Dynamics of a tagged monomer: Effects of elastic pinning and - - PowerPoint PPT Presentation

Dynamics of a tagged monomer: Effects of elastic pinning and harmonic absorption Shamik Gupta

Laboratoire de Physique Th´ eorique et Mod` eles Statistiques, Universit´ e Paris-Sud, France

Joint work with

Alberto Rosso Christophe Texier Ref.: Phys. Rev. Lett. 111, 210601 (2013)

Shamik Gupta Dynamics of a tagged monomer

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Shamik Gupta Dynamics of a tagged monomer

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Shamik Gupta Dynamics of a tagged monomer

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Shamik Gupta Dynamics of a tagged monomer

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Shamik Gupta Dynamics of a tagged monomer

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Shamik Gupta Dynamics of a tagged monomer

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Shamik Gupta Dynamics of a tagged monomer

The model

1

Rouse polymer of L monomers immersed in a solvent:

Shamik Gupta Dynamics of a tagged monomer

The model

1

Rouse polymer of L monomers immersed in a solvent:

2

hi: displacement of the i-th monomer from equilibrium.

Shamik Gupta Dynamics of a tagged monomer

The model

1

Rouse polymer of L monomers immersed in a solvent:

2

hi: displacement of the i-th monomer from equilibrium.

3

Elastic energy Eel = (1/2)

i(hi+1 − hi)2.

Shamik Gupta Dynamics of a tagged monomer

The model

1

Rouse polymer of L monomers immersed in a solvent:

2

hi: displacement of the i-th monomer from equilibrium.

3

Elastic energy Eel = (1/2)

i(hi+1 − hi)2.

4

Langevin Dynamics:

∂hi(t) ∂t

= − ∂Eel

∂hi + ηi(t) = j ∆ijhj(t) + ηi(t).

Shamik Gupta Dynamics of a tagged monomer

The model

1

Rouse polymer of L monomers immersed in a solvent:

2

hi: displacement of the i-th monomer from equilibrium.

3

Elastic energy Eel = (1/2)

i(hi+1 − hi)2.

4

Langevin Dynamics:

∂hi(t) ∂t

= − ∂Eel

∂hi + ηi(t) = j ∆ijhj(t) + ηi(t).

5

∆: discrete Laplacian, {ηi(t)} → independent Gaussian white noise: ηi(t) = 0, ηi(t)ηj(t′) = 2T δi,j δ(t − t′).

Shamik Gupta Dynamics of a tagged monomer

The model

1

Rouse polymer of L monomers immersed in a solvent ≡ L-dim. discrete Edwards-Wilkinson interface

hi i

2

hi: displacement of the i-th monomer from equilibrium.

3

Elastic energy Eel = (1/2)

i(hi+1 − hi)2.

4

Langevin Dynamics:

∂hi(t) ∂t

= − ∂Eel

∂hi + ηi(t) = j ∆ijhj(t) + ηi(t).

5

∆: discrete Laplacian, {ηi(t)} → independent Gaussian white noise: ηi(t) = 0, ηi(t)ηj(t′) = 2T δi,j δ(t − t′).

Shamik Gupta Dynamics of a tagged monomer

The model

1

Rouse polymer of L monomers immersed in a solvent ≡ L-dim. discrete Edwards-Wilkinson interface

hi i

2

hi: displacement of the i-th monomer from equilibrium.

3

Elastic energy Eel = (1/2)

i(hi+1 − hi)2.

4

Langevin Dynamics:

∂hi(t) ∂t

= − ∂Eel

∂hi + ηi(t) = j ∆ijhj(t) + ηi(t).

5

∆: discrete Laplacian, {ηi(t)} → independent Gaussian white noise: ηi(t) = 0, ηi(t)ηj(t′) = 2T δi,j δ(t − t′).

6

Set T = 1.

Shamik Gupta Dynamics of a tagged monomer

Diffusion and Subdiffusion

1

L-dim. discrete Edwards-Wilkinson interface:

hi i

Shamik Gupta Dynamics of a tagged monomer

Diffusion and Subdiffusion

1

L-dim. discrete Edwards-Wilkinson interface:

hi i

2

Centre of mass (1/L) L

i=1 hi(t) → Markovian dynamics, normal

diffusion: Mean-squared displacement ∼ 2(1/L)t.

Shamik Gupta Dynamics of a tagged monomer

Rouse polymer: Diffusion and Subdiffusion

1

L-dim. discrete Edwards-Wilkinson interface:

hi i

2

Tagged monomer → Non-Markovian dynamics, anomalous diffusion: Mean-squared displacement ∼

• 2

πb0

√t.

Shamik Gupta Dynamics of a tagged monomer

Rouse polymer: Diffusion and Subdiffusion

1

Tagged monomer Mean-squared displacement ∼

• 2

πb0

√t.

