Eric Soccard
email: eric.soccard@eads.net

Arnaud Poitou

Anaïs Barasinski
email: anais.barasinski@ec-nantes.fr

Adrien Leygue

A Poitou Gem

Identification of non uniform thermal contact resistance in automated tape placement process

Keywords: Anaïs Barasinski, Adrien Leygue, Automated tape placement, Thermal contact resistance, Thermal modeling, Thermoplastic composite material

Introduction

Following its high manufacturing potential, there is a great interest in the Automated Tape Placement (ATP) process. Among its advantages one can identify the various shape or geometry that can be achieved and the potentially high deposition velocity. The ability to manufacture large, complex shaped and thin laminated parts makes it suitable for aeronautical applications or wind turbine blades. The deposition of up to 60 m 2 of material per hour has been achieved [START_REF] Hamlyn | nd. International conference SEICO 11-New Material Characteristics to cover New Application Needs[END_REF][START_REF] Lukaszewicz | [END_REF].

The ATP process involves a mould and a deposition head. The deposition head comprises a heating source and a compaction system. A possible configuration consisting in a laser beam and two rollers and corresponding to the configuration used in this work is depicted in Fig. 1. The principle of the process is to lay down successive layers of material by welding with the deposition head an incoming tape on a substrate. This process is already well developed for thermosets materials and is divided in two steps: a laying step and a curing step. Recent technology developments aim at adapting this process to thermoplastic matrix composite materials, and develop a one step process by reaching in-situ consolidation of the material during the deposition step [3].

In order to reach in-situ consolidation with this process, the material has to undergo the several steps depicted in Fig. 1: heating, consolidation, and cooling. During these three stages, different physical phenomena occur:

-The heating step allows the matrix fusion which is a prerequisite for adhesion. -In the consolidation step, the interply adhesion takes place. This is also where the final porosity level is fixed. At the molecular scale, the interply adhesion of the thermoplastic matrix is achieved through the reptation of the polymeric macromolecules [4]. -During the cooling step the crystallization of the polymer matrix might occur if the cooling rate is appropriate [5]. The first heating step is crucial because the material is a semi-crystalline thermoplastic material. Indeed, on one hand, if the crystalline phase does not fuse, the adhesion is compromised. On the other hand, the matrix is not thermally stable: chain branching, and crosslinking occur at high temperature and lead to the thermal degradation of the matrix [6,7]. This must be avoided at any price: if the matrix is degraded, the adhesion isn't possible anymore. For a given material and roller, only the heat source and placement velocity play significant roles in this first step of heating to ensure fusion and avoid degradation. The consolidation step starts when the roller enters the heated zone and the compaction force is applied. The intimate contact between the two plies is achieved and interply adhesion takes place [8]. From the combination of temperature and pressure, a micro matrix flow can occur, and intraply porosities [START_REF] Pitchumani | [END_REF] can evolve.

As we can clearly understand, this process is complex because of the many different parameters that come into play. The different physical phenomena have to occur at the right time, and in the right order. Moreover, the important point is that all these phenomena take place at material interfaces and are governed by temperature. It is therefore critical to understand and control the evolution of the temperature field in the whole part during the forming steps. In this work, we develop a numerical model allowing the accurate prediction of the temperature field for this process. The key ingredient is the introduction of thermal contact resistances (TCR) to model the non perfect thermal transfer from one ply to the other.

Although the material is not homogeneous, one can reasonably assume homogenized thermal properties at the scale of the ply. This assumption does not hold at the scale of the laminate as conduction depends on the degree of intimate contact between the plies and is therefore imperfect [10]. This is why we postulate interply TCR: the vertical heat flux between two plies is proportional to the temperature difference between the two sides of the interface. For a deeper analysis of the interply thermal resistance the interested reader can refer to [11].

This hypothesis and the resulting thermal model are validated through the comparison between experimental and numerical results. The comparison shows the existence of TCR and the importance of taking them into account for the quantitative prediction of the temperature in the adhesion zone.

In the following sections, the experimental and numerical tools are first presented. Then, experimental results are analyzed to highlight the TCR existence. Finally we show the predictive capabilities of the model.

Material, method and model description

Temperature measurements

In order to carry out this study, we use a tape placement machine, located in the Technocampus EMC2, in Bouguenais, near Nantes (France).

The machine, called 'Banc 1D', is a prototype bench developed by EADS IW. This system allows laser assisted unidirectional draping of 8 tapes in parallel. The heat source is applied on the upper surface of the laminate to generate adhesion at the interface. The heat source is located between two rollers: the first roller is located before the laser and ensures the contact between the incoming tape and the substrate while the compaction roller is located after the laser, see Fig. 1. The laser system used is a continuous wave solid state laser manufactured by the TRUMPF Company. It has a wavelength of 1064 nm, and a maximum power of 4 kW. An optic system is mounted on the laser head to create a rectangular laser sheet of 50 × 25 mm cross section.

The material is a prepreg tape material (APC2) supplied by Cytec. APC2 consists approximately of 60 % impregnated carbon fibers in a PEEK matrix. The matrix has a fusion temperature of T f ≈ 340 • C and a glass transition temperature of T g = 143 • C [START_REF] Lamethe | Etude de l'adhesion de composites thermoplastiques semi-cristallins; application à la mise en oeuvre par soudage[END_REF]. The homogenized thermal properties are reported in Table 1, where K xx and K zz are the longitudinal and transverse conductivities. The tapes have a thickness of 0.135 mm and a corresponding width of 6.35 mm.

A unidirectional laminate made of 6 layers is manufactured and thermal acquisition is performed using K type thermocouples positioned at the ply interfaces, as depicted in Fig. 4. A diameter inferior to 100 μm has been chosen in order to limit their intrusivity. The laser power is fixed at 1000 W, and the deposition velocity v = 5 m min -1 .

Table 1 Homogenized thermal properties of the APC2 tape material used in the model [START_REF] Lamethe | Etude de l'adhesion de composites thermoplastiques semi-cristallins; application à la mise en oeuvre par soudage[END_REF]

ρ kg m 3 C p J kg K K xx W m K K zz W m K 1560 1700 5 0.5
Figure 5 (left) shows a typical measurement set for three sensors positioned at the three upper interfaces.

One directly foresees that this type of measurements somehow contradicts the hypothesis of the existence of TCR. Indeed, the sensors measure only one interface temperature while we postulate that the interface heat flux depends on the temperature jump a the interface. We will therefore assume that the measured temperature is the average of the surface temperatures of the two plies at the interface. As strong as this hypothesis might seem, it does not affect the qualitative analysis to prove the existence of evolving TCR and the very good quantitative agreement with the simulation will validate the assumption that the measured temperature is some average of the plies surface temperatures.

The heat transfer model

In the numerical model, we choose to neglect the effects of thermal degradation and crystallization of the PEEK matrix. These phenomena would modify the thermal properties of the material, but because of the considered time scales and of the importance of the laser source with respect to other thermal phenomena, we can use constant material parameters.

Although the process is transient from the point of view of the laminate, we can simplify its modeling, far from the edges, by computing a steady state solution of the heat transfer problem in a euclidian reference frame attached to the laser. In this frame, the deposition head is fixed while the laminate is moving under the laser spot. Furthermore we consider that as we are manufacturing huge parts, it is reasonable to consider that the incoming substrate has had the time to reach thermal equilibrium since its last exposition to the laser.

As all the plies are oriented along the same direction, parallel to the deposition direction, the symmetry of the problem allows us to solve the 2D case only. Within each ply we solve the following heat transfer problem:

ρC p (v • ∇T ) = ∇ • K • ∇T , ( 1
)
where T is the temperature, ρ is the volumetric mass, C p is the heat capacity, K is the anisotropic conductivity tensor, and v is the velocity of the laminate with respect to the laser spot. The vertical heat flux between two plies labelled i and j is modeled as follows:

q Zij (x, t) = 1 R th,ij (x) × T + j (x, t) -T - i (x, t) , ( 2
)
where q Z is the normal heat flux, R th,ij (x) (K m 2 W -1 ) is the value of the thermal resistance between plies i and j , and T i and T j respectively are the temperature in the two layers at the considered interface.

In order to assess the importance of the TCR we will consider three different models for the parameter R th,ij (x) and its x-dependence. These three models are illustrated in Fig. 2.

-The first and simplest model (Fig. 2 top right), consists in neglecting the TCR between plies. We consider the heat transfers are perfect and there is consequently no temperature discontinuity from one ply to the next. -The second model (Fig. 2 bottom left) assumes a nonzero but constant TCR along x. We consider poor heat transfers between plies that are not influenced by the changes occurring at the interface when heated and compacted. -The third model (Fig. 2 bottom right) assumes that the healing process taking place during the compaction of the plies by the roller improves the quality of the heat transfer. Consequently we consider a piecewise constant value for R th,ij (x), which decreases each time the interface is heated and compacted. From the experimental measurements presented below in Fig. 5 (left), we observe that the temperature at the third interface is too low (w.r.t T g & T f ) to induce any significant change of the interface upon compaction. Consequently we consider the RTC to be constant from the third interface on.

