Temperature-Humidity-Bias Testing and Life Prediction Modeling for Electrochemical Migration in Aerosol-Jet Printed Circuits

Additive manufacturing (AM) technologies, including direct-write printing methods such as aerosol-jet printing (AJP) and syringe printing (SP), bring new possibilities to the fabrication of printed hybrid electronics (PHE) for the electronics manufacturing industry [1–3]. This type of technology is capable of printing conductive traces as thin as 30 $μm$ in width, free-standing structural features ranging from a few microns to 4–5 mm in height, and circuit elements related to high-density, complex wiring [1–3]. It also provides more options for electronic device design and manufacturing, which enables engineers and designers to significantly expand circuit form factors and fabrication processes using novel conductive and dielectric ink materials, including but not limited to metal-nanoparticle (such as silver-nanoparticle (AgNP) and copper-nanoparticle (CuNP)) ink materials [4,5], polymer ink materials [6], and other types of functionalized ink materials, such as carbon-nanotube (CNT) inks [7,8]. However, the utilization of this new technology and novel functional ink materials evokes several new unknowns. Unlike the cases for conventional electronics where standards are mature, new standards associated with AM are still not mature, particularly for electronic products that fully or partially utilize additive manufacturing technologies. This stems from insufficient maturity of the following aspects: (i) optimum printing parameters that ensure smaller extent of defects and high yield rates, (ii) performance compared to conventional counterparts with identical functionalities, and (iii) the reliable behavior of additively manufactured or printed components when subjected to different types of imposed loading or environmental stresses.

Aerosol-jet printing is a commonly used technology for additive manufacturing of electronics [1,2]. There are many factors that need to be carefully considered for developing an optimized AJP printing process. These include (i) printing parameters, such as the sheath and carrier gas flow rates and the printing speed, (ii) in situ process parameters, such as bubbler and build plate temperature; and (iii) postprocessing parameters, including curing or sintering profiles. Researchers have studied the effect of those parameters on AJP. For example, Chen et al. employed computational methods to study the effects of different gas flow rates on the dynamics of aerosolized drops and the resulting overspray formation during AJP [9]. Poliks and coworkers employed a combination of experiments and simulations to probe the fluid dynamics associated with the AJP process [10,11]. Sivasankar et al. computationally probed the effect of in situ curing on the dynamics of AJ-deposited polymeric drop [12]. Additionally, Dalal et al. compared the quality and microcracking behavior of conductive metal nanoparticle ink trace AJ printed under different gas flowrate combinations [13]. Several of those considerations and related studies have been summarized and reviewed in multiple articles by Secor [14], Wilkinson et al. [15], Lutfurakhmanoy et al. [16], and Hines et al. [1].

Besides the optimization of printing processes used to produce high-quality, printed electronic components, the expected useful life of the resulting PHE circuits and components needs to be understood when they are subjected to real-world use conditions. The failure modes (like short, open circuit, dielectric breakdown, adhesive delamination, etc.) and the failure mechanisms (like electrochemical migration, corrosion, and fatigue cracking, etc.) that are observed in conventionally fabricated electronic products subjected to different kinds of operational and environmental stresses (such as mechanical shock, vibration, temperature, humidity conditions) should also be studied for PHE devices [17–22]. Although a few studies have evaluated the mechanical and thermo-mechanical fatigue of AJP printed traces [23,24], there are fewer number of publications or research studies on the electrochemical migration (ECM) failure mechanism of AJ printed traces and components. In conventional electronic packaging, ECM becomes one of the most critical failure mechanisms when a product is subjected to combinations of high temperature, high humidity, and voltage bias. ECM causes conductive metal dendrites to grow between neighboring conductors, thereby degrading the insulation resistance, and resulting in unexpected electrical shorts in electronic products [25,26]. Therefore, it is very important to evaluate, understand, and model the ECM propensity in PHEs.

