# 物理代写|热力学代写thermodynamics代考|AMME2262

## 物理代写|热力学代写thermodynamics代考|Thermodynamics of Surfaces

Following the thermodynamics approach we have learned so far, a surface or an interface will be treated as:

1. A 2D phase as an approximation. This is based on the fact that the thickness of the surface or interface is negligible in comparison with its lateral dimensions.
2. A uniform phase, and all surface properties $\left(\mathrm{U}, \mathrm{S}, \mathrm{N}1, \ldots \mathrm{N}{\mathrm{r}}\right.$ ) are considered as the average values over the interfacial thickness. Because the properties of a real interface is non-uniform only in the thickness direction.
3. In this way, a 3D, heterogeneous interfacial phase is approximated as a 2D, homogeneous, simple thermodynamic phase. All the thermodynamic principles and the equations for simple systems are applicable to such a surface phase. For a 3D bulk phase system, volume $\mathrm{V}$ is used to characterize the size of the system, and the volume change is used to characterize the mechanical work (compression or expansion). However, for a 2D surface phase, there is no volume, instead, the surface area A is used to characterize the size of the system, and the surface area change is used to characterize the mechanical work. Therefore, the fundamental equations of a surface phase are given by:
4. $$5. \mathrm{S}{\mathrm{A}}=\mathrm{S}{\mathrm{A}}\left(U_A, A, N_{1 A}, \ldots N_{r A}\right) 6.$$
7. and
8. $$9. \mathrm{U}{\mathrm{A}}=\mathrm{U}{\mathrm{A}}\left(S_A, A, N_{1 A}, \ldots N_{r A}\right) 10.$$
11. In the above equations, the variable $A$ is the surface area $\left(\mathrm{m}^2\right)$, and the subscript $A$ indicates the surface phase.
12. Generally, surfaces or interfaces may not be flat and they are curved. It should be mentioned that there are no curvature variables in these fundamental equations; therefore, these fundamental equations are valid only for moderately curved interfaces where the curvature effects of the interface are negligible. What is a moderately curved interface? Generally, when the radius of curvature of an interface is comparable to the thickness of the interface, such an interface is highly curved. If the radius of curvature is at least 100 times greater than the thickness of the interface, such an interface is moderately curved. For a moderately curved interface, the only geometrical variable required to describe the interface is the surface area. However, for a highly curved interface, certain curvature variables are also required to describe the interface, in addition to the surface area. This is because the energy associated with the shape change (such as shape change due to bending or twisting) of the interface is significant part of the total energy of the interface.

## 物理代写|热力学代写thermodynamics代考|Thermodynamics of Three-Phase Contact Lines

The mutual boundary of three immiscible bulk phases or the intersection of three interfaces is called the three-phase contact line. For example, as illustrated in the figure below, a three-phase contact line is formed by a sessile drop on a solid substrate surrounded by a vapor phase. As we have already known that each interface is a nonhomogeneous interfacial region with a finite thickness, the three-phase contact line that is formed by the intersection of three interfaces is also a non-homogeneous zone with a finite thickness. Material properties changes sharply through this line zone from one bulk phase/interface phase to another. However, the thickness of the three-phase contact zone is negligible in comparison with its length. Therefore, in the surface thermodynamics, similar to the treatment of interfaces, the three-phase contact line is treated as an one-dimensional, uniform linear phase.

The side view (upper figure) and the top view (lower figure) of a sessile drop resting on a flat solid surface.
The fundamental equation in the energy form for a line phase is given by:
$$U_L=U_L\left(S_L, L, N_{1 L}, \ldots N_{r L}\right)$$
where the variable $L$ is the length of the line phase, and the subscript $L$ indicates the line phase. This equation is valid only for moderately-curved line phases, as no curvature variables are introduced in the fundamental equation. The differential form of the fundamental equation is:
$$d U_L=T d S_L+\sigma d L+\sum \mu_i d N_{i L}$$
where the new parameter $\sigma$ is the line tension, defined by
$$\sigma=\left(\frac{\partial U_L}{\partial L}\right)_{S_L, N_L}[\mu \mathrm{J} / \mathrm{m}]$$

# 热力学代考

## 物理代写|热力学代写thermodynamics代考|Thermodynamics of Surfaces

1. 作为近似值的 2D 相位。这是基于这样一个事实，即与其横向尺寸相比，表面或界面的厚度可以忽略不计。
2. 均匀相和所有表面性质被认为是界面厚度的平均值。因为真实界面的性质仅在厚度方向上是不均匀的。
3. 以这种方式，3D、异质界面相近似为 2D、均匀、简单的热力学相。简单系统的所有热力学原理和方程式都适用于这种表面相。对于 3D 体相系统，体积在用于表征系统的大小，体积变化用于表征机械做功（压缩或膨胀）。然而，对于二维表面相，没有体积，而是使用表面积 A 来表征系统的大小，而表面积变化则用于表征机械功。因此，表面相的基本方程由下式给出：
4. $$5. \mathrm{S}{\mathrm{A}}=\mathrm{S}{\mathrm{A}}\left(U_A, A, N_{1 A}, \ldots N_{r A}\right) 6.$$
7. $$8. \mathrm{U}{\mathrm{A}}=\mathrm{U}{\mathrm{A}}\left(S_A, A, N_{1 A}, \ldots N_{r A}\right) 9.$$
10. 在上述等式中，变量一个是表面积(米2), 和下标一个表示表面相。
11. 通常，表面或界面可能不平坦，而是弯曲的。应该提到的是，这些基本方程中没有曲率变量；因此，这些基本方程仅对适度弯曲的界面有效，其中界面的曲率效应可以忽略不计。什么是适度弯曲的界面？通常，当界面的曲率半径与界面的厚度相当时，这样的界面是高度弯曲的。如果曲率半径至少是界面厚度的 100 倍，则这样的界面是适度弯曲的。对于适度弯曲的界面，描述界面所需的唯一几何变量是表面积。然而，对于高度弯曲的界面，还需要一定的曲率变量来描述界面，除了表面积。这是因为与界面的形状变化（例如由于弯曲或扭曲引起的形状变化）相关的能量是界面总能量的重要部分。

## 物理代写|热力学代写thermodynamics代考|Thermodynamics of Three-Phase Contact Lines

$$U_L=U_L\left(S_L, L, N_{1 L}, \ldots N_{r L}\right)$$

$$d U_L=T d S_L+\sigma d L+\sum \mu_i d N_{i L}$$

$$\sigma=\left(\frac{\partial U_L}{\partial L}\right)_{S_L, N_L}[\mu \mathrm{J} / \mathrm{m}]$$

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