Chinese Chemical Letters  2019, Vol. 30 Issue (6): 1282-1288   PDF    
Critical importance of current collector property to the performance of flexible electrochemical power sources
Fandi Ninga,b, Yangbin Shenb, Chuang Baia,b, Jun Weia,b, Guanbin Lub, Yi Cuia,d, Xiaochun Zhoua,b,c     
a Nano Science and Technology Institute, University of Science and Technology of China, Suzhou 215123, China;
b Division of Advanced Nanomaterials, Suzhou Institute of Nano-tech and Nano-bionics, Chinese Academy of Sciences(CAS), Suzhou 215123, China;
c Division of Nanomaterials, Suzhou Institute of Nano-Tech and Nano-Bionics, Nanchang, Chinese Academy of Sciences(CAS), Nanchang 330200, China;
d Vacuum Interconnected Workstation, Suzhou Institute of Nano-tech and Nano-bionics, Chinese Academy of Sciences(CAS), Suzhou 215123, China
Abstract: Flexible electrochemical power sources are attracting increasing attentions for their unique advantages like flexibility, shape diversity, light weight and excellent mechanical properties. In this research, we discover that the current collector can dramatically affect the performance of flexible electrochemical power sources with large size. For flexible air-breathing proton exchange membrane fuel cell (PEMFC), the performance could have more than 8 times increase by only adjusting the directions of current collectors. The different performances of different current collection types are mainly attributed to the diverse lengths of the electron transfer pathways. In addition, the conductivity of current collector can dramatically affect the capability of flexible PEMFCs with large-size. The flexible PEMFCs with thicker carbon nanotube membrane as current collector (low electric resistance) show higher ability. A mathematic model is successfully built in this work to further understand the performance. Moreover, the model and simulation are also applicable to other flexible power sources, such as flexible Li-ion battery and supercapacitor.
Keywords: Flexible     Fuel cell     CNT membrane     Current collector     Li-ion battery     Supercapacitor     Large-size    

Owing to the unique advantages, like flexibility, shape diversity, light weight and excellent mechanical properties [1-3], flexible devices including wearable devices and roll up displays are attracting increasing attentions [4, 5]. However, conventional power sources, e.g., solar cells, lithium-ion batteries and supercapacitors, are usually too heavy and rigid to fit the flexible electronics. Therefore, power sources with flexible ability are highly demanded for the future flexible devices.

Recently, some flexible power sources [6, 7], such as flexible solar cells [8-10], flexible lithium batteries [11, 12], flexible fuel cells [13-15] and flexible super capacitors [16-19], as most promising candidates were widely investigated [20, 21]. Moreover, these flexible power sources have shown a series of impressive performance in smaller volume, lighter weight, higher energy density, preferable stability, longer endurance time and so on [6-21]. The high performance is usually achieved from rational design and synthesis of admirable materials.

For all flexible power sources, they have a common characteristic in structure. In general, the thickness of flexible power sources is as thin as several millimeters or hundreds of micrometers, and the thickness is much smaller than the length and/or the width. Consequently, electrons need to transfer along the current collector in a long distance, and the current is at last extracted from the edges [6-21]. The effect of current collector will be stronger if the current collector is longer and the current is higher. To avoid the effect, most of the flexible power sources usually work with low current and small size.

However, higher power, higher current and larger size flexible power sources are usually required in practical applications. At such situation, an obvious ohmic drop will occur along the current collector, and could strongly affect the performance of flexible power sources if the ohmic drop is not properly suppressed. It is well known that the ohmic drop is mainly affected by the electric resistance of the current collector, which is related to the resistivity of material, length and area of cross section. Moreover, the ohmic drop also could be strongly affected by the geometric shape and current collection direction. However, such important problem has not attracted much attentions of researchers. Only very few works have been done on the related research [22-24].

In this research, the effect of current collector on the performance of flexible power sources is systematically studied. Especially, current collection direction, electric resistance of current collector and length of current collector are carefully investigated through both experiment and mathematic model. The research not only shows the significance of current collector but also provides some strategies to improve the performance of large sized flexible power sources, which includes flexible proton exchange membrane fuel cell (PEMFC), Li-ion battery and supercapacitor.

