[0001] This application claims the benefit of priority under 35 U.SC. § 1 19 of U.S.

Provisional Application Serial No. 61/564974 filed on November 30, 201 1 the content of which is relied upon and incorporated herein by reference in its entirety.

FIELD

[0001] The present invention relates to an apparatus and a method for producing localized thinning of a continuously moving glass ribbon, and more particularly for removing edge portions of the glass ribbon by propagating a crack along the thinned portion.

BACKGROUND

[0002] Glass sheets for use in display systems such as liquid crystal displays or organic light emitting diode display technologies, as components on photovoltaic devices or as cover plates of various shapes for hand held devices and televisions have been produced by well known high volume, continuous glass sheet manufacturing processes such as the float process, or the fusion downdraw process but can, under certain particular circumstances, be produced using a slot draw process.

[0003] The foregoing processes produce ribbons of glass that exhibit an increased thickness near the edges of the ribbon generally referred to as "beads". It is common to observe bead thicknesses of 3 to 4 times the nominal thickness of the central part of the glass ribbon.

When manufacturing very thin ribbons of glass, this ratio tends to increase, and may reach values as high as 10 time the thickness of the central portion of the glass ribbon when a central thickness of the glass ribbon on the order of 0.1 mm is considered.

[0004] The presence of these beads may be desirable in the early stage of the manufacturing process where the glass ribbon is formed and stretched by viscous deformation by aiding in sheet stabilization, sheet width loss control, thickness control contribution and so on.

However, their impact on internal stress and sheet shape later in the process may not be desired and, in particular cases, can be detrimental to the process and final product.

[0005] To reach low levels of internal stress in the glass ribbon, careful control of the cooling rates during the forming process is required. Significant thickness differences between different regions of the ribbon lead to different cooling rates, thus producing temperature gradients that reduce the ability to achieve low stress. This is the case in the bead region, where the large thickness gradients induce large temperature and stress gradients.

[0006] During the production of very thin sheets, it may be desirable to wind the glass ribbon on spools instead of cutting discrete glass sheets to accommodate a high draw speed. The presence of these thicker beads limits the ability to curve the sheet with a sufficiently small radius of curvature without inducing crack propagation and product loss.

SUMMARY

[0007] Disclosed herein is a method and apparatus to continuously remove the beads from a continuously moving glass ribbon by selectively thinning portions of the ribbon by the application of localized heating.

[0008] An optimal location to maximize thinning of the glass ribbon is near the root of the forming body for a fusion down draw process. For a slot draw process the optimal position is near the slot. Local modification of the glass viscosity is performed using a local heat generator within the viscous region of the process. Being subject to heat exchange by radiation and convection while traveling in a downward direction, this thin section of the ribbon will develop thermo-mechanical stresses induced by thermal gradients that can be used to propagate a crack initiated within the elastic region (usually below the pulling rolls) up to the top of the visco-elastic region, thus effectively separating the bead from the rest of the sheet. Once initiated, this separation can be sustained and controlled by adjusting the local viscosity near the root as well as adjusting the cooling rate down the draw in the thinned area from the root. Adjustment of the local viscosity near the root of the forming body can be done using a forced-air heating nozzle.

[0009] The heating nozzle comprises a compact heat generator that can be used to transfer energy to the glass near the root, mainly by convection, and to some extent, by radiation. Heat transfer efficiency is achieved through a high velocity hot air jet impinging on the glass surface. The hot air can provide a local and tunable viscosity gradient by controlling air flow, air velocity, air temperature and air flow direction to the glass.

[0010] While crack initiation can occur spontaneously, control of the initiation at a given location in the draw by, for example, local heating/cooling (C02 laser for example followed by air jet or air/water mist) to promote very high stress gradients or by damaging the glass surface (for example mechanically with a glass cutter or by applying very local twist by pairs of rollers).

[0011] Accordingly, disclosed herein is an apparatus for forming a glass ribbon comprising a forming body comprising converging forming surfaces that join at a bottom of the forming body and a heating nozzle comprising a refractory tube comprising: a plurality of passages extending longitudinally between first and second ends of the refractory tube, wherein at least one passage of the plurality of passages is in fluid communication with a flow of gas directed through the at least one passage, the first end being proximate the bottom of the forming body; and a heating element disposed about the refractory tube configured to heat the flow of gas. The refractory tube is preferably positioned within a refractory sleeve wherein the heating element is then positioned between the refractory tube and the refractory sleeve.

