Patent Publication Number: US-2012035899-A1

Title: Pigment orientation estimating method

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to a pigment orientation estimating method that estimates the orientation of pigment presents in a coating. 
     2. Description of the Related Art 
     In order to ensure stable coating quality, there is a need for understanding how a coating appearance varies when the characteristic of a paint and/or a painting condition are changed. Within the coating appearance, the orientation of pigments in the coating is the most important. 
     Japanese Patent Application Publication No. 2005-147875 (JP-A-2005-147875) describes a method of evaluating the orientation of pigments. In the method, a plurality of painted plates that are painted under different paint conditions and/or different painting conditions are prepared. Then, the positional information of pigments in the coating is acquired from a cut piece of each painted plate, and the inclinations of pigments to a coating surface are calculated from the positional information. 
     Japanese Patent Application Publication No. 2006-218425 (JP-A-2006-218425) describes a method of estimating a coating result. In the method, a plurality of painted test panels are prepared, and the visual characteristic value of each test panel is measured and stored in a database. Then, the database is used and subjected to multiple regression analysis. 
     In addition, Japanese Patent Application Publication No. 2004-152000 (JP-A-2004-152000) describes a technique of an image simulation method for a painted appearance. In the image simulation method, the arrangement of coloring materials in a coating is input to calculate the colored state of a virtually painted object, and the calculated colored state of the virtually painted object is displayed in image. 
     In the case of the technique described in JP-A-2005-147875, there are many combinations of paint conditions and painting conditions, and, in addition, it takes time to adjust the evaluation condition, adjust the temperature and humidity in a painting booth or adjust a paint. Thus, a method of evaluating the appearance by preparing a painted plate one by one requires much expense in time and effort, resulting in an increase in cost. 
     When the number of pieces evaluated is limited as in the case of the technique described in JP-A-2006-218425, there is a problem that it is difficult to detect a malfunction under an unexpected environmental condition. When the number of pieces evaluated is increased, there is a problem that it is necessary to take much time for evaluation. 
     Then, the technique described in JP-A-2004-152000 is to input the specified arrangement of coloring materials (the orientation of pigments), but the technique is not to estimate the orientation of pigments. As described above, the process of drying a coating is important to the orientation of pigments; however, there is no method for estimating the orientation of pigments in consideration of the drying process. 
     SUMMARY OF THE INVENTION 
     The invention provides a pigment orientation estimating method that is able to accurately and easily estimate the orientation of pigments in a coating depending on the characteristic of a paint and/or a painting condition. 
     A first aspect of the invention relates to a pigment orientation estimating method that estimates the orientation of pigments in a coating. The method includes: setting the orientation of the pigments when a paint is applied; and calculating the orientation of the pigments at time, at which a predetermined period of time has elapsed from when the paint is applied, on the basis of a shrinkage rate gradient of the paint in a thickness direction of the coating in the process of drying the paint. 
     With the above aspect of the invention, the orientation of the pigments when a paint is applied is set, and the orientation of the pigments at time at which a predetermined period of time has elapsed from when the paint is applied is calculated on the basis of a shrinkage rate gradient of the paint in a thickness direction of the coating in the process of drying the paint. Thus, for example, by assuming the state where the pigments are dispersed in the paint as the state where spheroids are present in viscous fluid, and by applying the motion model in which the spheroids present in the fluid rotate by a difference in velocity of flow of the fluid, it is possible to accurately and easily estimate the orientation of the pigments depending on the characteristic of a paint and a painting condition. 
     A second aspect of the invention relates to a pigment orientation estimating method that estimates the orientation of pigments in a coating. The method includes: a first step of setting the characteristic of a paint and a painting condition; a second step of setting the orientation of the pigments when the paint is applied and setting zero for time at which the paint is applied; a third step of advancing the time by one unit time; and a fourth step of calculating the orientation of the pigments at the time that is advanced by one unit time on the basis of a shrinkage rate gradient of the paint in a thickness direction of the coating in the process of drying the paint. 
