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Thermomechanical analysis and modeling of the extrusion coating process.

INTRODUCTION

Extrusion coating is commonly used to coat a substrate with a polymer film. As shown in Fig. 1, it combines a film extrusion and stretching with a "calendering" step to apply the film on the substrate.

The first step (film extrusion) is similar to the cast film process: a molten polymer is extruded through a flat die and then stretched in air. During the second step, the film is coated on a substrate and then goes through a nip constituted of two rolls: a chill roll cooled down by cold water and a nip roll covered with a rubber layer. In the nip the polymer is pressed against the substrate and simultaneously cooled by the chill roll. This passage through the nip is aimed at creating a good wetting of the substrate by the molten polymer, to ensure a good adhesion between the two materials. After the nip, the web is wrapped around the chill roll using a third roll (the stripper roll), for additional cooling. This second step is thus composed of two zones: the nip and an additional "cooling zone."

In this article, we focus on the second step of the process. The first one has been widely studied. both experimentally and numerically (see for example Agassant et al. (1)). Despite an important development of the extrusion coating process in the last decades, very few studies have been devoted to the coating step, and especially to the link between the thermomechanical phenomena occurring in the nip and the final properties of the coated substrate. This is mainly due to the fact that there are numerous variables that need to be considered during processing, which makes it difficult to control the process. As pointed out by Karszes (2), the process variables can be broken down into two categories:

1. Independent variables, specified by the equipment and the products: laminator configuration (roll diameters and lengths), chill roll roughness, substrate roughness, and chemical composition, web width, etc. In our work, these variables are fixed and are those of the pilot line of Arcelor Research, presented in the "Presentation of the pilot line and the used materials" subsection.

2. Dependent variables, which can be changed by the operator during the process: polymer melt temperature, die gap, draw ratio, nip pressure, chill roll temperature, line speed, nip length, etc. In this article, we explore the effect of some of these variables on the thermomechanical phenomena during the coating step of an extrusion coating process.

It is important to notice that, during the coating step, two main phenomena occur:

1. Development of the real contact area between the polymer film and the substrate, which is determinant for final adhesion or barrier properties.

2. Cooling of the molten polymer and the substrate which is generally preheated. Cooling on the chill roll, imposed by an internal flow of cold water, begins in the nip and it is very important for the release: if the polymer is not solidified, it will stick on the chill roll and there will be release problems at the contact point with the stripper roll. In most cases, the residence time in the nip is insufficient to solidify the polymer before its exit from the nip; this is the reason why the polymer is kept in contact with the chill roll by using a stripper roll, as shown in Fig. 1. The position of the stripper roll is adjustable, so that the length of this coating step is variable.

[FIGURE 1 OMITTED]

A zone of particular interest and importance is the nip, i.e. the contact surface formed by the soft nip roll (covered with a rubber layer) pushed against the chill roll. Complex thermomechanical phenomena occur in a very short time (the mean residence time in the nip is some milliseconds): what is heat transfer and viscous dissipation between the polymer film, the substrate, the metal of the chill roll and the rubber strip around the nip roll? What is exactly the nature of the mechanical contact between the two rolls? Is it a purely hydrodynamic contact, during which the polymer flow induces a pressure peak counterbalanced by the loading force? Is it a purely elastic contact equivalent to the dry contact between the two rolls or is it a mixed elasto-hydrodynamic contact? In other words, does a macroscopic polymer flow occur in the nip, with a thickness reduction of the polymer film? To answer all these questions, an experimental investigation is presented in the "Experimental investigation" section. The thermomechanical analysis of the coating step, deduced from this experimental investigation, is used for the model developed in the "Thermomechanical modeling" section. This model allows to predict the thermomechanical phenomena in the nip, which is of prime importance for the final properties of the coated substrate, especially the adhesion between the steel and the polymer film and the defects of the coated film. It is thus possible to link the thermomechanical phenomena with the final properties.

STATE OF THE ART

Very few studies in the literature are devoted to the study of the coating step in the extrusion coating process. In this section, we first explore some processes similar to the extrusion coating process, and then we present the different models dealing with thermal and thermomechanical phenomena in the coating step.

Comparison with Other Similar Processes

There are many processes in which melt polymer passes through the nip between two cylindrical rolls: calendering and all kind of coating processes such as roll coating, blade coating, kiss coating (see for example Booth (3)). Most of these processes have been widely studied both experimentally and numerically. However, as pointed out by Takase et al. (4), the conditions of the extrusion coating process are significantly different from, for example, those of the calendering process.

It is thus interesting to understand the specificities of the extrusion coating process, in comparison with two very similar processes: the calendering process and the roll coating process.

The Calendering Process. Calendering allows to produce sheets of elastomers or highly viscous polymers like PVC. The equipment consists of a train of several polished rolls forming sequential nips through which the material passes, reducing progressively the sheet thickness,

The different characteristics of this process are the following:

1. The gap between the driven rolls is constant and no load force is applied to the rolls;

2. Viscous dissipation is important so the process is highly nonisothermal;

3. The polymer accumulates upstream of the nip, to form what is called a bank, whose dimension can be 50 times the gap thickness. This bank acts as an intermediate reservoir between entry and exit flows, with very complex recirculating vortices (see, for example, Agassant et al. (5)).

The basic principle of calendering is quite different from the extrusion coating process. The aim is the thickness reduction: the feed thickness is greater than the gap thickness. A bank is created at the nip entrance, creating a transversal flow that leads to a widening of the polymer sheet. The polymer kinematics in the nip generates stresses, which induces a pressure peak between the rolls. Some experimental measurements show that this peak increases with higher roll speeds and thinner gaps.

