A unified theoretical model for Triboelectric Nanogenerators

R.D.I.G. Dharmasenaa, K.D.G.I. Jayawardenaa, C.A. Millsa,b, R.A. Doreyc, S.R.P. Silvaa*a Advanced Technology Institute, Department of Electrical and Electronic Engineering, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom. Email: s.silva@surrey.ac.ukb Advanced Coatings Group, Tata Steel Research Development and Technology, Voyager Building 9, Sir William Lyon Road, Coventry CV4 7EZ, United Kingdom.c Centre for Engineering Materials, Mechanical Engineering Sciences, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom.

Abstract

A unified theoretical model applicable to different types of Triboelectric Nanogenerators (TENGs) is presented based on Maxwell’s equations, which fully explains the working principles of a majority of TENG types. This new model utilizes the distance-dependent electric field (DDEF) concept to derive a universal theoretical platform for all vertical charge polarization TENG types which overcomes the inaccuracies of the classical theoretical models as well as the limitations of the existing electric field-based model. The theoretical results show excellent agreement with experimental TENGs for all working modes, providing an improved capability of predicting the influence of different device parameters on the output behaviour. Finally, the output performances of different TENG types are compared. This work, for the first time, presents a unified framework of analytical equations for different TENG working modes, leading to an in-depth understanding of their working principles, which in turn enables more precise design and construction of efficient energy harvesters.

Graphical Abstract

Triboelectric Nanogenerators are becoming increasingly popular as an energy harvesting technology, however there still a limited understanding about their output behavior. A unified theoretical platform to fully understand the working principles of the different working modes of triboelectric generators is presented, which allows their precise design and performance optimization for different applications.

Keywords: energy harvesting, triboelectric nanogenerators, universal model, distance-dependent electric field

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1. Introduction

Energy harvesting using ambient sources including solar energy[1–6], heat[7,8], mechanical motion[9–11], and biochemical energy[12] has been a major research focus in recent years.[9] Triboelectric nanogenerators (TENGs) are in the forefront of mechanical energy harvesting technologies, benefitting various low power applications.[13–18] The currently accepted theoretical models for TENGs are based on equivalent circuits composed of parallel-plate capacitors [19–22], that do not comprehensively describe their working principles or outputs.[11] These models are restricted to parallel-plate configurations, and contain limitations in modelling TENG layer separations.[11] Therefore, a fundamental analytical model describing different TENG types is a key requirement in this research area for its wider use and better applications.[9,11,16,17,23]

Recently, we introduced the distance-dependent electric field (DDEF) concept[11] for the vertical contact-separation TENG (VCSTENG), completely describing its working principles and output behaviour. Although this model has a broader scope than any previously described models[11,19], its applicability is restricted to dielectric-dielectric VCSTENGs. Herein, we present a novel unified theory based on the DDEF concept, which accurately predicts the output behaviour of all vertical charge polarization TENGs[19] (VCPTENGs). Originating from Maxwell’s equations, the unified DDEF model spans across VCSTENG, single electrode TENG (SETENG), free-standing triboelectric layer TENG (FSTENG), towards multi-layer TENG structures. The theoretical predictions from the models are compared with experimental and finite element model outcomes.

2. Classical theoretical models

The prevailing theoretical models assume a uniform electric field between charged triboelectric surfaces (parallel-plate capacitor assumption), therefore, are incapable of explaining the electric field propagation, polarization of TENG layers, induced voltage and charges in TENGs.[11] Furthermore, the use of bespoke configurations of parallel-plate capacitors to match the outputs introduces specific incompatibilities and drawbacks to each model.

Previous theoretical model for SETENG[21] states that the electric field lines must directly connect between Node 1 and Node 3 (Fig. 1a), without being shielded by Node 2, giving rise to a cyclic 3-capacitor circuit model (Fig. 1b). As per the capacitor approximation, electric field is uniform between Nodes 1 and 2, and a residual electric field cannot propagate towards Node 3 (equal and opposite electric field lines in Fig. 1c)[11]. Similar criteria apply to the electric field between Nodes 2 and 3. Hence, the assumption of direct field line connections between Node 1 and 3 becomes contradictory. Furthermore, the analytical equations of the previously accepted model[21] suggest C2=0 for all z (t) (triboelectric layer separation) >0, hence the derived equations[21] cannot explain the observed TENG outputs. The previous model[21] therefore implicitly depends on finite element modelling to evaluate C2, and accordingly the other output parameters.