     With the above aspect of the invention, the characteristic of a paint and a painting condition are set, the orientation of the pigments when a paint is applied is set, and the orientation of the pigments at time that is advanced by one unit time is calculated on the basis of a shrinkage rate gradient of the paint in a thickness direction of the coating in the process of drying the paint. Thus, for example, by assuming the state where the pigments are dispersed in the paint as the state where spheroids are present in viscous fluid, and by applying the motion model in which the spheroids present in the fluid rotate by a difference in velocity of flow of the fluid, it is possible to accurately and easily estimate the orientation of pigments at the time that is advanced by one unit time and depending on the characteristic of a paint and a painting condition. 
     The pigment orientation estimating method according to the above second aspect may further include a fifth step of determining whether the coating has been dried, subsequent to the fourth step, wherein a process may return to the third step when it is determined in the fifth step that the coating has not been dried, and the third step and the fourth step may be repeated until it is determined in the fifth step that the coating has been dried. 
     By so doing, it is possible to calculate the orientation of the pigments in the dried coating. Particularly, the third step and the fourth step are repeated until it is determined in the fifth step that the coating has been dried, so the orientation of the pigments may be calculated in consideration of a situation, such as the viscosity of the paint and the positions of the pigments, that varies momentarily from when the paint is applied to when the coating has been dried. Therefore, it is possible to accurately estimate the orientation of the pigments in the dried coating. 
     The shrinkage rate gradient of the coating may be calculated on the basis of the characteristic of the paint and a painting condition. 
     Then, the orientation of the pigments may be calculated using a mathematical expression that determines the rotating state of a spheroid in viscous fluid. 
     By so doing, the orientation of the pigments is calculated using a mathematical expression that determines the rotating state of a spheroid in viscous fluid. Thus, it is possible to calculate a variation in rotation angle of each pigment depending on a shrinkage of the coating in the thickness direction of the coating. Hence, it is possible to accurately and easily estimate the orientation of the pigments. 
     According to the aspects of the invention, by assuming the state where the pigments are dispersed in the paint as the state where spheroids are present in viscous fluid, and by applying the motion model in which the spheroids present in the fluid rotate by a difference in velocity of flow of the fluid, it is possible to accurately and easily estimate the orientation of the pigments depending on the characteristic of a paint and a painting condition. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing and further objects, features and advantages of the invention will become apparent from the following description of example embodiments with reference to the accompanying drawings, wherein like numerals are used to represent like elements and wherein: 
         FIG. 1  is a flowchart that illustrates a pigment orientation estimating method according to an embodiment of the invention; 
         FIG. 2A  and  FIG. 2B  are views that illustrate a change of orientation of pigments due to drying shrinkage of a coating; 
         FIG. 3  is a vertical cross-sectional view that illustrates force acting on a spheroid in viscous fluid; 
         FIG. 4  is a view that illustrates the shrinkage rate gradient of a coating in the thickness direction; and 
         FIG. 5  is a graph that shows the data of pigment orientation when a paint is applied and the data of pigment orientation after the coating is dried, estimated through the pigment orientation estimating method. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     As described above, there is a need for understanding how a coating appearance varies when the characteristic of a paint and/or a painting condition are changed. Then, within the coating appearance, the orientation of pigments is the most important. 
     For example, in the case of aluminum pigments, the more the orientation of the pigments is aligned parallel to a painted surface, the better the appearance of metallic color is. Therefore, by estimating the orientation of pigments on a computer, it is possible to omit the time and effort for testing the appearance of an actually prepared painted plate. The pigment orientation estimating method according to the present embodiment is implemented by executing a program for estimating the orientation of pigments on a computer. 
       FIG. 1  is a flowchart that illustrates a pigment orientation estimating method according to an embodiment of the invention. First, in step S 101 , parameters of the characteristic of a paint, a painting condition, and the like, are set (first step). Then, in step S 102 , a state at the time when a paint is applied (coating initial state) is set, and the time t at which a paint is applied is set at zero (t=0) (second step). 
     Table 1 is a listing of an example of parameters, and the like, set in step S 101  and step S 102 . For example, (1) reflective material size range, (2) reflective material size (m), (5) the amount of paint per unit area, (8) solvent specific gravity, (11) reflective material mean diameter (μm), and the like, are set as parameters. Then, (3) orientation angle range (from 1), (4) orientation angles (deg) of pigments, and the like, indicating the orientation of pigments when a paint is applied are set as the paint initial state. 
     