The air-breathing flexible PEMFCs were prepared through the method reported in our previous work [15]. Briefly, the preparation of flexible PEMFC includes four steps. First, we prepared CNT membrane with ordered through-hole structure by floating catalyst method and laser marking technology [25-29]. The pore size of CNT membranes varies with current of laser marking and increases with the increasing of current. Pore spacing can be set by adjusting the coordinates. Second, we prepared flexible composite electrode with an ordered through-hole array CNT membrane and carbon paper electrode by bonding of PTFE and carbon powder (Vulcan-X-72R) slurry. The flexible composite electrode serves as current collector, collecting network and electrode [15, 30]. Third, we prepared membrane electrode assembly (MEA) with electrocatalyst (70% Pt/C), flexible composite electrode and Nafion 115 membrane by a hot pressing process [15, 31, 32]. Fourth, polydimethylsiloxane (PDMS) membrane was used to form a space with the electrode via adhesive on the side of anode. The space acts as the flow field of the flexible PEMFC. Capillary (with 1 mm in outer diameter) was used to connect the hydrogen source and fuel cell to feed the PEMFC with hydrogen. The cathode side of the fuel cell is exposed directly to the air. An edge of the composite electrode was exposed to connect the sheets of copper for the current collection of loading [15].

The propertyofmaterials is directlyrelatedtothe performanceof fuel cell. Material thickness and square resistance are the characteristics we are most concerned about. The square resistance was measured by usingthe four-point probetechnique. Each sample was measured twenty times, and the parameters are the average of the measured data. The thickness of the carbon nanotubes film was measured by a screw micrometer and field emission scanning electron microscope (SEM), and the average value is obtained after measuring twenty times at different positions. The electrochemical performance was measured byan electrochemical workstation (CS), and copper sheets were connected to electrodes to derive current. The contact resistance is reduced by fixing copper plates and electrodes through fixtures. The operating conditions in our performance test were dry air and hydrogen though bottle-washer at the ambient temperature(20 ℃)andpressure.Additionally, onthe cathode side, the supply of oxygen is passive which means that the fuel cell is air-breathing PEMFC.

In order to eliminate the influence of different current collector characteristics on fuel cell performance, we first classify the current collection types under the same collector. There are usually four edges (1, 2, 3 and 4 on anode side; 1', 2', 3' and 4' corresponding on cathode side) for the current collection on each side as shown in Fig. 1a. When the flexible power sources work, it is needed to select one of the edges from anode and cathode sides firstly. For instance, we can choose the 1-1' or 3-3' pairs, which can be classified to Type-1 current collection (Fig. 1b). Further, according to the difference of the electron transfer path, the current collection pairs can be divided into five types named Type- 1, Type-2, Type-3, Type-4, and Type-5 (Figs. 1b-f). If we consider the current collection direction, we have another three types of collections, which are parallel, cross and opposite collection types. Type-1 and Type-4 can be classified to parallel collection type; Type-2 can be classified to cross collection type; Type-3 and Type-5 can be classified to opposite collection type. Diverse current collection types bring different electronics transport channels, and then lead different performances.

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Fig. 1. Current collection types of flexible air-breathing PEMFC. (a) Schematic illustrating of a flexible PEMFC with four current collectors on both anode and cathode. (b) Type-1 current collection including 1-1' and 3-3' pairs. (c) Type-2 current collection including 1-2', 1-4', 3-2' and 3-4' pairs. (d) Type-3 current collection including 1-3' and 3-1' pairs. (e) Type-4 current collection including 2-2' and 4-4' pairs. (f) Type-5 current collection including 2-4' and 4-2' pairs. Type-1 and Type-4 can be classified to parallel collection type; Type-2 can be classified to cross collection type; Type-3 and Type-5 can be classified to opposite collection type.

