Abstract:
A flexible secondary battery includes an electrode assembly having a first electrode plate, a second electrode plate, and a separator between the first and second electrode plates; and a pouch comprising a plurality of outwardly extending protrusions, the pouch sealing the electrode assembly and an electrolyte therein, wherein the pouch is configured such that a degree of strain of the electrode assembly resulting from bending of the pouch is lower than a degree of strain on the pouch from formation of the protrusions.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application claims priority to and the benefit of Korean Patent Application No. 10-2015-0022716, filed on Feb. 13, 2015, in the Korean Intellectual Property Office, the entire disclosure of which is incorporated herein by reference. 
       BACKGROUND 
       [0002]    1. Field 
         [0003]    One or more exemplary embodiments relate to a flexible secondary battery. 
         [0004]    2. Description of the Related Art 
         [0005]    Along with the development of electronics, markets for cellular phones, portable multimedia players (PMPs), and mpeg audio layer-3 (MP3) players, as well as markets for other various portable electronic devices such as smartphones, smart pads, electronic book readers, flexible tablet computers, and portable medical devices attachable to the human body have grown markedly. 
         [0006]    The growth of markets for such portable electronic devices has increased the demand for batteries suitable for operating such portable electronic devices. In addition, such portable electronic devices are increasingly required to be flexible to endure transportation, storage, and impact, and thus the demand for flexible batteries has also increased. 
       SUMMARY 
       [0007]    One or more exemplary embodiments include a flexible secondary battery which is stable even when repeatedly flexed or bent. 
         [0008]    Additional aspects will be set forth in part in the description which follows and, in part, will be apparent from the description, or may be learned by practice of the presented embodiments. 
         [0009]    According to one or more exemplary embodiments, a flexible secondary battery includes: an electrode assembly including a first electrode plate, a second electrode plate, and a separator between the first and second electrode plates; and a pouch including a plurality of outwardly extending protrusions and sealing the electrode assembly and an electrolyte therein, wherein a degree of strain of the electrode assembly resulting from bending of the pouch is lower than a degree of strain given to the pouch by formation of the protrusions. 
         [0010]    A flat base portion may be formed between the protrusions. 
         [0011]    Each of the protrusions may include: a first curved portion protruding outward; and a second curved portion formed along an edge of the first curved portion. 
         [0012]    The second curved portion may be separate from a center of the first curved portion by a constant distance and may completely surround the first curved portion. 
         [0013]    The first curved portion may be convex, and the second curved portion may be concave. 
         [0014]    The first curved portion may have a greater radius of curvature than a radius of curvature of the second curved portion. 
         [0015]    The first curved portion and the second curved portion may have radii of curvature satisfying the following formula: 
         [0000]    
       
         
           
             
               
                 
                   ( 
                   
                     x 
                     + 
                     t 
                     + 
                     a 
                   
                   ) 
                 
                 2 
               
               + 
               
                 b 
                 2 
               
             
             = 
             
               
                 ( 
                 
                   x 
                   + 
                   y 
                 
                 ) 
               
               2 
             
           
         
       
       
         
           
             y 
             = 
             
               t 
               + 
               H 
               + 
               a 
             
           
         
       
       
         
           
             x 
             = 
             
               
                 
                   b 
                   2 
                 
                 
                   2 
                    
                   
                       
                   
                    
                   H 
                 
               
               - 
               
                 H 
                 2 
               
               - 
               t 
               - 
               a 
             
           
         
       
     
         [0000]    where t=thickness of the pouch, x=radius of curvature of the second curved portion, y=radius of curvature of the first curved portion, b=radius of each of the protrusions, H=height of each of the protrusions, and a=y−H−t. 
         [0016]    The pouch may have a thickness of about 0.05 mm to about 0.1 mm. 
         [0017]    The radius of each of the protrusions may range from about 0.05 mm to about 1.8 mm. 
         [0018]    The height of each of the protrusions may range from about 0.05 mm to about 1.0 mm. 
         [0019]    A ratio of the vertical cross-sectional area to the surface area of each of the protrusions may be greater than about 1:1 but less than about 1:3. 
         [0020]    The number of protrusions per unit area (1 cm 2 ) of the pouch may be about 5 to about 125. 
         [0021]    The flexible secondary battery may further include: a first electrode tab connected to the first electrode plate; and a second electrode tab connected to the second electrode plate. 
         [0022]    According to one or more exemplary embodiments, a flexible secondary battery includes: an electrode assembly including a first electrode plate, a second electrode plate, and a separator between the first and second electrode plates; and a pouch including an outwardly protruding pattern unit and sealing the electrode assembly and an electrolyte therein, wherein the pattern unit includes: a protrusion; a sub-protrusion disposed close to the protrusion; and a base portion disposed between the protrusion and the sub-protrusion. 
         [0023]    The protrusion may include: a first curved portion protruding outward; and a second curved portion formed outside the first curved portion and connected to the base portion. 
         [0024]    The second curved portion may be separate from a center of the first curved portion by a constant distance and may fully surround the first curved portion. 
         [0025]    The first curved portion may be convex, and the second curved portion may be concave. 
         [0026]    The first curved portion may have a greater radius of curvature than a radius of curvature of the second curved portion. 
         [0027]    The pattern unit may have a first direction, a second direction perpendicular to the first direction, and a third direction perpendicular to the first and second directions, and the protrusion may protrude in the third direction, wherein the pattern unit may have a cross-section perpendicular to the first direction and parallel to the second and third directions, wherein the cross-section may have a second-direction length that is longer than a second-direction unit length of the pattern unit. 
         [0028]    The second-direction length of the cross-section of the pattern unit may vary along the first direction. 
         [0029]    The first curved portion and the second curved portion may have radii of curvature satisfying the following formula: 
         [0000]    
       
