Introduction. Theory: Anisotropy and Polymer Physics

Transcription

1 UNDERSTANDING AND CONTROLLING PRODUCT ANISOTROPY AND ITS EVOLUTION IN HOTMELT COATING OF PSA MATERIALS METHODOLOGY OF AND RESULTS FROM FINITE-ELEMENTS SIMULATIONS Thilo Dollase, Ph. D., tesa AG, Hamburg, Germany Bernd Dietz, tesa AG, Hamburg, Germany Prof. Manuel Laso, Ph. D., Universidad Polytéchnica de Madrid, Madrid, Spain Bernd Lühmann, Ph. D., tesa AG, Hamburg, Germany Introduction Material properties not only depend upon the chemical nature and structure of that material, or its constituents, but also on the specific way the material is processed. High molecular weight polymers represent the basis of most pressure-sensitive adhesives (PSAs) and, thus, play an important role in defining their performance profile. Many coating processes applied for PSAs, particularly solvent-free variants, involve substantial deformation of polymers causing molecular orientation and chain stretching to a degree that depends distinctly on specific details of process design and process parameters. Shear and elongational deformation (Figure 1) may exist in different process elements to a different extent and individually contribute to the evolution of product anisotropy. Since the degree and type of anisotropy present in a material may significantly affect materials properties, it is of central importance to control it. In a recent investigation we used results from finite-elements (FE) simulations to evaluate the influence of various process parameters and designs on the evolution of product anisotropy in a hotmelt PSA coating process. Applying a technique to simulate the behavior of the material during processing provides several advantages over exclusively analyzing experimentally available data one of which being the opportunity to easily modify process designs or, moreover, to check parameters that cannot experimentally be realized. Another major advantage of employing simulations is the fact that anisotropy can be monitored at any desired point along the process and, as a consequence, the influence of each single process segment can be evaluated. In order to gain reliable information from such simulations a comprehensive set of experimental data of the materials of interest has to be generated and used as an input for the computations. These data mainly include results from rheological testing which should reflect the behavior of the material under the desired process conditions as closely as possible. Applying an FE simulation technique allows us to establish correlations between modifying particular process elements and the resulting degree of product anisotropy. Such information provides implications on the way a proper coating process for exemplary materials may be designed in order to obtain a product of desired anisotropy. In our contribution we present the procedure of our simulation concept to monitor the evolution of product anisotropy for hotmelt PSAs furnished by results for exemplary cases. Theory: Anisotropy and Polymer Physics The phenomenon of anisotropy stems from differences in the structure of a molecule, a molecular segment or any other molecular building block along at least two directions. While a sphere has the

