INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul. Mater. Sci. Eng. 12 (2004) 91–107 PII: S0965-0393(04)69064-4

Modelling dielectric heterogeneity inelectrophotography

Andrew Cassidy1, Martin Grant1 and Nikolas Provatas2

1 Department of Physics, McGill University, 3600 University, Montreal, Quebec H3A 2T8,Canada2 Department of Materials Science and Engineering, McMaster University,1280 Main Street West, Hamilton, Ontario L8S 4L7, Canada

Received 18 May 2003, in final form 15 September 2003Published 7 November 2003Online at stacks.iop.org/MSMSE/12/91 (DOI: 10.1088/0965-0393/12/1/009)

AbstractWe introduce a continuum model of electrophotographic printing of paperand other disordered composite films. The model allows simulation of anydielectric distribution within a toner transfer gap. It is used to elucidate the roleof mass density and paper thickness variations on toner transfer efficiency inXerographic printing. Our simulations show that spatial variations in inorganicfiller distribution within paper as well as paper thickness variations will stronglyaffect toner transfer uniformity by controlling the point-to-point variation of thelocal dielectric constant of paper. This variation will manifest itself as a mottlein toned images. It is specifically shown that paper thickness variations in paperlead to toner density variation on the scale of 1 mm, similar to those found inXerographic printing of commercial papers.

1. Introduction

Recent advances in the science of electrophotography have made Xerographic printing [3,7,12]a commercially viable alternative to conventional printing in the area of short-run colourprinting. One of the crucial requirements for high quality printing in Xerography lies in theproduction of uniform solid prints. Since toners penetrate very little into the paper, even afterthermal fusing, print density mottle or toner thickness variation is primarily the result of uneventoner transfer onto the paper surface. An important, and poorly understood, cause of uneventoner transfer is that of paper structure heterogeneity, and its effect on the electrostatics of tonertransfer. The most common stochastic components of paper structure include mass density,surface roughness, thickness, moisture and filler distribution.

During electrophotographic printing, electrically charged toner particles are transferred toa printing medium (e.g. paper) by an electrostatic field. It is common in designing this processto treat the printing medium as a dielectric of uniform thickness and of a uniform consistency.Inspection of paper, however, reveals a very non-uniform structure [9]. This gives rise to solidXerographic printed images that contain a significant degree of non-uniformity.

0965-0393/04/010091+17$30.00 © 2004 IOP Publishing Ltd Printed in the UK 91

92 A Cassidy et al

Figure 1. Print density variations in a solid black image printed on commercial paper.

Figure 2. Print density variations in a solid black image printed on a pure cellulose fibre network.

Figure 1 (reprinted from [9]) shows a solid black Xerographically printed square on aportion of commercial paper. The data represents reflected intensity from a white light source,which is mapped onto a grey scale. For clarity of exposition, we present here the data on ablack and white scale, with white representing all intensities below a certain threshold andblack all the intensities above. The threshold was chosen in the middle range of the originalgrey scale map. The image shows variations on several length scales, ranging from 100 µm(close to the threshold discernible to the human eye under regular conditions) up to 1 mm.Figure 2 (also reprinted from [9]) shows a printed image under the same condition but ontoa laboratory-made hand-sheet, pure cellulose fibre network with isotropic fibre orientations.In contrast to hand-sheets, commercial paper also contain, in addition to cellulose fibres,precipitated calcium carbonate (PCC) filler particles and a high degree of anisotropy in fibreorientations. The figures show that hand-sheet prints are more uniform on longer scales thanprints on commercial paper sheets.

This work will investigate how spatial variations in the dielectric properties of paper (orother similar thin-film composites) affect spatial variations in the long-range electrostatic fieldresponsible for toner transfer in Xerography. In particular, we will focus on elucidating themechanisms for print density variations at the 100 µm to 1 mm scale that have been reportedexperimentally [9].

Dielectric heterogeneity in electrophotography 93

(fibre+filler)

(fibre)

Figure 3. The toner–air–paper gap treated as a four-layer dielectric material between two parallelplates at a fixed voltage V .

