Assessing the Impact of Twisting and Bending Deformations on Flexible Interconnect Performance

Flexible interconnects are essential components for signal transmission for foldable and wearable electronics. As such, they are exposed to a variety of mechanical deformations that can degrade their electromagnetic performance. This article analyzes the impact of bending and twisting deformations on the transmission properties of a variety of commercially available polymer and elastomer-based interconnects and compares with the performance of flat interconnects fabricated on rigid substrates. The analysis of the impact of deformations also takes into account the degradation in the conductivity of the printed lines due to reported mechanical deformations.


I. INTRODUCTION
W EARABLE electronic devices are a rapidly expanding area for a range of applications from sport to biomedicine. Flexible interconnects are essential components for signal transmission. A large volume of research has been focused on novel dielectric and conductive materials of the flexible interconnect that can effectively withstand the deformations these components are exposed to. E-textile, woven, and knit textile structures are inherently flexible but suffer from implementation uncertainty in large-scale production [1], [2]. The most popular dielectric materials are those made from polymers such as polyethylene terephthalate (PET) and polyimide (PI) [3], [4], [5], and elastomers such as polydimethylsiloxane (PDMS) and polyurethane (PU) [6], [7], [8]. The electrical properties, dielectric constant, and loss tangent for the operating range 1-10 GHz and key mechanical properties of PET, PI, PDMS, and PU substrates are summarized in Table I. According to the mechanical properties in Table I [9] (density, Young modulus, and elongation), elastomers (PDMS and PU) provide higher deformability compared to polymers (PET and PI) substrates; however, polymers offer higher electrical and thermal stabilities in practical environmental conditions. It can also be seen that the PU material has the highest dielectric losses while the PI material has the lowest dielectric losses.
A wide range of novel electrically conductive materials that are compatible with polymer and elastomer-based substrates have been reported, namely, nanowires (NWs) [10], Manuscript  graphene [11], Ag stretchable conductors [7], liquid metals [12], conductive polymers [6], [8], and conductive fibers [13]. However, their typical conductivity depends on the particle and ink composition and is typically poorer than that of traditional rigid metals [4], [5]. Additionally, the methodology of depositing metallic traces influences their thickness which can range from 15 to 25 μm for screen-printed metal flake polymer compounds [4], [14] to 1-3 μm for inject printed traces [5], and 0.20 mm for filament printing [15]. The thickness of the signal trace also limits the inherent flexibility of the components [9]. The limited structural stability of conductive prints under deformations [14] is typified by the appearance of nano-and microgaps that cause loss of current continuity [4], [6], [7], [14], [16]. Our earlier study of flat flexible interconnects showed that interconnects fabricated on PET and PI substrates have the smallest transmission losses compared to PDMS and PU substrates and that interconnects fabricated on PU substrates have the highest losses [17]. The losses are predominantly due to increased substrate losses, as shown in Table I.
Flexible interconnects also need to effectively withstand a wide range of complex deformations such as, for example, crumpling, bending, wrinkling, and twisting. An example of extreme twisting deformation on a flexible interconnect is shown in Fig. 1. However, the analysis of the impact of deformations on the performance of flexible interconnects is limited. Commercial simulation packages provide means for generating 1-D bending, i.e., bending over a cylindrical plane. For this reason, the flexibility of RF interconnects is evaluated both experimentally and computationally for the case of simple cylindrical bending [4], [11], [18], and for more complicated 2-D deformations, such as twisting, only experimental results for the interconnects on the PI and textile substrates are available [5]. Furthermore, these limited experimental and computational results have been reported for interconnects fabricated on PDMS [8] and PI [5], each under different conditions so that comparative evaluation across a comparable range of components is difficult to make. In this article, we extend the analysis to evaluate the impact of bending and twisting on the performance of a variety of realistic interconnects designed for commercially available polymer and elastomer substrates. The electromagnetic (EM) performance of interconnects under deformation and fabricated on a range of substrates, namely, PI, PDMS, PU, and PET, is systematically analyzed and compared to interconnects designed on a rigid substrate with the aim of ascertaining the most resilient platform for flexible electronics. All interconnects are assumed to have realistic dielectric and metallic losses.
