Chapter 4 Principles of solidification

Chapter 4 Principles of solidification
Crystallization of molten metals Nucleation Growth Homogeneous nucleation Heterogeneous nucleation

Objectives: Understand and Master: homogeneous nucleation, heterogeneous nucleation, constitutional undercooling, solidification of single-phase alloy and regular eutectic. Know：Application of thermodynamics For example: Growth of silicon single crystal

Main contents 4.1 Thermodynamics of solidification for pure metals 纯金属凝固的热力学条件 4.2 Nucleation 晶体的形核 4.3 Growth 晶体的生长 4.4 Plane front solidification of single-phase alloys 单相合金的平界面凝固 4.5 Cellular solidification 胞状凝固 4.6 Plane front solidification of eutectic alloys 共晶合金的平界面凝固

4.1 Thermodynamics of solidification for pure metals 纯金属凝固的热力学条件
4.1.1 The curves of free energy versus temperature of a liquid and a solid 液态和固态的自由能随温度的变化曲线 Two curves intersect at the equilibrium melting point Tm. 两曲线相交于平衡熔点Tm free energy n. 自由能 melting point 熔点 thermodynamics n. 热力学

Note: the free-energy curves and their intersection are changed by changes in ambient pressure. 液固两相（体积）自由能与温度的关系曲线及其交点位置随环境压力变化. G refers to the free-energy change on forming a solid from a liquid. 定义G为液固相变的自由能差. When a pure liquid cooled to its equilibriumrium melting point Tm, the solid may not form. 液态纯金属冷至平衡熔化温度时不会形成固相，即液态金属不会在无过冷度的情况下结晶. molar adj. [化][物]摩尔的 enthalpy n. [物]焓 entropy n. [物]熵

4.1.2 Free-energy change G on forming a solid from a liquid 液固相变过程中自由能的变化G
The free-energy change on forming a solid from a liquid at temperature T is 温度T时液固相变自由能差G为： Where G, H, S are molar changes in free energy, enthalpy, and entropy, respectively G、H和S 分别为液固相变时的摩尔自由能差、摩尔焓变和摩尔熵变. How to obtain the molar change in entropy?

(1) At the equilibrium transformation temperature Tm
The free-energy change is zero (2) At temperature T different from Tm undercooling. 过冷（度） heat of fusion n. 熔化热 molecular weight n. 分子量

Where T is the undercooling (Tm-T) T 为过冷度
Neglecting the small-temperature dependence of H and S and combining above two equations yields 忽略H和S随温度的微小变化，结合4.1式和4.2式，有： Where T is the undercooling (Tm-T) T 为过冷度 H is negative for solidification H为摩尔焓变。凝固时取负值。过冷引起的摩尔自由能差G是熔体结晶的热力学条件!

4. 2 Nucleation 晶体的形核越过一个势垒ΔGd 凝固驱动力ΔGm GL GS
Classical nucleation theory 越过一个势垒ΔGd 凝固驱动力ΔGm GL GS 克服势垒的能量是金属原子通过内部温度起伏，即能量起伏来实现的。

Nucleation and growth

4. 2.1 Homogeneous nucleation 均质形核
Classical nucleation theory Nucleation is the process of the aggregation of clusters of atoms that represent the first appearance of the new phase. 形核是原子团聚合以形成新相的起初阶段. As the temperature falls, the thermal agitation of the atoms of the liquid reduces, allowing small random aggregations of atoms into crystalline region. 温度降低，液态原子的热运动下降，使得原子聚合形成晶坯. aggregation n.集合, 集合体, 聚合 thermal agitation 热搅动，热波动，温度起伏 nucleation 形核 homogeneous nucleation 均质形核

We expect a material to solidify when the liquid cools to just below its freezing temperature, because of the energy associated with the crystalline structure of the solid is then less than the energy of the liquid, i.e. G0. 当金属液温度刚好冷至凝固点以下时，金属液应该要凝固。因为此时具有晶体结构的固相的能量比液相的能量低，即G0。 Classical nucleation theory The driving force for solidification is the energy difference between the liquid and the solid, i.e. the free energy change per unit volume Gv. 凝固驱动力是液固相的单位体积自由能差 Gv. How to obtain the total free-energy change G on forming a solid from a liquid?

Free-energy change G on forming a solid from a liquid:
The total free energy of the solid-liquid system changes with the size of the solid. When a small spherical embryo of radius r forms, a solid-liquid interface is created. 当半径为r的小球形晶坯生成后，晶坯与液相间就形成了一个液固界面. A surface free energy LS is associated with this interface; the larger the solid, the greater the increase in surface energy. 与此液固界面相应的界面自由能为LS。晶坯越大，液固界面自由能越大. Thus, the total change in energy G is: 总自由能变化G 为：

Gv is a negative since the phase transformation is assumed to be thermodynamically feasible. 由于凝固相变在热力学上是可行的，因此单位体积自由能变化 Gv 为负值. Figure 5.15 Surface and volume energies of an embryo of solid growing in a liquid give the total energy as shown. 液相中晶坯的表面能和体积自由能以及总自由能随晶坯半径r的变化. r* The top curve shows the parabolic variation of the total surface energy. 最上面的曲线代表表面自由能随晶坯尺寸呈抛物线变化。

The curve in the middle shows the variation of G. 中间的曲线代表总自由能变化。
The bottom most curve shows the total volume free-energy change term. 最下面的曲线代表体积自由能变化。 The curve in the middle shows the variation of G. 中间的曲线代表总自由能变化。 Below the critical size r*, any embryos will tend to shrink and disappear. Above r* increasing size reduces the total energy, so growth will be increasingly favoured, becoming a runaway process. 晶坯尺寸小于临界尺寸r*时，晶坯缩小并消失；大于临界尺寸时，晶坯长大对降低能量有利，为一自发过程.

1. The size of the critical radius r* 晶核临界半径 r*
The critical radius r* is the minimum size of the crystal that must be formed by atoms clustering together in the liquid before the solid particle is stable and begins to grow. （晶核）临界尺寸r*是液相中通过原子聚合形成的、可稳定存在并开始生长的晶体的最小尺寸. 1. The size of the critical radius r* 晶核临界半径 r* Where T is undercooling, means the equilibrium freezing temperature minus the actual temperature of the liquid, T = Tm-T .

As the undercooling increases, the critical radius required for nucleation decreases. 过冷度增大，临界晶核半径减小. Homogeneous nucleation occurs when the undercooling becomes large enough to cause the formation of a stable nucleus. 当过冷度足够大，可形成稳定的晶核时，发生均质形核. The temperature at which homogeneous nucleation can occur is called the homogeneous nucleation temperature. 均质形核发生时的液相温度称为均质形核温度. surface free energy 界面自由能 phase transformation 相变

2. The number of atoms in the critical nucleus n* 临界晶核中的原子数n*
Va—the volume of unit atom 单个原子的体积 parabolic approximation 抛物线逼近法 runaway n.逃跑, 逃跑者, adj.逃亡的, 逃走的 runaway effect [电子]失控效应 nucleus [nuclear的复数, 见nuclear] n.核子 rate of nucleation 形核速率

3. The free energy change to form a spherical nucleus of radius r 形核功（形成临界球形晶核所需的能量）
4. Rate of nucleation 形核速率 The number of nucleus formed per unit time and unit volume. 单位时间、单位体积液相中形成的晶核数. The rate of nucleation is a function of temperature. 形核速率与温度有关.

At the temperatures above the point Tm, the rate of nucleation is zero
过冷度较小，形核率很小;过冷度增加，形核功减小，形核率急剧增大;过冷度太大，形核率减小.存在最大形核率 As the temperature becomes lower, atom diffusion becomes slower, hence slow the nucleation process. 随温度下降，原子扩散变慢，因此形核过程变慢. A typical rate of nucleation reaches a maximum at some temperature below the transformation temperature. 在凝固温度以下，存在形核速率最大的形核温度.

Table: Values for freezing temperature Tm, latent heat of fusion H, surface energy  , and maximum undercooling T for selected materials. 不同材料的凝固温度、凝固潜热、界面能和最大过冷度. Metal Tm, ℃ H, J/cm3 , J/cm2 T, ℃ Ga 30 488 5610-7 76 Bi 271 543 5410-7 90 Pb 327 237 3310-7 80 Ag 962 965 12610-7 250 Cu 1085 1628 17710-7 236 Ni 1453 2756 25510-7 480 Fe 1538 1737 20410-7 420 H2O 40

4.2.2 Heterogeneous nucleation 异质形核
As is well know, metals and most other liquids are rarely undercooled by more than a few degrees before beginning to crystallize. 实际上，金属和其它液体开始结晶时的过冷度均很小，不超过几度. The crystallization begins on impurity particles, i.e., nucleating agents or mould walls, and by so doing avoids the very large thermodynamic barrier to homogeneous nucleation. 结晶可从杂质颗粒，如形核剂或型壁开始，从而可避免均质形核所需的很大的热力学能垒. Imagine a cluster as in figure at equilibrium on a flat substrate. 假定晶胚（原子团）在固体的平面基底上形成一个球冠 .