The Roll Coating Process. In this process, fluid (paint, lubricant, etc.) is applied on a substrate by passing it through a coating nip formed by a rigid roll and a flexible roll, covered with an elastomeric layer. The flexible roll is pressed against the rigid roll using an external load. The nip is thus totally similar to the one in the extrusion coating process. The flow in the inlet region determines the flow rate through the nip or the coating thickness. This flow induces pressure and viscous stresses, which will deform the elastomeric layer. The complete analysis of a Newtonian fluid flow in a deformable contact requires the use of the theory of elasto-hydrodynamic lubrication.

But. in the classical roll coating process, the amount of fluid available in the entry zone of the contact is very big and only a small fraction will flow through the contact, depending on the nip load, the roll speeds, the Young's modulus of the roll, and the fluid viscosity. On the contrary, in the extrusion coating process the feed flow rate is determined by the extrusion flow rate.

The Thermal Models of the Extrusion Coating Process

We focus now on the studies that concern only the thermal phenomena in the extrusion coating process.

The first category of models (Karszes (6), Blethen (7), Scannel (8)) concern some simple thermal exchange considerations, allowing to optimize the cooling capacity of chill rolls. They show the importance of the choice of the materials, the design of the chill roll, and the thermal resistance due to scaling of the water passages.

The second category (Van Ness (9) and Alheid (10)) solves the heat balance equation, leading to the temperature profiles through a multilayered structure constituted of the outer shell of the chill roll, a polymer layer, and a substrate, each layer having its particular thermal constants. This results in the evolution of the average temperature in the different layers as a function of the position around the chill roll.

Van Ness (9) studied the effect of different parameters on the "stripping temperature" (temperature of the melt at the end of the cooling zone, when it leaves the chill roll). The more significant factors for reducing the stripping temperature are a lower water temperature, a thinner outer shell for the chill roll, the use of materials with better thermal conductivity, a larger chill roll diameter, a lower line speed, or a higher heat transfer coefficient.

Alheid (10), using only analytic methods, points out the existence of a temperature gradient in the polymer thickness and the effect of some variables on the chill roll cooling capacity.

The third category of models is proposed by Trouilhet et al. (11), who tried to elucidate some extrusion problems, such as poor adhesion to the substrate or curling due to postcrystallization, using a thermal model. The authors proposed an unsteady, one-dimensional heat conduction model to calculate the temperature profiles in the polymer and the substrate. Typical results give the evolution of the temperature profiles in the substrate (paper layer) and in the polymer (LDPE layer) at different time steps. Knowing the polymer solidification temperature allows to determine the solidification time (noted [t.sub.s]), which can be compared with the residence time in the nip ([t.sub.n] = 2a/v, where 2a is the nip width and v the line speed) and can thus explain the origin of some defects. For example, to obtain good final properties (good adhesion, no bubbles), they stated that these two characteristic times should verify the following inequalities: [t.sub.n]/2 < [t.sub.s] < [t.sub.n], such that solidification occurs under high pressure.

The authors conclude that a simple heat transfer model can help explain and correct problems occurring in extrusion coating process.

The Thermomechanical Models of the Extrusion Coating Process

Only two studies couple the fluid flow and the heat transfer in the nip.

El Youssef (12) proposed a penetration model of the polymer melt during the extrusion coating of paper or paper boards substrates. Paper is considered as a porous medium, constituted of flat areas and cylindrical pores. The model considers a cylindrical volume of polymer, located in the flat area between the pores and traveling with the polymer film. When it goes through the nip, the cold chill roll exerts a compression force, which induces a radial polymer flow, considered as an unsteady nonisothermal squeeze flow between two disks, into the pores of the paper substrate.

The thickness reduction of the cylindrical volume of polymer, which depends on the nip load, the material temperatures, and the polymer rheology, is related to the final penetration depth in the pores, which is the relevant parameter to estimate the adhesion between the polymer and the substrate: the higher this penetration length, the better are the adhesion and the coating quality, and this explains why the adhesion increases with the polymer melt, web, and chill roll temperatures and decreases for higher line speeds and thinner polymer films. The author concluded that these results are in agreement with the experimental evidence found in Karszes (6).

Takase et al. (4) studied the effect of processing conditions and material properties on the cooling and melt flow in the nip. They solved the equations of continuity, motion, and energy using a Galerkin finite element method on a volume of polymer at the nip entrance. They checked the accuracy of their numerical model by comparing their temperature profiles with those found by Van Ness (9) and Karszes (6) and obtained a good qualitative agreement. Concerning the flow field, the computations reveal the formation, at the nip entrance, of a bank whose height (20-80 [micro]m) depends on coating conditions, in agreement with the experimental values. It appears that the bank height increases when increasing the extrusion temperature and melt thickness and decreases when decreasing the line speed and the shear thinning sensitivity. Nevertheless, velocity and pressure fields are not provided. The authors concluded that a good quality coating requires a small bank, like in calendering process.

Conclusions

The different thermal models developed until now mainly deal with coating on paper substrate. Transposition to metal coating must be done carefully, especially because thermal exchanges are much more important.

Concerning modeling of the polymer flow in the nip, the existing literature considers a macroscopic flow leading to a thickness reduction. Still, there are two different approaches:

1. A "squeeze flow" generated by the nip force which forces the polymer towards the pore of the substrate surface in El Youssef (12);

2. A flow similar to the calendering process, with the formation of a bank at the nip entrance in Takase et al. (4).

None of these approaches is based on an experimental approach and a thermomechanical analysis of the nip region, and this is the purpose of this article.

EXPERIMENTAL INVESTIGATION

The aim of this section is to investigate the thermomechanical phenomena in the nip.

