The nanodrum 1: shape in small gradient approximation

Solution of the linearized membrane shape equation

h[ρ_] := 
      h1 + h2*Log[(ρ/λ)] + h3*BesselI[0, (ρ/λ)] + 
        h4*BesselK[0, (ρ/λ)]

D[h[ρ], ρ]

hder[ρ_] := 
      (h2/ρ) + ((h3 BesselI[1, (ρ/λ)])/λ) - ((h4 BesselK[1, (ρ/λ)])/λ)

D[D[h[ρ], ρ], ρ]

hderder[ρ_] := -(h2/(ρ^2)) + ((h3 (BesselI[0, (ρ/λ)] + BesselI[2, (ρ/λ)]))/(2 (λ^2))) - ((h4 (-BesselK[0, (ρ/λ)] - BesselK[2, (ρ/λ)]))/(2 (λ^2)))

Determining the constants h1,...,h4 and the contact point c

1. boundary condition: h[Rp]==0

h[Rp]

Solve[h[Rp] == 0, h1]

h[ρ] /. 
      Part[Solve[h[Rp] == 0, h1], 1]

2. boundary condition: hder[R]==0

hder[Rp]

Solve[hder[Rp] == 0, h2]

(h[ρ] /. {Part[Part[Solve[h[Rp] == 0, h1], 1], 
              1]}) /. {Part[Part[Solve[hder[Rp] == 0, h2], 1], 
          1]}

3. boundary condition: hder[c] == (c/Rt)

hder[c]

hder[c] /. Part[Solve[hder[Rp] == 0, h2], 1]

Solve[(hder[c] /. Part[Solve[hder[Rp] == 0, h2], 1]) == 
        (c/Rt), h3]

FullSimplify[((h[ρ] /. {Part[
                    Part[Solve[h[Rp] == 0, h1], 1], 1]}) /. {Part[
                Part[Solve[hder[Rp] == 0, h2], 1], 1]}) /. 
        Part[Part[
            Solve[(hder[c] /. 
                    Part[Solve[hder[Rp] == 0, h2], 1]) == 
                (c/Rt), h3], 1], 1]]

hpre[ρ_] := (1/(Rt λ (c BesselI[1, (c/λ)] - 
                  Rp BesselI[1, 
                      (Rp/λ)]))) (h4 Rt (λ^2) - 
            c h4 Rt λ BesselI[1, (c/λ)] BesselK[0, 
                (Rp/λ)] + 
            c h4 Rt λ BesselI[1, (c/λ)] BesselK[
                0, (ρ/λ)] - 
            h4 Rp Rt λ BesselI[1, (Rp/λ)] BesselK[
                0, (ρ/λ)] - 
            c λ BesselI[0, 
                (Rp/λ)] (c λ + 
                  h4 Rt BesselK[1, (c/λ)]) + λ BesselI[
                0, (ρ/λ)] ((c^2) λ + 
                  c h4 Rt BesselK[1, (c/λ)] - 
                  h4 Rp Rt BesselK[1, (Rp/λ)]) + 
            (c^2) Rp λ BesselI[1, (Rp/λ)] Log[
                (Rp/λ)] + 
            c h4 Rp Rt BesselI[1, (Rp/λ)] BesselK[1, 
                (c/λ)] Log[(Rp/λ)] - 
            c h4 Rp Rt BesselI[1, (c/λ)] BesselK[1, 
                (Rp/λ)] Log[(Rp/λ)] - 
            c Rp (BesselI[1, 
                      (Rp/λ)] (c λ + 
                        h4 Rt BesselK[1, (c/λ)]) - 
                  h4 Rt BesselI[1, (c/λ)] BesselK[1, 
                      (Rp/λ)]) Log[(ρ/λ)])

4. boundary condition: h[c] == -h0 + ((c^2)/(2  Rt))

Part[
      Part[FullSimplify[
          Solve[hpre[c] == -h0 + ((c^2)/(2*Rt)), h4]], 1], 
      1]

