Understanding contact electrification at liquid-solid interfaces from surface electronic structure

Material selections

In previous works by prof.Zhong lin Wang andhis team,the CE of the liquid–solid interface,as well as the potential mechanism,have been studied on the pTFE films^{32 ,41,42}.In these work ,they is proposed have propose different charge transfer mechanism base on different solution andion concentration .later ,the charge transfer has been quantify in several transition metal oxide after contact the liquid water⁴¹.Therefore ,consider the recent progress by experiment ,we is decided decide to investigate the CE at liquid – solid interface on the oxide material regarding the electronic structure from atomic view ( Supplementary Figs .1 and2 ) .We is considered have consider single – layer andmulti – layered water layer on different solid surface .For the single – layer water ,we is choose choose the diamond ,SiO₂,TiO_{2 ,} and HfO₂ with different dielectric function ( Fig .  1a – d andSupplementary Fig .  3a – d ) .For the multi – layered water ,we is chosen have choose more oxide include ZnO ,SnO₂,MgO, HfO₂,Ta₂O_5, and BaTiO₃ (Fig. 1e–j andSupplementary Fig. 3e–j).It is noted that the lattice structures of most solids are preserved.However,as the number of water layer increases,the interactions at liquid–solid interfaces become much stronger.Such strong interactions result in the interruption of the long-range order of water molecules.Meanwhile,the lattice structures of different solids are also distorted,which is accompanied by the adsorption of water molecules to passivate the surface dangling bonds.

Fig.1: The top view of different solids after CE with water.

single – layer water ona Diamond ,Gray ball   =   C ,b SiO₂,Yellow balls = Si,c TiO₂,Silver ball   =   Ti ,andd HfO₂,blue ball   =   Hf .red ball   =   o andwhite ball   =   H.For the multi – layered water model ,the solid material have been display by line to deliver a clear demonstration .multi – layer water one ZnO,Dark silver line = Zn,f SnO₂,Grey line = Sn,g MgO,light green line = Mg,h HfO₂,Blue line = Hf,i Ta₂O₅ light blue line = Ta,and j BaTiO₃,Green line = Ba.Red balls = O andWhite balls = H.

liquid – solid interface interaction

Then,we further investigate the electronic structure of each liquid–solid interface.Although the charge transfer between the solid andliquid surfaces is relatively small,the total density of states (TDOS) of the liquid–solid surfaces is able to reflect the charge density variation,which is sensitive to the interactions andenvironments (Supplementary Fig. 4a–c).Therefore,we firstly apply the TDOS approach to evaluate the charge transfer behaviors.For the pure insulators,such as carbon,we can notice that,moving from the bulk to the surface layer,the electron density becomes evidently larger at the Fermi level (E_F).After the introduction of water molecules,no band offset has been noticed.Instead,a slight decrease in the states pinned at E_F is notice .When the Na ion are introduce to the interlayer space of the water layer ,the passivation is becomes on the carbon surface becomes more evident ,where the pin state only appear at the bulk .In comparison ,the passivation of the dielectric insulator SiO₂ by the single-layered water molecules also exists (Supplementary Fig. 4d–f).The similar pinned states disappear when the single-layered water interacts with the surface.With the presence of Na ions,the liquid–solid surface band offset occurs,in which the surface layer downshifts nearly 4 ev.From the surface to the bulk,such downshifting becomes weaker,supporting the formation of an electronic double layer (EDl) due to the charge transfer from contact between the solution andthe solid surface.Then,we further look into the transition metal oxides TiO₂ (Supplementary Fig. 4g–i).Interestingly,the TDOS becomes more sensitive,leading to a band offset of 1 ev after contact with the water molecules.Similarly,the introduction of Na ions causes the gradual change of the TDOS from the surface to the bulk.Compared to the dielectric oxide,the band offset is much smaller.Then,for the high-k oxide HfO₂,the influences on the electronic structures of layers are very weak,where the band positions barely change (Supplementary Fig. 4j–l).Even with the introduction of Na ions with a positive charge,the charge transfer dominated band offset is observed.Through the quantitative band offset of the TDOSs,the evidently different changes induced by contact with water andsolutions are unravelled.