Presentation of the Pilot Line and the Used Materials

The polymer film (PET or PP) is extruded through a flat die (35 cm in width and 1 mm in gap). To minimize the geometrical defects (width reduction and "dog bone" defect), the die is approached as close as possible to the laminator. The stretching distance is typically around 200 mm and the final thickness after stretching is e = 25-30 [micro]m. The laminator is constituted of two rolls of width W = 35 cm and diameters 2[R.sub.1] = 40 cm for the chill roll and 2[R.sub.2] = 30 cm for the press roll. Both are designed with an internal water cooling channel. The chill roll is rigid whereas the press roll is covered with a rubber layer of thickness b = 2 cm. Hydraulic jacks on the press roll allow to load the chill roll with a typical linear load F/W between 5 and 35 kN/m. The rolls are not motorized and their drive is supplied by the substrate drawn by the winder, which generates a line tension close to [T.sub.1] = 1 kN: the line speed v can reach 60 m/min in industrial conditions, still the classical speed line in this study will be 10-20 m/min. The release of the coated substrate is possible by using the stripper roll, whose position is adjustable (see Fig. 1).

The substrate is a Tin Free Steel (17.5 cm in width and 200 [micro]m in thickness), produced by applying electrolytic chromic acid treatment over steel sheets. Before the chromium coating, the temper rolling imposes the final surface roughness to the steel sheet. It presents two kinds of surface topology (see Fig. 2):

[FIGURE 2 OMITTED]

1. Case 1: the steel sheet shows classical rolling scratches;

2. Case 2: the steel sheet has more isotropic roughness with plateaus and valleys.

To increase the welting of the polymer film on the substrate, the steel sheets are preheated by infrared ovens, so that their temperature before coaling is 170-200[degrees]C.

Mechanical Experiments

The Dry Contact. As the nip roll is covered with a rubber layer, a rectangular surface contact is created when it is loaded against the chill roll. The dimension of this surface is a key parameter since it determines the time during which the polymer is forced against the substrate. A paper, sensitive to pressure, is used to measure these dimensions. Figure 3 gives the evolution of the nip width with the nip load. It is to notice that the dimensions of the nip are not modified when the coated substrate runs between the two rolls.

Effect of the Passage in the Nip on the Film. The purpose of this experiment is to investigate the effect of the nip force on the film thickness and its surface roughness.

[FIGURE 3 OMITTED]

In roll coating processes, the nip force is a key parameter and the thickness of the coated polymer decreases when the nip force is increased. Figure 4 shows clearly that, in extrusion coating, the nip force does not influence the polymer thickness coated on the substrate. This means that, whatever the value of the nip force, the amount of coated polymer is the same. Similarly, the preheating of the substrate does not modify the thickness of the coated polymer film. The nonsymmetric film thickness profiles shown Fig. 4 are typical of the "dog bone" defect, with over-thickness on the film edges, obtained in cast film process.

[FIGURE 4 OMITTED]

On the other hand, Fig. 5 compares the state of the polymer film surface before and after the contact with the steel sheet. It shows that the roughness of the steel sheet is transferred to the polymer film surface. During the passage in the nip, we can even observe the rolling scratches of the steel sheet on the film surface.

[FIGURE 5 OMITTED]

Investigation of the Nip Entrance. The purpose of this section is to investigate what is occurring in the nip entrance. If the polymer flows in the nip, then thickness difference should exist between the film in the nip and outside it. which means that a bank is created at the nip en trance, like the calendering process.

At a first glance, no bank is visible to the naked eye at the nip entrance. Furthermore, the extrusion coating line was stopped quickly, the film was cooled, the laminator was opened, and a film with a part in the nip and outside was cut and observed with a microscope as in Takase et al. (4). There is clearly no thickness difference between the film in the nip and outside (caliper thickness measurements). Microscope observation shows a clear separation line between the film in the nip and outside it. Observation with a grazing light allows to distinguish a slight over-thickness (Fig. 6), on the separation line with an extension of several tens of microns. It shows that a temporary nonuniform flow exists at nip entry in the polymer film. We can expect that this over-thickness zone which corresponds to a temporary decrease of the polymer mean velocity is in relation with the increase of its pressure required for insuring its entry in the nip (see the subsection "The contact with the sheet and the polymer film").

[FIGURE 6 OMITTED]

Thermal Experiments

Thermal phenomena are of prime importance in the extrusion coating process. In order to better understand the different thermal phenomena, temperature of the different materials is measured at different key points of the extrusion coating line, as shown on Figs. 7 and 8. These measurements were done after achievement of steady state conditions using either a contact thermocouple or an infrared camera. The effect of the process line speed (v) was investigated: Fig. 7 shows the measurements at 10 m/min and Fig. 8 at 15 m/min.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

The steel sheet is preheated before its entrance in the nip and the measurements show that the temperature is quite homogeneous along the steel sheet width, before its entrance in the nip.

These temperature measurements have led to the following conclusions:

1. The coated substrate is subjected to very strong cooling in the nip. It is to notice that we have no direct access to the polymer temperature in the nip region. Nevertheless, we can speculate high temperature gradients in the polymer thickness. This is due to the fact that the polymer surface in contact with the chill roll cools immediately, whereas the opposite side, in contact with the preheated steel sheet will cool much more slowly.

2. The cooling is less efficient as the line speed increases: for a given cold water flow rate through the chill rolls, the heat removal decreases as the line speed increases. If we want to obtain a more efficient cooling of the cooling device, it is necessary to increase the cold water flow rate.

Conclusions

We are now able, thanks to this experimental investigation, to describe precisely the main thermomechanical phenomena:

1. The film thickness is not reduced in the nip; it means, in particular, that there is no macroscopic flow of the molten polymer and that no bank is created at the nip entrance.