         
           
             
               
                 
                   ( 
                   
                     x 
                     + 
                     t 
                     + 
                     a 
                   
                   ) 
                 
                 2 
               
               + 
               
                 b 
                 2 
               
             
             = 
             
               
                 ( 
                 
                   x 
                   + 
                   y 
                 
                 ) 
               
               2 
             
           
         
       
       
         
           
             y 
             = 
             
               t 
               + 
               H 
               + 
               a 
             
           
         
       
       
         
           
             x 
             = 
             
               
                 
                   b 
                   2 
                 
                 
                   2 
                    
                   
                       
                   
                    
                   H 
                 
               
               - 
               
                 H 
                 2 
               
               - 
               t 
               - 
               a 
             
           
         
       
     
         [0000]    where t=thickness of the pouch, x=radius of curvature of the second curved portion, y=radius of curvature of the first curved portion, b=radius of the protrusion, H=height of the protrusion, and a=y−H−t. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0030]    These and/or other aspects will become apparent and more readily appreciated from the following description of the exemplary embodiments, taken in conjunction with the accompanying drawings in which: 
           [0031]      FIG. 1  is an exploded perspective view schematically illustrating a flexible secondary battery according to an exemplary embodiment; 
           [0032]      FIG. 2  is a plan view illustrating the flexible secondary battery depicted in  FIG. 1 ; 
           [0033]      FIG. 3  is a perspective view illustrating a portion of a pouch depicted in  FIG. 2 ; 
           [0034]      FIG. 4  is a cross-sectional view illustrating a region B in  FIG. 3 ; and 
           [0035]      FIG. 5  is a strain-curvature graph of flexible secondary batteries. 
       
    
    