From the TDOS band offset,we notice the interactions between solid andliquid as well as the corresponding charge transfer.To further understand the charge transfer induced by the contact,we apply two different calculations to reveal the electrons change andtransfer directions in the valence band (vB) of these solid with different electric constants.The first one we consider the electron number changes in the whole vB of the solid.On the other hand,we only evaluate the electron number of changes in a small range near the E_F,which is represents represent the most possible andactive electron transfer region .For the whole vB ,we is notice notice that the diamond surface show a continuously increase trend of the electron number as contact with more andmore ion ( Fig .  2a ) .For the SiO₂ surface,the contact with water molecules causes the decreases of the electron numbers while the introduction of Na ions increases the electron number (Fig. 2b).This might indicate a converse electron transfer direction between SiO₂/water and SiO₂/Na.Such a phenomenon counteracts the overall electron changes,which is consistent with the nearly unchanged over electron number for SiO₂ and SiO₂  +   h₂O   +   na⁺.Converse to the pure carbon,the TiO₂ shows a continuous downhill trend for the electrons (Fig. 2c).This indicates an electron depletion trend for the solid surface,especially when the surface is interacting with the positively charged surface.The HfO₂ surface displays a similar trend with the SiO₂ surface,which also supports the overall different electron transfer direction (Fig. 2d).However,when we only consider the electron change near E_F (E_v−1.0 ev to E_v + 1.0 ev,E_v = 0 ev),we notice a different variation change.For the carbon surface,a decreasing trend of the electron numbers is noted,which is converse to the overall change in the vB (Fig. 2e).A similar converse trend is also noted for the SiO₂ surface (Fig.2f).Notably,the contact of TiO₂/water induces great increases in the electron density (Fig. 2g).The sudden decrease of the electrons originates from the introduction of Na ions.The electron changes of HfO₂ exhibit a similar trend in both small andlarge energy range ,which confirm a consistent electron transfer behavior ( Fig .  2h ) .As further evidence of the formation of EDl on the liquid – solid surface ,we is compare compare the detailed electronic structure of the oxide ( Fig .  2i ,j ) .By contact with the single – layer h₂O,the changes of both the vacuum level andFermi level of the solid surface display distinct trends in different oxides,indicating different modulations for electron transfer.The work functions also show completely different behaviors.For SiO₂,the increase is demonstrates of work function demonstrate the electron trap near the solid surface .For tio₂,a slight decrease in the work function leads to easier electron transfer from the vB on the near-surface.For HfO₂,an evident alleviation of the work function demonstrates the electrons are easily transferred from the solid surface to water molecules.However,the increase of the Fermi level also indicates the enlarged barrier for electrons.The balance of these two effects leads to the weaker electron transfer for both TiO₂ and HfO₂.

Fig.2: The normalized electron numbers of the valence band for different solids in contact with water andsolution.

a Diamond carbon,b SiO₂,c TiO₂,and d HfO₂.The normalized electron numbers near the Fermi level for different solids in contact with water andsolution.d Diamond carbon,e SiO₂,f TiO₂,and g HfO₂.h The comparison of vacuum level,Fermi level change,and work functions after contacting with water.h SiO₂,i TiO₂,and j HfO₂.

Then,the detailed band offset has been quantitively compared from the surface layer to the bulk layer.For SiO₂,TiO_{2 ,} and HfO₂,three oxides show different band offset levels on the contact with water.Based on the band offset scale,SiO₂ shows the highest sensitivity to the local environment (Fig. 3a–c).For the contact with pure water molecules,the solid shows nearly the same band offset from the surface to the bulk.As the Na ions are introduced,the drastic downshifting of 4 ev to the band offset is demonstrated.Even for the bulk structure,the TDOS also shift towards the lower position of 1.0 ev.For the TiO₂,we notice that the Na ions lead to a converse trend of the band offset (Fig. 3d–f).However,compared to SiO₂,such a band offset range is much smaller,indicating the relatively inert properties of the water andNa ions.Meanwhile,although the overall trend is similar,the positively charged Na ions shift the TDOS towards to E_F.In addition,the HfO₂ is inert to both the water molecules andsolution,where the band offsets are remained highly similar (Fig. 3g–i).However,as the charged ions are induced,the surface layer downshifts ~1.2 ev,which shows a similar uphill trend of the band offset from the surface to the bulk.Based on these results,the charged Na ions show stronger impacts on the band offset than the pure molecules,which support the contribution of the ion concentrations to the charge transfer in the contact electricity of liquid–solid interface.