As shown in Fig. 1, there are five current collection types. Type-1 and Type-4 belong to the parallel collection type (Fig. 2a), while Type-3 and Type-5 belong to the opposite collection type (Fig. 2b). These two current collection types (parallel and opposite) are commonly used in flexible devices in practice [6-21]. In this work, two models were built for the parallel and opposite collection types, respectively (Figs. 2c and d). In order to make these models more popular, these two models only consider the working condition far below the mass transfer limit. Therefore, these two models were built on the electric circuit model, which does not include the items of the mass transport overvoltage. The establishment and mathematical solution of the circuit model are as follows.

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Fig. 2. Models for different current collection types. (a) Scheme of parallel type. (b) Scheme of opposite type. (c) Circuit model for parallel type. (d) Circuit model for opposite type.

For the model of parallel current collection type in Fig. 2c, the potential between the points A and B:

(1)

where En is the potential of segment n, jn is the current in segment n, in is the current in current collector of one segment, r1 is the inner resistance in one segment, and r2 is the resistance of current collector in one segment. Then, En, r1, and r2 are deduced and replaced (Please refer the Supporting information for the details of the derivation process).

(2)

Where η is the overpotential, dx is the length of one segment, J is the current density in one segment, ρ1, S is the resistant in unit surface area, w is the width of electrode, ρ2, S is square resistant.

According to Tafel equation

(3a)
(3b)

Submit Eq. (3b) into (2), then

(4)

For the model of opposite current collection type in Fig. 2d, the modeling process refers to the above.

(5)

Based on the established model and the deduction of mathematical expression, the relationship between voltage, current density and current transfer length, collector square resistance is built.

For the parallel connection type, we can get the over potential η and current in collector i along the electrode. Then the voltage of fuel cell is

(6)
(7)

where Ecell is the cell potential of PEMFC, JN is the current density in the last segment near the current collection point (Fig. 2c).

For the model of opposite current collection type, similarly, we can get the over potential η and current in collector i along the electrode. Then the voltage of fuel cell is

(8)
(9)

Under the same collector, different current collection types will bring different fuel cell performances. Fig. 1a shows that there are commonly four edges (1, 2, 3 and 4 on anode side; 1', 2', 3' and 4' on cathode side) for the current collection on each side. According to the difference of the electron transmission path, the current collection pairs can be divided into five types named Type-1, Type-2, Type-3, Type-4, and Type-5 (Figs. 1b-f). Each current collection type includes several equivalent pairs. For example, Type-1 includes pairs 1-1' and 3-3' in Fig. 1b. In this research, we first deeply studied the effect of current collection types on the performance of air-breathing flexible PEMFC under a certain electrode conductivity.

Figs. 3a and b show that the type of current collection could greatly affect the performance of air-breathing flexible fuel cell. A series of pairs of current collection directions were studied to find the optimal one (Figs. 3a and b). The performance order for the pairs is Type-1> Type-2 > Type-3 > Type-4 > Type-5. For a same flexible fuel cell, the performance of Type-1 current collection can reach 33.2 mW/cm2, while the performance of Type-5 current collection is only 4.0 mW/cm2. The performance has more than 8 times distinction between these two types of current collections. Therefore, it is necessary to select a proper type of current collection if we want to have a higher performance of flexible fuel cell.

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Fig. 3. Effect of current collection types on the performance of flexible PEMFC. (a) Polarization curves of air-breathing PEMFCs for five different current collection types. (b) Power density of air-breathing PEMFCs. The PEMFCs were fabricated with a 2 cm (length) × 1 cm (width) × 5 μm (thickness) CNT membrane. The experiment was performed with 50% hydrogen at 20 ℃ and atmospheric pressure. (c–f) Polarization curves of PEMFCs for the CNT membranes with four different thicknesses including 5, 10, 15, and 20 μm, respectively. Symbols are from experimental data, and curves are from fitting results. (g–j) Power density curves of PEMFCs for four different thickness of CNT membrane. Red symbols and curves belong to Type-1 (1-1') current collection type, and black symbols and curves belong to Type-3 (1-3') current collection type. PEMFC size is 2 cm (L) × 1 cm (w).