     DETAILED DESCRIPTION 
       [0036]    Reference will now be made in detail to exemplary embodiments, examples of which are illustrated in the accompanying drawings. Effects and features of the exemplary embodiments, and implementation methods thereof will be clarified through the following descriptions given with reference to the accompanying drawings. In this regard, the exemplary embodiments may have different forms and should not be construed as being limited to the descriptions set forth herein. As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items. Expressions such as “at least one of,” when preceding a list of elements, modify the entire list of elements and do not modify the individual elements of the list. 
         [0037]    Hereinafter, the exemplary embodiments will be described in detail with reference to the accompanying drawings. In the drawings, like reference numerals denote like elements, and overlapping descriptions thereof will be omitted. 
         [0038]    In the following descriptions of the exemplary embodiments, although the terms “first” and “second” are used to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another element. 
         [0039]    In the following descriptions of the exemplary embodiments, the terms of a singular form may include plural forms unless referred to the contrary. 
         [0040]    In the following descriptions of the exemplary embodiments, the meaning of “include,” “comprise,” “including,” or “comprising” specifies a property, a region, a fixed number, a step, a process, an element, a component, and a combination thereof but does not exclude other properties, regions, fixed numbers, steps, processes, elements, components, and combinations thereof. 
         [0041]    It will be understood that when a film, a region, or an element is referred to as being “above” or “on” another film, region, or element, it can be directly above or on the other film, region, or element, or intervening films, regions, or elements may also be present. 
         [0042]    Sizes of elements in the drawings may be exaggerated for convenience of explanation. In other words, since sizes and thicknesses of components in the drawings are arbitrarily illustrated for convenience of explanation, the following embodiments are not limited thereto. 
         [0043]    In the following description of exemplary embodiments, the directions of a flexible secondary battery  100  are defined for clarity in illustration as follows: the length direction of the flexible secondary battery  100  is defined as an x-axis direction, the width direction of the flexible secondary battery  100  is defined as a y-axis direction, and the thickness direction of the flexible secondary battery  100  is defined as a z-axis direction. 
         [0044]      FIG. 1  is an exploded perspective view schematically illustrating the flexible secondary battery  100  according to an exemplary embodiment, and  FIG. 2  is a plan view illustrating the flexible secondary battery  100  depicted in  FIG. 1 . 
         [0045]    Referring to  FIGS. 1 and 2 , the flexible secondary battery  100  may include a pouch  110 , an electrode assembly  130 , and electrode tabs  150  such as first and second electrode tabs  151  and  152 . 
         [0046]    The pouch  110  may accommodate the electrode assembly  130  together with an electrolyte and seal the electrode assembly  130  and the electrolyte. The pouch  110  may be flexible and prevent permeation of moisture or oxygen into the flexible secondary battery  100 . The pouch  110  may include: a first sealing material  111  covering a surface of the electrode assembly  130 ; and a second sealing material  112  covering an opposite surface of the electrode assembly  130  and connected to the first sealing material  111 . 
         [0047]    The pouch  110  may have a stacked structure formed by an insulation layer, a barrier layer, and an insulation layer. For example, the pouch  110  may be formed by sequentially stacking insulation and barrier layers. The pouch  110  may include a plurality of barrier layers, and the barrier layers may have different thicknesses. 
         [0048]    For example, a first barrier layer formed at a side of the pouch  110  may be thinner than a second barrier layer facing the first barrier layer. When the pouch  110  is formed in a curved shape, elongation of the pouch  110  increases in a direction away from the center of curvature of the curved shape. For example, the first barrier layer of the pouch  110 , relatively distant from the center of curvature of the curved shape, may be easily elongated, compared to the second barrier layer. Therefore, the breakage or deformation of the pouch  110  may be minimized, and the durability of the pouch  110  may be improved. 
         [0049]    In addition, the first barrier layer of the pouch  110  may be formed of a material having a relatively high degree of elongation, and the second barrier layer of the pouch  110  may be formed of a material having a relatively low degree of elongation compared to the first barrier layer. In this case, when the pouch  110  is formed in a curve shape, the first barrier layer of the pouch  110 , relatively distant from the center of curvature of the curved shape, may be easily elongated, compared to the second barrier layer. Therefore, the breakage or deformation of the pouch  110  may be minimized, and the durability of the pouch  110  may be improved. 
