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Lithographically defined interconnects with filamentary, serpentine configurations have been widely used in various forms of stretchable electronic devices, owing to the ultra-high stretchability that can be achieved and the relative simple geometry that facilitates the design and fabrication. Theoretical models of serpentine interconnects developed previously for predicting the performance of stretchability were mainly based on the theory of infinitesimal deformation. This assumption, however, does not hold for the interconnects that undergo large levels of deformations before the structural failure. Here, an analytic model of serpentine interconnects is developed starting from the finite deformation theory of planar, curved beams. Finite element analyses (FEA) of the serpentine interconnects with a wide range of geometric parameters were performed to validate the developed model. Comparisons of the predicted stretchability to the estimations of linear models provide quantitative insights into the effect of finite deformation. Both the theoretical and numerical results indicate that a considerable overestimation (e.g., > 50% relatively) of the stretchability can be induced by the linear model for many representative shapes of serpentine interconnects. Furthermore, a simplified analytic solution of the stretchability is obtained by using an approximate model to characterize the nonlinear effect. The developed models can be used to facilitate the designs of serpentine interconnects in future applications.
During the last decade, stretchable electronics has been attracting increasing attention, owing to their combination of high performance and extreme deformability that cannot be achieved by traditional electronics (Lacour et al., 2006; Kim et al., 2008; Sekitani et al., 2008; Sekitani et al., 2009; Sekitani and Someya, 2010; Li et al., 2011; Kim et al., 2012; Li et al., 2013; Lin et al., 2015; Lu and Yang, 2015; Xu et al., 2015; Yao and Zhu, 2015). Promising applications enabled by such technologies have been reported, such as spherical-shaped digital cameras (Ko et al., 2008; Song et al., 2013), “epidermal” health/wellness monitors (Lanzara et al., 2010; Kim et al., 2011; Jung et al., 2014) and sensitive electronic skins (Someya et al., 2004; Wagner et al., 2004; Mannsfeld et al., 2010; Kim et al., 2014; Yang et al., 2015). A key challenge in the design of stretchable electronics involves the balance of ultra-stretchability with high-performance, due to the brittleness of traditional high-quality electronic materials (Sun et al., 2012). An efficient solution employs the island-bridge architecture (Kim et al., 2008; Ko et al., 2008; Kim et al., 2009; Lee et al., 2011), in which the functional elements sit on the islands and the electrical interconnects form the bridges. Because of the considerable difference in the stiffness, the islands keep almost undeformed upon stretching, while the bridges unravel to accommodate the applied strain (Lacour et al., 2006; Lee et al., 2011). Therefore, a variety of design strategies have been proposed for the electrical interconnects, including the curvy configurations formed through postbuckling of straight ribbons (Sun et al., 2006; Ko et al., 2008; Xu et al., 2011), the planar filamentary configurations in the serpentine or fractal patterns (Gonzalez et al., 2009; Kim et al., 2009; Xu et al., 2013; Fan et al., 2014; Huang et al., 2014; Xu et al., 2014; Zhang et al., 2014; Zhu et al., 2014), and the planar configurations involving Kirigami patterns (Filipov et al., 2015; Song et al., 2015; Shyu et al., 2015; Zhang et al., 2015b). Of these design strategies, the serpentine interconnects have been widely exploited in various functional systems (Kim et al., 2008; Kim et al., 2011; Yang et al., 2015; Zhang et al., 2015a), because of the efficiency in the stretchability and relative simple geometry that facilitates the design and fabrication. In practical applications, the functional device systems interconnected by serpentine traces are usually integrated with an elastomeric substrate, as illustrated in Fig. 1a. Upon stretching, the serpentine ribbons might buckle out of plane or simply deform in the initial plane, depending on the magnitude of thickness/width ratio (Li et al., 2005; Lanzara et al., 2010; Zhang et al., 2013b). For relatively large thickness/width ratios (e.g., > 1.0) that are demanded in applications (e.g., radio-frequency coils, photovoltaic modules) (Yoon et al., 2008; Xu et al., 2013) where low electrical resistances are essential, the serpentine interconnects are governed by in-plane bending deformations. Another class of wide, ultra-thin interconnects that undergo buckled, out-of-plane deformations is also widely used to accommodate large levels of stretching without inducing failure in the materials, because of the characteristic of the wet etching lithography process (Hsu et al., 2011). Due to the complex, spatial deformations during postbuckling, the analytical modeling of buckled serpentine interconnects is very challenging, which is out of the scope of the current study. For the non-buckled serpentine interconnects, theoretical models have been developed to analyze the deformation and stretchability (Zhang et al., 2013a; Widlund et al., 2014; Yang et al., 2014). These models offer insights into the relationships between the stretchability and the various geometric parameters, on the premise that the effect of nonlinear deformation can be neglected. This assumption, however, does not hold for the interconnects that undergo large levels of deformations before the structural failure, where the prediction based on the theory of infinitesimal deformation could lead to an evident overestimation (e.g., > 50% relatively) of the stretchability.
