Discrete Ink Model and Ink Flow

The black ink in Chinese ink painting is a dilute mixture of water and colloidal carbon particles. The carbon particles are much smaller than those of watercolor paints such that they can diffuse into paper along with the absorbed water. Besides the capillarity, the forces acting on the motion of ink involve the interaction between waters and waters, waters and carbons and the gravity, etc. To simulate chaotic and complex motion of ink in fibers, we separate the ink into two kinds of particles: water particles and carbon particles. [26]

Water Particle

In Huang’s [13] definition of water, the water volume is divided into particles for the characteristic of discrete-computing ability of computers. All of the water particles are defined as objects that have uniform volume and uniform mass. The only different property of the water particles is their positions, which recorded the index of the papel they are at.

With the definition of water particles, we can decide the movement of these particles on paper. When the water in a certain papel is decided to flow out, the quantity and direction of the water to flow out are the two characters must be figured out. Based on some physical knowledge and hypothesis, the approximate equation of K(p), the ratio of the quantity of water to flow out to the quantity of water contained in the papel p is represented as

( )

(

)

)

(p Base Var Absorbency p

K = + × − − , (6)

where Base is a real number between zero and one to represent the basic flow rate of p, and

Var is a real number between zero and one to represent the difference between the highest flow rate and the lowest flow rate.

The reason for the ability of ink diffused in paper is the capillarity only on water particles, not on carbon particles. Based on this phenomenon, we have to decide the direction of the diffused water flowing out, which is decided by predicting the probabilities of the water particles. This method will be discussed in section 4.2.4.

Carbon Particle

Carbon particles don’t move like water particles. Instead, if the force produced by the energy of the movements of the water particles is larger than the force produced by the friction of the carbon particles, the carbon particles move with the water particles. According to Huang’s [13] definition, carbon particles have some attributes such as mass, position, diameter and color. But all of these attributes are not uniform, which is different from water particles. The diameter and mass of a carbon particle is decided according to the fineness when grinding the ink initially. If the initial ink grinding is coarse, the ink contains small and large particles which produce an observable change of color intensity along the border line of the initial brush area. On the other hand, if the carbon particles are homogeneous, small, and uniform, most carbon particles move with water unhindered by the fibers, such that a continuous and smooth intensity change appears across the diffusion area. This occurs because only sufficiently small carbon particles can seep into the fiber mesh and flow along with water particles, large carbon particles are left without flowing away.

Under the influence of the motion of water particles, suspended carbon particles move in a manner called Brownian motion [21]. According to this physical mechanism, the quantity of carbon flow is proportional to the quantity of water flow. A phenomenon occurs because only

those carbon particles whose granule size is smaller than the space between fibers can seep into the mesh along with water. Particles whose granule size is bigger than the space remain at the initial position, as shown in Figure 4.8. This phenomenon is referred as a “filtering effect”

[11, 16] of the fiber mesh, and can be represented as the following, where p is the papel the carbon particle lays in and Hole_Diameter is defined as the average length of diameter of the holes between fibers in the papel.

if ( Carbon_Diameter > Hole_Diameter(p)) then Carbon_Position Å p

else

Carbon_Position Å Water_OutFlow_Direction(p),

(a)

(b) Two adjacent cubes represent two neighbor papels. Black grains in papels are carbon particles in different sizes. The chaos strings on the face between two papels represent fibers. The arrow represents the direction of water flow which carbon particles are moving along with.

(c) Larger diameter of carbon particles can not pass the holes between fibers results in a filtering effect and be left in the original papel.

Figure 4.8 An illustration to explain the phenomenon called “filter effect” (Courtesy of Huang).

Besides the diameter-filtering mechanism, the mass-filtering mechanism is proposed.

Suppose is the velocity of the carbon c suspended in the water in papel p, and is the quantity of out-flowing water in p. The relationship between , and the diameter of

holes in p, Hole_Diameter(p), is described as

Vc W_p

If the carbon c is too heavy to moved out from papel p and then deposited in p, then . On account of this physical phenomenon [21], we define an upper-bounded eshold T_p for papel p, to decide whether the carbon particle can move out or not. If the mass of carbon c is larg than T 0