The Dynamic Steady State of an Electrochemically Generated Nanobubble

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Louise Underwood
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1 Supporting Information. Part 2 The Dynamic Steady State of an Electrochemically Generated Nanobubble Yuwen Liu, Martin A. Edwards, Sean R. German, Qianjin Chen and Henry S. White * The following is a detailed description of the finite element model as created by the Comsol report function for: Case 1, a H 2 nanobubble on a 27 nm disk electrode that is coplanar with its surround. The bubble has a pinned radius, r nb, ranging from 25.5 to 26.2 nm, an inner contact angle of 160 o and the solution contains 0.5 M H +. Case 2, a H 2 nanobubble residing on a recessed disk nanoelectrode. The geometric parameters were width of exposed Pt, w = 1 nm; electrode radius, a = 27 nm; recession ratio 5, implying a depth of L = 5a = 135 nm; the height of the bubble pillar was varied over the range 0.1L to 0.9L (13.5 to nm). The report details the creation of the geometry and the mesh, the parameters used within the model, the equations solved (partial differential equations, including boundary and initial conditions).

2 Case 1: A spherical cap nanobubble residing on a nanoelectrode that is coplanar with its glass surround 1 Global Definitions 1.1 Parameters 1 Parameters Name Expression zr 0 0 charge of H 2 zo +1 1 charge of proton uo DO/R_const/T E 12 s mol/kg mobility of protons theta degree*pi/ contact angle, in rad T [K] K temperature r_pin_gas 25[nm] 2.5E 8 m radius of pinned nanobubble r_curv_gas a_pt*(1 - (a_pt E 8 m r_pin_gas)/a_pt)/sin(theta) p_in 2*gamma/r_curv_gas + p_ex E6 N/m² internal pressure of the nanobubble p_ex 10^5[Pa] 1E5 Pa external pressure (1 bar) n_elec 2 2 electron transfer number Length 10000*a_Pt 2.7E 4 m radius of the simulation domain Hcp 7.8*10^(-6) [mol/(m^3*pa)] 7.8E 6 s² mol/(kg m²) Henry's constant at T in mol/(m^3 Pa) gamma [N/m] N/m surface tension of water at T F [C/mol] C/mol Faraday's constant DR 4.5e-5[cm^2/s] 4.5E 9 m²/s Diffusion Coefficient of H 2 DO 9.3*10^ - 5[cm^2/s] 9.3E 9 m²/s diffusion coefficient of H + degree contact angle, in degrees cr0 0[mol/L] 0 mol/m³ initial concentration of H 2 cr_interface Hcp*p_In mol/m³ solution concentration of H 2 at the gas-water interface co0 0.5[mol/L] 500 mol/m³ initial concentration of H + a_pt 27[nm] 2.7E 8 m radius of the Pt electrode

3 2 Component Definitions Variables Variables 1 Geometric entity level Entire model Name Expression Unit current intop1(comp1.tds.tflux_coz)*f_const A flow_h2 intop2(nr*tds.dflux_crr + nz*tds.dflux_crz) mol/s Component Couplings Integration 1 Coupling type Operator name Integration intop1 Source selection Boundary 5 Integration 2 Coupling type Operator name Integration intop2 Source selection Boundaries 11, s Mass transport domain type Explicit Domains 1 4 Pt Electrode type Explicit

seen here were used to aid meshing and were set to allow free diffusion

4 Boundary 5 Pt Electrode water-gas interface type Explicit Boundaries 11, 16 water-gas interface Note: the additional concentric part spheres seen here were used to aid meshing and were set to allow free diffusion across them, i.e., continuous concentration and flux. bulk type Explicit

5 Boundary 15 bulk 2.2 Geometry 1 Geometry 1 Units Length unit Angular unit nm deg

6 2.2.1 Nanobubble (c1) Position Position {0, -r_curv_gas*cos(theta)} Rotation -90 Radius r_curv_gas Sector angle Mass transport (c2) Position Position {0, 0} Radius Length Sector angle For mesh (c3) Position Position {0, -r_curv_gas*cos(theta)} Rotation -90 Type Curve Radius 1.2*r_curv_gas Sector angle For mesh 1 (c4) Position Position {0, -r_curv_gas*cos(theta)} Rotation -90 Type Curve Radius 2*r_curv_gas Sector angle For mesh 2 (c5) Position Position {0, -r_curv_gas*cos(theta)} Rotation -90 Type Curve Radius 5*r_curv_gas Sector angle Difference 1 (dif1) Compose

7 Relative repair tolerance 1.0E Rectangle 1 (r1) Position Position Width Height {0, -10*r_curv_gas} 10*r_curv_gas 10*r_curv_gas Difference 2 (dif2) Compose Relative repair tolerance 1.0E Polygon 1 (pol1) Object type Type Solid Coordinates r {2.6E-8, 2.7E-8} z {0, 0} Form Union (fin) Relative repair tolerance 1.0E Transport of Diluted Species Geometric entity level Domain Domains 1 4 Concentration Compute boundary fluxes Apply smoothing to boundary fluxes Linear