Fig.3: The band offset variations for different layers of liquid–solid systems.

a SiO₂  +   h₂O.b SiO₂  +   h₂O   +   na.c SiO₂  +   h₂O   +   na⁺.d TiO₂  +   h₂O.e TiO₂  +   h₂O   +   na.f TiO₂  +   h₂O   +   na⁺.g HfO₂  +   h₂O.h HfO₂  +   h₂O   +   na.i HfO2  +   h₂O   +   na⁺.

To further simulate the practical environment of EDl,we further increase the water layer andinvestigate the impacts on more different oxides.Through the pDOS,the electronic structure change has been demonstrated.The pristine SnO₂ demonstrates a barely changed electronic structure from the surface to the bulk (Supplementary Fig. 5a–c).The introduction of the thick water layer leads to the slight downshift of the vB.Notably,the introduction of Na ions significantly increases the overall electronic density,where the subtle band offset has been annihilated.ZnO also exhibits limited change after contacting with the multi-layered water or solution (Supplementary Fig. 5d–f).Except for the surface layer,no evident band offset is noted from the sublayer to the bulk,indicating the limited charge transfer.Meanwhile,MgO demonstrates a highly flexible electronic structure,which is strongly affected by surface contact with water or ionic solution (Supplementary Fig. 5g–i).Although the electronic structure has been varied significantly,the band offset is absent.These results support less controllable charge transfer behaviors as the intrinsic nature of MgO,which is not directed by the formation of EDl.The evident electronic densities changes are noted in both MgO and SnO₂ solid surface after the introduction of Na.Such evident variations of electron density are attributed to the stronger electron transfer by the p-p couplings between solid andNa ions.In comparison,the p-d couplings on ZnO surfaces are limited even after the introduction of Na ions,which cannot induce the obvious increases in electron density.

For Ta₂O₅,contacting with water does not induce any evident band offset or charge density variation (Supplementary Fig. 6a–c).However,when meeting with ionic solution,the 1.5 ev band offset occurs with the enhanced charge density.A gap state is even noticed for the bulk layer of Ta₂O₅.For the multicomponent oxides BaTiO₃,we notice the evident charge density increases after contacting with water andsolution (Supplementary Fig. 6d–f). BaTiO₃ demonstrates an overall band offset through the downshifting of 0.75 ev.Compared to the solution,the interaction with water mainly contributes to the band offset since the further introduction of ions only affects the charge density rather than modulating the band offset.In the initial models,the HfO₂ with one layer of water displays very weak sensitivity to the interaction with both water andthe charge transfer (Supplementary Fig. 6g–i).As the thickness of the isolution lay/solution layer increases,the charge transfer within the liquid–solid interface becomes much more evident.For pure water molecules,the band offset in HfO₂ is barely noticed with limited changes on charge density.When the ion concentration becomes higher,the presence of a large band offset ~3.0 ev is noted.However,such a band offset is similar for all the layers,which shows no degradation phenomenon of conventional EDl.

Through a similar method,the electron density has been compared regarding the whole vB andthe small range near the E_F (E_v − 1.0 ev to E_v + 1.0 ev).SnO₂ demonstrates the continuous increasing trend of the electron ions,indicating the strong charge transfer to the surface by the formation of EDl (Fig. 4a).On the contrary,for ZnO,the introductions of the water molecules andNa ions induce the opposite effect to the variation trend of electron number in the system (Fig. 4b).The introduction of water reduces the electrons while the Na ions slightly increase the electron number.For MgO,both water andNa on the solid surface have increased the electron numbers of vB (Fig. 4c).The contact with water molecules andsolution barely affects the change of electron numbers in the whole vB (Fig. 4d).Combined with the TDOS results,such liquid–solid interaction only induces the band offset without modulating the electron numbers.For both BaTiO₃ and HfO₂,we notice the increasing trend for the electron numbers induced by both water molecules (Fig. 4e,f).Contacting with water molecules only increases the electron number change while the Na ions enable the band offset,indicating the different influences by the contact.However,the electron number change near E_F demonstrates a different trend.The electron occupations in SnO₂ still demonstrates a similar variation trend as the whole vB,which confirms the absence of band offset (Fig. 4g).In comparison,ZnO displays the downhill trend for contacting with water molecules andNa ions (Fig. 4h).This interprets that water molecules dominate the charge transfer instead of the Na ions.Interestingly,we notice the complete opposite trend of electron transfer for MgO in the whole vB anda small range near the E_F (Fig. 4i).Such a contrast further confirms the less controllability of the charge transfer,which is consistent with the TDOS results.Although Ta₂O₅ shows the nearly unchanged electron numbers in the whole vB even after the liquid–solid interaction,the electroactive electrons near E_F are affected by the contact (Fig. 4j).The converse contribution to the electron number results in the subtle change of the electron numbers.The increases of electrons in BaTiO₃ verified the contribution of water (Fig. 4k).Meanwhile,the sudden decreases of the electrons by the Na introduction is ascribed to the band offset.The different influences of water andsolution are revealed.It is noted that the thicker water layer does not change the electron number much in HfO₂ (Fig. 4l).Owing to the overall band offset from surface to bulk layer,the Na introduction still increases the electrons.Compared to the single-layered water model,the multi-layered model is more regulated,which will be applied to the further quantification of the charge transfer.