In this study, the ECM propensity of PHEs has been studied through accelerated temperature-humidity-bias (THB) testing of AJ printed material-level test coupons (MLTCs). These MLTCs were comprised of a silver nanoparticle-based “comb” structure (design guided by IPC-9201 standard) printed onto a printed polymer base dielectric layer where no protective coating was added over the printed Ag traces. The matrix of test conditions for these MLTCs was designed based on conditions recommended in the IPC-TM-650 standard. The leakage current was in-situ monitored, and the surface insulation resistance (SIR) was estimated based on the recorded data. Detailed inspections and analysis were performed both before and after testing, and an empirical model was developed based on the test results. Varying levels of dendritic density and morphology were observed in this THB study, at different levels of electrical potential gradient between neighboring traces. This model provides good estimate of acceleration factors for use conditions where the temperature, humidity, and electrical field are within the tested range, but should be extrapolated with care beyond the tested range. The experimental setup of this work is introduced in Sec. 2, the testing results and failure analysis are presented in Sec. 3, and the empirical modeling approach is explained in Sec. 4.

In this section, the design of test sample geometry, methodology for in situ failure monitoring (leakage currents), and the determination of the testing conditions are presented.

Commercially available silver nanoparticle-based inks (DOWA) were used for printing test specimens in this study. In our previous works [9,13], we optimized the conditions for performing aerosol-jet printing with the DOWA ink. We use these parameters in this study as well.

In our previous work, we have optimized the process conditions of aerosol-jet printing with DOWA ink and studied the effect of different printing attributes [9,13]. One of the most critical concerns in AJP is that ink materials are being deposited onto unintended areas from overspray of emanating ink streams. Extensive overspray can impact on ECM rates by affecting effective distance between printed traces, surface area-to-volume ratio, and current density. In our previous studies, we measured the morphology of printed traces using a laser scanning confocal microscopy and developed a matlab code to quantify the effective line width and overspray region by applying specific thresholds. It was noticed that overspray can be optimized by interplaying the gas flow rates (i.e., sheath gas flow rate and carrier gas flow rate) [9,13]. While there is no “written” criterion on what should be characterized as a printed trace/component with “large” overspray, based on our extensive research on the topic (see Refs. [9,13]), for most practical purposes, we can identify an aerosol-jet printed specimen to have excessive overspray if the width of a printed trace with overspray is more than 10% of the width of the desired (with no overspray) trace (which is equivalent to the alternative guidance that overspray is considered excessive when the material in the overspray exceeds 1% of the total metal content). In this study, we followed these guidelines and used optimized process parameters. No excessive overspray was noticed in the specimens received for THB testing.

FR4 was selected as the substrate material, and a NEA 121 (Norland Electronic Adhesive) layer was printed on top of the substrate. The atomizer gas flow rate used in this study is 800 sccm (standard cubic centimeters per minute) and the sheath gas flow rate used here is 50 sccm. The NEA 121 layer is photocurable and was in situ cured by exposing it to ultraviolet (UV) light of an intensity of 1.2 W/cm². The AJ-printed NEA layer has good adhesion to both the FR4 and the printed silver trace and can prevent handling damages such as printed traces being accidentally peeled off. The printed specimens (including substrate, NEA, copper, and printed silver materials) were put into an oven for sintering. The sintering profile has 5 stages, which include (a) ramp from room temperature to 80 °C in 30 min, (b) hold at 80 °C for 30 min, (c) ramp from 80 °C to 150 °C in 30 min, (d) hold at 150 °C for 3 h, and (e) ramp back to room temp in 30 min.

