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Membrane and Bulk Metamaterials

Robert V. Kohn Courant Institute, NYU

(1) Membrane metamaterials: work with Jens Jorgensen, Jianfeng Lu, Michael Weinstein (2) Bulk metamaterials: to put part (1) in context. PICOF , Paris, April 2012

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Outline

(1) Membrane metamaterials

• The phenomenon
• A simplified model
• An exactly-solvable special case
• Further discussion

(2) Bulk metamaterials

• negative dynamic mass, as a substitute for damping
• the mechanism: resonance in the microstructure

Are these topics related? Yes: both involve classical wave theory; both have appln to sonic insulation. No: membrane metamaterials are not materials; mechanism is very different.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Membrane metamaterials – the phenomenon

Basic phenomenon: a thin (suitably structured) elastic membrane can achieve near-total reflection of time-harmonic acoustic waves

incident wave −

→ − → transmitted wave

reflected wave ←

−

membrane

The membrane: circular (or square, etc); thin elastic under tension; with a heavy weight in the middle. Different weights achieve total reflection at different frequencies.

Initial work: Z. Yang et al, Membrane-type acoustic metamaterial with negative dynamic mass, PRL 101 (2008) Further work: Z. Yang et al, Appl Phys Lett 96 (2010); also Naify et al, J Appl Phys 108 (2010) and 109 (2011).

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

A simplified model

Waveguide section Ω; membrane is at x = 0 − → − → ← − V(y, t) = transverse displacement of membrane U(x, y, t) = pressure in waveguide ρ(y)Vtt − divy(σ(y)∇yV) = ∂xU+ − ∂xU− at x = 0 (the membrane) Utt − ∆U = 0 for x < 0 and x > 0 (the waveguide) U(0, y, t) = V(y, t) continuity at membrane suitable bc at ∂Ω and as x → ±∞ Membrane pde is just balance of forces; associated Hamiltonian is

• x<0

U2

t +|∇U|2 dx dy+

• Ω×{0}

ρV 2

t +σ|∇V|2 dy+

• x<0

U2

t +|∇U|2 dx dy

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Time-harmonic scattering

− → − → ← − Take U(x, y, t) = u(x, y)e−iωt, V(y, t) = v(y)e−iωt Separation of vars in waveguide: U is superposn of special solns eiknx−iωtϕn(y) where −∆ϕn = λnϕn (with waveguide bdry conds) and −ω2 + λn + k 2

n = 0.

Choose sign convention kn(ω) = √ ω2 − λn if λn < ω2 (propagating) i √ λn − ω2 if λn > ω2 (evanescent) so that ei(knx−ωt)ϕn(y) is

• utgoing as x → ∞, if propagating

exp decaying as x → ∞, if evanescent Then: if incident wave is assoc n = 1, scattering expt has ux<0 = eik1xϕ1(y) +

∞

• n=1

rne−iknxϕn(y); ux>0 =

∞

• n=1

tneiknxϕn(y).

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

A simple (but boring) exactly-solvable case

Let membrane and waveguide share same bdry cond (say, Dirichlet) at ∂Ω, and take ρ = 1 and σ = 1 (uniform). Then modes decouple. Time-harmonic membrane pde is −ω2v − ∆v = [ux]x=0

• n Ω

while in waveguide ux<0 = eik1xϕ1(y) +

∞

• n=1

rne−iknxϕn(y); ux>0 =

∞

• n=1

tneiknxϕn(y). By continuity: 1 + r1 = t1; rn = tn for n ≥ 2; v(y) =

∞

• n=1

tnϕn(y). RHS of membrane PDE becomes [ux]x=0 = −2ik1φ1 + 2i

∞

• n=1

kntnϕn Modes decouple, so (−ω2 + λ1 − 2ik1)t1 = −2ik1 and tn = 0 for n ≥ 2. In particular, t1(ω) = 0, so total reflection is not achieved at any ω.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

A more interesting exactly-solvable case

As before, let membrane and waveguide share same bdry cond (say, Dirichlet) at ∂Ω, and let σ = 1. But now take ρ = 1 + mδy0, i.e. add mass m to membrane, at location y0. This makes sense only in 1D (we need v → v(0) to be continuous in H1 norm), so take Ω = (0, 1). Waveguide is a strip. Weak form of membrane eqn −ρω2v − vyy = [ux]x=0 is −ω2

• vw dy − ω2mv(y0)w(y0) +
• vywy dy =
• [ux]x=0w dy.

