晶体生在作业翻译

晶体生长翻译晶体生长翻译 应用物理应用物理 10 2 钱冬伟钱冬伟 I10690210 2 2 2 The equilibrium state 2 2 2 平衡状态平衡状态 Before going further it is instructive to consider the equilibrium state In an undersaturated system the equilibrium state gives the equilibrium distribution of the homophase density fluctuations Frenkel 1955 In a supersaturated system the equilibrium state can be realized near enough to the phase equilibrium Obviously the latter will be a metastable equilibrium and will give the equilibrium distribution of the heterophase fluctuations Far from the line of phase equilibrium an equilibrium state cannot be realized However we can write an expression even in this case which will serve as a convenient reference Under the condition Jo 0 from 2 47 follows 考虑到平衡状态 在之前它是有益的 在一个欠饱和系统的平衡状态给出的同位的密 度波动的平衡分布 弗伦凯尔 1955 在饱和系统的平衡状态 可实现足够近的相平衡 显 然 后者将是一个亚稳平衡和将得到的平衡分布的多相波动 远离线相平衡 的平衡状态不能被实现 但是 我们可以写一个表达式 即使在这种情况下 这将作为一 个方便的参考 Jo 0 的条件下 从式 2 47 如下 where Nn denote now the equilibrium concentrations of clusters of class 7t in the absence of molecular flux in the system Equation 2 51 is known as the equation of detailed balance It can be rewritten in the form 其中 Nn表示现在的平衡浓度中类 7 吨团簇的系统中不存在的分子通量 方程 2 51 是已知的详细的平衡方程 它可以被重写成 Multiplying the ratios Nn Nn l from n 2 to n gives NN NN L从 n 2 到 n 的比值连乘给出 As seen the rate constant ratio on the right hand side appears in the expressions for both the equilibrium concentration of clusters Nn of class 90 Nucleation n and the steady state rate of nucleation 2 50 Obviously the problem is reduced to finding expressions for the rate constants n and n We will do that for the simplest case of homogeneous formation of liquid nuclei in supersaturated vapors adopting the idea of Gibbs that the nuclei represent small liquid droplets The rate constant of the growth reactions n is given by the number of collisions of atoms from the vapor phase on the surface of the droplets Then for the rate constant of growth of a cluster of class n 1 to form a cluster of class n we have 正如看到的右手侧上的速率常数比出现 90 类晶核 的平衡浓度和稳定状态下的成核率 在两个团簇 NN 2 50 的表达 很显然 该问题被减小的速率常数瓦特找到表达式 n 和 n 我们将做到这一点的最简单的情况 液体的饱和蒸汽采用吉布斯的理想核 核代表 小液滴的形成均匀 速率常数的生长反应 n 从汽相的液滴的表面上的原子碰撞的数目 由下式给出 然后我们的速率常数的一类 n 1 个集群 形成产业集群类增长 where P 2 mkT 1 2 is the number of collisions per unit area P being the vapor pressure available in the system and n 1 is the surface area of a cluster of class n 1 The rate constant of the reverse reaction of decay of a cluster of class n to form a cluster of class n 1 can be evaluated as follows Volmer 1939 In equilibrium with the vapor phase the number of atoms leaving the cluster in a fixed interval of time is equal to the number of atoms arriving at its surface Hence the flux of atoms leaving the cluster is equal to the equilibrium flux of atoms arriving at its surface On the other hand condensation takes place when the center of mass of the molecule joining the droplet crosses the sphere of action of the interatomic forces Fig 2 10 The radius of the droplet increases just after that event The evaporation of a molecule is the reverse process When it leaves the droplet its