         [0050]    The barrier layers may be formed of a material such as aluminum, steel, stainless steel, or a polymer, and the Insulation layers may be formed of a material such as chlorinated polypropylene (CPP), polyethylene terephthalate (PET), or nylon. However, the barrier layers and the Insulation layers are not limited thereto. 
         [0051]    The electrode assembly  130  may include a first electrode layer  131 , a second electrode layer  132 , and a separator  133  located between the first electrode layer  131  and the second electrode layer  132 . The electrode assembly  130  may be formed by repeatedly stacking the first electrode layer  131 , the separator  133 , and the second electrode layer  132 , or rolling the first electrode layer  131 , the separator  133 , and the second electrode layer  132  in the form of a jelly roll. For ease of description, the case in which the electrode assembly  130  has a stacked structure formed by the first electrode layer  131 , the separator  133 , and the second electrode layer  132  will be primarily described in detail. 
         [0052]    The first electrode layer  131  may be one of a positive electrode film and a negative electrode film. If the first electrode layer  131  is a positive electrode film, the second electrode layer  132  may be a negative electrode film. On the other hand, if the first electrode layer  131  is a negative electrode film, the second electrode layer  132  may be a positive electrode film. 
         [0053]    The first electrode layer  131  may include a first metal collector, a first active material-coated portion formed by coating the first metal collector with a first active material, and a first non-coated portion which is a portion of the first metal collector that is not coated with the first active material. Similarly, the second electrode layer  132  may include a second metal collector, a second active material-coated portion  124  formed by coating the second metal collector with a second active material, and a second non-coated portion which is a portion of the second metal collector that is not coated with the second active material. 
         [0054]    If the first electrode layer  131  is a positive electrode film, the first metal collector may be a positive electrode collector, and the first active material-coated portion may be a positive electrode active material-coated portion. If the second electrode layer  132  is a negative electrode film, the second metal collector may be a negative electrode collector, and the second active material-coated portion may be a negative electrode active material-coated portion. 
         [0055]    The positive electrode collector may be a metal collector formed of aluminum, stainless steel, titanium, copper, silver, or a combination thereof. The positive electrode active material-coated portion may include a positive electrode active material, a binder, and a conducting agent. 
         [0056]    The positive electrode active material may be a material capable of reversibly intercalating and deintercalating lithium ions. For example, the positive electrode active material includes at least one material selected from the group consisting of lithium transition metal oxides such as lithium cobalt oxide, lithium nickel oxide, lithium nickel cobalt oxide, lithium nickel cobalt aluminum oxide, lithium nickel cobalt manganese oxide, lithium manganese oxide, and lithium iron phosphate; nickel sulfides; copper sulfides; sulfur; iron oxides; and vanadium oxides. 
         [0057]    The binder may include at least one material selected from the group consisting of polyvinylidene fluoride-containing binders such as polyvinylidene fluoride, vinylidene fluoride/hexafluoropropylene copolymer, and vinylidene fluoride/tetrafluroethylene copolymer; carboxymethyl cellulose-containing binders such as sodium-carboxymethyl cellulose and lithium-carboxymethyl cellulose; acrylate-containing binders such as polyacrylic acid, lithium-polyacrylic acid, acrylic, polyacrylonitrile, polymethyl methacrylate, and poly(butyl acrylate); polyamide-imides; polytetrafluoroethylene; polyethylene oxide; polypyrrole; lithium-Nafion; and styrene butadiene rubber-containing polymers. 
         [0058]    The conducting agent may include at least one material selected from the group consisting of carbon-containing conducting agents such as carbon black, carbon fiber, and graphite; conductive fiber such as metal fiber; metal powder such as carbon fluoride powder, aluminum powder, and nickel powder; conductive whisker such as zinc oxides and potassium titanate; conductive metal oxides such as titanium oxides; and conductive polymers such as polyphenylene derivatives. 
         [0059]    The negative electrode collector may include at least one metal selected from the group consisting of copper, stainless steel, nickel, aluminum, and titanium. The negative electrode active material-coated portion may include a negative electrode active material, a binder, and a conducting agent. 
         [0060]    The negative electrode active material may be a material capable of forming an alloy together with lithium or capable of reversibly intercalating and deintercalating lithium. For example, the negative electrode active material may be include at least one material selected from the group consisting of metals, carbon-containing materials, metal oxides, and lithium metal nitrides. 
         [0061]    The metals may include at least one selected from the group consisting of lithium, silicon, magnesium, calcium, aluminum, germanium, tin, lead, arsenic, antimony, bismuth, silver, gold, zinc, cadmium, mercury, copper, iron, nickel, cobalt, and indium. 
