DEM Simulations of toner behavior in the development nip of the Océ Direct Imaging print process

I.E.M. Severens

The Océ color technology relies on three innovations. The first is called Direct Imaging (DI). In Direct Imaging, seven DI-units are used to build a seven-color toner image. In each of these DI-units, a bitmap image is transferred directly into a visible toner image on a DI-drum. The second innovation is Océ's Color Copy-Press system, which compiles all partial color images from each DI-drum and transfers and fuses the complete full-color image to the paper. The third innovation is a color version of Océ's image processing technology Image Logic. The print quality of the Direct Imaging technology is primarily determined by the toner flow in the region between the DI-drum and an imaging roller. The collection of toner between the DI-drum and the imaging roller is called the DI toner assembly.

The discrete element method has been employed to get insight in the toner flow in the DI-unit of an Océ Direct Imaging color printer. In the discrete element method (DEM), all toner particles as well as the development rollers, are considered discrete elements. Each element interacts with its neighboring elements and its surroundings, i.e. electric and magnetic fields. These interactions are modeled on a microscopic scale: the motion of each particle is tracked numerically. Every time step, the forces that act on a particle are summed and, from this, the speed and the displacement of the particle are calculated by integration of Newton 's second law of motion. The macroscopic behavior of the toner flow and print output is then simulated using DEM. The model that is developed is a two-dimensional description of the DI toner assembly: a cross-section of the DI-unit in the plane of the ring electrodes of the DI-drum is described. The forces that act on particles in the DI toner assembly are due to collisions, friction, adhesion, and electromagnetic actions.

All toner particles and development rollers have been provided with a geometry to mimic the shape of the real object. To this end, a toner particle is described by clustered spheres. The mechanical roughness of the imaging roller is modeled as a periodic structure of line pieces. More realistic toner geometries can be achieved by increasing the number of clustered spheres that form one toner particle. However, high numbers of clustered spheres puts high demands on computational power. In the model, each collision between toner particles, and between toner particles and the development rollers, is modeled. During a collision, having a certain contact time, particles deform, energy is dissipated in the form of heat, and particles restore to their original shape. In the simulation, a collision is modeled by penetration of the objects during collision. The normal dissipation in a collision is characterized by the coefficient of normal restitution, which is defined as the ratio between the normal component of the relative velocity before and after the collision. The normal coefficient of restitution of toner is determined experimentally by recording a collision between a spherical toner particle with a counter-material (toner, imaging roller, or DI-drum). The collision duration time is estimated from the collision theory of Hertz. Substitution of measured values for the restitution coefficient and collision duration time lead to a strong numerical time step constraint.

Collisions amongst particles or between particles and objects are, in general, not head-on, and the particles have angular velocity. Therefore, shear is also taken into account. The shear contact force is modeled with a Coulomb friction model. Since Coulomb friction is a discontinuous force model, adjustments have been made to the model to avoid numerical instability of the force law in a simulation. The friction coefficient of toner is determined experimentally by moving a toner resin sample over a substrate (toner, imaging roller, or DI-drum) and measuring the force that is required to do this. The motion of the measuring device consists of two parts. In the first part, where the device moves at low speed, a so-called stick-slip motion is observed. The force that is measured is the static friction force. In the second part, the sample moves at a higher speed over the substrate. Then, the force represents the dynamic friction force.

When two materials are brought into each other's vicinity, they exert an attracting force onto each other. This force is referred to as the adhesion force. Hamaker's theory is used to describe this attractive force. The adhesion force is characterized by the Hamaker coefficient, which is a material property. For toner, its value is measured by atomic force microscopy and the centrifugal detachment method. Furthermore, adhesion forces are estimated from the macroscopic behavior of the DI toner assembly, i.e. in the reduction experiment.

In the DI toner assembly, the magnetic field originates from two sources: the field from the magnets within the imaging roller and the field from the magnetized surrounding toner particles. The magnetic field of the magnet within the imaging roller is calculated with the finite element method. The inter-particle magnetic force, also called dipole-dipole force, is calculated as if a single magnetizable particle were present in each toner particle with the same magnetic dipole moment as the whole toner particle. The magnetic properties of color and black toner are determined with a vibrating sample magnetometer.

To enable electric field toner development, the electric force exerted on the toner particle by the externally applied field strength must be stronger than the magnetic force on that particle. Unfortunately, because of their quadratic nature, electric forces cannot be determined by superposition. Thus, complex problems cannot easily be reduced to a sum of more tractable problems. As a result of this, and of the complicated geometries involved, a comprehensive theory of the forces on charged particles does not exist. The use of the Galerkin finite element method to solve the Laplace equation for the field distribution (with the appropriate re-meshing) at every time step is computationally not achievable in a DEM simulation. It has been shown that the local defect correction (LDC) technique can be used to solve Maxwell's quasi-static equations for the electric potential. With the developed method, it is possible to calculate electric fields in a domain that contains holes, here representing toner particles, with a prescribed integral charge condition. The applied LDC method uses a coarse global grid on which a numerical solution is calculated. The global coarse grid solution is iteratively refined with the help of fine local solutions around the holes. In case the holes are separated such that the local domains do not overlap, analytical local solutions can be used for the local boundary problems. In case the local domains overlap, analytic local solutions are not available any more. In that case, the Schwartz alternating process can be applied. If the toner particles move even closer, such that the local domains overlap the holes, also the use of Schwartz's method is not possible any more. Then, the method of finite differences is used to solve the local boundary value problems.

