Mechanism of the Action of Charge Control Agents(CCA)III:
A Tribocharging Model for Two-Component Developers.The Relations of Toner-to-Carrier Ratio (m_t/m_c=C_t) to Carrier Charge-to-Mass Ratio (q_e^c) and to Charging-Rate Constants (k_obsd)

Abstract

A model for two-component developers, where the charging process is an equilibrium of the transfer of charge between a toner and a carrier, was refined, giving the relations of the toner-carrier mass ratio (m_t/m_c=C_t) to the carrier charge-to-mass ratio (q_e^c) and to the apparent tribocharging-rate constants (k_obsd):
   1/q_e^c=-1/q_lim^t[(k_-1s_t/k₁s_c)+1/C_t]  and  k_obsd=k₁[1+(k_-1s_t/k₁s_c)C_t],
where q_lim^t is the limiting charge (the tribocharge when no charge transfers to the toner from the carrier, k_-1=0), s_t and s_c are specific surface areas of the toner and the carrier, respectively, k_obsd is the apparent tribocharging-rate constant, and k₁ and k_-1 are pseudo-first order rate constants of the transfer of charge from the toner to the carrier and from the carrier to the toner, respectively.
These relationships were confirmed to apply in our experiment using a model toner: The values of q_lim^t and k₁s_c/k_-1s_tobtained from the two equations were in good agreement with those obtained previously from the relationship between the toner charge-to-mass ratio (q_e^t) and m_t/m_c=C_t. Furthermore, the 1.0% CCA toner-to-0.5% CCA toner ratios in k₁ and k_-1 were not only comparable with each other but also with the CCA weight ratio and with the ratio of the frequency factors, Z, for the two toners estimated from the Arrhenius equation. These results show that our model consistently joins the dynamic and the static states and is valid.