2. Models of electrostatic toner transfer

2.1. An effective capacitor model

A Xerographic image (for the case of black and white images) is created when negativelycharged toners deposited on a photoconducting surface (drum) are transferred to a paper sheetusing a transfer corotron, a wire held at a positive voltage bias that creates an electric fieldbetween the photoconducting surface and the wire. This field causes negatively charged tonersto detach from the drum and be transferred to the paper substrate. This toned image is thenpassed through heated fuser rollers which melt it onto the paper substrate. For a uniform papersubstrate the simplest, and most common, design model for this process [12] is an effectivecapacitor model of the toner layer, paper substrate and the bias wire as shown in figure 3.In the figure, the voltage V represents the bias voltage that is applied between the paper andthe photoconducting drum. The height hp represents the thickness of the paper, hg is theheight of the entire paper–air–toner transfer region, ht is the height of the bottom of the tonerlayer, which is of thickness hg − ht . In this work, we further introduce a height hc, withinthe paper itself. This represents a height that separates a purely cellulose/pre-section of thepaper from one that is a mixture of fibre, pores and PCC filler. The inclusion of the heighthc within the paper is used to model the effect of surface-biased filler deposition, a situationcommon in commercial paper. Electrostatically, the only difference between these regions istheir dielectric constant. The dielectric constant of pure cellulose fibre regions is denoted εc,while that of the fibre + filler portion is denoted as εpf . Finally, the dielectric of the toner layeris denoted by εt . We denote by ρ the constant coarse-grained charge density of a toner layerand qt , the corresponding charge of an individual spherical toner.

Typical toner sizes are of the order of 10 µm and with a charge of 1.0 × 10−14 C [11]. Asthe focus of the work is to examine the role of paper structure, we will hereafter consider toneras a layer 10 µm in thickness with a charge density of ρ = 1×10−4 C m−2 (in two dimensions).For this size and charge of toners, the typical force required to detach a 10 µm toner from thedrum is roughly 250 nN [11].

The voltage applied across the transfer gap of figure 3 is equal to the electric field integratedacross the transfer gap from z = 0 to hg. The potential is thus split among the various layersof the system according to

V =[

σ

εc+

ρ(hg − ht)

εc

]hc +

[σ

εpf+

ρ(hg − ht)

εpf

](hp − hc)

+

[σ

ε0+

ρ(hg − ht)

ε0

](ht − hp) +

∫ hg

ht

[σ

εt+

ρ

εt(z − ht)

]dz, (1)

94 A Cassidy et al

where σ is the induced surface charge density on the top plate of the effective capacitor system.The quantities in the brackets [· · ·] correspond to the electric field in the cellulose, PCC-filledpaper, air gap and toner layer, respectively. Since the electric field in the toner layer at positionz is given by Gauss’s law [8],

E = σ

εt+

ρ

εt(hg − z), (2)

we solve equation (1) for the charge σ on the top surface, giving rise to a simple expressionfor the force per unit charge at position z within the toner layer. This is given explicitly as

Fz = − qt

{(V − ρ

ε0

[(hg− ht)hc

εc+

(hg− ht)(hp− hc)

εpf+ (hg− ht)(ht− hp) +

(hg− ht)2

2εt

])

×(

εt

[hc

εc+

(hp − hc)

εpf+ (ht − hp) +

(hg − ht)

εt

])−1}

− qt

[ρ(hg − z)

ε0εt

]. (3)

Equation (3) is a mean field expression, strictly speaking applicable when the paper substrateis uniform compared with the other elements of the system, such as the toner layer and thephotoconductor surface (represented here by the top parallel capacitor plate). In real systemsthe toner layer and photoconductors vary on length scales below the micrometre range [3],while paper itself can fluctuate both in its mass density and thickness on the scale ranging from100 µm (resolution of human reading vision) to tens of millimetres.

It is instructive to consider equation (3) qualitatively. It suggests which paper structurescan effect toner transfer. First, there can be significant effects due to variations in papermass density, which changes its effective dielectric constant. For instance, it has been shownexperimentally [13] that PCC filler (≈3 times a higher dielectric constant than a pure cellulosefibre network) increases the effective dielectric of paper as a function of filler concentration.Furthermore, equation (3) shows that the thickness of paper, hp, plays a critical role. A thickerpaper clearly increases the toner transfer strength (i.e. higher electric field in the toner layer)while a thinner paper will reduce toner transfer strength. Most importantly, point-to-pointvariations in both these quantities are expected to control the spatial variations of toner transferand print quality in Xerography.

