Hydrodynamics of Ink Transfer. Martin Dubé 1, François Drolet 2, Claude Daneault 1, and Patrice J. Mangin 1

Transcription

1 Hyroynamics of Ink Transfer Martin Dubé 1, François Drolet, Claue Daneault 1, an Patrice J. Mangin 1 1 Centre Intégré an Pâtes et Papiers, Université u Québec à Trois-Rivières, 3351 boul Des Forges, C.P. 5 Trois-Rivières, Québec, Canaa G9A 5H7. Pulp an Paper Research Institute of Canaa, 57 boul St-Jean, Pointe-Claire, Québec, Canaa H9R 3J9 Keywor : Ink Transfer, Tack, Network, Porosity Summary We iscuss the hyroynamic flow of a half-tone ot presse onto a moel porous substrate in a range of parameters representative of printing processes. We first escribe the lubrication approximation of the Navier-Stokes equations an iscuss its implications for flui tack. We then show how this approximation can be extene to situations in which the porous meium contains one or more vertical pores. The approximation is valiate through numerical solutions of the complete hyroynamics problem obtaine with a Diffuse Interface technique. One of the great avantages of that metho is that it naturally escribes complex phenomena such as the spreaing of a flui on a soli substrate or the breakup of a liqui filament. We use it to simulate the transfer of a flui ot onto a two-imensional network of pores. One of our main results is that, at constant porosity, flui transfer increases with ecreasing pore size. We also iscuss how tack relates to the pore size an how the istribution of flui insie the network epens on the relative sizes of the vertical an horizontal pores. 1. Introuction The transfer of a flui from a cyliner to a porous substrate, such as paper, is at the heart of the printing process. The amount of flui transferre is usually escribe by empirical relations such as the Walker-Fetsko equation, which ecomposes the process into three separate phases: contact, immobilization an splitting [1-3]. In general, these relations provie limite insight into the relative influence of the ifferent parameters affecting ink transfer such as printing pressure, press spee, ink viscosity, an substrate properties. In aition, when consiering the transfer of a single half-tone ot, the volume of ink that is transferre is very small, an the iscrete nature of the porous meium becomes important [4-6]. In such cases, ink transfer cannot be escribe in terms of macroscopic quantities or concepts such as the Darcy permeability. The main purpose of this article is to propose a microscopic approach to ink transfer that takes into account the etaile structure of the porous substrate an can etermine its influence on flui flow. Specifically, we consier a rop of Newtonian liqui being presse onto an iealize porous meium an solve the corresponing Navier-Stokes equations in a range of parameters relevant to the printing process. The moel oes not take into account the changes in pore structure occurring uner compression or in presence of polar liquis. A further limitation is that the viscoelastic behavior of the ink an its separation into separate constituents (vehicle + pigments) are ignore. However, the ynamics of the contact line at the air/liqui/soli interfaces, giving rise to capillary forces, is explicitly inclue, as is the breakup of the filament connecting the ink on the plate to that on the substrate. It is also possible to ajust the properties of the flui (viscosity, surface tension, wetting properties) as well as the operating conitions of the press. Despite its limitations, we believe that this moel can provie valuable insights into the influence of pore morphology an press parameters on flui transfer. The remainer of this paper is structure as follows. The moel setup use in our analysis is escribe in Section, which also inclues a iscussion of the various phenomena that affect ink transfer. In Section 3, we consier the case of a non-porous substrate with a smooth surface. In that case, it is possible to fin an approximate solution to the Navier-Stokes equations an to preict how the tack of a flui varies with its viscosity as well as with the

2 parameters of the press. In particular, we show that Stefan's law, often criticize in the literature, correctly preicts how tack changes with the thickness of the ink layer. We then generalize our analysis to inclue a single vertical capillary which opens at the surface of the substrate. Expressions preicting how the epth of penetration an the raius of the rop evolve in time are erive in Section 4a. Preictions from these expressions are compare with results from numerical simulations in Section 4b. The transfer of a flui rop to a substrate with several pores is iscusse in Section 5. For a collection of ientical vertical pores, we fin that, at constant porosity, the amount of flui transferre to the substrate is largest when the pores are small. This result can be explaine in terms of capillary forces: although more flui initially enters the larger pores, the smaller capillaries are more efficient at retaining the flui when the plate recees from the substrate. The same result hols when horizontal pores are ae to the pore structure, although to a lesser extent. We show how the tack of the ink epens on pore size an how the istribution of flui insie the network is affecte when pores of several sizes are present. A etaile escription of the hyroynamic moel an of its numerical implementation is finally given in Appenix A. substrate. The halftone ot has initial raius R (). Since both the cyliner an the paper have the same tangential velocity v, with the ot size much smaller than the raius of the printing cyliner, we neglect the horizontal component of the motion an consier a flat plate subject to uniform vertical acceleration, with velocity vp (t) = a(t t ) (.1) as illustrate in Fig..1. The plate initially moves ownwar, reaches its lowest point at time t an is then pulle upwar. In a conventional printing unit with a rotating image cyliner, the acceleration is given by a=v /R c, where R c is the raius of the cyliner. For a cyliner with raius R c =5 cm an tangential velocity v = 1 m/s, the acceleration a = 4 m/s. The time evolution for the plate height 1 h(t) = h min + a(t t ) (.) introuces h min, the lowest point reache by the plate, at time t. The value of h min can be ajuste to yiel pressures typical of a variety of printing processes. We consier that t= is the time at which the ot touches the substrate. The value h() then correspons to the initial thickness of the ot. At this point, we o not consier any eformation of the plate or of the substrate an we treat the ink as an incompressible Newtonian flui. The velocity fiel of the flui, v(x,t) is then governe by the Navier-Stokes equations: v ρ + v t ( v) = p + η v (.3) Fig..1: Moel setup: a thin flui rop is presse between a substrate an a flat plate that moves at a prescribe velocity v p (t). The liqui-air interface meets the substrate at a contact angle θ.. Moel escription of printing nip We are intereste in the transfer of a single halftone ot from a printing cyliner onto a moel porous where ρ is the ensity of the flui, η its viscosity an p is pressure. The incompressibility conition v =, must also be impose. EQ. (.3) must be solve not only in the region separating the plate from the substrate but also within the pore volume invae by the ink. The wetting properties of the flui have a large impact on the amount of flui that is transferre. Uner equilibrium conitions, the contact angle θ at