The heating source is modeled as a surface heat flux with a gaussian distribution, applied on the upper face of the top ply. The flux boundary conditions with the mould, the roller and the air are modeled as the heat transfers at the interface between two plies but with fixed temperatures for those external elements. The values of all transfer coefficients are taken from the literature [START_REF] Lamethe | Etude de l'adhesion de composites thermoplastiques semi-cristallins; application à la mise en oeuvre par soudage[END_REF][13][14] while details of the boundary conditions are reported in Fig. 3. The numerical solution of the model is computed using the Proper Generalized Decomposition (PGD) method [15] using a separation of the x and z coordinates. The solution is sought under the following separated form:

T (x, z) = N i X i (x) × Z i (z).
(

) 3
This method has proved its efficiency for computing thermal and elasticity problems in such thin geometries [16][17][18] where it allows the use of a very thin mesh in the thickness direction while maintaining reasonable computation times.

Along the deposition direction we did simulate a length of 0.5 m discretized with 1000 finite elements while in the thickness direction we have 6 plies of 0.135 mm, each discretized using 20 elements. The solution procedure is

Experimental highlighting of the existence of TCR

In order to assess the quality of the thermal contact between the layers and its evolution during the process, we used the following set up: First, several prepreg layers are deposited on the laminate part, with thermocouples at the interfaces, as shown in Fig. 4. The temperature is recorded using the thermocouples during the deposition of the last layer.

Then, an additional consolidation step is performed with the placement head, during which the laminate is heated and compacted but no ply is added to the part. The temperature is again recorded during this step. The measurements for both steps are reported in Fig. 5.

As we can observe in Fig. 5, the temperature profiles of the last deposition step and of the consolidation step are very different while the thermal properties of the material have not changed. During the last deposition step, the temperature, reaches about 330 • C at the first interface, 170 • C Fig. 3 Detail of the boundary conditions of the thermal model at the second and 130 • C at the third while during the consolidation step, the temperature at the three first interfaces are respectively about 260 • C, 200 • C, and 150 • C. A simple interpretation is that during the deposition pass, the heat is confined in the first layer, because the thermal contact is of poor quality, but during the consolidation pass, the bonding is better, so is the thermal contact. The heat can flow more easily through the substrate, making the temperature lower at the first interface, but therefore significantly higher at the next two interfaces. The only quantity capable of inducing such changes between the two passes are the interface characteristics which in our model corresponds to the RTC.

These measurements point towards the existence of an imperfect thermal contact between the different layers of the laminate which evolves during heating and compaction. These hypothesis will be validated later using the numerical model.

Influence of TCR values on the thermal field

As shown above, the healing and the degree of thermal contact between the different layers of the laminate evolves during the process, and has a significant influence on heat transfer. Using the numerical tool and model presented in Section "The heat transfer model", we simulate the temperature field in a laminate part during the process for the three different models of RTC (Fig. 2), but keeping material and process parameters constant. The simulation results are shown in Fig. 6.

One can observe that in the first case (Fig. 6, top left), i.e. no TCR taken into account, the temperature field is continuous, and the maximum value of the temperature is about 400 • C. In the second case (Fig. 6, top right), i.e. taking a constant value of the TCR, the thermal field becomes discontinuous. More precisely, there are discontinuities in the temperature between each ply. In that case the maximum temperature in the laminate reaches 600 • C as most of the energy is concentrated in the first ply and flows poorly to the next plies. Finally in the third case (Fig. 6, bottom), i.e. taking into account an evolving value of the TCR, the temperature field is still discontinuous, but the discontinuity is localized under the heating source, here the temperature reaches a maximum of 600 • C as well. Downstream of the laser, as the value of TCR decreases, heat flows further down which yields higher interface temperatures. With this comparison, we can develop a first understanding of the importance of taking into account TCR, and its evolution, in the numerical model of the tape placement process. Considering perfect heat transfers between plies will result in lower temperatures, while taking into account a TCR to model the imperfect thermal contact concentrates the heat in the first ply. This may promote the inter-ply adhesion phenomena, which are strongly driven by temperature.

Comparison between experimental and numerical results

Analysis of the different scenarios

In this section we use the numerical model to identify the values of the thermal contact resistance for the three modeling scenarios presented in Section "The heat transfer model". Starting from an initial guess, the value of the TCR is slightly adjusted to obtain the best possible simultaneous fit for the temperature at the two upper interfaces of the laminate during the deposition of an additional ply. As the presence of TCR induces a discontinuous temperature field at the interface we used for the fitting procedure the average of the two interface temperatures. The results of this procedure are shown in Figs. 7 to 9.

For the first model of TCR, we see in Fig. 7 that because of the important heat transfers through the interfaces the temperature at both interfaces cannot be predicted simultaneously. The simulated temperature peak at the first interface is not as high as the recorded one while the second interface temperature is clearly underestimated. Additionally, the slope of the temperature curve at the second interface is qualitatively wrong in the heating phase. The cooling phase however shows a good agreement between experiments and simulation.

For the second model of TCR, we see in Fig. 8 that the heating part in both curves shows an improved qualitative and quantitative agreement between the measured and simulated temperature for both interfaces. Yet, the cooling part exhibits large discrepancies. Clearly, the constant value of TCR concentrates the heat in the first ply, and does not Fig. 7 Comparison of experimental results with numerical ones calculated with a perfect interply thermal contact allow the thermal transfers occurring during the cooling phase.

For the third model of TCR, which accounts for an evolution of TCR, we see in Fig. 9 an overall agreement between experiments and simulations. Both the heating and cooling steps match quantitatively. Furthermore, we see that the numerical tool can predict fine details of the temperature curves such as the slight overshoot at the second interface at the end of the heating phase. This feature is actually caused in the model by the sudden evolution of the TCR under the compaction roller which allows the heat to flow from the first to the second ply. Accounting for an evolving TCR, corresponding to the qualitative evolution of the interply Fig. 8 Comparison of experimental results with numerical ones calculated with a constant TCR Fig. 9 Comparison of experimental results with numerical ones calculated with an evolving TCR properties, allows one to have a quantitative prediction of the temperature evolution in the laminate during its manufacturing. Finally in this last scenario, it should be noted that the simulation results are not very sensitive to the exact values of the RTC. The correct magnitude combined to the sudden decrease of the RTC under the compaction roller are enough to obtain a very good fit of the data.

Validation of the thermal model on the consolidation pass

Using the previously identified values of the TCR we validate the model by simulating the consolidation pass described in Section "Experimental highlighting of the existence of TCR". In Fig. 10, we compare the experimental interface temperatures of the three first interfaces with the Fig. 10 Comparison of experimental results with numerical ones calculated with an evolving TCR for the consolidation pass numerical predictions. Despite the simplicity of the model and the complexity of the experimental measurements, we find a surprisingly good quantitative agreement between experiments and simulations. The inset shows a detail of the interface temperatures where we observe that, due to the heat flow in the compaction roller, the temperature at the second interface becomes higher than the temperature at the first interface. This experimental feature is correctly predicted by the numerical model. These results suggest that the very good quantitative fit obtained in Fig. 9 does not involve as much over-fitting of the data as one might think and that the model can be used for prediction purposes.

Conclusion

In this work we presented an experimental and numerical study of the automated tape placement composite forming process applied to thermoplastics. The experiments suggest that degree of interply thermal contact is evolving during the manufacturing of the laminate and that this evolution has a dramatic effect on the thermal transfers in the laminate. We therefore propose a numerical model which accounts for interply thermal contact resistances. The model indeed shows that the value and the evolution of the thermal contact resistance is a critical parameter in order to compute accurate interface temperatures.

In particular we show that it is sufficient to consider that the thermal contact resistance decreases suddenly during the compaction step to quantitatively and accurately predict the experimental results. Moreover the PGD is an appropriate simulation tool for this problem as it allows the high resolution solution of the model in a very short time thanks to the separation of physical coordinates.

These result are very promising for thermal modeling of the automated tape placement process. Further work should focus on the accurate modeling of the heat source and the 3D geometry of the laminate. A more physically detailed model of crystallization, melting and degradation phenomena should also be considered.