2.2. A continuum model of electrostatic transfer

In this subsection, we introduce a continuum model of electrostatic transfer. We assumethat the toner transfer process occurs on timescales sufficiently long that a quasi-steady stateelectrostatic field is established in the transfer gap. This is a reasonable assumption as thetoner transfer process occurs on the order of milliseconds. We begin with the usual equationof electrostatics (Poisson’s equation) within a material (ε∇2φ = −ρ(�x)) where ρ(�x) is thefree charge density in the region. In the case of electrophotography this corresponds to thetoner charge distribution. We assume that there are no other free charges within the transfergap. The constant ε represents the local dielectric constant, while the function φ(�x) is theelectrostatic potential at any point in space.

If the edges between different domains representing paper, air and toner were sharp, onecould, in principle, solve Poisson’s equation in each material and match the solutions accordingto the well-known boundary conditions of electrostatics. Such a sharp-interface approach isextremely difficult to solve numerically, particularly for two- and three-dimensional disorderedand porous systems, involving complex front-tracking which would limit our investigation tounrealistically small system sizes due to the stochastic nature of paper structure.

Dielectric heterogeneity in electrophotography 95

To eliminate the problem of front tracking, we introduce a continuous dielectric functionε(�x) that continuously interpolates between any two states (materials) within the transfer gap(fibre, filler and air). With this definition of the dielectric, the solution of the electrostaticpotential everywhere in the transfer gap is given by

∇ · (ε(�x)∇φ) = −ρ(�x). (4)

This approach allows for very efficient numerical simulation of the electric field within thetransfer gap, without the need for tracking of material boundaries and manual implementationof boundary conditions. It is analogous to the use of thin-interface or boundary-layer modelsin the study of free surface motion and solidification [1,5,10]. One requirement of this methodis that the interface width between adjoining materials is sufficiently smaller than the fibreand toner dimensions. When this is the case, it can be rigorously shown [6] that the model inequation (4) satisfies all the usual boundary conditions of electrostatics.

2.2.1. Finite element analysis of the continuum model. We simulated equation (4) usinga simple relaxation technique applied to a uniform finite element mesh. In this case, theconvergence of the electrostatic field evolves as a pseudo-diffusion equation

∂φ

∂t= ∇(ε(�x)∇φ) + ρ(x), (5)

where t is a fictitious time. Describing equation (5) on a finite element mesh of linearisoparametric quadrilateral elements [4], and using the Galerkin weighted residual methodgives the following system of numerical matrix equations,

φn+1 = φn + �t[C−1]∗(−[K](φn)T − [R]T + [B.C.]T), (6)

where [K], [C] are analogous to the usual stiffness and mass matrices in diffusion problems,while [B.C.] and [R] are the boundary condition and source matrices. To avoid matrix inversionand large storage problems the matrix [C] is diagonalized using consistent mass lumping [4].Time efficiency of the code was made possible using a new C++ MPI-parallel FEM code wedeveloped and executed on a 16-node Beowulf cluster.

3. Paper heterogeneity and toner transfer variations

In this section we use the models described above and analytical methods to examine the roleof paper structure variation on toner transfer forces in the transfer gap depicted in figure 3.

3.1. Effect of local filler distribution

We first used the continuum model to investigate how local filler concentration and penetrationwithin the bulk of a paper sheet controls local toner transfer force. We simulated theelectrostatic field in the system depicted in figure 4. The transfer gap dimensions are 250 µmwide by 120 µm high, with a 2.5 µm resolution. In the figure, toner comprises a layer of 10 µmin thickness, dielectric constant εt = 4 and of charge density of σ = 1 × 10−4 C µm−2. Paperhas a thickness of 102.5 µm and dielectric constant of cellulose (εc ≈ 3), which is its mainconstituent. A uniformly random distribution of PCC filler is deposited in a ‘patch’ 100 µmwide and located near the top surface of the cellulose portion of the paper (see figure 4). Fillerparticles have a dielectric constant εf = 10. An air gap of 7.5 µm separates the two layers, anda potential of +1560 V is applied across the system shown (the bottom plate is held at ground).This would result in an electric field of 1.3 × 107 V m−1 in a capacitor filled only with air and

96 A Cassidy et al

micrometres

micr

omet

res

Figure 4. System used to study the effect of the filler distribution in electrophotography. The fillerpatch has dimensions 100 × 25 µm2 and a uniformly random distribution of filler particles, eachof dielectric constant εf = 10. The density of fillers is 0.0192 particles µm−2. Other parametersare found in table 1.

Table 1. Corresponding parameters in equation (3) for the system of figure 4.