[0012] The apparatus may further comprise a cooling door positioned below the bottom of the forming body, and wherein the heating nozzle is positioned between the bottom of the forming body and the cooling door. The cooling door functions to adjust a thickness of the glass ribbon across a width of the glass ribbon by directing a cooling gas against a thermal plate positioned proximate the descending glass ribbon.

[0013] The heating nozzle is preferably positioned to direct the heated flow of gas at a portion of the glass ribbon equal to or less than about 100 mm from an edge of the glass ribbon. For example, the heating nozzle may be positioned to direct the heated flow of gas at a portion of the glass ribbon equal to or less than about 50 mm from an edge of the glass ribbon. Preferably, the refractory tube is positioned within a thermal insulating shield.

[0014] In another embodiment, a method of locally thinning a continuously moving glass ribbon is described comprising flowing a molten glass from a forming body comprising converging forming surfaces that join at a root, the molten glass forming a continuously moving glass ribbon that is drawn from the root, directing a flow of heated gas from a heating nozzle at the glass ribbon, the heated gas impinging on the glass ribbon proximate the root, the impinging heated gas producing a localized thinned portion of the glass ribbon that extends along a length of the glass ribbon and separating an edge portion from the glass ribbon by propagating a crack along the thinned portion. A temperature of the heated gas is preferably in the range from about 1450°C to about 1650°C. Preferably, the crack is propagated by heating the thinned portion with a laser followed by cooling the thinned portion with a cooling fluid.

[0015] In some embodiment the heated gas impinges on the glass ribbon between an edge director and a centerline of the glass ribbon. For example, the heated gas may impinge closer to the edge director than to the centerline. Preferably, the heated gas impinges on the glass ribbon within about 100 mm of an edge of the glass ribbon, such as within about 50 mm of an edge of the glass ribbon. Preferably the thinned portion comprises a tensile stress bounded by a thickened portion comprising compressive stress.

[0016] Additional features and advantages of the invention will be set forth in the detailed description which follows, and in part will be readily apparent to those skilled in the art from that description or recognized by practicing the invention as described herein, including the detailed description which follows, the claims, as well as the appended drawings.

[0017] It is to be understood that both the foregoing general description and the following detailed description present embodiments of the invention, and are intended to provide an overview or framework for understanding the nature and character of the invention as it is claimed. The accompanying drawings are included to provide a further understanding of the invention, and constitute a part of this specification. The drawings illustrate various embodiments of the invention and, together with the description, serve to explain the principles and operations of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[0018] FIG. 1 is a schematic view of an exemplary fusion glass making apparatus;

[0019] FIG. 2 is a front view of a molten glass forming body comprising the apparatus of FIG. 1;

[0020] FIG. 3 is a perspective view of the forming body of FIG. 2 showing the edge directors;

[0021] FIG. 4 is a cross sectional view across the width of a glass ribbon drawn from the forming body of FIG. 2

[0022] FIG. 5 is a cross sectional view of the forming body of FIG. 2 as seen from an end thereof showing the placement of heating nozzles according to an embodiment of the present invention; [0023] FIG. 6 is a cross sectional side view of a heating nozzle disposed within a protective refractory sleeve and thermal shield;

[0024] FIG. 7 is a cross sectional view of a portion of the glass ribbon of FIG. 4 showing the effect of the heating nozzle positioned proximate a bead of the glass ribbon;

[0025] FIG. 8 is a representation of a unit volume of molten glass depicting the forces acting on the unit volume of molten glass

[0026] FIG. 9 is a plot representing ribbon thickness across a center portion of the glass ribbon;

[0027] FIG. 10 is a plot representing ribbon thickness near a bead of the glass ribbon when impinged upon by heated gas from the heating nozzle.

DETAILED DESCRIPTION

[0028] In the following detailed description, for purposes of explanation and not limitation, example embodiments disclosing specific details are set forth to provide a thorough understanding of the present invention. However, it will be apparent to one having ordinary skill in the art, having had the benefit of the present disclosure, that the present invention may be practiced in other embodiments that depart from the specific details disclosed herein. Moreover, descriptions of well-known devices, methods and materials may be omitted so as not to obscure the description of the present invention. Finally, wherever applicable, like reference numerals refer to like elements.