Despite having a comparable 3-layer structure as the SETENG, the circuit model for FSTENG (Fig. 1d) contains three capacitors in series (Fig. 1e), assuming the electric field lines between non-adjacent

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nodes are blocked by the intermediate node.[22] Consequently, the dependability between the theoretical models for SETENG and FSTENG is impaired. Furthermore, using the simplest FSTENG structure where similar metals are used in Nodes 1 and 4, reconciling the output behaviour using the previously accepted model[22] is challenging. Firstly, assuming similar contacting conditions of the dielectric sheet with both the metal electrodes, the identical quantity of similar triboelectric charges are transferred to Nodes 2 and 3. With no net charge variation, the electric field inside C2 is zero and the model is reduced to a two-capacitor system equivalent to the conductor-to-dielectric VCSTENG[24], which is significantly different from a FSTENG.[19,22] Secondly, the electric fields inside the capacitors are uniform[11], therefore, there will be no change in the resultant electric fields acting on metal 1 and metal 2, even if z (t ) is changed. The overall capacitance between the metal sheets is constant regardless of z (t), indicating that the device would not produce a net output, which contradicts previously reported observations.[22]

3. Theory

3.1 Triboelectric Charging

TENGs depend on the triboelectric effect[25]; static charge separation between contacting surfaces. Although the exact mechanism of triboelectricity is not yet established, it is commonly attributed to electron[26,27], ion[28] or charged material[29] transfer. When two triboelectric surfaces are contact-charged, the triboelectric charge density (σ T) increases and saturates within few initial contacts[30–32], which is almost insignificant compared to the practical usage of a TENG. Thereafter, σ T remains unchanged for long durations.[19,29,31] In our case, we consider the charges at their saturated value

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Fig. 1: (a) Graphical representation, and (b) corresponding equivalent circuit model for a SETENG with a parallel secondary electrode arrangement[21]. (c) The overall electric field behaviour of the SETENG when the triboelectric layers are separated by an air gap. The electric fields are indicated by E, depicted to act on a positive test charge where the colour of the arrow represents the triboelectric surface generating the field. (d) Graphical representation, and (e) the corresponding equivalent circuit model of a FSTENG with a dielectric free standing layer[22]. (f) Schematic representation of a TENG with m number of triboelectric surfaces.

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after the initial dynamic period, in examining the implications of a fundamental theoretical model for TENGs.

This work assumes that the triboelectric charges are evenly distributed, with a charge density of similar magnitude and opposite polarity on their contacting surfaces, which remains constant during TENG operation. [11,19,21,22,24] The uniform charge distribution is commonly observed in fundamental triboelectric charging experiments, and has been proven to be reasonably accurate via theoretical and experimental evidence [28,33–36]. Moreover, this has been reported as an accurate assumption for real world TENGs, considering the typical behavior of TENG contact surfaces [11,19,22,24,37]. Furthermore, we note that all the theoretical models presented for TENGs in the past have also followed the same assumptions stated above [11,19,20,22,24,37–39].

3.2 Distance-dependent electric field

The average electric field (E z) over the midpoint of a finite charged surface with L (length) and W (width), and a uniform surface charge density σ, can be derived using Gauss’s law:[11]

E z=σπεarctan (

LW

2( zW )√4( z

W )2

+( LW )

2

+1 )= σπε

f ( z )

(1)

where z is the distance from the charged surface and ε is the permittivity of the medium within which the electric field propagates. Magnitude of E z decreases as z increase.

We note that considering the fields over the midpoint of the finite charged surface significantly reduces the complexity of the mathematical derivation and the solving of output equations due to symmetry, while providing an electric field which represents the overall electric field behaviour of the surface[11].