FullSimplify[
      hpre[ρ] /. 
        Part[FullSimplify[
            Solve[hpre[c] == -h0 + ((c^2)/(2*Rt)), h4]], 
          1]]

hpre2[ρ_, h0_, Rp_, Rt_, λ_, 
        c_] := (2 (c^2) (λ^3) BesselI[0, (c/λ)] + 
            2 (c^2) (λ^3) BesselI[0, (Rp/λ)] - 
            4 (c^2) (λ^3) BesselI[0, (ρ/λ)] - 
            (c^3) (λ^2) BesselI[1, (c/λ)] + 
            2 c h0 Rt (λ^2) BesselI[1, (c/λ)] + 
            (c^2) Rp (λ^2) BesselI[1, (Rp/λ)] - 
            2 h0 Rp Rt (λ^2) BesselI[1, (Rp/λ)] + 
            c λ BesselI[1, 
                (c/λ)] (-2 (c^2) λ BesselI[0, 
                      (c/λ)] + 
                  2 (c^2) λ BesselI[
                      0, (ρ/λ)] + ((c^2) - 
                        2 h0 Rt) (c BesselI[1, (c/λ)] - 
                        Rp BesselI[1, (Rp/λ)])) BesselK[0, 
                (Rp/λ)] + 
            2 (c^3) (λ^2) BesselI[1, (c/λ)] Log[
                (c/λ)] - 
            2 (c^2) Rp (λ^2) BesselI[1, (Rp/λ)] Log[
                (c/λ)] - 
            2 (c^2) Rp (λ^2) BesselI[1, (Rp/λ)] Log[
                (Rp/λ)] + λ (c BesselI[1, (c/λ)] - 
                  Rp BesselI[1, (Rp/λ)]) BesselK[
                0, (ρ/λ)] (2 (c^2) λ BesselI[0, 
                      (c/λ)] - 
                  2 (c^2) λ BesselI[0, 
                      (Rp/λ)] - ((c^3) - 2 c h0 Rt) BesselI[1, 
                      (c/λ)] + 
                  Rp BesselI[1, 
                      (Rp/λ)] ((c^2) - 2 h0 Rt + 
                        2 (c^2) (-Log[(c/λ)] + 
                              Log[(Rp/λ)]))) + 
            Rp BesselK[1, 
                (Rp/λ)] (2 (c^2) (λ^2) BesselI[0, 
                      (Rp/λ)] BesselI[
                      0, (ρ/λ)] + (c BesselI[1, 
                            (c/λ)] - 
                        Rp BesselI[1, 
                            (Rp/λ)]) (λ BesselI[
                            0, (ρ/λ)] ((c^2) - 2 h0 Rt + 
                              2 (c^2) (-Log[(c/λ)] + 
                                    Log[(Rp/λ)])) + 
                        c ((c^2) - 2 h0 Rt) BesselI[1, 
                            (c/λ)] (Log[(Rp/λ)] - 
                              Log[(ρ/λ)])) + 
                  2 (c^2) λ BesselI[0, 
                      (c/λ)] (-λ BesselI[0, 
                            (Rp/λ)] + (-c BesselI[1, 
                                  (c/λ)] + 
                              Rp BesselI[1, (Rp/λ)]) (Log[
                                (Rp/λ)] - 
                              Log[(ρ/λ)]))) + 
            2 (c^2) Rp λ BesselI[1, (Rp/λ)] BesselK[0, 
                (c/λ)] (-λ BesselI[0, 
                      (Rp/λ)] + λ BesselI[
                      0, (ρ/λ)] + 
                  Rp BesselI[1, 
                      (Rp/λ)] (Log[(Rp/λ)] - 
                        Log[(ρ/λ)])) - 
            2 (c^3) (λ^2) BesselI[1, 
                (c/λ)] Log[(ρ/λ)] + 
            4 (c^2) Rp (λ^2) BesselI[1, 
                (Rp/λ)] Log[(ρ/λ)] + 
            c (-2 (c^2) λ BesselI[0, 
                      (c/λ)] + ((c^2) - 
                        2 h0 Rt) (c BesselI[1, (c/λ)] - 
                        Rp BesselI[1, (Rp/λ)])) BesselK[1, 
                (c/λ)] (λ BesselI[0, 
                      (Rp/λ)] - λ BesselI[
                      0, (ρ/λ)] + 
                  Rp BesselI[1, 
                      (Rp/λ)] (-Log[(Rp/λ)] + 
                        Log[(ρ/λ)])))/(2 Rt (c BesselI[1, (c/λ)] - 
                Rp BesselI[1, 
                    (Rp/λ)]) (λ (-2 λ + 
                      c BesselI[0, (Rp/λ)] BesselK[1, 
                          (c/λ)] + 
                      Rp BesselI[0, (c/λ)] BesselK[1, 
                          (Rp/λ)]) + 
                Rp BesselI[1, 
                    (Rp/λ)] (λ BesselK[0, (c/λ)] + 
                      c BesselK[1, 
                          (c/λ)] (Log[(c/λ)] - 
                            Log[(Rp/λ)])) + 
                c BesselI[1, 
                    (c/λ)] (λ BesselK[0, (Rp/λ)] + 
                      Rp BesselK[1, 
                          (Rp/λ)] (-Log[(c/λ)] + 
                            Log[(Rp/λ)]))))