Equilibrium pertains when the following equation exists

Where LM , MS , LS are the surface energies of substrate-liquid, substrate-cluster, and liquid-cluster interfaces, respectively. contact angle 接触角 Comparison of heterogeneous nucleation with homogeneous nucleation ： 1. The size of the critical radius rh* 异质形核的晶核临界半径 r*：

2. The free energy to form a spherical of radius r0 形成一曲率半径为r0的球冠晶胚的自由能（即形核功）为：
Where 3. The number of atoms in the critical nucleus nh* 异质形核时临界晶核中的原子数

4. 均质形核与非均质形核比较： f()的意义：相同半径时，球冠与球的体积比形核半径一样；
4. 均质形核与非均质形核比较： f()的意义：相同半径时，球冠与球的体积比形核半径一样；非均质形核的形核功和临界晶核的原子数均比均质形核的小；形成相同半径的晶核时，非均质形核的过冷度小；均质形核在均质（单质）的金属中形成，而非均质形核须依靠外来质点形核。 pertain v.适合, 属于 nucleating agent 形核剂 heterogeneous nucleation 异质形核 cobalt [化]钴(符号为Co) inoculating agent 孕育剂 fine-grained 细小晶粒的 zirconium n.锆 titanium n.[化]钛 boron n.[化]硼

4.2.3. Inoculating agents 孕育剂（形核剂）
In commercial practice, inoculating agents are added to many molten alloys to produce fine-grained materials. 生产中，通过向金属液中添加孕育剂（形核剂）旨在获得细晶材料. For example, titanium and boron for aluminum alloys, impure ferrosilicon for cast iron (to nucleate graphite), carbon for certain magnesium alloys, and zirconium for others. 铝中加入钛、硼；铸铁中加入硅铁；镁合金中加入碳；其它合金中加入锆以达到细化合金晶粒的作用. Cobalt, zinc and other materials sprayed on mould walls are effective in nucleating ferrous metals. 在铸型壁上喷施钴、锌等可促进黑色金属形核.

例：钛对锌铝合金的细化作用结论：铸态锌铝合金中加入钛可细化晶粒，获得细小等轴晶。 30

The general characteristics of a good grain refiner: 好的晶粒细化剂的一般特性：
(1) The refiner should be one that produces a small contact angle  between the nucleating particles and the growing solid. This implies high surface energy LM and low surface energy MS 晶粒细化剂和生长晶体间的接触角要小。即高的LM 和低的 MS. (2) A successful nucleating agent should be as stable as possible in the molten metal, possess a maximum of surface area, and have optimum surface character (perhaps be rough or pitted). 好的晶粒细化剂（或形核剂）要有高的热稳定性；大的表面积和良好的表面特性（如粗糙或凹凸不平）.

The quantity MS should decreases with decreasing lattice mismatch (or lattice disregistry) between particle and solid and with increasing chemical affinity between particle and solid MS随外来粒子和生长晶体间的点阵失配度减小而减小;随它们之间的化学亲合性增加而减小. 当5％时，通过点阵调整（畸变）可以实现晶面两侧的原子对应，这种界面可谓完全共格的界面（见图（a）），其界面能最低，促进非均质形核的能力最强，形核率最大。当5％<<25％时，通过点阵畸变和位错调节，可以实现部分的共格对应，衬底仍有部分的形核能力。 diffuse v.散播, 传播, 漫射, 扩散 adj. 散开的, 弥漫的 atomic adj.原子的, 原子能的, 微粒子的 cast iron 铸铁 zinc n.锌 vt.涂锌于 ferrous metal 黑色金属 ferrosilicon 硅铁，硅铁合金 lattice mismatch 晶格失配，晶格不匹配 disregistry 错合度 chemical affinity 化学亲合性 lattice disregistry 点阵失配度，晶格错配度 affinity n.密切关系, 吸引力,亲合力 pitted adj.有凹痕的, 去核的

共格对应理论在选择细化剂的理论中，应用最广的是共格对应理论。非均质形核时，要求基底晶面与结晶相界面两侧的原子具有共格对应的关系。严格地说，要求界面两侧的原子尺寸、原子间距和点阵类型均应一一对应。但其中最基本的对应关系是要求原子间距相近，或在某一范围内彼此呈比例关系。由点阵失配度来衡量原子间距的对应程度。当5％时，完全共格的界面。其界面能最低，促进非均质形核的能力最强，形核率最大。当5％<<25％时，部分共格对应。衬底仍有部分的形核能力。当>25％时，形核能力逐渐降低，甚至完全丧失形核能力。

界面共格对应理论(theories of coherence)模型
coherent interface semi-coherent interface

aM aS 衬底晶核（b）部分共格对应（a）完全共格对应点阵失配度

Example 1: Calculate the size of the critical radius and the number of atoms in the critical nucleus when solid copper forms by homogeneous nucleation. Comment on the size of the nucleus and assumptions we made while deriving the equation for radius of nucleus. Valus for Cu: Freezing temperature Tm=1085℃, Heat of fusion H=1628J/cm3, Solid-liquid interfacial energy LS=177×10-7J/cm2, atom distance 3.615×10-8cm, Typical undercooling for homogeneous nucleation T=236℃.

SOLUTION: ΔT=236℃ Tm= =1358K ΔH=1628J/cm3 LS=177×10-7 J/cm3 VUnit cell = (a0)3=(3.615×10-8)3=47.24×10-24 cm3

The number of unit cells in the critical nucleus is
Since there are four atoms in each unit of FCC metal, the number of atoms in the critical nucleus must be: (4atoms/cell)(174cells /nucleus) =696 atoms/ nucleus Note: In the calculations, we assume that a nucleus that is made from only a few hundred atoms still exhibits properties similar to those of bulk materials. This is not strictly correct and as such considered to be a weakness of the classical theory of nucleation.

4.3 Growth 晶体的生长 3.26结束。剩余时间做习题 The ease with which atoms can attach themselves to a growing solid interface depends on interface structure.原子在生长着的固相界面上堆砌的难易程度取决于液固界面的结构. There are two different types of interface structures, that is (1) Diffuse interface and (2) Atomically flat interface. 有两种界面结构类型，即弥散界面（粗糙界面）和（原子尺度上的）光滑界面。 Diffuse interfaces grow much more easily than those with an atomically flat interface and they grow in a different way. 粗糙界面生长比光滑界面容易，二者生长方式也不同.

Two different types of interface structures I
粗糙界面：界面固相一侧的点阵位置只有50％左右的堆砌位置（阵点）被原子所占有。这些原子散乱地随机分布在界面上，形成一个坑坑洼洼、凹凸不平的界面层。光滑界面：界面固相一侧的几乎全部被固相原子所占有，只留下少数空位；或是有界面上有很多点阵位置未被原子占据，留下大量空位。从而形成了一个平整光滑的界面。单原子层界面模型弥散界面（粗糙界面）光滑界面

Two different types of interface structures II
多原子层界面模型

4.3.1 Continuous growth 连续生长
连续生长的结果，晶体的棱角不很分明。金相观察时，晶体的表面是光滑的。据估计，1约为 cm/(s,K)数量级，因此在很小的过冷度下就可达很大的生长速率。通常，铸锭凝固时晶体生长速率约为10-2 cm/s，界面前沿的动力学过冷度 K。 The widely used kinetic law for continuous growth is: 广为使用的连续生长的动力学规律如下: R—growth velocity 晶体生长速度 TK—kinetic undercooling to drive the interface 生长所需的动力学过冷度 -a constant 动力学系数，为常数 For a typical metal, the kinetic coefficient  is the order of 100cm/(℃.s). 对于典型的金属。 的值约为100cm/(℃.s)的量级。

4.3.2 Growth by two-dimensional nucleation 二维晶核生长
结束 Crystal growth by two-dimensional mechanism

Growth velocity by two-dimensional nucleation
Gallium did not grow at observable rates when TK was less than about 0.5℃. 镓在过冷度低于0.5℃时几乎不会生长. The perfect (dislocation-free) ice crystals do not grow at measurable rates at less than about 0.03℃ undercooling. 过冷度低于0.03℃时，无位错的完整冰晶体不会明显生长.

4.3.3 Growth by screw dislocations 螺型位错生长机制
A crystal has a screw dislocation emerging at its liquid-solid interface, developing a spiral step structure. 液固界面上螺型位错露头，由于晶体生长发展成螺旋型台阶结构.

