Method of predicting optical properties and physical characteristics to formulate optimum coating system
Method of predicting optical properties and physical characteristics of coated paper or other sheet products for the purpose of formulating enhanced coating or sheet forming systems, and in particular to a method, which uses a diffusion approximation model derived from a radiative transfer theory employing various inputs of the properties and/or characteristics to predict coating or sheet system performance.
1. A method of providing at least one optical property of a coating system having a coating adjacent a base sheet comprising:
a) providing a radiative transfer equation that relates a number of physical characteristics of the coating system selected from the group consisting of coating thickness, coating pore size distribution, coating pigment particle size, a coating pigment particle size distribution, a coating roughness, scattering coefficient, absorption coefficient, anisotropy factor, and optical properties of a base sheet to one or more optical properties for a coating system selected from the group consisting of gloss, brightness, and opacity, wherein providing the radiative transfer equation comprises application of diffusion approximation to transport of light, said diffusion approximation defining average radiance U(r,t) as follows:
D∇ 2 U ( r,t )−μ a U ( r,t )−∂ U ( r,t )/∂ t=S ( r,t ),
wherein isotropic source density is denoted by S(r,t), r and t are the variables of position and time, and D is the diffusion coefficient which is defined in units of length as
D=1/3[μ a +μ s (1−g)] in terms of the absorption and scattering coefficients μ a and μ s , respectively, and g is the anisotropy factor;
b) solving the equation by inputting a number of the physical characteristics of the coating system to obtain at least one optical property output of the coating system; and
c) formulating a pigment coating including the obtained at least one optical property output.
2. The method of claim 1 , further comprising:
(i) selecting at least one target optical property,
(ii) comparing the at least one optical property output to the at least one target optical property, adjusting one or more of the physical characteristics and repeating step (b) to obtain another optical property output, and
(iii) repeating comparing step (ii) until the optical property output corresponds generally to the target optical property.
3. The method of claim 1 , wherein the providing of the radiative transfer equation comprises a) using a coating system including a pigment selected from the group consisting of clay and calcium carbonate, and combinations thereof, and b) using a base sheet comprising a paper product.
4. The method of claim 1 , further comprising coating the pigment coating upon a paper base sheet effective to form a coated paper.
5. The method of claim 1 , wherein the solving the equation by inputting a number of the physical characteristics of the coating system obtains the optical properties including brightness, gloss, and opacity.
6. The method of claim 1 , comprising the solving step utilizing a computer.
7. The method of claim 1 , comprising inputting at least the scattering coefficient, the absorption coefficient, and the anisotropy factor to obtain a brightness output of the coating system.
8. The method of claim 1 , further comprising coating the pigment coating upon a base sheet effective to form a coating thereon.
9. The method of claim 1 , further comprising coating the pigment coating upon a base sheet comprising a paper product effective to form a coating thereon.
10. The method of claim 1 , wherein the formulating of the pigment coating comprises including pigment selected from the group consisting of clay and calcium carbonate, and combinations thereof.
11. The method of claim 1 , wherein solving the equation further comprises determining diffuse reflectance, R(Q), using the mathematical expression:
R
(
Q
)
=
3
Q
(
1
+
Q
+
Qg
)
(
Q
3
-
3
g
+
3
Q
)
-
1/2
-
3
[
(
1
+
Q
)
2
+
g
(
-
1
+
Q
+
Q
2
)
]
(
-
1
+
g
-
Q
)
(
3
+
2
(
3
Q
1
-
g
+
Q
)
1/2
)
;
Q
=
μ
a
μ
s
.
12. A method of predicting at least one physical characteristic of a coating system having a pigment coating adjacent a base sheet comprising:
a) providing a radiative transfer equation that relates a number of physical characteristics of the coating system selected from the group consisting of coating thickness, coating pore size distribution, a coating pigment particle size distribution, a coating roughness, and optical properties of a base sheet to one or more optical properties for a coating system selected from the group consisting of gloss, brightness, and opacity, wherein providing the radiative transfer equation comprises application of diffusion approximation to transport of light, said diffusion approximation defining average radiance U(r,t) as follows:
D∇ 2 U ( r,t )−μ a U ( r,t )−∂ U ( r,t )/∂ t=S ( r,t ),
wherein isotropic source density is denoted by S(r,t), r and t are the variables of position and time, and D is the diffusion coefficient which is defined in units of length as
D=1/3[μ a +μ s (1−g)] in terms of the absorption and scattering coefficients μ a and μ s , respectively, and g is the anisotropy factor; and
b) solving the equation by inputting a number of the physical characteristics and at least one optical property to obtain the at least one physical characteristic output based on a coated system employing the inputted physical characteristics and the at least one optical property; and
c) formulating a pigment coating including the at least one physical characteristic output.
13. The method of claim 12 , further comprising:
(i) selecting a target optical property;
(ii) using the target optical property in step (b) as the at least one optical property to obtain at least one physical characteristic output.
14. The method of claim 12 , comprising the physical characteristic output corresponding to coating pore size distribution.
15. The method of claim 12 , comprising using a computer for performing the solving step.
16. The method of claim 12 , further comprising coating the pigment coating upon a paper base sheet effective to form a coated paper.
17. The method of claim 12 , comprising the optical property of the base sheet being one of brightness, gloss, or opacity.
18. The method of claim 12 , wherein the providing of the radiative transfer equation comprises using a coating system including a pigment selected from the group consisting of clay and calcium carbonate, and combinations thereof, and using a base sheet comprising a paper product.
19. The method of claim 12 , wherein solving the equation further comprises determining diffuse reflectance, R(Q), using the mathematical expression:
R
(
Q
)
=
3
Q
(
1
+
Q
+
Qg
)
(
Q
3
-
3
g
+
3
Q
)
-1/2
-
3
[
(
1
+
Q
)
2
+
g
(
-
1
+
Q
+
Q
2
)
]
(
-
1
+
g
-
Q
)
(
3
+
2
(
3
Q
1
-
g
+
Q
)
1/2
)
;
Q
=
μ
a
μ
s
.
20. A method of predicting at least one physical characteristic of a coating system having a coating adjacently contacting a substrate comprising:
a) providing a radiative transfer equation that relates a number of physical characteristics of the coating system selected from the group consisting of coating thickness, coating pore size, a coating pigment particle size distribution, a coating roughness, and optical properties of a substrate comprising metal or plastic to one or more optical properties for a coating system selected from the group consisting of gloss, brightness, and opacity, wherein providing the radiative transfer equation comprises application of diffusion approximation to transport of light, said diffusion approximation defining average radiance U(r,t) as follows:
D∇ 2 U ( r,t )−μ a U ( r,t )−∂ U ( r,t )/∂ t=S ( r,t ),
wherein isotropic source density is denoted by S(r/t), r and t are the variables of position and time, and D is the diffusion coefficient which is defined in units of length as
D=1/3[μ a +μ s (1−g)] in terms of the absorption and scattering coefficients μ a and μ s , respectively, and g is the anisotropy factor; and
b) solving the equation by inputting a number of the physical characteristics and at least one optical property to obtain the at least one physical characteristic output based on a coated system employing the inputted physical characteristics and the at least one optical property; and
c) formulating a pigment coating including the at least one physical characteristic output.
21. The method of claim 20 , further comprising coating the pigment coating upon a substrate effective to form a coating thereon.