Computational geometrical models of cylindrical deformation are made using standard Boolean computer-aided design (CAD) approach based on constructive solid geometry (CSG). However, this CAD approach is not reliable for generating twisting deformation as it may result in the formation of microscopic gaps between constitutive layers of the interconnect (i.e., substrate/ground plane/metallic trace) which can undermine EM simulations, as discussed in [19]. To overcome this difficulty, we apply a computer graphics approach based on Green coordinate (GC) technique for spatial manipulation of objects to create desired twisting deformation [20], [21]. The full description of how the GC methodology is implemented for the purpose of EM simulations is discussed in detail in [17] and [19].
To characterize the EM performance of the interconnects, we use an in-house 3-D numerical time-domain transmission line modeling (TLM) method [22] based on unstructured tetrahedral Delaunay mesh [23], [24], [25], [26]. The unstructured mesh variant of the TLM algorithm is second-order accurate with respect to the wavelength [25]. Compared to similar timedomain EM solvers, like finite difference time domain (FDTD) [27], the main advantage of the unstructured TLM method is that all field components are co-located in space and time and stability is provable on a cell-by-cell basis without invoking the Courant condition. The unstructured TLM method has been industrially characterized for a range of applications including electromagnetic compatibility (EMC) [28], [29] and aerospace [30], [31], and full details on unstructured TLM methodology are available in [23], [24], [25], and [26]. This article is structured as follows. Section II summarizes the computational models of interconnects designed for PET, PDMS, PI, and PU substrates. Section III analyzes the impact of convex and concave bending on a variety of realistic interconnects and compares it with the performance of the flat rigid interconnect. Section IV analyzes the impact of twisting on the EM performance of a variety of interconnects. In both cases, the conductive losses with and without the impact of deformation are considered. Section V summarizes the main conclusion of this article.

II. COMPUTATIONAL MODELS
This section summarizes the design parameters of a range of interconnects designed for commercially available substrates, namely, Sylgard 184 (PDMS) [6], Ninja-Flex (PU) [15], Kapton (PI) [5], Melinex (PET) [4], and RO4350 [30]. The substrate thickness of PI and PET substrates is taken to be 0.2 mm in accordance with [5]. The thicknesses for PDMS and PU substrates are considered to be 1 mm [6]. A realistic measurement setup is considered where interconnects are connected to the 50-coaxial cable via a standard subminiature A (SMA) connector.
In order to minimize reflections at the cable-interconnect interfaces, all interconnects are designed to have 50-characteristic impedance. This implies that different substrate thicknesses will dictate the width and length of the signal trace according to microstrip line synthesis equations [31]. Table II summarizes the widths (W ) of the signal traces, typical dielectric constant, ε r , dielectric loss, and tanδ for PI, PDMS, PET, and PU interconnects. The typical conductivity, σ , of metallic prints commonly fabricated on respective substrates is also given in Table II. For all cases, the coaxial cable has an impedance of 50 with relative permittivity, ε r = 2. The inner, r i , and outer radius, r o , of the coaxial cables for relevant substrate thicknesses are also summarized in Table II.
Unstructured tetrahedral meshing is ideally suited to sampling a problem that does not align to a Cartesian coordinate system. A 3-D Delaunay Mesher [24], [25], [32] is used to discretise the problem space. For the case of the flat and bent interconnects, the overall computational domain is set to be 1.21λ × 0.94λ × 1.21λ, where λ is the free-space wavelength at 5 GHz. For the case of the bent and twisted interconnects, the overall computational domain is set to be 1.67λ × 1.03λ × 1.67λ. Note that in the case of the flat interconnect, the region under the interconnects' ground plane is reduced to optimize the computational resources without affecting the accuracy of the results. The meshed computational domain of the bent and twisted interconnects is given in Fig. 2(a) and (b), where it can be seen that the discretization deploys a hybrid mesh that combines a cubic mesh around uniform regions and a tetrahedral mesh around the curved regions.
The near field of the interconnect is meshed with cubic cells of 0.75 mm (for 1-mm substrates) or 0.375-mm size (for thinner substrates), and a bigger mesh of 3-mm cubic cell size is used for the region further away from the interconnect. The computational domain is truncated with free-space impedance boundary conditions.