3.3 The unified DDEF model

As per eqn (1), the influence of a charged layer on a TENG electrode depends on σ and z parameters of the charged layer. Assuming a VCPTENG with m triboelectrically charged surfaces (Fig. 1f) and considering the electric field components originating from each surface and the electrodes, the potential of the electrodes a (Φa) and b (Φb);

Φa=−σu

π εa∫0

y

f ( x )dx+ 1π∑i=1

m

(σ T ,i

εa∫xa,i

∞

f ( x )dx ) (2)

Φb=σu

π εb∫0

y

f ( x )dx+ 1π ∑i=1

m

(σT ,i

εb∫xb , i

∞

f ( x )dx¿)¿ (3)

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where, ε aand ε b represent the permittivity of the medium related to the electrode interfaces, y the separation between the electrodes, xa ,i, xb ,i the distance between ithtriboelectrically charged surface and the respective electrode, σ T ,i the triboelectric charge density of ithtriboelectric surface, and , σ u the output charge density. The polarity of each σ T ,i component should be considered when applying eqn (2) and (3). During typical TENG operation xa ,i, xb ,i are time varying parameters similar to z (t). The voltage (V ) between the electrodes,

V=Φa−Φb (4)

The open circuit voltage V oc can be obtained by substituting σ u=0 to eqn (4).

In the short circuit configuration V=0, hence

σ u=∑i=1

n

(σT ,i

ε a∫xa, i

∞

f ( x )dx−σT ,i

ε b∫xb , i

∞

f ( x )dx)

( 1εa + 1εb )∫0y

f ( x )dx

(5)

and the short circuit current I sc=LWd σu

dt.

Considering a load resistor with resistance R connected between the electrodes, and using Ohm’s law,

LWRd σu

dt+σu

π ( 1εa+ 1εb )∫0y

f ( x )dx−1π

¿ (6)

Using eqn (6), output current at time t , I ( t )=LWd σu

dt, and power at time t , P ( t )=I( t )2R. The above

equations (1) – (6) allow for the full evaluation of performance for all TENG devices.

3.4 Vertical contact-separation TENG (VCSTENG)

Eqn (2) – (6) can be used to model the output behaviour of a metal-dielectric VCSTENG by defining the electric fields and output charge behaviour for a metal substrate, as detailed in Supplementary Note 1. This extends the applicability of the DDEF concept from a dielectric-dielectric VCSTENG [11], to all major material types and parameters.

3.5 Single electrode mode TENG (SETENG)

Considering the SETENG (Fig. 1a), primary electrode (Node 2) is not grounded, hence the electric field originating from Node 1 is not shielded by Node 2, and propagates towards secondary electrode (Node 3).[21,40] The fields from Node 2 and 3 also contribute to the potential of secondary electrode (Φ2). The potential of primary electrode (Φ1) is influenced by similar electric fields. When z (t) varies, the

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influence of the electric field from Node 1, on Nodes 2 and 3 varies corresponding to their positions, creating a potential difference between the electrodes. When electrodes are connected, output charges redistribute to nullify the potential difference, and this current is guided through a load to harvest energy.

Using eqn (2) and (3),

Φ1=−σu

πε0 [∫0d

f (x )dx ]+ σT

πε0 [∫0z(t )

f (x)dx ] (7)

Φ2=σu

πε0 [∫0d

f (x )dx]+ σT

πε0 [ ∫dz (t ) +d

f (x )dx ] (8)

ε 0 is the permittivity of free space,d is the electrode separation.

In the short circuit condition, using eqn (5),

σ u=

σ T

2[∫0

z (t )

f (x )dx− ∫d

z (t )+ d

f (x)dx ]

∫0

d

f (x )dx

(9)

Considering a load resistor R connected across the output terminals of the SETENG, using eqn (6),

LW Rd σu

dt+2σu

π ε0∫0

d

f (x)dx+σT

π ε 0¿ (10)

The secondary electrode is typically grounded via a resistor or placed in a non-parallel configuration in most practical TENGs[41–44], which can be represented using Eqn (2) – (6). Assuming a non-parallel secondary electrode identical to the primary electrode (Fig. S2a), and using eqn (2) and (3),

Φ1=−σu

πε0 [∫0∞

f ( x )dx ]+ σT

πε0 [∫0z( t )

f (x)dx ] (11)

Φ2=σu

πε0 [∫0∞

f (x )dx] (12)

For the short circuit condition, using eqn (5),

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σ u=

σ T

2[∫0

z (t )

f (x )dx ]

∫0

∞

f (x)dx(13)

Considering TENG power generation, using eqn (6),

LWRd σu

dt+2σ u

π ε 0∫0

∞

f ( x )dx−¿σT

π ε0∫0

z (t )

f ( x )dx=0¿ (14)

When the primary electrode is grounded via a resistor the output charge flow results in its potential being equal to the ground potential, which is modelled similarly, as shown in Supplementary Note 2.