3 which the liqui/air interface meets a soli surface satisfies the Young-Dupré equation σ σ σcosθ (.4) sl sa = where σ sl an σ sa are the surface tensions of the soli with the flui an air respectively an σ is the tension of the liqui/air interface. In general, the contact angle on the surface of the substrate will be ifferent from that on the plate. Furthermore, because the contact line is moving, the contact angle is ifferent from its equilibrium value an epens on the flui velocity [7]. As a result of the liqui/soli/air interactions, the surface of the liqui rop is curve (see Fig..1). In turn, this creates a capillary pressure ifference p c across that surface, relate to the curvature of the interface. For a flui in a cylinrical capillary of raius R, p c =σcosθ /R. a) b) c) ) Fig.. Numerical simulations of a half-tone ot, of raius R ()=1 μm an initial height h()= μm presse onto a twoimensional network of capillaries. The following parameter values were use: acceleration of the plate a=1 m/s, surface tension of the liqui σ= nn/m, flui viscosity η=. Pa-s. The equilibrium contact angle was set to θ=6 on both the plate an substrate. The vertical pores have size R =.5 μm, separate by a istance of 1 μm. The topmost horizontal pore has also a size of.5 μm, while the lower horizontal pore has size 1 μm. The figure shows the evolution (from inset a) to )) of the flui configuration uring the transfer process. The separation between the plate an substrate corresponing to each configuration is: a) h=1 μm in the pressing stage, b) h=h min =.5 μm, c) h=. μm in the pulling stage an ) h= 5. μm, right after the filaments break. The moel we use to stuy the flui transfer problem escribe above is explaine in etails in Appenix A. It involves solving the Navier-Stokes equations in both the liqui an air phases together with an equation for a fiel c(x,t), which takes value c=1 in the liqui an c= in the air. Interfaces separating the two fluis correspon to regions over which c goes from to 1. The equation for c is erive from a free energy functional which etermines the with ζ an surface tension σ of the liqui/air interfaces an escribes the interactions between the two fluis at soli surfaces. With this metho, the ynamics of the contact angle an the breakup of a liqui filament occur naturally from the ynamics an thermoynamics of the fluis. Another avantage is that the finite with of the interface regularizes any singularity that might occur when there are sharp iscontinuities in the soli surfaces. In orer to reuce the computational time, we performe our numerical simulations in two imensions. However, we expect the main results an conclusions of this paper to be vali also in three imensions. Figure. shows four snapshots obtaine at ifferent times uring a typical simulation. In that particular example, a liqui rop of initial thickness microns was presse into a two-imensional network consisting of sixteen vertical an two horizontal capillaries. In the initial stages of the flui transfer process, the rop of liqui is presse onto the porous structure, forcing liqui into the pores (Figures.a an.b). Then, as the plate is pulle back (after the time t ), some of the flui is remove from the pores an filaments form between the substrate an plate (Figure.c). Eventually, these filaments break, leaving a fraction of the flui transferre to the substrate (Figure.). The problem of flui transfer into twoimensional networks such as that shown in Figure. will be iscusse in Section 5-b, after substrates with a simpler pore structure are consiere.

