# 物理代写|流体力学代写Fluid Mechanics代考|PHYSICS7546

## 物理代写|流体力学代写Fluid Mechanics代考|Fractal Structures

Near the critical point of gas-to-liquid phase transition, $C_n(r)(9.27)$ showed long range correlation. At the critical point, where the correlation length $\xi$ diverges, $C_n(r) \sim r^{-\alpha}$, where $\alpha=d-2+\eta$, and $d$ is space dimension. Using (9.41)
$$S(\boldsymbol{q}) \sim \int d^d r e^{-i q \cdot \boldsymbol{r}} r^{-\alpha}=q^{-d+\alpha} \int d^d(q r) e^{-i q \cdot r}(q r)^{-\alpha} \sim q^{-d+o} .$$
The power laws both in the correlation and the structure implies an absence of any characteristic length scales, i.e., the structure looks the same at any magnification. This scale-invariant self-similar structure is called fractal.

Fractals are ubiquitous in nature (in the systems at thermodynamic critical points as well as in complex systems, e.g., polymers, snowflakes, colloidal aggregates, coastlines), and also can be artificially designed. Application of the concept of fractal nature may be valuable when measuring the properties of irregular biological structures, such as living organs (Fig. 9.5b). Consider a fractal of size $R$ that contains $N$ particles or units. The structure of a random fractal is characterized by the fractal dimension, which is defined by the way in which $N$ changes with $R$. For ordinary compact structures in $3 D, N \sim(R / l)^3$, where $l$ is inter-particle distance. For isotropic fractals,
$$N \sim(R / l)^{D_j}$$

where $D_f$, called the fractal dimension, is less than 3 and can also be a non-integer number. For example, $D_f$ of an ideal polymer chain is 2, as shown below. The fractal dimension is related to the radial distribution $g(r)$ and its Fourier transform $S(q)$ as following. Consider the number $N(r)$ of particles within a radius $r(l \ll r \ll R)$ from a central particle deep within the fractal. By the definition of radial distribution function, the number of particles within a shell of thickness $d r$ located at distance $r$ (Fig. 4.4) is
$$d N(r) \sim g(r) r^{d-1} d r .$$
This, along with $N(r) \sim(r / l)^{D_f}$, leads to $g(r) \sim(r / l)^{-d+D_f}$, so $(9.43)$ yields
$$S(q) \sim(q l)^{-D_f} \sim q^{-D_f},$$
for the region of moderate $q\left(R^{-1} \ll q \ll l^{-1}\right)$, in which scale-invariance is expected. The fractal dimension $D_f$ can be read from the power law decay of the structure factor (9.57) tells us $D_f=d-\alpha=2-\eta$ for a fluid at the critical point. The scattering for very small $q$, on the other hand, senses the large lengths beyond the finite size of the system $R$, on which the structure factor depends. A flexible chain studied below serves as another example and allows an analytical understanding of the features mentioned above.

## 物理代写|流体力学代写Fluid Mechanics代考|Structure Factor of a Flexible Polymer Chain

The polymer structure can be probed by scattering experiments, such as small angle $\mathrm{x}$-ray scattering (SAXS) and small angle neutron scattering (SANS). The scattering intensity for a single chain is proportional to the structure factor $S(\boldsymbol{q})=$ $N^{-1}\left\langle\sum_{n, m}^N e^{-i q \cdot r_{m m}}\right\rangle(9.36)$, where $N$ is the number of beads that compose the polymer, and $\boldsymbol{r}{n m}=\boldsymbol{r}_n-\boldsymbol{r}_m$ is the distance between the $n$th and $m$ th beads. Averaging over the orientations of the vector $\boldsymbol{r}{n m}$ yields
$$\left\langle e^{-i q \cdot r_{r_m}}\right\rangle=\frac{1}{4 \pi} \int_0^{2 \pi} d \varphi \int_{-1}^1 d \cos \theta\left\langle e^{-i \cos \theta q r_{m m} \mid}\right\rangle=\left\langle\frac{\sin \left(q\left|\boldsymbol{r}{n m}\right|\right)}{q\left|\boldsymbol{r}{n m}\right|}\right\rangle .$$
For small $q$, or small scattering angle $\theta(q=2 k \sin (\theta / 2)$, Fig. 9.1),