2. Nevertheless, a roughness transfer appears to happen in the nip, that is to say a microscopic flow, a kind of "redistribution" of the polymer at the interface between the substrate and the polymer film. This flow is induced by the loading of the nip force thanks to the fluidity of the polymer melt.

3. Cooling in the nip is very strong as indicated by the temperature measurements. The chill roll tends to impose its temperature to all the materials present in the nip and this results in a polymer viscosity increase, that slows down and may even stop the roughness transfer at the interface. If this occurs, the contact between the two antagonists is not optimal, which may lead to some defects such as poor adhesion, or bubble defects at the interface between the two antagonists.

THERMOMECHANICAL MODELING

Previous experiments have pointed out that there is no coupling between thermal and mechanical phenomena at the macroscopic level. This allows to develop separately the mechanical and thermal models. In this section, several models are proposed to account for the polymer flow within the roughness of the substrate surface. This flow is induced by the pressure profile developed in the nip, given by the Hertz contact theory whose adaptation to our configuration is made in the "Mechanical modeling of the nip" subsection. Besides, the thermal model giving the temperature profiles in the laminator, and especially the temperature at the polymer/substrate boundary, is presented in the "Thermal modeling of the nip" subsection. Finally, the flow within the various roughness of the steel sheets is modeled in the "Model of the polymer penetration in the substrate valleys" subsection.

Mechanical Modeling of the Nip

We will first consider the dry contact between the two rolls (pressure and chill rolls) and then, investigate the effect of the polymer film and the substrate on the characteristics of the dry contact. In all these subsections, the nip is supposed to be isothermal, which means that the polymer temperature equals the substrate temperature.

Investigation of the Dry Contact. It is of prime interest for our model to determine precisely the nip-width and the pressure profile developed in the nip. The first investigation of a dry contact between two deformable solids is due to Hertz (13). His study gives the dimensions of the contact surface, the pressure profile, and the deformation of the two solids. Considering two cylinders of width [R.sub.1] and [R.sub.2], Young's modulus [E.sub.1] and [E.sub.2], and Poisson's ratio [v.sub.1] and [v.sub.2], we can define:

(1.) A reduced elastic modulus:

E* = [([1-[v.sub.1.sup.2]/[E.sub.1]+1-[v.sub.2.sup.2]/[E.sub.2])].sup.-1]] (1)

(2.) An equivalent reduced radius:

[R.sub.eq] = [(1/[R.sub.1]+1/[R.sub.2]).sup.1]] (2)

The main results are the following, when pressing one cylinder against the other with a normal force F, the half-width of the contact zone is

a = 2/[square root of][pi]][[square root of][R.sub.eq.]F/E*W (3)

The pressure distribution in the contact is elliptic:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [p.sub.0] is the maximal pressure in the contact and can be written as

[p.sub.0] = 2F/[pi]aW = [(EF*/[pi][WR.sub.eq].sup.1/2]] (5)

The mean pressure in the contact can be written as

[p.sub.m] = [pi]/4[p.sub.0] = F/2aW (6)

For a given extrusion coating line, Eq. 4 gives the variation of the nip half-width a with the nip force F. Still, this calculated value must be corrected because, in our laminator, the press roll is covered with a rubber layer, whose thickness b appears to be a critical parameter. As shown by several authors such as Johnson (14) or Nowell and Hills (15), if the thickness of the rubber layer decreases, the effect of the core material cannot be neglected. The core material being stiffer than the rubber layer, it tends to reduce the nip width and to increase the maximal pressure. The variation of the corrective coefficients as a function of the ratio b/a is given in Nowell and Hills (15). It appears that the variations of the half width and the maximal pressure are negligible when the ratio b/a is higher than 1.

Figure 9 shows that the calculated corrected values of the nip width are in good agreement with the experimental values obtained in the "Mechanical experiments" subsection.

[FIGURE 9 OMITTED]

Knowing the nip dimensions, it is now possible, using Eqs. 4-6, to compute the pressure profile in the nip. Such pressure profiles are given in Fig. 10 for two different nip loads, as well as the mean pressure values.

[FIGURE 10 OMITTED]

The Contact with the Steel Sheet and the Polymer Film. In our case, the rolls are not motorized but are driven by the substrate passage, a very thin (about 200 [micro]m) and rigid steel sheet. The resulting line tension created tends to separate the two rolls and may modify the nip characteristics (its dimensions and the pressure profile).

Trouilhet et al. [11] tried to evaluate the depression generated by the line tension. Considering a force balance, they obtained:

p = [T.sub.1]/[WR.sub.2] (7)

where p [N/[m.sup.2]] is the pressure generated by the tension line, [R.sub.2][m] is the chill roll radius, W [m] the substrate width and [T.sub.1] [N] the tension line.

The resulting pressure computed by the authors is 0.004 MPa, which is about 1000 times less than the maximum pressure. A similar evaluation for our pilot line gives a value of 0.03 MPa, which is as well negligible before the pressure developed by the nip load.

It is thus reasonable to consider that the substrate does not modify the nip characteristics given by the Hertz theory.

Let us investigate now the effect of the polymer film on the nip characteristics. We considered the equivalent system shown on Fig. 11, constituted by a rigid plane and an equivalent roll, whose elasticity and radius are given by Eqs. 2 and 3. The deformation of the equivalent cylinder is given by the Hertz theory and can be written as

[DELTA][epsilon] = [a.sub.2]/2[R.sub.eq] (8)

Given the dimensions and the nip loads of our pilot line, the calculated value [DELTA][epsilon] is higher than 500 [micro]m. The coated polymer film thickness e is about 25-30 [micro]m and is thus negligible compared with the deformation of the cylinder. This geometrical argument allows us to conclude that the polymer film passage in the nip does not disturb significantly the dry Hertz contact.