3.6 Free-standing triboelectric layer mode TENG (FSTENG)

Triboelectric charging of FSTENG occurs between Nodes 1 and 2, and between Nodes 3 and 4 (Fig. 1d). The electric fields from Nodes 1, 2, and 3 propagate towards Node 4 without being shielded. Therefore, the potential of Node 4 depends on these electric fields, and on the electric field originating from the overall charge on itself. This potential varies with z (t), as the contribution from the electric fields originating from Nodes 2 and 3, towards the potential of the bottom electrode (Node 4) changes. Similar mechanisms apply to the potential of the top electrode (Node 1). An output charge flow results between the electrodes to equilibrate the potential difference, which is directed through a load to harvest energy.

Consider the dielectric free-standing layer FSTENG depicted in Fig 1d. For convenience of calculations, Nodes 1 and 4 are assumed to be constructed using the same metal. Using eqn (2) and (3) for potential Φ1 (Node 4) and Φ2 (Node 1);

Φ1=−σu

πε0∫0

g

f (x )dx+σT

πε0∫0

g−z ( t )−k

f (x)dx−σ T

πε0∫

g− z(t )¿

¿¿ g f ( x )dx (15)

Φ2=σu

πε0∫0

g

f ( x )dx+σ T

πε0∫0

z (t )

f (x )dx−σT

πε0∫

k +z (t)

g

f ( x )dx (16)

where g and k represent the electrode separation and thickness of the free-standing layer, respectively.

For short circuit configuration, using eqn (5),

σ u=

σ T

2[ ∫z (t ) +k

g

f ( x )dx−∫0

z (t )

f ( x )dx+ ∫0

g− z (t )−k

f ( x )dx− ∫g−z (t )

g

f (x)dx ]

∫0

g

f (x)dx (17)

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For TENG power generation, using eqn (6),

LWRd σu

dt+2σ u

π ε 0∫0

g

f ( x)dx+σ T

π ε0¿

(18)

Similar analysis can be applied for the metal free-standing layer FESTENG.

4. Experimental Method

4.1 Fabrication of the dielectric-metal VCSTENG

The dielectric contact surface of the dielectric-metal VCSTENG was constructed by coating a 20 µm thick Sylgard 184 poly(dimethyl siloxane) (PDMS) layer on a PET/In2O3/Ag/Au sheet (L=50 mm, W=50 mm, thickness=0.2 mm), fabricated using the previously reported methods[11]. A sheet of AISI 316 stainless steel (L=50 mm, W=50 mm, thickness=0.1 mm, GoodFellow (UK)) was used as the metal contact surface, and the TENG layers were arranged with a configuration similar to that depicted in Fig. S1. Additional device parameters are summarised in Table S1.

4.2 Fabrication of the SETENG

A non-parallel secondary electrode SETENG was constructed using identical triboelectric surfaces as the dielectric-metal VCSTENG (section 4.1), where a PDMS coated side of the PET/In 2O3/Ag/Au layer was used as the dielectric contact, and both primary and secondary electrodes were stainless-steel. The secondary electrode was placed away from the primary electrode in a non-parallel configuration, as shown in Fig. S2a. The electrical outputs were obtained across the stainless-steel electrodes.

4.3 Fabrication of the FSTENG

A dielectric free-standing layer FSTENG was constructed with a structure similar to that shown in Fig. 1d. The metal contact surfaces (electrodes) were fabricated using stainless steel sheets identical to those described in section 4.1, and were placed with a separation (g) of 8.2 mm between them. The dielectric free-standing layer was a polymethylmethacrylate (PMMA) sheet (L=50 mm, W=50 mm, thickness=0.5 mm, GoodFellow (UK)).

