Consideration of the Counter Charge Traveling
Distance on the Charge Transport Equilibrium
Mechanism with a Dynamic Procedure
Yoshihiro HATTORI

要旨

Abstract

電子写真の歴史において二成分磁気ブラシ現像法は，

In the history of the electrophotography, two-component

その品質，耐久性，および現像特性の制御性の高さから，

magnetic brush development method has been adopted in

多くのカラー MFP およびプロダクションプリンターで

many color MFPs and production printers because of its qual-

採用されてきた。栗田が提案した電荷輸送平衡メカニズ

ity, durability, and controllability in developing characteristics.

ムは，電子写真の二成分現像法における現像特性の解析

The charge transport equilibrium mechanism proposed by

モデルとして有用であり，多くのMFP設計に適用されてKurita is useful as an analysis model of development character-いる。このモデルでは，カウンターチャージの移動距離

istics in two-component development method of electropho-

を定数として扱い，その値を実験的に推定している。著

tography and has been applied in many MFP designs. In this者は，拡散モデルと電荷移動度モデルを適用することにmodel, the traveling distance of the counter charge is treatedより，カウンターチャージ輸送プロセスの反復計算を試

as a constant, and its value is estimated experimentally. The

みた。各計算モデルにおいて，現像特性は所定時間経過

author tried an iterative calculation of the process of counter

後緩やかな時間依存性を示し，電荷輸送平衡メカニズム

charge transport by applying a diffusion model and an electricが概ね支持されているという結果が得られた。

charge mobility model. In each calculation model, the devel-

拡散モデルは，DC バイアス電圧が印加された現像条

opment characteristics showed a gradual time dependence件に適合した。
「トナー供給制限」の条件を除いて，電荷after a lapse of a predetermined time, and the results obtained

輸送平衡メカニズムが適用可能である。電荷移動度モデthat the charge transport equilibrium mechanism was gener-

ルは，AC バイアス電圧を重畳した現像条件に適合した。

ally supported.

この場合，現像時間が交流電圧の振幅などによって決定

The diffusion model met the developing condition that DC

されるカウンターチャージの通過時間よりも長い場合，

bias voltage was applied. The charge transport equilibrium

電荷輸送平衡メカニズムが適用可能である。これらの結

mechanism can be accepted except the condition under the

果は，経験的ノウハウを解析的理解に置き換え，技術的‘limitation of toner supply’. The electric charge mobility model

知識の伝承に役立てるという価値をもたらす。電子写真

met the developing condition that AC bias voltage was super-

技術の向上のために，更なる応用が期待される。

imposed. In this case the charge transport equilibrium mechanism can be accepted if the developing time is longer than
the transit time of counter charge which is determined byamplitude of AC voltage and so on. These results provide thevalue of replacing empirical know-how with analytical understanding and helping to transfer technical knowledge. Further
applications are expected for improving electrophotographic
technology.

＊ EP Process R & D Division 1, EP Image Engineering Development Center, R & D Headquarters Business Technologies

38

KONICA MINOLTA TECHNOLOGY REPORT VOL. 19 (2022)

Fig. 2 shows comparison between experimental results

1 Introduction

and the calculated results reported in the author’s pre-In the history of the electrophotography, two-com-

vious paper [2]. Equation (1) and Equation (2) can meetponent magnetic brush development method has

good correlation with experimental results by estimat-

been adopted in many color MFPs and production

ing some constant value on the term including ‘counter

printers because of its quality, durability, and control-charge’. However, there is no suggestion in these mod-lability in developing characteristics. Several impor-

els why or how the counter charge is driven to the bal-tant theories and models helped in designing devel-

anced position. Okada [3] suggested a useful reference

oping systems and controlling algorisms for image

as a mathematical solution for both IMB (Insulative

stabilization. Kurita’s Charge transport equilibrium

Magnetic Brush) and CMB (Conductive Magnet Brush)model [1] is one of the useful models for analyzing

development method by focusing conductivity of theexperimental data. Equation (1) shows Kurita’s original

magnet brush. In this study, the author focused thecharge transport equilibrium model determining

IMB method and applied a numerical, iterative solu-

developed toner charge Q;

tion for toner deposition and counter charge transportation to understand the mechanism that thecounter charge term seems to be a constant.