FullSimplify[hpre2[Rp, h0, Rp, Rt, λ, c]]

FullSimplify[
      D[hpre2[ρ, h0, Rp, Rt, λ, c], ρ] /. ρ -> 
          Rp]

FullSimplify[
      D[hpre2[ρ, h0, Rp, Rt, λ, c], ρ] /. ρ -> 
          c]

Apart[FullSimplify[hpre2[c, h0, Rp, Rt, λ, c]]]

Plot[
        FullSimplify[N[hpre2[ρ, 3, 33.5, 12, 4.69, 0.7]]], {ρ, 2, 
          33.5}, AxesLabel -> {"ρ", "h(ρ)"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 400];

5. boundary condition: hderder[c] == (1/Rt) - (Sqrt[((2*w)/κ)])

FullSimplify[
      D[D[hpre2[ρ, h0, R, Rt, λ, c], ρ], ρ], 
      Assumptions -> {ρ > 0, Rp > 0, Rt > 0, 
          Rp > Rt, λ > 0, c > 0}]

hpre2derder[ρ_, h0_, R_, Rt_, λ_, 
        c_] := (((2 (c^3) (λ^2) BesselI[1, (c/λ)])/(ρ^2)) - ((4 (c^2) R (λ^2) BesselI[1, (R/λ)])/(ρ^2)) - 
            2 (c^2) λ (BesselI[0, (ρ/λ)] + 
                  BesselI[
                    2, (ρ/λ)]) + (((c^2) R BesselI[1, (R/λ)] (2 R λ BesselI[1, (R/λ)] + (ρ^2) (BesselI[0, (ρ/λ)] + BesselI[2, (ρ/λ)])) BesselK[0, (c/λ)])/(ρ^2)) + 
            (c^3) BesselI[1, 
                (c/λ)] (BesselI[0, (ρ/λ)] + 
                  BesselI[2, (ρ/λ)]) BesselK[0, 
                (R/λ)] + 
            c (-2 (c^2) λ BesselI[0, 
                      (c/λ)] + ((c^2) - 
                        2 h0 Rt) (c BesselI[1, (c/λ)] - 
                        R BesselI[1, 
                            (R/λ)])) (-((R BesselI[1, 
                            (R/λ)])/(ρ^2)) - ((BesselI[0, (ρ/λ)] + BesselI[2, (ρ/λ)])/(2 λ))) BesselK[1, (c/λ)] + (1/(2 λ))((c BesselI[1, 
                        (c/λ)] - 
                    R BesselI[1, (R/λ)]) (BesselK[
                      0, (ρ/λ)] + 
                    BesselK[
                      2, (ρ/λ)]) (2 (c^2) λ BesselI[0, (c/λ)] - 2 (c^2) λ BesselI[0, (R/λ)] - 
                    c ((c^2) - 2 h0 Rt) BesselI[1, (c/λ)] + 
                    R BesselI[1, 
                        (R/λ)] ((c^2) - 2 h0 Rt + 
                          2 (c^2) Log[(R/c)]))) + 
            R BesselK[1, 
                (R/λ)] ((c^2) BesselI[0, 
                      (R/λ)] (BesselI[0, (ρ/λ)] + 
                        BesselI[
                          2, (ρ/λ)]) + (1/(λ (ρ^2)))((c BesselI[1, (c/λ)] - 
                          R BesselI[1, 
                              (R/λ)]) (-2 (c^2) (λ^2) BesselI[0, (c/λ)] + 
                          c ((c^2) - 2 h0 Rt) λ BesselI[1, 
                              (c/λ)] + λ ρ BesselI[
                              1, (ρ/λ)] (-(c^2) + 
                                2 h0 Rt + 
                                2 (c^2) (Log[c] - 
                                      Log[R])) + (ρ^2) BesselI[
                              0, (ρ/λ)] ((c^2) - 2 h0 Rt + 
                                2 (c^2) Log[
                                    (R/c)])))))/(2 Rt (c BesselI[1, (c/λ)] - 
                R BesselI[1, 
                    (R/λ)]) (λ (-2 λ + 
                      c BesselI[0, (R/λ)] BesselK[1, (c/λ)] + 
                      R BesselI[0, (c/λ)] BesselK[1, 
                          (R/λ)]) + 
                R BesselI[1, 
                    (R/λ)] (λ BesselK[0, (c/λ)] + 
                      c BesselK[1, (c/λ)] (Log[c] - Log[R])) + 
                c BesselI[1, 
                    (c/λ)] (λ BesselK[0, (R/λ)] + 
                      R BesselK[1, (R/λ)] Log[
                          (R/c)])))