4.4 Device characterization

The contact-separation movement of the TENG layers was achieved using a bespoke linear motor setup (Fig. S3). The test procedures used for the TENGs are identical to the methods discussed in our previous work[11], as detailed in Supplementary Note 3. The voltage and charge outputs were measured using a Keithley 6517B electrometer, whereas the current measurements were conducted using a Keithley 6487 picoammeter.[11] The current, charge and power outputs were obtained under sinusoidal contact-

separation movements described as, z (t )=H sin(2πnt+ 3 π2 )+H , where H is the amplitude and n is

the frequency of the movement.

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5. Results and Discussion

5.1 Vertical contact and separation mode TENG (VCSTENG)

The experimental short circuit output charge (Qsc) and I sc of the dielectric-metal VCSTENG matched the DDEF predictions at σ T=30 µC/m2 (Supplementary Note 4). The output trends of the dielectric-metal VCSTENG are similar to dielectric-dielectric VCSTENGs[11] (Fig. 2). The I sc increases with a decreasing gradient when H is increased (while n is fixed), as indicated by both DDEF predictions and experimental outputs (Fig. 2a). A similar match is observed for Qsc (Fig. S4b).

A comparison was carried out between the predictions from the DDEF model and the previously accepted model for metal-dielectric VCSTENGs[19] (Supplementary Note 4), which shows the better accuracy of the DDEF model (Fig. 2b, Fig S4d). The major difference between the two models is observed for V oc predictions, where the DDEF model predicts a decreasing gradient against increasing z (t ), agreeing with the experimental trend, whereas the previous model[19] predicts a linear relationship (Fig. 2b). The mismatch in the magnitude of the DDEF model and experimental V oc is attributed to the relatively limited impedance of the voltage measurement unit.[11]

The power output for the VCSTENG was evaluated under a sinusoidal movement of H=1 mm and n=1 Hz.[11] The DDEF model predicts a characteristic three-working-region behaviour[11] where the power

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Fig. 2: Comparison of the DDEF model predictions against metal-dielectric VCSTENG outputs. (a) The experimental peak I sc

against increasing H (n=1 Hz), compared with DDEF mode predictions. (b) Comparison of the experimental V oc against the predictions from the DDEF model and the previously accepted model for VCSTENGs[19]. (c) The experimental peak output power measured through different load resistors (n=1 Hz, H=1 mm), compared with DDEF predictions. (d) Output charge for Al – Polyolefin TENG reported previously (indicated as previous study 1 - ref [45]), compared with the corresponding DDEF predictions.

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is a maximum around 10-4 W, between 108 Ω - 109 Ω. The experimental outputs coincide with the theoretical predictions (Fig. 2c), providing an accurate prediction of the power output.

The applicability of the DDEF model was further verified by comparing the outputs of a metal-dielectric VCSTENGs reported previously[45,46] against the corresponding predictions from the DDEF model (Fig. 2d, S4e), under identical device parameters (Table S2). Fig. 2d shows the charge output reported by Yang et al.,[45] using Aluminium and Polyolefin VCSTENG surfaces (Previous Study 1). The output power reported by Zhong et al.,[46] for a VCSTENG (Previous Study 2) is presented in Fig. S4e. In both cases, outputs from the previous studies agree well with DDEF predictions.

5.2 Single electrode mode TENG (SETENG)

A SETENG with a non-parallel secondary electrode (Fig. S2a) was used to validate the DDEF model for a generalized SETENG architecture. Identical triboelectric contact surfaces (stainless steel, PDMS) were used for VCSTENG and SETENG with similar test conditions (section 4), hence, the σ T approximated for VCSTENG (30 µC/m2) is used for the SETENG in DDEF simulations.

