roll  10 . 
     A die opening may include one or more channels through which the coating fluid can flow towards the back-up roll, where the one or more channels are arranged in the machine direction (for example, the slot dies described in chapter 4 of Jaewook Nam, Analysis of Tensioned-Web-over-Slot Die Coating, PhD Thesis, 2009, University of Minnesota) or in the cross-web direction (for example, the slot dies described in U.S. Pat. No. 7,846,504). The width of the die opening may be, for example, 0.05 mm, 0.1 mm, 0.25 mm, or any other suitable number. 
     In some embodiments, the die slot may also be angled relative to a radial projection of the back-up roll, with this angle being about 0 degrees, 2 degrees, 5 degrees, or 10 degrees, and with either positive or negative angles both being acceptable. It is to be understood that various configurations of slot die can be applied herein. Exemplary slot dies are described in, for example, Ian D. Gates, Slot Coating Flows: Feasibility, Quality, PhD Thesis, 1999, University of Minnesota. 
     In the embodiments depicted in  FIGS.  2 A and  2 B , the die lip  22  provides an upstream coating surface at  22   a  and a downstream coating surface at  22   b , separated by the die opening  25 , through which one or more coating liquids are applied to the flexible web  3 . The die lip  22  have its coating surface(s) in contact with the coating liquid. The coating surface(s) can take on various shapes. 
     In the present disclosure, at least a portion of a die lip described herein can be substantially concave and be faced towards a deformable back-up roll. A portion of a die lip surface may be considered concave when a vector normal to that portion of the die lip surface and pointing toward the center of curvature for the die lip is directed away from the die lip towards the back-up roll. As an example, the die lips in  FIG.  6 A  may be considered concave. Conversely, a die lip may be considered convex if a vector normal to the die lip surface and pointing toward the center of curvature for the lip is directed into the die. As an example, the die lips in  FIG.  6 C  may be considered convex. In some embodiments, a die lip may contain one or more sections which are concave and one or more sections which are convex. 
     In some embodiments, a substantially concave coating surface may have a radius of curvature in the range, for example, from about 0.1 mm to about 800 mm, from about 0.5 mm to about 600 mm, from about 1 mm to about 400 mm, from about 10 mm to about 400 mm, from about 20 mm to about 200 mm, from about 50 mm to about 150 mm, etc. In some embodiments, a substantially concave coating surface may have a radius of curvature in the range, for example, no less than about 1.0 mm, no less than about 2.0 mm, no less than about 5.0 mm, or no less than about 10.0 mm, etc. 
     It is to be understood that a coating surface that is substantially concave may have at least a portion with an infinite radius of curvature, in which case it may be considered approximately straight. It is to be understood that a coating surface may not be composed of a purely circular arc, and may have any desired concave-shaped geometries. In some embodiments, a coating surface that is substantially concave may include multiple facets having various radius of curvatures. In some embodiments, the radius of curvature of a substantially concave coating surface may also be infinite, indicating that the coating surface of a die lip can be approximately straight. In some embodiments, it might be beneficial to have a concave section of a downstream die lip (for example, a downstream lip  222   b  in  FIG.  6 A ) located nearest to the downstream edge of the surface (for example, the location G 1  in  FIG.  6 A ). A concave section of a downstream die lip adjacent to a downstream edge thereof along the coating surface may have a length of, for example, about 0.25 mm or greater, about 0.5 mm or greater, about 1.0 mm or greater, about 5.0 mm or greater, or 10.0 mm or greater. A concave section of a downstream die lip may have a length, for example, about 20% or greater, 
     A die lip may have a length both upstream and downstream from the opening(s), and this length may be, for example, from about 0.01 mm to about 25 mm, from about 0.1 mm to about 25 mm, from about 0.25 mm to about 25 mm, from about 0.5 mm to about 25 mm, from about 1.0 mm to about 25 mm, from about 2.0 mm to about 25 mm, from about 5 mm to about 25 mm, or any other suitable number. 
     At least a portion of the concave coating surface may have a radius of curvature greater than that of an undeformed back-up roll. In some embodiments, the coating surface may have a concave radius curvature, for example, about 5% greater, about 10% greater, about 20% greater, about 50% greater, or about 100% greater than that of the undeformed back-up roll. In some embodiments, the concave coating surface may have a radius of curvature of, for example, at least about 1.01 times, at least about 1.05 times, at least about 1.1 times, at least about 1.2 times, at least about 1.5 times, or at least about 2 times of the radius of curvature of an undeformed back-up roll. In some embodiments, the concave coating surface may have a radius of curvature of, for example, from about 1.01 to about 50 times, from about 1.1 times to about 20 times, from about 1.1 times to about 10 times, or from about 1.1 to 2.0 times, that of an undeformed back-up roll. The back-up roll may have a diameter in the range, for example, from about 50 mm to about 600 mm, from about 100 mm to about 400 mm, or from about 120 mm to about 300 mm. 
     Not to be bound by theory, when the radius of curvature of a concave coating surface or a concave portion of a coating surface is larger than the radius of the undeformed back-up roll, then as the die lip is engaged with the backup roll, the surface of the deformable back-up roll may deform, and a local radius of curvature for the surface of the back-up roll may increase, creating a more parallel fluid channel between the die lip coating surface and the surface of the deformed back-up roll. In contrast, for a rigid back-up roll, the use of a concave die lip with a larger radius of curvature than the radius of the backup roll may result in either an unacceptable level of convergence in the coating gap, or in divergence of the coating gap, with both cases potentially producing coating defects such as weeping, widening, ribbing, etc. It may be possible, therefore, to have a coating configuration that includes some initial divergence (or excessive convergence) between the die lip and the undeformed surface of the backup roll, and simply by increasing the engagement depth D of the die lip into the deformable backup roll, which locally flattens the roll, this initial divergence (or excessive convergence) may be eliminated, producing a uniform coating. 
       FIGS.  6 A-C  illustrate slot dies including coating surfaces with various geometries. A flexible web (not shown) is conveyed in the machine direction  5  into the coating zone formed by engaging the respective slot dies  200 ,  202  and  204  and the back-up roll  10 . The slot channels  252 ,  254  and  256  each with a die opening are provided for the respective slot dies  200 ,  202  and  204 . 
     In the embodiment depicted in  FIG.  6 A , a die lip  222  of a slot die  200  includes an upstream lip  222   a  and a downstream lip  222   b  which provide an upstream coating surface and a downstream coating surface, respectively. The upstream lip  222   a  and the downstream lip  222   b  each have a concave coating surface. 
     In the embodiment depicted in  FIG.  6 B , the die lip  224  of a slot die  202  includes an upstream lip  224   a  and a downstream lip  224   b  which provide an upstream coating surface and a downstream coating surface, respectively. The upstream lip  224   a  has a convex coating surface and the downstream lip  224   b  has a concave coating surface. In some embodiments, the upstream lip  224   a  can have a concave coating surface and the downstream lip  224   b  can have a convex coating surface. A convex coating surface may have a radius of curvature in the range, for example, from about 0.01 mm to about 200 mm, from about 0.1 mm to about 100 mm, from about 1 mm to about 100 mm. 
     As a comparison, as shown in  FIG.  6 C , the die lip  226  of a slot die  204  includes an upstream lip  226   a  and a downstream lip  226   b  which provide an upstream coating surface and a downstream coating surface, respectively. The upstream lip  226   a  and the downstream lip  226   b  each have a convex coating surface. 
     In the present disclosure, a rate of convergence or convergence rate of the gap can be controlled or adjusted to expand the coating window while significantly mitigating undesired coating effects. The rate of convergence is defined as the rate of change of the gap with respect to down-web position along the coating bead. The rate of convergence Rc can be expressed as: 
         Rc =( g   m   −g   n )/( L   Gm-Gn )×100%  [1a]
 