Growth velocity by screw dislocations 螺型位错生长时的晶体生长速率
The close-packed face has a step in it, such a step obviates the need for two-dimensional nucleation and permits growth at much lower undercoolings. 密排面上形成的生长台阶避免了二维形核，从而使生长可在较小的过冷下进行. No matter how many layers of atoms are deposited on the face, the step will persist. 生长过程中，台阶不会消失. Growth velocity by screw dislocations 螺型位错生长时的晶体生长速率

4.3.4 Growth by propagation of twin planes 晶体通过孪晶面提供的台阶生长
面心立方晶体反射孪晶及其凹角边界 There are still another source of steps at liquid-solid interfaces: the reentrant angle resulting from the emergence of a twin plane at a crystal surface. 在固液界面上还可能存在另一种形式的台阶，即在晶体生长晶面上由于孪晶的出现导致形成台阶。凹角可提供生长台阶，台阶也不会在晶体生长过程中消失。 A quantitative growth law for growth by the twin-plane mechanism has not been formulated. 孪晶生长机制的晶体生长速率公式有待研究. Example: 铸铁中片状石墨形成机制（旋转孪晶）

The reentrant angle resulting from the emergence of a twin plane at a crystal surface
面心立方晶体反射孪晶及其凹角边界

举例：铸铁中片状石墨形成机制（旋转孪晶）
根据缺陷的种类，小平面晶体可以有不同的宏观形态：若是线缺陷，则为针状；面缺陷则呈片状。石墨晶体是六角形晶格为基面的层状结构，基面之间的结合较弱。在结晶过程中原子排列层错好象使上下层之间旋转产生一定的角度，这样就在旋转边界周围提供若干生长位置，使石墨晶体沿着侧面<1010 >方向很快长大成为片状。

4.3.5 Discussion: Growth rate versus interface undercooling according to the three classical laws 不同生长模式下晶体的生长速率与动力学过冷度的关系. Figure: Growth rate versus interface undercooling according to the three classical laws 三种生长模式下晶体的生长速率与动力学过冷度的关系图

At high undercoolings, the rate of two-dimensional nucleation becomes so high that many nuclei form for each plane that grows, the growth rate approaches the continuous growth law as an upper limit. 当过冷度很大时，二维晶核生成速率很高，其生长速率与连续生长速率相同. For growth by screw dislocations, growth approaches the curve for continuous growth as an upper limit too.螺型位错晶体生长速度小于连续生长；当过冷度很大后，二者生长速度相同。 twin plane 孪晶面 reentrant angle 凹角 reentrant n.再进去, 凹角 adj.再进去的, 凹角的 51

Computer Simulation: Homogeneous nucleation and growth 1

Computer Simulation: Homogeneous nucleation and equiaxed growth

4.4 Plane front solidification of single-phase alloys 单相合金的平界面凝固
合金分类（凝固时晶体形成特点）单相合金：凝固时只析出一个固相的合金。如固溶体、金属间化合物等。多相合金：凝固时同时析出两个相的合金。如共晶、包晶或偏晶转变的合金。只讨论单相合金的溶质再分配问题。 The most important commercial applications of this type of solidification are for growth of crystals for semiconductors. 平界面凝固最重要的商业应用是半导体的晶体生长. Another important application is in growth of oxides for laser system and other optical applications. 另一重要应用是为激光器和其它光学器件生长氧化物. Growth of oxide crystals for jewels is another, much older commercial application of single crystal growth.珠宝等氧化物晶体的单晶生长.

提拉法晶体生长设备单晶生长过程提拉法晶体生长设备单晶硅

蓝宝石晶体Al2O3- Fe,Ti 宝石界将红宝石之外的各色宝石级刚玉都称为蓝宝石。刚玉中因含有铁（Fe）和钛（Ti）等微量元素，而呈现蓝、天蓝、淡蓝等颜色，其中以鲜艳的天蓝色者为最好。红宝石是指颜色呈红色、粉红色的刚玉，它是刚玉的一种，主要成分是氧化铝(Al2O3)，红色来自铬(Cr)。作为一种优良的红外透过材料，它可以用于制作红外探测器的红外窗口，保证红外光的透过和收集，从而实现了光电的最佳转换，在红外分析仪、夜视仪、制导仪中都有重要应用。

Crystal 水晶 Crystal is a term with dual meanings - one crystal refers to the naturally occurring colorless quartz (also called a semi-precious stone), which is termed as crystal quartz and the other is man-made crystal. Crystal is mostly glass mixed with lead - the magic ingredient that gives crystal its sparkle. Lead makes the glass heavier and allows the glass to be cut and given facets, making it glint and twinkle - like a diamond. Even today, crystals are synonymous with class and elegance.

Swarovski Crystals 施华洛世奇水晶
Although Crystal beads are manufactured by several companies throughout the world, the best crystal beads, the most popular and possibly the most expensive are made by Swarovski. And rightly so, since the crystal beads from Swarovski are the most perfect and accurately faceted beads - making every bead dance and shine like a diamond.

Swarovski Crystals

Melt-grown, plane front solidified crystals of metals and alloys find extensive use in research, but there is as yet no commercial application for these materials. 金属熔体中平界面生长出的金属在理论研究中有用处，但迄今尚未发现商业用途. 合金分类（凝固时晶体形成特点）单相合金：凝固时只析出一个固相的合金。如固溶体、金属间化合物等。多相合金：凝固时同时析出两个相的合金。如共晶、包晶或偏晶转变的合金。只讨论单相合金的溶质再分配问题。 (1) The assumption of equilibrium at the interface during growth：生长过程中的界面区域平衡假设： If solidification is occurring at temperature T*, the assumption of equilibrium at the interface requires that if either T*, CL* or CS* is specified, the other two are fixed by the phase diagram. 界面区域平衡假设：在凝固界面处，如果温度T*（℃）, 液相溶质浓度CL* （质量浓度，wt%）和固相溶质浓度 CS* （质量浓度，wt%）有一个参数固定，则其它两个参数可由相图确定.

k1 k<1 Equilibrium partition ratio k : 平衡分配系数

(2) Some common sense about k:
When the liquidus and solidus are straight lines emanating from the composition of the pure solvent, k is a constant. 如果液相线和固相线为直线，k为常数. In deriving various expressions below, k is assumed constant to simplify the mathematics. 为简化计算，下面的推导均假设k为常数. In the following discussion, k is assumed to be less than unity. However, the expressions are valid for alloys in which k is greater than unity. 通常讨论k1的情况，但相关分析方法和公式同样适用于k1的情形.

4.4.1 Equilibrium solidification
Consider a crucible of liquid alloy of length L and initial composition C0 freezing from one end. 考虑长为L，初始成分为C0的熔体从一端凝固的情形. crucible n.坩锅 emanate from v. 放射, 发源 liquid-solid interface 液固界面

The first solid begins to form at TL and is of composition kC0, lower in solute than initial liquid composition. 温度TL 时析出的固相成分为kC0，低于初始液相的成分. During subsequent cooling and solidification, both the liquid and solid become enriched in solute. 在此后的冷却和凝固过程中，液相和固相的溶质浓度均越来越高.

At temperature T. , the solid of composition CS
At temperature T*, the solid of composition CS* is forming in equilibrium at the liquid-solid interface with liquid of composition CL* 温度T*时，凝固界面两侧液、固相的平衡溶质浓度分别为CL*和CS*. fs和fL分别为固相和液相的重量分数。分三种情况讨论溶质再分配（忽略固相扩散）。 Because diffusion in the solid and liquid is complete, the entire solid becomes of uniform composition CS=CS* and the entire liquid of uniform composition CL=CL*. 由于扩散完全，液相和固相的成分是均匀的，分别等于CL*和CS*.

At temperature T*, a general materials balance (conserving solute atoms) is written: 任一温度T*时，固相质量分数和液相质量分数的关系如下： fs和fL分别为固相和液相的重量分数。分三种情况讨论溶质再分配（忽略固相扩散）。 Cs fs+CL fL=C (4.17) Cs fs+CL fL=C0(fs+ fL) Where fS and fL are weight fractions of solid and liquid, respectively fS和fL 分别是固相和液相的质量分数.

This is simply the equilibrium lever rule, which can be readily obtained for fraction solidified at a given temperature since fS+fL equal unity 任一温度T时，总有fS+fL=1成立，据此可以求得平衡杠杆定律，从而计算出任一温度下的凝固分数。 The composition of solid CS* and composition of liquid CL* at the liquid-solid interface: 凝固界面两侧固、液相的成分与凝固分数（或液相分数）的关系为： equilibrium lever rule 平衡杠杆定律

In spite of the equilibrium nature of solidification, substantial solute redistribution occurs during solidification 平衡凝固过程中，也存在着溶质再分配现象. The material is homogeneous only before and after solidification. 平衡凝固时，凝固开始和完成后材料成分均匀，但在凝固过程中材料成分不均匀.