Why does the type of current collection affect the performance of flexible fuel cell so much? Firstly, the current collection location means different electron transfer pathway. Unlike traditional PEMFCs, flexible PEMFCs collect current from the edges of the flexible electrode, as shown in Fig. 1a. The electron transfer pathway for locations 1-1', 3-3' (Type-1) is shorter than that for locations 2-2', 4-4' (Type-4). Since the current density of fuel cell is up to ampere level, shorter length of the electron transfer pathway will induce much lower ohmic trop. As a result, the performance of Type-1 current collection is much higher than that of Type-4 current collection, and Type-3 is better than Type-5. However, Type-2 current collection is a mixture type of the other four types of current collections, which is rarely used in real applications. Consequently, directions 1-1', 3-3' (Type-1) have the possibility of being good candidates for current collection directions.

However, we also found that the performance of Type-1 current collection (33.2 mW/cm2) is much higher than that of Type-3 current collection (8.7 mW/cm2). From Fig. 1a, we know that both Type-1 and Type-3 current collections have an equivalent length of the electron transfer pathway. Similarly, the performance of Type- 4 current collection is much higher than that of Type-5 current collection. Here, we define Type-1 and Type-4 as parallel current collection type, Type-3 and Type-5 as opposite current collection type. From Figs. 3a and b, we know that parallel current collection type is much better than opposite current collection type. Why they have so different performances? How can we get better performance by only select a proper type of current collection? These problems are also significantly important for other flexible power supplies, such as supercapacitor and Li-battery [33, 34], because they have similar problems of current collection direction.

For different current collectors, is there any difference in fuel cell performance due to the current collection types? In order to study the effect of the conductivity of current collector on the performance of flexible PEMFCs, we made a series of flexible PEMFCs with different conductivity (thicknesses) of CNT membranes (Figs. 3c-j). As shown in Table 1, the square resistance of the CNT membranes decreases from 0.474 Ω/sq to 0.394 Ω/sq when the thickness increases from 5 μm to 20 μm. For these PEMFCs, the performance of both Type-1 (pair 1-1') and Type-3 (pair 1-3') current collection types is studied, and shown with red and black symbols in Figs. 3c-j. Figs. 3c-j show that the performance of both current collection types was improved with the increase of CNT membrane thickness. For Type-1 current collection type, the performance increases from 33.2 mW/cm2 to 51.6 mW/cm2, when the CNT membrane thickness increases from 5 μm to 20 μm. For Type-3 current collection type, the performance increases from 8.7 mW/cm2 to 50.2 mW/cm2. Therefore, increasing the CNT membrane thickness and actually increase conductivity can greatly advance the property of flexible PEMFC.

Table 1
Parameters from measurement and fitting.

Why can thicker CNT membrane greatly improve the performance of flexible PEMFCs? Because the electron transfer occurs in the plane of the flexible composite electrode, the electrical resistance of the current collector can vastly affect the performance of flexible PEMFCs. Table 1 shows that the square resistance of composite electrodes decreases with the increase of CNT membrane thickness. With the decrease of square resistance, the corresponding flexible PEMFC will have much lower inner resistance, and exhibit much higher performance (Figs. 3c-j). For example, the flexible PEMFC with Type-3 current collection type made by 20 μm CNT membrane has a peak power density 50.2 mW/cm2, which is 5.8 times higher than the one made by 5 μm CNT membrane.

It is notable that the increment of performance becomes less and less with the increase of CNT membrane thickness. Figs. 3c-j show that the peak power density (Pmax) of the PEMFCs prepared with the 20 μm CNT membrane is only 3.1 mW/cm2 higher than that with 15 μm CNT membrane. The performance of PEMFC with size 2 cm (L) × 1 cm (w) will not have obvious improvement in this research if the thickness of CNT membrane is thicker than 20 mm. Therefore, there is economic thickness of CNT membrane for a certain size of PEMFC, due to the thickness of the electrode leads to competition between conductivity and diffusion of reactive gases. Hence, we did not further investigate the effect of increasing the thickness of CNT membrane on the performance of fuel cell.