V hg ht hp hc σ

(V) (µm) (µm) (µm) (µm) εt εc εf (C m−2)

1560 120.0 110.0 102.5 77.5 4 3 10 1 × 10−4

Distance (micrometres)

Elec

tric F

ield

(Volt

s/micr

omet

re)

Figure 5. Point-to-point variation of the electrostatic force field corresponding to the conditionsshown schematically in figure 4. Different curves correspond to the field at several positions withinthe thickness of the toner layer.

is a typical field strength in electrophotography [11]. These specific system parameters, whichare summarized in table 1, are approximate and can vary in real systems.

Figure 5 shows the electric field within the toner layer depicted in figure 4. The differentcurves correspond to the field at several vertical positions within the toner layer. Spatialvariations in the field correspond to variations in the local dielectric caused by the distributionof the fillers within the paper. The average electrostatic force on toners (over the horizontal

Dielectric heterogeneity in electrophotography 97

Figure 6. The magnitude of force on a 10 µm toner as a function of filler number concentrationfor the three different filler patch configurations. The bottom, middle and top curves correspond tothe 100 × 25 µm2, 100 × 50 µm2 and 100 × 102.5 µm2 filler patches, respectively.

extent of the filled region) versus the filler concentration is shown in figure 6 (bottom-mostdata points). Also shown in figure 6 is the average electrostatic force versus concentration fora filler distribution extending into the z-direction of the paper by 50 µm (middle data points)and 100 µm (top data points).

Figure 6 shows an increase of average toner transfer force with both filler concentrationand filler extent into the paper bulk. This is due to an increase in the local effective dielectricconstant of paper within the filled region. We also found that increasing filler depth ofpenetration into the paper tends to average out spatial fluctuations in the electrostatic fieldwithin the toner layer. This is due to the long-range nature of the electric field.

These results suggest that optimal toner transfer corresponds to a uniform distribution offiller through the bulk of the paper. Conversely, filler deposited near the surface will decreasetoner transfer efficiency and increase spatial variations in local toner transfer. The latter effectis only of practical interest when the filler distribution contains correlations on length scales>100 µm (the threshold of normal vision), as has been reported for certain commercial papers,which contain spatial correlations in their surface filler the order of 1 mm [9].

We also tested a result of [13] that the effective dielectric of filled paper varies linearlywith filler concentration by fitting the data of figure 6 to equation (3). The form for the localdielectric of filled paper was fit to εfp = α〈C〉+β, where 〈C〉 is the average filler concentrationthrough the filled paper region, and where α and β are different for the three filler geometriesshown in figure 6. The fits are shown by the solid lines in figure 6. We note that in order toobtain these fits, we adjusted the toner charge density to 3.28 × 10−14 C m−2, a value higherthan that of our FEM simulations. We suspect that this is due to the small size of the filledpatch relative to the overall transfer gap width being simulated.

3.2. Effect of a general dielectric distribution

To understand the effect of more complex distributions of filler and other constituents (i.e.fibres, porosity and moisture), it is necessary to derive the electrostatic field response thatarises as a result of a generalized, random dielectric distribution within the paper substrate.Such a random dielectric distribution may be inferred experimentally from surface potentialmeasurements of paper or any other printing medium.

98 A Cassidy et al

We begin by denoting a random distribution of dielectric by

ε(�x) = εu(z) + δε0ζ(�x⊥)θ(hp − z), (7)

where εu(z) = εp, the average dielectric constant of paper for z < hp and, εu(z) = εt , theaverage dielectric constant of toner for z > hp (variables refer to figure 3, with ht = hp, acommon situation in electrophotography). The function ζ(�x⊥) (�x⊥ = (x, y)) represents adimensionless in-plane random distribution that describes the deviation of the dielectric froma uniform value defined by εu(z). Pre-multiplying ζ(�x) by δ � 1 indicates that the deviationfrom the average value is small. We also neglect paper thickness in this analytic calculationby pre-multiplying by the step function θ(hp − z). Paper thickness effects will be consideredin the next section.

We characterize spatial variations in the electrostatic transfer field at a given height z inthe toner layer via the two-point correlation function, a common approach in the examinationof spatial effects in condensed matter systems [2]. This correlation function is denoted as

C(�r) = 〈(Ez(�x) − 〈Ez〉)(Ez(�x ′) − 〈Ez〉)〉E2

0

, (8)

where Ez is the z-component of the electric field, and the angled brackets denote averagesover all in-plane positions satisfying �x⊥ − �x ′

⊥ = �r (z is assumed to be fixed at z = (hp)+,

just above the paper–toner interface). The constant E0 is the electric field at z = (hp)+ in the

absence of any dielectric variation (i.e. that due to εu(z) alone). The function C(�r) is thus ameasure of how correlated two in-plane points in the electric field separated by a distance �rare with each other. For a general in-plane dielectric function ζ(x, y) equation (8) is given byequation (A21), derived in the appendix (in frequency space).