Shamik Gupta Dynamics of a tagged monomer

Rouse polymer: Diffusion and Subdiffusion

1

Tagged monomer Mean-squared displacement ∼

• 2

πb0

√t.

b0 encodes memory of polymer configuration at t = 0. Equilibrium at t = 0 → Tagged monomer exhibits fractional Brownian motion (correlated increments), b0 = √ 2. Out of equilibrium flat configuration at t = 0 → Correlated increments drawn from a Gaussian distribution with a time-dependent variance, b0 = 1 (Krug et al. (1997)).

Shamik Gupta Dynamics of a tagged monomer

What we are after....

Two specific situations of practical relevance:

Shamik Gupta Dynamics of a tagged monomer

What we are after....

Two specific situations of practical relevance:

1

Elastic pinning of the tagged monomer (cf. optical tweezers).

κ >0 t =0 t

h

i

i

Shamik Gupta Dynamics of a tagged monomer

What we are after....

Two specific situations of practical relevance:

1

Elastic pinning of the tagged monomer (cf. optical tweezers).

κ >0 t =0 t

h

i

i

2

Absorption of the tagged monomer on an interval. Example: Reactant attached to a monomer encounters an external reactive site fixed in space.

Shamik Gupta Dynamics of a tagged monomer

What we are after....

Two specific situations of practical relevance:

1

Elastic pinning of the the tagged monomer.

2

Absorption of the tagged monomer in an interval.

Shamik Gupta Dynamics of a tagged monomer

What we are after....

Two specific situations of practical relevance:

1

Elastic pinning of the the tagged monomer.

2

Absorption of the tagged monomer in an interval.

Questions: Dynamics of the tagged monomer, Steady state, Approach to the steady state, Memory of the initial condition.

Shamik Gupta Dynamics of a tagged monomer

What we are after....

Two specific situations of practical relevance:

1

Elastic pinning of the the tagged monomer.

2

Absorption of the tagged monomer in an interval.

Questions: Dynamics of the tagged monomer, Steady state, Approach to the steady state, Memory of the initial condition. Our work: Exact analytical results for elastic pinning and harmonic absorption. In particular, strong memory effects in the relaxation to the steady state.

Shamik Gupta Dynamics of a tagged monomer

Elastic pinning

κ >0 t =0 t

h

i

i

∂Wt[h|h0] ∂t

=

i ∂2 ∂h2

i +

i,j ∂ ∂hi Λijhj

• Wt[h|h0];

−Λij = ∆ij − κ δi,jδi,0.

Langevin approach (Vi˜ nales and Desp´

• sito (2006,2009), Grebenkov (2011))

Shamik Gupta Dynamics of a tagged monomer

Elastic pinning

κ >0 t =0 t

h

i

i

∂Wt[h|h0] ∂t

=

i ∂2 ∂h2

i +

i,j ∂ ∂hi Λijhj

• Wt[h|h0];

−Λij = ∆ij − κ δi,jδi,0.

1

Replace matrix Λ by number λ: 1d Ornstein-Uhlenbeck process.

Shamik Gupta Dynamics of a tagged monomer

Elastic pinning

κ >0 t =0 t

h

i

i

∂Wt[h|h0] ∂t

=

i ∂2 ∂h2

i +

i,j ∂ ∂hi Λijhj

• Wt[h|h0];

−Λij = ∆ij − κ δi,jδi,0.

1

Replace matrix Λ by number λ: 1d Ornstein-Uhlenbeck process.

2

Wt[h|h0] =

• det
• Λ

2π(1−e−2Λt)

• exp
• − 1

2(h −e−Λth0)T Λ 1−e−2Λt (h −e−Λth0)

• .

Shamik Gupta Dynamics of a tagged monomer

Flat initial condition

t = 0 :

t > 0 : T = 1 + elastic pinning with spring constant κ.

Shamik Gupta Dynamics of a tagged monomer

Equilibrated initial condition

t = 0 : Equilibrated at temp. T0

t > 0 : T = 1 + elastic pinning with spring constant κ.

Shamik Gupta Dynamics of a tagged monomer

Elastic pinning: Exact results

2 4 6 1 10 100 1000 10000

Time t T0 = 0 κ = 0.25 L = 200 T0 = 4 h2

0(t)

T0 = 1 h2

0(t) ≃ 1 κ

• 1 + T0−1

κ

• 2

πt − T0c1 κ2t + · · ·

• .

Shamik Gupta Dynamics of a tagged monomer

Absorption in an interval

Absorbing boundaries t > 0 : ∂Wt[h|h0] ∂t

=

i ∂2 ∂h2

i −

i,j

• ∂

∂hi ∆ijhj

• Wt[h|h0].

