The space tribo-charge region and equivalent charge plane model in triboelectric nanogenerators

Abstract

Recently, triboelectric nanogenerators (TENGs) have emerged as a rapidly growing technology for energy harvesting and self-powered sensing. Enhancing the output performance of TENGs requires a thorough understanding of their physical mechanisms. To clarify the inherent mechanisms of TENGs, formal electrostatic frameworks have been introduced to predict the output voltage of TENGs. However, these approaches still have certain limitations, because the triboelectric charge has been treated as a zero-thickness surface charge, which has been demonstrated to be overly idealistic by experimental data. Here, we propose a space tribo-charge region (STCR) model, which captures the finite thickness distribution of the triboelectric charge, replacing the traditional zero-thickness surface charge model, to more accurately describe the electrostatic behavior of contact–separation mode TENGs (CS-TENGs). As CS-TENGs typically operate under low frequencies, the electric field generation by the STCR is treated as an electrostatic phenomenon. An equivalent charge plane (ECP) model was also proposed to analytically derive the electric field distribution generated by the STCR. The formation process of the STCR model and its induced electrostatic field/potential distribution are investigated using a series of CS-TENGs. The dielectric layers of the examined CS-TENGs consist of fixed-thickness polydimethylsiloxane (PDMS) on one side and variable-thickness fluorinated ethylene propylene (FEP) layers ranging from 15 μm to 100 μm on the other side. The polarity-swapping phenomenon predicted by the STCR model was also examined. Overall, this work introduces a new perspective on how triboelectric surface charges should be treated and demonstrates how the STCR model can explain the open-circuit voltage output characteristics of CS-TENGs.

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Introduction

Contact electrification, triboelectricity, and electrostatic induction are phenomena that have been documented for thousands of years.^1–3 When two dissimilar materials without any net charge come into contact, opposite charges are generated on the contact surfaces.^4,5 In 2012, a triboelectric nanogenerator (TENG) to convert mechanical energy into electrical energy was developed based on the coupling effect of triboelectricity and electrostatic induction.^6–9 For a contact–separation mode TENG (CS-TENG), a typical design features dielectric materials sandwiched between two electrodes. The dielectric layers contact and separate in the vertical direction relative to the contact surface.^10–12 The electric field produced by the transferred charges generated in the contact and separation process gives rise to an output voltage on the electrodes attached to the dielectric layers.^12,13 Given the critical role of this output voltage in TENG operation, researchers have extensively studied the output performance of TENGs by examining material properties,^14–19 geometric configurations,^19–23 motion characteristics,²⁴ and environmental conditions.^11,25–27 Over the past decade, TENGs have emerged as a prominent technology for energy harvesting across diverse applications, including wearable electronics,^28,29 blue energy,^30,31 and self-powered sensors.^28,32–34

The working mechanism of TENGs and their corresponding theoretical analyses are based on fundamental principles of classical electrodynamics.^10,12 There are two main processes affecting output voltage. One is the generation of tribo-charges on the contact surfaces^35–38 and the other is the subsequent electrostatic induction.^13,39 Researchers have proposed various theoretical frameworks to explain these processes and mechanisms, which fall into two main categories. The first widely adopted approach is the parallel-plate capacitor model, introduced by Z. L. Wang et al. in 2013,⁶ which treats the triboelectric charge as uniformly charged, zero-thickness and infinitely large planes during the derivation of the induced electric field and potential distribution. The electric field is assumed to be constant with respect to distance and oriented perpendicular to the charged planes in this parallel-plate capacitor model. This method of deriving the electric field and potential distribution induced by triboelectric charge might have overly idealized the charge distribution. The second approach, a quasi-electrostatic model for TENGs, was proposed by the same research group in 2019.^40,41 This model proposes the concept of a distance-dependent electric field (DDEF), accounting for finitely large charge planes to construct a three-dimensional electric field distribution model. Subsequently, a redefinition of the Gaussian surface based on the DDEF model provided a quantitative description of the electric field between two friction layers grounded in Maxwell's displacement current theory.⁴² According to Gauss's law, the DDEF model indicates that the electrostatic field is not uniformly distributed in the air gap of CS-TENGs, making the description of the electrostatic field distribution more specific and realistic. In previous studies, theoretical models and explanations of experimental results implicitly treated the contact-electrification charges as two-dimensional, uniformly charged planes with no thickness.⁴¹ However, in typical measurements of CS-TENGs, the voltage and current output often exhibit an alternating current (AC) pattern, showing output voltages in opposite directions during the contact and separation stages. The traditional surface charge density model without thickness cannot adequately explain this AC pattern in voltage during the working cycle of the TENG. To develop a more accurate explanation of the TENG working mechanism and to better understand the nature of triboelectric charge, it is crucial to propose a more precise model of triboelectric charge.^{2,5,12,43–46} Therefore, we present a new STCR model that more accurately describes the distribution of triboelectric charge. Based on the STCR model, we also introduce an ECP model to facilitate theoretical analysis of the electrostatic mechanisms in CS-TENGs.