8 type when using splitting of complex variables Adsorption in porous media Dispersion in porous media Volatilization in partially saturated porous media Convection Migration in electric field Streamline diffusion Crosswind diffusion Equation residual Crosswind diffusion type for free flow Convective term Real Approximate residual Do Carmo and Galeão Non - conservative form Transport Properties 1 Geometric entity level Domain Domains 1 4 Material Diffusion coefficient Diffusion coefficient Diffusion coefficient Diffusion coefficient None User defined {{DO, 0, 0}, {0, DO, 0}, {0, 0, DO}} User defined {{DR, 0, 0}, {0, DR, 0}, {0, 0, DR}} Axial Symmetry 1 Boundaries No Flux 1 Boundaries 6 10

9 2.3.4 Initial s 1 Initial s 1 Geometric entity level Domain Domains 1 4 Concentration {co0, cr0} Flux 1 Boundary 5 Flux type Species co Species cr General inward flux Weak expressions Weak expression Integration frame 0 Material Boundary 5 -tds.tflux_coz*test(cr)*pi*r Material Boundary 5

10 2.3.6 Concentration 1 Boundaries 11, 16 Species co Species cr Concentration Apply reaction terms on Use weak constraints Constraint method {0, cr_interface} All physics (symmetric) Elemental Concentration 2 Boundary 5 Species co Species cr Concentration {0, 0} Apply reaction terms on All physics (symmetric) Use weak constraints Constraint method Elemental Concentration 3 Boundary 15

11 Species co Species cr Concentration Apply reaction terms on Use weak constraints Constraint method {co0, cr0} All physics (symmetric) Elemental 2.4 Mesh Size (size) Maximum element size Minimum element size 81 Curvature factor 0.3 Maximum element growth rate Distribution 1 (dis1) Boundary 5 Number of elements Distribution 2 (dis2) Boundaries 11, 16 Number of elements r_curv_gas*theta/0.5[nm] Free Triangular 1 (ftri1) Geometric entity level Domain Geometry geom Refine 1 (ref1) Geometric entity level Domain Geometry geom1

12 Elements to refine Number of refinements Refine 2 (ref2) Geometric entity level Domain Domains 1 3 Elements to refine Number of refinements 1 3 Study 1 Computation information Computation time 3 min 33 s CPU Intel(R) Core(TM) i7-6820hq 2.70GHz, 4 cores Operating system Windows Parametric Sweep Parameter name Parameter value list Parameter unit r_pin_gas range(25.5,0.1,26.2) nm 3.2 Stationary Study settings Include geometric nonlinearity Physics and variables selection Physics interface Transport of Diluted Species (tds) Discretization physics Mesh selection Geometry Geometry 1 (geom1) Mesh mesh1

13 Global: (mol/s) Current ~ r plot Global: (A)

14 Case 2: A bubble residing on a recessed disk nanoelectrode (the bubble is described as a hemispherical cap atop a cylindrical column) 1 Global Definitions 1.1. Parameters 1 Parameters Name Expression zr 0 0 charge of dihydrogen zo +1 1 charge of proton uo DO/R_const/T E 12 s mol/kg mobility of proton theta degree*pi/ contact angle, in rad T [K] K temperature r_pin_gas a_pt - w 2.6E 8 m radius of the pinned nanobubble r_curv_gas a_pt*(1 - (a_pt - 2.6E 8 m r_pin_gas)/a_pt)/sin(theta) p_in 2*gamma/r_curv_gas + p_ex E6 N/m² internal pressure of the nanobubble p_ex 10^5[Pa] 1E5 Pa external pressure (1 atmosphere) n_elec 2 2 electron transfer number Length 10000*a_Pt 2.7E 4 m radius of the simulation quarter circle Hcp 7.8*10^( 6) [mol/(m^3*pa)] 7.8E 6 s² mol/(kg m²) Henry's constant at T in mol/(m^3 Pa) gamma [N/m] N/m surface tension of water at T DR 4.5e-5[cm^2/s] 4.5E 9 m²/s Diffusion Coefficient of dihydrogen (H2) DO 9.3*10^ - 5[cm^2/s] 9.3E 9 m²/s diffusion coefficient of protons (H+) degree contact angle, in degree cr0 0[mol/L] 0 mol/m³ initial concentration of dihydrogen (H2) cr_interface Hcp*p_In mol/m³ solution concentration of dihydrogen at the gas-water interface co0 0.5[mol/L] 500 mol/m³ initial concentration of protons (H+) a_pt 27[nm] 2.7E 8 m radius of the Pt electrode L recess_ratio*a_pt 1.35E 7 m recession depth recess_ratio 5 5 recession ratio H 1.5*a_Pt 4.05E 8 m height of the cylindrical bubble pillar with a spherical cap w 1[nm] 1E 9 m width of exposed Pt

15 2 Component Definitions Variables Geometric entity level Entire model Name Expression Unit current intop1(comp1.tds.tflux_coz)*f_const A flow_h2 intop2(nr*tds.dflux_crr + nz*tds.dflux_crz) mol/s Component Couplings Integration 1 Coupling type Operator name Integration intop1 Source selection Boundary 6 Integration 2 Coupling type Operator name Integration intop2 Source selection Boundaries 5, s Mass transport domain type Explicit Domains 1 3 Pt Electrode type Explicit Boundary 6

water-gas interface type Explicit Boundaries 5, 10 bulk type Explicit Boundary 12 1.3.