Fig. 1

 1 Fig. 1 Schematic configuration of the ATP placement head and corresponding steps of the process

Fig. 2

 2 Fig. 2 Description of the three models of thermal contact resistance (K m 2 W -1 ) at the different interfaces

Fig. 4

 4 Fig. 4 Localization of the sensors during de deposition pass (left) and additional pass (right)

Fig. 5 Fig. 6

 56 Fig.5 Measured temperature in the plate during the deposition of last layer (left) and additional pass (right)


N Delhaye

A Poitou

M Chaouche

Squeeze flow of highly concentrated suspensions of spheres

Keywords: Suspensions, Squeeze flow, Flow-induced heterogeneity, Filtration, Pastes

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Squeeze tests are often utilised in practice as a straightforward technique to determine the flow properties of highly viscous materials such as concrete, molten polymers, ceramic pastes etc. The material is squeezed out between two parallel surfaces at either controlled normal force or squeezing speed, and assuming a rheological model, the flow parameters of the material are inferred by fitting the model with the experimental measurements. For small gaps and if the no-slip boundary condition is fulfilled, the flow is dominated by shear and the lubrication-type approach can be used to determine the relationship between the applied force (or velocity) and the rheological parameters of the material. In an early study, Stefan and Sitzungber [1] solved the case of a Newtonian fluid and Scott [2] considered the case of power law fluids. On the other hand, if the material slips along the solid surface, the kinematics is dominated by the extensional component of the flow. This gives a particularly interesting set-up to study the bi-axial elongation properties of such materials. More recently, Laun et al. [3] considered the case of partial wall slip. In particular, they reported a method, similar to that of Moony [4], for a capillary rheometer, to make a separation between the bulk shear and the wall slip from a single squeeze test.

In all the above studies, the material is supposed to have a pre-defined rheological behaviour. In particular, even complex materials are often assumed to remain homogeneous during the squeezing process and behave as simple power law fluids. Many complex fluids become heterogeneous when subjected to flow; this can induce a radical change in their rheology. For instance, this is the case for polymer melts that partially crystallise in flow fields. To our knowledge, the problem of flow-induced crystallisation in squeeze flows and its consequences on such a rheological technique is still open.

The flow-induced heterogeneities in multi-component systems have also been reported in the literature. Racineux [5] considered the rheological behaviour of highly concentrated ceramic pastes in both capillary and squeeze geometries. He showed that his experimental results could be correctly modelled only by assuming a phase separation in which the particle concentration increases in the low shear regions. However, the physical origins of this phase separation are not well understood.

In our study, squeeze flow experiments are conducted on a model, well-defined fluid-particulate system (suspension). Our objective is to highlight the physical mechanisms responsible for this flow-induced heterogeneity. Concentrated suspensions often become heterogeneous in complex flows. For example, it is well known that non-uniform shear flows induce particle migrations in concentrated suspensions [6]. The origin of this phenomenon is well understood and attributed to irreversible interactions (collisions) among the particles [7]. This process is diffusive [6,8] and is shown to be negligible in our experiments.

The experimental parameters and procedures are described in Section 2. In Section 3, we present the results along with a qualitative interpretation demonstrating a transition between a viscous flow of a suspension and a solvent filtration through a porous media made up by the particles.

Experiments

Squeezing set-up

The suspensions are squeezed out between two parallel discs mounted on a compression-traction machine. The upper disc can be displaced at controlled velocities, while the lower one is maintained stationary. The latter experiences a normal force that is recorded as a function of time for each fixed velocity. In all experiments, the initial disc separation is taken to be 5 mm, which is much smaller than the disc diameter (40 mm). In order to avoid a significant deformation of the particles, the squeezing experiment is stopped when the normal force exceeds 800 N. Since the compression-traction machine used here is designed for studying rather solid materials, the uncertainty in the force measurements is quite high (±2 N). On the other hand, the disc separation (velocity) is measured with a much higher accuracy, namely ±1 mm.

In the visualisation experiments, the lower disc is transparent (glass). Hence, owing to a CCD camera associated with an image analyses software (optimas), the evolution of the microstructure due the squeeze flow can be followed, at least qualitatively, since at high concentrations, the suspensions are opaque.

Suspensions

The suspensions consist of hard spherical particles dispersed in a Newtonian fluid. The particles are polymethylmethacrylate (PMMA) (kindly supplied by the company 'GAZECHIM matières plastiques'). A photograph of a sample of the particles is represented in Fig. 1. The shape of the particles is fairly spherical, but there is quite a strong polydispersity in their diameter distribution (Fig. 1). Four batches of particles are used, A, B, B1 and B2, corresponding to different average diameters, respectively: 117, 79, 69, and 89 m. Batches B1 and B2 are obtained by sieving batch B; they are then less polydisperse. Samples of about 100 particles are analysed, and the particle size distribution functions are found to present approximately a Gaussian shape. The breadth of this distribution is about 50% in the case of batches A and B, and 20% in the case of the two other batches.

Three different suspending fluids are used: a 20%/80% water/glycerol solution, pure glycerol and silicone oil. Their viscosities are measured using a Couette device (Stresstech from Rheologica). The viscosity of the solution water/glycerol is 0.085 Pa s. At 25 • C, the glycerol and the silicone oil are found to be Newtonian up to a shear rate of 10 2 s -1 , with a viscosity of 0.86 and 12.5 Pa s, respectively. The density of the particles is sufficiently close to that of the solvents so that we can ignore sedimentation effects. Indeed, in the worst case (highest sedimentation rate), the characteristic mean sedimentation time (the time required for a particle to sediment over its diameter) can be estimated as [START_REF] Russel | Colloidal Dispersions[END_REF] τ s = (18μ/( ρgd)) (1φ) -6.55 ≈ 2000 s. In this expression, μ is the viscosity of the solvent, ρ the difference between the density of the particle and that of the solvent, d the average diameter of the particles, and where φ is the volume fraction. This sedimentation time is higher than the largest corresponding flow characteristic time (the time required for the particle to move over its diameter due to the flow) involved in our experiments, which can be estimated as τ f = d/U ≈ 7 s. U is the squeezing velocity. Finally, the largest particle Reynolds number in our experiments is Re = (ρUd/μ) ≈ 2.5 × 10 -3 1; inertia is then negligible.

Experimental results and discussion

Viscous flow behaviour

Fig. 2 represents the evolution of the squeezing force versus the gap distance h for different velocities V. This corresponds to the case of a suspension in silicone oil with a volume concentration of 56.6%. The particles are taken here from Batch B1. The suspension behaves as a viscous fluid: the force is an increasing function of the velocity for a given gap thickness and diverges when the gap approaches 0. Nevertheless, the behaviour of the suspension is not Newtonian, contrary to that one would expect for non-Brownian hard spheres in a Newtonian fluid undergoing only hydrodynamic interactions. Moreover, the suspension can not be modelled as a simple power law with the same exponent for all the squeeze speeds considered. Indeed, in Fig. 2, we also represent the predicted temporal evolution of the force for the Oswald de Waele power law fluid. We used the Scott result for the normal force [2]:

F = 2π 2m + 1 m m A m + 3 V m h 2m+1 R (m+3) .
The rheological parameters m and A involved in this expression are related to the constitutive equation for a power law fluid which can be written in the case of a shear dominated flow as τ = A γ m , where τ is the shear stress and γ the shear rate. The best fit of Scott's equation with the experimental results gives A=594.6 Pa s m and m=0.5. For a Newtonian fluid, m=1 and A is its Newtonian viscosity; since m<1, the suspension behaves as a shear thinning fluid. The shear thinning effect can not be attributed to the solvent since the silicone oil is Newtonian in the shear rate interval involved in this experiment ( γ <5 s -1 ). This can rather be attributed to non-hydrodynamic close contact interactions among the particles. As we can see in Fig. 2, the experimental results for the other velocities can not be fitted using the same rheological parameters. The suspension can not be then modelled as a simple power law fluid. Other squeeze experiments have been conducted at smaller velocities for which similar behaviour has been obtained.

Filtration

In Fig. 3, the temporal evolution of the normal force is represented for different velocities, for the same batch (B1) and concentration as that given above, but with a less viscous solvent (glycerol). The squeeze behaviour of the suspension is radically different from that with silicone oil (see Fig. 2). Above a certain velocity (U c =0.5 mm s -1 ), the results are similar to those obtained with a suspension in silicone oil: for a given gap thickness, the force increases with the velocity. Below U c , the trend is reversed: the force strongly increases when the velocity decreases. Note also the appearance of very large force fluctuations at small velocities. This suggests that, in this very low velocity flow regime, the suspension deforms rather like a dry granular material.

Interpretation

The most significant parameters that control the transition reported above can be inferred by assuming a diphasic behaviour of the suspension. A model based on such an assumption has been successfully applied to describe the rheological behaviour of ceramic pastes [9,10]. In this approach, the suspension is assumed to deform with two flow fields v p and v f , associated, respectively, with the particles and the fluid. This vision is different from what is generally assumed for flow of suspensions, in which the particles are supposed to move affinely with the fluid. The diphasic approach is rather similar to the Brinkman model [11], but with a deformable porous media.