center of mass should cross the same sphere of action and at that moment the droplet shrinks and its surface area becomes equal to n 1 This area namely should be taken into account when considering the rate constant of the reverse reaction and 其中 P 2 mkT 1 2表示每单位面积的数量的碰撞 P 是在系统中可用的蒸汽压力 N 1是一个群集的类别 n 1 的表面面积 可以计算如下 Volmer 方程 1939 衰减的 N 类团簇 形成链团簇的类别 n 1 的逆反应速 率常数 与蒸汽相在平衡状态下 离开团簇的原子 在一个固定的时间间隔的数目是等于 到达在其表面上的原子的数目 因此 离开团簇的原子通量等于到达在其表面上的原子的 平衡通量 另一方面 缩合时发生分子加入液滴的质量中心的原子间力 图 2 10 的行动 球体相交 只是该事件发生后的液滴的半径增加 一个分子的蒸发是相反的过程 当它离 开的液滴 其质量中心应穿过相同的球体的行动 并在那一刻的液滴收缩和其表面积变得 等于 N 1 此区域 即 应该考虑到当考虑的逆反应的速率常数和 where P is the equilibrium vapor pressure of the cluster of class n equation The pressures P and Pn can be expressed through the Thomson Gibbs equation 其中 P 是类别 n 团簇的平衡蒸气压 方程 通过汤姆逊 吉布斯方程可以表示的压力 P 和 Pn and 和 Fig 2 10 For the determination of the surface area of a liquid droplet containing n molecules upon detachment of a single molecule shown by the small circle The molecule leaves the droplet when its center of mass crosses the surface of action of the intermolecular forces given by the dashed circle Precisely at that moment the surface area of the droplet is n 1 as given by solid circle after Volmer 1939 图 2 10 对于包含 n 分子的小圆圈所示的一个单分子脱离后的液滴的表面面积的测定 分 子离开时 其质量中心穿过给定的由虚线圆中的分子间力的作用的表面的液滴 正是在那 一刻的液滴的表面面积是 N 1后沃尔莫方程给出由实心圆 1939 where rn is the radius of a droplet consisting of n molecules Then 其中 rn个分子组成的液滴的半径 然后 We replace then the radii n and by the number of atoms through n 1 4 r3 3 and for every term in 2 52 and also in the sum of 2 50 one obtains 然后我们替换的半径 N和 由的原子数通过 n 1 4 r3 3 并为每一个术语中 2 52 也包括在 2 50 的总和 得到 Assuming n 1 the capillary approximation we replace the sum by an integral and carrying out the integration yields 假设 n 1 毛细管逼近 我们更换一个进行整合的产量总和的整体 A comparison of the above equation with Eqs 2 6 and 2 4 shows immediately that the expression in the curly brackets in the right hand side is simply the function G n kT Hence 上述方程式的比较 2 6 和 2 4 立即表达在花括号中的右手边是简单的功能的 G N KT 故 Or 而 i e every term in the sum in the denominator of 2 50 represents one point of the dependence exp G n kT Substituting 2 55 into 2 52 gives for the equilibrium concentration of clusters of class n 即每一个术语的分母中的总和 2 50 表示的一个点的依赖进出口 G n KT 2 55 式代入 2 52 N 类集群的平衡浓度 The reader can find a more rigorous derivation of 2 56 in the monograph of Frenkel 1955 see also Toschev 1973 读者可以找到一个更严格的推导 2 56 在弗伦凯尔的专着 1955 见 也陶斯崔恩 1973 2 2 3 Steady state nucleation rate 稳态形核率稳态形核率 Replacing the sum in the denominator of Eq 2 50 by an integral gives 更换式的分母的总和 2 50 由一个不可分割的式子给出 The function in the exponent in the right hand side displays a maximum at n n Fig 2 1 and can be expanded in Taylor series in the vicinity of the maximum 中的函数中的指数的右手侧显示的和最多在 n n 图 2 1 并且可以扩展在泰勒级数 在附近的最大 Then the sum attains the form 然后总和的形式 In order to carry out the integration we make the following approximations As shown in Fig 2 11 the exponent under the integral displays a sharp maximum in the vicinity of n and the limits of integration can be extended to and without making a significant error Also the 为了进行整合 我们提出以下近似 如图中所示 2 11 的指数下的积分显示在周边的 