       
         
           
               
               
             
               
                   
                 TABLE 1 
               
             
            
               
                   
                   
               
               
                   
                 Associated Functions 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                   
                   
                 Reflective 
                   
                   
                 Presence 
               
               
                   
                   
                   
                 Material 
                   
                   
                 or Absence 
               
               
                   
                   
                 Coating 
                 High 
                   
                 Incomplete 
                 of 
               
               
                   
                 Input Items 
                 Shrinkage 
                 Density 
                 Leveling 
                 Leveling 
                 Convection 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 0 
                 Version 
                   
                   
                   
                   
                   
               
               
                 1 
                 Reflective Material Size 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 Range 
               
               
                 2 
                 Reflective Material Size 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 (μm), Indicating Size 
               
               
                   
                 Section Range 
               
               
                   
                 Size Range + 1 
               
               
                 3 
                 Orientation Angle Range 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 (From 1) 
               
               
                 4 
                 Orientation Angles (deg), 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 Angle Section Range 
               
               
                 5 
                 Paint Amount (Weight 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 Of Paint Used) Per Unit 
               
               
                   
                 Area (g/m 2 ) 
               
               
                 6 
                 Paint Weight Residual 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 Percentage (NV to 1) in 
               
               
                   
                 Spraying Process 
               
               
                   
                 1 When No Volatilization 
               
               
                 7 
                 Solid Content NV (0 to 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 1) 
               
               
                 8 
                 Solvent Specific Gravity 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 (Density of Solvent 
               
               
                   
                 100%) (kg/m 3 ) 
               
               
                 9 
                 Dried Thickness (μm) 13 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 μm 
               
               
                 10 
                 Time Required for 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 Drying (s) 
               
               
                 11 
                 Aluminum Foil NJ 
                 ◯ 
                 ◯ 
                 ◯ 
                 ◯ 
               
               
                   
                 Reflective Material 
               
               
                   
                 Mean Diameter (μm) 
               
               
                   
                 About 14 μm (up to This 
               
               
                   
                 Item Indispensable) 
               
               
                 12 
                 Reflective Material 
                   
                 ◯ 
               
               
                   
                 Diameter Distribution 
               
               
                   
                 Logarithmic Normal 
               
               
                   
                 Distribution (Currently 
               
               
                   
                 Not Used) 
               
               
                   
                 Effective Flag of 
               
               
                   
                 Shrinkage of Coating 
               
               
                   
                 (Density of Reflective 
               
               
                   
                 Material Is High) 
               
               
                 13 
                 (Following Three Items 
                   
                 ◯ 
               
               
                   
                 Are Required When 
               
               
                   
                 Density of Reflective 
               
               
                   
                 Material Is High) 
               
               
                   
                 Weight Ratio of 
               
               
                   
                 Reflective Material to All 
               
               
                   
                 Solid Content (PWC) 
               
               
                 14 
                 (0 to 1) (Required 
                 Δ 
                 ◯ 
               
               
                   
                 When Reflective 
               
               
                   
                 Material Density Is 
               
               
                   
                 Derived As Well) 
               
               
                   
                 Reflective Material 
               
               
                   
                 Thickness μm About 0.5 
               
               
                   
                 μm 
               
               
                 15 
                 (Required for Reflective 
                   
                   
                 ◯ 
                 ◯ 
               
               
                   
                 Material High Density 
               
               
                   