Since the particles and the fluid do not necessarily have the same velocity, the rate of the viscous energy dissipation can be divided into two terms. The first term is related to the flow of the suspension as a whole (the particles have the same velocity as the fluid), and the second term is related to the damping force caused by the filtration of the fluid through the porous media made up by the particles. Hence, this rate of energy dissipation can be written as

W = V σ γ + η 0 k (v f -v s ) 2 dV .
In the above expression, k is the permeability of the porous media formed by the particles and σ the shear stress experienced by the suspension (which is a function of its effective viscosity η s ). The integration is made over the volume of the sample. The permeability scales with the particle size squared, and is a rather weak function of the concentration (cf. the Kozeny-Carman relationship [12]). This is in contrast to the effective viscosity (shear stress), which is a strong function of the concentration but independent of the particle size. As discussed above, the suspension presents a shear thinning flow behaviour; thus, the shear stress can be written as σ = A γ m ≈ A (U/ h) m , where m is the shear thinning index, and A is the consistency index of the suspension.

The 'critical' velocity for which the filtration competes with the suspension flow can be estimated by assuming that this occurs when the two terms of the viscous dissipation are equal:

U c = Ak η 0 h m+1 1/(1-m) .
According to this expression, the non-Newtonian character of the suspension is required to obtain a filtration/suspension flow transition.

The permeability of the porous media made up by the particles scales with the particle diameter squared; we expect then that the appearance of the filtration would depend strongly on the particle size, in particular for highly shear thinning suspensions. For example, we found experimentally that increasing slightly (from 0.069 to 0.089 mm) the mean particle diameter, and keeping the other suspension parameters unchanged, the velocity for which the filtration starts up is multiplied by a factor of 2.

The influence of the particle volume fraction results in two opposing effects. The effective suspension viscosity is an increasing function of the concentration (η s = η 0 (1 -(φ/φ max )) -2 for a Newtonian hard sphere suspension [13]). On the other hand, the permeability is a decreasing (although slightly) function of the concentration (k ∝ ((1φ) 3 /φ 2 ), according to the Kozeny-Carman model [12]). The resulting effect of the concentration increase would be the increase in the squeeze velocity for which the filtration (phase separation) appears. This is confirmed experimentally by conducting squeeze experiments for different volume fractions. For instance, increasing the volume fraction from 53.6 to 56.6% results in an increase in the transition velocity by a factor of 5 (0.1 to 0.5 mm s -1 ).

Due to the quite narrow interval of the suspension's parameters accessible in our experiments, it is not possible to go farther in the comparison between our qualitative model and the experiments. Moreover, the main objective of this model is only to bring to the fore the main parameters controlling the appearance of the phase separation. Numerical simulations similar to those of Racineux and Poitou [10] are needed to make quantitative predictions.

It is to be noted that the origin of this flow-induced concentration heterogeneity can not be attributed to the well-studied shear-induced particle migration in non-homogeneous shear flows [6,8]. Indeed, this phenomenon is diffusive and occurs on time scales (hours) [6] much larger than those involved in our experiments (seconds or few minutes). The length scale of the flow-induced heterogeneity observed in our experiment is the disc radius R, and the diffusion coefficient for shear-induced particle migrations can be estimated as a 2 γ [8]. The characteristic time corresponding to this phenomenon can be written then as R 2 /a 2 γ . This diffusive time is much larger than the squeezing time (1/ γ ) which is actually required to induce the observed heterogeneity.

Visualisation experiments

Visualisation experiments are carried out in order to examine, at least qualitatively, the temporal evolution of the concentration during a squeeze experiment. The experimental set-up is depicted in Fig. 4. In these experiments, a colouring is added to the suspending fluid in order to allow observations. In Fig. 5a, we represent a sequence of two images corresponding to the beginning (h=5 mm) and the end Fig. 5. (a) Visualisation of the initial and the final state of the suspension for a squeeze velocity (U=3 mm h -1 ) for which phase separation is not expected according to the force behaviour. The concentration remains homogeneous during the experiment. (b) A sequence of images showing the increase in the particle concentration in the central part of the discs. The dark regions correspond to high particle concentration. In this experiment, the velocity is 0.02 mm s -1 , and the phase separation is expected to appear at U=0.5 mm s -1 according to the force measurements.

(h=0.7 mm) of the experiment, at a velocity (U=3 mm s -1 ) for which phase separation is not expected. Qualitatively, it can be seen that the concentration remains homogeneous on sufficiently large scales.

In Fig. 5b, we represent an image sequence corresponding to a velocity (U=0.02 mm s -1 ) for which phase separation is expected according to the force behaviour. Clearly, it can be seen in Fig. 5b that the concentration significantly increases in the central region of the discs, a result of a phase separation between the particles and the fluid.

Conclusion

The squeeze test is often used in practice to determine the rheological properties of highly viscous materials. Such a rheometer is quite simple in its design, but the interpretation of its measurements is not straightforward since the materials it is used for are in general complex fluids. In this work, the flow of concentrated model suspensions in a squeeze rheometer has been considered. Our purpose was to help in understanding the results reported in the literature regarding the rheological behaviour of ceramic industrial pastes [5]. Our squeeze force results were qualitatively similar to those obtained with industrial pastes. At relatively high squeeze speeds, the suspension presented a usual, although non-Newtonian, viscous flow behaviour. At low squeeze velocities, the suspension responded much like a sponge-form media. This behaviour has been attributed to a phase separation between the fluid and the particles due to the competition between the shear flow of the suspension as a whole and the solvent filtration through the particles. This is confirmed by visualisation experiments, which clearly made evidence of a concentration increase in the central part of the squeeze device at sufficiently low velocities. Our investigation was rather qualitative, and more quantitative studies in order to test models based on the above assumptions are clearly needed, including for instance the concentration measurements in refractive index-matched suspensions.

Fig. 1 .

 1 Fig. 1. Visualisation of a sample of the particles using an optical microscope.

Fig. 2 .

 2 Fig. 2. Temporal evolution of the normal force for different velocities in the case of a concentrated suspension (56.6%) for a relatively high viscous fluid (silicone oil of 12.5 Pa s). The experimental results (dotted lines) are compared to a theoretical calculation assuming that the suspension has a power law fluid behaviour.

Fig. 3 .

 3 Fig. 3. Temporal evolution of the normal force for different velocities, in the case of a concentrated suspension (56.6%) in a less viscous fluid (glycerol) than in Fig. 2.

Fig. 4 .

 4 Fig. 4. Sketch of the set-up for the visualisation experiments.


S Bourrigault

Gérard Marin

Arnaud Poitou

S Bourrigaud

Shear Modification of long-chain branched polymers : a theoretical approach using POM-POM model

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

When working on extrusion coating, extrusion casting, or film blowing processes, one meets difficulties to extrapolate results from a laboratory scale to semiindustrial and industrial scales. One explanation of those difficulties is the great differences in thermomechanical histories that the melt undergoes. Mainly branched polymers like LDPE are affected by a very important change in there rheological behavior. 1,7 This well-known effect called "shear modification" or "shear refining" has been studied for a long time on an experimental point of view. 1,2 This effect is usually neglected but is in fact of main importance. Actually, one who deals with processing needs to know which are the effective rheological properties of the material to extrapolate its behavior under process. On the other hand, a good understanding of this effect can be a very powerful tool to adjust material properties to the process in order to increase process and product performances. 15 Shear modification is described to be a highly significant modification of rheological properties in both linear and nonlinear viscoelasticity domain and concerns mainly melts of long-chain branched molecules and high polydispersity. 10,14,15 The most significant modifications are observed on the storage modulus (G′) in linear viscoelasticity at low frequency (decreasing of about 20% in some extreme cases), on the recoverable shear strain, 6 transient shear, and normal force in nonlinear viscoelasticity, 6,14 and in the strain hardening behavior in uniaxial elongation. 15,16 Loss of die swell had been also reported extensively. [1][2][3][4][5]10,11,14,17 Viscous properties undergo a small decrease, whenever the melt flow index is reported to be significantly increased. All those considerations gave authors arguments to conclude on a decrease in melt elasticity and strain hardening behavior.

The molecular origin of this phenomenon has been believed to be a disentanglement mechanism of long-chain branched molecules for a long time, 1,2,9 but this idea was refuted by Leblans and Bastiaansen, 15 considering the annealing time value. As the modification is fully reversible, one can define a time the modified melt needs to recover its initial rheological properties. For these authors this time was measured to be of the order of the maximum relaxation time which is a piece of evidence that deepest levels of side chains and molecular backbone are involved in the shear modification dynamics. A second piece of evidence lays in the analysis of solution treatment data. When samples are dissolved and reprecipitated in order to produce molecular disentanglement, rheological properties modifications are quite similar to those caused by shear modification; however, the corresponding annealing time is lower by 2 orders of magnitude.