So in spectral Galerkin representation, mass at y0 introduces a rank-one perturbation to an otherwise-diagonal problem. Formally: v = ∞

n=1 tnϕn

where t = (t1, t2, · · · )T solves (D − ω2m ssT)t = b with D = diag matrix with entries dn = −ω2 + λn − 2ikn ssT = rank one matrix with n, mth entry ϕn(y0)ϕm(y0) b = (−2ik1, 0, 0, 0, · · · )T. Exactly solvable using Sherman-Morrison formula.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

The membrane with a point mass, cont’d

Recall: if Ω = (0, 1) and ρ = 1 + mδy0 then v = ∞

n=1 tnϕn where t solves

(D − ω2m ssT)t = b with D = diag matrix with entries dn = −ω2 + λn − 2ikn, s = (ϕ1(y0), ϕ2(y0), · · · )T, and b = (−2ik1, 0, 0, 0, · · · )T. Now suppose there is just one propagating mode, i.e. λ1 < ω2 < λ2 so that k1 is real and kn = i|kn| for n ≥ 2. Then (after some algebra) t1 = d−1

1 b1

1 − mω2 ∞

n=2 d−1 n |sn|2

1 − mω2 ∞

n=1 d−1 n |sn|2

The fraction has real numerator (since dn = −ω2 + λn + 2|kn| for n ≥ 2) and non-vanishing denominator (since d1 is not real). Moreover numerator = 1 − mω2

∞

• n=2

|sn|2 λn − ω2 + 2 √ λn − ω2 is a decreasing function of ω2, and a linear function of m. So There is at most one ω ∈ (√λ1, √λ2) at which total reflection occurs. For each ω, m can be chosen st t1(ω) = 0.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Some numerics

Recall: t1 = d−1

1 b1 1−mω2 ∞

n=2 d−1 n

|sn|2 1−mω2 ∞

n=1 d−1 n

|sn|2 where dn = −ω2 + λn − 2ikn,

sn = ϕn(y0), and b1 = −2ik1 = −2i

• ω2 − λ1.

Example: transmission |t1| graphed against ω2, when Ω = (0, 1), the bndry cond is Dirichlet, and the mass is midpoint y0 = 1/2:

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Is this an accident?

Success is initially surprising: t1(ω) is complex; why can we make it vanish by varying a real parameter (ω, or m)? Answer: actually, if there is just one propagating mode, then t1(ω) is restricted to a circle in the complex plane.

1/2

In fact: when there is one propagating mode, far-field reflected wave r1e−ik1xϕ1(y) and far-field transmitted wave t1eik1xϕ1(y) are related by |r1|2 + |t1|2 = 1. But continuity at membrane gives r1 + 1 = t1. So |t1 − 1|2 + |t1|2 = 1. (This holds for any ρ(y) and σ(y)). A consequence: The phenomenon is robust – perturbing a system will perturb the freq where t1(ω) = 0, but it won’t eliminate the phenomenon.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Stepping back

What is the essential mechanism?

• Effect of the mass is to couple modes. Clearly visible in our

example, where (D − ω2mssT)t = b with D diagonal.

• If coupling is suitable, t1(ω) will pass 0 as it moves on the circle

as a function of ω ∈ (√λ1, √λ2). How general is this?

• Hypothesis of one prop mode seems essential. For our solvable

example with λ2 < ω2 < λ3 (two prop modes), would need t1(ω) = t2(ω) = 0 for two indep incoming waves. Many (nonlinear, simultaneous) eqns!

• Hypothesis that membrane and waveguide share same bdry

condition is not physical. Closer to acoustics: Dirichlet bc for the membrane, Neumann for the waveguide. Effect persists numerically; can it be analyzed?

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Stepping back, cont’d

How useful is this?

• Effect is intrinsically narrow-band: if t1(ω) passes through 0 at

ω = ω0, then |t1(ω)| ∼ c|ω − ω0|.

• To get broader-band effect, Yang et al (J Appl Phys 2010)

explored using different membranes in series, and Naify et al (J Appl Phys 2011) explored using different membranes in parallel. − → − → − → ← − ← −

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Transition

(1) Membrane metamaterials

• The phenomenon
• A simplified model
• An exactly-solvable special case
• Further discussion

(2) Bulk metamaterials

• negative dynamic mass, as a substitute for damping
• the mechanism: resonance in the microstructure

Are these topics related? Yes: both involve classical wave theory; both have appln to sonic insulation. No: membrane metamaterials are not materials; mechanism is very different.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Bulk metamaterials

Key points: (a) Suitable microstructures can achieve negative dynamic mass. Key mechanism: microstructural resonance. Explored e.g. by

• Liu et al (Science 2000, “locally resonant sonic materials”)
• Bouchitte & Felbacq (NJP 2005, “neg magnetic permeability”)
• Milton & Willis (Proc Roy Soc 2007, “modifns of Newton’s law”)

(b) Neg dyn mass ⇒ lossless mechanism for sonic insulation

• Mathematically elementary
• Focus of many papers (expts + theory) by Hong Kong group

(leaders include C.T. Chan and P . Sheng).