2 same structure along all possible directions and thus represents an isotropic entity, for an ellipsoid three orthogonal semiaxes of different value are required to describe its geometry. An ellipsoid, hence, resembles an intrinsically anisotropic body. For polymers the envelope of a typical equilibrium chain conformation is represented by such an ellipsoid. Under equilibrium conditions an ensemble of polymers exhibits no net macroscopic orientation, i.e., the orientations of ellipsoids are statistically distributed as is shown in Figure 2. Since each ellipsoid, however, represents an anisotropic entity, a potential ability to be oriented exists for an ensemble of ellipsoids. Such an orientation process can be due to shear or elongational deformation [1,2]. The molecular and also net macroscopic orientation are beneficially expressed by the configuration tensor. The eigenvectors associated with this tensor give insight into the molecular orientation relative to a Cartesian coordinate system of the process comprising machine direction, cross direction and normal direction. The magnitude related to the eigenvectors, the eigenvalues Λ 1, Λ 2 and Λ 3, define the degree of orientation along all three axes of this coordinate system. Roughly speaking, Λ 1 is correlated with the machine direction, while Λ 2 is linked to the cross direction and Λ 3 is related to the normal direction. A valuable indicator for orientation within the PSA film is given by the ratio Λ 1 /Λ 2. In the nonlinear flow regime polymer chains may experience substantial deformation from their equilibrium conformation. The ability of being deformed to a particular degree is quantified by the maximum extension of the chain, λ max. For the equilibrium, chain conformation, λ, is defined to be 1. For systems with λ >> 1, chains are significantly deformed and the system tends to relax (Figure 3). In general, although extension and orientation can in principle be treated separately, a process incorporating deformation fields will usually lead to a superposition of both events and, as a result, extension will not be found without orientation. On a qualitative level, the flow behavior of entangled polymer melts which involves reorientation and relaxation processes is largely governed by their dynamics. A theoretical model that appropriately describes many polymer dynamics related aspects in an appropriate way is known as the reptation model [3] or tube model [4]. According to this model polymer chain motion breaks up into four distinct regimes separated by characteristic times, τ i. Furthermore, the effect surrounding chains impose on one selected chain are portrayed as a tube in which the chain can move. On some of the different timescales motion is restricted by the constraining tube. As one feature of this model, the reorientation of polymer sections can be correlated with stochastic fluctuations of these sections on different length and time scales. As such, it is able to predict the relaxation behavior of polymer melts. Short sections such as segments consisting of only a few repeat units relax faster than the longer segments like, e.g., those between to adjacent entanglements. The longest relaxation time, τ d, is related to the complete reorientation of an entire polymer chain. It expresses the motional process the chain undergoes encompassing the event of leaving one tube and entering another which is why it is termed disengagement time or disentanglement time. FE Simulations: Procedure, Selected Results and Discussion Altogether seven steps have to be undertaken in order to obtain results from FE simulations and, as a matter of fact, only a few of these steps are directly associated with the FE calculation itself. The complete workflow of an FE simulation of the type performed in our project involves (1) definition of process design, process conditions and type of materials

3 (2) rheological and thermophysical characterization of materials (3) selection of an appropriate constitutive equation and fit of rheological data (4) generation, refinement and convergence tests of FE meshes (5) numerical determination of temperature, velocity and velocity gradient fields (6) calculation of evolution of polymer structure along a process trajectory (7) extraction of molecular information (anisotropy) In the following we will carry out the procedure of the above given list of individual steps (see also the flow chart in Figure 4) and demonstrate its application for one exemplary process segment. (1) Definition of Process Design, Process Conditions and Type of Materials Typical equipment to coat hotmelt PSAs comprises a material supply device or container, a die, a conveyed web which is given by a moving medium that is to be coated, and a device that may optionally be used to cure the coated PSA material. Such a process is schematically illustrated in Figure 5. Each segment of the coating process may be different with respect to temperature and flow profile and, in particular, the influence of shear and extension. As a consequence, depending on its chemistry and physics, the effect of process conditions on the response of the material may differ from one process segment to another. It is of key importance to analyze the process with respect to the flow profile present in each segment. In sections where the processed material experiences parallel walls shear flow exists (Figure 1). The extent of shear depends on throughput but also on geometrical parameters such as channel diameters. It is typically quantified by the shear rate, γ&. Process segments with converging or diverging boundaries as is illustrated in Figure 1 additionally impart elongational flow within the processed material. As for shear, elongational flow is affected by throughput and the exact geometry of the local environment. Elongational flow is mainly characterized and quantified by the elongation rate, ε&. In general, at a given point in the process deformational flow of the material is given by a superposition of both, a contribution of shear and a contribution of elongation. In the present paper we will focus on the section adjacent to the die's exit as one selected segment of the coating process and discuss some of the aspects related to the flow behavior of a PSA material and the evolution of product anisotropy in this stage of the process. One goal of our simulations was to check the influence of the slit height of the die's exit, h, at a given length of the lip of the die, L, on the product anisotropy of the PSA material. As numbers for these quantities we chose h = 150 µm, and L = 60 mm. A sketch of the respective process segment is shown in Figure 6. As model adhesives we selected non-branched polyacrylates which, depending on their composition, can be used as PSAs in their pure form. The simplicity of such a system renders it suitable for a polymer theoretical description and thus structure/property relations can easily be established once results from FE simulations are available. From the chemical point of view, the physical properties of a linear polyacrylate, especially its rheological characteristics, which are directly related to its behavior under flow conditions during processing, depend on the type and amount of comonomers used and the nature of its molecular weight distribution. These structural details affect properties like glass transition temperature, entanglement molecular weight, disentanglement time and the relaxation spectrum which distinctly define the rheological profile along with its temperature and time dependence of the material in the melt. From the physicist's point of view, the rheological behavior of the polymer melt is mainly characterized by intermolecular interactions on different length and time scales. Such interactions become prominent in phenomena such as monomeric friction coefficients or transient chain entanglement [4,5]. All chemical and physical properties summarized in this paragraph are reflected by the rheological behavior of the material of interest.