\begin{aligned} S(\boldsymbol{q}) &=N^{-1} \sum_{n, m}^N\left\langle\frac{\sin \left(q\left|\boldsymbol{r}{n m}\right|\right)}{q\left|\boldsymbol{r}{n m}\right|}\right\rangle \ & \approx N^{-1} \sum_{n, m}^N\left\langle 1-\frac{1}{6} q^2\left|\boldsymbol{r}{n m}\right|^2\right\rangle=N\left(1-\frac{1}{3} q^2 R_G^2\right) \end{aligned} where the radius of gyration $R_G$ defined by $$R_G^2=\frac{1}{2 N^2} \sum{n, m}^N\left\langle\left(\boldsymbol{r}n-\boldsymbol{r}_m\right)^2\right\rangle=\frac{1}{N} \sum{n=1}^N\left\langle\left(\boldsymbol{r}n-\boldsymbol{R}{c m}\right)^2\right\rangle$$
represents the chain size $R$ and $\boldsymbol{R}{c m}$ is the center of mass position. Therefore, from the data of small $q$ or small angle scattering, one can get information about the radius of gyration. For a chain in which $\boldsymbol{r}{n m}$ is distributed in Gaussian,
\begin{aligned} S(\boldsymbol{q}) &=N^{-1} \sum_{n, m}^N\left\langle e^{-i \boldsymbol{q} \cdot \boldsymbol{r}{\boldsymbol{m}}}\right\rangle=N^{-1} \sum{n, m}^N \exp \left(-\frac{1}{2}\left\langle\left(\boldsymbol{q} \cdot \boldsymbol{r}{n m}\right)^2\right\rangle\right) \ &=N^{-1} \sum{n, m}^N \exp \left(-\frac{1}{6} q^2\left\langle\boldsymbol{r}_{n m}^2\right\rangle\right) \end{aligned}

## 物理代写|流体力学代写Fluid Mechanics代考|Fractal Structures

$$S(\boldsymbol{q}) \sim \int d^d r e^{-i q \cdot r} r^{-\alpha}=q^{-d+\alpha} \int d^d(q r) e^{-i q \cdot r}(q r)^{-\alpha} \sim q^{-d+o} .$$

$$N \sim(R / l)^{D_j}$$

$$d N(r) \sim g(r) r^{d-1} d r .$$

$$S(q) \sim(q l)^{-D_f} \sim q^{-D_f},$$

## 物理代写|流体力学代写Fluid Mechanics代考|Structure Factor of a Flexible Polymer Chain

$$\left\langle e^{-i q \cdot r_{r_m}}\right\rangle=\frac{1}{4 \pi} \int_0^{2 \pi} d \varphi \int_{-1}^1 d \cos \theta\left\langle e^{-i \cos \theta q r_{m m} \mid}\right\rangle=\left\langle\frac{\sin (q|\boldsymbol{r} n m|)}{q|\boldsymbol{r} n m|}\right\rangle .$$

$$S(\boldsymbol{q})=N^{-1} \sum_{n, m}^N\left\langle\frac{\sin (q|\boldsymbol{r} n m|)}{q|\boldsymbol{r} n m|}\right\rangle \quad \approx N^{-1} \sum_{n, m}^N\left\langle 1-\frac{1}{6} q^2|\boldsymbol{r} n m|^2\right\rangle=N\left(1-\frac{1}{3} q^2 R_G^2\right)$$

$$R_G^2=\frac{1}{2 N^2} \sum n, m^N\left\langle\left(\boldsymbol{r} n-\boldsymbol{r}m\right)^2\right\rangle=\frac{1}{N} \sum n=1^N\left\langle(\boldsymbol{r} n-\boldsymbol{R} c m)^2\right\rangle$$ 表示链大小 $R$ 和 $\boldsymbol{R} \mathrm{cm}$ 是质心位置。因此，从小数据 $q$ 或小角度散射，可以获得有关回转半径的信息。对于 一个链，其中 $r n m$ 以高斯分布， $$S(\boldsymbol{q})=N^{-1} \sum{n, m}^N\left\langle e^{-i \boldsymbol{q} \cdot \boldsymbol{r m}}\right\rangle=N^{-1} \sum n, m^N \exp \left(-\frac{1}{2}\left\langle(\boldsymbol{q} \cdot \boldsymbol{r} n m)^2\right\rangle\right) \quad=N^{-1} \sum n, m^N \exp \left(-\frac{1}{6}\right.$$

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