The DDEF model predicts an increase in I sc with a decreasing gradient when H is increased (n=1 Hz) saturating towards 58 nA, and the experimental outputs depict a comparable trend reaching 60 nA (Fig

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Fig. 3: Comparison of DDEF predictions against experimental outputs for (a) peak I sc against increasing H (n=1 Hz), and (b) peak output power against increasing resistances (H=1 mm, n=1 Hz), considering a SETNG with a non-parallel secondary electrode. (c) – (d) Comparison of the previously reported results[47] with the corresponding predictions from the DDEF model for a parallel secondary electrode SETENG; (c) V oc, and (d) output charge, against increasing H for the previous study 3 (ref [47]). Error bars indicate the standard deviation of 10 readings. For (a), the solid markers indicate the simulated points and the line indicates the interpolated trend.

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3a). A similar trend with a gradually saturating output is observed for Qsc (Fig. S5a-b). The slight mismatch in the experimental and DDEF predictions is due to quicker saturation of experimental outputs, possibly due to interfering electric fields and leakage currents, in the high impedance non-parallel secondary electrode SETENG setup. Further experiments are currently conducted under more controlled environments to study these effects. The DDEF model suggests a 3-working-region behaviour for the SETENG power output, where the power maximizes around 100 GΩ - 1 TΩ (Fig. 3b). The experimental outputs closely follow these predictions for the range of resistances tested. The V oc trend for the SETENG is similar to VCSTENG, where the voltage increases with z (t) and gradually plateaus, as evident from the experimental and DDEF outputs (Fig S5c). The mismatch between the magnitude of the V oc is again due to the relatively limited impedance of the voltage measurement unit.[11]

Applicability of the DDEF model for a parallel-secondary electrode SETENG is demonstrated using the outputs reported in the literature[21,47]. Fig. 3c-d shows the finite element model (FEM) outputs reported by Chen et al.[47] (Previous Study 3), compared with corresponding DDEF predictions under identical parameters (Table S3). The FEM results for V oc (Fig. 3c) and Qsc (Fig. 3d) display an increase in their magnitude with a decreasing rate against increasing z (t), and the DDEF predictions agree with these trends. A similar analysis is conducted for V oc and Qsc reported by Niu et al.[21] (Previous Study 4) (Fig. S5d-e), further confirming the accuracy of the DDEF model. Additionally, a theoretical prediction on the influence of gap between the electrodes (d ) towards SETENG outputs is presented using the DDEF model (Fig. S7 and Supplementary Note 5), presenting the relationship between the parallel and non-parallel SETENG configurations.

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Fig. 4: The output comparison between a dielectric-metal VCSTENG and a comparable non-parallel secondary electrode SETENG (PDMS - Stainless Steel triboelectric surfaces). Comparison of peak (a) Qsc (b) I sc, and (c) power output, predicted by the DDEF model. (d-f) Corresponding experimental outputs. Error bars indicate the standard deviation of 10 readings.

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Previous studies have highlighted the importance of the inter-comparison between different TENG types in assessing their suitability for different applications.[21,24,48] Fig. 4 depicts a comparison of theoretical (Fig. 4a-c) and experimental (Fig. 4d-f) outputs between a SETENG (section 4.2) and a dielectric-metal VCSENG (section 4.1) composed of comparable TENG surfaces (stainless steel-PDMS). The DDEF model suggests the Qsc (Fig. 4a), I sc (Fig. 4b), and V oc (Fig. S6) from the VCSTENG are significantly higher than the corresponding SETENG outputs, as verified via the experimental results. The peak power output is higher for the VCSTENG compared to the SETENG, dissipated through a relatively lower resistance (Fig. 4c, f). The differences in their output behaviour can be understood using the DDEF concept (Supplementary Note 5).

5.3 Free standing triboelectric layer mode TENG (FSTENG)

The FSTENG was fabricated using stainless-steel and PMMA (section 4.3), considering the convenience of fabrication and testing. A dielectric-metal VCSTENG was constructed using similar triboelectric layers

as a control device, to evaluate σ T for these surfaces. The short circuit outputs for the control device indicate σ T~ 2 µC/m2 for the two materials (Supplementary Note 6), which is used to simulate the outputs of the FSTENG.