         [0062]    The carbon-containing materials may include at least one material selected from the group consisting of graphite, graphite carbon fiber, coke, mesocarbon microbeads (MCMBs), polyacene, pitch-derived carbon fiber, and hard carbon. 
         [0063]    The metal oxides may include at least one selected from the group consisting of lithium titanium oxides, titanium oxides, molybdenum oxides, niobium oxides, iron oxides, tungsten oxides, tin oxides, amorphous tin oxide composites, silicon monoxide, cobalt oxides, and nickel oxides. 
         [0064]    The binder and the conducting agent may be the same as the binder and the conducting agent included in the positive electrode active material. 
         [0065]    The positive electrode film and the negative electrode film may be formed by applying active materials to metal collectors by various methods. That is, methods of applying electrode active materials are not limited. 
         [0066]    The separator  133  may be a porous polymer film such as a polyethylene film or a polypropylene film. The separator  133  may be woven fabric or non-woven fabric including polymer fabric. The separator  130  may include ceramic particles and may be formed of solid polymer electrolyte. The separator  133  may be an independent film or may be a porous layer formed on the first electrode layer  131  or the second electrode layer  132 . 
         [0067]    The separator  133  electrically separates the first electrode layer  131  and the second electrode layer  132  from each other, and the separator  133  is not required to have the same shape as the shape of the first electrode layer  131  or the second electrode layer  132 . 
         [0068]    The electrode tabs  150  may be connected to the electrode assembly  130  for electric connection with an external device. The first electrode layer  131  is electrically connected to the first electrode tab  151 , and the second electrode layer  132  is electrically connected to the second electrode tab  152 . The first electrode tab  151  may include a first insulator  153 , and the second electrode tab  152  may include a second insulator  154 , so as to prevent a short circuit between the first and second electrode layers  131  and  132 . Materials that may be used to form the first insulator  153  and the second insulator  154  are not limited as long as the materials are insulative. 
         [0069]      FIG. 3  is a perspective view illustrating a portion of the pouch  110  depicted in  FIG. 2 , and  FIG. 4  is a cross-sectional view illustrating a region B in  FIG. 3 . The size and arrangement of protrusions  116  of the pouch  110  will now be described in detail with reference to  FIGS. 3 and 4 . 
         [0070]    As described above, the pouch  110  may include: the first sealing material  111  covering a surface of the electrode assembly  130 ; and the second sealing material  112  covering an opposite surface of the electrode assembly  130  and connected to the first sealing material  111 . The protrusions  116  may be formed on one or more of the first sealing material  111  and the second sealing material  112 . One or more protrusions  116  may be formed. For ease of description, the following description is presented under the assumption that each of the first and second sealing materials  111  and  112  includes a plurality of protrusions  116 . 
         [0071]    Each of the protrusions  116  may include a first curved portion  113  protruding outward from the pouch  110  and a second curved portion  114  formed along the edge of the first curved portion  113 . A curvature change portion  115  may be formed between the first curved portion  113  and the second curved portion  114 . The pouch  110  may include a base portion  117  among the protrusions  116 . 
         [0072]    The pouch  110  may include pattern units U as shown in  FIG. 2 . Each of the pattern units U may include a protrusion  116 , sub-protrusions formed close to the protrusion  116 , and the base portion  117  formed among the sub-protrusions. The sub-protrusions are portions of protrusions  116 . In other words, in combinations of the pattern units U, sub-protrusions combining with other sub-protrusions may form protrusions  116 . 
         [0073]    Each of the pattern units U may include four sub-protrusions located close to a central protrusion  116 . In one embodiment, the sub-protrusions are substantially equal to the quadrisections of one protrusion  116 . Therefore, if four pattern units U are combined, a protrusion  116  may be formed by a combination of sub-protrusions. Thus, the following description will be primarily directed to the protrusions  116 . 
         [0074]    In each of the protrusions  116 , the first curved portion  113  may have a convex shape, and the second curved portion  114  may have a concave shape. The second curved portion  114  may be separate from the center (O) of the first curved portion  113  by a constant distance and may fully surround the first curved portion  113 . The radius of curvature (y) of the first curved portion  113  may be greater than the radius of curvature (x) of the second curved portion  114 . 
         [0075]    Referring to  FIG. 4 , Formula 1 for calculating the radius of curvature (y) of the first curved portion  113  and the radius of curvature (x) of the second curved portion  114  may be derived as follows. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             x 
                             + 
                             t 
                             + 
                             a 
                           