This paper aims to formulate an analytic model of serpentine interconnects in the framework of finite deformation, which provides a more accurate prediction of stretchability than the linear models (Zhang et al., 2013a; Widlund et al., 2014; Yang et al., 2014). To characterize the limit of elastic deformation for the serpentine interconnects made of metals, we introduce the elastic stretchability that is defined as the magnitude of applied strain needed to induce plastic yielding in the interconnects (Zhang et al., 2013b). For the serpentine interconnects made of relatively brittle materials (i.e., silicon, PMMA), we are usually interested in the total stretchability that is defined as the applied strain needed to induce fracture in the interconnects. In the current study, we consider suspended serpentine interconnects that can deform freely during unraveling, such that the constraint of substrate can be neglected. The developed model, as validated by finite element analyses (FEA), is exploited to analyze the nonlinear geometric effect on the stretchability for a wide range of geometric parameters and material systems. Furthermore, an analytic solution of the stretchability is obtained by using an approximate model to account for the nonlinear effect, which agrees well with the results of precise model and FEA. The results suggest that the neglect of such nonlinear effect can result in a considerable overestimation (e.g., > 50% relatively) of stretchability for relatively slender geometries, which might be detrimental in practical applications.
A typical serpentine interconnect consists of three straight wires (length L or L/2) and two arcs with identical radius (R) and arc angle (α), as illustrated in Fig. 1b. Three dimensionless parameters, the width/radius ratio = w/R, the arm length/radius ratio = L/R and the arc angle α, can be then adopted to characterize its shape. The width/radius ratio () of the non-buckled interconnect is usually much smaller than 0.5, such that it can be modeled as a curved, Euler-Bernoulli beam.
Consider a general planar, curved beam AB (width w) in the Cartesian coordinates (X,Y), which is subjected to forces or displacement loading at the ends, as shown in Fig. 1c. Let E1 and E2 denote the unit vectors along the tangential and normal directions of central axis, and Θ the angle by rotating the X axis anticlockwise to E1. After deformation, the material point P(X,Y) moves to p(x, y), and the unit vectors E1 and E2 rotate to e1 and e2. Accordingly, the angle Θ becomes θ in the deformed configuration. The stretch along the central axis is
where S and s is the arc-length coordinate in the undeformed and deformed configurations, respectively. The basic geometric equations include
where is the curvature change corresponding to the work conjugate of bending moment (Su et al., 2012), and K = −dΘ/dS is the initial curvature. The equilibrium equations can be written in terms of the internal forces (t = t1e1 + t2e2) and moment (m) as
In practical applications, the serpentine interconnects can be made of plastic metals (e.g., Au, Cu, Al, etc.) (Brosteaux et al., 2007; Gonzalez et al., 2008), or relatively brittle materials [e.g., silicon, polymethyl methacrylate (PMMA)] (Kim et al., 2008; Kim et al., 2014). Here, a linear, elastic constitutive relation is adopted to characterize the material behavior:
where EA and EI are the tensile and bending stiffnesses, respectively. Such linear relation is valid before the plastic yielding (or fracture) occurs in the metals (or brittle materials).