                 Calculation, Optionally 
               
               
                   
                 for Shrinkage of 
               
               
                   
                 Coating) 
               
               
                   
                 Wave Amplitude during 
               
               
                   
                 Painting, Ratio to Mean 
               
               
                   
                 Thickness (0 to 1) 
               
               
                 16 
                 (This Item &gt; 0 When 
                   
                   
                   
                 ◯ 
               
               
                   
                 There Is Leveling) 
               
               
                 17 
                 Effective Flag of 
                   
                   
                   
                 ◯ 
               
               
                   
                 Incomplete Leveling 
               
               
                 18 
                 Wave Period during 
                   
                   
                   
                 ◯ 
               
               
                   
                 Painting (mm) 
               
               
                 19 
                 Paint Surface Tension 
                   
                   
                   
                 ◯ 
               
               
                   
                 (N/m) 
               
               
                 20 
                 Time Range (Viscosity 
                   
                   
                   
                 ◯ 
                 ◯ 
               
               
                   
                 Sampling) 
               
               
                 21 
                 Time (s), Starting from 0 
                   
                   
                   
                 ◯ 
               
               
                   
                 (Viscosity Sampling) 
               
               
                 22 
                 Viscosity (Pa · s), Value at 
                   
                   
                   
                 ◯ 
                 ◯ 
               
               
                   
                 Low Shear Rate 
               
               
                 23 
                 Effective Flag of 
                   
                   
                   
                   
                 ◯ 
               
               
                   
                 Presence or Absence of 
               
               
                   
                 Convection 
               
               
                 24 
                 Thermal Expansion 
                   
                   
                   
                   
                 ◯ 
               
               
                   
                 Coefficient of Paint 
               
               
                 25 
                 Thermal Conductivity 
                   
                   
                   
                   
                 ◯ 
               
               
                   
                 (W/m/K) of Paint 
               
               
                 26 
                 Specific Heat Capacity at 
                   
                   
                   
                   
                 ◯ 
               
               
                   
                 Constant Pressure 
               
               
                   
                 (J/kg/K) of Paint 
               
               
                 27 
                 Baking Temperature - 
                   
                   
                   
                   
                 ◯ 
               
               
                   
                 Room Temperature (K) 
               
               
                   
               
            
           
         
       
     