− ε0V0Q=
ds /ks + dt /k t + dc /kc θ

 (1)
1

where V0 is given potential difference, θ is speed

ratio between developing roller and photoconductor,
the photoconductor, developed toner layer and thecounter charge. In this model, developability is domi-

Q/Q0

and ds /ks, dt /kt and dc /kc are equivalent thickness of

0.5

nated by the counter charge thickness and θ.
Equation (1)Equation (2) shows Hattori’s revised charge trans-

Equation (2)port equilibrium equation (2019) [2]. In this equation,
the term of the counter charge thickness is modified
considering relative movement of developer from theviewpoint of developed photoconductor.

Q= d

s

ks

dt
kt

 (2)

dc

+

0.5

1

1.5

2

2.5

θ

Fig. 2 Comparison between experimental results and the calculation;solid lines are by Equation (1) and dotted lines are by Equation (2).

− ε0V0+0

Q/M
28.1 μC/g
36.3 μ C/g

k c ∙ | θ | ∙ (1 + | θ − 1 | )2 Charge transport model and calculating
methodFig. 1 refers above model [2] showing the counter

A calculation model is shown in Fig. 3. It is a ‘multi-

charge is condensed or diluted to 1/|θ| and |θ-1| islayered model’ consists of developer layer, air gap,

ejected to the vertical direction. In this figure the dis-developed toner layer and photoconductor layer, wheretance of the counter charge is constant.

developer layer including counter charge is divided inn sublayers (L1, L2 … Ln) to reproduce counter charge

Photoconductor

Δx

distribution. At the step 1, developed toner charge Qt

Q

and newly generated counter charge in the top sub-ks

Developed toner|θ-1|ΔxCounter chargedslayer of the developer ΔQdn are calculated so as the

kt

|θ|Δx

|θ-1|:1vs (θ-1)

ΔV

dt

ΔV

kc θ
dc

electric field in the imaginary air gap layer to be zeroelse toner density in the top sub-layer to be empty.

ΔQdn is condensed or diluted from Qt by Equation (2);Q

Fig. 1 Schematic illustration and equivalent circuit of the revised chargetransport equilibrium model at the condition of 0<θ<1.

∆ Qdn =Qt (t) − Qt (t − ∆t)θ (1 + θ − 1 )KONICA MINOLTA TECHNOLOGY REPORT VOL. 19 (2022)

 (3)

39

•Developing bias voltage VB = -500 V

At the same time, amount of counter charge trans-•Latent image potential on photoconductor Vi = -100 V

port among neighboring sub-layers is calculated by
the determined model. After Δt, at the step 2, elec-•Toner charge per weight q/m = -30 μC / gtric field in all sub-layer are re-calculated on the basis•Permittivity of the air gap (vacant)ε0 = 8.85×10 -12 F/mof the new charge distribution calculated at the step•Relative permittivity of the developer layer εd = 2.5

2. Step 1 and step 2 are alternately repeated.

•Relative permittivity of the toner layer εt = 2.0•Relative permittivity of the photoconductor εs = 2.7
Developer layer L D =nΔx(including counter charge)

VB

Air gap

Developed
toner•Counter charge mobility μ = 1.0×10-7 m2 / Vs

Photoconductor

Unit time Δt is tuned asμEn (0) doesn’t exceed LD /n/Δt.

L n-1 L n L a
L t = M t /ρt /σt − L sεd εd εa = ε0εtεs−Dd Dd Da = L a / ε0 Dt = L t / ε t
Ds = Ls / εs
Q dn-1Q dn
Q t (in layer) − Qs (on surface)=Vi / Ds−
E a=・・・
E n= Ea ε0 /εd−En-1= En -Qdn /εd−En-2=En-1 -Qdn-1 /εd・・・・・・−
E i = Ei +1 -Qdi +1 /εdEi -1 =E i -Qdi /εd−・・・・・・

L 1 L 2・・・
L i = ΔxεdεdDd = Δx / ε
Dd
Q d1 (in layer) Q di

Development efficiency η is evaluated;

Q
Qt
ηη =
= Qt
Q00

Fig. 3 Iterative calculation model by alternating development and chargetransport.

