Multi-scale modeling of delamination through fibrillation
Introduction
Adhesion is a subject of large interest, both from an industrial and scientific point of view. A recent problem is the adhesion in copper–rubber interfaces, such as applied in stretchable electronics. Stretchable electronics are typically found in electronic textiles and biomedical applications (Coosemans et al., 2006, Li et al., 2005, Khang et al., 2006, Song et al., 2008, Gonzalez et al., 2008). In general, stretchable electronic devices are constructed from small rigid semiconductor islands interconnected by metal, usually copper, conductor lines. These interconnects are embedded in a highly compliant substrate, typically a rubber material. It has been shown that delamination of the copper–rubber interface is a precursor to (mechanical and/or electrical) failure of the device (Hsu et al., 2010). Experimental observations have led to the conclusion that in these samples, delamination occurs through fibrillation of the rubber. This process involves the formation, elongation and failure of rubber fibrils (Hoefnagels et al., 2010).
Because of the detrimental effect of delamination on the performance and life time of the product, knowledge of the interface properties is of key importance for the development of new products. Usually, interface properties are determined through macroscopic (finite element) modeling using cohesive zone (CZ) elements, where a pre-defined constitutive relation between the interface opening and traction (the traction–separation law, or TSL) is employed. By fitting this model on experimental data the CZ parameters are obtained. In earlier work on delamination in copper–rubber interfaces this macroscopic approach proved to be successful (Hoefnagels et al., 2010, Van der Sluis et al., 2011). Good correspondence between the fitted model and the experiments was observed in terms of load-displacement curve, local peel geometry, and the location of the onset of delamination.
However, several issues limit the applicability of these models. The first concern is the lack of a clear quantitative relation between the obtained cohesive zone parameters and the observed fibrillation micromechanics. In fact, it was observed by Hoefnagels et al. (2010) that the occurring fibril length was about 6 times smaller than the obtained cohesive zone critical opening length. This makes it difficult to scrutinize the physical meaning of the established cohesive zone parameters. This problem is caused by the fact that adhesion is a multi-scale phenomenon, whereas it is projected on a single scale when pursuing a macroscopic approach as illustrated in Fig. 1. Evidently, the micromechanics of fibrillation in the experiment is not accounted for in the macro-scale model. In particular, all micro-mechanisms in the vicinity of the interface are lumped into the CZ model, leading to the mentioned mismatch between microscopic experimental observations and macro-scale interface parameters.
Because of the fibrillation process, the macroscopic interfacial dissipation (work-of-separation) consists not only of the thermodynamic work of de-adhesion between the two materials (i.e. resulting from the intrinsic adhesion), but also of the energy dissipated in the fibrillation process (Crosby and Shull, 1999, Lakrout et al., 1999, Creton et al., 2001, Brown et al., 2002). This can lead to work-of-separation values that are several orders of magnitude larger than the thermodynamic work of de-adhesion (Hoefnagels et al., 2010, Van der Sluis et al., 2011).
Associated to this, and severely limiting the practical use of the obtained interface parameters, is the fact that the fibrillation micromechanics, and thus the associated dissipation, depend on the loading conditions (Brown et al., 2002, Van der Sluis et al., 2011). As a result, the obtained macro-scale interface properties are system properties instead of interface properties. This problem can only be resolved by taking into account the fibrillation micromechanics in the macro-scale interface description. This encompasses the development of a model of the fibrillation micromechanics and a method to couple this model to the macroscopic interface description.
To address the above-mentioned issues, a multi-scale interface method is used. Micromechanical modeling of crack growth in viscoelastic media was achieved through an analytical description of the fibrillation process by Allen and Searcy (2001). In the present paper, the multi-scale interface concept introduced by Matousˇ et al. (2008) is used. The method has later been used to describe a variety of interfaces, see e.g. Hirschberger et al. (2009), Verhoosel et al. (2010) and Cid Alvaro et al. (2010). The method was initially developed to describe adhesive layers with known thickness. However, since there is no adhesive layer in the copper–rubber interface, some modifications need to be made.
In the multi-scale method the macro-scale interface is still described by a cohesive zone formulation. However, the TSL is no longer defined a priori, but is obtained from a micro-model through a numerical homogenization scheme. Hence, this method allows the explicit modeling of the fibrillation micromechanics and simultaneous application of the obtained response at the macro-scale. Clearly, in this way the obtained TSL is quantitatively coupled to the micromechanical quantities, e.g. maximum fibril length. Since no closed form solution is aimed for in the present paper, it is more appropriate to refer to a ‘traction–separation response’ rather than a TSL.