[0029] FIGS. 1 and 2 illustrate an exemplary embodiment of a fusion glass making apparatus 10 for forming a glass sheet comprising melting furnace 15, fining vessel 20, stirring vessel 25, receiving vessel 30, downcomer 35, inlet 40 and forming body 45 from which a thin, continuously moving ribbon 50 of a molten glass-forming material descends. Glass making system 10 further comprises various other vessels or conduits for conveying the molten glass-forming material, including a melter-to-fining vessel connecting tube 55, a fining vessel-to-stirring vessel connecting tube 60, and a stirring vessel-to-receiving vessel connecting tube 65. While the melting furnace and forming body are typically formed from a ceramic material, such as ceramic bricks comprising alumina or zirconia, the various vessels and piping therebetween often comprise platinum or an alloy thereof, such as a platinum- rhodium alloy. Although the following description relates to an exemplary fusion downdraw process, such as the process illustrated in FIG. 1, embodiments enclosed herein may be equally applicable to other variations of downdraw glass making processes such as a single sided overflow process or a slot draw process, which processes are well known to those skilled in the art.

[0030] In accordance with the exemplary fusion process of FIG. 1, melting furnace 15 is provided with a batch material 70, as indicated by arrow 75, that is melted by the melting furnace to produce a glass-forming material (hereinafter molten glass 80). Molten glass 80 is conveyed from melting furnace 15 to fining vessel 20 through melting furnace-to-fining vessel connecting tube 55. The molten glass is heated in fining vessel 20 to a temperature in excess of the melting furnace temperature, whereupon multivalent oxide materials contained within the molten glass release oxygen that rises through the molten glass. This high- temperature release of oxygen in fining vessel 20 aids in removing the small bubbles of gas within the molten glass generated by the melting of the batch material.

[0031] The molten glass then flows from fining vessel 20 through fining vessel-to-stirring vessel connecting tube 60 into stirring vessel 25 where a rotating stirrer mixes and homogenizes the molten glass to ensure an even consistency. The homogenized molten glass from stirring vessel 25 then flows through stirring vessel-to-receiving vessel connecting tube 65 and is collected in receiving vessel 30. From receiving vessel 30 the molten glass is routed to forming body 45, through downcomer 35 and inlet 40, and formed into glass ribbon 50 by drawing the molten glass from the forming body.

[0032] Forming body 45 comprises an open channel 90 positioned on an upper surface of the forming body and a pair of converging forming surfaces 95, best seen in FIGS. 2 and 3, that converge at a bottom or root 100 of the forming body. The molten glass supplied to the forming body flows into open channel 90 formed in an upper surface of forming body 45 and overflows the walls thereof, thereby separating into two individual flows of molten glass that flow over the converging forming surfaces. When the separate flows of molten glass reach the root, they recombine, or fuse, to form glass ribbon 50 that descends from the root of the forming body. Edge directors 106 positioned on forming body 45, best seen in FIG. 3, function to effectively extend the width of the root and thereby aid in widening the glass ribbon, or at a minimum act to minimize narrowing of the glass ribbon. FIG. 3 is a perspective view of a portion of forming body 45 showing an edge director 106. In operation there are typically four edge directors, two edge directors opposing each other at one end of the forming body, and another pair of opposing edge directors positioned at the opposite end of the forming body.

[0033] As the glass ribbon descends from root 100, pulling rolls 1 10 contact the viscous glass ribbon along the edges of the ribbon and aid in drawing the ribbon in a downward path with a velocity vector V having both direction and speed. Pulling rolls 1 10 comprise opposing, counter-rotating rollers that grip the glass ribbon at edge portions thereof and draw the glass ribbon downward. Additional driven or non-driven rolls positioned above or below the pulling rolls (not shown) may also contact the edges of the ribbon to aid both in guiding the ribbon and maintaining a width of the ribbon against naturally occurring surface tension effects that work to otherwise reduce the width of the ribbon.