According to the equilibrium equations, the internal forces (t1 and t2) can be correlated to those (t1R and t2R) of right end (S = Ls) by:
where the subscript ‘R’ denotes the right end (x = xR and y = yR), and θR is the angle at this point. All of the segments in the serpentine interconnects have a constant initial curvature, such that dK/dS = 0 holds within each segment. Then the insertion of Eqs. (5) and (3) into (4c) gives
and the symbol ‘ ± ’ is determined by the sign of dθ/dS. If there is one point [S* (0, LS)] within the deformed beam that satisfies (dθ/dS)|S=S* = 0, or equivalently, F1 (θ*) = 0 , then the continuity requires that the sign of dθ/dS keeps unchangedknthe domain [0, S*) or (S*, Ls], i.e.,
If the solution of S* is out of the domain [0, LS], then dθ/dS takes the same sign as the right end throughout the beam.
Integration of Eq. (8) from the left to the right end gives an equation for determination of the angle θL at the left end, i.e.,
With θL solved, the internal forces (t1L and t2L) and moment (mL) can be then obtained from Eqs. (3), (5b), (6) and (8). The coordinates of the left end after deformation can be calculated from Eqs. (2), (5a) and (6a),
So far, we have established a set of equations to determine all of the displacement and force components (t1L, t2L, mL, θL, xL, yL) at the left end from those (t1R, t2R, mR, θR, xR, yR) at the right end. The components (t1, t2, m, θ, x, y) of an arbitrary point in the beam can be determined in a similar manner.
The above model of a general beam with a constant initial curvature is then utilized to analyze the finite deformation of a serpentine interconnect. For the island-bridge design under an external stretching, the serpentine interconnect is typically subject to the displacement-type boundary conditions, as shown in Fig. 1b. The effective applied strain (εapp) of the serpentine interconnect is defined as
where Uapp is the applied tensile displacement. Due to the large stiffness of islands, the clamped boundaries (i.e., vanishing rotational angle and transverse displacement component) can be adopted to model the serpentine interconnect at the two ends. Considering the anti-symmetric geometry with respect to the center point, we only analyze a half of the interconnect as illustrated in Fig. 2, where the center point is pinned, corresponding to a zero moment. This model consists of two straight segments (denoted by ‘ I ’ and ‘ III ’) that are connected by an arc segment (denoted by ‘ II ’). Local coordinate systems (Xi, Yi) and (xi, yi) (i = I, II, III) can be adopted for each of the three segments under the undeformed and deformed configurations, as shown in Fig. 2b-d. Because of the discontinuity of curvature at the two joints, the analytic model discussed above is applicable only within each segment. The continuity conditions of force and displacement components require that:
The boundary conditions can be written as
By using the equilibrium equation of the interconnect in the deformed configuration and the boundary condition mIIIL = 0, the bending moment (mIR) at the right end can be given in terms of the internal forces (t1IR and t2IR), i.e.,
Then the boundary condition mIIIL = 0 can be replaced by Eq. (17). For a prescribed displacement (Uapp), a wide range of trial solutions of t1IR and t2IR at the right end can be sought to calculate the local coordinates (xIIIL and yIIIL) at the left end. The calculations terminate when the displacement conditions (xIIIL = −L/4 and yIIIL = 0) are satisfied with a sufficient accuracy. This process can be implemented numerically using the commercial software MATLAB or FORTRAN. Once the internal forces (t1IR and t2IR) are solved, the deformed configuration of the entire interconnect can be determined using the theoretical model introduced above. Although the maximum principal strain cannot be obtained in an explicit form, its peak value in the serpentine interconnect can be related to the applied strain and the geometric parameters by:
where the membrane strain is neglected, F2(α,,εapp) is a function that can be determined numerically using the above model, and the subscript ‘ max – nonlinear’ indicates the results of finite deformation theory.
To provide a clear understanding of the finite deformation effect on the stretchability, the analytic models (Zhang et al., 2013a; Widlund et al., 2014; Yang et al., 2014) of infinitesimal deformation are revisited for comparison. By neglecting the strain energies due to membrane and shear deformations, the Castigliano's energy method (Widlund et al., 2014) can be adopted to obtain the relationships between the constraint forces and the applied strain. According to this linear model, the peak value (εmax-linear) of maximum principal strain might occur at the interconnect end (IR in Fig. 2b), or the arc segment (II in Fig. 2c), and can be given by
is the maximum principal strain at the interconnect end, and
is the peak strain in the arc segment. The expressions of f1(α, ) and g1(α, ) are given in Appendix, and θpeak = arctan [f1(α, )/g1(α, )] [0, α].