findcontact[h0_, Rp_?NumberQ, Rt_, λ_, κ_, w_] := 
      Part[Part[
          FindRoot[(hpre2derder[c, h0, Rp, Rt, λ, 
                  c] - ((1/Rt) - (Sqrt[((2*w)/κ)]))), {c, 
              0.000001, 0, Rp}], 1], 2]

findcontact[3, 33.5, 12, 4.69, 1.1, 0.0]

The shape h(ρ)

hprimefinal[ρ_, h0_, Rp_, Rt_, λ_, κ_, w_] := 
        hpre2[ρ, h0, Rp, Rt, λ, 
          findcontact[h0, Rp, Rt, λ, κ, w]];

Plot[
        hprimefinal[ρ, 3, 33.5, 12, 4.69, 1.1, 0.0], {ρ, 2, 30}, 
        AxesLabel ->  {"ρ / nm", "h(ρ)/ nm"}, 
        Background -> GrayLevel[1.0], 
        ImageSize -> 500];

Influence of adhesion

h0a = 3; Rpa = 33.5; 
    Rta = 12; λa = 4.69; κa = 1.1;
\(\(p1 = 
        Plot[\((1\/\(2*Rta\)*ρ\^2 - 3)\), {ρ, 0, Rpa}, 
          PlotStyle -> {Hue[0.6]}, PlotRange -> {\(-3.1\), 2}, 
          ImageSize -> 800, DisplayFunction -> Identity, 
          AxesLabel -> \ {"ρ / nm", "h(ρ)/ nm"\ }];\)\)
p2 = 
      Plot[hprimefinal[ρ, h0a, Rpa, Rta, λa, κa, 
          0.08], {ρ, 
          findcontact[h0a, Rpa, Rta, λa, κa, 0.08], Rpa}, 
        ImageSize -> 500, PlotRange -> {-3.1, 1}, 
        PlotStyle -> Dashing[{0.01}], DisplayFunction -> Identity]; 
    p3 = Plot[
        hprimefinal[ρ, h0a, Rpa, Rta, λa, κa, 
          0.06], {ρ, 
          findcontact[h0a, Rpa, Rta, λa, κa, 0.06], Rpa}, 
        ImageSize -> 500, PlotRange -> {-3.1, 1}, 
        PlotStyle -> Dashing[{0.02}], 
        DisplayFunction -> Identity];
p4 = 
        Plot[hprimefinal[ρ, h0a, Rpa, Rta, λa, κa, 
            0.04], {ρ, 
            findcontact[h0a, Rpa, Rta, λa, κa, 0.04], Rpa}, 
          ImageSize -> 500, PlotRange -> {-3.1, 1}, 
          PlotStyle -> Dashing[{0.03}], 
          DisplayFunction -> Identity];
p5 = 
        Plot[hprimefinal[ρ, h0a, Rpa, Rta, λa, κa, 
            0.02], {ρ, 
            findcontact[h0a, Rpa, Rta, λa, κa, 0.02], Rpa}, 
          ImageSize -> 500, PlotRange -> {-3.1, 1}, 
          PlotStyle -> Dashing[{0.04}], 
          DisplayFunction -> Identity];
p6 = 
        Plot[hprimefinal[ρ, h0a, Rpa, Rta, λa, κa, 
            0.01], {ρ, 
            findcontact[h0a, Rpa, Rta, λa, κa, 0.01], Rpa}, 
          ImageSize -> 500, PlotRange -> {-3.1, 1}, 
          PlotStyle -> Dashing[{0.05}], 
          DisplayFunction -> Identity];
p7 = 
        Plot[hprimefinal[ρ, h0a, Rpa, Rta, λa, κa, 
            0.0001], {ρ, 
            findcontact[h0a, Rpa, Rta, λa, κa, 0.0001], Rpa}, 
          ImageSize -> 500, PlotRange -> {-3.1, 1}, 
          DisplayFunction -> Identity];
p8 = 
        Plot[hprimefinal[ρ, h0a, Rpa, Rta, λa, κa, 
            0.0], {ρ, 
            findcontact[h0a, Rpa, Rta, λa, κa, 0.0], Rpa}, 
          ImageSize -> 500, PlotRange -> {-3.1, 1}, 
          PlotStyle -> {Hue[0.8]}, 
          DisplayFunction -> Identity];