The fibrillation micromechanics has received considerable attention, especially for viscoelastic pressure-sensitive adhesives (PSAs). Zosel (1998) and later Creton and co-workers used tack tests to gain insight into the parameters that control the fibrillation process, see e.g. Lakrout et al. (1999), Brown et al. (2002) and Shull and Creton (2004). In their tests a rigid cylindrical punch was brought into contact with a layer of PSA, and after establishing good contact the punch was retracted. Their experiments show that the fibrillation process consists of several phases. The first phase is homogeneous deformation of the film, after which cavities nucleate at the interface between the two materials. Increasing the loading leads to the growth of these cavities. Consequently, a fibrillar structure is formed. Finally the interface fails either by fracture of the fibrils or by debonding of the foot of the fibril from the substrate. It was observed that the combination of interface and material parameters determines whether the PSA fibrillates during retraction of the punch, however, a quantitative prediction of the fibrillation process parameters (e.g. dissipation, maximum stress) could not be provided (Brown et al., 2002).
One of the reasons why the process is quantitatively not properly understood is the (extremely) large deformations in the fibrils. Indeed, strains of 1450% have been observed by Horgnies et al. (2007), which need to be taken into account (Creton et al., 2001). Other authors attempted to model fibrillation in PSAs by assuming the fibrils to extend through the entire thickness of the adhesive layer (Yamaguchi et al., 2006, Zhang and Wang, 2009, Lin et al., 2002). However, in stretchable electronics there is no adhesive layer, since the rubber itself adheres to the copper, meaning that the fibrils are actually drawn from the rubber material. Therefore the highly inhomogeneous deformation associated with the drawing of the material from the bulk layer into the fibril needs to be considered. Krishnan and Hui, 2008, Krishnan and Hui, 2009 studied the deformation of a soft solid in contact with a rigid substrate. The observed deformations closely resembled the shape of a fibril, but an analysis towards macroscopic interface properties was not carried out.
Failure during fibrillation occurs either by fracture of the fibril or by debonding of the fibril from the substrate. In this work, we focus on debonding only, which is considered hard to describe (Creton et al., 2001, Creton and Lakrout, 2000), since it requires an accurate description of the local geometry of the fibril near the interface. Glassmaker et al. (2008) described a particular debonding mechanism, namely frictional sliding of the foot of a viscoelastic fibril. However, they note that fibrils of elastic materials, such as rubber in stretchable electronics, basically fail without foot sliding (Glassmaker et al., 2008), making this failure mode unlikely to occur in rubbery materials.
This introduction emphasizes that fibrillation during delamination is a subject of great interest, and that several mechanisms have been studied separately. In this work, a micro-model is developed that addresses the complex interplay between several of these individual mechanisms. To this end, the large deformations associated with the formation of fibrils are incorporated. Furthermore the intrinsic adhesion between copper and rubber is taken into account. This permits us to study the relation between the rubber material, the intrinsic adhesion and the resulting fibrillation, which gives insight into some fundamental aspects of the fibrillation process. By exploiting the multi-scale interface method, a clear relation between the fibrillation micromechanics and the resulting macroscopic interface properties is recovered. Eventually, once all relevant microscopic mechanisms are identified and incorporated in the micro-model, this method will also reduce the need to do separate time consuming interface characterization experiments for each loading condition and microstructural variation. This, however, has not yet been achieved in the current work.
The outline of this paper is as follows: first, the multi-scale interface model is summarized. Then the micro-model is introduced in detail. Next, numerical examples are shown for the micro-model, especially focusing on the fibrillation process. The relation between the micro-scale phenomena and the macro-scale interface properties is presented. The influence of several physical micro-scale parameters on the results is analyzed. A multi-scale model of a peel test is carried out, incorporating the obtained micro-model results and demonstrating the suitability of the method. Finally, the implications of the results and limitations of the model are discussed.
Section snippets
Multi-scale interface model
Since the first order multi-scale interface method has been extensively described in the recent literature, only the main aspects are summarized in this section and illustrated in Fig. 2. For more detailed information, the reader is referred to Matousˇ et al. (2008), Hirschberger et al. (2009), Verhoosel et al. (2010) and Cid Alvaro et al. (2010). The physical problem is discretized using the finite element method. At the macro-scale the interface (with predefined geometry) is modeled with
Results
In this section, first the interplay between intrinsic adhesion and fibril deformation is illustrated. In addition, the influence of the intrinsic adhesion properties on the macroscopic interface properties is discussed.
Conclusion
A multi-scale interface model was developed to describe the fibrillation micromechanics in a delaminating copper–rubber interface. The macroscopic interface description uses a traction separation law, which is obtained directly from the micro-model through a consistent computational homogenization scheme. Using this approach, a concurrent macro-scale and micro-scale analysis is possible.
At the micro-level, several essential aspects of the fibrillation process were included. Departing from a
Acknowledgments
This research is supported by the Dutch Technology Foundation STW, which is the applied science division of NWO, and the Technology Programme of the Ministry of Economic Affairs under Project No. 10108. The authors acknowledge Jan Neggers from Eindhoven University of Technology for providing the SEM images.
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