Two types of charge transport model were tested:
the Fick’s laws of diffusion and the electric charge

3 Results and discussion
3. 1 Diffusion-model-based analysis

mobility model. By Fick’s laws, counter charge density Qdi (t) in sublayer Li is expressed in the following
Equation (4);

 (6)

−
− εε00 (V(VBB −−V
Vii ))Q
Q00 == L /ε
Lss /εss ++L
Ltt /ε/εtt

Fig. 4 shows an example of the result supposing
charge transport was given by the Fick’s laws of diffusion at the condition δ = 0.1 and θ = 0.25. In casethat diffusion model was supposed, counter charge

Qdi (t) = Qdi (t −∆t)generated at L40 gradually moved slowly and kept

− δ{Qdi (t −∆t) − Qdi −1 (t −∆t)}where δ is the diffusion coefficient. This model
assumes the counter charge transportation is dominated by mechanical dispersion such as vibration or
collision rather than electrostatic force.
In another model, the counter charge transportation is dominated by electrostatic force received fromthe electric field;

stagnation around the photoconductor side.
1.0E-05

Counter charge density (C/m2 )+ δ{Qdi +1 (t −∆t) − Qdi ((t −∆t)} (4)9.0E-06
8.0E-06
7.0E-06
6.0E-06
5.0E-06
4.0E-06
3.0E-06
2.0E-06
1.0E-06
0.0E+00
t=4.15ms
t=3.32ms
t=2.49ms
t=1.66ms
t=0.83ms
t=0.00ms∆ Qdi (t)= μQdi (t − ∆ t)Ei (t − ∆ t)∆t

1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334353637383940

 (5)where μ is the counter charge mobility and Ei is theelectric field in the i-th sublayer of the developer layer.

Sub-layer number

Fig. 4 Calculated result of the counter charge distribution in the developer
layer by applying diffusion model (at θ = 0.25, δ = 0.1).

Fig. 5 is the time dependence of development efficiency by supposing the diffusion model. The figure

List of the parameters for calculation is as follows;

shows rapid increment of efficiency from t = 0 to 0.1ms•Developer layer thickness LD = 0.5 mm

and nearly static state after that. During the period of•Number of divisions of the developer layer n = 40

above rapid increment, the calculation result indi-•Imaginary Air gap layer La = 0.1μm

cated a state like ‘limitation of toner supply’. Even if•Packing ratio of toner layer σt = 0.5•Specific gravity of toner ρt = 1000 kg/m

3

•Photoconductor layer thickness Ls = 25μm

40Ea was not canceled to zero, toner density at L40 became

empty and development did not progress until toner
was dispersed from the neighboring sub-layer.

KONICA MINOLTA TECHNOLOGY REPORT VOL. 19 (2022)

Development efficiency

4.5%

consecutively from t=0.4ms to 2.2ms, developing

4.0%

efficiency was increased according to counter charge

3.5%transport form the photoconductor side to the devel-

3.0%

oping roller side, and finally developing efficiency

2.5%

became constant. The result can be understood thattransport of counter charge induces toner deposition

2.0%

and that it requires ‘transit time’ for developing prop-

1.5%

erty to be in the stable situation. The ‘transit time’

1.0%

may be a factor for the process speed limitation of

0.5%

the system.

0.0%
0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-045
1.0E-05

Fig. 5 Calculated result of development efficiency by applying diffusion
model (at θ = 0.25, δ = 0.1).