4 (2) Rheological and Thermophysical Characterization of Materials The simulation technique described here requires input of experimental data covering the flow behavior of the PSA material of interest as closely as possible. An arbitrarily chosen coating process for hotmelt PSAs typically comprises a number of different flow regimes that have to be accounted for in collecting such data. In addition to linear viscoelastic properties it is of major importance to probe nonlinear behavior under shear and elongation. A set of experiments that mimics the flow conditions of a given process is summarized in Table 1. Linear viscoelasticity under shear is reflected by the storage and loss moduli, G' and G'', respectively, determined by small amplitude oscillatory rheometry at various frequencies and temperatures. Data regarding steady state nonlinear viscoelasticity under shear, namely viscosity, η, and first normal stress coefficient, N 1, are obtained from capillary viscosimetry or via rotational rheometry at different shear rates, γ&, and temperatures. The nonlinear viscoelastic behavior under transient shear can be determined by using rotational rheometry and, last but not least, the viscoelasticity under transient elongation is probed by an elongational rheometer such as, e.g., the Meissner rheometer [6]. Besides rheological data information on the temperature dependence of density, heat capacity and heat conductivity should also be available. All experimental data are used to introduce materials properties relevant to the processing behavior into the molecular simulation model. (3) Selection of an Appropriate Constitutive Equation and Fit of Rheological Data to it As a tool for the FE computations we applied the CONNFFESSIT approach (Calculation of Non- Newtonian Flow: Finite Elements and Stochastic Simulation Technique) developed by Öttinger and Laso [7,8] employing the GENERIC formalism (General Equation for the Non-Equilibrium Reversible- Irreversible Coupling) to describe flow processes of entangled polymer melts [9,10]. Generally speaking, FE simulations are a means to correlate the resulting stress present in a polymeric fluid at a given point of a flow process with its strain history. Within this procedure the FE calculation takes on the task to provide a solution to the viscoelastic flow problem within the process geometry and, thus, combines data resulting from computations on the molecular behavior with geometrical details of the process. As a principal outcome FE simulations offer information on the physical state in particular with respect to stress at any desired point of the process. In order to be able to accomplish the above mentioned job, a suitable molecular model, the so called constitutive equation, has to be implemented into the FE computation. A constitutive equation provides the formalism to describe the mass and transport balances and the molecular dynamics of a material. For this purpose the GENERIC approach has been selected as an advanced set of constitutive equations since it proves suitable to properly account for double reptation, chain stretching, convective constraint release and anisotropic tube cross sections and to avoid independent alignment [10]. The GENERIC formalism represents a sophisticated and improved variant of the tube model introduced by Doi and Edwards [4]. In particular, in contrast to the tube model the GENERIC approach is able to account for chain extension. As another advantage of the applied constitutive equation GENERIC is not a single chain model but averages over an ensemble of approximately polymer chains and correspondingly the same number of reptation processes. If the molecular model is fit to the rheological data, four different fit parameters are obtained along with their temperature dependence. These four parameters are given by the number of entanglements per chain, Z, the maximum chain extension, λ max, the plateau modulus, G N 0, and the disentanglement time,