4.4.2 No solid diffusion, strong convection in liquid 固相无扩散，液相均匀混合
It is assumed that no diffusion at all takes place in the solid. 假设固相无扩散，液相均匀混合. As in equilibrium solidification, the first small amount of solid to form is of composition kC0 at temperature TL. 与平衡凝固一样，TL时，最初形成的少量固相成分为kC0。 During subsequent cooling and solidification, the liquid becomes richer in solute and so the solid that forms is of higher solute content at later stages of solidification. 随凝固进行，液相溶质富集，因而后凝固的固相比先凝固的固相的溶质含量高.

Since there is no diffusion in the solid state, the composition of the solid formed in the initial stages of freezing remains unchanged. 由于固相无扩散，早期形成的固相的成分不会发生改变.

Ts时，液相数量较多，固相平均成分低于C0。随温度下降，凝固继续进行。
TE时，还剩余有一定数量的液相，液相为共晶成分。液体凝固成共晶组织，成分为CE。

Nonequilibrium eutectic 非平衡共晶。
C0虽然远离共晶成分，但有共晶体析出— nonequilibrium eutectic 非平衡共晶。 How to get the composition of the solid at the liquid-solid interface CS* as a function of fraction solid fS? master alloy 中间合金，母合金 peritectic reaction 包晶反应 periodic table 周期表 alkaline earth metal 碱土金属

Normal segregation equation 正常偏析方程
A quantitative expression is easily obtained by equating the solute rejected when a small amount of solid forms with the resulting solute increase in the liquid. The balance is: 设凝固某时刻，凝固界面上液相、固相的成分各为CL*和Cs*，相应的质量分数分别为fS和fL。当固相增量为dfS，析出溶质使液相浓度升高dCL。据质量平衡关系，凝固时排出的溶质量应等于使液相浓度升高的溶质量.

Equating the solute rejected when a small amount of solid forms with the resulting solute increase in the liquid 质量平衡关系：凝固排出溶质量=扩散进入液相溶质量（凝固排出溶质量）（扩散进入液相溶质量）

Primary condition：初始条件

Thus, the composition of the solid at the liquid-solid interface CS
Thus, the composition of the solid at the liquid-solid interface CS* as a function of fraction solid fS: 凝固界面上固相成分Cs*与固相质量分数fS的关系为： Or, in terms of liquid composition and fraction liquid 凝固界面上液相成分CL*为 Equation (4.20) and (4.21) are termed the nonequilibrium lever rule, or the Scheil equation. 非平衡杠杆定理或夏尔（Scheil ）方程.

根据固相和液相的质量分数，可以求出凝固界面两侧的固相或液相的浓度及凝固过程中固相成分的变化规律。
Example for using Scheil equation to solve actual problems: molar adj.[化][物]摩尔的 solute redistribution 溶质再分配 phase diagram 相图 nonequilibrium eutectic 非平衡共晶 binary phase diagram 二元相图 ternary phase diagram 三元相图 enriched solute boundary layer 溶质边界富集层

Example 2: Al-1%Cu alloy solidifies in strong convection condition
Example 2: Al-1%Cu alloy solidifies in strong convection condition. If the diffusion in solid is neglected,CE=33wt％Cu, Csm=5.65wt％Cu, Tm=660℃, TE=548℃, then: 1）When the weight fractions of solid fs is 0.5, what are the solid composition Cs* and liquid composition CL* at the solid-liquid interface? 2）After complete solidification , what are the weight fractions of eutectic? Al-1%Cu合金在强烈对流条件下凝固。Al-Cu合金相图上，CE=33wt％Cu，Csm=5.65wt％Cu，Tm=660℃，TE=548℃。不考虑固相扩散，问：1）当固相质量分数fs=0.5时，界面上液、固相溶质浓度CL*和 Cs*各为多少？ 2）凝固后组织中共晶相所占比例是多少？

Analysis：Al-Cu binary phase diagram

解：根据非平衡杠杆定理，有：则：

2) 凝固后的共晶相是由TE温度时的残余液相转变而来的，即共晶相所占比例即为TE温度时液相的比例.
或： 2) 凝固后的共晶相是由TE温度时的残余液相转变而来的，即共晶相所占比例即为TE温度时液相的比例. 根据非平衡杠杆定理

或：即为共晶相的体积分数为1.5%。

Example 3：In a binary phase diagram, the liquid metal (composition: WB =10%) is placed in a ceramic boat and freezing from left side to right side gradually. The temperature gradient at the front of solid-liquid interface is large enough to maintain growth with planar interface front. It is assumed that no solid diffusion and the composition of the entire liquid are uniform. Then: (1) Prove that the average composition of the solid is

(2) During solidification, the liquid becomes richer in solute and so the liquidus temperature will reduce. Prove the liquidus temperature TL as a function of fraction solid fs is Where Tm is melt point of the pure metal , m is the slop of the liquidus line. (3) Mark the average solid composition in the phase diagram when TL is 750℃, 700℃, 600℃ and 500℃ respectively. What are the weight fractions of eutectic after solidification?

Example 3：某二元合金相图1所示。合金液成分为wB=10％，置于长瓷舟中并从左端开始凝固。温度梯度大到足以使固-液界面保持平面生长。假设固相无扩散，液相均匀混合。
（1）证明已凝固部分(fs)的平均成分为（2）当试棒凝固时，液相成分增高，而这又会降低液相线温度。证明液相线温度TL与fs之间关系如下式。式中Tm为纯金属的熔点（900℃），m为液相线斜率（-13.3）,k=0.5 （3）在相图上标出TL分别为750℃、700℃、600℃与500℃下的固相平均成分。问试棒中将有百分之几按共晶凝固?

解：（1）wB=10%，C0=0.1 所以（2）液相线温度TL与fs之间关系

（3） TL,℃ 750 fs ？ 6.9 700 0.89 7.5 600 0.95 8.1 500 0.97 8.6 T=500℃，fL=1-0.97=0.03=3%，故试棒中将有3%按共晶结晶.

4.4.3 Limited liquid diffusion, no convection 液相有限扩散，无对流或搅拌
Solidification begins exactly as in the previous examples, with the initial solid forming of composition kC TL以下时，最初形成的少量固相成分为kC0。 The solute rejected into the liquid is transported only by diffusion and so an enriched solute boundary layer forms and gradually increases in solute. 排入液相的溶质仅依赖扩散传输，从而在液固界面前沿形成一个溶质富集层。且。随着凝固的进行，溶质不断富集，CL*和Cs*不断升高。

If the crystal is sufficiently long, a steady state approached, the composition of the solid forming is exactly the overall alloy composition C 如果晶体长度足够，凝固可达到稳态，此时，形成的固相成分与晶体的整体成分相同，为C0.

steady state 稳定生长阶段—界面上凝固排出的溶质量等于溶质富集层内液相扩散走的溶质量。溶质富集层内液相的溶质浓度不随时间变化.
The composition in the liquid during steady-state solidification. 稳定生长阶段下溶质富集层内液相的溶质浓度分布的计算 sandpaper 砂纸 magnesia n. 氧化镁，镁砂 strontium n. 锶

The composition in the liquid during steady-state solidification
注：由于假设液固相的密度相等，因此溶质浓度无论是体积浓度还是质量浓度，本章的计算过程和计算结果是一样的。体积分数就是质量分数。只要一致即可取一活动坐标系，原点设在固液界面上。 x是以界面为原点沿其法向伸向熔体的动坐标；设R为液固界面的生长速度。 CL(x)为液相中沿x方向的浓度分布，界面处液相中的浓度梯度为。 CL* 和CS* 分别为固液界面两侧液相中的溶质质量浓度和固相中的溶质质量浓度，其单位为wt%。设液固相的密度相等，则溶质的质量-体积浓度（kg/m3）等于溶质的质量百分浓度与密度（kg/m3）之积.