Show[p1, p2, p3, p4, p5, p6, p7, p8, 
        DisplayFunction -> $DisplayFunction, 
        Background -> GrayLevel[1.0]];

Profiles for different values of indentation

Rpi = 33.5; Rti = 12; λi = 4.69; κi = 1.1; 
    wi = 0;
\(\(pl1 = 
        Plot[\((1\/\(2*Rti\)*ρ\^2 - 3)\), {ρ, 0, Rpi}, 
          PlotRange -> {\(-20.1\), 2}, ImageSize -> 800, 
          DisplayFunction -> Identity, 
          AxesLabel -> \ {"ρ / nm", "h(ρ)/ nm"\ }];\)\)
pl2 = 
      Plot[((1/(2*Rti))*(ρ^2) - 10), {ρ, 0, Rpi}, 
        PlotRange -> {-20.1, 2}, ImageSize -> 800, 
        DisplayFunction -> Identity, 
        AxesLabel ->  {"ρ / nm", "h(ρ)/ nm" }]; 
    pl3 = Plot[((1/(2*Rti))*(ρ^2) - 20), {ρ, 0, Rpi}, 
        PlotRange -> {-20.1, 2}, ImageSize -> 800, 
        DisplayFunction -> Identity, 
        AxesLabel ->  {"ρ / nm", "Eh(ρ)/ nm" }];
pl4 = 
        Plot[hprimefinal[ρ, 3, Rpi, Rti, λi, κi, 
            wi], {ρ, 
            findcontact[3, Rpi, Rti, λi, κi, wi], Rpi}, 
          ImageSize -> 500, PlotRange -> {-20.1, 1}, 
          PlotStyle -> Thickness[0.004], 
          DisplayFunction -> Identity];
pl5 = 
        Plot[hprimefinal[ρ, 10, Rpi, Rti, λi, κi, 
            wi], {ρ, 
            findcontact[10, Rpi, Rti, λi, κi, wi], Rpi}, 
          ImageSize -> 500, PlotRange -> {-20.1, 1}, 
          PlotStyle -> Thickness[0.004], 
          DisplayFunction -> Identity];
pl6 = 
        Plot[hprimefinal[ρ, 20, Rpi, Rti, λi, κi, 
            wi], {ρ, 
            findcontact[20, Rpi, Rti, λi, κi, wi], Rpi}, 
          ImageSize -> 500, PlotRange -> {-20.1, 1}, 
          PlotStyle -> Thickness[0.004], 
          DisplayFunction -> Identity];

Show[pl1, pl2, pl3, pl4, pl5, pl6, 
        DisplayFunction -> $DisplayFunction, 
        AspectRatio -> Automatic, 
        Background -> GrayLevel[1.0]];