[0034] Once the descending ribbon has cooled through the glass transition temperature range and a portion thereof has been transformed from a viscous liquid to an elastic solid, a discrete glass sheet may be produced from the ribbon. The production of discrete glass sheets from a continuous and continuously moving glass ribbon typically involves first scoring the glass ribbon across a width of the ribbon, or a portion of the width. A tensile stress may then be applied across the score, indicated by dashed line 105, to create a crack that propagates through the thickness of the ribbon and across the width of the glass ribbon. Score 105 may be formed by any conventional method. For example, score 105 may be produced by contacting the ribbon with a scoring wheel, a scribe or an abrasive member that creates surface damage on the ribbon. The tensile stress may be applied by bending the glass ribbon in a direction that places the scored side of the glass ribbon in tension across the score line. The tension in turns drives the crack formed at the score line through the thickness of the ribbon and across the width of the ribbon.

[0035] A consequence of a relatively free-hanging ribbon of a glass forming material in a downdraw process such as a fusion or slot draw process is that surface tension and high flow density near an edge portion 1 15 of the ribbon may cause the glass ribbon to be thickened near the extreme edges of the ribbon, as shown in FIG. 4. These thickened regions are commonly referred to as beads 120. FIG. 4 is an illustration of a cross section of an edge portion 115 of a glass ribbon formed by a fusion process, such as the fusion process described supra, comprising beads 120. As the intent of the present process is to form glass sheets of high purity having a pristine surface and substantially parallel major surfaces (substantially uniform thickness), the existence of beads 120 within the ribbon edge portions 1 15 is detrimental to the commercial value of glass sheets cut from the ribbon. Thus, the beads are typically removed.

[0036] While current practice is to remove a glass sheet from the continuously moving glass ribbon and then remove the beads from the glass sheet, this approach has significant drawbacks. One drawback in particular is a difficulty in reliably scoring the glass ribbon across the entire width of the ribbon while maintaining a clean, straight break. The nonuniform thickness of the edge portions of the glass ribbon can result in uncontrolled cracking of the glass ribbon, where the score line, or more often the subsequent separating crack, wanders from the intended path. To overcome this tendency, scoring is often performed over the interior quality portion 125 of the glass ribbon (between dashed lines 130) without scoring the beads. Quality portion 125 is positioned between the two flanking edge portions 1 15 of the ribbon, and is generally that portion of the ribbon that serves as saleable product. However, the energy needed to propagate the separating crack across the unscored beads can produce large perturbations in the glass ribbon during the separation. These perturbations can propagate into the glass transition region of the glass ribbon and have an undesirable affect on the glass ribbon. For example, stresses may be frozen into the glass ribbon that affect the final shape of the glass sheet.

[0037] FIG. 5 illustrates a cross sectional end view of an exemplary forming body 45 for a fusion down draw process such as the fusion down draw process illustrated in FIG. 1. In accordance with FIG. 5, forming body 45 is contained within enclosure 135 that maintains a consistent thermal environment about the forming body. Heating elements 140 are used to control a temperature within enclosure 135. Heating elements 140 may be, for example, electrical resistance-heated metallic coils or bars. Interior walls 145 diffuse the heat produced by heating elements 140 and help provide more even heating of the forming body and molten glass. Interior walls 145 may be formed, for example, from silicon carbide. As the glass ribbon is drawn from forming body 45 by pulling rolls 110, a thickness of the glass ribbon, such as within quality portion 125, is controlled by cooling doors 150. Cooling doors 150 are configured to be movable so that the cooling doors can be extended in a direction toward the glass ribbon, or retracted away from the glass ribbon. The cooling doors extend across all or a significant portion of the glass ribbon width (i.e. in a direction perpendicular to draw vector V, which denotes a direction and speed at which the glass ribbon is drawn).

[0038] Contained within each cooling door are a plurality of cooling nozzles 155 that are supplied with a cooling gas, typically air. The air may be cooled prior to its delivery to cooling nozzles 155. Cooling gas exiting cooling nozzles 155, as represented by arrows 160, is directed against a front plate 165 of each cooling door. Front plates 165 may be formed, for example, from silicon carbide. The ability to obtain localized cooling of the cooling door front plate can result in a variable temperature distribution across a width of the front plate. The localized cooling of the front plate affects the viscosity of the glass ribbon, and therefore the thickness of the glass ribbon directly adjacent to that specific portion of the front plate. Thus, thickness control of the glass ribbon across a width of the glass ribbon can be obtained by varying the temperature and/or flow of cooling gas through cooling nozzles 155. By directing the cooling gas against a plate located between the cooling nozzles and the glass ribbon, the impact of the cooling nozzles can be smoothed across the draw.