Fig. 6 shows effect of θ at the diffusion model calculation. By giving sufficient θ, ‘limitation of toner supply’ could be avoided. Anzai et al. [4] suggested thattoner to develop latent image was provided from theCounter charge density (C/m2 )Time (s)69.0E-066
8.0E-06
6
7.0E-06
6
6.0E-06
6
5.0E-06
6
4.0E-06
6
3.0E-06
6
2.0E-06
6
1.0E-0600.0E+00
ms
t=4.15ms
2ms
t=3.32ms

uppermost carrier layer, and that mechanical force=2.49mst=2.49ms

by magnetic brush should be included in developing

t=1.66ms
t=0.83ms

forces acting on toner. The diffusion model seems tot=0.00ms
1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334353637383940

correspond to their investigation in case of IMB system that DC bias voltage is applied.

Sub-layer number

Fig. 7 Calculated result of the counter charge distribution in the developer layer by applying electric charge mobility model (at θ = 0.25,
μ = 1.0×10 -7 m2/Vs).

20%
18%

12.0%

14%
10.0%

12%
10%
8%
6%
4%
2%

θ=0.25

θ=0.5

θ=1.0

θ =1.50%0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

Time (s)Development efficiency

Development efficiency

16%

8.0%

6.0%

4.0%

2.0%

0.0%

Fig. 6 Calculated result of development efficiency by applying diffusion
model (at θ = 0.25, 0.5, 1.0 and 1.5, δ = 0.1).

3. 2 Electric-charge-mobility-model-based analysis
Fig. 7 shows another example of the result suppos-

0.0E+00

1.0E-032.0E-033.0E-03

4.0E-03

Time (s)Fig. 8 Calculated result of development efficiency by applying electriccharge mobility model (at θ = 0.25, μ = 1.0×10 -7 m2/Vs).

ing charge transport is given by electric charge mobil-

Fig. 9 shows effect of θ at the electric charge mobil-

ity model. Counter charge generated at the photo-

ity model calculation. As can be seen from the figure,conductor side of the developer layer was transported

developing efficiency was relatively increased byto the developing roller side in about 2ms and wasapplying larger θ but it does not have effects to accel-

stayed around the neighboring area of the developing

erate the developing speed. On the contrary, Fig. 10

roller. Fig. 8 shows a corresponding result of develop-

shows effect of μ. By giving the larger μ, the devel-

ment efficiency. At the start of development, ‘limita-

oping efficiency reached to the saturated level in ation of toner supply’ was observed from t=0 to 0.4ms,

shorter time, but the saturated levels were the same.

KONICA MINOLTA TECHNOLOGY REPORT VOL. 19 (2022)

41

In rough approximation, supposing the process50%

Development efficiency

45%

speed limitation of the single-headed two-component

40%

development method utilized in color MFPs to be

35%
30%

around 500 mm/s and contact width between magnet

25%

brush and photoconductor to be 1mm, developing

20%

time is calculated as 2.0 ms. Developing property

15%

should be saturated until the end of the developing10%
5%

θ=0.25

θ=0.5

θ=1.0

θ=1.5

0%
0.0E+00

1.0E-032.0E-033.0E-03

4.0E-03

Time (s)Fig. 9 Calculated result of development efficiency by applying electriccharge mobility model (at θ = 0.25,0.5,1.0 and 1.5, μ = 1.0×10 -7
m2/Vs).

region if the unit were designed with sufficient robustness. When the developer layer thickness is 0.5 mm
and developing bias is 500 V, mobility of the countercharge is approximated to be 5×10-7m2/Vs. It is assame order as mobility of organic semiconductor.

Sin wave
1.5 kVPP
1.0 kVPP
0.5 kVPP
DC

60%

10.0

40%
30%
20%
10%

μ =-0.5e-7

μ =-1.0e-7

μ =-1.5e-7

M/A on paper (g/m2)

Development efficiency

50%

5.0

0%
0.0E+00

1.0E-032.0E-033.0E-03

4.0E-035.0E-03

Time (s)0Fig. 10 Calculated result of development efficiency by applying electriccharge mobility model (at θ = 1.7, μ = 0.5×10 -7, 1.0×10 -7 and1.5×10 -7 m2/Vs).