The DDEF simulations for FSTENG suggest a linear variation of Qsc (Fig. 5a) and I sc (Fig. 5b) against increasing H , deviating from SETENG and VCSTENG output trends. The experimental results agree with

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Fig. 5: Analysis of the output behaviour from the FSTENG. Comparison of the experimental results and DDEF predictions for peak (a) Qsc , and (b) I sc, at increasing H (n=1 Hz). (c) The experimentally measured power output through different resistors (n=1 Hz, H=1 mm), compared with the predictions from the DDEF model and the previously accepted model[22] for FSTENG. (d) Experimental Qsc reported in Previous Study 5 (ref [22]) compared against the corresponding predictions from the DDEF model. Error bars indicate the standard deviation of 10 readings.

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the theoretical predictions (Fig. 5a-b). V oc also shows a linear trend against increasing z(t), for simulated and experimental outputs (Fig. S8c). The linearity of the FSTENG outputs has previously been highlighted in literature.[22] However, DDEF simulations indicate that the FSTENG outputs deviate from the linear behaviour at sufficiently large g (Supplementary Note 6).

The DDEF model predicts a three-working-region behaviour for FSTENG peak power output, maximizing between 5 GΩ - 100 GΩ (Fig. 5c). The experimental results coincide with the predictions, showing the onset of saturation around 5 GΩ. The corresponding predictions from the previously accepted model for FSTENG[22] are also depicted in Fig. 5c, highlighting the relatively higher accuracy of the DDEF model. The experimental Qsc reported by Niu et. al.,[22] (Previous Study 5[22]) is compared with corresponding DDEF predictions (Fig. 5d) as detailed in Table S4. The simulated outputs coincide with the experimental results[22] at σ T=2 µC/m2, verifying accuracy of the DDEF model.

A comparison between the FSTENG and the corresponding VCSTENG (control device) is depicted in Fig. 6. The main difference between the devices is the linear output response of the FSTENG, compared to the gradually saturating output of the VCSTENG (Fig. 6a-b, d-e). The Qsc for the FSTENG is initially lower but surpasses the corresponding Qsc values of the VCSTENG at higher H , as per the DDEF model (Fig.

6a) and experimental outputs (Fig. 6d). However, I sc from the VCSTENG is higher than the FSTENG (Fig. 6b, e).

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Fig. 6:The output comparison between the dielectric-metal VCSTENG and the comparable dielectric free-standing layer FSTENG (PMMA – Stainless Steel triboelectric surfaces). The DDEF predictions for peak (a) Qsc, (b) I sc, at n=1 Hz and increasing H, and the (c) peak power output at n=1 Hz, H=1 mm, for increasing resistances. (d-f) Comparable experimental outputs. Error bars indicate the standard deviation of 10 readings. σ T=2 µC/m2 was used for the DDEF simulations.

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According to the DDEF model, the peak power output of the FSTENG is lower than the VCSTENG for most of the load resistances, initially increasing linearly (Fig. 6c). However, the power output of the FSTENG maximizes around 5 GΩ - 100 GΩ, whereas the VCSTENG power output peaks around 1 GΩ - 5 GΩ. The experimental outputs agree well with the predictions (Fig. 6f), where the VCSTENG output maximizes around 1 GΩ, and the FSTENG output starting to saturate towards 5 GΩ. The difference in the output behaviour of the VCSTENG and the FSTENG can be understood with the DDEF concept (Supplementary Note 6).

These experiments demonstrate the capability of the DDEF model to accurately predict the output behaviour of the VCPTENG structures for multiple TENG layer configurations, starting from the basic two triboelectric layer scenario (VCSTENG), and expanding up to three triboelectric layers for the FSTENG which evaluates the effect of electric field components up to six different charged surfaces (Fig. 1).

In summary, the above results verify the accuracy and versatility of the unified DDEF model in explaining the working principles and modelling the outputs of different VCPTENG structures, hence, offering precise guidance in designing TENGs for different applications. As a simple illustration (Fig. 7), higher power output of VCSTENGs is useful in energy scavenging from ambient sources, the versatility and mobility of SETENGs in wearable and portable electronics, and the linearity of FSTENG outputs in self-powered sensing.

6. Conclusions

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Fig. 7: The perspective applications for Vertical Charge Polarization TENGs; VCSTENG – Energy harvesting from wind, ocean waves, walking and ambient vibration sources. SETENG – Wearable and portable electronic devices for different activities. FSTENG – Position sensing platforms, smart buildings and tracking systems.

