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Polymers 2021, 13, 1258 numerical model, we characterize the ionic channel network of proton exchange membranes with different amounts of water uptake. We also extract quantitative information relating the ionic channel network to the proton exchange membrane. Furthermore, we attempt to provide a general model for interpreting changes in the morphology of a proton exchange membrane. 4 of 13 2. Experimental Setup and Model Development °imCeonvtes.rnTighhet.firTshtemsemcobnrdanme,ecmalblreadnteh,ecadlrleydmtehmebwreatnem,ewmabsrbaankee,dwiansasnoaokveedniantw80ateCr overnight. TBheefosrecomnedamsuermembreante,,tchaelldedrythmeewmetbmraenmebwransee,xwpaossseodaktoedaminbwieantetrcovnedrintiognhst. fBoerfo2rhe,mwehaisleurtehmeewnett,tmhemdrbyramnemwbarsansoeawkeadseinxpwoasetedr.toambientconditionsfor2h,while thewBeottmhemebmrbanraenwesaswseoraekmedaipnpwedataenr.danalyzedsystematicallyinseveralsteps.First, eachBmoethmmbreamnebrwanaessswcaenrneemdaaptpaedfraenqdueancaylyozfed1sHyszteamsathtiecasllaymipnlseebviearsalvsotletpasg.eFwirasts, cehacahngmeedmfbroramne–w3aVsstcoan3nVediant1afVreqinutenrvcyalosfb1yHuzsaisngthePasarkmpsylestbeimassvXolEta-1g5e0wAasFMcha(nPgaerdk Sfryosmtem−s3,VSutow3oVn,iKn o1rVeai)n.tPerhvaasles ibmyaugseinsganPdartkospyosgtermapshXyEw-1e5r0eAsiFmMu(ltPaanrekoSuyssltyemsa,pSpuewdoin, tKhoisresat)e.pP.hTasheimaegaens apnhdasteoploagravpahluyewoefreasicmhuilmtaangeeouwslaysmsuapbpseqduienntthlyis cstaelpcu. lTahtedmaenadn 2. Experimental Setup and Model Development Nafion 212 membranes were studied under two different conditions in our experNimafienotns.2T1h2emfiermstbmraenmebsrwaneere,csatluleddietdheunddryermtwemobdriafnfer,ewntascobnadkietdioinsainovuernexapte8r0- ◦ phase lag value of each image was subsequently calculated and plotted. Finally, these plotted. Finally, these mean phase values were analyzed using an approximation model mean phase values were analyzed using an approximation model based on Shen et al.’s based on Shen et al.’s study [19]. study [19]. An electrical interaction occurs when a bias voltage is applied between the tip and An electrical interaction occurs when a bias voltage is applied between the tip and the sample surface, as the dielectric sample becomes polarized. The capacitive force that the sample surface, as the dielectric sample becomes polarized. The capacitive force that is is induced between the tip and the sample surface can be expressed as [24] induced between the tip and the sample surface can be expressed as [24] F = 1 𝜕𝐶 𝑉 (1) 12∂C𝜕𝑧 2 F = 2 ∂z V (1) where F is the capacitive force, C is the capacitance of the space between the tip and the sample, V is the applied voltage, and z is the distance between the tip and the sample where F is the capacitive force, C is the capacitance of the space between the tip and the surface. The capacitance of the tip (Ctip), which is modeled as a plate, is [19] sample, V is the applied voltage, and z is the distance between the tip and the sample surface. The capacitance of the tip (Ctip), which is modeled as a plate, is [19] cwahpearceitaRnceiesqtuhaetiorand,ituhsecohfatrhgetaicpcuamndulεateidsitnhethpeetripmistt[i1v9it]y of free space. tip 0 capacitance equation, the charge accumulated in the tip is [19] 𝑄 = 𝐶𝑉 𝐶 ≅ 𝜋𝜀 𝑅 (2) CTip ∼= πε0R2tip (2) where Rtip is the radius of the tip and ε0 is the permittivity of free space. From the From the (3) Figure 1. Configuration of the conductive tip and Nafion attached to the sample holder, explaining Figure 1. Configuration of the conductive tip and Nafion attached to the sample holder, QTip = CTipV The tip and sample create a nanosized parallel-plate capacitor filled with air and Nafion, (3) The tip and sample create a nanosized parallel-plate capacitor filled with air and Nafion, as shown in Figure 1. as shown in Figure 1. the origin of the capacitor model. explaining the origin of the capacitor model. The capacitance of this parallel-plate capacitor is calculated as [24,25] C = Qtip (4) VPDF Image | Ionic Domains on a Proton Exchange Membrane Electrostatics
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