     Subsequently, in step S 103 , the time t is advanced by one unit time (third step). In the present embodiment, one unit time is set to one second. Then, in step S 104 , the orientation of pigments at the time that is advanced by one unit time is calculated on the basis of the shrinkage rate in the thickness direction in the process of drying a paint (fourth step). Here, the time evolution differential equation of the pigment orientation, which will be described later, is used to calculate the orientation of pigments. 
     Then, in step S 105 , it is determined whether the coating has been dried (fifth step). In the present embodiment, the above determination is made on the basis of a period of time elapsed. When a predetermined period of time has elapsed from application of a paint (YES in step S 105 ), it is determined that the coating has been dried. 
     On the other hand, when the predetermined period of time has not elapsed from application of a paint (NO in step S 105 ), it is determined that the coating has not been dried yet, and the process returns to step S 103  again. After that, until it is determined in step S 105  that the coating has been dried, the process in step S 103  and the process in step S 104  are repeatedly executed. 
     Then, when it is determined in step S 105  that the coating has been dried, the process proceeds to step S 106 . In step S 106 , the orientation of pigments after the coating has been dried is output. 
     Next, the processes in the pigment orientation estimating method will be described in detail.  FIG. 2A  and  FIG. 2B  are schematic views that illustrate a change of orientation of pigments due to drying shrinkage of a coating.  FIG. 2A  is a cross-sectional view that shows the state of a paint immediately after application of a paint.  FIG. 2B  is a cross-sectional view that shows the state of the dried coating. 
     In the process of drying a paint (curing process), as a solvent included in a paint  12  evaporates from the state shown in  FIG. 2A , a coating  10  shrinks, and the thickness of the coating  10  reduces. Then, with a reduction in the thickness of the coating  10 , the orientation of pigments  11  dispersed in the paint varies, and, as shown in  FIG. 2B , the orientation of the pigments  11  varies so as to be parallel to a painted surface Wa of a painted object W. 
     Immediately after application of the paint  12 , as shown in  FIG. 2A , the pigments  11  are dispersed in the paint  12 . If this is assumed that spheroids are present in viscous fluid, a variation in the orientation of the pigments  11  due to a reduction in the thickness of the coating  10  as shown in  FIG. 2A  and  FIG. 2B  may be modeled as the rotational motion of the spheroid  11  due to a difference in velocity of flow of fluid  12 , exerted on portions of the pigments  11 . Note that the spheroid is a solid made by rotating an ellipse, and may be classified into an oblate spheroid, a spherical spheroid, and a prolate spheroid. 
       FIG. 3  is a vertical cross-sectional view that illustrates force acting on the spheroid with a reduction in the thickness of the coating. The spheroid  11  has a major axis  11   a  and a minor axis  11   b . It is assumed that the spheroid  11  is oriented in position such that the inclination angle of the minor axis  11   b  with respect to a direction parallel to the painted surface (x-axis direction in the drawing) is φ and the inclination angle of the minor axis  11   b  with respect to a direction perpendicular to the painted surface (y-axis direction in the drawing) is θ. 
     Then, with a reduction in the thickness of the coating, flow occurs in the paint  12  from an upper layer side in the thickness direction, located adjacent to the coating surface, toward a lower layer side in the thickness direction, located adjacent to the painted surface (in  FIG. 3 , from the upper side toward the lower side). Within the flow in the thickness direction in the paint  12 , flow Vu at the upper layer in the thickness direction is faster than flow VI at the lower layer in the thickness direction (Vu&gt;Vl), and force acts on the spheroid  11  to rotate the spheroid  11  along the painted surface Wa. Thus, the spheroid  11  rotates toward a direction in which the spheroid  11  is parallel to the painted surface Wa. 
     According to Jeffery&#39;s theoretical formula, the rotational motion of the spheroid in fluid may be expressed by the mathematical expression (1). 
     
       
         
           
             
               
                 
                   
                     
                        
                       ϕ 
                     
                     
                        
                       t 
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           γ 
                           . 
                         
                         2 
                       
                     
                      
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             
                               
                                 r 
                                 2 
                               
                               - 
                               1 
                             
                             
                               
                                 r 
                                 2 
                               
                               + 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 
                                   sin 
                                   2 
                                 
                                  
                                 ϕ 
                               
                               - 
                               
                                 
                                   cos 
                                   2 
                                 
                                  
                                 ϕ 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where
 
φ is an angle made between the direction of the axis of the spheroid and the direction along the painted surface,
 
r is an aspect ratio (major axis/minor axis) of the spheroid (in the case of an oblate spheroid, the spheroid becomes a complete disk at the limit of the major axis&gt;&gt;the minor axis), and
 
γ is a shear rate (shrinkage rate gradient).
 
     Here, the shear rate γ (shrinkage rate gradient) may be obtained under the following concept. 
       FIG. 4  is a view that illustrates the shrinkage rate gradient of a coating in the thickness direction. Where a shrinkage of the coating  10  in the thickness direction is Δd and a period of time until the coating  10  is cured is Tdry, the shrinkage rate in the thickness direction is Δd/Tdry if it is regarded as invariable. Then, the shrinkage rate gradient is Δd/Tdry/(d(t)tan φ) as shown in  FIG. 4 . Here, the thickness d(t) is expressed by d(t)=d 0 −Δd/Tdry×t. 
     The pigment orientation is a probability P(θ) of occurrence of an orientation angle θ, and is expressed by Gaussian distribution (for example, see  FIG. 5 ). A variation over time in the probability P(θ) of the pigment orientation is expressed by the time evolution differential equation shown in the following mathematical expression (2). 
     