Our STCR model posits that the triboelectric charge on the contact surface is randomly distributed within two STCRs of finite thickness. In CS-TENGs, the friction layers are typically made of insulators.^6,10,36 As the movement of electric charges is greatly restricted in insulators,⁴⁷ in the separation stage, the STCR remains beneath the contact surface after contact electrification. Thus, the STCR model is applicable only when triboelectric charges can retain their distribution beneath the contact surface, which means that it is only suitable for insulating solid dielectric materials as friction layers. By offering a more precise description of triboelectric charge on friction-layer contact surfaces, the STCR model also explains why CS-TENGs typically exhibit an AC output pattern.

In CS-TENGs, the electrostatic field generated by electric charge inside the STCR raises the voltage between the two electrodes attached to the outer side of each dielectric layer.^6–9 This voltage is typically measured by connecting the electrodes to an external circuit. Based on the STCR model, we propose an ECP model, which simplifies the random-depth distribution of triboelectric charge and allows the distribution of the induced electrostatic field and potential to be analytically calculated according to Coulomb's law and the superposition principle. Derivation using the ECP model offers a more specific description of how the electric field and potential are distributed from one electrode to the other, thereby enabling an analytical explanation of the observed output voltage patterns.

Theory and analysis

Formation and effects of STCR on the contact-electrification stage

The STCR, which describes the distribution of triboelectric charge, forms when two materials are brought into close contact under a certain pressure, as demonstrated in previous studies.^48–50 Under this circumstance, the materials are caused to exchange electrons, resulting in the spontaneous transfer of electrons from the higher occupied energy states of one material to the lower empty energy states of the other.^2,5,36,51 As illustrated in Fig. 1a, net positive charges and negative charges are generated due to the electron transfer in the contact stage.

Fig. 1 Schematic of the STCR and electrostatic analysis at its formation and stabilized stages. (a) Schematic of the STCR and randomly distributed tribo-transferred charge. Negative charge and positive charge result from transferred electrons.^2,5,46 (b(i)) Spontaneous transfer of electrons from one material to the other and (ii) stabilized STCR and balanced spontaneous and electric-field-driven electron transfer.

During the contact stage, the initial accumulation of net positive and negative charges in the STCR is believed to hinder further electron transfer and to stimulate electron movement in the opposite direction, thereby creating a negative feedback mechanism in this two-way transfer of electrons. Through this mechanism, a dynamic equilibrium state is eventually established, in which spontaneous electron transfer from higher occupied to lower empty energy states is offset by backward transitions driven by the electrostatic field in the STCR. This equilibrium is viewed as a balance between two transfer mechanisms: (i) spontaneous transfer, during which electrons transfer from higher to lower energy states between two dissimilar materials, as shown in Fig. 1b(i) and (ii) electric-field-driven transfer, during which electrons are pushed back by the electric field generated by the positive and negative charge in the STCRs, as illustrated in Fig. 1b(ii). In the contact stage, this balanced state is reached in an extremely short period, primarily because the triboelectric charge density is orders of magnitude smaller than the total number of electrons in any given material.^1,36,52

By employing this STCR model, a qualitative description can be provided to illustrate how the two transfer processes attain balance and how the resulting net positive/negative charges inhibit further accumulation of similar charges. This approach to describe the distribution of triboelectric charge may offer valuable insights for future studies regarding the limitations of charge density at friction-layer interfaces.