     where Gm and Gn represent an upstream and a downstream position of a coating surface of a slot die, respectively; g m  and g n  represent the gap values at the positions Gm and Gn, respectively; and L Gm-Gn  represents the distance along the lip coating surface between the positions Gm and Gn. 
     When the gap between a die lip and the backup roll is smaller at a downstream portion (e.g., at position Gn) of a die lip relative to an upstream portion (e.g., at position Gm) of a die lip, the rate of convergence Rc is positive, or the gap is said to be convergent, between those two portions. When the gap between a die lip and the backup roll is larger at a downstream portion of a die lip relative to an upstream portion of a die lip, then the rate of convergence is negative, or the gap is said to be divergent, between those two portions. For the purposes of this document we consider the convergence rate Rc to be between the die lip and the backup roll in its undeformed state, though as stated previously it is to be understood that the backup roll may deform during coating, and so the gap between the die lip and the backup roll surface in a deformed state may be different from the gap when the roll is in its undeformed state. 
     In some embodiments, a convergence rate Rc of gaps between the coating surface of the slot die and an outer surface of the deformable outer layer of the back-up roll under an undeformed state may have an absolute value no greater than about 30%, no greater than about 20%, no greater than about 15%, no greater than about 10%, or no greater than about 5%. The absolute value of a convergence rate Rc may be no less than about 0%, no less than about 0.1%, no less than about 0.5%, no less than about 1%, or no less than about 1.2%. In some embodiments, the absolute value of a convergence rate Rc may be in the range, for example, from about 0.1% to about 20%, from about 0.1% to about 10%, from about 0.5% to about 10%, or from about 1.0% to about 5.0%. 
     The convergence rate can be linear, or non-linear, across all or parts of a die lip. The rate of convergence has an impact on the force that is generated by the coating material in the coating area. A preferred rate of convergence is application-dependent and may depend on the rheology of the coating material, the back-up roll and the die design. 
     The convergence rate can be controlled by adjustments to the position of a die lip (e.g., elevation, shift, rotation, or any combinations thereof) with respect to a back-up roll, by changing the geometry of the die lips, and/or by a combination thereof. 
     In some embodiments, a slot die can be designed with its die slot intersecting a radial projection of a back-up roll to form an angle therebetween such that desired range of convergence rates can be obtained. In some embodiments, the die body can be rotated, twisted, and/or shifted such that an angle is formed between the die slot and a radial projection of a back-up roll. The convergence rate of the die lips relative to the undeformed back-up roll can be controlled or adjusted by changing the angle between the die slot and a radial projection of a back-up roll in any suitable manner. 
     For example, as shown in  FIG.  6 A , the die lip  200  is positioned with a shift or rotation with respect to a radial projection  10   a  or  10   b  of the back-up roll  10 , resulting in an angle  212  between the direction  200   a  of the die slot and the radial projection  10   b  that faces toward the die slot. In the depicted embodiment of  FIG.  6 A , the gaps g 1 , g 2 , g 3 , and g 4  at the positions G 1 , G 2 , G 3  and G 4  can be approximately equal (i.e., a relatively small convergence rate is shown in this figure). 
     In some embodiments, the angle  212  can be adjusted by rotating the die lip  200  to control the gaps g 1 , g 2 , g 3  and/or g 4 , and thus control the resulting convergence rate of the die lips. For example, for the embodiment of  FIG.  6 A , rotating the die lip  200  counterclockwise by some amount might result in an increase in the angle  212 , a decrease in the gap g 1 , and an increase in the gap g 4 . This may have the net effect of increasing the convergence rate of the die lip  200 . 
     While not wanting to be bound by theory, it is believed that the convex coating surfaces of the slot die  204  in  FIG.  6 C  may result in a rapidly converging channel where the coating material is located when engaging the back-up roll  10 , which may significantly narrow the coating window. A coating window described herein refers to the range of engagement depths D over which an acceptable coating can be produced. In some cases, sufficiently high levels of convergence may result in the generation of a lower total force by the coating material, thereby reducing the coating window for some sets of process conditions and materials. 
     Referring to the embodiments of  FIGS.  6 A-B , reducing the convergence rate by employing a concave die lip geometry allows for the generation of larger forces by the coating material in the coating channel. The optimal die lip geometry can be determined by considering factors including, for example, the characteristics of the back-up roll (e.g., diameter, compression/force relationship, etc.), geometry of the die lips, process conditions, coating material, etc. 
     The die coating processes described herein are referred to as pre-metered coating processes. In some embodiments, the coating apparatuses described herein can further include a pump and a control system for the pump. The pump can provide a predetermined flow rate of the fluid coating material into the internal manifold  24 . The predetermined flow rate, along with other factors such as, for example, the web speed, can largely define the thickness of the coating layer. The pump can be, for example, a high bandwidth precision pump that is in fluid communication with an input port of the die body. The pump is configured to supply the coating material  7  into the internal manifold  24  at an adjustable flow rate such that the coating material  7  can be dispensed onto the moving web  3  through the die lip  22  to form a coating  9  with a desired thickness. In some embodiments, the coating thickness can be controlled in a range, for example, about 1 to about 500 micrometers. 
     The coating material  7  can be any coatable material including, for example, water- or solvent-based solutions, radiation curable solutions primers, adhesives, inks, dispersions, emulsions, etc. The coating material may be Newtonian or non-Newtonian. In some embodiments, the coating solution may have a shear-sensitive viscosity or may shear thin and have a viscosity below about 100,000 centipoise (cPs), optionally below about 1,000 cPs. For example, a typical fluid may have a viscosity of about 10,000 cPs at a shear rate of 10 l/s and a viscosity of about 3,000 cPs at a shear rate of 2,000 l/s. The wet coating on the web can be dried, cured, or solidified to form a coating layer on the web. A uniform coating  9  is formed on the surface  31  of the web  3  that faces the slot die  20 . A wet coating thickness refers to the coated thickness on the web immediately after the slot die. After drying, curing, or solidification, the coating thickness can be reduced. That reduction of coating thickness is due to a loss of volatile materials during drying, and/or shrinkage of the polymer. Curing can be accomplished by, for example, exposure of the coating to elevated temperature, or actinic radiation. Actinic radiation can be, for example, in the UV spectrum. 
     The back-up roll  10  has a deformable inner layer  12  with a surface thereof covered by an outer layer  14 . The inner and outer layers  12 ,  14  may be permanently bonded together in some embodiments and may not be permanently bonded together in other embodiments. It is to be understood that the “outer layer” does not necessarily mean an outermost layer; and the “inner layer” does not necessarily mean an innermost layer. The outer layer  14  has a substantially uniform thickness about the periphery of the inner layer  12 . The deformable inner layer  12  is mounted onto a rigid central core  11  (e.g., a metal core, a fiberglass core, a fiberglass shell mounted on a metal core, etc.) with a substantially uniform thickness about the periphery of the rigid central core  11 . In some embodiments, the thickness ratio between the deformable inner layer  12  and the outer layer  14  can be about 3:1 or greater, about 5:1 or greater, about 7:1 or greater, or about 10:1 or greater. In some embodiments, the outer layer  14  has a thickness in the range from about 0.005″ to about 0.300″, optionally from about 0.005″ to about 0.120″. As used herein, 1″ equals to 2.54 cm. In some embodiments, the deformable inner layer  12  has a thickness in the range from about 0.125″ to about 3″, optionally from about 0.4″ to about 1.0″. In some embodiments, compressible rollers described in U.S. Pat. No. 5,206,992 can be used to make the back-up roll herein. 
     In some embodiments, the material used for the inner layer  12  can be softer than the material used for the outer layer  14 . That is, an identical compressive force applied to an identically sized block of each material can result in a larger deformation in the direction of applied force with the softer material than with the harder material. This softness may be provided in several ways, for example by choosing a material with a lower hardness (as indicated using any appropriate hardness scale, such as Shore A or Shore OO), by choosing a material with a lower elastic modulus, by choosing a material with a higher compressibility (typically quantified via a material&#39;s Poisson&#39;s ratio), or by modifying the structure of the softer material to contain a plurality of gas inclusions, such as a foam or an engraved structure, etc. For example, when the outer layer  14  includes a material having a hardness of 60 Shore A (as measured using ASTM D2240), then the hardness of the inner layer  12  may be less than 60 Shore A. It should be noted that in some cases the hardness may be most appropriately measured using different scales for the inner and outer layers (e.g., Shore A durometer for the outer layer and Shore 00 for the inner layer). In some embodiments, the compressibility of the inner layer  12  may be measured via Compression Force Deflection Testing per ASTM D3574 when the inner layer is foam; and via Compression-Deflection Testing per ASTM D1056 when the inner layer is a flexible cellular material such as, for example, sponge or expandable rubber. The inner layer  12  may have a compressibility of less than about 45 psi at 25% deflection, optionally less than about 20 psi at 25% deflection. As used herein, 1 psi equals to 6.89 kPa. 
     In some embodiments, the outer layer  14  can be made of material(s) that are substantially incompressible, e.g., the relative volume change of the material in response to a contact pressure is less than 5%, less than 2%, less than 1%, less than 0.5%, or less than 0.2%. The inner layer  12  is configured to be elastically deformable, e.g., being capable of substantially 100% (e.g., 99% or more, 99.5% or more, or 99.9% or more) recovering to its original state after being deformed. In some embodiments, the inner layer  12  can be compressible to provide the desired deformability. In some embodiments, the inner layer  12  may be substantially incompressible, but sufficiently soft to provide the desired deformability. In some embodiments, the inner layer  12  may be a layer made of substantially incompressible material which has been patterned, 3D printed, embossed, or engraved to provide the desired deformability. 
     In some embodiments, the deformable inner layer of the back-up roll has a hardness less than that of the deformable outer layer of the back-up roll. In some embodiments, the hardness of the deformable outer layer  14  can be greater than about 40 Shore A, optionally greater than about 50 Shore A. In some embodiments, the hardness of the deformable inner layer  12  can be less than about 20 Shore A, optionally less than about 10 Shore A. 
     In some embodiments, the inner layer  12  may have a higher compressibility than the outer layer  14 . In some embodiments, the outer layer  14  can have a Poisson&#39;s ratio greater than about 0.1, greater than about 0.2, greater than about 0.3, or preferably greater than about 0.4. In some embodiments, the deformable inner layer  12  can have a Poisson&#39;s ratio less than about 0.5, less than about 0.4, less than about 0.3, or preferably less than about 0.2. In some embodiments, the deformable inner layer  12  can have a negative Poisson&#39;s ratio. 
     In some embodiments, the deformable outer layer  14  can include one or more materials of an elastomer, a metal, a fabric, or a nonwoven. In some embodiments, the outer layer  14  can be a substantially incompressible elastomer having a hardness greater than about 40 Shore A, or optionally greater than about 50 Shore A. The thickness of the outer layer  14  of the back-up roll  10  can be less than about 10 mm, less than about 5 mm, or less than about 2 mm. Suitable elastomers may include thermoset elastomers such as, for example, Nitriles, fluoroelastomers, chloroprenes, epichlorohydrins, silicones, urethanes, polyacrylates, EPDM (ethylene propylene diene monomer) rubbers, SBR (styrene-butadiene rubber), butyl rubbers, nylon, polystyrene, polyethylene, polypropylene, polyester, polyurethane, etc. 
