Rouse model

Schematic representation of the Rouse model as a chain of mass points (blue circles) and connecting springs (gray) with N = 13 mass points and the mean distance l between two mass points

The Rouse model is one of the simplest models used in polymer physics for the dynamics of polymers .

description

The Rouse model describes the polymer as a perfect chain of mass points (often called English beads called), which are connected by springs. The change in the conformation over time is then realized by Brownian movement of the individual mass points, in that a random (thermal) force acts on each. No volume exclusion effects are taken into account, so the polymer can cross over with itself. The model was proposed by Prince E. Rouse in 1953 .

The Rouse model leads to the following stochastic differential equation ( Langevin equation ) for the position of the -th mass point: ${\ displaystyle {\ vec {R}} _ {n}}$ ${\ displaystyle n}$

{\ displaystyle {\ frac {\ mathrm {d} {\ vec {R}} _ {n}} {\ mathrm {d} t}} = \ underbrace {{\ frac {k} {\ zeta}} \ cdot \ left ({\ vec {R}} _ {n-1} - {\ vec {R}} _ {n} + {\ vec {R}} _ {n + 1} - {\ vec {R}} _ {n} \ right)} _ {\ text {Interaction with neighboring members}} + \ underbrace {{\ vec {f}} _ {n} (t)} _ {\ text {Random force}}}

Here is the spring constant of the springs in the model, the coefficient of friction of a mass point in the solvent and the number of chain links. The term is the linear restoring force from the spring to the predecessor and to the successor (one of the two terms is omitted at the two free ends of the polymer). The term describes a thermal random force (Brownian molecular dynamics) that has no preferred direction and is spatially and temporally uncorrelated. The following properties of the polymer result from this approach: ${\ displaystyle k}$ ${\ displaystyle \ zeta}$ ${\ displaystyle N}$ ${\ displaystyle k ({\ vec {R}} _ {n \ pm 1} - {\ vec {R}} _ {n})}$ ${\ displaystyle n-1}$ ${\ displaystyle n + 1}$ ${\ displaystyle {\ vec {f}} _ {n} (t)}$

Diffusion coefficient of the center of gravity: ( : temperature : Boltzmann's constant ) ${\ displaystyle D_ {G} = {\ frac {k _ {\ mathrm {B}} T} {N \ zeta}}}$ ${\ displaystyle T}$ ${\ displaystyle k _ {\ mathrm {B}}}$

Rotational relaxation time:

{\ displaystyle \ tau _ {R} = {\ frac {\ zeta N ^ {2} l ^ {2}} {3 \ pi ^ {2} k _ {\ mathrm {B}} T}}}

mean square displacement of a single segment:

{\ displaystyle \ left \ langle {\ vec {R}} _ {n} ^ {2} (\ tau) \ right \ rangle = \ left \ langle \ left ({\ vec {R}} _ {n} ( t + \ tau) - {\ vec {R}} _ {n} (t) \ right) ^ {2} \ right \ rangle \ approx {\ frac {2Nl ^ {2}} {\ pi ^ {3/2 }}} {\ sqrt {\ frac {\ tau} {\ tau _ {R}}}}}

Extension: The Zimm model

Hydrodynamic interaction: The force F _n (red) acts on a segment n . This leads to a local flow shown in green, which in turn affects the neighboring segments (forces as small black arrows).

An important extension was published by Bruno Zimm in 1956 : his model (often referred to simply as the " Zimm model ") also takes into account hydrodynamic interactions between the mass points of the chain. These are interactions (forces) that are mediated by the solvent molecules around the polymer: The mass points pull the solvent molecules with them as they move, which also leads to a force on neighboring chain links (see figure on the right). The Zimm model leads to a better description of real polymers than the Rouse model, which also agrees with experimental data for dilute solutions of certain polymers.

