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ResultsSummaryAppendix

Molecular Dynamics and the Rouse ModelAn Introduction to the Physics of Polymers

Daniel Bridges1 Dr. Aniket Bhattacharya2

1Department of Physics & AstronomyMiddle Tennessee State University

2Department of PhysicsUniversity of Central Florida

July 28, 2009

Daniel Bridges, Dr. Aniket Bhattacharya Molecular Dynamics and the Rouse Model

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Why am I here?Polymer physics

Polymers are essential to life.

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Don’t sneeze; you might lose some polymers.

We can model a simplified DNAstrand as a beaded necklace.

Rouse model: ”Bead-springmodel”

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Don’t sneeze; you might lose some polymers.

We can model a simplified DNAstrand as a beaded necklace.

Rouse model: ”Bead-springmodel”

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Polymers sit in solvents!

Polymer/solvent interaction (varies with temp.) dictates threesituations:

1 polymer ”stretched” ⇒ ”good solvent”

2 ideal random walk ⇒ ”theta condition”

3 collapses to form compact globule

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modeling this behavior ...

~F = −~∇U

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1 polymer ”stretched” ⇒ ”good solvent”2 ideal random walk ⇒ ”theta condition”3 collapses to form compact globule

~F = −~∇U − ~FR(t)

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~F = −~∇U − ~FR(t)− ζ d~rdt

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Langevin dynamics

~F = −~∇U − ~FR(t)− ζ d~rdt = m d2~r

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Langevin dynamics

~F = −~∇U − ~FR(t)− ζ d~rdt = m d2~r

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Relaxation time = MD equilibriation

Want polymer to be ”at rest”.

Important for MD accuracy and clarity

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Verlet AlgorithmThe Rouse ModelThe Zimm ModelSome Topics Considered

Verlet Algorithm is ”numerical integration” of motion eqns.

Next position ~r(t + δt) ”future” determined by

current position ~r(t) and acceleration ~a(t): t ⇒ ”now”

previous position ~r(t − δt): t − δt ⇒ ”past”

~r(t + δt) = ~r(t) + δt~v(t) + 12δt2~a(t) + ...

~r(t − δt) = ~r(t)− δt~v(t) + 12δt2~a(t) + ...

The Verlet Algorithm

~r(t + δt) = 2~r(t)−~r(t − δt) + δt2~a(t) + ...

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~r(t + δt) = ~r(t) + δt~v(t) + 12δt2~a(t) + ...

~r(t − δt) = ~r(t)− δt~v(t) + 12δt2~a(t) + ...

~r(t + δt) = 2~r(t)−~r(t − δt) + δt2~a(t) + ...

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~r(t + δt) = ~r(t) + δt~v(t) + 12δt2~a(t) + ...

~r(t − δt) = ~r(t)− δt~v(t) + 12δt2~a(t) + ...

~r(t + δt) = 2~r(t)−~r(t − δt) + δt2~a(t) + ...

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~r(t + δt) = ~r(t) + δt~v(t) + 12δt2~a(t) + ...

~r(t − δt) = ~r(t)− δt~v(t) + 12δt2~a(t) + ...

~r(t + δt) = 2~r(t)−~r(t − δt) + δt2~a(t) + ...

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Basic assumptions of the the ”Bead-Spring” Model

N ”beads” (monomers are points) with

spring constant ksp = 3kBTb2

No hydrodynamic interactions betweenbeads

Proposed by P. E. Rouse, J Chem Phys 21,1272 (1953).

Drawbacks: Does not accurately express diffusion coefficient orrelaxation time.

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Building on Rouse: the Zimm model

addresses hydrodynamicinteractions between beads

the diffusion coefficient andrelaxation times agree withexperiment.

Proposed by B. H. Zimm, J ChemPhys 24, 269 (1956).

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Interesting Quantities (Rouse)

Average Radius of Gyration:

R2g = 1

∑npartn=1

⟨(~Rn − ~RG )2

⟩End-to-end Distance:

∥∥∥~R1N

∥∥∥ =∥∥∥∑N

n=1~rn

∥∥∥Diffusion Constant:DG = kBT

Nζ = 16t

⟨‖~ri (t)−~ri (0)‖2

⟩Relaxation Time: τr ' Nb2

ζ, (viscous) frictioncoefficient

N, number ofmonomers

kB , Boltzmannconstant

T , temperature

τ , relaxation time

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Diffusion Constant

DG = kBTNζ = 1

⟨‖~ri (t)−~ri (0)‖2

⟩Rahman: 2.43× 10−5 cm2

sMine: 2.47× 10−5 cm2

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Future Outlook

Future goals:

understanding forced translocation

manipulate polymer movement

gene therapyvirus injectionprotein sequencing

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Future Outlook

Future goals:

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Future Outlook

Future goals:

gene therapy

virus injectionprotein sequencing

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Future Outlook

Future goals:

gene therapyvirus injection

protein sequencing

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Future Outlook

Future goals:

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Future Outlook

Thank you!

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For Further Reading

For Further Reading I

Allen, M.P., Tildesley, D.J. (1987).Computer Simulation of Liquids.Oxford: Clarendon Press.

Doi, M. (1995).Introduction to Polymer Physics.Oxford: Clarendon Press

Jones, R. A. L. (2002).Soft Condensed Matter.Oxford: Oxford University Press.

Teraoka, I. (2002).Polymer Solutions: An Introduction to Physical PropertiesJohn Wiley and Sons, Inc.

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For Further Reading

For Further Reading II

Rahman, A. (1964).Correlations in the Motion of Atoms in Liquid Argon.Physical Review, 136, Number 2A

molecular dynamics and the rouse modelww2.che.ufl.edu/reu/student_presentations/2009/... ·...

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