Definition and equations of motion

One obvious way to improve on the Rouse chain, is by introducing hydrodynamic interactions  between the beads. The chain so defined is called the Zimm chain . The equations describing hydrodynamic interactions have been obtained in section 5.4.

Let us now derive the equations of motion of the beads in a Zimm chain. Let $\Psi =\Psi ( \vec{R}_{0},\ldots ,\vec{R}_{N};t)$ be the probability density of finding particles $0,\ldots ,N$ near $\vec{R}_{0},\ldots , \vec{R}_{N}$ at time t. The time development of $\Psi$ obeys

$$\frac{\partial \Psi }{\partial t}=-\sum_{j=0}^{N}\vec{\nabla}_{j}\cdot \vec{ \mathcal{J}}_{j}$$ (7.1)

where $\vec{\mathcal{J}}_{j}$ is the flux of particles j. This flux may be written as

$$\vec{\mathcal{J}}_{j}=-\sum_{k}\bar{\mu}_{jk}\cdot (\vec{\nabla}_{k}\Phi )\Psi -\sum_{k}\bar{D}_{jk}\cdot \vec{\nabla}_{k}\Psi$$ (7.2)

The first term results from the forces $-\vec{\nabla}_{k}\Phi$ felt by all the beads. On the Smoluchowski time scale , these forces make the beads move with constant velocities $\vec{v}_{k}$. This amounts to saying that the forces $-\vec{\nabla}_{k}\Phi$are exactly balanced by the hydrodynamic forces acting on the beads k, i.e. the forces exerted by the fluid on the beads k are equal to $\vec{ \nabla}_{k}\Phi$. Introducing these forces into Eq. (5.54), we find the systematic part of the velocity of bead j:

$$\vec{v}_{j}=-\sum_{k}\bar{\mu}_{jk}\cdot (\vec{\nabla}_{k}\Phi )$$ (7.3)

Multiplying this velocity by $\Psi$, we obtain the systematic part of the flux of particle j. The second term in Eq. (7.2) stems from the random displacements of all beads, which result in a flux along the negative gradient of the probability density.

Combining Eqs. (7.1) and (7.2) we find

$$\frac{\partial \Psi }{\partial t}=\sum_{j}\sum_{k}\vec{\nabla... ...k}\cdot \vec{\nabla} _{k}\Phi +\vec{\nabla}_{k}\ln \Psi \}\Psi$$ (7.4)

where $\bar{D}_{jk}^{-1}$ is the inverse of $\bar{D}_{jk}$. In the stationary state the right hand side must be zero, and $\Psi =c\exp \{-\beta \Phi \}$. From this it follows that

$$\bar{D}_{jk}=k_{B}T\bar{\mu}_{jk}$$ (7.5)

which is a generalization of the Einstein equation .

The equation of motion of the probability density, i.e. the Smoluchowski equation , now reads

$$\frac{\partial \Psi }{\partial t}=\sum_{j}\sum_{k}\vec{\nabla... ... \{\vec{\nabla}_{k}\Phi +k_{B}T\vec{\nabla}_{k}\ln \Psi \}\Psi$$ (7.6)

The Langevin equations  corresponding to this Smoluchowski equation are

 

$$\frac{d\vec{R}_{j}}{dt} -\sum_{k}\bar{\mu}_{jk}\cdot \vec{\nabla}_{k}\Phi +k_{B}T\sum_{k}\vec{\nabla}_{k}\cdot \bar{\mu}_{jk}+\vec{f}_{j}$$ = (7.7)

$$\langle \vec{f}_{j}(t)\vec{f}_{k}(t^{\prime })\rangle 2k_{B}T\bar{\mu} _{jk}\delta (t-t^{\prime })$$ = (7.8)

The reader can easily check that these reduce to the equations of motion of the Rouse chain when hydrodynamic interactions are neglected.

Let us note at this point that for the approximation of the mobility tensor we make use of

$$\vec{\nabla}_k \cdot \bar{\mu}_{jk} = 0,$$ (7.9)

which greatly simplifies Eq. (7.7).