n 急剧最大 积分限为 和 不进行显着的错误的情况下 可以延长 另外 Fig 2 11 Dependence of the Gibbs free energy change G n kT dashed line exp G n kT and the reciprocal of the rate constant of the forward reaction on the cluster size As seen exp G n kT displays a sharp maximum and its width is confined to the near vicinity of the critical size The reciprocal of the rate constant l wn is a weak function of n and can be taken as a constant at the critical size 图 2 11 依赖的吉布斯自由能变化 G N KT 虚线 进出口 G N KT 团簇 的大小正向反应速率常数的倒数 可以看出 exp G N KT 显示一个尖锐最大的 其 宽度被限制在附近的临界尺寸 倒数的速率常数 l WN 的 n 是一种弱的功能 并且可以被 视为恒定的临界尺寸 rate constant wn is not a sensitive function of n and can be replaced by n const and then taken out before the integral Finally carrying out the integration from minus infinity to plus infinity one obtains for the steady state nucleation rate 速率常数关于 n 是不敏感的功能 可以取代 N const 和然后取出之前的积分 最后 进行整合 从负无穷到正无穷大 得到的稳态形核率 where 21 is the steady state concentration of single molecules in the vapor phase and w is the frequency of the attachment of molecules to the critical nucleus The parameter in the expression 2 58 for the steady state nucleation rate 其中 图 21 是在汽相和 w 的单分子的稳态浓度是附着分子的临界核的频率 参数 在表达的稳态形核率 2 58 is known in the literature as a factor of Zeldovich 1943 see also Frenkel 1955 In fact it was derived for the first time by Farkas 1927 Its physical meaning can be revealed if one inspects more closely the definition of the steady state nucleation rate 2 50 Zeldovich 1943 argued that the steady state distribution function Zn deviates perceptibly from the equilibrium one Nn only in the vicinity of the critical size n In other words the processes taking place in an interval An nr n1 Fig 2 12 around the critical size determine the overall rate of nucleation According to Zeldovich the width of this interval is determined by the condition that the free energy change G n varies by kT around the maximum at n n1 Zn Nn and at n nr N ZN 0 i e 文献中已知的作为泽利多维奇 1943 也见弗伦凯尔 1955 的一个因素 事实上第一 次得到的是法卡斯 1927 可以发现如果一个系统更密切稳态的成核速率 2 50 的定义及 其物理意义 泽利多维奇 1943 认为 稳态分布函数锌偏离可察觉从平衡 1 中 Nn 只在 周边临界大小为 n 换言之 在一个间隔中发生的各种过程 An nr n1 的 图 2 12 在临界点附近的大小确定的整体成核率 根据泽利多维奇由条件确定该时间间隔的 宽度的自由能的变化 G n 的变化的最大值 在 n n1的 Zn kT n nr N ZN 0 即 Fig 2 12 For the determination of the interval n nr n1 around the critical size According to Zeldovich 1943 the processes which take place in this interval determine the overall rate of the nucleation The width of the interval is in fact the reciprocal to the nonequilibrium Zeldovich factor 图 2 12 用于测定的时间间隔的 n nr n1的周围的临界尺寸 根据泽利多维奇 1943 发生在这个区间的确定过程的整体成核率 间隔的宽度实际上是非平衡泽利多维奇因子 的倒数 Bearing in mind 2 57 it follows that the width of the interval n reads 记住 2 57 的宽度读取的时间间隔 N In other words the factor of Zeldovich is simply a reciprocal of n and thus accounts for the deviation of the system from the equilibrium state A more rigorous treatment of this problem is given by Kashchiev 1969 Bearing in mind Eqs 2 2 and 2 3 it turns out that is directly proportional to the square of the supersaturation 2 8 1 3kT 1 2 In other words 0 at the phase equilibrium 0 and increases steeply when deviating from it In the cas