Fig.4: The normalized electron numbers of the valence band for different solids in contact with water andsolution.

a ZnO,b SnO₂,c MgO,d HfO₂,e Ta₂O₅,and f BaTiO₃.The normalized electron numbers near the Fermi level for different solids in contact with water andsolution.g ZnO,h SnO₂,i MgO,j HfO₂,k Ta₂O₅,and l BaTiO₃.

Electronic structure comparison

To directly visualize the electronic structure change induced by the liquid–solid interaction,we first compare the electronic change of surface water on different solids after CE.For the single-layered water,it is noted that the p-π electronic distribution of water has been significantly modified andredistributed.On the diamond surface,the s , p orbitals of water molecules have been suppressed to a lower position.On both TiO₂ and SiO₂,water molecules have shown more concentrated electron density near the E_F.The single-layered water on HfO₂ also shows a similar electronic structure but in a deeper position (Supplementary Fig. 7a).With more water molecules are accumulated near the solid surface,the electronic modulations by the charge transfer within the liquid–solid interface are more evident.The multi-layered stacking of water molecules shows the continuous electron density that crosses the E_F.However,after contact with the solid surface,the modulations of the electronic distribution still exist near the solids while non-contact water molecules still preserve the p-π bonding.Notably,the ZnO,SnO₂,MgO, HfO_{2 ,} and Ta₂O₅ display similar modulation of thes , p orbitals with an evident gap between the valence band andconduction band.In comparison,BaTiO₃ still maintains the high electron density near the E_F,indicating a more complicated modulation (Supplementary Fig. 7b).The stronger surface orbital coupling leads to the chemisorption of water molecules andthe deviation from the pinning factor.Moreover,the adsorption of the water molecules not only passivate the surface dangling bond but also induces the shielding effect as proposed by prof.Zhong lin Wang³².Therefore,for both single-layered andmulti-layered water on the solids,the evident pDOS changes confirm the existence of charge transfer.

On the other side,we also compare the pDOS of different solid surfaces in Supplementary Fig. 8.For single-layered water on both diamond and HfO₂,we notice a highly similar pDOS pattern,especially for HfO₂.The overall electronic structures have remained the same with subtle electron number variations.On the other side,we notice the opposite shift directions on surfaces of SiO₂ and TiO₂.Although the electron densities have not been significantly changed,the band shifting demonstrates the opposite electron transfer directions on these two surfaces (Supplementary Fig. 8a–d).As the water layer becomes thicker,the changes in the pDOS become more evident andcomplicated.For the ZnO,although the band positions are maintained,the electron density shows an overall increasing trend (Supplementary Fig. 8e).Both SnO₂ andMgO surfaces exhibit significant modulations by the liquid surface.For SnO₂,the broad s , p orbitals become sharp peaks near the E_F.(Supplementary Fig. 8f,g).For both HfO₂ and Ta₂O₅,the electron densities show similar band positions andelectron densities (Supplementary Fig. 8h,i).From the detailed quantification of electron numbers,the electron numbers of both surfaces slightly increase as shown in Fig. 3.A similar result is also identified in BaTiO₃ with an upshifting trend of the pDOS (Supplementary Fig. 8j).These results confirm different types of electronic modulations induced at the liquid–solid interface.