The standard test specimen geometry, as per the IPC-B-24 and IPC-B-25 copper test coupons in IPC-9201, consists of two interdigitated “finger-like” or “comb-like” structures, in which the two interlaced neighboring electrodes act as the cathode and anode, respectively (as shown later in Fig. 1). These specimens have been widely used in research studies focusing on different factors in the ECM process [27]. The geometric specimen design in this study was guided by the standard B-24 and B-25 structures, with the dimensions appropriately scaled to adapt to the inherent limitations of the AJP process. Silver nanoparticle conductive ink (DOWA, Japan) was used to AJ print 200 $μm$ width, 10 $μm$ thick such interdigitated electrodes on top of a previously AJ-printed NEA121 polymer base dielectric layers on an FR4 substrate. The gap area between electrode traces (where the electric field is distributed) was designed to have an 8000 $μm$ overlap length and a 200 $μm$ spacing. Two versions of the sample design were utilized in this study with (i) one printed test specimen pattern per substrate with short copper pads containing a drilled hole and (ii) six printed test specimen patterns on one substrate, with one shared terminal and extended-length copper pads for surface-mount electrical connections. The MLTC designs used in this study are shown in Fig. 1.

Three environmental stress factors were considered in the THB study: (i) temperature, (ii) relative humidity (RH), and (iii) electric field (the average E-field is defined here as the applied voltage bias divided by the nominal spacing). The effect of at least two combinations of each factor was considered. The testing conditions were selected based on IPC-TM-650 industry standards (2.6.14.1, 2.6.3.3, and 2.6.3.4) [21], which suggested testing temperatures of $65 °C$ and $85 °C$⁠, a relative humidity level of $85±3%$⁠, and voltage levels from 45 V to 50 V (E-field ranging from 50 V/mm to 300 V/mm based on the applied standard testing board). To obtain the acceleration factor under different types of environmental stresses, a second humidity condition of $75±3%$ RH was introduced, and the voltage levels were adjusted to between 10 V and 30 V (E-field of 50 V/mm to 150 V/mm) based on revised sample dimensions. Twelve specimens were tested under each selected condition with the implemented test matrix presented in Table 1. Samples in each testing group were mixed (i.e., each testing group has samples from multiple print runs) to deconvolute the effect of AJP process drift on failure rate.

Temperature-humidity-bias (THB) testing was performed in programable chambers with controllable temperature and humidity. Small through holes were drilled on the MLTC substrates that enabled them to be vertically suspended in the test chambers from the wire racks, to reduce the possibility of water droplets condensing in the area where the E-field is distributed between the electrodes. The chambers were first programed to ramp from the initial ambient condition to the set point at a constant ramping rate, and the voltage bias was applied to the testing specimens only after the temperature and humidity conditions inside the chamber reached equilibrium.

All the samples were carefully inspected, cleaned, and connected to failure-monitoring wires prior to testing. All specimens were photographically documented using optical microscopy before being them in the test chamber, and results from initially defective samples (samples with holes on dielectric layer, or cracks on printed silver traces) were excluded from the data analysis. Isopropyl alcohol (IPA) and fine cotton swabs were used to clean the samples before they were loaded into the chamber, and the samples were electrically connected to the power supply using either mechanical fasteners (for Version 1 of sample design containing small copper pads with drilled holes) or soldering (for Version 2 with a design that minimized the effect of solder flux by using larger copper pads and thorough sample cleaning method). Inspections were repeated before and after the cleaning and soldering process.

1 $MΩ$ resistors were used as voltage dividers to provide desired voltage bias across each specimen and two configurations were used in this setup to simultaneously test under multiple E-field values. Dataloggers were connected to specific resistors during the testing and provided in situ recording of the bias voltage across the reference resistors. The leakage current for each specimen was determined from the in situ voltage monitoring. Terminal strip blocks were used to connect 1 $MΩ$ resistors, datalogger, power supply, and the specimens inside the chamber (see Figs. 2(b) and 2(c)). The schematic of the testing circuit and an example photograph of a prepared specimen is presented in Fig. 2.

The actual resistance for each reference resistor was measured prior to connecting to the circuit, and the surface insulation resistance (SIR) was determined using the following equations (Eqs. (1) and (2)).