It is instructive to apply the formalism derived in the appendix to the case of a uniformlyrandom dielectric distribution (i.e. the ‘best case’ situation in commercial paper, where overallfiller density and formation is uniform from point-to-point in the sheet). In this case, thedistribution ζ(�x) (dropping the subscript ⊥) is uniformly random, exhibiting a two-pointdensity–density correlation function satisfying 〈ζ(�x)ζ(�x ′)〉 = δ(�x − �x ′) [2]. The explicitform for the correlation function is given by equation (A23) in the appendix. Figure 7 plotsC(r) = C(|�x − �x ′|) in frequency space as function for three paper thickness values. The decayof C(q) represents the (inverse) scale of spatial variations of toner (print) density on the papersurface.

The results of figure 7 are interesting in that they show that even in the case of uniform massdistribution within paper there will be spatial variations in toner transfer force on length scalesproportional to the thickness of the paper or, more precisely, the thickness of the paper relativeto the thickness of the toner layer. Specifically, toner density variations become uniform onlonger length scales with increasing paper thickness.

3.3. Effects of paper thickness heterogeneity on toner transfer forces

This subsection uses the continuum model to numerically simulate the electrostatic transferforces within a transfer gap simulated using a digitized surface electron microscope (SEM)map of a real section of paper substrate, which exhibits non-uniformity in both mass densityand profile thickness. The bottom half of figures 8 and 9 show a cross-section of a paper sheetobtained using a SEM. The section is 7460 µm in length and it is shown in its entirety in thebottom of figure 8, where the height has been increased in scale by 7.5 times. Figure 9 shows asection of the paper at 1 : 1 scale. White pixels in the bottom halves of figures 8 and 9 representfiller, grey is fibre and the solid black is air. The top images of figures 8 and 9 represent thesimulated magnitude of the z-component of the electric field when the paper is inserted between

Dielectric heterogeneity in electrophotography 99

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

C(q)

0.2 0.4 0.6 0.8q

Figure 7. Two-point correlation functions of the electric field in frequency space at z position justinside the toner layer. The bottom, middle and top curves correspond to paper of thickness 105 µm,95 µm and 75 µm, respectively.

Figure 8. Electric field magnitude (top) and SEM cross-section (bottom) for a 7460 µm filled-papersubstrate. Note the height has been increased in scale by 7.5 times.

Figure 9. Electric field magnitude (top) and SEM cross-section (bottom) for theregion 3730–4662 µm of the 7460 µm filled-paper strip, shown in figure 8.

100 A Cassidy et al

Figure 10. Local thickness of the SEM cross-section of the paper sheet.

two capacitor plates (i.e. a model transfer gap), where the top plate is grounded and the bottomis held at a potential of 1625 V. For the moment we ignore the presence of toner. The grey scalerepresents the magnitude of the field, where darker shades correspond to the lowest strengthand lightest correspond to the highest field strength. Note that due to shielding, the field insidethe sheet is negligible compared to the air gaps above and below the sheet.

Using Photoshop and a custom program, the SEM images were processed such thatcellulose, filler and air could be identified and then converted into a dielectric map at a 2.5 µmresolution. In our simulations, the dielectric constant of cellulose was set to εc = 3 while thatof filler was εf = 10 and ε0 = 1 for air. In what follows we label ‘side 1’ as the bottom sideof the sheet cross-section and all in-plane positions x are taken from right to left. ‘Side 2’corresponds to the top side and the positional results are also taken from right to left. Thetop and bottom plates (not shown) which establish a potential across the gap would be at thetop/bottom boundary of the figures.

The digitized data of figures 8 and 9 were analysed, yielding information about thicknessand filler density variations, as well as their corresponding two-point correlation functions.(Two-point correlation functions were calculated as in equation (8), with Ez replaced with themeasure of interest.) Figure 10 shows a plot of the local thickness of the paper sample versusposition. The corresponding two-point thickness correlation function, shown in figure 11,exhibits a correlation length (decay length in the correlation function) of about ∼550 µm.