4 3. Lubrication approximation an flui tack We first consier the situation in which the halftone ot is in contact with a non-porous flat substrate. The problem then becomes equivalent to a layer of flui being squeeze or pulle between two moving plates or isks. Because that layer is thin (h/r << 1), the flow is preominantly parallel to the plate (v x >> v z ) an the velocity graient is much more pronounce in the thickness irection ( v x / z >> v x / x ). At sufficiently low Reynols numbers, the inertial term in EQ. (.3) (i.e., the left-han sie of the equation) can also be neglecte. In that limit, the so-calle lubrication approximation, the Navier- Stokes equations reuce to v z x η p = x (3.1) Equation (3.1) must further be solve subject to the continuity equation: vx vz v = + = (3.) x z an the following bounary conitions, vali for a ot centere at x=, with the substrate s surface locate at z=: at z =, v x = v z = at z = h, v =, v = v (3.3) at x = ± R, p = x p z In the last bounary conition, p is the atmospheric pressure which we set to zero for convenience. In EQ. (3.3) we have neglecte capillary effects at the eges of the rop. These effects will be consiere in the next section. The solution to EQ. (3.1) reas (x,z) v x 1 p = z(z h) (3.4) η x The flow rate parallel to the plate is given by: 3 h h p Q(x) = z v (x,z) = (3.5) x 1η x p Integrating EQ. (3.) over z, using EQ. (3.3) an EQ. (3.5) then gives a Poisson equation for the pressure fiel: p 1η = v 3 p, (3.6) x h subject to the bounary conition EQ. (3.3). The pressure fiel is then obtaine: ( x ) 6η p(x) = v 3 p R (3.7) h The total force exerte by the halftone ot on the printing plate is foun by integrating the pressure over the surface of the ot, giving ηvp(t) 3 F(t) = R (t) (3.8) 3 h (t) Because inertial effects were neglecte, the time epenence appears only through the position an velocity of the plate, a through the raius of the rop. Equation (3.8) implies that the force exerte on the plate increases with velocity an size of ot but ecreases with increasing plate-substrate separation. The time epenence is also antisymmetric aroun t, ie., F(t-t )=-F(t -t). The erivation presente above is easily generalize to three imensions. In that case, the force is given by: 3π ηvp(t) 4 F(t) = R (t) (3.9) 3 h (t) an is known as Stefan's law. Its applicability to ink transfer has been controversial for a long time [8,9]. One argument often use to iscount EQ. (3.9) is the fact that Stefan's original set-up, involving the separation of two plates immerse in a flui, is not representative of the conitions foun in a printing nip. However, the erivation presente above shows that the same result also applies to the more relevant case of a thin flui rop squeeze or pulle between two plates. It is also often argue that Stefan's law cannot apply

5 to ink transfer because it preicts that the tack of ink shoul ecrease as the ink film thickness increases, contrarily to what is seen in most experiments. In general, however, that argument is incorrect, as we now show by calculating the value of the tack explicitly. Tack is efine as the maximum force exerte by the flui on the plate. After some straightforwar algebra we fin, in three imensions, 4. Transfer to a single capillary We now consier the transfer of a flui rop from a flat plate to a substrate that contains a single vertical pore, of raius R, centere on the flui rop as shown in Figure 4.1. Capillary effects, both at the sie of the rop an insie the pore are inclue in the calculation. F max / = π η ar ()h ()hmin (3.1) 1 The epenence of tack on ot thickness epens on the type of experiment that is performe. i) When the ot is presse onto an then pulle from the substrate, as in a printing process, h() an h min are inepenent, with h min <h(). EQ. (3.1) then preicts that whereas tack ecreases as the minimum istance between the plate an substrate is mae larger (F max ~ h min -9/ ), it increases with the initial thickness of the halftone ot ( F max ~ h ()). ii) The situation is ifferent if the ot simply rests on the substrate an the plate is pulle upwar. This is similar to the setup use by Stefan in his original experiment. In that case, the minimum separation h min is equal to its initial value h() which also correspons to the initial ot thickness. Equation (3.1) now preicts F max ~ h () h min -9/ = h -5/ () implying that tack ecreases with the initial thickness of the ink layer. These results explain why tack is foun to increase with film thickness when measure between two rollers [1] (which squeeze the flui before pulling on it) while the opposite tren is observe when measure with a parallel plane tackmeter (which only involves pulling of the flui layer between two plates) [11,1]. The trens preicte by Stefan's law are vali as long as the conitions leaing to the lubrication approximation are met. However, it is clear that Stefan's law, as presente above, oes not quantitatively apply to the printing process, since it neglects the effects of capillary pressure an cavitation [13,14]. Fig. 4.1: Schematic representation of a rop presse onto a substrate containing a single vertical capillary, of size R. The flui has penetrate into the capillary up to a istance l. In the calculations presente in Section 4-a, both R an l are assume to be much larger than the curve meniscus giving rise to capillary pressures at the flui bounaries. 4-a Analytical Solution As long as the lubrication approximation hols, it is again possible to fin an approximate solution to the Navier-Stokes equations an to preict how the amount of flui insie the substrate changes as the plate first approaches an then recees from its surface. In orer to fin that solution, we ivie the volume of liqui above the substrate's surface into three regions, as illustrate in Fig. 3. In regions I an III, the pressure satisfies once again EQ. (3.6), with bounary conition p( ± R ~ ) = σcosθ/h p c at the eges of the rop. We assume that θ remains equal to the equilibrium contact angle at all times. In orer to be vali, this assumption requires that the contact line moves very slowly, as will be the case if the velocity of the plate is small. The secon bounary conition is the pressure at x= ±R which we enote by p 1. It can be shown that in the limit h 3 (t)l(t) >> R 4, the pressure fiel is constant over region II so that p(-r < x < R, < z < h) = p 1. This