                           ) 
                         
                         2 
                       
                       + 
                       
                         b 
                         2 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           x 
                           + 
                           y 
                         
                         ) 
                       
                       2 
                     
                   
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                     y 
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                       t 
                       + 
                       H 
                       + 
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                     = 
                     
                       
                         
                           b 
                           2 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           H 
                         
                       
                       - 
                       
                         H 
                         2 
                       
                       - 
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                       a 
                     
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
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         [0076]    In  FIG. 4 , O denotes the center of curvature of the first curved portion  113 , O′ denotes the center of curvature of the second curved portion  114 , t denotes the thickness of the pouch  110 , x denotes the radius of curvature of the second curved portion  114 , and y denotes the radius of curvature of the first curved portion  113 . Further, b refers to the radius of the protrusion  116 , which is a distance from the center of the first curved portion  113  to an edge of the second curved portion  114 . H refers to the height of the protrusion  116 , which is a distance from a surface of the base portion  117  to the apex of the first curved portion  113  while a refers to a value calculated by subtracting the height (H) of the protrusion  116  and the thickness (t) of the pouch  110  from the radius of curvature (y) of the first curved portion  113  (a=y−H−t). 
         [0077]    Referring to  FIG. 4 , the first equation of Formula 1 may be derived using the Pythagorean theorem. In addition, the second equation of Formula 1 may be derived from a length relationship, and the third equation of Formula 1 may be derived by combining the first and second equations. 
         [0078]    The thickness (t) of the pouch  110  may range from about 0.05 mm to about 0.1 mm. If the thickness (t) of the pouch  110  is less than about 0.05 mm, the pouch  110  may easily be deformed or broken, and the electrolyte contained inside the pouch  110  may leak. In contrast, if the thickness (t) of the pouch  110  is greater than about 0.1 mm, the flexibility of the pouch  110  may be low, and thus, it may be difficult to form the protrusions  116  on the pouch  110 . 
         [0079]    The radius (b) of the protrusions  116  may range from about 0.05 mm to about 1.8 mm. If the radius (b) of the protrusions  116  is less than about 0.05 mm, the height (H) of the protrusions  116  is not sufficiently large, and thus the amount of deformation of the pouch  110 , caused by the formation of the protrusions  116 , may be insufficient. If the radius (b) of the protrusions  116  is greater than about 1.8 mm, the number of protrusions  116  per unit area is reduced, and thus the amount of deformation of the pouch  110 , caused by the formation of the protrusions  116 , may be insufficient. If the amount of deformation of the pouch  110 , caused by the formation of the protrusions  116 , is insufficient, when the pouch  110  is bent, stress may not be effectively distributed in the pouch  110  (described later in more detail), and thus concentration of stress may not be effectively prevented. 
         [0080]    The height (H) of the protrusions  116  may range from about 0.05 mm to about 1.0 mm. If the height (H) of the protrusions  116  is less than about 0.05 mm, the amount of deformation of the pouch  110 , caused by the formation of the protrusions  116 , may be insufficient, and thus when the pouch  110  is bent, stress may not be effectively distributed (described later in more detail). If the height (H) of the protrusions  116  is greater than about 1.0 mm, the amount of deformation of the pouch  110 , caused by the formation of the protrusions  116 , may be excessive, and thus the strength of the pouch  110  may be decreased. In addition, since the volume of the pouch  110  increases in proportion to the height (H) of the protrusions  116 , the pouch  110  may not be easily disposed in an electronic device, and the spatial efficiency of the pouch  110  may be lowered. 
         [0081]    The radius of curvature (y) of the first curved portion  113  of each protrusion  116  may be adjusted to be within the range of about 0.1 mm to about 24.05 mm according to the thickness (t) of the pouch  110  and the height (H) and radius (b) of the protrusions  116 . In addition, the radius of curvature (x) of the second curved portion  114  of each protrusion  116  may range from about 0.0 mm to about 23.95 mm. If the second curved portion  114  is not formed (x=0), a time necessary to form the pouch  110  may be reduced owing to omission of the second curved portion  114 . 
         [0082]    If the protrusions  116  are formed in accordance with the above-described ranges of the thickness (t) of the pouch  110 , the height (H) and radius (b) of the protrusions  116 , the ratio of the vertical cross-sectional area to the surface area of the protrusions  116  may be greater than about 1:1 but less than about 1:3. The vertical cross-sectional area of each protrusion  116  may be expressed as the product of the square of the radius (b) of the protrusion  116  and π (i.e., vertical cross-sectional area=πb 2 ), and the surface area of each protrusion  116  may be expressed as the sum of the surface area of the first curved portion  113  and the surface area of the second curved portion  114 . 
         [0083]    If the protrusions  116  are formed in accordance with the above-described ranges of the thickness (t) of the pouch  110 , the height (H) and radius (b) of the protrusions  116 , the number of the protrusions  116  per unit area (1 cm 2 ) of the pouch  110  may be about 5 to about 125. 
         [0084]    The amount of deformation of the pattern units U will now be described according to the formation of the protrusions  116  with reference to  FIG. 3 . Portions of the pouch  110  are the same as the pattern units U. Hereinafter, the x-axis, y-axis, and z-axis directions will be referred as first, second, and third directions, respectively. 
         [0085]    Each of the pattern units U has a cross-section (e) perpendicular to the first direction and parallel to the second and third directions. The length (second-direction length) of the cross-section (e) measured in the second direction may be longer than a unit length (d) of the pattern unit U. In addition, the second-direction length of the cross-section (e) may vary along the first direction. 
         [0086]    As shown in Table 1 below, a pattern unit U was divided into ten parts in the first direction to calculate variations of the second-direction length of the cross-section (e) and resulting strain levels. In the measurement, the unit length (d) of the pattern unit U was 3.0 mm, the thickness (t) of the pouch  110  was 0.07 mm, and the height (H) of the protrusions  116  was 0.3 mm. 
         [0000]    
       
         
               
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
               
                   
                 Dividing 
                 Second-direction 
                   
                   
               
               
                   
                 positions (mm) 
                 length (mm) 
                 Ratio (%) 
                 Strain (%) 
               
               
                   
                   
               
             
             
               
                   
                 0.0 
                 3.1439 
                 104.80 
                 4.80 
               
               
                   
                 0.3 
                 3.0965 
                 103.22 
                 3.22 
               
               
                   
                 0.6 
                 3.0037 
                 100.12 
                 0.12 
               
               
                   
                 0.9 
                 3.0050 
                 100.17 
                 0.17 
               
               
                   
                 1.2 
                 3.0965 
                 103.22 
                 3.22 
               
               
                   
                 1.5 
                 3.1439 
                 104.80 
                 4.80 
               
               
                   
                 1.8 
                 3.0965 
                 103.22 
                 3.22 
               
               
                   
                 2.1 
                 3.0050 
                 100.17 
                 0.17 
               
               
                   
                 2.4 
                 3.0037 
                 100.12 
                 0.12 
               
               
                   
                 2.7 
                 3.0965 
                 103.22 
                 3.22 
               
               
                   
                 3.0 
                 3.1439 
                 104.80 
                 4.80 
               
               
                   
                 Average 
                 3.0759 
                 102.53 
                 2.53 
               
               
                   