Shamik Gupta Dynamics of a tagged monomer

Absorption in an interval

Absorbing boundaries t > 0 : ∂Wt[h|h0] ∂t

=

i ∂2 ∂h2

i −

i,j

• ∂

∂hi ∆ijhj

• Wt[h|h0].

Absorbing boundary conds. for the tagged monomer.

Shamik Gupta Dynamics of a tagged monomer

Harmonic absorption

t > 0 :

Absorption probability ∝ µh2

0(t). ∂Wt[h|h0] ∂t

=

i ∂2 ∂h2

i −

i,j

• ∂

∂hi ∆ijhj + hiAijhj

• Wt[h|h0];

Aij = µ δi,j δi,0.

Shamik Gupta Dynamics of a tagged monomer

Harmonic absorption: Exact results

6 12 1 10 100 1000

Time t T0 = 0 T0 = 4 4µ = 0.0025 L = 200 T0 = 1 h2

0(t) abs

T0 = 0 T0 = 1 10 10−3 103 104 Time t δh2

Shamik Gupta Dynamics of a tagged monomer

Survival probability

1

S(t): survival probability of an initial configuration h0.

Shamik Gupta Dynamics of a tagged monomer

Survival probability

1

S(t): survival probability of an initial configuration h0.

2

∂Wt[h|h0] ∂t

=

i ∂2 ∂h2

i −

i,j

• ∂

∂hi ∆ijhj + hiAijhj

• Wt[h|h0].

Shamik Gupta Dynamics of a tagged monomer

Survival probability

1

S(t): survival probability of an initial configuration h0.

2

∂Wt[h|h0] ∂t

=

i ∂2 ∂h2

i −

i,j

• ∂

∂hi ∆ijhj + hiAijhj

• Wt[h|h0].

3

∂tS(t) = −µ h2

0(t) S(t).

Shamik Gupta Dynamics of a tagged monomer

Survival probability

1

S(t): survival probability of an initial configuration h0.

2

∂Wt[h|h0] ∂t

=

i ∂2 ∂h2

i −

i,j

• ∂

∂hi ∆ijhj + hiAijhj

• Wt[h|h0].

3

∂tS(t) = −µ h2

0(t) S(t).

4

S(t) = exp

• −µ

t

0 dτ h2 0(τ)

• .

10-3 10-2 10-1 1 1500

L = 200 4µ = 0.0025 T0 = 0 T0 = 1 T0 = 4 Time t S(t)

Consistent with simulations (Kantor and Kardar (2004)).

Shamik Gupta Dynamics of a tagged monomer

Conclusions

1

Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results.

Shamik Gupta Dynamics of a tagged monomer

Conclusions

1

Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results.

2

Strong memory effects:

Shamik Gupta Dynamics of a tagged monomer

Conclusions

1

Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results.

2

Strong memory effects:

Pinning case: Relaxation as 1/√t, unless evolution at the same temp. as that of the initial eqlbm. when relaxation as 1/t. Non-monotonic relaxation depending on the initial eqlbm. temp.

Shamik Gupta Dynamics of a tagged monomer

Conclusions

1

Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results.

2

Strong memory effects:

Pinning case: Relaxation as 1/√t, unless evolution at the same temp. as that of the initial eqlbm. when relaxation as 1/t. Non-monotonic relaxation depending on the initial eqlbm. temp. Absorption case: Relaxation always as 1/t. Non-monotonic relaxation, except for T0 = 0.

Shamik Gupta Dynamics of a tagged monomer

Conclusions

1

Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results.

2

Strong memory effects:

Pinning case: Relaxation as 1/√t, unless evolution at the same temp. as that of the initial eqlbm. when relaxation as 1/t. Non-monotonic relaxation depending on the initial eqlbm. temp. Absorption case: Relaxation always as 1/t. Non-monotonic relaxation, except for T0 = 0.

3

Analysis may be generalized to a Rouse chain in d dimensions or a d-dimensional EW interface, by using the corresponding Laplacian matrix in place of ∆.

Shamik Gupta Dynamics of a tagged monomer

Conclusions

1

Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results.

2

Strong memory effects:

Pinning case: Relaxation as 1/√t, unless evolution at the same temp. as that of the initial eqlbm. when relaxation as 1/t. Non-monotonic relaxation depending on the initial eqlbm. temp. Absorption case: Relaxation always as 1/t. Non-monotonic relaxation, except for T0 = 0.

3

Analysis may be generalized to a Rouse chain in d dimensions or a d-dimensional EW interface, by using the corresponding Laplacian matrix in place of ∆.

4

Hydrodynamic effects for the chain or long-range elastic interactions for the interface may be included by replacing ∆ with the corresponding fractional Laplacian −(−∆)z/2.

Shamik Gupta Dynamics of a tagged monomer