In the above equations, $R1$ is the equivalent resistance (including the resistance of in-series connected resistors and internal resistance of the meter) in series with the specimens under test, $R2$ is the equivalent resistance in parallel with the specimens under test, and $Vsupply$ and $VDAQ$ represent the input voltage bias and the voltage measurements from in situ monitoring by connected datalogger, respectively.

The SIR values for pristine samples were documented at the beginning of each test, which were larger than 10 M$Ω$ and typically at the level of Giga-Ohms (G$Ωs$⁠). The failure criterion in this study is that SIR drops to below $1 MΩ$⁠, which is at least one order of magnitude drop from the initial SIR of pristine samples. Intermittent SIR recovery due to dendrite breakdown was observed in some cases, and the time-to-failure (TTF) was documented when the specimen's SIR dropped below $1 MΩ$ for the first time. Example plots for SIR calculated from this in situ monitoring setup are presented in Fig. 3.

In this section, experimental data and the results from the post-test analysis for each selected condition are presented. The statistical failure data are shown in Sec. 3.1, followed by the failure analysis and inspections of the tested specimens in Sec. 3.2.

Time-to-failure (TTF), which refers to the first time that surface insulation resistance (SIR) of a specimen dropped below 1 $MΩ$⁠, was documented for all the tested specimens. Individual variations were found among the samples received, and printing defects, such as holes on the polymer dielectric layer or cracks on printed silver traces, were noticed in a few samples. Those printing defects may lead to premature TTFs and were addressed as a separate subpopulation in the data analysis.

Two-parameter Weibull probability distributions were used to quantify the failure data that excluded extreme early failures (from defective specimens, TTF less than 10 h). Plots of the cumulative failure distribution $(F(t)=1−e(−tη)β)$ and the corresponding probability density function $(pdf,f(t)=βη−βtβ−1e−(tη)β)$⁠, are presented in Figs. 4–6. In order to facilitate simple elucidation and illustration of the effects of the various governing factors on the TTF, the failure data are presented as follows: Fig. 4 compares the TTF under different E-field conditions, for each temperature-humidity combination, Fig. 5 shows the TTF under different temperature conditions, and Fig. 6 shows the TTF under different relative humidity conditions.

More detailed subsequent analysis revealed that the failure data followed a bimodal distribution, with specimens containing printing defects (as defined earlier) demonstrating infant mortality (<10 h). In a subsequent section, this ‘infant mortality’ subpopulation was treated separately in the statistical analysis (using a “competing failure-modes” analysis technique in the reliability software [28]) so that the wear-out statistics of the ‘main population’ could be effectively quantified.

The parameters in the Weibull distribution were determined using ranked regression on x (RRX) method, since this method works better than maximum likelihood estimation (MLE, works better with censored data sets) for data sets with small sample sizes (sample sizes less than 30) and containing only complete failure data (i.e., all samples failed with known time-to-failures). These parameters (presented in Table 2) are the scale parameter, η, (which represents the characteristic life for failure of 63.2% of the samples, i.e., the comparative metric of durability) and shape parameter, β, which represents the slope of the hazard rate function (metric of variability). The scale parameter, shape parameter, and mean time to failure (MTTF), along with the standard deviations of TTFs for each testing group, are presented in Table 2.

In the appendix, the uncertainties of Weibull parameters, due to limited sample sizes, are presented. Estimations of those parameters and CDFs with confidence bounds (2-sided at 90% confidence level) have also been provided.

The relationship of the E-field to characteristic Weibull life (TTF) can be seen to be different between the conditions 65 °C, 85% RH, and 85 °C, 85% RH. In both test groups, a nonmonotonic behavior was observed, i.e., in some cases specimens tested under higher voltage and E-field demonstrated larger TTF than those with smaller E-field (see Fig. 7). In contrast, the temperature among different testing conditions remained monotonic, i.e., if other conditions are the same, higher temperatures always result in faster sample degradation and shorter TTF. Our hypothesis to explain this unusual E-field acceleration, along with corresponding empirical modeling approaches, is introduced in Secs. 3.2. and 4.2. Moreover, to better understand the modes, i.e., to separate premature failure subpopulation from the main population in the TTF results, the entire experimental dataset is normalized to a reference stress condition using the established empirical model, and a 3-parameter Weibull distribution is used, as discussed in Sec. 4.2.3. The plan for establishing a physics-based model for TTF is presented in Sec. 5.