As rheological properties of a processed melt are similar to those of linear polymers in elongation (without any strain hardening behavior), a new mechanism was proposed: long side chain branches align each other along the backbone in a single tube. According to chain reptation theories of branched polymers, the longest time is associated with the backbone dynamics, which reptates since side chain branches have relaxed. 8,12,20 The side chain branches alignment is therefore in contradiction with reptation theory because the shear modification lifetime is associated with a time of the order of the backbone relaxation time. The molecular explanation of shear modification is still today an open issue. The aim of this work is to discuss the side chain branches alignment hypothesis in the light of recent molecular dynamics picture on long-chain branched polymers using the pom-pom polymer model.

Theory

Pom-Pom Model.

The pom-pom model was developed by McLeish and Larson 18 to describe the rheological behavior of melts with long-chain branching and appears to be a generalization of the H-polymer structure. The molecules are composed of two "poms" considered as stars of q arms of dimensionless length s a ) M a /M e connected by a backbone of dimensionless length s b ) M b /M e , where M a , M b and M e are respectively the molecular weights of arms, backbone, and entanglements. The main idea of this model is to consider three different hierarchical relaxation mechanisms. In a first step molecular orientation of the crossbar segment is considered. In a second step, as the branch points cause frictional drag force, McLeish and Larson introduced a backbone stretching parameter λ with entropic elastic force of the stretched segment balanced by the frictional drag force of branch points. This mechanism occurs until the limiting stretching value of q is reached which is determined by Gaussian chain statistics and equilibrium considerations. At last, branch point withdrawal is considered to occur when the flow is strong enough to induce backbone maximum stretching value of q. s c is then introduced as the M c / M e ratio with M c the molecular weight of the withdrawn arm. This last mechanism is taken into account by adding an extra contribution to the stress tensor and by considering that the dangling arms length is reduced. The authors of the pom-pom model do not take into account the increase in tube length, and therefore the entanglement network remains basically controlled by the initial backbone volume fraction. They argue that the star-arm relaxation remains much faster on the time scale of their restricted flow conditions. We must notice here that the major effect of branch point withdrawal is to induce a decrease of the relaxation time scale.

We consider in our approach that, when branch point withdrawal occurs, the relaxation times are modified and have to account for both the reduction of the arm length and the increase of the backbone length. As was shown by McLeish and Larson, the local deformation rate within a tube segment surrounding a withdrawn arm segment is much larger than the bulk deformation rate. Therefore, even under the restricted flow conditions of these authors, the withdrawn arm fraction acts as a topological obstacle. This also means that the tube edges act as new branch points. We postulate therefore that under strong flow conditions the branch point withdrawal mechanism is very significant and induces some important changes in the molecular entanglement network so that the effect of chain retraction is to induce some significant changes, both increase and decrease, in the relaxation time spectrum.

2.1.1. Pom-Pom Arms Dynamics. In the original pom-pom model, the relaxation time of a segment of length s a -s c is where φ b the fraction of backbone material. Ball and McLeish 13 defined the characteristic time τ as where τ 0 is the Rouse time and ν is a structural parameter relative to the chemical nature of the material, and its value was experimentaly measured to be 0.6 for polyethylene. 19 The exponential is derived from the breathing modes of stars, and the subtraction inside the exponential is derived from the dynamic dilution effect which has a strong accelerating effect on the time scale of arms relaxation. (1φ b ) is the volume fraction of material involved in relaxing arms. In the case of flows considered by McLeish and Larson, 18 branch point withdrawal has a minor effect and can be reasonably neglected as was argued by these authors. Nevertheless, in the case of stronger flows, a significant fraction of arm material is withdrawn into the backbone tube. To derive consistently our model from the pom-pom model, we have to take into account the reduction of arm length. Following this molecular picture, the definition of the characteristic time τ for the arm relaxation becomes Furthermore, the fraction of arm material withdrawn into the backbone tube does not contribute to the volume fraction of relaxing arms involved in the dynamic dilution effect. Therefore, when branch point withdrawal occurs, the effective volume fraction of relaxing arms becomes 1φ bφ c , where φ c is the volume fraction of withdrawn material. We take into account this effective volume fraction of material outside the backbone tube by modifying expression 1 into where We can notice here that taking into account the material outside the tube is a small conceptual difference from the original pom-pom model which can have however a significant effect on the relaxation time scale. Nevertheless, s c must be of the order of s a so that the fraction of withdrawn material φ c cannot be neglected (0.1 < φ c < 1), which means that shear modification occurs only in strong flows situations (high strain rates and high strains). As we can see, if s c remains close to 0, our model is identical to the original pom-pom model.

Backbone Orientation. The branch point friction constant ζ b is defined as

where a eff is the effective tube diameter and τ a (s c ) the relaxation time of the arms outside the tube. The entanglement molecular weight is assumed to be proportional to the reciprocal of the concentration of entangled strands, which is φ b in the case where no arms retraction takes place. Under strong flows, branch point withdrawal occurs which increases the length of the backbone tube. Actually, q retracted arm parts s c define along the backbone chain a single tube of length s b + 2s c , reminding that tube concentration is defined as the ratio of the tube length to the total length of

τ a (s c ) ) τ exp [ 15 4 s a ( (1 -s c /s a ) 2 2 -(1 -φ b ) (1 -s c /s a ) 3 3
)]

(1)

τ ) τ 0 s a 2 π ν(s a ) (2) τ(s c ) ) τ 0 (s a -s c ) 2 π ν(s a -s c ) τ a (s c ) ) τ(s c ) exp[ 15 4
s a ( ( 1 -s c s a ) 2 2 - (1 -φ b -φ c ) ( 1 -s c s a ) 3 3
) ] (3)

φ c ) 2qs c 2qs a + s b ζ b ) 2kT τ a (s c ) a eff 2 q (4)
material. The concentration of entangled strands becomes therefore

In that way the dilated tube primitive path φ b s b becomes Thus, the final equation for the backbone relaxation time is

The orientation dynamics is described by the following differential equation using the tensor S 2.1.3. Backbone Stretch. As we consider that only the backbone segment is stretched, the equilibrium separation between branch points is still s b a eff and the effective elastic constant κ ) kT/s b a 2 . Therefore, no effect of branch point withdrawal has to be taken into account on the stretch relaxation time

The stretch parameter λ dynamics is described by the following differential equation: 2.1.4. Arm Retraction. The relaxation time of arm retraction is defined as the relaxation time of a segment of length s a -s c , which is τ a (s c ). Therefore, the differential equation describing the retraction dynamics remains 2.1.5. Expression for the Stress. For strain rates between τ b -1 and τ a (s c ) -1 the main contribution to the stress becomes from the backbone. The "pom" arms are relaxed and are seen as a swelling purely viscous solvent. Reminding that for strong flows s c contributes to backbone orientation but not to backbone stretching, McLeish and Larson dissociate the stress into two parts. To be consistent, we must take into account in the definition of the stress that the concentration of topological obstacles increases when branch point withdrawal occurs. We therefore derive the following equation for the stress tensor where the first coefficient φ b + 2s c /(2qs a + s b ) takes into account the dependence the tube diameter on concentration of topological obstacles as discussed in section 2.1.2. Here again we recall that the dilated tube diameter is proportional to the reciprocal of tube concentration, which is 2qs c /(2qs a + s b ) + φ b . φ b accounts for stretched segments concentration and 2qs c /(2qs a + s b ) the volume fraction of withdrawn arm segments that contribute to orientation.

Global Set of Differential Equations and

s b + 2s c 2qs a + s b '(s b + 2s c ) (6) τ b (s c ) ) 4
π 2 (s b + 2s c ) 2 ( φ b + 2s c 2qs a + s b ) τ a (s c )q (7) ∂S ∂t ) K'S + S'K T -2(S:K)S - 1 τ b ( S -1 3
I ) (8) τ s ) s b τ a (0)q (9) ∂λ ∂t ) λ(K:S) - 1 τ s (λ -1) ( 10
)
∂s c ∂t ) ( q s b 2 + s c) K:S - 1 2τ a (s c ) (11) σ ) 3G 0 ( φ b + 2s c 2qs a + s b )( φ b λ 2 + 2qs c 2qs a + s b ) S (12) ∂S ∂t ) K'S + S'K T -2(S:K)S - 1 τ b ( S -1 3
I ) τ a (s c ) ) τ(s c ) exp[ 15 4
s a ( ( 1 -s c s a ) 2 2 - (1 -φ b -φ c ) ( 1 -s c s a ) 3 3
) ]

τ(s c ) ) τ 0 (s a -s c ) 2 π ν(s a -s c ) φ c ) 2qs c 2qs a + s b τ b (s c ) ) 4 π 2 (s b + 2s c ) 2 ( φ b + 2s c 2qs a + s b ) τ a (s c )q ∂λ ∂t ) λ(K:S) - 1 τ s (λ -1) τ s ) s b τ a (0)q ∂s c ∂t ) ( q s b 2 + s c) K:S - 1 2τ a (s c ) σ ) 3G 0 ( φ b + 2s c 2qs a + s b )( φ b λ 2 + 2qs c 2qs a + s b ) S
decreases when arm retraction occurs, so that arm relaxation slows down. Thus, τ a increases. The other one is the branch length reduction which induces a decrease of τ a . Actually, we must point out here that in some cases the Weissenberg number (with respect to arm dynamics) can be larger than unity depending on the higher transient value reached by τ a before it decreases. In this case one would have to include arms contribution to the stress. This will be the topic of a later work. Those small modifications of the initial pom-pom model without any further assumptions introduce naturally the effect of flow strength on the rheological properties of the material. As one will see in the following part, the behavior of the modified (i.e., processed) material can differ significantly from that of the native one.