[0039] In accordance with various embodiments described herein, a plurality of heating nozzles 170 are positioned above cooling doors 150 and are configured to direct heated air at specific portions of the continuously moving glass ribbon. The following description will be directed to one such heating nozzle 170, with the understanding that the description is equally applicable to the remaining heating nozzles 170.

[0040] As shown in FIG. 6, each heating nozzle 170 comprises a refractory body 180 comprising a plurality of passages 185. A heating gas 190, such as air, is delivered to at least one passage of the plurality of passages 185 at first end 195 of refractory body 180, and exits from the passage at second end 200 of refractory body 180. Second end 200 is positioned proximate to glass ribbon 50. Refractory body 180 may be positioned within refractory sleeve 205 such that refractory sleeve 205 surrounds and is substantially concentric with refractory body 180. Refractory sleeve 205 may comprise, for example, AI2O3.

[0041] A high temperature heating element 210, such as a wire coil, is disposed around refractory body 180. For example, if heating element 210 is a coil, heating element 210 may be wound around refractory body 180. Heating element 210 is preferably positioned between refractory body 180 and refractory sleeve 205. Heating element 210 may be formed, for example, from a platinum-containing wire, or other suitable high temperature metals. For example, the wire may be a platinum alloy such as platinum-rhodium. Heating element 210 is supplied with an electric current that heats heating element 210 and therefore refractory body 180 and heating gas 190 traveling within the at least one passage 185. For example, a single heating nozzle may require equal to or greater than about 400 watts in electrical power to sufficiently heat the heating gas flowing through refractory body 180. Heating gas 190 flowing in the at least one passage 185 is preferably heated to a temperature equal to or greater than 1450°C. For example, the heating gas may be heated to a temperature in a range from about 1450°C to about 1650°C, in a range from about 1500°C to about 1650°C, in a range from about 1550°C to about 1650°C, or in a range from about 1600°C to about 1650°C. To ensure sufficient heating gas flow, the at least one passage 185 should be of adequate internal diameter. For example, the inside diameter of the at least one passage 185 may be equal to or greater than 1 mm. Other passages of refractory body 180 may contain instruments or other devices for measuring a temperature of the heating gas. For example, as shown in the embodiment of FIG. 6, other passages 185 contained within refractory body 180 may contain a thermocouple element 220. Refractory sleeve 205 may be positioned within a suitable thermal insulating shield 225 that is disposed about refractory sleeve 205.

[0042] As best seen with the aid of FIGS. 2 - 3, heating nozzles 170 are preferably positioned at or near root 100 inward of an edge 226 of glass ribbon 50 (e.g. between an edge 226 and the centerline 230 of the glass ribbon). For example, heating nozzles 170 may be positioned vertically between root 100 and cooling door 150 and laterally so that the heated gas emitted by the nozzles is directed to a position where edge portions 115 will be removed from the quality portion 125. Preferably a heating nozzle 170 is positioned such that heating gas 190 is directed at a location between an inside edge of an edge director 106 and centerline 230 of glass ribbon 50. The heated gas emitted by the heating nozzles 170 impinges on the glass ribbon and locally reduces the viscosity of the glass, thereby causing a local thinning of the glass ribbon. As the continuously moving glass ribbon continues to descend from forming body 45, the localized thinning forms a narrow thinned region 235 running longitudinally along a length of the glass ribbon (see FIG. 7).

[0043] FIG. 8 represents an elementary volume of glass for an idealized infinitely wide ribbon subject to a vertical downward draw force F. Under equilibrium conditions, two sets of companion forces appears, each force equal to F/2: one set of forces perpendicular to and thinning the ribbon, and another set contained within the glass plane and in equilibrium with a horizontally adjacent elementary volume of glass. This last force is responsible for a loss in ribbon width, as the ribbon edge cannot be in equilibrium because there is no immediately adjacent volume of glass. With such a force distribution, glass thinning is only vertical (thickness variation due to drawing only occurs vertically).