5 τ d. In combination with the constitutive equation these data are sufficient to describe the materials flow behavior. This information will be filled into the FE mesh and for each segment of the grid the state of the material will be known. (4) Generation, Refinement and Convergence Tests of FE Meshes Mesh generation represents the key step of the FE method for numerical computation. For a given domain in the present case a polygon describing the flow domain generating a mesh means to partition it into simple "elements", e.g., triangles. In addition to efficiency, mesh generation should take into account that some portions of the domain may need small elements for increased accuracy of computation results. All elements should also be "well shaped" which in general means that individual elements are never highly elongated. In the present calculations unstructured meshes were computed using an efficient Delaunay triangulation with post-processing refinement [11,12]. In addition, element (triangle) quality was usually controlled by its aspect ratio which was between 0.44 and 0.8. Furthermore, the meshes were locally refined as is required in order to capture rapid spatial variations in the field variables (temperature, velocity). It was also checked by a series of numerical tests that the calculations converged with mesh refinement. An optimized FE mesh for the die section discussed in this contribution in shown in Figure 7. (5) Numerical Determination of Temperature, Velocity and Velocity Gradient Fields By a parameter field the map of the regarded parameter such as temperature, velocity or velocity gradient within the coordinate system of the process is meant. In other words, the field describes the distribution of possible values the quantity adopts for each point of the space taken up by the PSA material in the process. For each triangle of the FE mesh generated in the simulation phase outlined above the temperature, velocity and velocity gradient fields are determined by solving the mass, momentum and energy balance. An example of a velocity field for the regarded process segment is given in Figure 8. As is known from polymer physics, temperature has a substantial effect on polymer dynamics and it, thus, plays an important role in the reptation model. This temperature dependence is, indeed, found for the fit parameters λ max, G N 0 and τ d given in workflow phase (3). Depending on the temperature in a given FE triangle all polymer dynamics related phenomena occur faster or slower. These phenomena mainly comprise orientation and relaxation where orientation leads to a build-up of anisotropy while relaxation causes its decline. Note that temperature affects the rate of both orientation and relaxation in a similar way since both processes are governed by τ d. The velocity field controls the motion of a volume element of PSA material, the statistic ensemble of the above mentioned polymer chains, throughout the process and, thus, defines the streamline this regarded volume element travels along. Of particular importance for orientation and chain stretching is the velocity gradient. While velocity itself only causes convection and mass transport, it is the velocity gradient that is coupled to the change in chain conformations and orientations. The connection between velocity gradient and chain deformation and orientation becomes obvious if one refers to Figure 1 where the basis of both shear and elongational flow are shown to be related to velocity gradients (see also [2]). (6) Calculation of Evolution of Polymer Structure along a Process Trajectory