In addition to the random distribution of triboelectric charge on the contact surface, the random distribution of depth beneath the defined contact surface has also been observed due to surface morphology and roughness.^1,53,54 As illustrated in Fig. 2a, the STCR is depicted as a cuboid under the contact surface, at the minimum thickness that encompasses all randomly distributed charges beneath the contact surface. In order to provide a quantitative description of the electrostatic field generated by each electric charge, the distance from a charge carrier to the contact surface is denoted as h. Within the friction layer, the distance from any point in the dielectric layer to the contact surface is denoted as r. The induced electric field intensity of each triboelectric charge, E_ind, can be expressed as:

where k and e denote the electrostatic force constant and the amount of charge carried by an elementary charge, respectively.

Fig. 2 Schematics and simulations of random-depth distributed charge and ECP. (a) 3D schematic of the triboelectric charge distribution at random depth beneath the contact surface in STCRs. (b(i)) Charge distribution at random depth in STCRs and (ii) in the ECP with an average depth of charges. (c(i and ii)) Simulation of electric potential distribution of randomly distributed charges at an average depth of 5 nm and surface charge density of 1.6 × 10⁻⁴ C m⁻², with a dielectric-layer thickness of 20 nm and 200 nm, and (iii and iv) simulation of the electric potential distribution of an ECP at the same charge density using the average depth of the randomly distributed charges, for dielectric-layer thicknesses of 20 nm and 200 nm.

It can be observed that when calculating the electric field intensity induced by each point charge, as the distance from the contact surface becomes sufficiently large, the induced electric field intensity E_ind becomes less dependent on the buried depth h of the charges and more dependent on the distance r from the contact surface. Compared to the thickness of the dielectric layers in TENGs—typically on the order of micrometres to millimetres—the variation in h can be considered negligible. Consequently, a new model, referred to as the ECP model, is proposed to mitigate the impact of the randomness of the buried depth h when calculating and quantitatively analysing the electrostatic potential induced by such tribo-transferred charges, as illustrated in Fig. 2b.

As the buried depth of the triboelectric charge is much smaller than the thickness of the dielectric layers, the charge distribution is approximated as an ECP. The ECP approximation replaces the actual distribution of different h values with their average value and thus, avoids randomness in h and conserves the fact that the triboelectric charge is distributed at various depths below the contact surface. As no prior research has definitively demonstrated how triboelectric charges are distributed in terms of depth at the contact surface, it is assumed that the buried depth distribution follows a normal distribution, making the average buried depth approximately half the maximum depth. Thus, the ECP is positioned at the midpoint of the STCR, as depicted in Fig. 2b(ii).

The electric field and potential induced by the triboelectric charge originally arise from individually distributed point charges. According to the superposition law, the overall electric field induced by triboelectric charges is treated as a sum of the electric fields induced by each individual electric charge inside the STCR. Consequently, at any point inside the dielectric layer, the magnitude of the electric field is considered to be inversely proportional to the square of the distance (∼r⁻²) from the STCR. When the induced electric field and potential of the ECP are calculated, this same ∼r⁻² dependence is retained. As shown in Fig. 2c, the potential distribution generated using randomly distributed point charges is approximately identical to that produced by an ECP when the dielectric-layer thickness is 10–20 times greater than average buried depth of triboelectric charges. Further details regarding the evolution of the electric potential distribution with increasing dielectric-layer thickness are provided in Note S2 of the ESI.†

By introducing the ECP, an analytical framework is established for analytically calculating the electric field and potential distribution of CS-TENGs.

Electrostatic analysis of the contact–separation cycle of TENGs

With the introduction of the ECP, a quantitative analysis of the electric field and potential distribution generated by the electric charges within the STCR can be performed. Inside the friction layers, these charges are assumed to induce an electrostatic field in a homogeneous dielectric medium, allowing Coulomb's law and the superposition principle to be applied to give an analytical calculation of electric field intensity. It should be noted that the analysis below is based on a stabilized triboelectric charge distribution, which means that the electrification stage is completed and the charge distribution remains stable inside the STCRs. The region that encloses all the randomly distributed charges, known as the STCR, has a thickness that is denoted as d. The electric field at a distance r away from the contact surface can be obtained:

In this expression, r₁ and r₂ are used to represent the distances to the contact surface inside dielectric materials 1 and 2, respectively, while d₁ and d₂ denote the thicknesses of the STCRs in those materials. The dielectric constants for materials 1 and 2 are indicated by ε₁ and ε₂, respectively. The symbols E₁ and E₂ refer to the absolute values of the induced electrostatic fields within each material. The surface charge density of the ECP on both sides of the STCRs is represented by ρ, which is defined as the sum of electric charges divided by the contact surface area. As illustrated in Fig. 1b, when forward and backward electron transfer reach a stable state, no further net charge movement is present inside the STCR. Consequently, the potential is considered to be uniform throughout the STCR, and a 0 V reference potential is assigned to it. A schematic of the electric field intensity is shown in Fig. 3a. Owing to the ∼r⁻² relationship and the mutual weakening effect of positive and negative STCRs—each inducing an electric field in the opposite direction—the overall electric field intensity decreases rapidly with increasing r. As a result, significant electric field values appear in proximity to the STCR, whereas values approach zero at larger distances. This distribution agrees well with the simulations mentioned in Note S2 of the ESI.†

Fig. 3 Schematic of STCRs and their electric field and potential distribution in the contact stage.⁴⁶ (a(i)) Schematic of STCRs and electric fields induced separately by positive and negative STCRs. Positive indicates the direction pointing right and negative pointing left; (ii) schematic of the overall electric field intensity at the contact stage. (b(i)) Schematic of the potential induced by STCRs and the output voltage V_contact observed by an external circuit; (ii) schematic of the electric potential distribution at contact stage.

By integrating the electric field intensity from eqn (2) and (3) over r, the electric potential at the boundary of the dielectric layers can be obtained, which is subsequently measured by conductive electrodes:

The voltage shifts between the contact surface and the border of the dielectric layers are denoted by V₁ and V₂, respectively. The thicknesses of dielectric layers 1 and 2 are denoted by t₁ and t₂, respectively. A schematic of the electric potential distribution induced by the triboelectric charge inside the STCR is shown in Fig. 3b. During the contact stage, the induced potential detected by the electrodes can be treated as an equivalent voltage source, generating a positive voltage on the negatively charged side and a negative voltage on the positively charged side, as illustrated in Fig. 3b(i). The voltage between two electrodes at the contact stage, V_c, can be calculated as:

In previous research, the voltage output at the contact stage has been treated as “baseline drift” or attributed to charge accumulation.^23,54–56 In conventional models, V_c is neglected, because according to Gauss's law and Coulomb's law, the positive and negative charge planes would overlap and produce a zero electric field outside their friction area. However, this apparently contradicts the phenomenon observed in several experimental works.^23,57,58 Our STCR model gives a clear explanation, supported by theoretical derivations, as to why a voltage output exists in the contact stage with theoretical derivations.

When the dielectric layer is sufficiently thin, the electric field induced by each single electric charge within the STCR remains significant at the electrode interface, as shown in Fig. 2c(i). Under this circumstance, carriers of opposite charge on the conductive electrodes can be drained into the STCR and recombine with the triboelectric charges, thereby reducing the overall charge density as equilibrium is reached during the contact stage. This phenomenon has been observed experimentally; below a certain threshold thickness, thinner dielectric layers lead to a lower TENG open-circuit voltage output.⁵⁷ The STCR model clarifies why carrier recombination happens when the dielectric-layer thickness falls below a certain threshold, in alignment with the previously recognized mechanism of carrier recombination in ultra-thin layers. Moreover, the electric field distribution derived from the STCR model indicates that carrier recombination at low dielectric thickness results from locally high electric field intensities, as shown in Fig. 2c(i) and (ii) and Note S2 in the ESI.†

When two dielectric layers separate, the stable STCRs formed during the contact stage remain beneath the contact surfaces.⁵⁹ Because the dielectric layers separate and move apart, the electric field and potential distribution induced by the STCRs change significantly. As illustrated in Fig. 4a(i), in addition to the electric field within the dielectric layers, the STCR also induces an electric field in the gap between the contact-charged surfaces.

Fig. 4 Schematic of STCRs and their electric field and potential distribution in the separated stage. (a) Schematic of electric fields induced separately by the positive and negative STCRs and the overall electric field. (b) Schematic of electric field intensity distribution. (c) Schematic of the electric potential distribution. (d) Schematic of overall direction of the voltage drop and observed output voltage from an external circuit.