The convergence of the LDC method and the Schwartz alternating process are proven. The application of the method of finite differences is shown for the Laplace equation with the integrated flux conditions. With help of these three types of solution methods for the local boundary problems, the LDC method can be applied to a domain with an arbitrary number of holes. Thus, a general numerical method has been constructed that for dilute systems, in which the convergence of the local problems is more or less independent of the other local solutions, is computationally efficient and that is able to calculate electric forces in DEM simulations accurately. For dense systems, such as the DI toner assembly, also an alternative approach can be chosen: bispherical coordinates can be used to calculate analytically the force on a conducting toner particle in the field of an infinitely large electrode.

The particles of the DI toner assembly that are pressed against the DI-drum, are charged when a voltage difference is applied between the imaging roller and the DI-drum. The charge of the toner particles is the result of a flow of charge from the imaging roller through the bulk of the DI toner assembly to the toner particles, pressed against the DI-drum. An SiO x layer, a dielectric layer above the conducting tracks, ensures that the electric charge on the toner does not leak to the conducting tracks. A necessary condition for the flow of charge to a particle is the existence of a conducting path from that particle to the imaging roller. The conducting path is provided by the conducting toner particles of the DI toner assembly. These particles form chains from the imaging roller to the DI-drum. Because the DI toner assembly is not a static, but a dynamic assembly, the chains that provide the conducting paths are constantly broken up and restored. Due to the charging and decharging of the DI toner assembly, the voltage difference between toner particles and the DI-drum changes in time, and, accordingly, the electric force.

The conducting paths consist of toner-toner contacts and toner-imaging roller contacts. The contact between a toner particle and another toner particle is treated as an ideal electric resistance. Similarly, the contact between a toner particle and the imaging roller is treated as an ideal electric resistance. Toner particles within a certain range of the DI-drum are treated as capacitors with respect to the DI-drum. The contact of a toner particle with the DI-drum is treated as a capacitor in parallel with an ideal resistance. By this routine, a simulation geometry is transferred into an electric circuit. One relevant characteristic of the charging process of the DI toner assembly is the $RC$-time; that is the time during which charging of a toner particle in the vicinity of the DI-drum takes place. Another point is that toner particles cannot be charged or decharged if they are not in conducting contact with the imaging roller, so that charge can only be distributed during collisions of toner particles that form conducting chains to the imaging roller. These characteristics are captured when assuming that the charging of toner particles is due to a flow of charge through the shortest conductive path formed to the imaging roller. This shortest path is calculated for every particle and is translated to a 1-dimensional electric circuit, which determines the total resistance of the conducting path to the printable particle. The equations of the 1-dimensional electric circuit can easily be solved. The number of equations that need to be solved is equal to the number of toner particles that are less than d 50 (the value d n indicates that n% of all toner particles have a volume that is bigger than ?d n 3 /6) away from the DI-drum. The computational costs of this approach are thus limited, so that the method can be applied in a discrete element method simulation.

The set-up, which is used to measure the resistance of the DI toner assembly and the capacity of toner with respect to the DI-drum, consists of a DI-unit with an electrical circuit for measuring currents. The DI-unit that is used has a DI-drum with a built-in measurement probe. When toner is printed on the region of the DI-drum above the probe, a current flows through the DI toner assembly and probe. The model parameters are calibrated by measuring the amplitude and the phase shift of the measured voltage for relevant frequencies of the applied input signal, and by matching the parameter values such that the calculated and measured amplitude and phase shift agree.

With the simulation tool, it is possible to calculate and visualize the behavior of the DI toner assembly, which consists of at most 10,000 particles, for a time frame of about 15ms in approximately one night on a PC. The discrete element method has been used for simulating the behavior of the DI toner assembly in the development nip of the é Direct Imaging print process. It is shown that by determining the appropriate interaction rules and the associated parameters, it is possible to gain qualitative and quantitative agreement between experimental and simulation results. The simulations have shown good agreement with experiments on the front position of the DI toner assembly. Observed quantitative differences can be ascribed to the modeling of toner particles by clustered spheres. In the DI toner assembly, shear between toner particles and the imaging rollers and between toner particles themselves plays an important role. Consequently, geometrical friction and thus toner shape play an important role in modeling the shear characteristics of the DI toner assembly. By clustering three spheres to form one toner particle, it is possible to describe the geometrical roughness of toner to a large extent. However, it also turned out that for describing the full geometry of toner, clustering spheres is not sufficient. If one wants to achieve a full resemblance, other geometry models such as convex particles may be more appropriate; but this will have its effect on computational costs.

The simulations have also been applied for determining print quality measures. Good quantitative predictions have been made with the model on normal edge sharpness. This not only confirms the quality of the model, but also opens opportunities for many applications of the model. The model can, for instance, be used to predict aspects of print quality for deviating settings of the DI-unit. Since it is relatively easy to do predictions with the DEM model for exotic settings of the DI-unit, new ideas can be tested fast and good ideas discovered more easily. The model can also be used as a design tool for fundamentally new ideas that are difficult to test experimentally. Furthermore, the DEM model provides a lot of insight in the relevant processes that take place in the DI toner assembly, and allows people to improve the system on basis of a fundamental insight in the physics playing a role in the image forming process.