Detailed failure analysis and inspection were conducted on the silver-nanoparticle (AgNP) based dendrites. Optical microscopy imaging and scanning electron microscopy (SEM) imaging were conducted to demonstrate the morphology and density of the grown dendrites. Energy dispersive spectroscopy (EDS) was used to perform elemental analysis. An example of a typical morphology of AgNP-based dendrites observed in this study is shown in Fig. 8, along with magnified observation under the SEM and the EDS.

Only one failure mode (i.e., dendrite growth) and only one consistent failure mechanism (i.e., electrochemical migration) were found in failure analysis in this study. Some failed specimens demonstrated a few dominant conductive dendrites, while other specimens demonstrated a much higher dendrite density (i.e., the number of dendrites initiated along the overlapped length of the traces). Overall, three different levels of dendrite density were noticed: (a) level 1: Single (or a few) dominant dendrites (see Fig. 9(a)); (b) level 2: moderate number of dendrites per unit length (see Fig. 9(b)); and (c) level 3: extensive number of dendrites per unit length (see Fig. 9(c)). The morphology of silver dendrites for each of the three levels is also found to be different as elucidated by the optical microscopic imaging (see Fig. 9).

The cases depicted in Figs. 9(b) and 9(c) are more commonly observed under higher E-field strengths, which could be a possible reason to explain the nonmonotonic TTF-versus-E-field behavior mentioned in the previous section. The hypothesis is that extensive ion nucleation with increase in dendrite density (number of dendrites/length) growing simultaneously under high E-field levels can lead to a slowdown of overall average growth rate of the dendrite population, leading to an effective increase of TTF. In the meantime, the difference noticed between Figs. 9(b) and 9(c) also implies that the temperature and the relative humidity condition may also affect the dendrite morphology and dependence on the E-field. Similar phenomenon has also been observed in a research study by Zhong et al. where they found longer TTF and more tin dendrite density under higher voltage bias in one of their experiments [27].

The literature on existing reliability models used to represent ECM behavior in conventional electronics is summarized in Sec. 4.1. Some of these models used an empirical approach while others attempted to modeling based on underlying physics of dendrite initiation and/or dendrite growth. A preliminary empirical reliability model, proposed based on the limited data collected in our study, to describe the observed nonmonotonic behavior (w.r.t., electrical field strength) and to demonstrate potential interaction among different types of stress terms, is introduced in Sec. 4.2.

Failures in conventional electronics due to dendrite growth have been studied for various materials under different testing conditions [21,22,25,26,29–36]. Reliability models considering individual dominant factors (temperature, humidity, electric field, and salt concentration) or multiple interactive factors have been established. The approaches for developing a reliability model for ECM failures in electronics include: (1) empirical approach, which relies on systematic curve fitting methods to determine the correlations between testing conditions and experimental data, (2) reliability-physics approach, which describes the dendrite initiation and/or growth rates in terms of the THB conditions, and describe failure in terms of evolution of relevant parameter (such as insulation resistance and flux density), and (3) hybrid approach, which is a combination of empirical and physics-based approaches.

In many classical empirical models, such as the Peck's model [31,32], exponential Arrhenius relationship has been used in THB testing to express the TTF dependence on temperature, while the inverse power law relationships has been used to express the dependence of TTF on relative humidity [25,30–32]. The dependence on voltage bias or electric field has also been introduced using inverse power law relationship in several existing models [25,33–36], but it can vary for different material compositions or microstructures. For example, as mentioned earlier, Zhong et al. noticed a longer TTF under higher electric field strength for cases with high Cl⁻ concentrations [29]. Recently, novel methods using machine learning algorithms to establish THB failure models have been studied: Zhou et al. compared three different regression methods to develop a THB model and compared their results with physics-based models, in which they claimed that machine learning-based empirical approaches (trained with hundreds of data points) can better predict specimen life than model-based approaches [33].