Shear Modification Mechanism.

Figure 1 shows the mechanism of "shear modification" for a pompom molecule with s a ) 30, s b ) 30, and q ) 5 under simple shear. The horizontal axis represents t/τ a (t)0), the dimensionless time with respect to the initial value of τ a . For τ a and τ b , results are expressed in terms of the ratio of the instantaneous relaxation times and their initial value τ a,b (t)/τ a,b (t)0). We adjusted the strength of the flow to We a ) 0.1 in order to respect the initial pom-pom assumption of We a < 1 (where We a is the Weissenberg number with respect to the arm relaxation time τ a ). On the contrary, the flow must be strong enough to induce chain stretching, We s > 1 (where We s is the Weissenberg number with respect to backbone stretching relaxation time τ s ). In such flow conditions, λ increases until it reaches its maximal value of q, and then arm retraction takes place, reaching a plateau value close to 5. As we can see, arm retraction induces a peak of τ a followed by a plateau value below the initial value.

Shear Modification Recovery.

A key feature of shear modification phenomena is the annealing time. We already said that this time was described to be longer than the arm relxation time of the native polymer. This can be well understood considering that τ a and τ b go through a maximum during the shearing process. To recover its initial properties, the material has to take the same path. This means that the recovery of the material is much longer than the arm relaxation time of the annealed material because τ a and τ b can reach respectively 2.5 and 3 times the equilibrium values. Figure 2 shows the shear modification process up to t/τ a (t)0) ) 1000 under simple shear. At that time the flow is stopped, and one can see the recovery phenomenon until the material reaches its initial properties at a time close to t/τ a (t)0) ) 1020. Therefore, the time the melt needs to anneal is 20 times the arm relaxation time.

Rheological Properties. a. Behavior under Simple Shear.

To investigate the rheological properties of a modified material and to compare results, we simulate some experiments from the literature. What has been done by many authors 3,9,11,15 is to shear modify a material using an extruder or a mixing device and to collect a sample and remold some tablets in order to make some rheological measurements. Authors are confident that the sample collection and preparation were fast enough to avoid recovery. We therefore assume that the global orientation of the molecules is lost, but the microscopic topological modifications still sur- vive. In a first step we characterize its rheological behavior under shear (Figure 3). According to experimental conditions, we assume that an overall isotropic state is reached during sample preparation, but we consider that arm retraction and backbone stretching still survive according to the slow recovery kinetics.

Then one has to enter the backbone stretching and branch point withdrawal values of the sheared material to initiate the calculation. We apply the same shear rate as during preshearing in order to make relevant comparisons with the native polymer. As τ a has a lower value, We a is still lower than unity, and the basic pompom model assumptions are still respected.

The viscosity curve of the native material shows a singularity (Figure 3), corresponding to the dynamics of s c whereas the modified material has a smooth behavior with an overshoot of the viscosity (Figure 4). As the backbone chains remain stretched, no further contribution of the stretch parameter λ on the evolution of the viscosity can occur. The behavior of the modified material is therefore radically different from that of the native one. Furthermore, as the relaxation times are lower, the dynamics of the backbone orientation and branch point withdrawal is different, too. It is now obvious that the level of arm retraction during the preshearing process controls the shear behavior of the flow modified material.

b. Uniaxial Elongation. The native material is highly branched, and as expected for strain rates higher than τ s -1 (but still lower than τ a -1 ), it shows the strain hardening behavior in uniaxial elongation (Figure 5).

The modified material exhibits a very different behavior under uniaxial elongation. Actually, for the strain rate value considered here (the same as for the native polymer), the melt shows no strain hardening behavior. As for shear flow, we can argue that if backbone chains remain stretched, no further stretch evolution can occur, annealing the strain hardening property (Figure 6). Furthermore, one can notice that rheological properties are more modified by elongation than by simple shear. In fact, with the same strain rate, "elongation modification" (Figure 7) achieves higher branch point withdrawal (s c is about 8 instead of 5 for shear) which induces lower values of the relaxation times than shear modification does (Figure 1), which is in good agreement with the previous work of Leblans and Bastiaansen. 15 2.3.4. Effects of Topology. Having designed a molecular model to explain the origin of the shear or elongation modification mechanism, it is interesting to study now the effect of molecular topology. In this analysis, we make the dimensionless arm length s a vary from 3 to 90, keeping the same number of arms (q ) 5) and the same backbone length (s b ) 30).

Samples are subjected to simple shear with a Weissenberg number relative to arm relaxation time We a ) 0.1, from t ) 0 to t ) 1000τ a . At that time, the flow is stopped, and the melt begins to recover its initial properties. For molecules with short-chain branches (s a ) 3), we can notice in Figure 8 that branch point withdrawal reaches a low equilibrium value of 0.1, which means that the volume fraction of retracted arm material φ c remains negligible compared to the volume fraction of material inside a tube φ b . Therefore, in eq 3, only the branch length reduction s a controls the evolution of τ a and τ b . Finally, the relaxation times reach a value lower by 15%. For material with very long-chain branches (s a ) 90), subjected to the same simple shear flow strength (We a ) 0.1), branch point withdrawal reaches a higher value close to 23, which induces a significant increase of φ c . φ c becomes nonnegligible compared to φ b . In eq 3 both arm length reduction and dynamic dilution control the evolution of τ a . the relaxation times increase in a first step by a factor of 200 for the arms and by a factor of 550 for the backbone. In this case, the recovery mechanism is very slow (3500τ a (t)0)).

In processing conditions, these very long-chain branched molecules are more affected by shear modification, and their recovery kinetics is so slow that the melt properties remain very different from the native state during the whole processing time. This is in qualitative agreement with the experimental results of Ritzau et al. 14

Discussion

Starting from these results, we can discuss the qualitative agreement between the shear modification effects derived from the molecular mechanism described by this model and the experimental evidence. First we must point out that only long-chain branched molecules are concerned by this phenomenon. For short-chain branched molecules, the modification is minor whereas long-chain branched molecules properties are strongly affected. This is in qualitative agreement with experiments, for which it was reported that material with broader molecular weight distribution and long-chain branching (such as LDPE from autoclave synthesis) are more sensible. We believe that those materials are more affected because they contain much more long-chain branched molecules compared to the other materials. Relaxation times of the modified material are lower, resulting in a smaller intrinsic elasticity and no strain hardening behavior. Another important experimental observation is the annealing time needed to reverse the modifications which is longer than expected. We can now say that the side chain branches alignment hypothesis is not in contradiction with the reptation theory, and it is now acknowledged that the annealing time is longer than the arm relaxation time of the native polymer because of the dynamic dilution effect which increases the global time scale of the material as side chain alignment occurs. Finally, as was assumed by the same authors, "elongation modification" is more effective than "shear modification" despite too few experimental results were reported in the literature.

Our approach has some similarities with the work of Blackwell et al. 21 in the sense that we also consider the effect of branch point withdrawal on the time scale of a pom-pom molecule. These authors investigated the effect of a coupling mechanism between a local branch point withdrawal and the backbone stretch. Their main conclusion is that the relaxation time for the stretch is reduced improving the fit to transient and steady-state data. Furthermore, this avoids the unphysical cusp in extensional viscosity caused by the sharp transition from the stretch dynamics to the branch point withdrawal dynamics when maximal backbone stretch is reached. Nevertheless, there are major differences between this approach and ours. We investigate some much larger branch points displacements which induce some significant changes in the entanglement network and therefore some modifications on all relaxation times. What is described here is a global change of the "apparent topology" of the molecule from an initial state where branched molecules entangle each other to a state where side chain branches and backbone are aligned in a same linear "supertube". This new "apparent topology" induces a strong modification of rheological properties for long-chain branched molecules. On an experimental point of view, reported results show less drastic modifications, and we believe that polydispersity may smooth topological changes because only long-chain branched molecules undergo branch point withdrawal.