[0044] The variation in amplitude for draw force F is a function of the viscosity, flow density and length of attenuation (which is inversely proportional to the cooling rate and draw velocity), and can be approximated by the following expression, where η is viscosity, Q is the flow density and L is the length of the attenuation. If a negative viscosity gradient is introducing locally but far from an edge using heating nozzles 170 for example, a reduction in the draw force F occurs and, consequently, a reduction of the horizontal force component contained within the ribbon plane. To maintain internal equilibrium, a horizontal glass flow appears in the direction of the adjacent glass volume elements, inducing localized thinning in the ribbon. However, the thinned region 235 is formed at the expense of local thickening (236) for adjacent glass volume elements, as shown by FIG. 9. The thickness response illustrated in FIG. 7 is indicative of what would happen if, for example, a heating nozzle were directed at a center of the ribbon.

[0045] On the other hand, if the negative viscosity gradient is introduced near an edge 226 of the ribbon, this horizontal flow will not induce a local thickening, or at least a reduced local thickening 236, as it will be (at least partially) compensated for by a slight increase in ribbon width. This is illustrated by FIG. 10. This occurs because the edge portion of the ribbon where the bead forms is normally brought to equilibrium in terms of horizontal forces by a reduction in ribbon width. If the horizontal force component is decreased glass ribbon width is increased.

[0046] When local thickness control is achieved close to the beads using heating nozzles 170, e.g. within about 100 mm of an edge 226 of the glass ribbon, the edge portion may then be separated from the glass ribbon. Thermo-mechanical stresses induced by thermal gradients can be used to propagate a crack initiated within the elastic region of the glass ribbon (usually below the pulling rolls) up to the top of the visco-elastic region, thus effectively separating the edge portions 115, and the beads, from the rest of the ribbon. Crack propagation terminates within the visco-elastic region of the thinned section of the glass ribbon, as most of the crack's energy of propagation is being absorbed by viscous shear. It should be understood that the location of the visco-elastic region of the thinned section of the glass ribbon is a function of the local temperature and cooling rate, and can be tuned upon demand with heating nozzle 170, that in turn controls the local thickness and the local glass temperature. Local cooling rates down the length of the draw (e.g. along the lengthy of the glass ribbon) can also be tuned using heaters below the forming body. It is also possible to use additional specific heating and/or cooling below the forming body to precisely tune the cooling rate.

[0047] It would be quite detrimental for the glass ribbon if the crack progresses out of the track of the thinned region 235. Propagation control may be enabled by controlling stress gradients in the thinned sections and in sections adjacent to the thinned sections. As noted above, this stress may be induced by the glass coefficient of thermal expansion of the glass ribbon and is mainly a function of the temperature gradient and ribbon thickness.

[0048] There being thick portions on both sides of the thinned section 235, the thinned section will be under tension while the adjacent thicker sections will be in compression. This will preferentially promote crack propagation in the center of thinned section 235 where the energy of propagation is the lowest.

[0049] Once initiated, separation of the edge portion 115 from glass ribbon 50 (i.e. from quality portion 125) can be sustained and controlled with heating nozzles 170 by adjusting the local viscosity modification near root 100 in terms of the viscosity distribution and amplitude as well as by adjusting the cooling rate longitudinally down the draw on the thinned region as a function of distance from the root. That is, by controlling the temperature of the air exiting heating nozzles 170, a localized viscosity of the glass ribbon can be controlled.

[0050] It is also advisable to ensure that the separated edge portion 115 does not touch the adjacent glass ribbon as the glass ribbon descends to the pulling rolls and below. Such contact can damage the quality region 125. This can be achieved for example by using additional rolls just above the pulling rolls that are offset from the plane of the glass ribbon by a few centimeters to ensure that the separated bead will travel out of the plane of the ribbon. [0051] While crack initiation can occur spontaneously, preferably cracking is induced at a predetermined location in the draw, for example, by localized heating and/or cooling. For example, thinned section 235 may be heated with laser 240, such as a CO2 laser, followed by cooling with a cooling fluid 245 (e.g. air jet or air/water mist) can promote very high stress gradients (see FIG. 2). Alternatively, a crack can be initiated by damaging the glass surface mechanically with a glass cutter or by applying a localized twist by pairs of rollers.

[0052] It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit and scope of the invention. Thus it is intended that the present invention cover the modifications and variations of this invention provided they come within the scope of the appended claims and their equivalents.