It is also prominent that the performance between the two current collection types. Type-1 (pair 1-1') and Type-3 (pair 1-3'), becomes less and less in the wake of the increasing CNT membrane thickness. For example, Fig. 3g shows that the Pmax difference between pairs 1-1' and 1-3' is up to 24 mW/cm2 for the 5 μm CNT membrane, while Fig. 3j shows that it becomes only 1.5 mW/cm2 for the 20 μm one. Therefore, the current collection type is more important for the thin CNT membrane, and becomes less for the thick CNT membrane. It also means that at lower conductivity, the current collection type has a greater impact on air-breathing flexible PEMFC performance than higher conductivity.

In order to further understand the effect of CNT membrane thickness on the performance of flexible PEMFCs, we use the mathematic model to fit the polarization data of PEMFCs in Figs. 3c-f, and calculate the current in collector (i) and overpotential along x direction (η). Symbols in Figs. 3c-f are from experimental data, and the curves are from the fitting with the mathematic model. We can see that the model in this work can fit the data very well. From the fitting, we can obtain the resistant in unit surface area ρ1, S (ohm cm2) and square resistant ρ2, S (Ω/sq) for each thickness of CNT membrane as listed in Table 1. The resistant in unit surface area, ρ1, S increases with the thickness of CNT membrane. This may be due to the much higher resistant in the vertical direction of CNT membrane than that in the plane, because the CNT in the membrane is parallel to the plane of CNT membrane.

Table 1 also shows that square resistant ρ2, S decreases with the increase of CNT membrane thickness. The trend is consistent with the measurement value of square resistance of composite electrode listed in the fourth column of Table 1. However, the square resistant ρ2, S is much larger than the measured square resistance of composite electrode. The value from measurement decreases from 0.474 Ω/sq to 0.394 Ω/sq, while the value of ρ2, S from model fitting vary from 19.3 Ω/sq to 0.9 Ω/sq, when the thickness of CNT membrane increases from 5 μm to 20 μm (Table 1). The huge difference may be due to the damage of CNT membrane during hot pressing, which induces the increase of square resistant ρ2, S. Accordingly, it is necessary to do more search on the interaction between the composite electrode and catalyst layer in future research.

We also calculated the current in collector (i) and overpotential (η) along x direction for both Type-1 (1-1') and Type-3 (1-3') current collection types. Figs. 4a and b show the scheme for the two different current collection types, Type-1 (1-1') and Type-3 (1-3') current collection types. Figs. 4c and d show the distribution of current in collector (i) along the length direction for Type-1 (1-1') and Type-3 (1-3') current collection types at different electric resistance of current collector (ρ2, S) respectively.

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Fig. 4. Effect of electric resistance of current collector ρ2, S and current collector length on the current in collector (i) and overpotential along x direction (η) of flexible PEMFCs at current density 50 mA/cm2. (a) Type-1 (1-1') current collection type. (b) Type-3 (1-3') current collection type. (c) Distribution of i along the length direction for Type-1 (1-1') current collection type at different ρ2, S. (d) Distribution of i for Type-3 (1-3'). Black symbol in (d) is the distribution of i for Type-1 (1-1') at ρ2, S = 0.01 Ω/sq. (e) Distribution of h for Type-1 (1-1'). (Distribution of η for Type-3 (1-3'). Black symbol in (f) is the distribution of h for Type-1 (1-1') at ρ2, S = 0.01 Ω/sq. The blue arrows point the variation direction when ρ2, S increases from 0.01 to 9.01 Ω/sq. The parameter ρ1, S is fixed at 1.00 Ω cm2; PEMFC size 2 cm (L) × 1 cm (w); current collection type is Type-1 (1-1'); ρ2, S includes 0.01, 1.01, 2.01, 3.01, 4.01, 5.01, 6.01, 7.01, 8.01 and 9.01 Ω/sq. (g) Photograph of flexible PEMFCs with different length (L). (η) Polarization and power density curves of PEMFCs with different electrode lengths. The CNT membranes in PEMFCs have 20 μm thickness and 1 cm width. (i) Peak power density (Pmax) for the PEMFCs with different electrode lengths at different electric resistance of current collector. The parameter ρ1, S is fixed at 1.00 Ω cm2; current collection type is Type-4 (2-2').