The amount of solute, q1, rejected by the solid at the unit solid-liquid interface per unit time is ：单位时间、单位液固界面处由固相排出的溶质量为q1 The amount of solute, q2, diffuse into the liquid from the unit solid-liquid interface per unit time is ：单位时间内通过单位液固界面扩散进入液相的溶质量为q2

During the steady-state solidification, q1=q2 稳态时， q1=q2
式4.23与浓度单位无关！ During the steady-state solidification, the composition of the liquid is independent of the time. That is 稳态时，液相成分动态稳定，不随时间变化，即有：

下面考虑凝固过程中液相溶质浓度随凝固时间的具体变化:
1) 菲克第二定律- 扩散 C0 C0/k dx x CL(x) L 2) 凝固界面推进造成液相浓度的变化这样，液相内溶质浓度随时间的变化可以写成下式:

The general solution to the equation (4.24) is 方程4.24的通解为：
稳态时，液相成分动态稳定，不随时间变化，则有： The general solution to the equation (4.24) is 方程4.24的通解为： Boundary condition: 边界条件为（1）x=，CL(x)=C （2）x=0，CL(x)=C0/k

The solution to Eq. (4. 24) can be readily obtained with the aid of Eq
The solution to Eq. (4.24) can be readily obtained with the aid of Eq. (4.23), is 方程4.24的最终解为：式4.25与溶质浓度无关都成立！在x=0时CL-C0达到最大值 C0/k-C0，在 x=DL/R时降低到最大值的1/e（大约三分之一）。这样稳定状态边界层的厚度可以定为DL/R，终了过渡的特征长度就是DL/R 。 This type of solidification results in a nearly uniform composition except for the initial and final transients. 对于液相有限扩散、无对流情形下试样的定向凝固，除了凝固初始段和终了段外，试样的成分几乎均匀.

当k1时，初始过渡成分的数学表达式为：
式中，z是到凝固起点的距离。C0-Cs在x=0时达到最大值 C0- k C0 ，在x=DL/(kR)时下降到最大值 1/e。这样初始过渡成分的特征长度是DL/(kR)，要比终了过渡的长度大得多，这是因为k1。

Example 4: For a Al-Cu alloy, during its steady-state solidification, the solute composition in the liquid ahead of the interface is as follows: Suppose the interface temperature is 624℃, the melting point of pure aluminum is 660℃. Try to calculate: (1) equilibrium partition ratio k; (2) slope of liquidus mL; (3) solute composition difference between in the solid phase and the liquid phase at liquid-solid interface C0; (4) maximum temperature difference in the liquid T0; (5) liquidus temperature of the Al-Cu alloy TL.

Example 4: 已知某一Al-Cu合金在稳态时，固液界面前沿液相中的溶质浓度分布为：
已知界面温度TL*为624℃，铝的熔点T0为660℃，求平衡分配系数k,液相线斜率m,液固界面上液固相的成分差C0,液相内最大温差T0，该Al-Cu合金的液相线温度TL。

解：在稳态下，固液界面前沿的溶质分布一般式为：
由题意知：因此有（1）C0=2wt%，k=0.14 界面处液相的浓度为：

C0=CL*-C0=14.3-2=12.3wt%Cu TL=T0+mLC0=660-5=655℃ （2）液相线斜率为：
（3）界面处液固相的成分差为： C0=CL*-C0=14.3-2=12.3wt%Cu （4）液相内最大温差为： T0=-mLC0=31K （5）液相线温度为： TL=T0+mLC0=660-5=655℃

4.5 Cellular solidification 胞状凝固
4.5.1 Constitutional supercooling and cell formation 成分过冷及胞晶形成. A solute-rich boundary layer builds up in front of a solidifying planar interface in solidification with limited liquid diffusion and no convection 液相有限扩散，无对流时在凝固平界面前沿形成了溶质富集边界层.

A solute-rich layer is present in front of a growing interface, in which liquid composition is a maximum CL* at the interface and decreases with increasing distance from the interface. 凝固界面前沿液相中形成的溶质富集层中，在界面处液相溶质浓度最高，为CL*；离界面越远，液相溶质浓度越低。 With the aid of the phase diagram, it is a simple task to plot the equilibrium liquidus temperature of the liquid TL as a function of distance from the interface. 借助相图，可画出液相中平衡液相线温度TL与液相中离开界面的距离间的关系曲线.

The equilibrium liquidus temperature of the liquid TL as a function of distance from the interface 右图为液相中平衡液相线温度随离开液固界面的距离间的关系曲线.

The equilibrium liquidus temperature increases with distance from the interface because the lower the solute content, the higher the liquidus temperature. 随离凝固界面越远，平衡液相线温度越高。原因是溶质含量越低，平衡液相线温度越高. The actual temperature in the growing crystal is TA, the curve must pass through T* at the interface x=0, but otherwise its shape is dictated by heat flow. The actual temperature in the liquid is TA, the curve must pass through T* at the interface x=0, but otherwise its shape is dictated by heat flow. 设液相中的实际温度分布为TA，在凝固界面（x=0）处液相的温度为T* ，其余部位的温度分布由热流情况决定. Constitutional supercooling 成分过冷 Cellular solidification 胞状凝固 phase diagram. 相图 solute-rich boundary layer. 溶质富集(边界)层

The figure below show a condition where the interface is exactly at the equilibrium liquidus temperature and where every point in front of the interface is at a temperature above the liquidus. The actual temperature in the growing crystal is TA, the curve must pass through T* at the interface x=0, but otherwise its shape is dictated by heat flow. 右图表示在凝固界面处的温度刚好为平衡液相线温度，而界面前沿液相中任一处的实际温度TA均高于液相线温度TL. This represents the condition necessary for stable plane front solidification. 这种情形就代表了稳定的平界面凝固的条件.

If an instability causes a protuberance to form on the flat interface, it will find itself in a superheated environment and will melt back. 如果由于失稳导致在平界面上产生突起，由于前方液相过热，突起将被重熔，界面仍保持为平面。 Figure below , on the other hand, represents an unstable case. 否则，平界面生长就会不稳定。见下图。 protuberance n.隆起, 突出 plane front solidification. 平面凝固

A protuberance to form on the flat interface will find itself in a superheated environment and will melt back

Here, liquid immediately in front of the interface is at an actual temperature TA that is below its equilibrium liquidus temperature TL. It is, therefore, supercooled. 此时，界面前沿液相中的实际温度TA低于平衡液相线温度TL，即液相出现“过冷”。 This kind of supercooling is termed as constitutional supercooling. The word constitutional indicates that the supercooling arises from a change in composition, not temperature. 这种过冷称为“成分过冷”，即由于液相成分发生变化而引起的过冷，而非温度因素。由液相实际温度所导致的的过冷称为“热过冷”。

热过冷：当凝固界面前液相中形成负的温度梯度时，在界面前方液相中也可形成过冷。这种仅由熔体实际温度分布所决定的过冷为“热过冷”。
Constitutional supercooling theory and Constitutional supercooling criterion 成分过冷理论以及成分过冷判据. This supercooling results in instability of the plane front since any protuberance forming on the interface would find itself in supercooled liquid and therefore would not disappear. 成分过冷导致平界面生长不稳定。平界面上形成的突起会伸入前方的过冷液相内，因而突起不会消失。

4.5.2 Constitutional supercooling theory 成分过冷理论
heat and mass flow 即考虑温度梯度和浓度梯度 To develop quantitatively the constitutional supercooling criterion, we need consider heat and mass flow only at the interface. 为了建立成分过冷判据，需要考虑液固界面处的热量传输和质量传输（注：即考虑温度梯度和浓度梯度）。 The gradient of solute in the liquid at the interface is as given by Eq. (4.23): 界面处液相的溶质浓度梯度已由式4.23给出，即：

Assuming equilibrium at the flat interface, the slope of the curve of the equilibrium liquidus temperature TL versus distance from the interface x is related to that of liquid composition CL by the slop of the liquidus line mL: 假设在平界面上的平衡情况。平衡液相线温度TL 随离开界面的距离x的关系曲线TL(x)的斜率与液相的成分CL (x) 和液相线的斜率mL有关，即 equilibrium liquidus temperature n.平衡液相线温度 temperature gradient. 温度梯度 Cell structure 胞状组织 protrusion n.伸出, 突出 undulatory adj.波动的

Constitutional supercooling is absent when that actual temperature gradient in the liquid at the interface GL is equal to or greater than dTL(x)/dxx= 固液界面上液相的实际温度梯度GL等于或大于平衡液相线温度曲线TL在凝固界面处的切线斜率dTL(x)/dxx=0时，不会形成成分过冷，即：

Combining Eq. (4. 27) with Eq. (4. 23) and letting CS. =kCL
Combining Eq. (4.27) with Eq. (4.23) and letting CS*=kCL* gives the general constitutional supercooling criterion; that is , a plane front is stable when: 将式4.23代入式4.27中，同时根据CS*=kCL*，即可得到成分过冷判据。即当下式成立时，平界面稳定： Equation (4.28) is applicable regardless of the presence or absence of convection since a laminar layer exists next to the solidifying interface regardless of degree of convection. 无论有无对流，成分过冷判据均成立。因为在凝固界面前沿总存在溶质富集边界层。