6 implies that the flow rate into the pore has the stanar form [15]: Q c 3 R p1 pc = (4.1) 3 η l(t) where p c =σcosθ/r is the capillary pressure in the pore an l is the epth of penetration of the flui insie the pore (see Fig. 4.1). The value of p 1 is etermine from volume conservation, which implies that the flow rate over all surfaces of the rop is zero, i.e., Q(R (t), t) + Q (t) + R (t)v (t) (4.) c p = where Q(R,t) is given by EQ. (3.5). This conservation law leas to an expression for p 1 from which it is then possible to extract the ynamical evolution of the penetration epth an of the ot size: 3 ~ l vpr (R R ) + h R (pc pc)/(3η ) = 3 3 t h l + 4R (R R ) (4.3) R 1 l = vpr R (4.4) t h t The complex hyroynamics problem has thus been reuce to a set of two couple ifferential equations, reaily solve numerically. The soli lines in Fig. 4. are preictions from EQ. (4.3) an EQ. (4.4) that show how the volume of flui insie the capillary, R l(t), evolves in time for two ifferent pore sizes. In both cases, the pore initially fills with liqui ue to both capillary an plate pressures. The volume insie the capillary reaches a maximum at roughly t=t, i.e., when the istance between the plate an substrate is minimum. As shown in Figure 4., that maximum in volume increases with R. Then, as the plate begins to recee, the pressure p 1 above the capillary becomes negative an there is a tenency to remove liqui from the pore. Depening on the value of the capillary pressure, which is inversely proportional to R, the pore can either be completely emptie or remain partially fille. The two scenarios are illustrate in Fig. 4., where the larger pore is shown to become empty aroun t=14 ms. These results show that there are no simple correlations between pore size an flui transfer: smaller pores usually take less flui, but can retain it better. Our results also suggest the existence of a critical pore raius above which the flui is completely remove from the capillary when the plate is pulle away from the substrate. Volume of Flui in Pore ( μm ) R =.55 μm R =.35 μm Time (ms) Fig. 4.: Time evolution of the flui volume insie a capillary of size R.Preictions from EQS. (4.3) an (4.4) (soli lines) are compare with the results of numerical simulations for the values R =.35 µm (circles) an R =.55 µm (squares). The following parameter values were use: R =5 μm, h()= μm, a =.5 m/s, h min =.5 μm, σ= mn/m, η=.1 Pa-s an θ=6. When the pore size R is much smaller than all other length scales in the problem, it is possible to fin an approximate solution to EQ. (4.3). It reas: R 1 1 l (t) = pc(t t ) + R R ()h () η h (t) h () (4.5) This equation clearly shows the interplay between capillary an nip pressures. The first term on the right-han sie of the equation is always positive an correspons to the Lucas-Washburn equation (l(t) t 1/ ) [15]. The secon term, which arises from the motion of the plate, can be either positive or negative epening on the separation between the plate an substrate.

7 Although the solution to EQ. (4.3) an EQ. (4.4) can give some insights into the flui transfer process, it is impossible to make any statement about the total amount of liqui transferre from the plate to the substrate. This requires a proper escription of the formation an subsequent breakup of liqui filaments between the moving plate an the porous surface. The non-equilibrium behavior of the contact angle must also be consiere. In the following section, we present results from numerical simulations of the full hyroynamic moel which take all of these elements into account. 4-b Numerical simulations The iffuse interface moel escribe in appenix A was first use to valiate the analytical treatment presente above. The circles an squares in Fig. (4.) are results from numerical simulations performe with the same parameter values that were use earlier in EQ. (4.3) an EQ. (4.4). As can been seen from the figure, the preictions from these two equations agree very well with the simulation results for the two values of R consiere. This confirms that the lubrication approximation escribes well the flow of the liqui (prior to the formation of filaments) when the acceleration of the plate is sufficiently small (a=.5 m/s in this case). equilibrium contact is the same on the plate an on the substrate (θ=6 ). At higher plate accelerations, we must rely on the iffuse interface approach. From the simulation results, it is possible to estimate the percentage of flui that is transferre to the substrate. It is efine as the fraction of flui that remains on the substrate after filament breakup. Figure 4.3 shows how that percentage changes with pore size for a plate acceleration a=1 m/s. The transfer is relatively constant at smaller pores, until a large rop occurs for a raius R 1.5 μm. After this point, transfer is small. The two insets in Figure 4.3 show that the flui istribution after filament breakup is ifferent in both cases. Whereas the flui is mostly confine within the capillary at small pore sizes, it is istribute on the outsie of the capillary at larger sizes. This istribution results from the creation of two filaments, with feet on both sies of the pore, followe by their subsequent breakup. It shoul be note that this situation occurs even though there is no cavitation as such. The sharp iscontinuities occurring at the pore opening may have an influence on the final flui configurations. However, we emphasize that these iscontinuities o not cause any computational problems, an we expect the results to be qualitatively similar in more realistic situations. 5. Transfer to a network of capillaries In orer to go further on the problem of ink transfer, we now consier a more complex pore structure consisting of a network of capillaries. The network approach is use because the volume of flui transferre correspons only to a small fraction of the total pore volume. As a result, typical homogenization techniques, leaing to macroscopic relations such as Darcy's law are not necessarily vali an the ynamics of the flui is greatly influence by the iscrete nature of the substrate. This was pointe out early by Lenorman [4] an iscusse in the context of printing in refs. [5] an [6] among others. Fig. 4.3: Flui transfer to a substrate with a single capillary of raius R. The rop has size R =1 μm an initial thickness h()=4 μm. The acceleration a=1 m/s an h min =. μm. The surface tension σ= mn/m an viscosity η=.1 Pa-s. The The proceure outline above in Section 4-a can be in principle generalize to a -imensional iscrete network (compose of pores connecte by vertical an horizontal throats), since it basically amounts to