     In some embodiments, the deformable inner layer  12  can include one or more materials of a foam, an engraved, structured, 3D printed, or embossed elastomer, a fabric or nonwoven layer, or a soft rubber. The inner layer  12  of the back-up roll  10  can have a hardness less than about 20 Shore A, or less than about 10 Shore A. A suitable foam can be open-celled or closed-celled, including, for example, synthetic or natural foams, thermoformed foams, polyurethanes, polyesters, polyethers, filled or grafted polyethers, viscoelastic foams, melamine foam, polyethylenes, cross-linked polyethylenes, polypropylenes, silicone, ionomeric foams, etc. The inner layer may also include foamed elastomers or vulcanized rubbers, including, for example, isoprene, neoprene, polybutadiene, polyisoprene, polychloroprene, nitrile rubbers, polyvinyl chloride and nitrile rubber, ethylene-propylene copolymers such as EPDM (ethylene propylene diene monomer), and butyl rubber (e.g., isobutylene-isoprene copolymer). A suitable foam inner layer  12  of the back-up roll  10  can have a compressibility, for example, less than about 45 psi at 25% deflection, or less than about 20 psi at 25% deflection. It is to be understood that the inner layer  12  may include any suitable compressible structures such as, for example, springs, nonwovens, fabrics, air bladders, etc. In some embodiments, the inner layer  12  can be 3D printed to provide desired Poisson&#39;s ratio, compressibility, and elastic response. 
     Referring again to  FIGS.  1  and  2 A /B, the flexible web  3  is conveyed along a web path and fed into the coating zone  120 . The back-up roll  10  can rotate about an axis thereof to transport the web  3  along the machine direction  5  and through the coating zone  120 . The back-up roll  10  can be rotated using a motor, or can be rotated simply due to frictional contact with the flexible web  3 . 
     The flexible web  3  can include any suitable flexible substrate, such as, for example, a polymer web, a paper, a polymer-coated paper, a release liner, an adhesive coated web, a metal coated web, a flexible glass or ceramic web, a nonwoven, a fabric, or any combinations thereof. The flexible web  3  is disposed between the back-up roll  10  and the slot die  20 , wrapping around the back-up roll  10  with various wrap angles. In some embodiments, the flexible web  3  can wrap the back-up roll  10  with a wrap angle in the range, for example, from about 1 to about 180 degrees, about 1 to about 120 degrees, about 1 to about 90 degrees, or about 1 to about 60 degrees. 
     In some embodiments, the flexible web  3  may exhibit distortions or non-flatness characteristics when it is conveyed along the web path as a baggy web. The non-flatness characteristics may include, for example, lanes, strips, bumps, ripples, etc.  FIG.  1   ′ illustrates exemplary non-flatness characteristics 43′ on the baggy web  3 ′, which can be located on any portions of the web (e.g., center or edge). In the free-span coating of  FIG.  1   ′, the surface portions of the web  3 ′ having such non-flatness characteristics 43′ may result in variations (e.g., coating defects  44 ′ over the non-flatness characteristics 43′) in coat weight across the baggy web  3 ′ that is conveyed along the machine direction  5 ′. The methods and apparatuses described herein can significantly mitigate the variations induced by the non-flatness characteristics of a baggy web. 
     As shown in  FIGS.  2 A-C , the slot die  20  is pressed against the back-up roll  10  to form the coating zone  120 , where the flexible web  3  at a contacting area  15  is impressed into a deformable surface of the back-up roll  10  with an engagement width W along the machine direction  5  and an engagement depth D. The deformation of the backup roll is due to the pressure that builds between the slot die  20  and the flexible web  3  such that the backup roll  10  deflects to the engagement depth D in the contacting area. In some embodiments, the machine-direction engagement width W may be in a range, for example, from about 0.1 mm to about 50 mm. In some embodiments, the engagement depth D can be within a range, for example, from about 0.01 mm to about 10 mm, from about 0.05 mm to about 5 mm, or from about 0.1 mm to about 1 mm. It is to be understood that the contacting area  15  may not be limited to the area or space between the back-up roll  10  and the slot die  20  (i.e. there may be some distance after the slot die  20  in the machine direction before which the back-up roll  10  recovers to its original shape). A contacting area might refer to an area where the surface of the back-up roll  10  is deformed upon the engagement with the slot die. 
     In some embodiments, the back-up roll  10  may not be perfectly cylindrical, with a departure from cylindricity quantified using a total indicated runout (TIR), which can be defined as the difference between the largest and smallest values of the radius on the roll. For example, a roll with a maximum radius of 150.100 mm in one location, and a minimum radius of 150.000 mm in another location, would have a TIR of 0.100 mm. When the back-up roll engages a slot die and rotates, the nonuniformities in roll radius may translate through the coating bead formed between the back-up roll and the slot die. The differences in radii can produce a difference in pressure within a coating (e.g., in a liquid phase), resulting in a nonuniform coating. The impact of this TIR can be diminished by increasing the softness of the back-up roll (thereby making it easier to deform under fluid or mechanical pressure), though it is well known in industry that soft materials can be more difficult to machine into precise shapes. One of the benefits of the present disclosure is that the thin, outer layer  14  can present a harder surface, and so is more practical to machine, without sacrificing the overall softness of the back-up roll construction. In some embodiments, the TIR of the back-up roll  10  may be, for example, no greater than about 100 micrometers, or no greater than about 50 micrometers. 
     Referring again to  FIG.  2 A /B, the portion of flexible web  3  at the contacting area  15  is impressed, via the slot die  20 , into the face of the back-up roll  10  with the engagement depth D. The slot die  20  can apply a uniform pressure at the contacting area  15  across the web. The flexible web  3  can spread evenly along the cross-web direction over the face of the back-up roll  10 . A non-baggy surface of the flexible web  3  can be formed when the web goes through the coating zone  120 . As shown in  FIG.  2 C , the non-flatness characteristics 43 are significantly reduced in the web  3  on the wrapping area around the back-up roll  10 . The coating material  7  is applied to form an even coating  9  on the non-baggy surface of the web  3  that contacts the slot die  20 . The non-flatness characteristics 43 on the baggy web may restore after the flexible web  3  leaves the contact with the back-up roll  10 , which may not affect the uniformity of the coating already formed on the web. 
     The coating  9  can have a substantially uniform thickness across the surface of the flexible web  3 . In addition, when the web  3  is conveyed through the coating zone  120  by, e.g., rotating the back-up roll  10 , the back-up roll  10  has sufficiently low total indicated runout (TIR, e.g., less than 100 micrometer, preferably less than 50 micrometer), which helps to maintain a uniform force to create uniform coating along the down-web direction. 
     In some embodiments, the engagement depth D between the slot die  20  and the back-up roll  10  can be adjusted. The engagement depth D can be adjusted to be within a range, for example, from about 0.01 mm to about 10 mm, from about 0.05 mm to about 10 mm, or from about 0.1 mm to about 5 mm. In some embodiments, the engagement depth D can be adjusted by positioning the slot die  20  and/or the back-up roll  10 . The relative position of the slot die  20  and the back-up roll  10  can be adjusted using a mounting and positioning mechanism. The engagement depth D can be adjusted by positioning the slot die  20  and/or the back-up roll  10  such that the die lip of the slot die  20  intersects the curved plane defined by the surface of the back-up roll  10  in its un-deformed state. It should be understood that the engagement depth D (defined as the displacement of the outer surface of the back-up roll from its undeformed state) may be increased by the presence of the coating liquid and may not be set solely by the position of the die. 
     In some embodiments, as the back-up roll is rotated, variations in the surface uniformity (TIR) and mechanical properties of the back-up roll may lead to variations in force in the coating bead. As the engagement depth D is increased, these variations in force may become small relative to the overall force experienced by the coating bead. This may lead to improvements in coating uniformity and stable coating operating windows, where the variance in the coating weight/thickness over the surface of the flexible web can be, for example, less than about 10%, less than about 5%, less than about 2%, less than about 1%, or less than about 0.5%. It is to be noted that this is despite the back-up roll having a TIR that is significant compared to the wet coating thickness. For example, the ratio of the TIR to the wet coating thickness may be up to 300%, up to 100%, up to 50%, or up to 25%. 
     In some embodiments, the engagement depth D can be controlled to be less than a critical value to avoid defects such as lateral (i.e., cross-web direction) spreading of a coating liquid, or web tension issues which may be caused by a large engagement depth D, which may be due to contact between the roll/substrate and any section of the coating die that is not wetted. 
     Not wanting to be bound by theory, the range of engagement depth D that produce an acceptable coating window (e.g., enough engagement to minimize the impact of TIR and material property variation of the back-up roll, but not so much to produce noticeable coating defects such as lateral spreading) is a complex function of the viscosity vs. shear behavior of the coating liquid, the mechanical deformation behavior of the back-up roll, and the geometry of the coating die. For example, when the viscosity of the coating liquid in the coating bead is too low and/or the modulus of the back-up roll is too high, the coating liquid may not be able to support enough pressure in the coating bead against the back-up roll to increase the engagement depth D, and the back-up roll may behave in a manner similar to a rigid back-up roll, and so the effect of TIR may be large. It is to be noted, then, that in the contacting area  15 , there needs to be a balance of the viscous force developed by the coating liquid (at the relevant shear rate in the coating process) to the elastic force developed by the back-up roll at a given engagement depth D to observe a successful coating window. 
     It is useful to provide a quantitative description of the qualities of the back-up roll covering that confer the unexpected performance advantages of this disclosure. For example, it has been found that solid rubber covers, even those having a very low modulus, may not perform as well as dual layer covers having a thin solid rubber outer layer over a compressible inner layer. Furthermore, even dual layer covers having a very thin compressible inner layer may not confer the desired coating uniformity over the entire length of the back-up roll. For example, U.S. Pat. No. 6,079,352 describes a roll with an inner compressible layer thickness between “about 0.3175 cm and about 1.27 cm” and often “about 0.635 cm” with an outer layer thickness between “about 0.0127 and about 0.1524 cm”. As shown in the example section below, a back-up roll D 1 , which has a compressible inner layer thickness of 0.404 cm and an outer layer thickness of 0.152 cm that fall within the ranges specified by U.S. Pat. No. 6,079,352, failed to confer desired coating uniformity over the entire length of the back-up roll. 
     The operation of the present disclosure will be further described with regard to the following detailed examples. These examples are offered to further illustrate the various specific and preferred embodiments and techniques. It should be understood, however, that many variations and modifications may be made while remaining within the scope of the present disclosure. 
     Examples 
     These Examples are merely for illustrative purposes and are not meant to be overly limiting on the scope of the appended claims. Notwithstanding that the numerical ranges and parameters setting forth the broad scope of the present disclosure are approximations, the numerical values set forth in the specific examples are reported as precisely as possible. Any numerical value, however, inherently contains certain errors necessarily resulting from the standard deviation found in their respective testing measurements. At the very least, and not as an attempt to limit the application of the doctrine of equivalents to the scope of the claims, each numerical parameter should at least be construed in light of the number of reported significant digits and by applying ordinary rounding techniques. 
     Examples of Back-Up Roll 
     Quantitative roll covering characterization was conducted on a selection of back-up rolls  10  described in Table 1 below. The back-up rolls have various roll-cover configurations mounted on a rigid core. The back-up rolls labeled R 1 , R 2 , D 1 , D 2 , and D 3  were used for mechanical testing. Diameters for the Test Roller and the Test Plate are provided for reference. The foam inner layers of rolls D 1 , D 2 , and D 3  and a separate roll (not listed in Table 1) with only a single foam layer and no outer rubber layer were all constructed of the same material, a closed-cell polyurethane foam provided by American Roller Company, with varying thicknesses. Roller R 1  was commercially available from Finzer Roller, Des Plaines, Ill. Rollers R 2 , D 1 , D 2  and D 3  were commercially available from American Roller Company, Union Grove, Wis. 
     