At steady state, with no convection, CS. =C0 and Eq. (4
At steady state, with no convection, CS*=C0 and Eq. (4.28) becomes:在无对流的稳定态时， CS*=C0，上式变为下式：（T凝固区间） A very large amount of qualitative and quantitative conformation of the constitutional supercooling theory has been obtained since its formulation in 1953 by Chalmers. 成分过冷理论于1953年由Chalmers提出，并已经大量证实. The agreement of this simple theory with experiment is excellent even though it neglects a number of other factors. 尽管成分过冷判据忽略了许多因素，但与实验吻合依然很好。

3 界面稳定性动力学理论的基点 2 成分过冷的不足它不能预见界面失稳后的尺度； 1 成分过冷的基点
界面的平衡受到固液界面前沿液相的温度梯度、浓度梯度、固相与液相的热传导、结晶潜热以及固液界面张力的影响；界面上无时不存在扰动；固液界面是由无穷小的正弦波所组成；界面的稳定性取决于正弦波的振幅随时间的变化率。正温度梯度浓度梯度正温度梯度和浓度梯度这两个相反效应的相互抵消之后而获得界面能扩散 2 成分过冷的不足界面能效应；结晶潜热潜热和溶质的横向扩散。它不能预见界面失稳后的尺度；

Example 5: An Al-1%Cu alloy is grown by normal freezing at 3×10-4cm/s with convection completely suppressed. (1) What’s the temperature at the planar liquid-solid interface at steady state? (2) What’s the minimum temperature gradient to be required to maintain the plane front solidification according to the constitutional supercooling criterion? (Note: In the phase diagram for Al-1%Cu alloy,CE=33%Cu, Csm=5.65%Cu, Tm=660℃, TE=548℃, and constant k and mL; DL=3×10-5cm2/s.) 例5、Al-1%Cu合金在正常凝固下以生长速度为3×10-4cm/s生长，对流完全被抑制。问（1）在稳态下，液/固平界面上的温度为多少？（2）按照成分过冷判据，需要保持平界面凝固所需的温度梯度应为多大？（Al-Cu合金相图上，CE=33％Cu，Csm=5.65％Cu，Tm=660℃，TE=548℃，k和mL为常数，DL=3×10-5cm2/s。）

解：（1）在稳态下，液/固平界面上的温度T*:
注意液相线斜率为温度差与溶质浓度之比，其中浓度只取百分浓度的分子数！！！液相线斜率mL可写成在稳态，无对流时，CS*=C0：

2）求需要保持平界面凝固时的温度梯度，直接将相关参数代入成分过冷判别式即可。
所以 T*=Tm-mLCL*= ×5.84=640℃ 2）求需要保持平界面凝固时的温度梯度，直接将相关参数代入成分过冷判别式即可。

Example 6: A Ge-Ga crystal is grown by normal freezing
Example 6: A Ge-Ga crystal is grown by normal freezing. Initial melt composition is 10 ppm Ga. Growth rate is 8×10-3 cm/s. Assume k=0.1, mL=-4℃/wt%, DL=510-5cm2/s. (1) If convection is completely absent, what thermal gradient is required to maintain a plane front when the ingot is 50 percent solidified? (2) If convection is sufficiently vigorous, what thermal gradient is required to maintain a plane front when the ingot is 50 percent solidified? 例6 Ge-Ga晶体以正常凝固生长，原始熔体成分为10ppmGa，生长速度为8×10-3cm/s，设k=0.1，mL=-4℃/wt%，DL=5×10-5cm2/s，问：（1）假如完全没有对流，当锭子的50%已凝固时，需要多大的温度梯度才可平界面凝固？（2）若对流相当激烈，当锭子的50%已凝固时，温度梯度应为何值方可保持平界面凝固？

解：C0=10ppm=1010-4% 1)假如完全没有对流，当锭子的凝固分数为50%时，凝固过程为稳定态，即保持平界面凝固的温度梯度为：

2)若对流相当激烈时，平界面凝固的稳定性判据为：
由此可以看出，无对流时，为要保持平界面凝固前沿，GL值需要大些；而在完全对流时，GL值可小些。

Example 7: For Al-2%Cu alloy, it is known that equilibrium partition ratio k is 0.14, the liquidus slope mL is -2.5, solute diffusion coefficient in melt DL is 3×10-5 cm²/s. According to the constitutional supercooling theory, what is the stability limit (GL/R) for the planar interface growth in steady-state solidification? 例7: 已知Al-2%Cu合金的平衡分配系数k=0.14，液相线斜率mL为-2.5， DL=3×10-5cm2/s 。根据成分过冷理论，稳态时保持平界面生长的稳定性极限GL/R值等于多少？

解：要保持平界面生长时，需使下式成立: 因而，稳态时保持平界面生长的稳定性极限GL/R值为：

4.5.3 Cell structure 胞状组织 Figure 3.2f Experiments on transparent organic liquids show that as a planar interface becomes unstable, it first becomes gently undulatory, with protrusions later developing into the fully formed cells shown in figure below. 在透明有机物液体上的实验表明，当平界面失稳后，首先在界面上出现小的突起，进一步发展成完整的胞状组织. qualitative adj. 定性的 decant vt. 轻轻倒出

Two ways to observe cellular structure in metallic alloys 有两种直接观察金属中胞状组织的方法：
Figure 3-9F One good way to observe cellular structure in metallic alloys is to interrupt the solidification process suddenly by a rapid quench. The cell structure can be shown on the metallographic sections of solidified specimens. 快淬法中断凝固过程。在凝固冷却后的金相试样截面上可显示出胞状组织. An alternative way of studying cellular structures that has been used by many investigators is to decant the bulk liquid during solidification is processing. The liquid-solid interface is then examined directly. 倾倒法。即在凝固进行过程中倒出金属液，直接观察液固界面.

One good way to interrupt the solidification process suddenly by a rapid quench 用快淬法中断凝固过程观察胞状组织。
Figure 3-9F <Constitutional supercooling > 成分过冷

An alternative way is to decant the bulk liquid during solidification is processing. 倾倒法-凝固过程中倒出金属液
A solute buildup could lead to instability of the planar front. 溶质富集导致平界面的不稳定.

Appendix: Cellular interface & cellular structure
胞状界面：长了许多胞或凸缘的固液界面；胞状组织：由于胞状生长，胞的中心溶质贫乏，胞间富集溶质。这种在晶体中由富集的溶质所勾画出来的亚组织称为胞状组织。也称为蜂窝状组织，网络组织。胞状组织特征：在垂直于生长方向的断面中呈为网络状；在平行于生长方向的断面中呈平行线。

随成分过冷的增加，胞晶的形成与发展过程：
Sn-0.05wt%Pb合金随成分过冷的增加，胞晶的形成与发展过程：无成分过冷，平界面成分过冷很小，界面上出现凹沉成分过冷稍大，凹沉增加并趋于连接成分过冷继续增大，凹沉连接成为沟槽成分过冷进一步增大，形成规则的“六角形胞晶” 成分过冷再增大，胞晶变得不规则。

凝固过程研究方法概述 √倾出法（ decant the bulk liquid ） √激冷法（液淬法）（ rapid quench ）
√模拟物质法彩色金相法 X-射线衍射法 √数学解析法（mathematical analysis method ） √温度场的实测法（measurement of temperature field ）数值模拟法（numerical stimulation method ）

4.5.4 Formation of dendrite 枝晶形成
When regular cells form and grow at relatively low rates, they grow perpendicular to the liquid-solid interface regardless of crystal orientation 胞晶低速生长时，无论晶体取向如何，胞晶均垂直于液固界面生长. However, when growth rate is increased, crystallographic effects begin to exert an influence and the cell-growth direction deviates toward the preferred crystallographic growth direction (for example, <100> for cubic metals). 当生长速度增大后，晶体学效应发挥影响，胞晶的生长方向就偏向优先的结晶学生长方向（如立方金属的优先的结晶学生长方向为 <100>方向）.

Simultaneously, the cross section of the cell generally begins to deviate from its previously circular geometry owing to effects of crystallography 同时，晶体学效应也会导致胞晶的横截面开始偏离其原先的圆形几何形状. As growth rate increases still further, the cross structure first becomes more apparent and then serrations begin to appear in the flanges of the cross; that is, secondary dendrite arms become discernible. 随生长速度进一步增大，截面形状变化更明显。先出现凸缘，然后在凸缘上出现锯齿状突起. 此时，可以辨别二次枝晶臂.