Meaning of negative dynamic mass: in our time-harmonic scalar setting, U(x, y, t) = u(x, y)e−iωt with −ω2u − ∆u = 0 −ρ∗ω2u − ∆u = 0 −ω2u − ∆u = 0 where ρ∗ < 0 and u, ux are conts at interfaces.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Negative dynamic mass = ⇒ sonic insulation

−ω2u − ∆u = 0 ω2u − ∆u = 0 −ω2u − ∆u = 0

x = 0 x = L

Modes separate. Though mode assoc eigenfunction ϕ may be propagating for x < 0 and x > L, it is evanescent for 0 < x < L. Thus: single mode scattering problem is u = eikxϕ + re−ikxϕ u = Aeαxϕ + Be−αxϕ u = teikxϕ with α > 0. After some algebra, one gets relation of form

µ + r 1 + µr = γe−2αL with |µ| = |γ| = 1.

As L → 0 this gives r → −µ. Since |r|2 + |t|2 = 1, this means |t| → 0. Conclusion: layer with neg dynamic mass provides sonic insulation.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Suitable microstructure = ⇒ negative dynamic mass

Example discussed by Zhikov (2000) & Bouchitte, Felbacq (2004): PDE: −ω2uε − ∇ · (σ(x/ε)∇uε) = 0, where x ∈ Rn The microstructure: a periodic array of inclusions in a uniform matrix Periodicity ε. Inclusions are translates of εD. Standard homogenization, for holes: if σ =

• in inclusions

1 in matrix then eff eqn is −ω2ueff − ∇ · (σeff∇ueff) = 0 The metamaterial: if σ =

• ε2

in inclns 1 in matrix then soln in matrix is −ω2ρeff(ω)uext−∇·(σeff∇uext) = 0 where σeff has same value as for holes. Key point: ρeff(ω) has poles.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

More explicitly...

We call ρeff(ω) the dynamic mass since soln (in the matrix) solves −ω2ρeff(ω)uext − ∇ · (σeff∇uext) = 0 It has a formula: ρeff = 1 + c

• j

ω2 λj − ω2

• φj

2 where φj, λj are eigenfunctions and eigenvalues of Dirichlet Laplacian

• n inclusion shape D, and φj = avg of φj.

Singular at ω =

• λj whenever φj = 0. As ω crosses such value, ρeff

changes sign. Thus ρeff can easily be negative! Note: “effective eqn” is for uext, not avg of u. Appropriate for waveguide problem, if inclusions don’t meet interface.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Effect comes from microstructural resonance

−ω2uε − ∇ · (σ(x/ε)∇uε) = 0 Structure of unit cell depends on microstructural length scale: σ = ε2 in inclusions, σ = 1 in matrix. This permits uη to have order-one oscillations in inclusions. In fact, at leading order: uε ≈ uext(x)m(x/ε) where uext solves the equation given earlier and −ω2m − ∆m = 0 in D m = 1 at ∂D and in matrix. More: at next order, correctors in matrix are same as for periodic holes, and correctors in inclusions are zero. ρeff(ω) =

• period cell

m(y) dy

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Resonant behavior of ρeff

Recall definition: ρeff =

• Y m(y) dy where Y = [0, 1]n and

∆m + ω2m = 0 in D m = 1 at ∂D and in matrix. Work with m∗ = m − 1, which vanishes outside D and solves ∆m∗ = −ω2m∗ − ω2 in D. Expand in eigenfns of Dirichlet Laplacian (−∆φj = λjφj): if m∗ = αjφj then −λjαj = −ω2αj − φj, 1ω2, whence αj = φj, 1ω2 λj − ω2 Thus

• Y

m(y) dy = 1 +

• D

m∗ dy = 1 +

• ω2

λj − ω2 φj, 12

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Remarks on bulk metamaterials

Rigorous analysis: two-scale convergence (Bouchitté & Felbacq,

Zhikov).

Actually very familiar: homogn with σ = ε2 in inclusions considered long ago (“double porosity model”). Among earliest applns of two-scale convergence. Singular behavior of ρeff opens the door to applications; first noticed by the physics community. Similar microstructural resonance is behind many examples of “metamaterials” with surprising physical properties. Applications of such metamaterials go far beyond sonic insulation.

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials

Perspective

Bulk metamaterials achieve exotic material properties through microstructural resonance. Therefore intrinsically narrow-band, very sensitive to intrinsic damping. Membrane metamaterials achieve sonic insulation through more classical wave effects. Not composites and not “materials.” No microstructural resonance, but still narrow-band. Many open questions. System design and analysis has thus far been example-driven. Can there be a more systematic approach? Are there limitations on what can be achieved?

Robert V. Kohn Courant Institute, NYU Membrane and Bulk Metamaterials