       
         
           
               
               
               
               
               
             
               
                   
                 TABLE 1 
               
             
            
               
                   
                   
               
               
                   
                 Diameter 
                 Rubber Layer 
                 Foam 
                 S-Factor 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                   
                 Outside 
                 Core 
                 Thickness 
                 Hardness 
                 Modulus 
                 Thickness 
                 Average 
                 Slope 
               
               
                 Roller Name 
                 (mm) 
                 (mm) 
                 (mm) 
                 (Shore A) 
                 (MPa) 
                 (mm) 
                 (10 5  · N/m 5/2 ) 
                 (10 8  · N/m 7/2 ) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 R1 - Medium rubber 
                 120 
                 95 
                 12.7 
                 60 
                 4.27 
                 — 
                 31.0 
                 61233 
               
               
                 R2 - Soft rubber 
                 120 
                 100 
                 10.1 
                 20 
                 0.45 
                 — 
                 6.3 
                 8868 
               
               
                 D1 - Dual layer thin 
                 110 
                 99 
                 1.52 
                 60 
                 4.27 
                 4.04 
                 21.6 
                 11517 
               
               
                 D2 - Dual layer medium 
                 120 
                 100 
                 2.54 
                 55 
                 3.21 
                 7.54 
                 54 
                 34 
               
               
                 D3 - Dual layer thick 
                 165 
                 127 
                 1.65 
                 49 
                 2.26 
                 17.3 
                 2.7 
                 −102 
               
               
                 Test Roller 
                 90 
               
               
                 Test Plate 
                 ∞ 
               
               
                   
               
            
           
         
       
     