Calculations based on the nonlinear FEA were carried out to validate the developed analytic model. Euler-Bernoulli beam elements with relatively large thickness/width ratios (e.g., > 1.5) were adopted to model the serpentine interconnects, in which the out-of-plane deformations were suppressed. Refined meshes were used to ensure the computational accuracy. Figure 3 presents the results of peak strain (εmax) and the associated deformed configurations for two representative serpentine interconnects, under different levels of stretching (εapp). The predictions based on the nonlinear model agree remarkably well the FEA results for both geometries. An evident discrepancy can be observed between the curves (Fig. 3) calculated from the linear and nonlinear theories, especially for relatively large applied strains. Taking the serpentine geometry in Fig. 3a as an example, the linear theory gives εmax = 1.07% for 800% stretching, which is nearly half of the prediction (εmax = 2.04%) using the nonlinear theory. Such underestimation of the maximum strain is dangerous in practical applications.
To characterize the total stretchability (εtotal–stretchability) of serpentine interconnects made of relatively brittle materials (i.e., silicon, PMMA), we introduce a simple fracture criterion based on the maximum principal strain, and takes the corresponding threshold (εth) as 2% (Gleskova et al., 1999). For the serpentine interconnects made of metals, we are usually interested in the elastic limit of applied strain that keeps the interconnects from plastic yielding, since the fatigue can be easily induced under cyclic loading, with the emergence of residual strains. The elastic stretchability (εelastic–stretchability) of metallic serpentine interconnects can be then defined as the applied strain to reach εmax = εyield, in which the yield strain εyield is taken as 0.3% (William et al., 1999). For the serpentine interconnect in Fig. 3a, the total stretchability and elastic stretchability can be then determined as ~ 793% and ~ 193%, respectively, according to the nonlinear model. These values are smaller than the predictions (~ 1495% and ~ 224%) of linear model. To quantify such nonlinear effect on the prediction of stretchability, we introduce two relative errors, δelastic–stretchability and δtotal–stretchability, given by
The effect of finite deformation depends highly on various geometric parameters (α, , ), as evidenced by the results of δelastic–stretchability and δtotal–stretchability in Fig. 4. The dependences of δelastic–stretchability and δtotal–stretchability on the length/radius ratio () and arc angle (α) are described by the contour plots in Fig. 4a,b,d,e. For relatively narrow interconnects (e.g., = 0.04 in Fig. 4d), the relative error of the total-stretchability δtotal–stretchability is always larger than 0.5 for > 2 and α ≥ 90°, and can even reach 0.8 for certain design regions. In the current study, the relative errors of stretchability arise directly from the geometrically nonlinear effect, and such effect becomes more evident under a larger stretching. Therefore, the relative errors of stretchability decrease monotonously with the increase of width/radius ratio (), because both the elastic and total stretchabilities decrease with increasing . Figure 4c,f illustrates such trend for δelastic–stretchability and δtotal–stretchability, in which P1, P2 and P3 correspond to the design points (, α)=(1,180°), (2,180°) and (3,180°) in the contour plots. For those representative shapes, the relative error of δelastic–stretchability is below 5% for > 0.1, suggesting the linear model is overall acceptable for the prediction of elastic stretchability of relatively wide interconnects. In contrast, the relative error of δtotal–stretchability is still ~ 30% at =0.1 for the three different shapes, which is not negligible in practical applications.