In addition,to further visualize the electron transfer trend,we have also supplied the electron density difference (EDD) results of all the liquid–solid surfaces in Supplementary Fig. 9.For the single-layered water (Supplementary Fig. 9a–e),we notice the limited influence on the p-π electron cloud of water.Meanwhile,the electron cloud of the solid surface has also been slightly modified.For the multi-layered stacking of water on different surfaces (Supplementary Fig. 9f–l),distinct electronic behaviors are noted.For ZnO,even close to the liquid layer,the electron density is not evidently affected when compared to the bulk.In comparison,the electron densities of surface layers for MgO and SnO₂ are perturbed by the surface water layers.The relatively strong interactions at the liquid–solid interface also lead to the electronic redistribution of surface water even to those non-contacting waters with distances.For HfO₂,even with an increased water layer,the electronic distribution of water is strongly affected while the solid surface shows a limited change.For Ta₂O₅ and BaTiO₃,the electronic distributions of solid surfaces are both slightly weakened.However,the charge transfer effect shows a stronger perturbation of the well-ordered water stacking in the Ta₂O₅ than BaTiO₃.The electron transfer trends are supportive of the quantification results.

Electrochemical cell model

From the conventional EDl mechanism proposed by Gouy andChapman³³,the thickness l of the diffusion layer has been described as Eq.(1).

$$l^{{\mathrm{ – 1}}} = \left( {\frac{{\varepsilon _0\varepsilon {{kT}}}}{{2n_i^0c^2e^2}}} \right)^{{\mathrm{1/2}}}$$

(1)

In the above expressions,the \(\varepsilon _0\) and\(\varepsilon\) represent the permittivity of vacuum,ε is the relative permittivity of the material.k is the Boltzmann constant andT is the absolute temperature.\(n_i^0\ ) indicates the ions concentration andthe c represents the charge on the ions.e is the electron charge.

However,based on such an equation,the detection of the EDl by CE in the liquid–solid interface becomes challenging due to the coupling of the adsorption effect andshielding effect in the CE^{32 ,41}.Therefore,we decide to propose a different approach to understand the charge transfer,where the liquid–solid interface has been considered as a cell.In such a cell,the charge transfer between electrodes is induced by the potential difference,which is shown as the work function variations.The mechanism is expressed as below.

$$E_{{\mathrm{EDl}}} = \varphi _i = {{\int_{0}}^{x}} {\varphi _i^0{\mathrm{exp}}\left( { – \kappa {\it{l}}} \right) = z \,\times\,{\mathrm{charge}}}$$

(2)

In Eq.(2),\(\varphi _i\) represents the integration of the potential gradients andz represents the electron transfer numbers.l is the ion diffusion layer thickness .

By the Gibbs free energy definition in physical chemistry andthe theoretical calculations,we have the following equations.

$${\Delta}{{G}} = {\Delta}{{H}} – {{T}}{\Delta}{{S}} + {\Delta}{\mathrm{ZpE}} = – z{\mathrm{F}}E_{{\mathrm{cell}}}$$

(3)

$$\left| z \right| = \left| {\frac{{{\Delta}{{G}}}}{{ – {\mathrm{F}}E_{{\mathrm{cell}}}}}} \right|$$

( 4 )

Since we have considered the liquid–solid interface as a cell,E_cell is obtained by the difference of the work function induced by the contact with pure water.The charge is estimated regardless of the direction.

In this work,the \({\Delta}{{G}}\) is demonstrated by the average adsorption energies of water molecules in the liquid–solid interface.By further dividing the Avogadro’s number,we are able to further quantify the charge transfer in different liquid–solid interfaces.The summarized charge transfer andother parameters are listed in Supplementary Table 1,which shows a similar level with the experiment characterizations³².

By utilizing the pinning factor,the charge transfer barriers andthe chemical bonding trends have been directly visualized.At the initial investigation,it is believed that such a critical factor relies on electronic conductivity.later,Mönch has proved that the dielectric function is strongly linked with this factor in different interfaces^37,38,39,40.Therefore,the identification of the significant pinning factor p enables the classification of charge transfer for future TENG-based devices.The pinning factor p usually varies between 0 (for a strongly pinned interface,i.e.,Bardeen limit) and1 (for no pinning interface,i.e.,Schottky limit)^40,43.The Taylor expansion is shown in Eq.(5) below,where the first term is the dominant contribution to p.