Reliability physics-based models have also been introduced to evaluate and compare the ECM propensity for different materials used in electronics [21,22,25,26]. Such models focused on one or several dominant stages that are hypothesized to occur during electrochemical migration, including (1) ion generation; (2) ion accumulation; (3) ion migration; and (4) dendrite growth. He et al. calculated the incubation and copper dendrite growth time first under a condensed water condition, based on Nernst–Planck equation, and the model was further extended to the case with noncondensed electrolytes. His results aligned more with the cases using DI water as electrolyte [15,16]. Herzberger et al. improved He's model by considering the dynamic increase of electric field strength as dendrites continue to grow between neighboring conductors, and thus quantitatively explained the empirically observed acceleration of dendrite growth rate [26]. Yang et al. found that in humid, noncondensed cases the dominant term in TTF is due to the significant SIR degradation that happens before dendrite initiation, and they developed a model focusing on the surface conductivity change during the ion accumulation stage [25]. A few existing reliability models are listed in Appendix Table 11.

Three important items are addressed in this section: (i) discussion of the inability of currently available ECM models to address the nonmonotonic E-dependence and interactions between E and T stresses, observed in this study; (ii) a proposed approach to overcome the limitations listed above in item (i); and (iii) analysis of the multimodal TTF distributions observed due to premature failures caused by fabrication variability.

where $AFRH$⁠, $AFT$⁠, and $AFE$ represent the acceleration factor considering different types of stresses (relative humidity, temperature, and E-field strength, respectively).

However, based on the nonmonotonic behavior observed in experimental results and discussed in previous sections, Eq. (9) is no longer suitable to demonstrate the E-field acceleration in our case. In the meantime, there is a significant difference among experimental $AFT$ under different E-field strengths was observed (see Table 3), suggesting interactions among the influences of the three different stress types. Thus modifications of Eqs. (7)–(9) are needed based on the results of our study.

This dependence of the temperature acceleration factor on the E-field suggests that the model constant A in Peck's model may need to be updated with a complex function $A(E, T)$ that depends on both E-field and temperature. This type of interaction is normally not included in commonly used ECM reliability models (see Eqs. (7) to (9)). In the following section, an empirical modification of Eq. (6) is presented, where A values are presented under specific testing conditions (T, RH, and E), in order to describe the interactions between the different stress types and also to describe the nonmonotonic E-field dependence reported in Fig. 7 in Sec. 3.1. Values can be interpolated with piece-wise linear interpolation, for stress T and E values that fall between the reported test conditions. As with any empirical model, extrapolation beyond the tested range should be done with extreme caution. In Appendix A.3, a closed-form quadratic equation is presented, to facilitate this interpolation. The quadratic function is, of course, purely an empiral fit, and physics-based modeling of this nonmonotonic E-dependence (and its dependence on T) is deferred to a future paper.

where $n$ is experimentally determined model constant, $R$ is the universal gas constant, and $Eσ$ is the empirically estimated activation energy. Values for $Eσ, n,$ and A(E, T) and estimated from curve fitting, are presented in Table 4.

The values for the TTF model constants given in Table 4 (and in Appendix A.3) reflect the process variability and the limited sample size of the tested specimens. Clearly, the model constants therefore have some uncertainty bounds (resulting in uncertainty bounds for the Weibull parameters, shown in the Appendix). Of all the model constants used in Eq. (6), the one with the most severe uncertainty bounds is usually the parameter A. Hence a common practice in the literature is to rely on acceleration factor (AF) models instead, since the influence of the uncertainty in A gets eliminated in most of the AF model. The AF allows us to estimate the Weibull characteristic value of the TTF expected under a set of use conditions different from those used in this study, where the ECM is still the driven failure mechanism. Therefore, extrapolating to use conditions well outside of the corner cases studied here should be done carefully due to the possibility of changes in failure mechanisms.

where A(E,T) is linearly interpolated from the values given in Table 4.