Remark: It is easy to understand why the polymer film passes through the nip without any significant thickness reduction. Let us consider the polymer flow just before the nip entrance, where its pressure must increase from the atmospheric pressure to the mean contact pressure [p.sub.m]. This increase in pressure is obtained by shearing the polymer in the inlet zone which may be considered as a converging channel with the attack angle (see Fig. 11): [gamma] ~ a/[R.sub.eq] - 0.058 rad. Integration of the Reynold's equation, starting from an infinite value of the material thickness (which means an infinite length of the inlet zone), provides that the film thickness [e.sub.0] of a material with viscosity [eta] is (see, for example, Wilson et al. (16))

[FIGURE 11 OMITTED]

[e.sub.0] = 6[eta]v cot[gamma]/[[rho].sub.m] or [eta]v/[e.sub.0] = tan[gamma]/6[p.sub.m] (9)

If the entry value of the material thickness is finite ([e.sub.i]), the same computation holds. It is to notice that, as far as [e.sub.i] > 1.5 [e.sub.0], there will be a reverse flow in the upstream region, which means the initiation of a bank. Equation 9 points out that the pressure increases as the film thickness decreases.

For typical values used on the pilot line (v = 10 m/min, [gamma] = 0.058 rad, [eta] = 100 Pa s, [p.sub.m] = 1 MPa), Eq. 9 leads to [e.sub.0] = 1.7 mm, which is two orders of magnitude bigger than the coated polymer thickness. As a conclusion, we can say that there is no reverse flow in the inlet zone and that the film passes through the nip without significant thickness reduction, as observed experimentally.

Thermal Modeling of the Nip

The experimental investigation of the temperature in the laminator, presented in the "Mechanical experiments" subsection, has shown the importance and the complexity of the thermal phenomena in the extrusion coating process. This complexity is due to the fact that cooling should be a compromise between two extreme situations:

1. If the cooling is too slow, the polymer is still molten when it comes at the release point, which could provoke some release problems (the polymer can remain stuck on the chill roll);

(2.) If the cooling is too fast, the wetting of the substrate by the polymer is poor, which could hinder a good adhesion between the two antagonists.

As a consequence, the determination of the cooling rate of the polymer film requires great attention. In particular, the polymer temperature in contact with the substrate, which is not accessible by experimental measurements as shown in the "Mechanical experiments" subsection, is of prime interest.

A thermal model will allow to calculate the temperature variations along the nip in the thickness of the different materials in contact (chill roll, polymer film, steel sheet, press roll). It will give access to the polymer tem-perature at the interface with the steel sheet and will allow to study the effect of different parameters on the cooling capacity of the laminator such as the line speed, the regulation temperature, the initial substrate, film temperatures, the nip dimensions, etc.

Presentation of the Thermal Model. The thermal model, developed at Arcelor Research, is presented briefly: the extrusion coating line is divided into several blocks, as shown on Fig. 12, and the resolution of the heat balance equation is made block after block, with the calculated temperature of the previous block as initial conditions for the computation in the current block. Each of these blocks is constituted of several material layers with different boundary conditions. For example, in the nip, denoted as Block 4 on Fig. 13, the following materials are in contact:

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

1. the outer shell of the nip roll in stainless steel,

2. the external rubber layer of the nip roll,

3. the substrate (steel sheet),

4. the polymer film, and

5. the outer shell of the chill roll in stainless steel.

The boundary conditions on each side are convective exchanges with the water of the cooling channels, which are taken into account through a heat transfer coefficient (h_awater).

The Heat Balance Equation. Each layer is considered as a slab of finite thickness e. As there is no macroscopic flow, there is no mechanical deformation of the slab and the thickness e is constant. The energy balance equation is

[rho](T)C(T)dT/dt = div(k(T)gradT (10))

where k is the thermal conductivity, [row] the density, and C the heat capacity.

We consider a fixed frame of reference. Let us assume that the heat flow is steady [sigma]T/[sigma[t = 0, that there is no temperature variation in the slab width, i.e. in the [gamma]-direction, T = T(x, z), and that the conductive term in x-direction is negligible compared with the convected term (i.e. the Peclet's number is high). As the slab moves at a velocity v, the heat balance Eq. 10 can be written as

[rho](T)C(T)(v[[partial derivative]/[partial derivative]]x) = [partial derivative]/[partial derivative]z(k(T)[partial derivative]T/partial derivative]z] (11)

Finally, if we consider a frame of reference moving with the slab, Eq. 11 can be written as

[rho](T)C(T)[partial.derivative]T/[[partial.derivative]t = [partial.derivative]/[partial.derivative]z(k(T)[partial.derivative]T/[partial.derivative]z)(12)

Equation 12 is solved using a classical finite volume method.

Results. The model provides the temperature variation through the thickness of the different materials along the extrusion coating line as a function of time (or distance in the nip). The adjustable parameters are the heat transfer coefficients (especially those between the outer shell and the cooling water), determined by fitting with the experimental temperature measurements presented in the "Thermal experiments" subsection.

The model takes into account the temperature decrease of the polymer film during the stretching path in air (for a detailed study of this temperature decrease, see Sollogoub et al. [17]) and of the steel sheet before entering the nip. Still, for this stretching step, only the mean temperature of the polymer film in the thickness is considered.

Typical results are shown Fig. 14 (temperature variation vs. the distance traveled by the polymer film from the die exit) for standard conditions: preheated steel sheet temperature 200[degrees]C, polymer thickness 25-30 [micro]m, line speed 10 m/min, extrusion temperature 270[degrees]C, air gap 215 mm.