     Test Methods 
     The following test methods have been used in evaluating some of the Examples of the present disclosure. 
     Shore A Hardness Measurements 
     The Shore A hardness measurement of the rubber layers in Table 1 was measured, on the ASTM D2240 type A scale, using a Model 306L durometer tester manufactured by Pacific Transducer Corporation of Los Angeles, Calif. The hardness values in the table are an average of individual hardness measurements obtained from three cross-web locations at three positions around the circumference of each roller. It is understood that the hardness measurement mainly reflects the material properties of the outer rubber layer of the roller, though it may also be affected by the properties of the underlying foam layer. 
     Shore OO Hardness Measurements 
     Using the same procedure described above, the hardness of the separate foam roller without an outer rubber layer was measured to be 35 on the ASTM D2240 type OO scale, using a Model 1600 durometer tester with a MS-OO indenter manufactured by Rex Gauge Company of Buffalo Grove, Ill. It was not possible to measure the hardness of the foam layers in rollers D 1 , D 2  and D 3  of Table 1 because of the presence of the outer rubber layer. As rollers D 1 , D 2 , D 3  and the separate foam roller were all manufactured by American Roller Company, using the same manufacturing process, it is assumed that the hardness of the foam layers in rollers D 1 , D 2  and D 3  is similar to that of foam roller, namely 35 on the OO durometer scale. 
     Modulus Measurements 
     The Young&#39;s modulus values in Table 1 were obtained from the measured hardness values using a formula presented in a paper by J. K. Good, “Modeling Rubber Covered Nip Rollers in Web Lines”, Proceedings of the Sixth International Conference on Web Handling, Oklahoma State University, 2001. 
     Mechanical Compression Testing 
     Mechanical compression testing using a mechanical testing machine, such as those manufactured by Instron Corporation, is well understood by those versed in the art. Referring to  FIGS.  3 A and  3 B , rolls, labeled  10  in the figures and designated R 1 , R 2 , D 1 , D 2 , and D 3  in Table 1, were first pressed into a Test Roller  40  having an outside diameter of 90 mm as shown in  FIG.  3 A  and second into a Test Plate  42 , corresponding to a flat plate having an essentially infinite outside diameter as shown in  FIG.  3 B  in an Instron (Model 5500R) universal mechanical testing machine. The mechanical testing machine engaged each roller over a range of engagement depths D or D′ and widths W or W′ at a constant speed of about 83.8 micrometers per second. The engagement depth and the contact force between the back-up roll  10  and the Test Roller  40  or Test Plate  42  were measured and recorded using the Instron&#39;s frame position sensor and force load cell. The force versus engagement curve was then plotted for each test. Two such representative force versus engagement curves for the back-up roll D 2  are shown in  FIG.  4   . 
     Referring to  FIG.  4   , data U 2  represents the force vs. engagement curve for the roller D 2  in Table 1 engaged with the Test Roller  40  of  FIG.  3 A , while U 1  represents the curve for the roller D 2  engaged with a flat surface Test Plate  42  of  FIG.  3 B . As can be appreciated from  FIGS.  3 A and  3 B , engaging the roller D 2  with the Test Plate requires the displacement and or compression of more cover material, and therefore more force F, than a comparable level of engagement of D 2  with Test Roll. Correspondingly the force vs. engagement curve U 1  rises more steeply than curve U 2 . As neither the Test Plate or Test Roller necessarily represent the condition of engaging a die lip of arbitrary diameter into roller D 2 , well established principles in the field of contact mechanics may be used to generate force vs. engagement data that are independent of the geometry used for mechanical testing, as described in the S-Factor determination. 
     S-Factor Determination 
     A formula was derived for the force F required to engage a roller having a cover with a deformable layer by a distance D into a rigid roller or flat surface. See Formulas 5.74 and 5.70 in Contact Mechanics; K. L. Johnson; Cambridge University Press 1985; Lib. of Congress catalog: 84-11346, which are valid for a deformable roller with a single deformable layer. 
     
       
         
           
             
               
                 
                   W 
                   = 
                   
                     2 
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             3. 
                             
                               F 
                               . 
                               
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                   D 
                   = 
                   
                     
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                   2 
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     where W is contact width, F represents the applied force, normalized to a unit length of roller contact, ν is Poisson&#39;s ratio, b is the cover layer thickness, and E is elastic modulus of the cover layer of the roller, D is the engagement of the deformable cover into a rigid roller or surface, and R E  is the effective radius given by 
     
       
         
           
             
               
                 
                   
                     R 
                     E 
                   
                   = 
                   
                     
                       
                         D 
                         1 
                       
                       · 
                       
                         D 
                         2 
                       
                     
                     
                       2 
                       · 
                       
                         ( 
                         
                           
                             D 
                             1 
                           
                           + 
                           
                             D 
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   3 
                   ] 
                 
               
             
           
         
       
     
     where D 1  and D 2  representing the diameters of the two rollers or surfaces in contact with each other, and a flat plate corresponding to an essentially infinite roller diameter (i.e., a flat plate can be treated as a roller with an infinite diameter).
 
Substituting Equation [1] into Equation [2] gives
 
     
       
         
           
             
               
                 
                   F 
                   = 
                   
                     K 
                     . 
                     
                       D 
                       
                         3 
                         / 
                         2 
                       
                     
                     . 
                     
                       
                         R 
                         E 
                       
                     
                   
                 
               
               
                 
                   [ 
                   4 
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   K 
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           2 
                         
                         . 
                         E 
                         . 
                         
                           
                             ( 
                             
                               1 
                               - 
                               v 
                             
                             ) 
                           
                           2 
                         
                       
                     
                     
                       3. 
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               v 
                               2 
                             
                           
                           ) 
                         
                         . 
                         
                           ( 
                           
                             1 
                             - 
                             
                               2 
                               ⁢ 
                               v 
                             
                           
                           ) 
                         
                         . 
                         b 
                       
                     
                   
                 
               
               
                 
                   [ 
                   5 
                   ] 
                 
               
             
           
         
       
     
     Equations [4] and [5] were derived for rollers with a single deformable layer. For the more general case of multi-layer rollers it may be possible to derive analogous equations, though in this case the exponent on the engagement D may no longer be 3/2, and the variable K may now be a function of the engagement D. Additionally, the variable K may represent the compressibility factor characterizing the elastic modulus, Poisson&#39;s ratio, and thicknesses of all layers present I the deformable cover. 
     The data represented by curves U 1  and U 2  in  FIG.  4    may be rendered into a geometrically invariant form by correcting for the geometry of the fixture used to obtain the data, namely Test Roller,  40  in  FIG.  3 A  or Test Plate,  42 , in  FIG.  3 B . Using the relationship between F and R E  in Equation [4], geometry corrected data C 1  in  FIG.  4    were obtained by dividing data U 1  by the square root of R E-Flat , equal to 60.1 mm and calculated using Equation [3], for engaging the roller D 2  into the Test Plate. A similar geometric correction was applied to obtain data C 2  from U 2  in  FIG.  4    by dividing by the square root of R E-Roll , equal to 25.8 mm, for engaging the roller D 2  into the Test Roller. To within a small experimental error, the curves C 1  and C 2  in  FIG.  4    are equal. This shows the corrected force vs. engagement data in C 1  and C 2  are in fact geometrically invariant, or in other words are not dependent on the original geometric differences between the Test Roller and the Test Plate used to obtain the uncorrected compression test data U 1  and U 2 . 
     To obtain force vs. engagement data from C 1  and C 2  for an application, for example engaging a die lip into roller D 1  from Table 1, the previously corrected force data can be multiplied by the square root of R E  that is appropriate for the application geometry. Using this procedure, the geometrically invariant data can be recast into a form that is appropriate for the application. It should be noted that this geometric correction procedure, transforming force vs. engagement data obtained from a compression testing apparatus to a geometrically invariant form and then transforming it again for modeling a contacting surface of a die lip is valid only if the parameter K in Equation [4] is held substantially constant. For the purposes of this application K is considered constant, even for back-up rolls having different diameters, if the roller covers are constructed in an equivalent manner, having the same layers, made of similar materials with the same layer thicknesses. 
     An experimentally obtained parameter, S-Factor, may be obtained for any roller system by dividing the geometrically corrected force vs. engagement data C 1  or C 2 , based on  FIG.  4   , by the roller engagement D, for each data point. 
     