Transformation from cellular growth to dendritic growth 胞状生长向枝晶生长的转变
Lateral侧面的，侧部 (a) regular cell growing at low velocity 低速生长的正常胞晶 (b) regular cell growing in <100> dendrite direction 延<100>枝晶方向生长的正常胞晶 (c) flanged cell 带凸缘的胞晶 (d) dendrite exhibiting the start of periodic lateral branching 带有周期侧面分枝的枝晶

总结：柱状树枝晶等轴树枝晶胞状树枝晶胞状晶平面晶 C C0 平面晶胞状晶胞状树枝晶柱状树枝晶等轴树枝晶（过冷度增大）对于C0成分的合金：

Computer Simulation: 定向凝固过程中的组织演变
Microstructure evolution in directional solidification

4.6 Plane front solidification of eutectic alloys 共晶合金的平面生长
规则共晶；金属-金属共晶；非小平面-非小平面共晶非规则共晶；金属-非金属共晶；非小平面-小平面共晶（组织形态）（构成组元）（界面类型） Lamellar eutectic growth 层状共晶生长 Rod eutectic growth 棒状共晶生长 Faceted-nonfaceted eutectic growth 小平面-非小平面共晶生长 Nonfaceted-nonfaceted eutectic growth非小平面-非小平面共晶生长 serration n.锯齿状, 锯齿状突起 flange n.边缘, 轮缘, 凸缘 discernible adj.可辨别的 lateral adj.横(向)的, 侧面的

Typical phase diagram of binary system eutectic alloys

4.6.1 Lamellar eutectic growth 层状共晶生长
非小平面-非小平面共晶，多由金属-金属相或金属-金属间化合物相组成。固液界面在原子尺度上是粗糙界面，宏观上固液界面基本上是平面，等温面近似平直。

Pb-Sn eutectic solidification Pb-Sn共晶凝固
Lamellar勒买乐 periodic adj. 周期的, 定期的 hexagonal adj. 六角形的, 六边形的 strontium n. 锶

The relation of interlamellar spacing  with growth rate R 规则共晶的层片间距与生长速度R 的关系

The relation of interlamellar spacing  with growth rate R

4.6.2 Rod eutectic growth 棒状共晶生长
The rod morphology is the stabler growth form than the lamellar morphology when the volume fraction of one phase is less than 1/. 当一相的体积分数小于1/时，棒状共晶生长比层状共晶生长稳定.

Rod eutectic growth

Computer Simulation: Eutectic growth 共晶生长的数值模拟
附：第三组元对共晶凝固方式的影响第三组元在共晶相和液相之间发生溶质的再分配，形成溶质富集区。在低的生长速率和高的温度梯度下共晶界面可维持平面凝固；但在高的生长速率下共晶凝固的平界面将失稳，从而形成胞状凝固界面。 Computer Simulation: Eutectic growth 共晶生长的数值模拟

4.6.3 Faceted-nonfaceted eutectic growth 小平面-非小平面共晶生长
platelike板状的，层状的 platelike crystal板状晶体 A number of commercially important eutectic alloys, including cast iron and aluminum-silicon, solidify such that one phase has a faceted liquid-solid interface. 许多重要的工业用共晶合金包括铸铁和铝硅合金，在凝固时其中一相具有小平面的液固界面.

See more about “共晶合金的生长方式”
The faceted phase grows rapidly out from the interface with a platelike structure, and the nonfaceting phase immediately following, apparently covering everything except the tip and leaving pools of liquid behind the tip of the solid. 小晶面相生长速度快，在界面前形成层片状结构。非小晶面相紧随其后，紧贴在其上生长，只留下小晶面相的生长尖端与液相接触，从而将大量液相抛在生长尖端后面. platelike板状的，层状的 platelike crystal板状晶体 Example: Transparent organic eutectic during growth. 非小平面-小平面共晶 See more about “共晶合金的生长方式”

附：共晶合金的生长方式根据共晶两相在析出过程中表现的相互关系不同，共晶结晶方式分为：共生生长和离异生长。 1、共生生长
图5-17， 1、共生生长共晶合金结晶时，后析出相依附于领先相表面析出，形成具有共同生长界面的双相核心，然后依靠溶质原子在界面前沿两相间的横向扩散，互相为对方提供生长所需的组元而以同样的速度共同生长。这种生长方式为共生生长。共晶长大时的原子扩散

1）共生生长的基本条件两相的生长能力相近，且后析出相要容易在先析出相上形核和长大。共晶合金两组元在界面前沿的横向传输能保证两相等速生长的需要。理论共生区共生区：共晶相图中，能满足共生生长条件，从而进行共生生长的区域。伪共晶组织

2）实际共生区共生区示意图 a）对称型 b）非对称型比理论共生区小可能不对称于共晶点，偏向高熔点组元一侧。共晶成分合金可能得不到共晶组织！

共晶合金两相生长时，并没有共同的生长界面，并以不同的生长速率进行结晶，没有共生生长的组织特点，称之为离异生长方式，形成的组织叫离异共晶。
2、离异生长共晶合金两相生长时，并没有共同的生长界面，并以不同的生长速率进行结晶，没有共生生长的组织特点，称之为离异生长方式，形成的组织叫离异共晶。小平面相：溶解熵大于23J/(mol.K)。非小平面相：溶解熵小于23J/(mol.K) 晶间偏析型领先相呈团球型

合金成分偏离共晶点很远，在初生晶长得很多、很大后才发生共晶反应。结果一相在初生枝晶上继续长出，而把另一相单独留在了枝晶间。
晶间偏析型：合金成分偏离共晶点很远，在初生晶长得很多、很大后才发生共晶反应。结果一相在初生枝晶上继续长出，而把另一相单独留在了枝晶间。图4-65， C0 A B 晶间偏析型

在金属-非金属共晶中，领先相往往是高熔点的非金属。此时领先相形成团球形态，另一相围绕其表面长大，形成晕圈。
领先相呈团球型：在金属-非金属共晶中，领先相往往是高熔点的非金属。此时领先相形成团球形态，另一相围绕其表面长大，形成晕圈。图2-39 小平面相：溶解熵大于23J/(mol.K)。非小平面相：溶解熵小于23J/(mol.K) 共晶结晶的晕圈组织 a)不完整晕圈的离异生长 a)封闭晕圈的离异生长

Example 8: A solid/liquid interface becomes unstable when relation GL0 (GLmGc) is obeyed in the case of a pure metal (alloy). Show that GL0 is implied by relation GLmGc. Discuss the differences. 例8 对于满足G0的金属或满足GmGc的合金，固液界面将发生失稳。证明后者中也隐含着前者，并讨论二者的差别。解： GL0 （纯金属） GLmGc （合金）稳态时，在平面状固液界面处，GcC0。随着合金纯度的增加，Gc不断减小、消失（Gc=0），这时后者就变为前者。但是这两种情况是有差别的。纯金属的失稳仅受热传导的制约，而合金则同时受到热传导和溶质扩散的控制。因此最终最终的界面形态也不相同。

解：如图所示，取一单位长度的体积元为研究对象。
Example 9: 用纯几何的方法计算相体积分数，在此分数下，棒状或层片状组织均具有较低的总/界面能。假定/界面能为各向同性，相间距在两种情况下相等，并为常数. 层状共晶棒状共晶解：如图所示，取一单位长度的体积元为研究对象。对单位长度的棒状共晶，总界面面积A与体积元的体积V之比为:

对单位长度的层片状共晶，总界面面积A与体积元的体积V之比为:
单位体积的棒状和层片状共晶的界面面积相等时，有：即：棒状共晶的体积分数为：

代入r=/，得出当两种组织的总界面面积相等时的临界体积分数：
即对于具有各向同性的/界面能的共晶系，当其中一相的体积分数小于1/时，棒状共晶由于其单位体积的界面能较低，因而是优选组织。

Assignments 复习思考题 Explain the following terms: Homogeneous nucleation Heterogeneous nucleation  Which factors can affect the formation of the constitutional undercooling or stability of the planar front solidification?  How do these factors affect the constitutional undercooling? To explain the essence and condition of crystallization (i.e. nucleation and growth). How the crystallization affect castings performance? 试述结晶（形核与生长）的实质与条件。结晶过程对铸件性能有何重大的意义?

How to explain the nucleation and growth mechanism by theory of phase transitions? 怎样从相变理论理解液态金属结晶过程中的生核、生长机理? Find out the difference and connection between homogeneous nucleation and heterogeneous nucleation, and explain how the wettability of the substrate influences the critical nucleation undercooling from two aspects, i.e. the critical radius and the free energy change. In order to satisfy the thermodynamics requirements for the heterogeneous nucleation, which two basic conditions must possess for the liquid metal? 试述均质生核与非均质生核之间的区别与联系，并分别从临界晶核曲率半径、生核功两个方面阐述外来衬底的湿润能力对临界生核过冷度的影响。要满足纯金属非均质生核的热力学要求，液态金属必须具备哪两个基本条件?