16 water-gas interface type Explicit Boundaries 5, 10 bulk type Explicit Boundary Geometry 1 Geometry 1 Units Length unit Angular unit nm deg Geometry statistics Space dimension 2 Number of domains 3 Number of boundaries 12 Number of vertices Nanobubble (c1) Position

17 Position {0, H - r_curv_gas*cos(theta)} Layers Rotation -90 Radius r_curv_gas Sector angle Rectangle 2 (r2) Position Position {0, 0} Layers Width r_pin_gas Height H Union 1 (uni1) Compose Keep interior boundaries Rectangle 1 (r1) Position Position {0, 0} Layers Width a_pt Height L Mass transport (c2) Position Position {0, L} Layers Radius Length Sector angle for_mesh (c3) Position Position {0, L} Radius 2*a_Pt Sector angle 90 Note: the additional concentric part sphere was used to aid meshing and were set to allow free diffusion across them, i.e., continuous concentration and flux.

18 1.4. Transport of Diluted Species Geometric entity level Domain Domains 1 3 Concentration Compute boundary fluxes Apply smoothing to boundary fluxes type when using splitting of complex variables Adsorption in porous media Dispersion in porous media Volatilization in partially saturated porous media Convection Migration in electric field Streamline diffusion Crosswind diffusion Equation residual Crosswind diffusion type for free flow Convective term Linear Real Approximate residual Do Carmo and Galeão Non - conservative form Transport Properties 1 Geometric entity level Domain Domains 1 3 Material Diffusion coefficient Diffusion coefficient Diffusion coefficient Diffusion coefficient None User defined {{DO, 0, 0}, {0, DO, 0}, {0, 0, DO}} User defined {{DR, 0, 0}, {0, DR, 0}, {0, 0, DR}}

19 Axial Symmetry 1 Boundaries 1 2, No Flux 1 Boundaries Initial s 1 Geometric entity level Domain Domains 1 3 Concentration {co0, cr0} Flux 1 Boundary 6 Flux type Species co Species cr General inward flux Concentration 1 Boundaries 5, 10

20 Species co Species cr Concentration Apply reaction terms on Use weak constraints Constraint method {0, cr_interface} All physics (symmetric) Elemental Concentration 2 Boundary 6 Species co Species cr Concentration {0, 0} Apply reaction terms on All physics (symmetric) Use weak constraints Constraint method Elemental Concentration 3 Boundary 12 Species co Species cr Concentration Apply reaction terms on Use weak constraints Constraint method {co0, cr0} All physics (symmetric) Elemental

21 1.5. Mesh Size (size) Maximum element size Minimum element size 81 Curvature factor 0.3 Maximum element growth rate Free Triangular 1 (ftri1) Geometric entity level Domain Domains Refine 1 (ref1) Geometric entity level Domain Domains 2 3 Elements to refine Number of refinements Distribution 1 (dis1) Boundary 6 Number of elements Distribution 2 (dis2) Boundary 10 Number of elements r_curv_gas*theta/0.5[nm] Distribution 3 (dis3)

22 Boundary 5 Number of elements H/0.5[nm] Distribution 4 (dis4) Boundary 7 Number of elements L/0.5[nm] Distribution 5 (dis5) Boundary 7 Number of elements a_pt/0.5[nm] Free Triangular 2 (ftri2) Geometric entity level Remaining Refine 2 (ref2) Geometric entity level Domain Domains 1 3 Elements to refine Number of refinements Refine 3 (ref3) Geometric entity level Domain Domains 1 2

23 Elements to refine Number of refinements 1 3 Study Parametric Sweep Parameter name Parameter value list H range(0.1,0.1,0.9)*l 1.7. Stationary Study settings Include geometric nonlinearity Physics and variables selection Physics interface Transport of Diluted Species (tds) Discretization physics Mesh selection Geometry Geometry 1 (geom1) Mesh mesh Solver Configurations Solution 1 Compile : Stationary (st1) Study and step Use study Study 1 Use study step Stationary Dependent Variables 1 (v1) General Defined by study step Stationary Initial values of variables solved for Solution Zero s of variables not solved for Solution Zero

24 Concentration (comp1.co) (comp1_co) General Field components comp1.co Concentration (comp1.cr) (comp1_cr) General Field components comp1.cr Stationary Solver 1 (s1) General Defined by study step Stationary Results while solving Probes None Fully Coupled 1 (fc1) General Linear solver Direct 1 Method and termination Initial damping factor 0.01 Minimum damping factor 1.0E-6 Maximum number of iterations 50 Direct 1 (d1) General Solver PARDISO 4 Results Study 1/Parametric Solutions 1 Solution Solution Parametric Solutions 1 Component Save Point Geometry 1

25 1.9. Plot Groups flow~ r plot Global: (mol/s) Current ~ r plot Global: (A)

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