The above Langevin equation for the Rouse model is extended by a tensor (matrix) that represents the hydrodynamic force between the -th and -th segment: ${\ displaystyle \ mathrm {H} _ {nm}}$ ${\ displaystyle n}$ ${\ displaystyle m}$

{\ displaystyle {\ frac {\ mathrm {d} {\ vec {R}} _ {n}} {\ mathrm {d} t}} = \ underbrace {k \ cdot \ sum \ limits _ {m} \ mathrm {H} _ {nm} \ left ({\ vec {R}} _ {n-1} - {\ vec {R}} _ {n} + {\ vec {R}} _ {n + 1} - {\ vec {R}} _ {n} \ right)} _ {\ text {Interaction with neighboring members + hydrodynamics}} + \ underbrace {{\ vec {f}} _ {n} (t)} _ {\ text {Random force}}}

It should be noted that the tensor depends on the positions of all segments. As a result, the above Langevin equation is non-linear and can no longer be easily solved. Bruno Zimm therefore replaced it with his equilibrium mean , which can be calculated. The following properties of a zimm polymer are derived from this: ${\ displaystyle \ mathrm {H} _ {nm}}$ ${\ displaystyle {\ vec {R}} _ {0}, ..., {\ vec {R}} _ {N-1}}$ ${\ displaystyle \ mathrm {H} _ {nm} ({\ vec {R}} _ {0}, ..., {\ vec {R}} _ {N-1})}$ ${\ displaystyle \ langle \ mathrm {H} _ {nm} \ rangle _ {eq}}$

Diffusion coefficient of the center of mass: ( : viscosity of the solvent) ${\ displaystyle D_ {G} = {\ frac {8k _ {\ mathrm {B}} T} {3 {\ sqrt {6 \ pi ^ {3}}} \ eta _ {s} {\ sqrt {N}} \ cdot l}}}$ ${\ displaystyle \ eta _ {s}}$

Rotational relaxation time:

{\ displaystyle \ tau _ {R} = {\ frac {\ eta _ {S} ({\ sqrt {N}} l) ^ {3}} {{\ sqrt {3 \ pi}} k _ {\ mathrm { B}} T}}}

mean square displacement of a single segment:

{\ displaystyle \ left \ langle {\ vec {R}} _ {n} ^ {2} (\ tau) \ right \ rangle = {\ frac {2 \ Gamma (1/3) Nl ^ {2}} { \ pi ^ {2}}} \ left ({\ frac {\ tau} {\ tau _ {R}}} \ right) ^ {2/3}}

Experimental observation

In this section we want to point out some real polymers that can be described well with one of the above models:

Single-stranded DNA is a relatively flexible polymer and shows zimmer-like dynamics for individual segments on short time scales in dilute solution, as could be shown with fluorescence correlation spectroscopy .
Double-stranded DNA is significantly stiffer than single-stranded DNA, so the hydrodynamic interactions play a much smaller role, so that their monomer dynamics in dilute solution can be described well with the Rouse model.

literature

Masao Doi: Introduction to polymer physics Oxford University Press, USA, 1996, ISBN 0-19-851772-6
Masao Doi, SF Edwards: The theory of polymer dynamics Oxford University Press, USA, 1988, ISBN 978-0-19-852033-7

Individual evidence

^ Prince E. Rouse, A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers , J. Chem. Phys. 21, 1272 (1953), cited over 1000 times by 2010.

↑ The mean square displacement ( english mean squared displacement or short MSD) is a good size to characterize the dynamics of a randomly moving particle (see random walk ). It measures the mean distance that a particle travels from its origin in a period of time. For a freely diffusing particle (in 3 dimensions) the relationship with the diffusion coefficient results . Deviations from this linear behavior (normal diffusion) are referred to as anomalous diffusion . Then the more general form often applies (as in the present case) . ${\ displaystyle \ left \ langle r (\ tau) \ right \ rangle}$ ${\ displaystyle \ tau}$ ${\ displaystyle \ left \ langle r ^ {2} (\ tau) \ right \ rangle = 6D \ cdot \ tau}$ ${\ displaystyle D}$ ${\ displaystyle \ left \ langle r ^ {2} (\ tau) \ right \ rangle = \ Gamma \ cdot \ tau ^ {\ alpha}}$

↑ Bruno H. Zimm, Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss , J. Chem. Phys. 24, 269 (1956).

↑ ^a ^b Roman Shusterman, Sergey Alon, Tatyana Gavrinyov, Oleg Krichevsky: Monomer Dynamics in Double- and Single-Stranded DNA Polymers . In: Physical Review Letters . tape 92 , no. January 4 , 2004, ISSN 0031-9007 , doi : 10.1103 / PhysRevLett.92.048303 ( PDF; 158kB ).

[1] Prince E. Rouse, A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers , J. Chem. Phys. 21, 1272 (1953), cited over 1000 times by 2010.