In typical CS-TENG geometries, the dielectric layers are less than 1 mm thick, while the maximum separation distance between the two contact surfaces can reach several centimeters.^7,8,54,60 Consequently, it is assumed that the distance between the two dielectric surfaces, denoted as d_g, is orders of magnitude larger than the thickness of the dielectric layers, t₁ and t₂, and the thickness of the STCR, d. A qualitative schematic of the electric field distribution in the separated state is provided in Fig. 4a. The electric field within a dielectric layer at a distance r from the separated contact surface (on the same dielectric layer) can be determined by applying Coulomb's law:

When the two contact surfaces are separated by a distance d_g, the electric field in each dielectric layer is primarily induced by the STCR within that layer, and the influence of the oppositely charged STCR can be neglected due to the relatively large separation distance. This simplification is applied in eqn (7) and (8) because for typical TENG geometries, d_g ≫ r (in typical TENG geometries, d_g ≫ t ≥ r).^2,5,7,8,54 Under these conditions, electric field inside the dielectric layers can be simplified as:

In this analysis, the STCR is treated as an equipotential body, and thus, the electric field within the STCRs is considered to be zero. The electric field in the gap between the two separated contact surfaces, located at distance l from the positively charged plane, is derived using Coulomb's law:

When calculating the voltage between the two electrodes, the midpoint of the air gap between the positively and negatively charged STCRs is defined as the 0 V reference plane, as shown in Fig. 4c. Focusing on the positively charged side, the potential of the positively charged STCR, V_pg, is determined by integrating E_g from the contact surface to the midpoint of the air gap:

In a similar fashion, the potential of the negatively charged STCR, V_ng, is obtained by integrating E_g:

The potential difference between the positive and negative STCRs is thus given by:

Although eqn (14) presents the potential difference of the positively and negatively charged STCRs, the voltage measured by the electrodes is not V_g, but instead the potential difference at the outer borders of the dielectric layers. By integrating E₁ over r inside the dielectric layers, the potential differences between each STCR and its respective dielectric-layer border are written as:

Here, V_pd and V_nd represent the absolute value of the potential differences from the STCR boundary to the conductive electrode. By simultaneously considering eqn (14)–(16), the voltage between the two electrodes at the separation stage, V_s, can be determined in the way shown in Fig. 4c:

Accordingly, the voltage differences in the contact and separation stages, derived in eqn (6) and (17), allow the V_OC to be expressed as the sum of voltage shifts during the contact stage, V_c, and separation stage, V_s:

Taking into account the magnitude of each term in this expression (t_d ≫ d), a further approximation of V_OC yields:

Therefore, over each contact–separation cycle, the maximum output voltage difference between contact and separation, V_OC, is governed by eqn (19), which depends on the following parameters:

(1) Surface charge density of triboelectric charges, ρ;

(2) Permittivity of two dielectric materials, ε₁ and ε₂;

(3) Thickness of dielectric layers, t₁ and t₂;

(4) Thickness of STCR, d₁ and d₂.

The analysis and derivations above are based on the assumption that the dielectric layers are homogeneous and isotropic, and that the thickness of dielectric layers is magnitudes of order larger than the thickness of the STCRs. The electric field and potential distribution can be much more complex in different material systems, which will require more detailed analysis according to the specific devices.

The electrostatic analysis in this article only accounts for the electrostatic potential change caused by the geometrical redistribution of the CS-TENGs in each contact–separation cycle. Another important source of the output of CS-TENGs is the displacement current caused by dynamic motions,^42,61,62 which is beyond the scope of this article, but our STCR and ECP model, with its corresponding electrostatic analysis, should provide a meaningful supplement to explaining the output performance of CS-TENGs.

Additionally, the STCR and ECP model are presented for describing the existence of triboelectric charges in a more realistic way. Although various types of TENGs exist, such as TENGs with solid–solid or solid–liquid contact modes,^16,31,63 the STCR and ECP models are theoretically available for analysing all kinds of TENGs that use insulating solid materials as friction layers, but the electrostatic field and potential distributions require case-specific reanalysis for each TENG structure.

In experimental section, the relationship between dielectric layer thickness and working cycle voltage was examined, and the same trend predicted by this model was observed.