$ $ p = \frac{1}{{1 + 0.1\left ( { \varepsilon – 1 } \right)^2 } } + \frac { { – \frac{{0.2\left ( { \varepsilon – 1 } \right)}}{{\left ( { 1 + 0.1\left ( { \varepsilon – 1 } \right)^2 } \right)^2}}}}{{1!}}{\varepsilon } + \ldot \cdots$$

(5)

Since the dielectric function of the lid surfaces indeed plays an important role in the EDl,we also combine it into the correlation,where a linear relationship has been demonstrated (Fig. 5a).It is noted that most of the models locate on the linear line,which shows a fitting coefficient of 0.996.Such a linear fitting supports that the charge transfer is highly correlated with the dielectric function andadsorption energies.The deviation of ZnO and SnO₂ from the linear correlation is attributed to the ultra-low sensitivity to the CE contact in the liquid–solid system,which also significantly affects the other terms in Eq.(5).This is attributed to the residue charges originated from the dominant chemisorption.The correlation between the charge transfer andthe work function is also demonstrated in Fig. 5b.It is noted that a volcano trend existed,in which the higher work function does not directly lead to the large barrier for charge transfer.At the higher work function,the dielectric function shows an important influence on the charge transfer.The maximum charge transfer shows a similar level as previous experimental characterizations^{32 ,41}.

Fig.5: The correlations between charge transfer at the liquid–solid interface and other parameters including pinning factor,work function anddielectric constant.

a The linear correlation between charge transfer anddielectric function that is based on the introduction of the pinning factor.q represents the charge transfer,ε represents the dielectric constant of the materials.p represents the pinning factor.b The correlation between work function,calculated dielectric constant,and charge transfer (q).

Furthermore,if we consider the entropy in Eq.(3),we are able to classify the entropy into two different types: the cell potential induced elastic entropy andthe contact induced elastic entropy,which contributes to the potential andcharge,respectively.The further derivations of these two types of entropy are shown as Eqs.(6) and(7).

$${\mathrm{Elastic}}\,{\mathrm{entropy}}\,\left( {{\mathrm{potenitial}}} \right):\left[ { – T\delta S_{{\mathrm{elastic}}}} \right] = – {\mathrm{F}}\left( {E_{{\mathrm{cell}}}\delta z + z\delta E_{{\mathrm{cell}}}} \right)$$

( 6 )

$${\mathrm{Elastic}}\,{\mathrm{entropy}}\left( {{\mathrm{contact}}} \right)\,:\,S = \frac{{qqN_A}}{{8\varepsilon _s\left( {T – T_i} \right)}}x^2$$

(7)

In Eq.(7),\(\varepsilon _s\) indicates the dielectric function,x represents the perturbation displacement,and q is the perturbation charge⁴⁴.

In the current work,we applied the work function change induced by the CE to represent the electrode potential difference in the cell for the estimation of the charge.However,since other potential parameters also are influenced by the CE such as the vacuum level andFermi level,we have considered a more complicated correlation between the charge andthe dielectric function by applying the different potential differences in Eq.(4) to estimate charge transfer induced by the contact with water.

To correlate the direct correlation between the charge transfer anddielectric function,we have proposed nine different situations.For the calculated charge,we have applied three different cell potentials derived from the work function,vacuum level,and band offset.However,between these calculated charge results andthe dielectric function of the solid surface,there are no obvious regulations,supporting more complicated parameter is needed for the charge transfer evaluation (Fig. 6a–c).When we apply the charge calculated by more complicated potential combinations,it is noted that the charge has been varied significantly,indicating that the applied potential is the key factor to influence the calculated charges (Fig. 6d–f).Even we further introduce more coefficients to the calculated charge,no evident correlation is identified between the charges anddielectric function.Among different conditions of the calculated charge transfer,we noticed that HfO₂ andMgO show a relatively high charge value under different correlation situations,supporting the higher potential to be applied in such CE systems.More importantly,the lack of direct correlation between the charge transfer andthe dielectric function indicates the further investigation of the charge quantification.Especially when the simple oxides become multicomponent oxide,more complicated factors impose an impact on the pinning factor (Fig. 6g–i).

Fig.6: The correlations between the dielectric functions andcharge transfer which is calculated by different potential parameters.

a The work function.b The vacuum level.c The band offset.d The vacuum level × the vacuum level.e The band offset × the band offset.f The band offset × the vacuum level.g The band offset calculated charge × the vacuum level calculated charge.h The band offset × the vacuum level calculated charge.i The band offset variation × the vacuum level charge.