The comparison (Tables 5–7) between the acceleration factors obtained from experimental results and empirically fitted model (values calculated using Eqs. (12)–(14); equation parameters are listed in Table 4) shows good agreement, indicating that the proposed empirical function provides a good quality of fit to the experimental results. Please note that such acceleration factors obtained from experimental results provide the ratio between characteristic lives under different selected testing conditions, and the determination of characteristic life, Weibull $η$⁠, for each condition has been discussed in Sec. 3.1 and the corresponding value has been provided in Table 2.

The parameters for the above distribution are shown in Table 9, where a large negative location parameter (⁠$γ$⁠) was found. This implies that some of the failures may have resulted from pretesting factors, such as printing defects or damage induced during handling storage, and specimen preparation.

Only one failure mode (i.e., dendrite growth) and only one consistent failure mechanism (i.e., electrochemical migration) were found in failure analysis in this study. However, a single Weibull statistical distribution cannot adequately describe all the failure data unless the manufacturing quality (piece-to-piece variability) follows a unimodal probability distribution function. The resulting negative location parameter raises the possibility that the manufacturing variability may in fact be multimodal. The rest of this section is devoted to an analysis of premature failures, using the same extrapolated dataset as an example.

Competing failure mode analysis was performed using a commercial data analysis software [28], which divides the failure data into two groups: (i) subpopulation of premature failures and (ii) main unimodal population of failures. The most likely number of premature failures was determined through a parametric exploration of the best fits for progressively different partitioning between the premature failure subpopulation and the main population. The shape parameter (⁠$β$⁠) showed a sharp increase when the number of failures assigned to the premature-failure subpopulation reached 16, as shown in Fig. 11. Therefore, the first 15 failures of this extrapolated dataset were assigned to the premature-failure subpopulation and the rest were considered to belong to the main life for well-printed samples. The resulting 3-P Weibull parameters are shown in Table 10, and the corresponding cumulative density function (CDF) and probability density function (pdf) plots are presented in Fig. 12.

The characteristic life $(η+γ)$ for the main population is about 240 h, while it is only about 88 h for the premature failures. The example presented here implies that despite the infant failures being removed from the dataset, multimodal distribution of fabrication quality and pretesting damage still need to be considered when performing THB testing on AJ-printed electronics. The data in the main population suggests that “well-made” PHE specimens should experience a failure-free operation period of about 68 h in the reference THB condition (⁠$85 °C, 85% RH, 50 V/mm$⁠) considered in this illustrative example.

Although there was a good fit between the proposed empirical model and experimental data, the current model still has the following limitations: (1) this empirical approach is not constrained by the physical principles or electrochemical processes that are active in ECM, and the accuracy of this model and the fitting uncertainty (see Appendix A.1) largely depends on the sample size; (2) The E-field dependence term $A(E, T)$ is a temperature-dependent empirical function, which brings more complexity to the model and makes it difficult to extrapolate to use conditions that are outside the range of the tested conditions; (3) The effect of the variations in the AJP process conditions on the variabilities of the individual specimens and the corresponding consequences on the ECM phenomenon were not discussed in detail in the present study. For example, overspray of the emanating ink streams is one of the critical concerns in AJP, and it can significantly affect effective distance between the printed traces, surface area-to-volume ratio, current density, and the ECM rate. The correlation between the ECM and overspray level in AJ printed PHEs have not been investigated here and needs to be quantified in future studies. (4) The sources of individual specimen variability due to AJP process variability are evident in the extrapolated multimodal TTF distribution but the root cause of this variability cannot be addressed without further study.