6 Once all parameter fields are filled into the FE mesh the actual simulation, i.e., the "experiment on the computer," can be performed. In the beginning of the computation an ensemble of chains is selected at a starting point of the process. During the CONNFFESSIT simulation the pathway of the ensemble through the process regulated by the velocity field is monitored and the effect of temperature and velocity gradient field on chain deformation and relaxation recorded. As the central result this procedure provides data on chain extension, λ, and the molecular orientation, Λ 1 /Λ 2, for each point of the streamline the ensemble has travelled along. Information on λ and Λ 1 /Λ 2 are separately obtained and can be visualized in streamline maps such as those given in Figure 9. The starting point selected for the trajectory shown in Figure 9 was situated in the center of the die channel with respect to normal direction and cross direction. The interpretation of the orientation/extension behavior is discussed in the following section. (7) Extraction of Molecular Information (Anisotropy) This final stage of the simulation workflow involves the interpretation of data obtained by the computation. At this point information on anisotropy, which is now available in terms of chain extension, λ, and molecular orientation, Λ 1 /Λ 2, can be correlated with the parameter modifications implemented for the different simulation cases. In our example we focussed on the design of the exit section of the die. We deliberately varied the exit height, h, from to 150 µm while the channel before the exit remained unchanged. Results for product anisotropy are given in Figure 9 for the two different die exit designs and additionally two different temperatures, T 1 and T 2. First of all, the extension map looks slightly, but noticeably, different from the orientation map, suggesting that the flow profile present in the process affects these anisotropic properties in a different way. Furthermore, the orientation map seems to be more pronouncedly superimposed by noise which is why we choose to concentrate on the chain extension map in the further discussion. In the beginning of the die exit section, just after L = 0 mm (indicated by an asterisk in Figure 9), an undershoot of anisotropy is detected indicating substantial relaxation of the previously extended polymer chains. This can be explained by changes in the die channel design just before position L = 0 mm, the details of which are beyond the scope of this contribution and, therefore, not shown here. The undershoot neatly demonstrates the nonlinear viscoelastic behavior of the processed material which is reflected by a flow history memory. In other words, the viscoelastic response of the material in a particular location of the process is not only affected by the process conditions at this point but also by the flow characteristics the material has experienced before. After the undershoot the polymer travels through a section constrained by parallel walls. In this segment the degree of anisotropy remains constant on a reproducible level for both simulation cases with modified die geometry. At L = 40 mm the die channel starts to taper. For h = 150 µm the convergence of channel walls is even more pronounced than for the h = case. As a consequence, the product anisotropy is slightly increased at the die exit for the 150 µm slit height. The same trend is unravelled for the molecular orientation map, even though data are more noisy. In order to rank the differences in product anisotropy as they result from the modification of die design we compare our data with those obtained for a second temperature which is 20 K lower than the first one. Lowering the temperature leads to a substantial increase in product anisotropy (see upper traces in Figure 9). On the basis of these data the influence of the die exit design can be considered to be of insignificant nature. In addition, besides being on a higher level, the evolution of product anisotropy at the second temperature,t 2, follows a course very similar to what was revealed for T 1. Other results obtained by the FE simulation technique imply that changing the length of the die section involving parallel walls close to the die's exit does not significantly affect the level of achievable anisotropy. As yet another aspect we found that increasing throughput leads to an increase of anisotropy for materials at

7 the point of the die's exit which is explained by increased velocity gradients, yet the effect is not as substantial as what is obtained by changing the temperature. Conclusions In this contribution we have presented an FE simulation procedure for treating a PSA coating process on a computer. The simulation technique allows us to study the influence of exemplary process modifications on product anisotropy given in terms of chain extension and molecular orientation. The design of the die exit within our presented parameter range does not have a pronounced effect on the anisotropy of the PSA. In contrast, temperature was found to represent an efficient control parameter for the generation of anisotropy. Literature Citations [1] I. M. Ward (ed.), Structure and Properties of Oriented Polymers, 2 nd ed., 1997, Chapman & Hall, London [2] I. M. Ward, D. W. Hadley, An Introduction to the Mechanical Properties of Solid Polymers, 1993, Wiley, New York [3] P. G. de Gennes, J. Chem. Phys., 1971, 55, 572 [4] M. Doi, S. F. Edwards, The Theory of Polymer Dynamics, 1988, Clarendon Press, Oxford [5] J. D. Ferry, Viscoelastic Properties of Polymers, 3 rd ed., 1980, Wiley, New York [6] J. Meissner, J. Hostettler, Rheol. Acta, 1994, 33, 1 [7] M. Laso, H. C. Öttinger, J. Non-Newtonian Fluid Mech., 1993, 47,1 [8] K. Feigl, M. Laso, H. C. Öttinger, Macromolecules, 1995, 28, 3261 [9] H. C. Öttinger, J. Rheol., 1999, 43, 1461 [10] J. Fang, M. Kröger, H. C. Öttinger, J. Rheol., 2000, 44, 1293 [11] G. F. Carey, M. Sharma, K. C. Wang, Int. J. Numer. Meth. Eng., 1988, 26, 2607 [12] M. Bern, D. Eppstein, J. R. Gilbert, Proc. 31st IEEE Symp. Foundations of Computer Science, 1990, 1, 231 Acknowledgments The authors appreciate the dedicated contributions by all coworkers in Hamburg and Madrid associated with this project, of which only a small fraction is presented in this paper. We further acknowledge the cooperation with Dr. K. Geiger of IKT Stuttgart who granted access to a Rheometrics RME rheometer.