For each side of the paper sample, the surface filler density was calculated and plotted asa function x, the position. Surface filler included all filler particles in a thin layer penetrating asmall depth into the paper. Filler layers varying from a thickness of 2 to 40 µm were examinedfor each side of the paper. Figure 12, shows the distribution of filler density for filler for athickness 20 µm below side 1 of the paper. Side 2 of the paper displays an analogous fillerdistribution. We note that filler density was measured near the surface since this is what isactually measured experimentally in many practical cases [9]. Figure 13 shows the two-pointcorrelation function for surface filler on side 1 at two different depths. Point-to-point spatialcorrelations for this sample are clearly uniform beyond 100 µm. This is in contrast to themeasurements of filler concentrations reported in [9], which found a strong correlation on the

Dielectric heterogeneity in electrophotography 101

Figure 11. Two-point correlation function of the point-to-point thickness variations of papercross-section.

Figure 12. Number of filler particles from a depth of 20 µm to the top of the surface of side 1.

scale of 1 mm in certain commercial papers. The sample shown here is from an unknowncommercial source.

The electrostatic field in the air gap below/above the two paper surfaces was simulatedusing the continuum model discussed above. The magnitude of this field at the position ofthe bottom plate is shown in figure 14 (side 1). The electric field at the position of the topplates (side 2) was analogous. The two-point correlation function (C(r) = C(|�x − �x ′|))was calculated at the position of the bottom and top plates of the transfer gap, and plotted infigure 15. The data show strong correlations upto 500 µm.

Simulations were also performed in a transfer gap that contained a toner layer (withuniform dielectric εt = 4) inserted under to the plate corresponding to side 1. The results oftoner force variations on side 1 of the paper are essentially identical to the case without a tonerlayer, except that exhibited a higher overall magnitude just inside the toner layer due to theshielding introduced by the toners.

The filler density from the simulated paper sample analysed in this subsection is uniformon scales above 100 µm. This fortuitous uniformity in the filler allows us to identify the electric

102 A Cassidy et al

Figure 13. Two-point correlation function of the filler density under the surface of side 1. The twodifferent curves correspond to the filler density of a depth of 5 µm from the surface (· · · · · ·) and20 µm from the surface (——).

Figure 14. Simulated electric field magnitude at the plate corresponding to side 1 of the papersheet.

field variations in the air gap as being created mainly as a result of paper thickness variations,consistent with the prediction in the previous section. The main result of this section is thatthe Xerographic print density variations, for paper of otherwise uniform mass density, aregoverned predominantly by paper thickness variations, which were found to be correlated onthe scale of ≈0.5 mm (a result typical of many commercial paper sheets). In the case whenboth paper thickness and filler display variations on visible length scales [9], both factors willcontribute to large toner transfer force variations. We note that in this work, as in [9], the overallvisual formation3 was measured and found to vary on scales larger than 5–10 mm, typical offlocs of fibres in paper. We thus conclude that a visual formation does not significantly affecttoner transfer variation.

3 Visual formation is a measure of mass coverage and is not a good indicator of mass density variations whenconstituents of different density are contained within the paper.

Dielectric heterogeneity in electrophotography 103

Figure 15. Two-point correlation functions of the simulated electric field magnitude at the platescorresponding to side 1 and side 2.

4. Conclusion

We examined the effects of paper structure on electrophotography by introducing a phase-field type continuum model that allows simulation of any general dielectric distribution in theelectrophotographic transfer gap. The model was used to elucidate the role of filler densityand paper thickness variations on toner transfer efficiency in Xerographic printing.

Our simulations showed that high-dielectric filler particle concentration affects the localstrength of the toner transfer field. We also found that spatial variations in filler density lead tocorresponding variations in the electrostatic transfer field. In some commercial papers, wherethe filler can exhibits spatial variations on scales of the order of 1 mm, our results suggest thatfiller density variations will play a major role in establishing the pattern of print density mottlein Xerographic solid images [9].

We also examined paper thickness variations, finding that these lead to a separate sourceof significant variations in the electrostatic field that transfers toner to paper. Simulations ofreal commercial paper subjected to an electrostatic transfer field revealed that paper thicknessvariations correlated well with electrostatic field variations on length scales of 0.5 mm.

Our results allowed us to separately identify paper thickness and filler density variationsas two attributes of print media that are each responsible for producing toner transfer non-uniformity in commercial Xerographic printing. Further work is required to characterize thesetwo effects when they are present in paper structure concurrently. This can be done easilyusing the continuum FEM model developed in this work combined with SEM data from awide range of paper samples.