[FIGURE 14 OMITTED]

There is a large temperature gradient in the thickness of the polymer: the melt in contact with the chill roll freezes instantly whereas the temperature of the polymer in contact with the steel sheet decreases more or less rapidly, depending on the coating conditions. This temperature gradient vanishes, as the coated substrate progresses in the cooling zone.

Two results are of particular interest for our study:

(1.) The temperature at the polymer/substrate boundary, which determines whether the polymer is able to flow in the asperities of the substrate or not;

(2.) The temperature at the polymer/chill-roll boundary, which determines the release temperature.

Influence of Several Parameters on the Film Cooling. This model allows to investigate the influence of several key parameters: the preheating substrate temperature, the extrusion temperature, the film thickness, the line speed, the thickness of the outer shell of the rolls, the roll dimensions, the flow rate of the cooling water, etc. Nevertheless, in this article, we fix the dimension and characteristics of the rolls and we test the effect of the preheated substrate temperature, the extrusion temperature, the melt thickness, and the line speed (when investigating the effect of one parameter, the others were kept constant).

Effect of the Preheated Substrate Temperature. Figure 15 presents the temperature variation in the polymer thickness for three substrate temperatures (25[degrees]C, 200[degrees]C, 350[degrees]C) and shows that the substrate temperature has a significant influence on the polymer temperature in the laminator. Actually, the hotter the steel sheet is, the hotter are the polymer and even the chill roll surface. For example, for a substrate temperature of 350[degrees]C, the chill roll temperature remains higher than 50[degrees]C, denoting that its cooling capacity is insufficient to maintain the temperature at 25[degrees]C, which is the temperature of the cooling water.

[FIGURE 15 OMITTED]

It is interesting to notice that the temperature gradient in the polymer thickness vanishes rapidly and that the polymer temperature levels off, as the maximal heat removal of the chill roll is reached. It means that a bigger wrapping angle is not able to reduce the release temperature.

Effect of the Extrusion Temperature. The computation shows that the extrusion temperature (and thus the polymer temperature when it enters the nip) has no effect on the thermal phenomena in the laminator. This is because the steel sheet imposes its temperature to the polymer film, independently on the temperature of the film before the coating. As soon as the polymer film enters the nip, its surface in contact with the steel sheet reaches instantly the steel sheet temperature, which means in particular that, if the preheated substrate temperature is higher than the polymer temperature at the nip entrance, the melt is heated by the substrate.

Effect of the Film Thickness. Figure 16 plots the mean temperature in the thickness along the contact for different melt thicknesses. The effect of this parameter is significant: the thicker the polymer film is, the more difficult the temperature gradient in the polymer disappears (particularly for thickness from 100 [Mm up). It appears that the wrapping angle is important contrary to what was observed for the influence of the preheated substrate. This is due to the speed of heat removal, much more rapid for the steel sheet than for the polymer film.

[FIGURE 16 OMITTED]

Effect of the Line Speed. It appears on Fig. 17 that the line speed is a key parameter for the thermal phenomena, as pointed out by experience. Actually, as the speed increases, the cooling capacity of the chill roll is lower and, for a given water flow and water temperature, the surface temperature of the chill roll increases.

[FIGURE 17 OMITTED]

[FIGURE 18 OMITTED]

Conclusions. The model allows to predict the temperature in the thickness of the different materials in the laminator. It shows that, when entering the nip, a temperature gradient is created in the polymer thickness, between the two limit temperatures fixed by the chill roll surface and the steel sheet temperatures. This temperature gradient vanishes as the coated substrate progresses in the laminator. This model is thus a good means to know whether the polymer at the interface between the steel sheet is able to flow in the valleys of the substrate or not.

The investigation of the influence of several process parameters in the laminator has pointed out that, when increasing the line speed and the temperature of the preheated substrate, all the temperatures in the laminator are increased. Similar results were observed experimentally for the influence of the line speed. In particular, it allows to delay polymer solidification, that is to say the moment at which the polymer flow is stopped. Nevertheless, since the temperature of the chill roll surface is higher, this may hinder the release of the coated substrate.

It is also shown that the wrapping angle has an important influence for melt thickness from 100/Mm up.

Model of the Polymer Penetration in the Substrate Valleys

Modeling Process. In the model developed by EI Youssef (12), the substrate surface is composed of large plate areas (of characteristic length 200 [micro]m) alternated with small pores (4 [micro]m). The surface of the plate areas is much greater than the surface of the pores. This explains why, when the coated substrate enters the nip. the dominant flow is the radial squeeze flow occurring on the top of the large plate areas between the pores, which in turn induces melt penetration in the pore; but this flow is not investigated by the author.

Our approach is different: the previous thermal model shows that the lower part of the film in contact with the substrate is still in a molten state, whereas the upper part, in contact with the chill roll, is solidified and cannot flow any more. This means that the flow can happen only at the interface with the steel sheet and not in all the polymer film thickness. Besides, the surface of the steel sheets is quite a succession of plateaus and valleys of rather equivalent dimensions. It implies that the dominant flow is not a radial one like that in El Youssef's study but a flow orthogonal to the steel sheet that may happen every where at the interface. As said before, this flow can be considered rather like a local polymer redistribution and has thus no effect on the film dimensions. It is a microscopic flow in the steel sheet roughness and it does not imply a significant decrease of the polymer film thickness. Figure 18 shows the differences between these two approaches.

Several physical phenomena can promote or hinder this flow in the steel sheets roughness:

1. First of all, the pressure generated in the nip by the load force tends to force the polymer to flow in the valleys. It has been shown that, outside the nip, the pressure generated by the line tension is too low to expect polymer flow in this region.