For Type-1 (1-1') current collection type in Fig. 4c, the current increases linearly from left end to the location of current collection at low ρ2, S, and increases faster and faster at high ρ2, S. We know that J is proportional to di/dx, Jdi/dx. Thus, J is almost same along x direction at low ρ2, S, indicating that every part of PEMFC works homogenous. However, J is small at the left end part and becomes much higher near the location of current collection at high ρ2, S, indicating that the part of PEMFC far from the location of current collection is not efficiently working during the discharging. The current distribution affects heat distribution, the unbalance of current and heat distribution could affect the performance of fuel cell even burn.

For Type-3 (1-3') current collection type in Fig. 4d, the current increases linearly from the end to the location of current collection at low ρ2, S. While it increases faster at two ends and slower in the mid part at high ρ2, S. Accordingly, J is almost same along x direction at low ρ2, S, indicating that every part of PEMFC works homogenous. However, J is high at the end part and becomes much smaller in the mid part at high ρ2, S, indicating that the mid part of PEMFC is not efficiently working during the discharging.

To compare the current distribution for the two current collection types, the distribution of current in collector for Type-1 (1-1') current collection type at ρ2, S = 9.01 Ω/sq is shown in Fig. 4d with black symbol. We can see the distributions of the current in collector (i) for Type-3 (1-3') current collection type are always around the straight line, while the distributions of i is far away from the straight line for Type-1 (1-1'), indicating that the distribution of the current in collector (i) for Type-3 (1-3') is better than that for Type-1 (1-1').

Figs. 4e and f show that the overpotential distribution is also more homogenous at low ρ2, S for both current collection types. Fig. 4f also shows that the distributions of the overpotential for opposite current collection type are always around the black straight line, indicating that the distribution of the overpotential for opposite current collection type is better than that for parallel current collection type.

All results above show that the opposite current collection type shows better distributions of current in collector (i) and overpotential. However, why the performance of parallel current collection type is better than that of opposite current collection type? When we compare Eqs. (6) and (9), we can find that the opposite current collection type has an additional part , which will dramatically decrease the cell potential. Consequently, the PEMFC with opposite current collection type will have lower performance.

When the size of flexible fuel cell increases, it means that the electron transfer path increases, and the performance of the fuel cell may decline. In addition, the performance of large-scale flexible fuel cell directly affects the integration and performance of the stack. Therefore, how to establish the relationship among fuel cell size, current collector square resistance and fuel cell performance is very important to the preparation of large-scale fuel cell and the integration of fuel cell stack.

It is needed to recognize the influence of electrode length (L in Fig. 1a) on the performance of flexible PEMFC if we intend to fabricate large size flexible PEMFC. In this research, we fix the width of electrode (w) at 1 cm, and vary the length of electrode from 1 cm to 4 cm (Fig. 4g). Type-4 (2-2') is selected for the current collection type. Fig. 4h shows that the length of electrode has a dramatic effect on the performance of PEMFCs. Pmax decreases with the increase of electrode length, indicating shorter length of electrode, e.g., L = 1 μm, has much higher specific power density than the longer one, e.g., L = 4 μm. The tremendous difference is due to that longer length of electrode has a longer transfer pathway of electron, and has higher inner resistance.