       
         
           
             
               
                 
                   S 
                   = 
                   
                     F 
                     
                       D 
                       · 
                       
                         
                           R 
                           E 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   6 
                   ] 
                 
               
             
           
         
       
     
     In the case of a roller composed of a single deformable layer, Equations [4] and [5] can be used to estimate the S-factor directly: 
     
       
         
           
             
               
                 
                   S 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           
                             2 
                           
                           . 
                           E 
                           . 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 v 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       
                         3. 
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 v 
                                 2 
                               
                             
                             ) 
                           
                           . 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 2. 
                                 v 
                               
                             
                             ) 
                           
                           . 
                           b 
                         
                       
                     
                     . 
                     
                       D 
                     
                   
                 
               
               
                 
                   [ 
                   7 
                   ] 
                 
               
             
           
         
       
     
     For natural and synthetic rubbers, the Young&#39;s modulus can be determined using the following expression between Shore A (i.e., IRHD) hardness and modulus presented in a paper by J. K. Good, “Modeling Rubber Covered Nip Rollers in Web Lines”, Proceedings of the Sixth International Conference on Web Handling, Oklahoma State University, Stillwater, O K, 2001, pp 159-177. 
         E= 0.145 e   0.0564.IRHD  (MPα)  [8]
 
     The calculation in Equation [6] is carried out individually for each data pair (F i , D i ) obtained from the mechanical compression test described previously. In addition, for rollers R 1  and R 2  in Table 1, which have single rubber covers, using a Poison&#39;s ratio of 0.47 for the rubber covers, the S-Factor is calculated from Equation [7] for the same experimental engagement ranges for these rollers. The S-Factor is related to the slope of the corrected force data C 1  and C 2  in  FIG.  4   , having the same units of measure, namely N/m 5/2 . It should be noted that this S-Factor is not a true local slope because it depends on the magnitude of the corrected force datum F i  and total engagement value D i  used to obtain that force. 
     S-Factors calculated for rollers R 1 , R 2 , D 1 , D 2  and D 3  in Table 1 are shown as a function of roller engagement D in  FIG.  5   . As indicated, for rollers R 1  and R 2  the S-Factor curves calculated from the experimental force-engagement data sets showed good agreement with the 5-Factor curves determined from closed form Equation [6] (indicated by R 1 ′ and R 2 ′). S-Factors quantitatively describe intrinsic design properties of the roller covers in Table 1 and are governed by the thickness, modulus, Poisson&#39;s ratio or compressibility of the various layers covering the rigid core of the back-up roll. Because of the aforementioned geometric correction procedure for experimentally obtained force data, S-Factors do not depend on the lengths or diameters of the Test Roller  40  in  FIG.  3 A  or Test Plate  42  in  FIG.  3 B . Likewise, when used to calculate cross-web engagement D and contact pressure F, S-Factors do not depend on the lengths or diameters of a die lip or back-up roll in contact with each other. We note that, generally speaking, if two rolls have different S-Factors, the roll with the lower S-Factor may be considered softer. 
     Referring to  FIG.  5   , rollers R 1 , R 2 , D 1 , D 2  and D 3  have qualitative and quantitative differences in S-Factor as a function of engagement depth D. Both rollers R 1  and R 2 , having a single layer solid rubber cover and roller D 1  having a solid rubber outer layer over a thin compressible inner layer have S-Factors that increase monotonically with engagement D. Rollers R 1 , D 2  and D 3  have S-Factors that are substantially smaller in magnitude to rollers D 1  and R 2 . Quantitatively, S factors averaged over a range of engagement D from 0 mm to 1 mm are tabulated in Table 1 along with the slope of the S-Factor for engagements D greater than 0.2 mm. It is to be understood that in some embodiments, the S factors can be averaged over a range of engagement D from 0.05 mm to 1 mm without significantly changing the result. It is important to note that there may be an upper engagement limit for some back-up roll constructions. For example, a compressible inner layer may be engaged to such an extent that the force begins to rise quickly with further engagement. When calculating the slope of the S-Factor it is understood that the range of engagement values used falls below an upper engagement limit wherein a compressible inner layer has been compressed beyond its design limit. The average S-Factor was calculated by averaging S-Factor data pairs (S i , D i ) for all engagement values D i  between 0 mm and 1 mm. The S-Factor slope was calculated by fitting a line to the S-Factor data pairs (S D) for engagement values D i  between 0.2 mm and 2 mm using the least squares method. 
     The S-Factor may be directly related to the uniformity of engagement D and contact force over the entire width in the cross-web direction of a slot die coating system. Consistent engagement pressure has been noted as a key element to obtaining uniform coating over the entire width of the web. A resilient back-up roll cover, having a low and consistent force response to changes in engagement D, can tolerate greater roller TIR or substrate thickness variation with minimal or no change to coating thickness or quality. In fact, a sufficiently resilient back-up roll cover can tolerate process upsets such as baggy web or splices with minor effect on coating quality. Such a resilient back-up roll cover can have an S-Factor, averaged over a range of engagement D from about 0 to 1.0 mm, or from 0.05 to 1.0 mm, that is less than 15 (10 6 ·N/m 5/2 ) and preferably less than 10 (10 6 ·N/m 5/2 ). Furthermore, a resilient back-up roll cover can have a slope in the S-Factor vs. engagement curve, for engagement values greater than 0.2 mm, that is less than 5000 (10 6 ·N/m 72 ), preferably less than 500 (10 6 ·N/m 72 ) and most preferably less than 50 (10 6 ·N/m 72 ). 
     Coating Examples 
     A web of 30.5 cm (12″) PET was prepared with a 5.08 cm (2″) wide baggy lane in the center (the level of bag was approximately 1%). A die with downstream land length of 0.53 cm (0.21″) and convex radius of 24.5 cm (9.625″) was used to coat an adhesive solution with a viscosity at 10 l/s of roughly 8 Pa-s and a viscosity at 2,000 l/s of roughly 3 Pa-s onto the prepared baggy web, using roll 1 from table 1 as a backup roll. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
             
            
               
                   
                   
               
               
                   
                 Materials 
                 Outer diameters (in) 
                 Shore A 
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Roll 
                 Inner layer 
                 Outer layer 
                 Steel core 
                 Inner layer 
                 Outer layer 
                 Durometer 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 1 
                 Urethane foam 
                 Urethane rubber 
                 3.937 
                 4.524 
                 4.724 
                 50 
               
               
                   
                   
                 Continuous, 
               
               
                   
                   
                 non-porous 
               
               
                 2 
                 Silicone rubber 
                 N/A 
                 3.937 
                 4.724 
                 N/A 
                 20 
               
               
                   
               
            
           
         
       
     
     Condition 1 
     The apparatus described in the previous section was used to produce coatings at a wet thickness of approximately 6.8 mils and a line speed of 0.381 meters per second (75 FPM). As used herein, 1 mil equals to 2.54E-3 cm. The web tension was set to 12 lbs, or 1 lb/linear inch (PLI) with this 30.5 cm (12″) wide film. The die was positioned in the free span (see e.g.,  FIG.  1   ′), with the downstream edge of the die located about 1″ below the center line of the backup roll. The die was engaged into the path of the web by approximately 50 mils, thereby deforming the web in an attempt to produce a uniform coating. The coating was continuous on either side of the baggy lane in the web, but was discontinuous and consisted of large bubbles in the baggy center lane. This would be considered a defective coating. 
     Condition 2 
     Using the same setup as condition 1, the die was engaged a further 50 mils (to a total of 100 mils) without any improvement in coating quality in the baggy lane. 
     Condition 3 
     Using the same setup as condition 2, the force on the web was increased to 24 lbs (2 PLI) without any improvement in coating quality in the baggy lane. 
     Condition 4 
     The coating setup and conditions from the previous examples were reused, with the exception that the die was now positioned such that the downstream edge of the lip was located along the centerline of the roll. The gap between the roll and the die was set to 10 mils using steel shims, after which the die was engaged into the roll by 30 mils. In this case a uniform coating was obtained across the web, including in the baggy center lane. 
     Condition 5 
     Using the same setup as condition 4, the force on the web was reduced to 12 lbs (1 PLI) without any degradation in the coating quality observed in condition 4. 
     Condition 6 
     The coating setup from condition 5 was used, but with roll 2 replacing roll 1. In this case the coating was generally uniform in the baggy center lane. However, significant additional defects were observed that coincided with defects due to nonuniformities in the rubber layer of roll 8. 
     Table 2 below summarizes Conditions 1-6. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                   
                   