Derivation of charge transfer mechanism

Besides the dielectric function,the geometry parameters of the liquid–solid interface are also crucial for the charge transfer.In experiments,the contact angle of the liquid droplet also affects the charge transfer,which is related to the contact area andangle⁴¹.However,two types of situations should be considered.When the droplet is ultra-small andthe surface tension dominates the contact,the contact area A is expressed as below.

$$A = {\uppi}\left( {\frac{{3v}}{{\pi {\mathrm{tan}}^2\frac{\theta }{2}\left( {\frac{3}{{{\mathrm{sin}}\theta }}{\mathrm{ – tan}}\frac{\theta }{2}} \right)}}} \right)^{\frac{2}{3}}$$

(8)

In the abovementioned expression,v represents the volume of the droplet andθ indicates the contact angle between the droplet andthe solid surface.On the other hand,when the volume of the droplet becomes larger or under pressure,the gravity results in the flattening droplet anddetermines the contact area of the droplet,which leads to a distinct derivation of the contact area A as shown in Eq.(9).

$${{A}} = \frac{{\sqrt {\frac{{\rho {{g}}}}{\gamma }} }}{{2{\mathrm{sin}}\frac{\theta }{2}}}\,{{v}}$$

(9)

In this equation,ρ indicates the liquid density,g is the acceleration of gravity,and the weak surface tension is expressed by γ.When the contact angle θ is fixed,the increase of liquid only enlarges the contact area A instead of the thickness of the liquid layer .

By considering the contact area,we derive the charge transfer from Eqs.(2) and(3).

$$E_{{\mathrm{EDl}}} = \varphi _i = {{\int_{0}}^{x}} {\varphi _i^0{\mathrm{exp}}\left( {{\mathrm{ – }}\kappa {{l}}} \right)} = \frac{{{\Delta}{{G}}}}{{z \times {\mathrm{e}}}}$$

(10)

$${\mathrm{ln}}\,\varphi _i^0 – \kappa {{l}} = \frac{{ – z {\mathrm{F}}E_{{\mathrm{EDl}}}}}{{z \times {\mathrm{e}}}}$$

(11)

By introducing Eq.(1) in Eq.(11),we have the following expression for c = ±1 charge on each ion such as OH⁻ andH⁺.

$$- \left( {\frac{{2n_i^0e^2}}{{\varepsilon _0\varepsilon {{kT}}}}} \right)^{\frac{1}{2}}{{l}} = {\mathrm{ln}}\frac{{ – {\mathrm{F}}E_{{\mathrm{EDl}}}}}{e} – {\mathrm{ln}}\varphi _i^0 = {\mathrm{ln}}\frac{{ – {\mathrm{F}}E_{{\mathrm{EDl}}}}}{{{\mathrm{e}} \times \varphi _i^0}}$$

(12)

For a certain liquid–solid interface,the E_EDl and\(\varphi _i^0\) should be constant andwe apply constant D to represent the right constant term.Thus,the diffusion length of the work is demonstrated in Eq.(13).

$${{l}} = – {\mathrm{D}} \times \sqrt {\frac{{\varepsilon _{\it{0}}\varepsilon kT}}{{2n_i^{\it{0}}}}} \times \frac{1}{e}$$

(13)

In previous work,the charge transfer by the contact has been calculated by Eq.(14) ³².

$$q_s = {\mathrm{2e}}N_{\mathrm{A}}{{cl}}A$$

(14)

Then,we introduce both the diffusion layer thickness andcontact area into Eq.(14) anddetermine the optimized description of charge transfer expression as Eq.(15).

$${{q}} = – {\mathrm{D}} \times \sqrt {\frac{{2\varepsilon _0\varepsilon kT}}{{n_i^0}}} \times \pi \left( {\frac{{3v}}{{\pi {\mathrm{tan}}^{\mathrm{2}}\frac{\theta }{2}\left( {\frac{3}{{\sin \theta }} – \tan \frac{\theta }{2}} \right)}}} \right)^{2/3} \times N_{\mathrm{A}}c$$

(15)

Therefore,the actual charge transfer within the liquid–solid system is much more complicated than previous works,which involves the contact angle,dielectric function,and temperature,ion concentration,etc.Based on our proposed electrochemical cell model,the evaluation of the charge transfer has been preliminarily investigated through different approaches,which demonstrates the complicated mechanism for the CE within liquid–solid systems.This work indicates the diverse factors involves in the liquid–solid interface,which explains why the CE within the liquid–solid interface cannot be well fitted by conventional EDl or charge transfer models.Further in-depth investigations are still needed to determine the key factors affecting the charge transfer in CE to promote the performances of TENG in an extra wider range of applications.