As mentioned in Section 2, the maximum strain might occur at two different locations (i.e., the arc segments or the two ends). According to the results of nonlinear theory and FEA, the nonlinear effect on the maximum strain of the arc segment is nearly negligible for εmax below 2%, with two examples shown in Fig. 5a,d. This might be attributed to the shift of maximum strain point within the arc, as the applied strain increases. Thereby, the maximum strain of the arc segment is approximately the same as the result [εmax2-linear in Eq. (20b)] of linear theory. Differently, the maximum principal strain of the interconnect ends show an evident nonlinear dependence on the applied strain (Fig. 5a,d). A square term of the applied strain can be introduced to characterize the nonlinear effect, such that the maximum strain (εmax1-nonlinear) of the interconnect ends can be approximated by
where εmax1-linear is the prediction based on the linear theory, as given by Eq. (20a); G(, α) is a function depending on the length/radius ratio and arc angle, and can be determined from the calculations using the precise nonlinear model in Section 2. Typical values of G(, α) are given in Table 1 for a wide range of and α. The maximum strain of the entire interconnect can be then written as
This approximate model offers a good accuracy (e.g., < 10% relatively) of predictions for most practical applications (e.g., εmax/ <1/2). Based on this approximate model, both the elastic and total stretchabilities can be obtained as
where the threshold (εth) of material failure corresponds to the fracture strain (2%) for the total stretchability of brittle materials, and the yield strain (0.3%) for the elastic stretchability of metals.
Figure 5a,d shows the maximum strains (εmax1 and εmax2) of interconnect end and arc segment for two representative serpentine interconnects under different levels of applied strain (εapp). The location of the maximum strain might shift in the serpentine interconnect (Fig. 5a), as the stretching proceeds. Since linear constitutive relations [Eq. (5)] are adopted in this paper, the results are accurate only when the maximum principal strain is below the fracture strain (2%) for the brittle materials or the yield strain (0.3%) for the metals. Figure 5b,c,e,f illustrates the effects of geometric parameters ( and α) on the elastic stretchability and the total stretchability. Good agreements can be found among the predictions of approximate model, precise model and FEA. These results suggest that a large length/radius ratio () or large arc angle (α) can be adopted to enhance the stretchability of serpentine interconnect substantially.
All of the above analyses are focused on the single-period serpentine interconnects that can be joined with islands in practical applications. In addition to this type of construction, the multiple-period filamentary serpentine microstructures are also widely used in stretchable electronics (Yang et al., 2014). For such multiple-period serpentine geometries, the finite-deformation model developed in this paper can be also extended to analyze their stretchabilities. Consider a long serpentine microstructure with (n) periodic cells, we still take half of microstructure to simplify the analyses, considering the anti-symmetric geometry with respect to the mid-point of the entire microstructure. This model can be then divided into (n) arc segments and (n+1) straight segments. For each segment, the finite deformation model [Eqs. (1) ~ (13)] can be still used to determine the components (t1, t2, m, θ, x, y) of one end from those at the other end. The continuity and boundary conditions of this multiple-period model are similar to Eqs. (15) and (16). Then, we can adopt the same numerical algorithm to solve the above equations and continuity/boundary conditions. For example, a wide range of trial solutions for the inner forces at the right-end boundary can be sought to calculate the two coordinates at the left-end boundary, and the calculations terminate until the two coordinates satisfy the displacement boundary conditions to a sufficient accuracy. Substituting the solutions of inner forces and displacements/angles into the finite deformation model then gives all of the relevant information during stretching.
This work presents a finite deformation model of the non-buckled serpentine interconnects under large levels of stretching. Both the predicted maximum strain and deformed configurations agree well with the corresponding FEA results. The comparison of predicted stretchabilities to the counterparts of linear model elucidates quantitatively the effect of nonlinear finite deformation. Systematic analyses suggest that the linear model can result in a considerable overestimation (e.g., > 50% relatively) of total stretchability for many representative shapes of serpentine interconnects. Differently, the linear prediction of the elastic stretchability is overall acceptable for most metallic serpentine shapes with moderately wide ribbons (e.g., >0.1). Furthermore, an approximate model is proposed to characterize the nonlinear effect of finite deformation, which provides good predictions of stretchabilities over a wide range of geometric parameters. These results can serve as design guidelines in the optimization of serpentine interconnects for achieving highly stretchable functional devices.
Y.Z. acknowledges support from the National Science Foundation of China (Grant No. 11502129) and the Thousand Young Talents Program of China. Y.H. acknowledges the support from NSF (CMMI-1300846 and CMMI-1400169) and the NIH (grant #R01EB019337). K.C.H. acknowledges the support from the National Basic Research Program of China (Grant No. 2015CB351900).
The expressions of f1(α, ) and g1(α, ) in Eq. (20) are given by
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