8 solving a set of couple equations for the pressures at the pores. However, the computational problem of free surface flows that occurs at filament breakup cannot be easily taken into account. For this reason, it is preferable to perform the numerical simulations of flui transfer with the Diffuse Interface technique, solving the complete set of Navier- Stokes equations. 5-a Collection of vertical pores We now consier flui transfer from the plate to an ensemble of vertical pores. In that case, the porosity of the substrate is given by φ=r /(L s +R ), where L s is the average istance between the capillaries. The presence of several pores reuces the amount of ot spreaing that takes place. That reuction is cause by flui flow into the pores [16] as well as by pinning of the contact line on the sies of the pores [5]. A etaile analysis of this last phenomenon is beyon the scope of this work an we concentrate here on flui transfer. ones. For a fixe value of pore size R, flui transfer however increases with porosity, as shown on Fig b Two-imensional network Horizontal pores in the porous structure have a ual role. At an intersection between vertical an horizontal pores, the liqui front in stoppe until the pressure is large enough to force it into the crosse pores. This is clearly apparent on Fig... Typically, if a pore has raius R, a pressure of the orer σ/r is necessary to push the flui past it. Figure. also shows that if some liqui has been transferre to a horizontal pore, it remains essentially trappe in place. The receing phenomenon observe in Fig. 4. is much less pronounce. Fig. 5.: Flui transfer as a function of porosity at a fixe pore size R =.55 μm. The other parameters of the simulations are as liste in Fig. 5.1 Fig. 5.1: Flui transfer as a function of pore size at fixe porosity φ=.4. The ot has initial raius R =15.5 μm an thickness h()=5 μm. The acceleration a=1 m/s an the minimal istance of approach h min = μm. The surface σ= mn/m an viscosity η=.1 Pa-s. The contact angle is the same on the plate an substrate (θ=6 ). Figure 5.1 shows the percentage of flui transferre as a function of pore size, for a uniform istribution of pores at a fixe porosity φ. It is clear that there is a larger transfer for smaller pores than for larger The overall transfer of flui into this twoimensional network is qualitatively very similar to the transfer into vertical pores. We escribe the network in terms of the surface porosity φ=r /(L s +R ), as for vertical pores, an the position of the horizontal pores. This is an effective escription, preferable to a more formal expression for the porosity in a -imensional network, since only a few horizontal pores are invae by the flow.

9 8 7 Force (N/m) 4 - latter phase. The maximum force in the pressing phase is relatively constant with pore sizes, inicating that it is mostly etermine by porosity. However, tack increases with smaller pore sizes, ue to the capillary suction of the pores. Tack (N/m) Time (ms) a) b) ,5 1 1,5 R (μm) c) ) 1 1 Fig. 5.3: Tack as a function of pore size at fixe porosity φ=.33 for the parameters escribe in Fig... The tack is efine as the maximum in the force uring the receing phase of the plate motion an increases with pore sizes. The units of force (N/m) reflect the two-imensional nature of the simulations. The inset shows the evolution of the force as a function of time for the case R =.4 μm (largest value of the tack),.5 μm, 7. μm, an 1. μm (smallest tack). The numerical simulations show again that flui transfer ecreases with pore size at constant surface porosity an increases with porosity at fixe pore size, an that these trens are inepenent of the position of the horizontal pores. Figure 5.3 iscusses the tack of the flui. The simulations are performe with the set of parameters use in Fig.., for a constant surface porosity φ=.33 an with a uniform istribution of pore sizes of ifferent values R. The force is calculate by integrating the pressure fiel over the with of the ot an tack is efine as the peak in force uring the receing phase of the plate motion. The overall form of the curve is ifferent from the force measure for a film between rolling nip [1] an is more in line with the behavior escribe by EQ. (3.8) (single half-tone ot presse on a non-porous substrate). The role of the pores becomes apparent in the large ifference that exists in the peak pressure between the pressing (plate moving towars the substrate) an the receing (plate moving away from the substrate) phases; there is a larger force on the plate in the Fig. 5.4: Flui configuration as a function of the pressure on the plate. The values of h min =.5 μm,.5 μm,.75 μm an 1. μm correspon to the inset a), b), c) an ) respectively. The numerical simulations also show that the local quantity of flui transferre epens to a large extent on the local pore structure. Smaller pores can lea to a large overall transfer an ten to keep the flui close to the surface of the porous meium. Figure 5.4 shows the flui istribution within the porous meium at various pressures, in a structure containing both large an small vertical pores, together with horizontal pores. The pressure is change by changing the parameter h min in EQ. (.), such that smaller values of h min lea to larger values of pressure. These figures show the flui configuration at h=1. μm, as the plate is being pulle. It is clear that there is less transfer to horizontal pores as the pressure is reuce. However, it is interesting to note that it is those horizontal pores ajacent to large vertical pores that remain fille. This is because the pressure for breakthrough ecreases as the pore size increases.