We must also discuss the increase of the plateau modulus values predicted by our model. On eq 12 one can argue that the shear modulus increases with s c which is in qualitative disagreement with the intuitive idea that disentanglement increases with arm retraction and with deformation. Nevertheless, in the original pom-pom model, arms do not contribute to the effective entanglement network for flow rates below τ a -1 . Therefore, the decrease in arms length does not induce any change in the molecular weight between entanglements M e . On the contrary, the increase of backbone length makes M e decrease and therefore the plateau modulus increase. Taking into account arms to the entanglement molecular weight is thus an important issue which will be treated in a future work.

Conclusion

The aim of this work was to use a theoretical model based on a recent molecular dynamics picture of LCB polymers to discuss the origin of the phenomenon of "shear modification". Small changes in the original pompom model introduce naturally a dependence of rheological properties on flow history. Side chain branches partly align each other within a single backbone tube. This change in apparent topology leads to an evolution of the global morphology which induces a very significant modification of relaxation times. This modification has a great effect on the recovery kinetics of the initial properties. The annealing time may be larger by several orders of magnitude compared with the longest relaxation time of the native material. This result explains why there is no contradiction between the side chain alignement hypothesis proposed by Leblans and Bastiaansen 15 and the reptation theory. As previously discussed by these authors, our results show that "elongation modification" is more effective than "shear modification" for the same flow strength. Furthermore, the effect of molecular topology is important, and results demonstrate that very long-chain branched molecules are highly affected by "shear modification", whereas short-chain branched molecules (linear polymer) undergo very small changes.

All our results are in qualitative agreement with previously reported experimental results. However, we have only studied flow conditions respecting an upper limit of strain rates such that We a < 1. This is an important restriction from an experimental point of view, especially for very long-chain branched molecules. We therefore want to include the arms contribution to the stress in order to study stronger flow conditions and try to achieve a quantitative comparison with experimental results. This will be the aim of our future work. Eventually, it will be of highest importance from an industrial point of view to study an effective way to control this "flow modification" in order to get a powerful control of material properties during the process and to achieve higher processability and performances.

  Relaxation Times. a. Dynamics of orientation tensor S arm relaxation time where and backbone relaxation time b. Dynamics of stretch parameter λ stretch relaxation time c. Dynamics of arm withdrawal d. Expression for the stress2.3. Results.From our set of equations we can first notice that under slow flow conditions such that branch point withdrawal can be neglected (s c ) 0) our model is identical to the original pom-pom model. Nevertheless, under stronger flows, the model exhibits a singular behavior in both simple shear and uniaxial elongation. The most interesting feature is its consequence on the relaxation time scale. As we can observe in eq 3, when s c increases, two competing effects control the value of τ a . The first one is the dynamic dilution effect that s b + 2s c 2qs a + s b(5)

Figure 1 .

 1 Figure 1. Shear modification of a pom-pom polymer.Figure 2. Shear modification recovery; flow is stopped at t/τa(t)0) ) 1000.

Figure 2 .

 2 Figure 1. Shear modification of a pom-pom polymer.Figure 2. Shear modification recovery; flow is stopped at t/τa(t)0) ) 1000.

Figure 3 .

 3 Figure 3. Transient shear viscosity before preshearing.

Figure 4 .

 4 Figure 4. Transient shear viscosity after preshearing.

Figure 5 .

 5 Figure 5. Uniaxial viscosity of the native polymer.

Figure 6 .

 6 Figure 6. Uniaxial viscosity of the modified polymer.

Figure 7 .

 7 Figure 7. Elongation modification.

Figure 8 .

 8 Figure 8. Recovery kinetics for a sa ) 3, sb ) 30, q ) 5 pompom molecule; flow is stopped at t/τ a(t)0) ) 1000.Figure9. Recovery kinetics for a sa ) 90, sb ) 30, q ) 5 pompom molecule; flow is stopped at t/τ a(t)0) ) 1000.

Figure 9 .

 9 Figure 8. Recovery kinetics for a sa ) 3, sb ) 30, q ) 5 pompom molecule; flow is stopped at t/τ a(t)0) ) 1000.Figure9. Recovery kinetics for a sa ) 90, sb ) 30, q ) 5 pompom molecule; flow is stopped at t/τ a(t)0) ) 1000.

† Universite ´de Pau et des Pays de l'Adour. ‡ Ecole Normale Supe ´rieure de Cachan. * Corresponding author: e-mail gerard.marin@univ-pau.fr.


Alexander Smirnov

Alexey Kashevnik
email: alexey@iias.spb.su

Nikolay Teslya
email: teslya@iias.spb.su

Nikolay Shilov

Andreas Oroszi

Mario Sinko

Michael Humpf
email: hump@de.festo.com

Jens Arneving

Knowledge Management for Complex Product Development

Keywords: product knowledge management, product design, complex product configuration

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Current trends in the worldwide economy and increasing competition require companies to propose not only new products but whole integrated solutions to their customers. Such solutions might consist of multiple physical devices as well as services. One of the consequences of this is appearance of "complex products", which consist of other products (both regular products and complex products) and often include software units using different services.

Another strategy that brings companies and their customers in a closer collaboration is innovation democratisation. This is a relatively new term standing for involvement of customers into the process of designing and creating new products and services. This makes it possible for companies to better meet needs of their customers. [START_REF] Hippel | Democratizing innovation[END_REF] Modularisation is one of the keys to carry out mass customisation and innovation democratization strategies with low cost [START_REF] Egan | Implementing a Successful Modular Design[END_REF][START_REF] Sharda | Information and Management Systems for Product Customization' Integrated Series in Information Systems[END_REF]. Module is a result of disassembling functions and architecture of the product. It can provide the end-products in wider scope while decreasing diversification in product components.

In this situation the previously developed PLM mechanisms and workflows supporting products from the design phase to putting them to the market and further has appeared to be inefficient. As a result, there was a need to design new, knowledge-based workflows and supporting software systems to increase efficiency of designing and maintaining new product ranges. Knowledge-based PLM is currently a critical and strategic issue (Bernard and Tichkievitch, 2008) and the company recognizes this (e.g., [START_REF] Smirnov | Collaborative Recommendation Systems for PLM: Approach and Technological Framework[END_REF].

However, implementation of such changes in large companies faces many difficulties because business process cannot be stopped to switch between old and new workflows, old and new software systems have to be supported at the same time, the range of products, which are already in the markets, has to be maintained in parallel with new products, etc. Another problem is that it is difficult to estimate in advance which solutions and workflow would be efficient and convenient for the employees. Hence, just following existing PLM implementation guidelines is not possible (also confirmed by [START_REF] Bokinge | PLM implementation guidelines -relevance and application in practice: a discussion of findings from a retrospective case study[END_REF], and this process has to be and iterative and interactive.

The paper represents an ongoing long-term project aimed at knowledge management for multi-level complex product development at Festo AG&Co KG company that has more than 300 000 customers in 176 countries supported by more than 52 companies worldwide with more than 250 branch offices and authorized agencies in further 36 countries.

Fig. 1 represents the developed multi-level knowledge management concept for complex product development, which is the core of the described project. The black rectangles have been already implemented and the appropriate tools are indicated (grey areas). The major product ontology is in the center of the model. It is used to solve the problem of knowledge heterogeneity and enables interoperability between heterogeneous information sources due to provision of their common semantics and terminology [START_REF] Uschold | Ontologies: Principles, methods and applications[END_REF]. It describes all the products (produced and to be produced) and their features (existing and possible). This ontology is supported by two tools addressed in sec. 2: NOC (New Order Code) and CONCode ("old code", CON is a prefix based on the reverse acronym of New Order Code). Building a knowledge map that connects this ontology to different knowledge sources of the company is yet to be addressed.

The configuration level has also been addressed only partially so far. The tool called CONSys (SYStems of products) supports creation of configuration models of complex products and solutions (sec. 3). Configuration models include their structures as well as different rules responsible for complex product consistency checks, technical data calculation, etc. The support of the configuration process design (the left lower part of the figure) has not been implemented yet.

Based on the built configuration model the process of complex product or solution configuration in accordance with given requirements can be automated. This is done via the tool called CONFig (CONFIGuration) addressed in sec. 4. So far this tool supports the configuration process in terms used within the company (Festo knowledge level). In reality, the customers are used to operate different terminology (Customer level), which doesn't correspond "one to one" to that used within the company. Besides, customers from different industries can also operate different terms. As a result, there is a need to create configuration tools that can map customers' requirements to those used in the company taking into account the context (customer's industry segment, history of customer's orders, etc.). This is the goal of the future research.

Product Ontology

The developed approach is based on the idea that knowledge can be represented by two levels. The first level describes the structure of knowledge. Knowledge represented by the second level is an instantiation of the first level knowledge; this knowledge holds object instances.