                   
               
             
          
         
       
     
         [0087]    Referring to Table 1, although the unit length (d) of the pattern unit U on which the protrusions  116  are formed is 3.0 mm, the second-direction length of the cross-section (e) is 3.0759 mm on average. In other words, the second-direction length of the cross-section (e) is greater than the unit length (d) by 2.53% on average. 
         [0088]    As shown in Table 2 below, pattern units U were each divided into ten parts in the first direction to calculate variations of the second-direction length of the cross-section (e) according to variations of the height (H) of the protrusions  116 . In addition, strain levels were calculated according to the variations of the second-distance length of the cross-section (e). When the strain levels of the second-direction length of the cross-section (e) were calculated, the unit length (d) of the pattern units U was 3.0 mm, the thickness (t) of the pouch  110  was 0.07 mm, and the height (H) of the protrusions  116  was varied to be 0.3 mm, 0.4 mm, and 0.5 mm. 
         [0000]    
       
         
               
               
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                   
                 TABLE 2 
               
             
             
               
                   
                   
               
               
                   
                 Dividing 
                 Strain (%) 
               
             
          
           
               
                   
                 positions (mm) 
                 0.3 mm 
                 0.4 mm 
                 0.5 mm 
               
               
                   
                   
               
             
          
           
               
                   
                 0.0 
                 4.7967 
                 8.4867 
                 12.9533 
               
               
                   
                 0.3 
                 3.2167 
                 6.4533 
                 10.3000 
               
               
                   
                 0.6 
                 0.1233 
                 0.5367 
                 0.8667 
               
               
                   
                 0.9 
                 0.1667 
                 0.5367 
                 0.8700 
               
               
                   
                 1.2 
                 3.2167 
                 6.4533 
                 10.3000 
               
               
                   
                 1.5 
                 4.7967 
                 8.4867 
                 12.9533 
               
               
                   
                 1.8 
                 3.2167 
                 5.4533 
                 10.3000 
               
               
                   
                 2.1 
                 0.1667 
                 0.5367 
                 0.8700 
               
               
                   
                 2.4 
                 0.1233 
                 0.5367 
                 0.8667 
               
               
                   
                 2.7 
                 3.2167 
                 6.4533 
                 10.3000 
               
               
                   
                 3.0 
                 4.7967 
                 8.4867 
                 12.9533 
               
               
                   
                 Average 
                 2.53 
                 4.77 
                 7.59 
               
               
                   
                   
               
             
          
         
       
     
         [0089]    Referring to Table 2, the strain of the pattern units U increases in proportion to the height (H) of the protrusions  116 . If the height (H) of the protrusions  116  increases, the strain of the second-direction length of the cross-section (e) increases. 
         [0090]    With the advance of electronic technology, the electronic device market has increased, and thus demand for batteries for electronic devices has increased. Particularly, demand for flexible secondary batteries having a high degree of efficiency in utilization of space has increased in various application fields. 
         [0091]    When flexible secondary batteries are manufactured, a bending reliability test is performed. In the test, pouches of flexible secondary batteries were repeatedly subjected to a bending force to check whether the flexible secondary batteries were damaged or broken. In addition, when a force is applied to a flexible secondary battery to place the flexible secondary battery in an installation space by bending the flexible secondary battery, a pouch of the flexible secondary battery may be damaged or broken. 
         [0092]      FIG. 5  is a strain-curvature graph of flexible secondary batteries. 
         [0093]    Electrode assemblies each having a length of 110 mm were used in a test performed to plot the strain-curvature graph of  FIG. 5 . The thicknesses of the electrode assemblies were 0.5 mm, 0.8 mm, 1.10 mm, and 1.35 mm, respectively, and the electrode assemblies were bent to radii of curvature of 25 mm, 40 mm, 100 mm, and 200 mm, respectively. Then, strain levels of the electrode assemblies were calculated according to the radii of curvature. If an electrode assembly is bent to have a certain radius of curvature, a portion of the electrode assembly is stretched into a convex shape, and the other portion of the electrode assembly is compressed into a concave shape. Therefore, the strain of the electrode assembly indicates the degree of expansion or compression of the electrode assembly in the length direction thereof. 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                      
                     
                       ( 
                       % 
                       ) 
                     
                   
                   = 
                   
                     
                       t 
                       
                         2 
                          
                         
                             
                         
                          
                         R 
                       
                     
                     × 
                     100 
                   
                 
               
               
                 
                   Formula 
                    
                   
                       
                   
                    
                   2 
                 
               
             
           
         
       
     