10 6. Conclusion The results presente above illustrate that flui flow in a printing nip is the result of a complex interplay between flui properties, nip pressure, capillary forces an network geometry. In particular, at constant porosity, there is more transfer into small pores than into larger ones. The role of small pores has alreay been highlighte in the context of ink setting on coate paper [17-19] an it is interesting to note the corresponence, even though the present context is ifferent. Although the results presente here o not allow us to fully assess the role of inertia [17], the metho can in principle be useful in this respect, since meniscus formation at the entrance of a pore occurs naturally. It shoul be note however that, since only a few pores are involve in the process, the use of porosity is ambiguous, as it is a statistical quantity. Our results show that ifferent network structures presenting the same overall porosity behave ifferently with respect to flui flow. Future stuies will concentrate on linking the variations in ink transfer to the variation in a given structure at fixe porosity. The methos evelope here can also be use to link the phenomenological Walker-Fetsko equations to the microscopic input parameters of the moel. Acknowlegement: The authors woul like to thanks J. Aspler, from Paprican, H. Reinius, from KCL an J.-F. Bloch, from EFPG, for useful iscussions. CD acknowleges financial support from the Canaa Research Chair in Value Ae Paper, an PJM acknowleges support from the National Science an Engineering Council of Canaa. A. Diffuse Interface Hyroynamics The hyroynamics of free surface flows is a notoriously ifficult computational problem [,1]. The ifficulty stems from the fact that it involves motion of the interface separating the liqui from the ambient air. The usual approach to this type of problem is to iscretize the volume occupie by the liqui an to solve the Navier-Stokes equations within that volume. The interface then simply marks the bounary of the computational omain. At that interface, two sets of bounary conitions are applie: i) There are no tangential stresses an ii) Across the interface there exists a pressure ifference p c = σκ (A.1) where σ is the surface tension of the liqui an Κ is the curvature of the interface. The ynamics of that interface then correspons to the normal velocity of the flui. Drawbacks of this approach inclue the fact that one must keep track of the position of the interface at all times an solve the Navier-Stokes equations in a complex geometry. It is also important to note that the interface has no intrinsic ynamics, it is only a mathematical bounary elimiting the computational volume. This can be a problem in situations where rastic changes in topology occur, or where the ynamics at the interfacial level becomes important. Examples of such cases inclue the break-up of a liqui filament [] or the spreaing of a rop [7], two situations which are at the heart of ink transfer problems. To circumvent these potential problems, the numerical simulations presente here are performe with a technique that consiers the interface ynamics explicitly [,1]. With this metho, use extensively to stuy phase separation in binary fluis [], the computational volume is increase to inclue both the air an liqui phases. To istinguish between the two, an orer parameter or phase fiel c(x,t) is introuce, such that c=1 in the bulk of the liqui phase an c= in the ambient air. The local flui ensity, now efine everywhere in space, is given by ρ( x, t) = ρlc( x, t) + ρa (1 c( x, t)) (A.) where ρ L an ρ A are the liqui an air ensity, respectively. A similar expression efines the local flui viscosity. The orer parameter c satisfies the convection-

11 iffusion equation, relate to the Cahn-Hilliar equation [3]: c + v c = M μ (A.3) t where the chemical potential δf μ = β δc 1 c(c 1)(c 1) K is erive from the free energy functional [3] c (A.4) 1 1 K F = x β c (1 c) + c (A.5) The parameter M is a mobility coefficient an K an β -1 escribe the molecular interactions of the fluis. In the absence of flow (v=), EQ. (A.5) amits the homogeneous solutions c= an c=1, which both minimize the free energy F, as well as the omain-wall profile 1 x x c(x) = 1+ tanh (A.6) ζ corresponing to a coexistence between the two phases. The interface separating the two is locate at some arbitrary position x where c=1/, an has thickness ζ = ( Kβ ) 1/. The introuction of an interface in the system increases its free energy by an amount corresponing to the surface tension σ of the liqui. For our moel, it can be shown that σ = K / β / 3. The constant M in EQ. (A.3)) is relate to the self-iffusion constant of the liqui through D=Mβ -1, a quantity well known from Brownian motion theory [4] an given by k BT D = (A.7) 6πη where k B is the Boltzman constant, η is the viscosity of the flui an is a characteristic molecular length scale of the flui. For water η=1-3 Pa-s an for =. nm, D=1-9 m /s. On the other han, for a light oil, such as linsee oil, η=1 Pa-s, =1 nm an D= x 1-11 m /s. In general, we consier a generic value D=1-1 m /s. When flui flow is ae to the problem, EQ. (A.3) must be solve together with the Navier-Stokes equations v ρ + v t ( v) = p + η v + μ c (A.8) which has been moifie through the introuction of the term μ c on its right-han sie []. That term is non-zero only in the interfacial regions where c goes from to 1 over a short istance. It can be shown that as long as the interfacial with is much smaller than the raius of curvature of the interface (the so-calle sharp interface limit), that term prouces a pressure ifference p c across the interface which is given by EQ. (A.1). Soli-liqui interactions, as escribe in EQ. (.4), can also be inclue in the moel. Within the iffuse interface approach, the soli-flui interactions are introuce by aing an extra term to the free energy functional [5] F [c] = s G(c) (A.9) w S w where the integration is performe over all soli surfaces. The function G(c) in EQ. (A.9) assumes one value at the soli-liqui interfaces an another when the soli is in contact with air. Here we choose G(c) =g(3c -c 3 )/6, where the parameter g controls the strength of the interaction, although the exact form of this function is not important. At the wall, equilibrium requires that c = n G c (A.1) which provies us with a bounary conition for the orer parameter c. One can show that the parameter g is relate to the contact angle through cosθ=g/6. In summary, we have transforme the hyroynamic problem in the following way:

12 (i) The computational volume has been enlarge to inclue both the liqui an air phases. This was one by introucing a phase fiel c(x,t) use to istinguish between the two fluis. (ii) That fiel, which has a thermoynamic origin, can escribe a state of coexistence between the liqui an air phases. The two fluis are separate by an interface of with ζ to which is associate a surface tension σ. Both ζ an σ can be varie in the calculations. (iii) The ynamics of the orer parameter is given by EQ. (A.3). The interface separating the two fluis is ynamically stable an the coupling to the Navier-Stokes equations automatically satisfies all the require bounary conitions. In particular, the contact angle θ arises naturally, an the ynamics of the contact line, at the junction of the soli, liqui an air phases follows irectly from the ynamics of the fiels c(x,t) an v(x,t). Hence, to the price of introucing one more equation, we can ispense with tracking the interface separating the liqui from the air phase. The computational volume also remains the same throughout the calculation, a istinct avantage over other methos. A.1 Dimensionless form of the equations For computational purposes, the Cahn-Hilliar an Navier-Stokes equations must first be written in imensionless form. The basic length scale of the problem is the interface length ζ, an the basic time scale is efine as: ζ τ D = (A.11) D Scaling the length by ζ an time by τ D yiels the imensionless form of EQ. (A-3) c + v c = μ (A.1) t with μ = c(c 1)(c 1) K c (A.13) an the Navier-Stokes equations v R e + v t ( v) = p + v + C μ c a (A.14) where R e = ρd/η is an effective Reynols number (it escribes the importance of inertia with respect to iffusion), an C a = σζ/ηd is an effective capillary number. Finally, p τdp/η is a scale pressure. The interfacial with is the most important parameter of the problems. Typically, it is of the orer of a few nanometers. In the present case, to reuce the computational space neee, this length is fixe ζ=.1 μm. This choice has no influence on the flui flow as long as ζ remains the smallest length scale of the problem. Together with the value of the iffusion constant, this fixes the times scale τ D = 1-4 s, the capillary number C a = 1 an the Reynols number R e = 1-7. A. Numerical Scheme There are several efficient numerical algorithms to compute the hyroynamics flow of a compressible liqui. A complication in the present case is that the top part of the simulation box moves own towars the substrate an then back up at some prescribe velocity. Numerically, this is one by changing the computational lattice spacing ynamically in time, with complete remeshing of the computational space when require. Space is iscretize in a lattice of size Δx an Δz(t), such that L x =N x Δx an L z (t)=n z Δz(t), with L x an L z the physical size of the computational volume an N z an N x the number of computational points. Spatial points x = (x,z) are then iscretise as x=iδx an z=kδz. The fiels at position (x,z) are c(x, z, t) = c(iδx, kδz(t), t) c (t). The calculations are performe in a moving frame, by letting the lattice spacing Δz vary in time. The ik