2. The air, entrapped in the valleys and which pressure increases as the polymer flows in the valleys, tends to block the flow. Still, if we look carefully at the surface roughness of the steel sheets (cf. Fig. 2), it appears that there is generally a way of venting for the air. It is obvious for the steel sheet 1, for which the air venting is possible through the rolling scratches. It is less obvious on the steel sheet 2, but a meticulous observation reveals the existence of "connections" between the valleys. We can thus assume that the counter pressure, created by the entrapped air, is negligible.

3. The capillary forces are assumed to be negligible in comparison with the load force. So, we can assume that the pressure profile developed in the nip (given by the dry contact model) is the driving force for the flow into the steel sheets roughness. The existence and the magnitude of this flow will depend on the polymer viscosity, that is, the polymer temperature at the film/substrate boundary, known by using the thermal model described in the "Thermal modeling of the nip" subsection. Finally, the flow will depend on the surface topography of the substrate, modeled in the next subsection.

Model of the Substrate State Surface. Modeling topographies of real surfaces is a difficult task because of their random structures. Roughness measurements on the two surfaces (cf. Fig. 2) were taken, giving a 3D reconstruction of the surface roughness of the two steel sheets (shown Figs. 19 and 20), as well as the characteristic roughness parameters. To model these surface topographies, we try to detect a dominant repeatable geometrical shape.

[FIGURE 19 OMITTED]

[FIGURE 20 OMITTED]

Surface 1 shows rolling scratches that can be assimilated with dihedrons of depth L and slope [alpha] (flank angle), [w.sub.1] and [w.sub.2] are lengths defining, respectively, the entry and the bottom of the dihedron (see Fig. 19). When the polymer reaches the depth l, it occupies a surface of width w (l) [member of] [[w.sub.2], [w.sub.1]], defined as follows:

w(l) = [w.sub.1] - ltanx (13)

More roughly, we consider the topography of the steel sheet 2 as a series of cylindrically shaped anfractuosities or pores, of various radii R and depths L, as shown Fig. 20.

Calculation of the Filling Time. For each asperity, we determine a filling time, assuming a Newtonian behavior for the polymer.

The filling time calculation is simply based on a flow rate balance:

Q = dl/dt [S.sub.e] = [Q.sub.rugosity] (14)

where [S.sub.e] is the entrance surface of the roughness and Qrugosity the flow rate in the roughness, generated by the pressure [p.sub.m] due to the nip load.

* Steel sheet 1: The flow of a Newtonian fluid in a dihedron is analyzed, for example, in Agassant et al. [18]; the surface of the entry section is given by

[S.sub.e] = 2[w.sub.1]2a (15)

and the relationship between flow rate and pressure drop can be written as

[Q.sub.dihedron] = 2a/[eta][p.sub.m]/([w.sub.1.sup.2] - [w.sub.2.sup.2)[w.sub.1.sup.2][w.sup.2]/[tan.sup.2][alpha](sin2[alpha]/2-[alpha] cos2[alpha]) (16)

* Steel sheet 2: In this case, the flow is a classical Pois-euille flow, with the following characteristics: and

[S.sub.e] = [pi][R.sup.2] (17)

[Q.sub.dihedron] = [pi]/8[eta][p.sub.m]/l[R.sup.4] (18)

To calculate the filling time, we must integrate the differential Eq. 14 for the two steel sheets.

[FIGURE 21 OMITTED]

We first assume that the polymer viscosity is constant:

[eta] = [[eta].sub.0]

The resulting filling time, called "constant viscosity filling time," [t.sub.r0] can be written as

Steel sheet 1:

[t.sub.r0] = 2[[eta].sub.0]/[p.sub.m]([[tan.sup.2[alpha] + [tan.sup.4][alpha]/tan[alpha] - [alpha] + [alpha][tan.sup.2[alpha]]) x [[w.sub.1]/[w.sub.1]tan[alpha] - L[tan.sup.2][alpha] - 1/tan[alpha] - L/[w.sub.1] (19)

Steel sheet 2:

[[t.sub.r0] = [4[eta].sub.0][L.sup.2]/[p.sub.m][R.sup.2]] (20)

So, whatever the surface geometry of the substrate, the constant viscosity filling time can be written as

[t.sub.r0][infinity][[eta].sub.0/[p.sub.m].f(geometrical parameters of the roughness) (21)

It depends on the following factors:

1. The load in the nip. which accelerates the filling;

2. The polymer viscosity (i.e. the temperature) at the interface between the film and the substrate;

3. The geometrical characteristics of the steel sheet (shape and size of the roughness of the sheet).

In the case of the steel sheet 2, the filling time (expression 20) is all the longer as the steel sheet asperity is deep and of small radius.

The influence of the geometrical parameters on the filling time is less obvious for the steel sheet 1 (Eq. 19). Figure 21 shows that the filling time increases when increasing the flank angle [alpha] and reducing the entry upper width [w.sub.1].

In fact, it does not seem reasonable to consider a constant viscosity, given the strong cooling of the polymer during coating. From the previously described thermal model, we can deduce the temperature evolution in the nip of the polymer in contact with the steel sheet, considering, for example, a mean temperature on a thickness of 10-[micro]m (cf. Fig. 22). Based on the Arrhenius law, we obtain the variation of the mean viscosity of the polymer near the substrate as a function of the residence time in the nip, as shown on Fig. 23.