To efficiently manage computational resources, the printed signal line and ground plane are not directly meshed but are embedded between computational cells as outlined in [26]. The 1-D analytic transmission line models are used to characterize the lossy conductive layer as described in [26].
In all simulations, the interconnect is excited with the fundamental TEM mode of the coaxial cable with a spectrum from 0.25 to 10.25 GHz. For all simulations, the timestep taken is 6.67128 ns and simulations are run for three million timesteps.

III. PERFORMANCE UNDER BENDING DEFORMATIONS
This section investigates the impact of bending on the transmission properties of interconnects for a range of commercially available flexible substrates.    Table II. According to Fig. 3(a) and (b) for thin substrates, the PETand PI-based interconnects exhibit similar performance with transmission losses of around 1.5-2 dB over the operating band. The results for PET interconnect agree very well with experimental measurements for a 30 • bend reported in [4]. The PU-based interconnects have the highest losses compared to PET-, PI-, and PDMS-based interconnects. Fig. 4 shows that the PDMS-based interconnect has the overall best performance in terms of transmission loss. In all cases, concave or convex bending does not significantly affect the transmission properties. However, when compared to the conventional rigid interconnect, the flexible interconnects are overall more lossy. This can be explained by the fact that flexible substrates and conductive material have higher material losses (tanδ) and conductive losses (σ ) compared to the conventional gold microstrip line on a rigid substrate.
To separate dielectric and conductive losses, Fig. 4(a) and (b) shows the comparison of the total system losses of flat interconnects for the case where in (a) only dielectric losses of transmission lines are considered and in (b) both dielectric and conductive losses are considered. Fig. 4 shows that in the absence of metallic losses, the best performing are PET and PI interconnects that are outperforming the conventional rigid microstrip line. The PU-based interconnect has the highest dielectric losses. Fig. 4(b) shows that when metallic losses are added, all flexible interconnects have increased system losses which is as expected. In the absence of metallic losses, the rigid transmission line has the lowest losses, which is primarily due to the low conductive losses of the metallic line. PDMSbased interconnects have the smallest system losses, and PI-and PET-based interconnects have similar losses while the PU-based interconnect remains the most lossy transmission system. Comparing the loss values from Table II, it can be seen that PI-based interconnect has smaller dielectric and conductive loss compared to PDMS line but higher overall loss. This can be explained by the fact that the width of the PI line is much smaller compared to the PDMS line meaning that field is less concentrated in the substrate and implying that the additional loss of the PI line is due to radiation losses.

IV. PERFORMANCE UNDER TWISTING DEFORMATIONS
In this section, the impact of twisting deformation on a range of interconnects is analyzed. Fig. 5 explores the impact of twisting deformation for two different lengths of the PDMS interconnect, namely, (a) 50 and (b) 25 mm. The substrate width is 12.7 mm. All other parameters of the interconnects and coaxial lines are given in Table II. Metallic losses are neglected. The inset of Fig. 5 shows the image of the interconnect under twisting deformation. The twisting is characterized by an angle ρ and is performed in the anticlockwise direction, as shown in Fig. 5.
Four different twisting deformations are considered, namely, ρ = 22.5 • , 45 • , 58.5 • , and 96 • . Fig. 5(a) shows the results for 50-mm-long PDMS and Fig. 5(b) for 25-mm-long PDMS interconnect. Results for the flat PDMS are also included for reference. The S 21 -parameter is unaffected by the twisting angle at lower frequencies while the number of peaks in VSWR depends on the length of the interconnect and the main operating wavelength. This implies that the reflections are caused by the combination of deformation that affects the impedance mismatch between the mode propagating along the interconnect and the coaxial line. The results for the flat case set a level for the connector mismatch. The fact that peaks are not at the same level illustrates that there exists radiative coupling to the connector in addition to the fundamental mode and this contributes to the increased VSWR. As the waveguide is twisted, the strength of the radiative coupling will vary. Fig. 6 shows the impact of twisting deformation for the angle of 58.50 • for PI-and PET-based interconnects fabricated on thin 0.2-μm substrate and compares it with the performance of flat interconnects. All interconnects are 50 mm long, and the realistic metallic and dielectric losses are assumed as given in Table II. The twisting deformation has similar impact on the transmission characteristic of PET-and PI-based interconnects, and the VSWR can increase above desired value indicating increased reflection losses.  Printing or fabrication process of a conductive line on a flexible substrate may not be perfect and can result in microcracks or nanoholes in the conductive trace [5], [7], [14]. Repeated torsional deformation on such interconnects can result in the expansion of these nonuniform microcracks which in turn causes reduction in the overall conductivity of the conductive trace [5], [7], [14]. According to [14], when the interconnect is exposed to repeated twisting deformation, conductivity of the Fig. 7. Comparison of S 21 and VSWR for the flat and twisted interconnects designed for the PET and PI substrates. Twisting angle is 265.17 • , and the conductivity of metallic line for deformed case has changed according to [14] and given in the figure. metallic trace can be reduced by 33.33% for PET interconnect and 16.66% for PI interconnect, respectively. To account for that, Fig. 7 demonstrates the impact of reduced conductivity for the case of deformation angle of 58.50 • .