Results and discussion

Formation of STCR

The formation of STCRs is attributed to a strong overlap of electron clouds at the surfaces of two dielectric layers.⁴⁶ This overlap is facilitated when a certain stress is applied to the contacting surfaces, allowing atoms in one material to approach those in the other closely enough to enter the microscopic repulsive zone, thereby enabling triboelectrification at the contact interface.⁴⁹ In practice, friction layer surfaces typically exhibit irregular hills and valleys, so a sufficient amount of pressure is required for the STCR to form and remain stable.^2,35,54,64

As shown in Fig. 5a, a certain pressure is required for the two contacting surfaces to achieve sufficient contact to maximize the effective contact area. Before contact, no electrification occurs, as shown in Fig. 5a(i). When minor pressure is applied to the contact surface, some areas experience strong overlap of electron clouds and undergo complete electrification, while others may not approach close enough to initiate electrification, as shown in Fig. 5a(ii). As the contact pressure intensity continues to increase, the contact surface will finally reach a stage at which the entire possible contact area experiences strong overlap of electron clouds, and the effective contact area reaches its maximum value, as shown in Fig. 5a(iii).

Fig. 5 Sketches of the relationship between pressure and the formation of the STCR. (a) Sketches of surface topology: (i) surface topology before contact; (ii) surface topology at low pressure, defined as insufficient contact; (iii) surface topology under saturated pressure, in which the effective contact area reaches its maximum value. (b) Device sketch: (i) separated stage; (ii) contact stage. (c) Pressure–working cycle voltage relationship of the CS-TENG with FEP and PDMS as dielectric layers. The thickness of the PDMS layer is fixed at 0.5 mm, and the FEP layers are 15, 20, 50 and 80 μm. (d) Simultaneous measurement of the compression pressure and voltage output on two electrodes. The pressure peak occurs during the contact stage and stays at zero during the separation stage.

This phenomenon and the dependence of voltage on pressure have also been studied and explained in previous research.³ This section reports the measurement of a series of CS-TENGs. PDMS, a flexible and stretchable material with a relative permittivity of 2.7, was used as the negative friction dielectric layer (which gains electrons and produces a negative STCR during the contact stage), while FEP, a flexible material with a relative permittivity of 2.1, served as the positive dielectric layer (which loses electrons and produces a positive STCR in the contact stage). The two metal electrodes attached to the outer sides of the CS-TENGs were made of a conductive cloth tape, which was produced by coating a high-strength polyester fiber cloth with the highly conductive metals—copper and nickel. The thickness of conductive metal electrodes was controlled to be 200 μm to avoid the impact of electrode thickness on output performances.^12,23,54,64 To avoid the impact of surface area on the output performance of the TENGs,^12,19,54 the maximum contact area of these CS-TENGs was controlled to be 4 cm², shaped as 2 cm × 2 cm squares.

The relationship between the pressure intensity and V_OC of the CS-TENGs was then examined, as shown in Fig. 5c. Different lines represent results from CS-TENGs having FEP dielectric layers of different thicknesses. In each CS-TENG consisting of FEP dielectric layers with different thickness, the V_OC saturates at a similar pressure, as the pressure required to achieve the saturation is determined by the surface topology of and the electronic structures of materials.³⁹ Consequently, the thickness of the friction layer has a relatively minor impact on the saturation pressure intensity. This regulation occurs due to the fact that the saturation pressure intensity only causes the contact surface to achieve the maximum microscopic contact area, as shown in Fig. 5a, which has also been studied and explained in previous research.³Fig. 5b shows a schematic of the experimental setup, in which a linear motor attached to the end of the platform and a spring were used to control the motion during each contact–separation cycle. The motion speed provided by the linear motor of CS-TENGs was set to 1 m s⁻¹, acceleration was set to 10 m s⁻², and the maximum separation distance was set to 10 cm to avoid the impact of speed on the output performance of the CS-TENGs. The maximum compression pressure was regulated by attaching the PDMS side to a supporting plate with a spring and adjusting the maximum compression deformation of the spring, as illustrated in Fig. 5b. During each contact–separation cycle, the compression pressure intensity and output voltage from the two electrodes were measured, as shown in Fig. 5d. It was observed that at the moment of separation, a negative pressure peak occurred due to the stickiness of PDMS and the close contact between the PDMS and FEP surfaces. The V_OC reached its maximum value once the pressure intensity approached 2000 Pa, which was used as the saturation pressure for subsequent experiments.

