Relationship Between Ink Quantity and Density Based on Nonlinear Regression Analysis

Abstract

Whether the amount of ink is suitable or not greatly affects the color reproduction of printed matter. In order to improve the quality of the printed matter, the mathematical relationship between the ink thickness and field density is obtained by Lambert-Beer law and a mathematical model is established accordingly. The traditional offset printing process is simulated by a printability tester, and nonlinear regression method and least square method are used to solve the parameters D_∞ and m to get the fitting curve. Finally, the effect of the fitted curve is evaluated according to the size of the coefficient of determination R². That is the closer R² is to 1, the better the fitting effect.

1 Introduction

In offset printing, the amount of ink has a great influence on the color reproduction process of the printed matter. When the amount of ink changes (the thickness of the ink layer changes), the color of the printed matter will change accordingly. The ink layer thickness refers to the average thickness of the ink layer attached to the paper surface in the vertical direction with the ink layer paper [1]. Low ink supply, thin ink layer, dim ink gloss, printing quality is not good. More ink supply, thicker ink layer and longer drying time will result in serious dot enlargement, dirty back, ink bar, paste and other phenomena, which will affect the quality of printed matter [2, 3]. Therefore, the control of the amount of ink is particularly important. In practice, by measuring the density of the ground, the amount of ink is determined, and then the amount of ink is adjusted to ensure the ideal reproduction of the image color and the quality of the print.

In this paper, the corresponding values of ink layer thickness and field density are obtained through experiments, and then their mathematical expressions are obtained. Then, the accuracy of the mathematical model established by MATLAB curve fitting analysis is used to guide production practice.

2 Analysis of the Relation Between Ink Thickness and Field Density

Practice has shown that within a certain range, the printed solid density increases with the increase of the thickness of the ink layer. When the thickness of the ink layer increases to a certain value, the solid density does not continue to increase, and a maximum density value, that is, the saturation density value is generated [4], as shown in Fig. 1. It can be seen from the figure that when the amount of ink is low, the ink layer is thin, the color is light, and the absorption light is less, so the reflectance is high and the density value is small; when the amount of ink is high, the ink layer is thick, the color is deep, and the amount of absorbed light is large. Therefore, the reflectance is low and the density value is large [5, 6]. The Lambert-Beer’s law is usually used to describe the relationship between the density of the field and the thickness of the ink layer. The expression is as follows:

Fig. 1.

Relationship between ink layer thickness and field density

$$ \text{D} = D_{\infty } (1 - \text{e}^{ - m \cdot l} ) $$

(1)

D is the solid density, D_∞ is the saturation density, m is the parameter related to the printing paper, and l is the thickness of the ink layer. Among them, D_∞ and m are obtained by experimental methods.

3 Experimental Part

3.1 Experimental Materials and Instruments

Printing material: SKYEYLION offset light fast dry blue ink 128 g double-sided coated paper. Printing instrument: ACT2-5IGT Printability meter Electronic balance Precision ink injector X-Rite Spectrodensitometer.

3.2 Experimental Procedure

• (1) Calculating Ink Density

The uniform ink was injected into the ink injector, the ink quality M₁ was measured with an electronic balance, and then the ink amount of 0.01 ml was extruded, and the mass at this time M₂ was measured. It is known that the ink quality is $\Delta \text{M} = M_{1} - M_{2}$ and the ink volume is $\text{V} = 0.01\,\text{ml}$, so the ink density $\uprho = 0.95\,\text{g}/\text{cm}^{3}$ is calculated by the formula $\uprho = \Delta \text{M}/\text{V}.$

• (2) Uniform Ink and Proofing

In this experiment, 18 groups of numerical values were selected in the range of 0.01–0.3 ml to homogenize the ink. After homogenization, color samples were printed at 625 N printing pressure and 0.2 m/s printing speed on the printability tester.

• (3) Calculating Ink Thickness

Measuring the quality m₁ of ink roll after inking with electronic balance and the quality m₂ of ink roll after printing, the ink consumption transferred to paper is $\Delta \text{m} = m_{1} - m_{2}$. The length and width of ink imprint on paper were measured and the imprint area s was calculated. Then, the ink thickness L_C was calculated by formula ${\text{d}} = \Delta {\text{m}}/\left( {\rho \cdot {\text{s}}} \right).$

• (4) Measuring Field Density

After the ink on the spline is dried, the solid density value D_C is measured by a spectrodensitometer in a portion where the amount of ink is uniform. Then establish the one-to-one relationship between the two, and select some data to form a table, such as Table 1, and finally draw a scatter map using 18 sets of data. As shown in Fig. 2.

Table 1. Some data of D_c and l_c

Fig. 2.

Scatter plot of field density as a function of ink layer thickness

The regression analysis was performed using the data in the table to determine $D_{\infty }$ and m in this experimental environment. And the relationship between ink layer thickness and field density is plotted by MATLAB, as shown in Fig. 3. From this, the relationship of the model is obtained.

Fig. 3.

Relationship between ink layer thickness and field density and fitting curve

$$ \text{D} = 2.0689\,(1 - \text{e}^{{( - 1.0723 \cdot \text{l})}} ) $$

(2)

3.3 Model Checking

The coefficient of determination R² is also called the goodness of fit or the correlation index, and the closer to 1 indicates the better the fitting effect. The calculation formula is:

$$ R^{2} = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{18} \left( {D_{i} - \overline{{D_{i} }} } \right)^{2} }}{{\mathop \sum \nolimits_{i = 1}^{18} \left( {D_{i} - \overline{{D_{i} }} } \right)^{2} }} $$

(3)

According to the data in the table and the fitting curve, R² = 0.8184 is calculated, which approaches 1. This shows that the fitting curve and the sample value have higher goodness of fit, and the model is more accurate, which accords with the relationship between ink layer thickness and field density under this condition. However, the mathematical model with high accuracy will also have errors. We analyze the reasons for the errors. There are the following points.

• 1. The acquisition of the experimental data is obtained by simulating the actual printing process on the printing suitability meter, and the printing suitability meter does not represent the actual working condition of the printing machine, which is the main cause of the errors.

• 2. The measurement of ink volume is not accurate due to air bubbles in ink injector when calculating ink density, and the quality of ink is measured by electronic balance, so there are inevitable errors in the measurement process.

• 3. There are human errors in density measurement and other calculations.

4 Conclusions

Since there are many factors affecting the ink density in the actual printing process, a mathematical model is established as much as possible for the specific printing conditions, so that it can better reflect the relationship between the amount of ink and the density. In this paper, the mathematical model established obtains the density law equation of the ink used in the laboratory, and through verification, the fitting value is in good agreement with the experimental value, which provides theoretical guidance for production practice. Therefore, the effective reduction of the color and the level can be achieved by effectively controlling the amount of ink to ensure the stability of the print quality. In the actual printing process, we can change the mathematical model according to the specific printing conditions, as long as the model is generated, the amount of ink can be effectively controlled.