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A unified theoretical framework is presented based on the DDEF concept, fully explaining the working principles of a majority of TENG structures, expanding from VCSTENG, SETENG, FSTENG to multi-layer VCPTENG modes. This new model accommodates metal and non-metal triboelectric surfaces and different motion profiles, encompassing the full range TENG parameters. The unified DDEF model output predictions are verified using experimental results and previously reported TENG outputs, proving the outstanding accuracy of the model. Finally, the relative performances of different TENG types are assessed in a comparable platform. The overall analysis highlights the effectiveness of the DDEF model predictions across different TENG working modes, and its higher accuracy in comparison to the previous models.

For the first time, this work provides a system of unified analytical equations for all VCSTENG types, eliminating the necessity of having different circuit model approximations, which not only overcomes the drawbacks of the conventional models, but also the limitations of the previous electric-field based model. The accuracy of the DDEF model in predicting TENG outputs will be critical in design and optimization of their structures for different applications, and for the inter-comparison of various TENG working modes.

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Acknowledgements

The authors acknowledge the support from the EPSRC Strategic Equipment Account EP/L02263X/1. R. D.I.G.D. acknowledges financial support for a URS studentship from the University of Surrey and Advanced Technology Institute. The authors would like to thank Jonathan Deane from the Department of Mathematics, University of Surrey, for the useful discussions regarding the solving of differential equations.

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Appendix A. Supplementary Information

Competing Interests

The authors declare no competing financial interests.

Ishara Dharmasena is a PhD research student at the Advanced Technology Institute, University of Surrey. He received his B.Sc. (Engineering) degree from the University of Moratuwa, Sri Lanka in 2014, where he was awarded as the best student in his discipline. Ishara worked at Sri Lanka Institute of Nanotechnology (SLINTEC) as a research scientist prior to moving to Surrey, developing nanotechnology enhanced textiles. His research interests include mechanical energy harvesting, fundamentals of triboelectric nanogenerators, wearable electronics and nanotechnology for wearable applications.

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Dr. Imalka Jayawardena received his B.Sc (Engineering) degree from the University of Moratuwa, Sri Lanka in 2008 and obtained his Ph.D. in Electronic Engineering from The Advanced Technology Institute at the University of Surrey in 2012, under the supervision of Dr Simon Henley. He is currently working as a postdoctoral researcher in Prof. Ravi Silva's group developing new high efficiency perovskite and organic solar cells and high sensitivity X-ray detectors and imagers. He is the author of over 30 research papers and was a recipient of one of the EPSRC Postdoctoral Prizes offered at the University of Surrey in 2012.

Dr. Chris Mills obtained his PhD in Electrical and Electronic Engineering from the University of Bangor, Wales in 2001. Following this, he worked as a postdoctoral researcher for over 12 years in Glasgow, Barcelona and Surrey, investigating nanotechnology applications of polymers and polymer semiconductors. He is the author of over 50 research papers and has also been the recipient of the Spanish “Ramon y Cajal” fellowship. He is currently working as a Research Scientist at Tata Steel.

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Prof. Robert Dorey holds the chair in Nanomaterials at the University of Surrey and is Fellow of the Institute Materials, Mining and Minerals (FIMMM) and Higher Education Academy (FHEA) as well as a Chartered Scientist and Engineer. Professor Dorey joined the University of Surrey from Cranfield University in 2014. Between 2003 and 2008 he held a prestigious Royal Academy of Engineering/EPSRC Research Fellowship. His research interests are focussed on the synthesis & manipulation of nanomaterials, through sustainable processing routes, for the production of functional devices for energy & environmental applications.

Prof. Ravi Silva is the Director of the Advanced Technology Institute at University of Surrey. He joined Surrey in 1995 after completing undergraduate and postgraduate degrees at Cambridge University. His research interests include nanoelectronics, large-area electronics and renewables, and resulted in over 550 presentations at conferences, and over 500 journal papers. He is the inventor of 30 patents, including a key patent on low temperature growth of carbon nanotubes, and fabrication of large area nanotube-organic solar cells. He is a Fellow of the Royal Academy of Engineering, UK and a Fellow of the National Academy of Sciences Sri Lanka.

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