The knowledge of the first level (structural knowledge) is described by a common ontology of the company's product families (classes). Ontologies provide a common way of knowledge representation for its further processing. They have shown their usability for this type of tasks (e.g., [START_REF] Bradfield | A Metaknowledge Approach to Facilitate Knowledge Sharing in the Global Product Development Process[END_REF][START_REF] Chan | A framework of ontology-enabled product knowledge management[END_REF][START_REF] Patil | Ontology-based exchange of product data semantics[END_REF] The first step to implementation of the approach is creation of the ontology. This operation was done automatically based on existing documents and defined rules of the model building. The resulting ontology consists of more than 1000 classes organized into a 4 level taxonomy, which is based on the VDMA classification (Verband Deutscher Maschinen-und Anlagenbau, German Engineering Federation, VDMA, 2013). Taxonomical relationships support inheritance that makes it possible to define more common attributes for higher level classes and inherit them for lower level subclasses. The same taxonomy is used in the company's PDM and ERP systems.

For each product family (class) a set of properties (attributes) is defined, and for each property its possible values and their codes are defined as well. The lexicon of properties is ontology-wide, and as a result the values can be reused for different families. Application of the common single ontology provides for the consistency of the product codes and makes it possible to reflect incorporated changes in the codes instantly.

The ontology is multilingual. Each class and attribute can be assigned names in different languages so that specialists from different countries could work simultaneously preserving consistency of the ontology.

The selectable properties can be communicated to the customers with so called configuration solutions or product selectors. These tools can be used in electronic catalogues and online shop systems. As a result, a new product codification scheme based on product families and characteristics has been developed [START_REF] Oroszi | Ontology-Driven Codification for Discrete and Modular Products[END_REF]. For companies with wide assortments of products (more than 30 000 -40 000 products of approx. 700 types, with various configuration possibilities) it is very important to ensure that customers can easily navigate among them. This is an important task for customer communication management because well defined and understandable product identification is mandatory for ensuring good corporate look for the company [START_REF] Baumeister | Customer relationship management for SMEs[END_REF]Fjermestad and Romano, 2002;[START_REF] Piller | Individualization Based Collaborative Customer Relationship Management: Motives, Structures, and Modes of Collaboration for Mass Customization and CRM[END_REF]. The system should be organised in such a way that would ensure the uniqueness of the code and its validity. The resulting code is supplied into the company's PDM and ERP system.

The described above ontology provides rules for the codification system in the following way. For each class a number of attributes is assigned in a certain se-quence. This sequence of attributes forms a template for codes of products belonging to the appropriate product family. For each product the properties are replaced with codes of their values corresponding to the particular product to generate its code.

For example, the DSBC series is a family of pneumatic drives with a single moving rod (Fig. 2). There are 35 different properties with each between 4 to 10 values. Customer and engineers can select from these properties and values. Some combinations of different properties and values are not allowed because they are technically impossible.

Typical properties of the DSBC series are "locking in end positions" and "special antifriction features". Their possible values are presented in Table 1.

Given code template consisting of a delimiter and the above two properties, the following codes can be built:

-Standard (no locking, no antifriction features): DSBC -"Extend / retract" (locking in both directions) and no antifriction features: DSBC-E1 -"Extend / retract" and "Low friction": DSBC-E1L Fig. 3 represents a fragment of the real code scheme for the DSBC series. Inheritance of more common properties from higher, more abstract classes ensures that for different branches of the classification the sequence of common properties will be the same that simplifies the code interpretation.

After finding the desired product based on its properties, the set of selected properties is translated into a product code based on the existing code scheme of its class (product type). After transmitting the product code to the company, the customer receives an order confirmation listing selected product properties. The confirmation is generated with the company's ERP system, which uses code schemes to translate product codes into the properties of product.

Complex Product Configuration Rules

As it was already mentioned, today, the customers tend to buy complete customized solutions (referred to as "complex products") consisting of numerous products, rather than separate products. Before, the complex products were configured by experts based on the customer requirements. Today, this process requires automation. However, inter-product relationships are very challenging. For example, the most common use case is the relationship between a main product and an accessory product. While both products are derived from different complex products there are dependencies which assign a correct accessory to a configured main product. The dependencies are related to the products' individual properties and values. E.g., "1x3/2 or 2x3/2-way valve" cannot be installed on a valve terminal if its size is "Size 10, deviating flow rate 1". The depth of product-accessory relationships is not limited, so accessory-of-accessory combinations have also to be taken into account. The relationships can be very complex when it comes to define the actual location and orientation of interfaces and mounting points between products.

Complex product description consists of two major parts: product components and rules. Complex product components can be the following: simple products, other complex products, and application data. The set of characteristics of the complex product is a union of characteristics of its components. The rules of the complex products are union of the rules of its components plus extra rules. Application data is an auxiliary component, which is used for introduction of some additional characteristics and requirements to the product (for example, operating temperatures, certification, electrical connection, etc.). They affect availability and compatibility of certain components and features via defined rules.

Product Configuration

When the configuration model is finished it is proposed to the customers so that they could configure required products and solutions themselves or with assistance of product managers.

An example configuration interface is presented in Fig. 5. The figure represents the mentioned above valve terminal VTUG with an electrical control module CTEU. The user can choose values (from drop down lists, outlined in Fig. 5 with bold dotted line) for different properties (rows in the table). Depending on the choices other properties either selected automatically, disabled or become hidden. This is done on the basis of the rules described during the complex product definition. The rules are processed "on the fly", i.e. as soon as the user chooses a value, all other values are checked for consistency immediately.

Integrated Workflow

So far developed integrated knowledge management workflow is presented in Fig. 6. At the first stage the major product ontology is filled with generic classifications of products and their components. This is done via two tools (NOC and CONCode) since recently developed order code scheme differs from that used before. However, since multiple customers are used to operate with the old classification it has to be maintained.

At the next stage the product managers and modelers design new products and solutions based on existing products and components (the CONSys tool). If a new product or component is needed, its implementation can be requested from the order code structure team. Together with new products and solutions, the appropriate rules and conditions are designed as well (e.g., acceptable load, size, compatibility constraints, etc.).

When the configuration model is finished it is proposed to the customers so that they could configure required products and solutions themselves or with assistance of product managers (the CONFig tool).

Conclusion

The paper presents a long-term ongoing project aimed at introduction of product knowledge management for different stages of its lifecycle. The systems significantly simplify and speed up the process of product development. The technical options presented by the product manager and developer are converted into order-Fig. 6. Integrated workflow relevant options. As most of the characteristics can be used again, only new options must be discussed and entered in the system. Besides this, the error risk has been significantly reduced.

The major advantages of the developed systems are: systematic order codes for all products; machine readability; quick orientation for selecting right products and services; security when selecting and ordering products and services.

One other advantage is the reusability of the data. The structured data are used in other processes such as:

-Automatically creating master data in SAP models; -Automatically creating data for the configuration models and services; -Automatically generating an ordering sheet for the print documentation (this ordering sheet was generated earlier with high expenditure manually); -Automatically generating a product and service list which is needed in the complete process implementing new products.

Fig. 1 .

 1 Fig. 1. Multi-level knowledge management for complex product development

Fig. 2 .

 2 Fig. 2. DSBC product: a pneumatic drive

Fig. 4 .

 4 Fig. 4. Solution rules example. Some example rules are shown in Fig. 4. The figure represents a valve terminal (VTUG) and compatibility of electrical accessories option C1 (individual connecting cable) with mounting accessories (compatible only with H-rail mounting) and accessories for input-output link (not compatible with 5 pin straight plug M12). These rules are stored in the database and can be later used during configuration of the valve terminal for certain requirements.

Fig. 5 .

 5 Fig. 5. Product configuration interface: an example

Table 1 :

 1 Compatibility table

	Value Code	Value Name	Description
	Locking in end positions
		Without (Standard) the most common choice, no locking in end positions
	E1	Extend / Retract	locking is applied in both directions of movement
	E2	Extend	locking is applied in the direction of extension
	E3	Retract	locking is applied in the direction of retraction
	Special antifriction features
		None / Normal type the most common choice, no special antifriction features
	L	Low friction	product with reduced friction
	S	Slow speed	product oriented to slow movement

Acknowledgments

The research presented is motivated by a joint project between SPIIRAS and Festo AG&Co KG. Some parts of the work have been sponsored by grants # 12-07-00298-a, # 12-07-00302-a of the Russian Foundation for Basic Research, project # 213 of the research program "Intelligent information technologies, mathematical modelling, system analysis and automation" of the Russian Academy of Sciences, and project 2.2 "Methodology development for building group information and recommendation systems" of the basic research program "Intelligent information technologies, system analysis and automation" of the Nanotechnology and Information technology Department of the Russian Academy of Sciences.