Distinction of the contact and separation stage voltage and discharge current

As shown in Fig. 6a(i) and (ii), when the CS-TENG is connected to an external circuit, the polarity of the output voltage alternates within each contact–separation cycle. When a resistor is connected between the two electrodes to form a closed loop, the electric potential induced by the tribo-transferred charge can be observed by detecting the voltage across the resistor and the current flowing through the external circuit.

Fig. 6 Schematic of the closed-loop circuit discharge experiment. The electrodes of the CS-TENG are connected in series with a 100 MΩ resistor as an external circuit. (a) Schematic of the TENG as an equivalent voltage source with an external circuit in the (i) contact stage and (ii) separation stage. (b) Voltage and current plots over multiple contact–separation cycles. (i) Voltage output over multiple contact–separation working cycles. (ii) Current output over multiple contact–separation working cycles.

In the contact stage, electrons are driven by V_contact, as depicted in Fig. 6a(i), causing electrons to leave the electrode on the side of the negative STCR and enter the positive STCR. In the separation stage, electrons are driven by V_s, as shown in Fig. 6a(ii). As the voltage between the two electrodes reverses compared to the contact stage, the electrons then flow in the opposite direction. In this experiment, a 100 MΩ resistor was connected in series with the two electrodes, as shown in Fig. 6a, and the voltage and current output across the resistor were measured, as shown in Fig. 6b. The use of a resistor connected to an external circuit instead of open-circuit conditions ensured that there would be no net charge accumulation on the electrodes, thus avoiding the voltage baseline drift in each contact–separation cycle.^12,19,54 Qualitatively, it can be observed that in both our previous derivation and the experimental setup of a typical CS-TENG (Fig. 6a), the voltage and current during the contact and separation stages flow in opposite directions, consistent with the predictions of the induced electric field and potential derived from the STCR and ECP models. Alternating output voltage polarity is a common phenomenon observed in various studies.^2,7,54,60 Another strong piece of evidence supporting the voltage polarity shift phenomenon predicted by the STCR model is found in ref. 58, which demonstrates that current in opposite directions are generated at the moments of contact and separation, corresponding to the formation of V_c and V_s. An analytical explanation for this polarity-shifting voltage output is provided by our STCR model from an electrostatic perspective.

Effects of dielectric-layer thickness on working cycle voltage

In this experiment, the permittivity of the FEP layer, ε₁, the PDMS layer ε₂, and the thickness of the PDMS layer, t₂, were kept constant, and the thickness of the STCR, d, was considered to be constant. According to eqn (19), the relationship between the thickness of the FEP layer t₁, thickness of the PDMS layer t₂ and the output open-circuit voltage V_OC can be expressed as:

By varying the thickness of the FEP layer, while keeping the thickness of PDMS layer unchanged, as shown in Fig. 7a and b, the relationship between dielectric thickness t₁ and output open-circuit voltage V_OC was examined. In each measurement of the CS-TENG open-circuit voltage V_OC with different thicknesses of the FEP dielectric layer, the maximum pressure was set to 5 kPa to ensure sufficient contact between the two layers, allowing the STCR to form over the entire effective contact area during each working cycle. In a similar fashion, we also examined the relationship between the thickness of the PDMS layer t₂ and output voltage V_OC.

Fig. 7 Schematic of the potential distribution in each working cycle. (a) Contact stage: (i) schematic of TENG structure; (ii) potential distribution for different FEP thicknesses. (b) Separation stage: (i) schematic of TENG structure; (ii) potential distribution for different FEP thicknesses. (c) Relationship between the FEP layer thickness and open-circuit output voltage. PDMS layers were fixed at 500 μm thick. (d) Relationship between the PDMS layer thickness and open-circuit output voltage. FEP layers were fixed at 250 μm thick. (e) Fitting of the voltage–dielectric thickness relationship from data in ref. 56. (f) Relationship between various combinations of PDMS and FEP layer thicknesses and open-circuit output voltage.