         [0094]    In Formula 2, S refers to the strain of an electrode assembly, t refers to the thickness of the electrode assembly, and R refers to the radius of curvature of the electrode assembly. 
         [0095]    When an electrode assembly having a length of 110 mm along a thickness centerline is bent, a portion of the electrode assembly may be stretched into a convex shape to have tensile strain, and the other portion of the electrode assembly may be compressed into a concave shape to have compressive strain. 
         [0096]    Since the pouch  110  of the flexible secondary battery  100  covers outer sides of the flexible secondary battery  100 , if the electrode assembly  130  is bent to a radius of curvature as shown in  FIG. 5 , the pouch  110  receives substantially the same bending force as that the electrode assembly  130  receives. If data shown in Tables 1 and 2 and  FIG. 5  is compared and analyzed, the following conclusion may be obtained. When the protrusions  116  are formed on the pouch  110 , at least portions of the pouch  110  are deformed with a first strain. Then, if the electrode assembly  130  is bent, the electrode assembly  130  is stretched and compressed with a second strain. In this case, if the electrode assembly  130  is bent within a range in which the second strain is less than the first strain, the pouch  110  may not be permanently deformed or broken even though an external force is applied to the flexible secondary battery  100 . In other words, permanent deformation or breakage of the flexible secondary battery  100  caused by an external force applied thereto may be minimized, and thus the durability of the flexible secondary battery  100  may be improved. 
         [0097]    For example, if a secondary battery having a thickness (t) of 1.1 mm and not including protrusions  116  is bent to a radius of curvature of 25 mm, the secondary battery may be deformed with a strain of 2.2%. Thereafter, although the secondary battery is returned to an original shape thereof by resilience or elasticity, the rigidity of the secondary battery may be reduced, and the secondary battery may be cracked or tom. 
         [0098]    In the case of Table 1, the pouch  110  has a thickness of 0.07 mm, and the protrusions  116  of the pouch  110  have a height (H) of 0.3 mm. In this case, if the pouch  110  covers an electrode assembly  130  having a thickness (t) of 1.1 mm, although the electrode assembly  130  is bent to a radius of curvature of 25 mm or greater, the pouch  110  may not be permanently deformed. 
         [0099]    In the case of Table 2, the pouch  110  has a thickness of 0.07 mm, and the protrusions  116  of the pouch  110  have a height (H) of 0.4 mm. In this case, if the pouch  110  covers an electrode assembly  130  having a thickness (t) of 1.35 mm, although the electrode assembly  130  is bent to a radius of curvature of 25 mm or greater, the pouch  110  may not be permanently deformed. 
         [0100]    In other words, since the pouch  110  is previously deformed to have a certain degree of strain, the deformation of the pouch  110  may be prevented within the certain degree of strain previously given to the pouch  110 . That is, the degree of strain previously given to the pouch  110  by the formation of the protrusions  116  is adjusted to be greater than a degree of strain of the electrode assembly  130  when the flexible secondary battery  100  in which the electrode assembly  130  and the pouch  110  are coupled is deformed by a force applied thereto. Therefore, although an external force is applied to the flexible secondary battery  100 , the pouch  110  may not be permanently deformed or damaged. 
         [0101]    When an external force is applied to the flexible secondary battery  100 , the external force may be distributed to the protrusions  116  of the pouch  110 , and thus the durability of the pouch  110  may be improved. When an external force is applied to the flexible secondary battery  100 , the pouch  110  may be flexibly deformed while the protrusions  116  and the base portion  117  formed between the protrusions  116  are stretched or compressed. Therefore, although an external force is applied to the flexible secondary battery  100 , the pouch  110  may not be permanently deformed or damaged. 
         [0102]    Since the pouch  110  used as a sealing material for the flexible secondary battery  100  includes the protrusions  116 , stress caused by an extemal force may be distributed to the protrusions  116 . Therefore, the durability of the flexible secondary battery  100  may be improved. 
         [0103]    The second curved portions  114  are formed in the protrusions  116  so that the protrusions  116  may have a gentle curvature, and thus the pouch  110  of the flexible secondary battery  100  may be less cracked or tom. 
         [0104]    As described above, according to the one or more of the above exemplary embodiments, the stability of the flexible secondary battery  100  may be maintained when the flexible secondary battery  100  is repeatedly flexed or bent. 
         [0105]    Other effects of the exemplary embodiments may be understood from the above description with reference to the accompanying drawings. 
         [0106]    It should be understood that the exemplary embodiments described herein should be considered in a descriptive sense only and not for purposes of limitation. Descriptions of features or aspects within each exemplary embodiment should typically be considered as available for other similar features or aspects in other exemplary embodiments. 
         [0107]    While one or more exemplary embodiments have been described with reference to the figures, it will be understood by those of ordinary skill in the art that various changes in form and details may be made therein without departing from the spirit and scope as defined by the following claims.