In order to further comprehend the influence of electrode length we used the mathematic model in this work to simulate the peak power density (Pmax) for the PEMFCs with different electrode lengths at different electric resistance of current collector (ρ2, S). Fig. 4i shows that Pmax always decreases with the increase of electrode length. This phenomenon is consistent with our results in Fig. 4h. But the decay shapes for different ρ2, S are very different. For example, the Pmax for large ρ2, S (wine red star in Fig. 4i) decays very fast within the length < 10 μm, and approaches 0 after 10 μm. The Pmax for small ρ2, S (black square in Fig. 4i) decays very slowly within the length < 10 μm, and becomes faster after 10 μm. Even if the length of electrode is as long as 20 μm, the Pmax for ρ2, S = 0.002 Ω/sq only decays 31%. In order to make large size of flexible PEMFC, it is needed to use highly conductive electrode, whose square resistance is lower than 0.002 Ω/sq.

In this research, the value of ρ2, S from model fitting vary from 19.3 Ω/sq to 0.9 Ω/sq, when the thickness of CNT membrane changes from 5 μm to 20 mm. Therefore, the curve with wine red star (1.2 Ω/sq) in Fig. 4i is close to the result of the composite electrode made with 20 μm thick CNT membrane in this research. The curve shows that the composite electrode made with 20 μm thick CNT membrane is not suitable for the PEMFC longer than 5 μm. We also can calculate that the small ρ2, S 0.002 Ω/sq is the equivalent of just 8.5 μm thick copper foil. Therefore, it is possible and necessary to further increase the conductivity of composite electrode when we make large size of flexible PEMFC and extralarge current.

Our research results and related models have important guiding significance to the preparation of large-scale flexible fuel cells. However, the preparation of other large-scale flexible power sources may also have such problems. Can our results also be applied to other large-scale flexible power sources? Although this work mainly focuses on the flexible fuel cell, the results from this work are also applicable to other flexible electrochemical power sources, such as Li-ion battery and supercapacitor. As mentioned in the introduction part, all flexible power sources have a common characteristic in structure. Therefore, the mathematic model is applicable to other flexible electrochemical power sources. As shown in Eqs. (3a) and (3b), the only differences for different flexible power sources are the parameters α and k. If other researchers want to use our model to fit their result, they only need to adjust the values of parameters α and k.

Researchers in other flexible power sources or even devices fields also need to consider the conductivity of flexible electrode when they make large size ones. Although the flexible supercapacitor and Li-ion battery can directly use high conductivity materials, such as copper foil, a proper thickness of the foils is still a problem that needs to be considered. If they want to obtain a low cost and high performance simultaneously, they need to optimize the thickness. The results and the models in this research will be very favorable in their optimization. Obviously, this research is also significant for other flexible devices and power sources, such as supercapacitors and Li-batteries [33, 34] which have similar problems of current collection.

In this research, we observed that under the same current collector, the current collection locations can dramatically affect the performance of air-breathing flexible PEMFCs. The optimal performance is obtained when current collection locations are on the same side of long edge. The different performance of different types of current collection is mainly attributed to the different lengths of electron transfer pathways. Besides, the conductivity of current collector can seriously affect the capability of air-breathing flexible PEMFCs even the current collection type is optimized. The air-breathing flexible PEMFCs with thicker CNT membrane (higher conductivity) show higher performance. Furthermore, in order to better understand the influence of current collection locations, a mathematic model is successfully built in this work. The simulation by the model shows the distribution of the current in current collector (i) and overpotential (η) along current collection direction of flexible PEMFCs, presents the proper electric resistance of current collector (ρ2, S) for large size flexible fuel cell. The results and knowledge from this research could also be applied into flexible Li-ion battery and supercapacitor and other flexible devices.

Acknowledgments

The authors are grateful for financial support granted by Ministry of Science and Technology of China (Nos. 2016YFE0105700, 2016YFA0200700), the National Natural Science Foundation of China (Nos. 21373264, 21573275), China Postdoctoral Science Foundation (No. 2018M632406), the Science and Technology Project of Nanchang (No. 2017-SJSYS-008). This work was sponsored by the Collaborative Innovation Center of Suzhou Nano Science and Technology.

Appendix A. Supplementary data

Supplementary material related to this article can be found, in the online version, at doi: https://doi.org/10.1016/j.cclet.2019.02.032.

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