       
         
           
             
               
                 
                   
                     
                        
                       P 
                     
                     
                        
                       t 
                     
                   
                   = 
                   
                     
                       
                         
                           ∂ 
                           P 
                         
                         
                           ∂ 
                           θ 
                         
                       
                        
                       
                         
                           ∂ 
                           θ 
                         
                         
                           ∂ 
                           t 
                         
                       
                     
                     + 
                     
                       P 
                        
                       
                         
                           ∂ 
                           
                             θ 
                             . 
                           
                         
                         
                           ∂ 
                           θ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Note that θ indicates a state where the pigment  11  is laid parallel to the painted surface Wa when the angle made between the minor axis  11   b  direction of the spheroid  11  and the direction perpendicular to the painted surface Wa (y-axis direction). The left-hand side of the above mathematical expression (2) indicates a variation in the probability P per unit time, and the second term in the right-hand side is a term for maintaining the area of Gaussian distribution at constant. As Gaussian distribution expands or contracts laterally, the height of distribution varies. 
     Then, the first term ∂θ/∂t in the right-hand side indicates a variation in scale θ of the probability P(θ), and is obtained by reversing the sign of a variation over time of the orientation angle θ calculated by Jeffery&#39;s equation. When the time evolution differential equation of the pigment orientation distribution, shown in the mathematical expression (2), is solved, the pigment orientation after the coating has been dried is obtained. In the present embodiment, the above differential equation is solved by Runge-Kutta method. 
       FIG. 5  is a graph that shows the data of pigment orientation when the paint is applied and the data of pigment orientation after the coating is dried, estimated through the pigment orientation estimating method. The thickness d of the dried coating  10  is set at about one third of the thickness at the time when the paint is applied (43 μm→13 μm). According to the results of calculation, the half-width of the orientation distribution changes from 30 degrees at the time when the paint is applied into 9 degrees after the coating has been dried, so the orientation is improved. 
     With the above described pigment orientation estimating method, the characteristic of the paint  12  and the painting condition are set, the orientation of the pigments  11  when the paint  12  is applied is set, and the orientation of the pigments at time t that is advanced by one unit time is calculated on the basis of the shrinkage rate gradient in the thickness direction in the process of drying the paint  12 . Thus, by assuming the state where the pigments  11  are dispersed in the paint  12  as the state where spheroids are present in viscous fluid, and applying the motion model in which the spheroids present in the fluid rotate by a difference in velocity of flow of the fluid, it is possible to accurately and easily estimate the orientation of the pigments  11  at time that is advanced by one unit time and depending on the characteristic of the paint  12  and the painting condition. 
     Then, the processes in step S 103  and step S 104  are repeated until it is determined in step S 105  that the coating  10  has been dried, so the orientation of the pigments  11  may be calculated in consideration of a situation, such as the viscosity of the paint  12  and the positions of the pigments  11 , that varies momentarily from when the paint is applied to when the coating has been dried. Therefore, it is possible to accurately estimate the orientation of the pigments  11  in the dried coating. 
     In the above described embodiment, the pigment orientation estimating method is implemented by a computer program, for example; however, the pigment orientation estimating method is not limited to this configuration. For example, the pigment orientation estimating method may be implemented by a combination of software and hardware or may be implemented by a device. 
     In addition, the aspect of the invention is not limited to the above described embodiment; the aspect of the invention may be modified in various forms without departing from the spirit of the invention. For example, in the above described embodiment, calculation is made from when the paint is applied to when the coating has been dried, for example; however, a period of time during which calculation is made is not limited to the period of time from when the paint is applied to when the coating has been dried. Instead, the period of time may be part of that period of time. 
     In addition, in the above described embodiment, the orientation of pigments is calculated every unit time, for example. Instead, as long as a period of time, or the like, during which, for example, the viscosity, or the like, of a coating varies proportionally to an elapsed time from when a paint is applied, the orientation of pigments at time at which a predetermined period of time has elapsed from when the paint is applied may be calculated.