In case of two-component development method
applying AC bias voltage superposition, AC ampli-

0

100

200

Developing potential (V)Fig. 11 Experimental result of the change in development property
depending on AC bias superposition reported in [5]. M/A (mass of
toner per unit area) was increased by applying AC bias superposition from 0 (= DC only) to 1.0 kVP-P and saturated at 1.0 kVP-P and
more.

tude is considered as a parameter of the countercharge mobility. In actually, counter charge is a partof charge on the carrier surface with opposite polarity to toner. Counter charge is appeared when a toner

4 Conclusion
Although the charge transport equilibrium model

particle is deposited from the carrier surface andis well-known as a useful model for analyzing experi-transferred and disappeared when another toner

mental data of the two-component development method,

moved from neighbor onto there. There exist many

there was no discussion why and how the counterkinds of barriers or traps for toner transition throughcharge layer was determined. In this study, an itera-

the developer layer. AC bias generates an alternatingtive calculation method was applied to consider theelectric field in the developer layer and is thought toprocess of counter charge transport. A diffusion model

affect the toner to overcome barriers and traps result-and an electric charge mobility model were discussed.

ing counter charge transport. Fig. 11 shows experi-

The diffusion model met the case that DC bias voltage

mental data of developing property under various

was applied. The charge transport equilibrium mech-condition of AC bias superposition reported by theanism can be accepted except the condition underauthor in 1997 [5]. In this figure, M/A (mass of tonerthe ‘limitation of toner supply’. The electric charge

per unit area) was increased by applying AC bias

mobility model met the case that AC bias voltage wassuperposition from 0 (=DC only) to 1.0 kVP-P and satu-

superimposed. In this case the charge transport equi-

rated at 1.0 kVP-P and more. It is understood that thelibrium mechanism can be accepted if the develop-

counter charge mobility was increased by AC bias

ing time is longer than the transit time of countersuperposition and the transit time came up at the

charge which is determined by amplitude of AC volt-

developing time when AC bias became 1.0 kVP-P.

age and so on.

42

KONICA MINOLTA TECHNOLOGY REPORT VOL. 19 (2022)

Although two-component development method has
been applied to many commercial products and has
contributed to progress in speed and quality of electrophotographic process, few analytic investigations
have been presented recently. In this paper, two-component development method was discussed by utilizing a charge transport model in multi-semi-conductive-layer. Such a model is worthful to improve technicalknowledge from empirical know-how to analyticalunderstanding. Further applications are expected for
improving electrophotographic technology.

References[1] T. Kurita, “Charge transport equilibrium mechanisms inelectrographic toner development system”, DENSHISHASHINGAKKAISHI, 30(2), pp. 131-142 (1991) [in Japanese].
[2] Y. Hattori, “Re-consideration of the speed ratio dependent
term on the charge transport equilibrium mechanism,”
Proceedings for the 6th fall meeting of Federation of
Imaging Societies, 2019 (2019) [in Japanese].
[3] H. Okada, “An analysis of two-component magnetic brushdevelopments considering development resistance andtoner-supply-limit (1) – with-mode development –“, Journalof the Imaging Society of Japan, 52(3), pp. 181-193 (2013)[in Japanese].
[4] M. Anzai, N. Hoshi, H. Sawada and A. Shimada, “High-speed
magnetic brush reversal development in electrophotographic
process”, DENSHISHASHIN GAKKAISHI, 24(2), pp. 95-101(1985) [in Japanese].
[5] Y. Hattori, “Technologies in four-color developing process
and image stabilizing system”, DENSHISHASHIN GAKKAISHI,
36(4), pp. 324-335 (1997) [in Japanese].

Acknowledgment

Dr. Inan Chen, former in Xerox Corporation, gave lectures andconsultations of physics and mathematics from 1999 to 2009
concerning electrophotographic process and materials including
multi-layered model for charge transport calculation. The
author expresses cordial gratitude for his worthful instructions.
Reprinted with permission of the Imaging Society of Japan, therepresentative host society of ICAI 2021, the International
Conference on Advanced Imaging 2021. The copyright of thisarticle is owned by the Imaging Society of Japan.

KONICA MINOLTA TECHNOLOGY REPORT VOL. 19 (2022)

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