In future work, a TTF model based on reliability physics should be developed for the AJP printed PHEs, where more detailed characterization and calculations based on ECM-related physical parameters (such as the flux density, dynamic change of field strength, and the surface conductivity) along with a normalized form of TTF-versus-E-field term will be introduced. Moreover, the defects and the individual specimen variations should be considered, where statistical process control (SPC) of the AJP process can be explored, to improve PHE reliability under THB conditions.

Aerosol-jet printing (AJP) technology, with its accurate spatial resolution, offers more possibilities for precise manufacturing of electronic products with high wire densities. However, the increased density of conductive traces will also result in higher risk of electrochemical migration (ECM) failures. Therefore, it is essential to study and understand the ECM propensity of PHEs developed using the AJP technology. In this paper, accelerated temperature-humidity-bias (THB) testing is performed under different selected conditions, and detailed experimental results and failure inspection results are presented. An empirical model is proposed for this case, which shows a good fit to the experimental data. Empirically observed variability is quantified with Weibull distributions (along with corresponding uncertainty bounds) to enable reliability estimates. Different levels of dendrite growth and a nonmonotonic relationship between the E-field and TTF were experimentally observed, which is represented by considering a temperature-dependent quadratic polynomial function (see Appendix) in the empirical model. The empirical fit is used to present models for acceleration factors for different use conditions (temperature, humidity, and electrical field gradients). Care must be exercised when using the acceleration factors beyond the conditions used in the current study. To the best of our knowledge, the present study is one of the first studies that established an ECM failure model for aerosol-jet printed electronics or as a matter of fact any electronics additively manufactured using printed conductive nanoparticle-based traces. However, the accuracy of the current empirical model largely relies on the sample size instead of on the representation of the underlying science of the ECM processes. Physics-based reliability models should be addressed in future work.

We are grateful for technical inputs from the scientists at the Laboratory for Physical Sciences and use of experimental facilities at the Center for Advanced Life Cycle Engineering (CALCE) at University of Maryland.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

In this section, the uncertainties of Weibull parameters used in the paper are presented. CDF plots with confidence bounds (2-sided at 90% confidence level) for all testing conditions are presented in the following Figs. 13–19. The lower and upper bounds for their Weibull parameters are accordingly listed with the plots accordingly.

In the following Table 11, a few existing time-to-failure models that are widely used in THB testing or failures related to ECM are listed.

where C, $a0(T)$⁠, $a1(T)$⁠, and $a2(T)$ are empirically determined temperature-dependent parameters, E is the tested E-field strength, and $Eref (T)$ is an estimated reference E-field strength used to indicate the electric field that has the strongest effect on ECM at a selected temperature (i.e., TTF is shortest). Values for the model parameters, determined from curve fitting, are presented in Table 12.

Moreover, as we were only able collect data for a limited number of stress combinations, the proposed preliminary quadratic function for A(E, T) can possibly represent overfitting. Other forms of A(E, T) may become more representative when new combinations of stress conditions are introduced to the testing matrix in future work. New ECM reliability models (i.e., more representative empirical forms of A(E, T)) for AJ-printed electronics should be established with a more comprehensive dataset. More detailed discussions on the limitation of the current work and suggested future researches are listed in Sec. 5. The main purpose of proposing the preliminary quadratic form of A(E, T) in this section is to provide an example of empirical ECM acceleration factor modeling (modified version of the well-known empirical Peck's model) that includes possible interactions of E-field strength and temperature accelerationin in ECM of AJ-printed AgNP-based circuits, and also empirically aligns with the nonmonotonic E-field dependence observed in our experiments. This empirical quadratic interpolation for A(E,T) presented in Eq. (A1), will facilitate interpolation of AFs between the tested stress conditions. As with any empirical fits, readers are cautioned to use extreme caution when extrapolating beyond the tested conditions. Future work will focus on models that are based on the underlying physics of ECM initiation and propagation.