[FIGURE 22 OMITTED]

[FIGURE 23 OMITTED]

To provide analytical integration of the differential Eq. 14 we consider an exponential dependence for the viscosity as a function of time:

[eta] = [[eta].sub.0] exp([beta]t) (22)

The parameters [[eta].sub.0] and [beta] are given by fitting the graph of the Fig. 24 (in this case, [[eta].sub.0] = 7500 Pa s and [beta] = 26 [s.sup.-1]).

We obtain thus a new expression of the filling time that takes into account the strong cooling of the polymer in the laminator. It writes, as a function of the constant viscosity filling time [t.sub.r0], as follows (with [beta] X [t.sub.r0] [less than or equal to]1, [t.sub.r0] calculated using Eqs. 19 or 20):

[t.sub.] = 1n(1-[beta][t.sub.r0]/[beta] (23)

It means that increasing [beta], that is to say considering a stronger cooling, for a given typical value of [t.sub.r0] = 100 ms, leads to a higher filling time [t.sub.r] (up to 300 ms).

It is to notice that these filling times are of the order magnitude of the residence time in the nip, which may explain the roughness transfer of the steel surface on the coated polymer (see the subsection "Mechanical experiments" and Fig. 5).

CONCLUSIONS

An experimental investigation of the coating step enables to characterize the leading thermomechanical phenomena. It is shown that there is no macroscopic polymer flow in the nip (like in other more classical coating processes) but a local flow within the valleys of the steel substrate surface. Several models are proposed, giving access to the temperature profile through polymer and substrate, the pressure distribution in the nip as well as the filling time of the steel roughness.

The microscopic flow, at the interface between the film and the substrate, is strongly influenced simultaneously by the temperature profile in the polymer near the substrate, by the pressure induced by the nip load and by the residence time in the nip. It is a subtle balance of these three parameters which will induce a good coating quality.

NOMENCLATURE

2a nip width

b rubber layer thickness around the press roll

C heat capacity

e polymer film thickness

[e.sub.0] polymer thickness which would results from the lubrication approximations

[e.sub.i] upstream polymer thickness

[E.sub.1], [E.sub.2] press roll, chill roll Young's modulus

E* reduced elastic modulus

F normal force

h heat transfer coefficient

k thermal conductivity

L depth of the steel sheet roughness

l penetration depth of polymer in the steel sheet roughness

p contact pressure

[p.sub.m] mean contact pressure

Q polymer flow rate in steel roughness

[R.sub.1], [R.sub.2] press roll, chill roll radius

[R.sub.eq] equivalent contact radius

R valley radius of the steel sheet roughness (case 2)

T temperature

[T.sub.1] line tension

t time

[t.sub.n] residence time in the nip

[t.sub.r] roughness filling time

[t.sub.r0] constant viscosity filling time

[t.sub.s] solidification time

v line speed

W line width

[w.sub.1], [w.sub.2] upper and lower widths of the steel sheet scratches (case 1)

x coordinate along the contact width

Greek Letters

[alpha] flank angle of the steel sheet scratches (Fig. 21)

[beta] time dependence of polymer viscosity in the nip

[gamma] attack angle

[eta] polymer viscosity

[rho] density

[v.sub.1] [v.sub.2] press roll, chill roll Poisson's ratio

REFERENCES

(1.) J.F. Agassant, Y. Demay, C. Sollogoub, and D. Silagy, Int. Polym. Process., 20, 136 (2006).

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(3.) G.L. Booth, Coating Equipment and Processes, Lockwood Publishing, New York (1970).

(4.) M. Takase, R. Katsumoto, T. Kegasawa, S. Kihara, and K. Funatsu, Polym. Eng. Sci., 42. 836 (2002).

(5.) J.F. Agassant, Le calandrage des matieres thermoplastiques, These de doctorat d'Etat, Universite Pierre et Marie Curie (Paris VI) (1980).

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(11.) Y. Trouilhet and B.A. Morris, Polymers, Laminations and Coatings Conference, TAPPI Proceedings, 457 (1999).

(12.) N. El Youssef, Modeling the Penetration of Polymer Melt During Extrusion Coating of Paper, Doctoral thesis. University of Maine (1995).

(13.) H. Hertz, J. Reine and Angewandte Mathematik, 92, 156 (1882).

(14.) K.L. Johnson, Contacts Mechanics. Cambridge University Press, Cambridge (1985).

(15.) D. Nowell and D.A. Hills, Int. J. Mech. Sci., 30. 945 (1988).

(16.) W.R. Wilson and J.A. Walowit, The Mechanics of Hydrody-namic Lubrication in Steady Deformation Processes. ASME Paper no. 70 LUBG (1970).

(17.) C. Sollogoub, Y. Demay, and J.F. Agassant, Int. Polym. Process., 18, 80 (2003).

(18.) J.F. Agassant, P. Avenas, J.P. Sergent, B. Vergnes. and M. Vincent, La Mise en Forme des Matieres Plastiques, Lavoisier, Paris (1996).

Correspondence to: Cyrille Sollogoub; e-mail: cyrille.sollogoub@cnam.fr

DOI 10.1002/pen.2l099

Published online in Wiley InterScience (www.interscience.wiley.com). [c] 2008 Society of Plastics Engineers

C. Sollogoub, (1) E. Felder,' Y. Demay, (1) J.F. Agassant, (1) P. Deparis, (2) N. Mikler (2)

(1) Ecole des Mines de Paris, Centre de Mise en Forme des Materiaux, UMR CNRS/Ecole des Mines 7635, BP 207-06904 Sophia Antipolis Cedex, France

(2) ARCELOR Research, Voie Romaine, 57280 Maizieres les metz, France
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Author:Sollogoub, C.; Felder, E.; Demay, Y.; Agassant, J.F; Deparis, P.; Mikler, N.
Publication:Polymer Engineering and Science
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Date:Aug 1, 2008
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