According to eqn (20), the STCR model predicts that V_OC should be inversely proportional to t₁ and t₂. The experimental V_OC–t₁ relationship was fitted in the 20–100 μm range using the Allometric1 model. The Allometric1 model fits the data on the y-axis and x-axis using the equation y = a × x^b. According to eqn (20), the exponential part b of t₁ and t₂ in the V_OC–t₁ and V_OC–t₂ relationships should be −1. We fitted the relationship between V_OC and t₁ and that between V_OC and t₂ separately to analyze how each variable influenced V_OC. The fitting curve of the V_OC–t₁ relationship reveals an exponential component b with a value of −0.901 ± 0.015, as shown in Fig. 7c, and the V_OC–t₂ relationship reveals an exponential component b with a value of −0.952 ± 0.007, as shown in Fig. 7d. These results are in close agreement with our model's predicted value of −1. Additionally, we performed similar fittings on other experimental datasets; for instance, the exponential component in ref. 56 was found to be −0.971 ± 0.211, as shown in Fig. 7e. Additionally, the relationship between various V_OC and t₁/t₂ combinations is shown in Fig. 7f and Note S5 of the ESI.†

When the thickness of the FEP layer exceeded 20 μm, strong agreement was observed between the theoretical predictions and experimental results, demonstrating that V_OC is inversely proportional to t₁. However, when the FEP thickness was 15 μm, the working cycle voltage exhibited a sharp drop, which can also be explained using the STCR model.

As shown in Fig. 2c(i) and Note S2 in the ESI,† the electric field intensity inside the dielectric layer rapidly increases when approaching the edge of the STCR. When the dielectric-layer thickness becomes comparable to the thickness of the STCR, the electric field that reaches the electrode remains sufficiently strong to cause charge carriers of opposite net charge to be drained into the STCR. This results in a weakened charge density in triboelectric charges, which leads to lower V_OC. This phenomenon has also been observed in other studies,^57,65,66 and similar results were observed in our experiment, as shown in Fig. 7c.

Our STCR model provides a theoretical explanation for this phenomenon from an electrostatic perspective. It also confirms that an optimum thickness exists for any dielectric material used in CS-TENGs. According to eqn (19), V_OC consists of two components that are inversely proportional to the thicknesses t₁ and t₂ of the dielectric layers. This indicates that a thinner dielectric layer leads to a larger V_OC, as long as the STCRs remain stable during the whole working cycle. However, this inverse relationship diminishes as the electrodes approach the strong electric field zone induced by the randomly distributed point charges inside the STCR. As a result, a limit exists beyond which further reduction in the dielectric layer thickness no longer increases V_OC but instead cause a sharp decrease. This limit is expected to correspond to the optimum thickness for the dielectric layers in CS-TENGs.

Conclusions

This study proposed a STCR model to more explicitly describe the presence of triboelectric charges. The formation process of the STCR, which is associated with the dynamic balance of electron transfer in contacting dielectric materials, has been demonstrated. The electric field and potential distribution in the CS-TENGs, which are induced by the triboelectric charges in the STCR, have been derived through electrostatic analysis based on Coulomb's law, the superposition principle, and the ECP model. Experimental investigations examined the shifting of the output voltage polarity and the relationship between voltage and dielectric layer thickness as predicted by the STCR model.

The STCR model illustrates that treating the triboelectric charge as point charges with random buried depths beneath the contact surface provides a clearer understanding of the working mechanism of CS-TENGs. Additionally, the STCR and ECP models help to quantitatively explain the relationship between output voltage and dielectric layer thickness in CS-TENGs, while also offering a qualitative explanation for the polarity-shifting voltage output characteristics.

By introducing the STCR model, this work advances the understanding of the nature of triboelectric charge. Moreover, it provides a framework for quantitatively describing the electrostatic process during the working cycle of CS-TENGs. This approach contributes to a deeper understanding of triboelectric phenomena and paves the way for improvements in the performance and future development of TENGs.

Data availability

The data supporting this article have been included as part of the ESI.†

Author contributions

X. C. Chen conceived the idea, designed the experiments, processed the data, and wrote the manuscript. H. Wu and X. Y. Chen discussed the experimental results and reviewed the manuscript. J. Li and W. Ou-Yang supervised the project and the manuscript. All authors discussed the results and contributed to the manuscript.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was financially supported by National Natural Science Foundation of China (Grant No. 61771198), Natural Science Foundation of Chongqing, China (Grant No. CSTB2024NSCQ-MSX0632) and Natural Science Foundation of Shanghai Municipality, China (Grant No. 17ZR1447000 and 25ZR1401101).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5ta01136a

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