                   
                 Line 
                   
                   
                   
                   
               
               
                   
                 Web 
                 Engagement 
                 speed 
                 Wet 
               
               
                   
                 Tension 
                 Depth 
                 m/s 
                 Thickness 
                 Uniform 
                 Die 
                 Backup 
               
               
                 Condition 
                 (pli) 
                 (mils) 
                 (fpm) 
                 (mils) 
                 coating 
                 Position 
                 Roll 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 1 
                 1 
                 50 
                 0.38 (75) 
                 6.8 
                 No 
                 Free span 
                 N/A 
               
               
                 2 
                 1 
                 100 
                 0.38 (75) 
                 6.8 
                 No 
                 Free span 
                 N/A 
               
               
                 3 
                 2 
                 100 
                 0.38 (75) 
                 6.8 
                 No 
                 Free span 
                 N/A 
               
               
                 4 
                 2 
                 30 
                 0.38 (75) 
                 6.8 
                 Yes 
                 On Roll 
                 1 
               
               
                 5 
                 1 
                 30 
                 0.38 (75) 
                 6.8 
                 Yes 
                 On Roll 
                 1 
               
               
                 6 
                 1 
                 30 
                 0.38 (75) 
                 6.8 
                 No 
                 On Roll 
                 2 
               
               
                   
               
            
           
         
       
     
     Examples of Impact of Skin Thickness on Coating Quality 
     Two foam rolls were prepared as back-up rolls for coating. Roller F 1  had only a single foam layer constructed of a closed-cell polyurethane foam provided by American Roller Company, with no outer layer. Roller F 1  was commercially available from American Roller Company, Union Grove, Wis. Roller F 2  was composed of a pourable water blown flexible foam commercially available from Smooth-On, Macungie, Pa., under a trade designation FlexFoam-iT VIII Pillow Soft. The foam was cast between a 3″ diameter stainless-steel roller and a 5″ diameter cylindrical mold mounted concentrically with the roller. The foam was used as-is after being unmolded and had a natural skin which formed during the foam casting process. 
     Shore OO Hardness Measurements 
     The Shore OO hardness values of rollers F 1  and F 2  were measured by following ASTM D2240, using a Model 1600 durometer tester with a MS-OO indenter manufactured by Rex Gauge Company of Buffalo Grove, Ill. The hardness value of roller F 1  was found to be 35 shore OO. The hardness value of roller F 2  was low enough that it did not register on the durometer tester and so was assigned a value of 0 shore OO. These hardness values are an average of individual hardness measurements obtained from three cross-web locations at three positions around the circumference of each roller. 
     Coating Conditions 
     A web of 30.5 cm (12″) wide, 1.8 mil (0.0457 mm) thick PET was prepared for coating. A die with a downstream land length of 0.53 cm (0.21″) and convex radius of 24.45 cm (9.625″) was positioned so that the 1.8-mil (0.0457 mm) PET was between the die and the foam roller. Web tension was held constant at 1 pli, and all run conditions were at a line speed of 0.127 meters per second (25 fpm). Adhesive was fed to the die via a gear pump at a rate to achieve a 16-mil (0.731 mm) wet coating thickness target. The conditions run can be found in Table 3 below. During coating the die was impressed into the roll to find the best coating condition and coating quality which was evaluated visually. During Conditions 3-6, once coating was established strips of stainless steel shim stock (approximately 8″ wide and 3 ft long, and commercially available from Precision Brand Products Inc., Downers Grove, Ill.) were passed through the coating zone on the back side of the web between the PET and the foam roller. 
     Both rollers F 1  and F 2  provided clearly nonuniform coatings when used as-is. Roller F 2  provided increasingly more uniform coatings when the stainless-steel shims of increasing thickness were passed through the coating zone. Coating onto roller F 2  covered by a shim with a thickness of 2 mils (Condition 4) provided clearly improved coating quality over coatings obtained with rollers F 1  and F 2  alone, as well as coatings obtained with roller F 2  and a 1 mil (0.0254 mm) thick shim. Coating onto roller F 2  covered by a shim with a thickness of 3 mils or greater (conditions 5 and 6) provided an even greater improvement in coating quality. 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 Coating conditions demonstrating impact of skin thickness 
               
               
                 on coating quality (+ = poor, ++ = moderate, +++ = good) 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                 Core 
                 Outer 
                 Foam 
                   
                   
               
               
                   
                   
                 Diameter 
                 Diameter 
                 Thickness 
                 Shim 
               
               
                 Condi- 
                 Foam 
                 cm 
                 cm 
                 cm 
                 Thickness 
                 Coating 
               
               
                 tion 
                 Roll 
                 (in) 
                 (in) 
                 (in) 
                 (mils) 
                 Quality 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 1 
                 F1 
                 10.0 
                 11.61 
                 1.00 
                 N/A 
                 + 
               
               
                   
                   
                 (3.937) 
                 (4.570) 
                 (0.3937) 
               
               
                 2 
                 F2 
                 7.37 
                 14.15 
                 3.39 
                 N/A 
                 + 
               
               
                   
                   
                 (2.900) 
                 (5.570) 
                 (1.335) 
               
               
                 3 
                 F2 
                 7.37 
                 14.15 
                 3.39 
                 1.000 
                 + 
               
               
                   
                   
                 (2.900) 
                 (5.571) 
                 (1.335) 
               
               
                 4 
                 F2 
                 7.37 
                 14.15 
                 3.39 
                 2.000 
                 ++ 
               
               
                   
                   
                 (2.900) 
                 (5.572) 
                 (1.335) 
               
               
                 5 
                 F2 
                 7.37 
                 14.16 
                 3.39 
                 3.000 
                 +++ 
               
               
                   
                   
                 (2.900) 
                 (5.573) 
                 (1.335) 
               
               
                 6 
                 F2 
                 7.37 
                 14.16 
                 3.39 
                 5.000 
                 +++ 
               
               
                   
                   
                 (2.900) 
                 (5.575) 
                 (1.335) 
               
               
                   
               
            
           
         
       
     
     Examples of Impact of Die Lip Geometry on Coating Quality 
     An adhesive was fed to a die via a gear pump, achieving a wet coating thickness 16-mil (0.406 mm) at a coating width of 11″, with coating conditions found in Table 4 below, using the backup roll D 3  described in Table 1. Over the course of each experiment, the engagement depth D of the coating die was adjusted, with the coating quality assessed visually, and a coating window defined as the range of engagement depths over which an acceptable coating was noted. The minimum engagement depth was defined as the point at which the coating bead failed, producing sections of the web that were not coated. The maximum engagement depth was defined as the point at which the coating width had spread to approximately 11.5″. Three fluids were used in these experiments. Fluid 1 has a viscosity at 10 l/s of roughly 18 Pa-s, a viscosity at 100 l/s of roughly 7.6 Pa-s, and a viscosity at 1,000 l/s of roughly 1.1 Pa-s. Fluid 2 has a viscosity at 10 l/s of roughly 10 Pa-s, a viscosity at 100 l/s of roughly 5.1 Pa-s, and a viscosity at 1,000 l/s of roughly 1.2 Pa-s. Fluid 3 has a viscosity at 10 l/s of roughly 5.7 Pa-s, a viscosity at 100 l/s of roughly 2.3 Pa-s, and a viscosity at 1,000 l/s of roughly 0.66 Pa-s. 
     1. Die Lip Example E1 and Comparative Example CE1 
     The die lip Example E1 is geometrically similar to the die lip  222  in  FIG.  6 A , and includes a concave upstream die lip portion and a concave downstream die lip portion, each having a radius of curvature of about 4.341″ and a length of about 0.375″. The die lip Comparative Example CE1 is geometrically similar to the die lip  226  in  FIG.  6 C . The die lip Comparative Example CE1 includes a convex upstream die lip portion and a convex downstream die lip portion, each having a radius of curvature of about 2″. The upstream die lip portion has a length about 0.5″, while the downstream die lip portion has a length about 0.375″. Results are summarized in Table 4. At all line speeds studied (27, 54, and 100 fpm), no acceptable coating window was observed using the convex lip construction CE1, while the concave lip construction E1 resulted in coating windows of 6 and 7 mils respectively. This demonstrates that concave die lips enable larger coating windows as compared to convex die lips. 
     2. Die Lip Example E2 and Comparative Example CE2 
     The die lip Example E2 is geometrically similar to the die lip  224  in  FIG.  6 B . The die lip Example E2 includes a convex upstream die lip portion having a radius of curvature about 2.0″ and a length about 0.5″ and a concave downstream die lip portion having a radius of curvature about 4.341″ and a length about 0.375″. The die lip Comparative Example CE2 is geometrically similar to the die lip  226  in  FIG.  6 C . The die lip Comparative Example CE2 includes a convex upstream die lip portion having a radius of curvature about 2″ and a convex downstream die lip portion having a radius of curvature about 1″. The upstream die lip portion has a length about 0.5″. The downstream die lip portion has a length about 0.2″. Results are summarized in Table 4. We observed a 4 mil coating window using the convex downstream die lip construction CE2, while the concave downstream lip construction E2 resulted in a penetration window of 20 mils, indicating that the downstream die lip geometry plays a critical role in increasing the die penetration range. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 4 
               