Acknowledgment

We wish to thank The National Science and Engineering Research Council of Canada (NSERC)for financial support of this work.

Appendix A. Linear response theory

In order to focus on variations of the dielectric of the paper, we examine Maxwell’s first equationof electrostatic (equation (4)) with the paper thickness constant. Let the total dielectric constant

104 A Cassidy et al

be denoted by

ε(x, y, z) = ε0(z) + δε1(x, y, z), (A1)

where δ is a small constant quantifying the deviation of the dielectric constant from a functionε0(z) for which we know how to solve equation (4). We note that ε(x, y, z) is in units of thepermittivity of free space εf

0 = 8.85 × 10−12 F m−1. Inspired from perturbation analysis, wecan assume that the perturbation of the ‘dielectric field’ away from ε0(z) will bring about aperturbation of the electric potential as an expansion in powers of the small parameter δ,

φ = φ0 + δφ1 + δ2φ2 + · · · . (A2)

The electric field has a similar expansion since E = −∇φ. Substituting the expansion for ε

(equation (A1)) and φ (equation (A2)) into equation (4), we obtain a set of equations whosesolutions in principle give all terms in the expansion of φ. The lowest order terms, φ0 and φ1

are given by the solution of the following two equations:

∇ · (ε0(z)∇φ0) = −ρ(�x) = ρt (A3)

and

∇ · (ε0(z)∇φ1) = −∇(ε1(�x)∇φ0). (A4)

We define ε0(z) to describe a uniform paper–toner double layer given by:

ε0(z) = εp + (εt − εp) (z − hp), (A5)

where (z) is the unit step function, εp is the dielectric of paper and εt is the dielectric of toner.The solution of equation (A3) is then

E0(z) = −∇φ0 =

−a, 0 < z < hp,

−εp

εta +

ρt

εt(z − hp), hp < z < hg,

(A6)

wherea = a(εp, εt, hp, hg, V ) is a constant that depends on toner and paper dielectric constants,the height of the paper layer, the thickness of the transfer gap and the bias voltage, respectively.Its form is not shown here as it will not be used here explicitly.

Equation (A4) is solved using a Green’s function method. Specifically,

φ1(�x) = −∫ ∫

Gap

∫G(�x; �x ′)∇(ε1(�x ′)∇φ0(�x ′)) dx dy dz, (A7)

where G(�x; �x ′) is a Green’s function. This is the solution of equation (A4) with the right-handside replaced by a delta function ‘point source’ δ(�x − �x ′), subject to the boundary conditionsφ1 = 0 at z = 0 and z = hg. We expand the three-dimensional Green’s function in atwo-dimensional Fourier transform

G(�x; �x ′) = 1

4π2

∫kx

∫ky

g(z; z′; �k)e−i�k·(�x⊥−�x ′⊥) dkx dky, (A8)

where �x⊥ ≡ (x, y) and �k = (kx, ky) is a two-dimensional wave-vector, i.e. |�k| = 2π/λ,corresponding to a wavelength λ. The function g(z; z′; �k) is similarly a one-dimensionalz-direction Green’s function corresponding to a given wavevector �k. Substituting equation (A8)into Green’s function equation, we arrive at a one-dimensional piecewise linear equation forg(z; z′; �k) since ε0(z) is piecewise constant. The solution requires three additional boundaryconditions. These require that: (1) g(z; z′; �k) be continuous when z′ = z and z′ = hp and(2) Gauss’s law [8] is satisfied at material interfaces, i.e.

ε1dg

dz

∣∣∣∣I+

− ε2dg

dz

∣∣∣∣I−

= 0, (A9)

Dielectric heterogeneity in electrophotography 105

where I± represents the top-sided and bottom-sided limit of the matching points z′ = z andz′ = hp. After some lengthy algebra, we arrive at:

g(z; z′, �k) =

sinh(|�k|(z − hg)) sinh(|�k|(z − hp)) sinh(|�k|z′)�

,

0 < z′ < hp,

sinh(|�k|(z − hg))

εt�

[c1

|�k| sinh(|�k|(z′ − hp)) + εt sinh(|�k|hp sinh(|�k|(z − z′))]

,

hp < z′ < z,

c1 sinh(|�k|(z − hp)) sinh(|�k|(z′ − hg))

εt|�k|� ,

z < z′ < hg,

(A10)

where,

� = c1 sinh(|�k|(z − hp)) + εt|�k| sinh(|�k|hp) sinh(|�k|(z − hg)), (A11)

c1 = εt|�k| sinh(|�k|hp) cosh(|�k|(z − hp)) + εp|�k| sinh(|�k|(z − hp)) cosh(|�k|hp). (A12)