13 time erivatives are moifie to v t t g z (A.15) where v g (z) correspons to the velocity of the computational gri at position z, ie., vp vg (z) = k (A.16) N z with v p efine in EQ. (.1). We further efine ϕ = cδz an write the moifie Cahn-Hilliar Equation (EQ. (A.3)) ϕ + ( v ϕ) = μ (A.17) t where -1-1 μ = ϕ( ϕ Δz )(ϕ Δz ) ϕ (A.18) The Navier-Stokes equations are moifie to: v R e + ( v v t g ) ( v) = p + v + C μ ϕ a (A.19) where v g =zv g (z), given by EQ. (A.16). Equations (A.17) an (A.19), together with the bounary conition EQ. (A.1), to inclue wetting an contact line, ynamics are the crucial equations of the problem. The algorithm to calculate the time evolution of the fiels ϕ an v is stanar. Euler's algorithm is use for the Cahn-Hilliar equation, while the velocity fiels are propagate forwar in time using a projection metho on a staggere computation gri [6]. The crux of the metho involves solving a Poisson equation for the pressure subject to appropriate bounary conitions at the limits of the computational box (but not at the air/liqui interface!). This is solve by Gauss-Sieel techniques. The iteration time step is usually restricte to t Re 1.1 Δx < (A.) To increase the time step, the value R e =.1 an R e =1-4 are chosen for the Reynols number of the liqui, an air respectively. It has been checke that this relatively high value oes not influence the ynamics of the flows. In general, Δx=1. an 1. <Δz(t)<1.5. When Δz falls outsie these limits, a new computational gri is generate, with linear interpolation for the values for the fiels from one gri to the next. References [1] J.M. Fetsko, an W.C. Walker, Measurements of Ink Transfer in Printing Coate Paper, Am.Ink Maker 33 (11), 38 (1955). [] J.H. De Grâce, an P.J. Mangin, A mechanistic approach to ink transfer.part I:Effect of substrate properties an press conitions, Avances in Printing Science an Technology, W.H. Banks E., Pentech Press (Lonon) 17, 31 (1984). [3] J.H. De Grâce, an P.J. Mangin, A mechanistic approach to ink transfer.part II: The splitting behaviour of inks in printing nips, Avances in Printing Science an Technology, W.H. Banks E., Pentech Press (Lonon), 19, 146 (1987). [4] R. Lenorman, C. Zarcone, an A. Sarr, Mechanism of the isplacement of one flui by another in a network of capillary ucts, J. Flui Mech. 135, 337 (1983). [5] T.J. Senen, M.A. Knackstet, an M.B. Lyne, Droplet penetration into porous networks, role of pore morphology, Noric Pulp Paper J.15, (). [6] R.J. Roberts, T.J. Senen, M.A. Knackstet an M.B. Lyne, Spreaing of aqueous liquis in unsize papers is by film flow, J. Pulp Paper Sci. 9, (3). [7] P.G. e Gennes, Wetting: statics an ynamics, Rev. Mo. Phys. 57, 87 (1985). [8] J. MacPhee, A unifie view of the film splitting process, part I, J. Am. Ink Maker 75 (1), 4 (1997). [9] J. MacPhee, A unifie view of the film splitting process, part II, J. Am. Ink Maker 75 (), 51 (1997). [1] Y.H. Zang, J.S. Aspler, M.Y. Boluk, an J.H. De Grâce, Direct measurement of tensile stress (tack) in thin ink film, J. Rheology 35(3), 345 (1991). [11] V. Kelha, M. Manninen, P. Oittinen, an J. Tiesmaki, Parallel plate tackmeter, a new tackmeter measuring fast ink film splitting in plane geometry, Graphic Arts Finlan, 1 (1973). [1] V. Kelha, M. Manninen, P. Oittinen, an J. Tiesmaki, Tack force measurement an picking, Tappi J. (4), 86 (1974). [13] M. Tirumkuulu an W.B. Russel, On the measurement of tack for ahesives, Phys. Fluis 15, 1588 (3). [14] S. Poivret, F. Nallet, C. Gay, J. Teisseire, an P. Fabre, Force response of a viscous liqui in a probe-tack geometry:

14 Fingering versus cavitation. European Phys. J. E 15, 97 (4). [15] E.W. Washburn, The ynamics of capillary flow, Phys. Rev. 17, 73 (191). [16] M.F.J. Bohan, T.C. Claypole, D.T. Gethin an M.M.H. Megat Ahme, A moel for ink impression in printing contacts, J. Pulp Paper Sci. 6, 414 (). [17] J. Schoelkopf, P.A.C. Gane, C.J. Rigway, an G.P. Matthews, Influence of inertia on liqui structure absorption into paper coating structures, Noric Pulp Pap. Res. J. 15 (5), 48 (). [18] Y. Xiang an D.W. Bousfiel, Influence of coating structure on ink tack ynamics, J. Pulp Paper Sci. 6,1 (). [19] J. S. Preston, N. J. Elton, A. Legrix, C. Nutbeem, an J. C. Husban, The role of pore ensity in the setting of offset printing ink on coate paper, Tappi J. 1 (3), 3 (). [] D. Jasnow an J. Vinals, Coarse-graine escription of thermo-capillary flows, Phys. of Fluis 7, 747 (1996). [1] D. M. Anerson, G. B. McFaen an A. A. Wheeler, Diffuse-interface methos in flui mechanics, Ann. Rev. Flui Mech. 3, 139 (1998). [] G. H. McKinley an T. Srihar, Filament-stretching geometry of complex fluis, Ann. Rev. Flui Mech. 34, 375 (). [3] J.W. Cahn an J.E. Hillar, Free energy of a nonuniform system. I: interfacial free energy, J. Chem. Phys. 8, 58 (1958). [4] P. M. Chaikin an T. C. Lubenski, Principles of Conense Matter Physics}, Cambrige University Press (1995). [5] D. Jacqmin, Contact-line ynamics of a iffuse flui interface J. Comput. Phys. 155, 96 (1999). [6] R. Peyret an T.D. Taylor, Computational Methos for Flui Flow, Springer-Verlag (New York) 1986.