Comparing the results of the flat and flexible PET and PI interconnects with degraded conductivity, it can be seen that the reduction in the conductivity has resulted in the increased transmission loss in the range 0.5-1 dB over the operating band as well as increased VSWR. This is as expected considering the degraded conductivity effectively means a more lossy structure.
When it comes to thicker substrates, Fig. 8 demonstrates the return and insertion losses of PDMS and PU interconnects for the case of twisting deformation of 58.50 • . This deformation does not significantly affect the line transmission performance, and the results agree with experimental ones reported in [5]. The VSWR remains below desired 1.2 value. Fig. 9 extends the analysis by taking into account the degradation of conductivity of the line under repeated deformation which, according to [14], results in conductivity being reduced by 33.33% for both PDMS and PU interconnects. Again, it can be seen that the reduction in the conductivity Fig. 9. Comparison of S 21 and VSWR for the flat and twisted interconnects designed for the PDMS and PU substrates. Twisting angle is 58.50 • , and the conductivity of metallic line for deformed case has changed according to [14] and given in the figure. Fig. 10. Comparison of S 21 and VSWR for the flat and twisted interconnects designed for the PDMS and PU substrates. Twisting angle is 265.17 • , and the conductivity of metallic line for deformed case has changed according to [14] and given in the figure. has resulted in increased insertion loss of the interconnects by 0.5-1 dB over the operating band. The PDMS has overall the smallest transmission losses compared to other types of flexible interconnects.
Finally, Fig. 10 includes the impact of conductivity degradation for the case of extreme twisting deformation of 265.17 • for the interconnects based on PDMS and PU substrates that have high elasticity. The change in conductivity is taken from [14] that predicts 50% reduction in conductivity for PDMS and PU substrates. Fig. 10 shows that PDMS-based interconnects provide better transmission performance compared to interconnects based on PU substrates and provide a good solution where higher deformability is expected.

V. CONCLUSION
The rise of flexible electronics demands tools that can assess the impact of deformations. This article presents a comprehensive analysis of the impact of twisting and bending deformation on the transmission properties of interconnects designed for the four most popular types of polymers and elastomer-based flexible substrates, namely, PET, PI, PDMS, and PU substrates. This article summarizes the electrical and mechanical properties of these substrates and typical conductivity achievable with modern fabrication technologies.
This article analyzes the impact of bending and twisting deformation in order to separate and independently quantify these two effects. In the case of bending, both concave and convex cases are considered. It is shown that the impact of bending deformation does not have significant impact on the line performance.
Twisting deformation has a small impact on the transmission loss at low frequencies [5] but the twisting deformation contributes to the radiative coupling to the connector causing worsening of the VSWR. This article confirms that even when the impact of deformation on the metal conductivity is considered, the performance of PI [5], PDMS, and PET interconnects remains stable [5]. For thin substrate, PI provides better performance compared to PET substrates.
Interconnects based on PDMS substrates have mechanical properties similar to PU substrates, but due to overall lower substrate and conductive losses, they are the better candidate for flexible electronics. This article shows that interconnects based on PDMS exhibit the best performance even under extreme deformation making them suitable for applications that require high deformability.