As an example of this formalism, we consider an in-plane sinusoidal perturbation of thedielectric, which is limited to a thin layer hc < z < hp within the paper. This perturbation iswritten as

ε1(�x) = 12ε0�θ(hp − z) + θ(z − hc) sin(�q · �x⊥). (A13)

Substituting ε1 into equation (A7), we obtain a first-order relation for the electric field in thetoner layer,

εf0Ez(x, y, z)

ρt�= εp

εt

[1 +

δεt

4πεp

(∂g(z, h−

p ; |�q|)∂z

− ∂g(z, hc; |�q|)∂z

)sin(�q · �x⊥)

]E0. (A14)

where � = 5 µm (half the size of a toner particle) and where E0 = E0(z = h−p ) (the notation

h−p implies that we use the relevant function on the interval (0, hp)) is in units of ρt�/ε0

f inequation (A14). The expression in the square brackets [· · ·] in equation (A14) can be usedto predict the linear response, within the toner layer, for any generalized in-plane dielectricvariation within the paper that can be spectrally decomposed.

Appendix A.1. Derivation of the two-point correlation function

We next consider the case where hc = 0 in equation (A13) and replace sinusoidal perturbationsin(�q · �x⊥) by

ζ(�x) = 1

4π2

∫ ∞

−∞

∫ ∞

−∞ζ�qe−i�q·�x d2 �q, (A15)

noting that ζ(�x) continues to be pre-multiplied by a small amplitude δ and adopting the notation�x = (x, y) for the remainder of this section. The (random) perturbation created by ζ leads toa perturbed three-dimensional field given by E ≈ −∇φ0 − δ∇φ1, as discussed above.

We characterize variations in the z-component of the electric field (at a height z > hp)through the two-point correlation function defined by

C(�r) = 〈(Ez(�x) − 〈Ez〉)(Ez(�x ′) − 〈Ez〉)〉E2

0

, (A16)

106 A Cassidy et al

where the angled brackets denote spatial averages over all in-plane vectors �x and �x ′ such that�x − �x ′ ≡ �x⊥ − �x ′

⊥ = �r . Alternatively, we can consider averages over different configurationsof ζ while �x and �x ′ remain fixed. Substituting the perturbed E-field into equation (A16), weobtain, after some straightforward algebra

C(�r) = δ2 〈∇zφ1(�x)∇zφ1(�x ′)〉E2

0

, (A17)

where ∇z denotes the z-direction derivative. In the above, we have assumed that theaverage of the electric field perturbation, is zero over the area of the paper cross-section,i.e. 〈∇zφ1(�x)〉 = 0. The field φ1 is given by substituting ε1(�x) into equation (A7),

φ1 = −E0εf

0

8π3

∫ ∞

−∞

∫ ∞

−∞ζ�q

∂g(z; h−p , |�q|)

∂ze−i�q·�x d2 �q. (A18)

Substituting for ∇zφ1(�x) into equation (A17) we obtain, after some straightforwardalgebra, an expression for a structure factor of the electric field (i.e. the Fourier transformof the two-point correlation function),

S(�k) =(

δεf

0

4π2

)2

〈ζ�kζ−�k〉[

∂g(z; h−p , |�k|)

∂z

]2

. (A19)

In obtaining equation (A19) we assume that the correlation function satisfies C(�x, �x ′) =C(�x − �x ′), which implies that electric field variations are homogeneous. In the case ofhomogeneous dielectric perturbation we can also write

〈ζ(�x)ζ(�x ′)〉 = Qζ(�x − �x ′). (A20)

This allows us to write the structure factor per unit area of the paper as

C(�k) =(

δεf

0

4π2

)2

Qζ (�k)

[∂g(z; h−

p , |�k|)∂z

]2

, (A21)

where Qζ(�k) is the Fourier transform of Qζ (�x).The special case discussed in the text for a uniformly random dielectric distribution

corresponds to a dielectric perturbation with correlation function given by

〈ζ(�x)ζ(�x ′)〉 = δ(�x − �x ′) (A22)

for which Qζ (�k) = 1, and

C(�k) =(

δεf

0

4π2

)2 [∂g(z; h−

p , |�k|)∂z

]2

, (A23)

which is the function plotted in the text.

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Dielectric heterogeneity in electrophotography 107

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