What influence the substrate curvature has on the heterogeneous nucleation? 衬底曲率对非均质生核过程有何影响？
What is characteristics of the nucleation rate curve of liquid metal? how does it change in the heterogeneous nucleation process? 液态金属生核率曲线特点是什么?在实际的非均质生核过程中这个特点又有何变化? What basic properties an ideal inoculating agent should have?一种理想的生核剂应该具备哪些基本条件? What's the main content of the coherent interface theory? What the reason of its limitation? 界面共格理论的主要内容是什么？其局限性原因何在?

From the atomic-scale view, what conditions determine the micro-structure of solid-liquid interface? Explain the relationship of micro-structure of solid-liquid interface with growth mechanism and growth rate? What are the characteristics of the growth surface and growth direction for the different micro-structures of solid-liquid interface? 从原子尺度看，决定固-液界面微观结构的条件是什么?各种界面结构与其生长机理和生长速度之间有何联系?它们的生长表面和生长方向各有什么特点? Which scale we based on when considering the crystallization process of single-phase alloy? What is the relationship between the scale and the atomic process of the crystallization and the grain structure formed? 我们是从什么尺度着眼讨论单相合金的结晶过程的?它与结晶的原子过程以及最后的晶粒组织之间存在什么联系?

 In one binary phase diagram, the liquid metal (composition: WB =40%) is placed in a ceramic boat and freezing from left side to right side gradually. The temperature gradient at the front of solid-liquid interface is large enough to maintain growth with planar interface front. It is assumed that no solid diffusion and the composition of the entire liquid are uniform. Then: 某二元合金相图1所示。合金液成分为wB=40％，置于长瓷舟中并从左端开始凝固。温度梯度大到足以使固-液界面保持平面生长。假设固相无扩散，液液相均匀混合。试求： (1) What the equilibrium partition ratio k0 between  phase and liquid phase? 相与液相之间的平衡分配系数k0； (2) What’s the weight fraction of eutectic formed after solidification? 凝固后共晶体的数量占试棒长度的百分之几?

某二元合金相图

(3) Draw the curve showing the compositions at different positions along the length of the sample, and indicate the characteristic compositions and corresponding locations. 画出凝固后试棒中溶质B的浓度沿试棒长度的分布曲线，并注明各特征成分及其位置。 What is the solute redistribution？Whether it is only determined by the balance redistribution coefficient k0？When the liquidus and solidus are straight on the phase diagram，try to prove k0 is a constant. 何谓结晶过程中的溶质再分配?它是否仅由平衡分配系数k0所决定?当相图上的液相线和固相线皆为直线时，试证明k0为一常数。

Assume that the composition of the alloy is WB=10% in exercise 13
Assume that the composition of the alloy is WB=10% in exercise 13. (1) Prove that the average composition of the solid is (2) During solidification, the liquid becomes richer in solute and so the temperature of liquidus will reduce. Prove the liquidus temperature TL as a function of fraction solid fs is Where T0 is the melt point of A , m is the slop of the liquidus line. (3) Mark the average solid composition in the phase diagram when TL is 750℃, 700℃, 600℃ and 500℃ respectively. What are the weight fractions of eutectic at different liquidus temperatures?

假设13题中合金成分为wB＝10％。（1）证明已凝固部分(fs)的平均成分；（2）当试棒凝固时，液体成分增高，而这又会降低液相线温度。证明液相线温度TL与fs之间关系。（3）在相图上标出TL分别为750℃、700℃、600℃与500℃下的固相平均成分。问试棒中将有百分之几按共晶凝固?

No diffusion in the solid and the diffusion in the liquid is complete
No diffusion in the solid and the diffusion in the liquid is complete. Assume that the PQ line in the drawing is the average of the Cs' (the solid phase component at T1) and the Cs*, which is the solid-phase component at the solid-liquid interface. Try to prove . 固相无扩散、液相均匀混合。假设右图PQ线是CS' (T1时固相成分) 与界面处固相成分CS*的算术平均值，试证

Explain the meaning of constitutional undercooling and thermal undercooling. What’s the differences and relation between them? 试述成分过冷与热过冷的涵义以及它们之间的区别和联系。 What is the criterion for constitutional undercooling? Which factors affect the magnitude of the constitutional undercooling? How does it affect the growth pattern of crystals and the crystallization state? Are all growth patterns just decided by the constitutional undercooling factors? 何谓成分过冷判据? 成分过冷的大小受哪些因素的影响?它又是如何影响着晶体的生长方式和结晶状态的？所有的生长方式仅仅由成分过冷因素决定吗？

The growth velocity R>2. 510-3cm/s in ingot and casting
The growth velocity R>2.510-3cm/s in ingot and casting. For most of the metals，DL≈10-5cm2/s, |m|>1 below liquidus temperature. Assume s=L, calculating the GL needed for planar growth when C0 =10%, 1%, 0.01% and k0 =0.4 and Taking into account that GL＜3~5℃/cm in the ingot or castings in general, what conclusion can be got based the calculation results? 已知在铸锭和铸件中R＞2.5×10-3cm/s；多数金属在液相线温度下DL=10-5cm2/s；|m|＞1。假设s=L，试分别求出当C0=10%（质量分数，下同）、1%、0.01%以及k0=0.4与0.1时确保平面生长所必需的GL值。考虑到铸锭或铸件中一般情况下GL＜3~5℃/cm，根据计算结果你能得出什么结论？

What is the relation between the refining dendrite arm spacing and improving the quality of castings? 细化枝晶间距与提高铸件质量之间有什么关系？ During eutectic crystallization, what are the basic conditions for the coupled growth and divorced growth? What the relation between the solid-liquid interface structure of two phases in eutectic and structural characteristics of coupled growth zone? What is their influence on the crystallization mode of eutectic alloys? 共晶结晶中，满足共生生长和离异生长的基本条件是什么？共晶两相的固-液界面结构与其共生区结构特点有何关系？它们对共晶合金的结晶方式有何影响?

The following is a schematic diagram showing the volume element for a two phase eutectic with coupled growth. Presume the volume element is a rectangle which all sides are unit. If the  phase is rod like, and its volume Vr=r2, the area between  and  phases is Sr=2r (r is the radius of the rod on the transverse section). If the  phase is flake like, its volume Vb=b, area between phases is Sb=2. Please prove: (1) If Vr=1/, Sr=Sb; (2) If Vr1/, SrSb ; (3) if Vr1/, SrSb. Explain the influence of interfacial area energy on the rod to flake structure transform in the coupled eutectic growth based the above calculation results

下图为某二元共生共晶体积元的示意图。设体积元是一个边长为1的立方体。若相为棒状，其体积为Vb=b，、相间面积为Sr=2r （式中r为棒横截面半径）；若为片状，则其体积为Vb=b ，相间面积为Sb=2。试证明（1）当Vr=1/ 时， Sr=Sb ；（2）当Vr1/ 时，SrSb ；（3）当Vr1/ 时， SrSb ；试用上述结果说明相间界面能对共生共晶中的棒状→片状组织转变的规律。

What‘s the most obvious characteristics of faceted-non-faceted eutectic growth？What’s the relationship between that and the principle of modification? 小面-非小面共晶生长的最大特点是什么？它与变质处理原理之间有什么关系？在熔点附近的温度下，液态金属结构有哪些特点?液态金属结构与结晶之间有何联系?液态金属性质对结晶有何影响? 试述均质形核与非均质形核有何联系与区别?非均质形核时，对形核剂有什么要求? 根据固—液界面的微观结构可将晶体分成哪两类?这两类晶体的生长各有什么特点?

铸铁中的石墨和Al-Si合金中的硅属于何种界面微观结构?为什么在通常的生长条件下总是呈片状生长的?
正、负温度梯度是怎样建立的? 什么叫做界面的不稳定性?它是什么原因引起的? 负温度梯度(GL＜0)时，界面总是不稳定的；正温度梯度(GL>0)时，界面总是稳定的，对不对? 纯金属和单相合金的过冷度的本质是否一致?为什么单相合金的过冷度叫做成分过冷? 溶质再分配是怎样引起的?它对单相合金的结晶有些什么影响？

为什么平衡分配系数的定义中可以对固-液界面作“平衡”的假设?
试述单相合金在几种不同的传质条件下的溶质在界面前沿的分布规律。何谓成分过冷?影响成分过冷的因素有哪些?它如何影响晶体的生长? 共晶体的生长方式分为哪几种?各有何特点? 什么叫共生区?它是怎样形成的? 层片状共晶体的层片间距受到哪些因素的影响? 试推导出平衡凝固及液相完全混合条件下凝固时，凝固界面温度 TL*与固相质量分数fs的关系。

Chapter 4 Principles of solidification

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