8 Tables Table 1. Overview of Rheological Experiments Used to Generate Experimental Input for Simulations Shear Elongation linear nonlinear steady state nonlinear transient nonlinear transient small amplitude oscillatory shear in plate/plate geometry capillary viscosimetry or rotational rheometry in cone/plate geometry rotational rheometry elongational rheometry log G log G log η log N 1 log η log η e log ω. log γ log t log t G' = f(ω,t) G'' = f(ω,t) η = f(γ&,t) N 1 = f(γ&,t) η = f(t,γ&,t) η e = f(t,ε&,t) ω exp = ˆ γ& process γ & = & exp γ process γ & = & exp γ process ε & = & exp ε process T exp = T process T exp = T process T exp = T process T exp = T process Illustrations Figure 1. Illustration of a building element in a process with parallel and converging walls. Deformation of a processed material for the former case is governed by shear whereas the latter element imposes an elongation field. orientation relaxation net macroscopic orientation isotropic anisotropic Figure 2. An ensemble of polymer chains depicted as a set of ellipsoids in an isotropic (without macroscopic orientation) and anisotropic state. Deformation of the isotropic ensemble induces ordering of the ellipsoids resulting in molecular and net macroscopic orientation.

9 extension relaxation a = b = c b c a b a a = b = c c Figure 3. An arbitrary chain conformation takes up the space of an ellipsoid. This ellipsoid is characterized by three semiaxes, a, b, c, which differ in value. Deformation leads to a distortion of the shape of the ellipsoid with new semiaxes a', b', c' and correspondingly of the polymer chain. define type of materials define process conditions collect rheological and thermophys. data select constitutive equation numerical computation of T, v, v- gradient fields fit exp. data to constitutive equation computation of molecular changes extract molecular information define process design generation refinement convergence of FE mesh Figure 4. Flow Chart Illustrating the Procedure of an FE Simulation for a Coating Process of PSA Materials. material supply coating device medium to be coated curing station Figure 5. Schematic Representation of a Coating Process for Hotmelt PSAs.

10 final section of die PSA exit h L Figure 6. Illustration of the process segment discussed in this paper. h mm L Figure 7. Result of the generation of an FE mesh for one section of a die. h mm L 0.35 m/s Figure 8. Velocity field for a die as an example for one process segment. Different values of the velocity are encoded by a color scale map (reproduced here as grey scale). (a) λ * 7 T 6 2 T µm T 150 µm mm (b) Λ 1 /Λ 2 3 T mm L L * Figure 9. Streamline maps for (a) chain extension and (b) molecular orientation.

11 Dr. Thilo Dollase tesa AG Quickbornstraße Hamburg phone: +49 (0) fax: +49 (0) Academic and Professional CV Name: Dr. Thilo Dollase Date of Birth: June 26, 1971 Place of Birth: Wolfsburg, Germany 10/1991 7/ /1994 8/1995 Chemistry Studies at Technische Universität Berlin 1st degree (Vordiplom) Graduate Studies at University of Oklahoma, USA Research work on a Master's Thesis entitled "NMR Investigations of the Orientational Ordering of Liquid Crystalline Phases" (Supervisor: Prof. Dr. B. M. Fung) /1995 6/ /1997 4/2000 1/1999 3/ /2000 Master of Science Completion of Chemistry Studies at TU Berlin 2nd degree (Diplom) Research work on a scientific dissertation at Max-Planck-Institut für Polymerforschung, Mainz. Title: "Influence of Rigid Confinements on Local Order and Chain Dynamics in Polymer Melts" (Supervisor: Prof. Dr. H. W. Spiess) Research stay at Ben-Gurion-University of the Negev, Israel Ph. D. awarded joined tesa AG, since recently holding a position as Technology Manager within R&D Dept. for Raw Materials & Polymer Physics