               
                   
               
               
                   
                   
                 Web 
                 Line 
                 Pump 
                   
                   
               
               
                   
                 Coating 
                 tension 
                 Speed 
                 Speed 
                 Window 
               
               
                 Examples 
                 solution 
                 (Pli) 
                 (FPM) 
                 (cc/Min) 
                 (Mils) 
                 Figure 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 CE1 
                 Fluid 1 
                 1.0 
                 27 
                 273 
                 0 
                 6C 
               
               
                 E1 
                 Fluid 1 
                 1.0 
                 27 
                 273 
                 6 
                 6A 
               
               
                 CE1 
                 Fluid 1 
                 1.0 
                 54 
                 546 
                 0 
                 6C 
               
               
                 E1 
                 Fluid 1 
                 1.0 
                 54 
                 546 
                 7 
                 6A 
               
               
                 CE1 
                 Fluid 2 
                 2.0 
                 100 
                 833 
                 0 
                 6C 
               
               
                 E1 
                 Fluid 2 
                 2.0 
                 100 
                 833 
                 7 
                 6A 
               
               
                 CE2 
                 Fluid 3 
                 1.5 
                 25 
                 50 
                 4 
                 6C 
               
               
                 E2 
                 Fluid 3 
                 1.5 
                 25 
                 50 
                 20 
                 6B 
               
               
                   
               
            
           
         
       
     
     Listing of Exemplary Embodiments 
     Exemplary embodiments are listed below. It is to be understood that any one of the embodiments 1-11 and 12-19 can be combined. 
     Embodiment 1 is a method of applying a coating onto a web, the method comprising: 
     providing a back-up roll having a deformable inner layer with a surface thereof covered by a deformable outer layer, the inner layer being softer than the outer layer; 
     providing a slot die having one or more die lips extending along a cross direction, the one or more die lips being positioned proximate to the back-up roll; 
     disposing a flexible web between the back-up roll and the one or more die lips; 
     dispensing a liquid coating material from the slot die onto the flexible web, 
     wherein the flexible web at a contacting area is impressed into the back-up roll with an engagement depth D. 
     Embodiment 2 is the method of embodiment 1, further comprising adjusting the engagement depth D in a range from about one micrometer to about 2 mm.
 
Embodiment 3 is the method of embodiment 1 or 2, further comprising metering a liquid flow through the slot die to control a wet thickness of the coating.
 
Embodiment 4 is the method of embodiment 3, wherein the wet thickness of the coating is adjusted to be within the range of about 1 to about 500 micrometers.
 
Embodiment 5 is the method of any one of embodiments 1-4, further comprising pressing the slot die against the back-up roll.
 
Embodiment 6 is the method of any one of embodiments 1-5, wherein the flexible web has one or more surface non-flatness characteristics.
 
Embodiment 7 is the method of embodiment 6, wherein the flexible web is a baggy web.
 
Embodiment 8 is the method of embodiment 6 or 7, wherein the one or more surface non-flatness characteristics includes a splice, a web thickness variation, or a web wrinkle.
 
Embodiment 9 is the method of any one of embodiments 1-8, further comprising wrapping the flexible web around the back-up roll.
 
Embodiment 10 is the method of any one of embodiments 1-9, wherein the back-up roll has an S-Factor, averaged over a range of the engagement D from about 0.05 mm to about 1 mm, optionally being less than about 15 (10 6 ·N/m 5/2 ), or less than about 10 (10 6 ·N/m 5/2 ).
 
Embodiment 11 is the method of any one of embodiments 1-10, wherein the liquid coating material has a viscosity lower than about 10,000 centipoise (cps).
 
Embodiment 12 is a coating apparatus comprising:
 
     a back-up roll having a deformable inner layer with a surface thereof covered by a deformable outer layer, the inner layer being softer than the outer layer; 
     a slot die having one or more die lips extending along a cross direction, the one or more die lips being positioned proximate to the back-up roll; and 
     a flexible web disposed between the back-up roll and the slot die, 
     wherein the flexible web at a contacting area is impressed into the back-up roll with an engagement depth D, and the slot die is configured to dispense a liquid coating material onto the web. 
     Embodiment 13 is the coating apparatus of embodiment 12, wherein the deformable inner layer of the back-up roll has a hardness less than 20 Shore A, optionally less than 10 Shore A.
 
Embodiment 14 is the coating apparatus of embodiment 12 or 13, wherein the inner layer of the back-up roll has a compressibility of less than about 45 psi at 25% deflection, optionally less than about 20 psi at 25% deflection.
 
Embodiment 15 is the coating apparatus of any one of embodiments 12-14, wherein the deformable outer layer of the back-up roll has a hardness greater than about 40 Shore A, optionally greater than about 50 Shore A.
 
Embodiment 16 is the coating apparatus of any one of embodiments 12-15, wherein the deformable outer layer includes one or more materials of an elastomer, a metal, a fabric, or a nonwoven.
 
Embodiment 17 is the coating apparatus of any one of embodiments 12-16, wherein the deformable inner layer includes one or more materials of a synthetic foam, an engraved, structured, 3D printed, or embossed elastomer, a fabric or nonwoven layer, a plurality of cavities filled with gas of a controlled pressure, or a soft rubber.
 
Embodiment 18 is the coating apparatus of any one of embodiments 12-17, wherein the back-up roll has an S-Factor, averaged over a range of the engagement D from about 0.05 mm to about 1 mm, optionally being less than about 10 (10 6 ·N/m 5/2 ), or less than about 5 (10 6 ·N/m 5/2 ).
 
Embodiment 19 is the coating apparatus of any one of embodiments 12-18, wherein the liquid coating material has a viscosity lower than about 10,000 centipoise (cps).
 
     Reference throughout this specification to “one embodiment,” “certain embodiments,” “one or more embodiments” or “an embodiment,” whether or not including the term “exemplary” preceding the term “embodiment,” means that a particular feature, structure, material, or characteristic described in connection with the embodiment is included in at least one embodiment of the certain exemplary embodiments of the present disclosure. Thus, the appearances of the phrases such as “in one or more embodiments,” “in certain embodiments,” “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily referring to the same embodiment of the certain exemplary embodiments of the present disclosure. Furthermore, the particular features, structures, materials, or characteristics may be combined in any suitable manner in one or more embodiments. 
     While the specification has described in detail certain exemplary embodiments, it will be appreciated that those skilled in the art, upon attaining an understanding of the foregoing, may readily conceive of alterations to, variations of, and equivalents to these embodiments. Accordingly, it should be understood that this disclosure is not to be unduly limited to the illustrative embodiments set forth hereinabove. In particular, as used herein, the recitation of numerical ranges by endpoints is intended to include all numbers subsumed within that range (e.g., 1 to 5 includes 1, 1.5, 2, 2.75, 3, 3.80, 4, and 5). In addition, all numbers used herein are assumed to be modified by the term “about.” 
     Furthermore, all publications and patents referenced herein are incorporated by reference in their entirety to the same extent as if each individual publication or patent was specifically and individually indicated to be incorporated by reference. Various exemplary embodiments have been described. These and other embodiments are within the scope of the following claims.