## 物理代写|流体力学代写Fluid Mechanics代考|Random Walk Model for a Flexible Chain

We consider a flexible polymer that has the contour length much longer than its persistence length, e.g., a $1 \mu$ long ss DNA fragment. We introduce the ideal chain model, in which the chain conformation is made by a random walk. In this model, a chain consists of a large number $(N)$ of freely-jointed links each with length $l$ that we studied in Chap. 3 (Fig. 10.2). This segmental length $l$, called the Kuhn length, is not necessarily the molecular bond length, but is introduced to represent the length over which the link orientation is uncorrelated, namely,

$$\left\langle\boldsymbol{l}i \cdot \boldsymbol{l}_j\right\rangle=l^2 \delta{i j},$$
where $l_i$ is the ith link vector. $\langle\cdots\rangle$ denotes the average over equilibrium ensemble of the chains of $N$ links, and $\delta_{i j}$ is the Kronecker delta function, which is 1 if $i=j$, and 0 otherwise.

Let us characterize the conformational state of the chain by its end-to-end distance (EED) vector,
$$\boldsymbol{R}=\sum_{i=1}^N \boldsymbol{l}_i .$$
Taking the ensemble averages, we have
$$\langle\boldsymbol{R}\rangle=0$$
and
$$\left\langle\boldsymbol{R}^2\right\rangle=N l^2 \equiv R_0^2 .$$
The root-mean-squared (rms) EED for the ideal chain
$$R_0 \equiv\left\langle\boldsymbol{R}^2\right\rangle^{1 / 2}=N^{1 / 2} l$$
is a measure of the natural size of the chain.

## 物理代写|流体力学代写Fluid Mechanics代考|The Entropic Chain

In light of the coarse-grained description described in Chap. 5, the relevant degree of freedom for the chain is $\mathcal{Q}=\boldsymbol{R}$, and its distribution is $P(\mathcal{Q}) \propto e^{-\beta \mathcal{F}(\mathcal{Q})}$ (5.5). Then, the (10.6) allows us to identify the chain’s effective Hamiltonian or the free energy function associated with $R$ as
$$\mathcal{F}(\boldsymbol{R})=\frac{3 k_B T}{2 N l^2} \boldsymbol{R}^2,$$
apart from a term $\sim k_B T \ln N$, which is independent of $R$ so is irrelevant. By virtue of the thermodynamic relations introduced in Chap. 2, the associated entropy function is
$$S(\boldsymbol{R})=-\frac{\partial \mathcal{F}(\boldsymbol{R})}{\partial T}=-\frac{3 k_B}{2 N l^2} \boldsymbol{R}^2$$
This demonstrates that as the chain is extended ( $R$ increases) the entropy decreases. When $\boldsymbol{R}=0$, the free energy is minimum, and the entropy is maximum; it is because the number of chain (random walk) configurations is maximal. Although (10.18) reasonably describes the entropy change associated with the extension, it neglects other contributions that are irrelevant to $\boldsymbol{R}$. To keep the EED of the chain at $\boldsymbol{R}$, a force

$$\boldsymbol{f}(\boldsymbol{R})=\frac{\partial \mathcal{F}(\boldsymbol{R})}{\partial \boldsymbol{R}}=K_e \boldsymbol{R}$$
must be applied along the direction in which the entropy decreases. Here
$$K_e=\frac{3 k_B T}{N l^2}$$
called the entropic spring constant (Fig. 10.3), increases with temperature but decreases with contour length. This remarkable behavior of chain entropy and flexibility is indeed the emergent behaviors of a long chain. This behavior was derived earlier using the freely-jointed chain model in Chap. $3 .$

## 物理代写|流体力学代写Fluid Mechanics代考|Random Walk Model for a Flexible Chain

$$\left\langle\boldsymbol{l i} \cdot \boldsymbol{l}j\right\rangle=l^2 \delta i j,$$ 在哪里 $l_i$ 是第 $\mathrm{i}$ 个链接向量。 $\langle\cdots\rangle$ 表示链的平均过平衡集合 $N$ 链接，以及 $\delta{i j}$ 是克罗内克三角函数，如果是 $1 i=j$, 否则为 0 。

$$\boldsymbol{R}=\sum_{i=1}^N \boldsymbol{l}_i .$$

$$\langle\boldsymbol{R}\rangle=0$$

$$\left\langle\boldsymbol{R}^2\right\rangle=N l^2 \equiv R_0^2 .$$

$$R_0 \equiv\left\langle\boldsymbol{R}^2\right\rangle^{1 / 2}=N^{1 / 2} l$$

## 物理代写|流体力学代写Fluid Mechanics代考|The Entropic Chain

$$\mathcal{F}(\boldsymbol{R})=\frac{3 k_B T}{2 N l^2} \boldsymbol{R}^2,$$

$$S(\boldsymbol{R})=-\frac{\partial \mathcal{F}(\boldsymbol{R})}{\partial T}=-\frac{3 k_B}{2 N l^2} \boldsymbol{R}^2$$

$$\boldsymbol{f}(\boldsymbol{R})=\frac{\partial \mathcal{F}(\boldsymbol{R})}{\partial \boldsymbol{R}}=K_\epsilon \boldsymbol{R}$$

$$K_e=\frac{3 k_B T}{N l^2}$$

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