12-04-2024

空气中的二氧化碳（CO ₂ ）或咖啡中的糖等化学物质，即使处于静止状态，一段时间后也会与周围的流体完全混合。热量也会通过静止的物体在同样的传输机制下传递，直到温度变得均匀为止。类似地，在存在摩擦的流体层中，作用于流体表面上力会在流体层的整个厚度上传递。在流体环境中，湍流能促进其组分的分散，但是这种分散并没有达到分子的尺度。正是这种分子的无序运动才完成了这一我们称之为扩散的输运过程。我们还发现，分子扩散是一个非常缓慢的过程，但湍流能显著促进扩散。

“ diffusion ”一词的含义不尽相同，但所有解释的共同点是：某种量通过介质基本组成粒子的搅动而分散到周围介质中的输运过程。在物理学中，我们要区分下面两种类型的扩散现象：

一种是基于介质中原子或分子无序运动属性而产生的 传输现象 ，其特征在于输运量的分布会趋于均匀。这个量可以是特定化学物质的浓度、热量或是动量。最终达到的均匀状态即为环境的热力学平衡状态。

另外一种是非均匀介质中 波的衍射现象 ，这种波可以是像光之类的电磁波，也可以是声波。这些现象也会在本百科全书中的其他文章中介绍，如 《天空的颜色》 和 《声音的发射、传播和感知》 。在英语中，这种波的衍射现象也被称为“散射”（scattering），但我们应将其与扩散（diffusion）现象本身相区别。

在日常用语中，“ diffusion ”一词也有不同的含义，它既可以指对公众的知识的传播，也可以指通过无线电（无线电广播）或电视（电视广播）实现各种类型的信息传递。

1. 静止的空气：一群躁动不安的分子

文章 《地球的大气层和气体层》 中提到，干燥空气的主要成分有氮气（71%）、氧气（28%）、氩气（少于1%），还有其他含量更少的气体。气体分子会充满所有可用的空间，这是气体有别于其他凝聚态物质（液体或固体）的一种属性。固体或液体有特定的体积，固体还有自己固定的形状。

首先，让我们以空气为例来描述相对稀薄气体中分子的主要参数。分子能搅动的尺度（纳米 [1] 尺度）远超我们的感官范围。根据玻尔兹曼（Ludwig Boltzmann） [2] 的一些假设，我们可以简单确定这种环境下参数典型的数量级：假设 分子 都是 直径相同、质量相同的球体 ，直径设为 σ ，质量设为 m ，并假设这些 球体具有相同的速度 c 。不过这样的假设会大大简化这一问题，因为实际上氮气（N ₂ ）和氧气（O ₂ ）都是双原子分子，状似哑铃而非球体，而且由于分子在不停地相互碰撞，其速度也不可能为常数。尽管如此，这种假设模型在宏观尺度描述分子及其相互作用上仍具代表性，且量级相当精确。

设单位体积中的分子数为 n ，我们来考虑其中的一个分子。在单位时间（1s）内，这个分子沿虚线轨迹行进了距离 c （如图1所示）。因为我们在数量级上也作了限定，不妨假设与之碰撞的其它所有分子都是静止的。它将遇到中心位于圆柱体区域内的其他分子，该圆柱体的直径是分子直径的两倍2 σ ，体积等于其底面（直径为2 σ 的圆盘）面积乘以单位时间分子移动的距离 c ，即 cπσ ² ，因此，该圆柱体中存在的分子数为 ncπσ ² （如图2所示）。分子连续两次碰撞之间通过的平均距离称为 平均自由程 ，用 λ 来表示 ， λ = 1/ nπσ ² 。“平均”的意思是说分子在碰撞之间的实际自由程是在 λ 左右变化的（如图1所示）。

2. 气体的分子参数与宏观属性之间的关系

通过感知和测量，我们可以很容易地将分子世界的这些参数与一些宏观量相关联。 《压强、温度和热量》 一文对压力 p ，温度 T ，单位体积的质量（密度） ρ 和粘度 [3] μ 这些物理量之间的关系已经做了分析，从中我们可以得到分子尺度与宏观尺度之间的 第一个关系 ： m = M / N _A ， N _A = 6.022×10 ²³ mol ^-1 ，即每个分子的质量 m 等于摩尔质量 M 除以阿伏伽德罗常数（每摩尔中的分子数）。并且密度 ρ 等于单位体积中的分子数量 n 乘以每个分子的质量，这样我们就得到了两种尺度参数之间的 第二个关系 ： ρ = nm 。

为了将气态介质中某一点的压力 p 与气体分子的参数联系起来，我们假设在该气态介质中的某处有一个单位面积的固平面（ S = 1，单位可任意选择）。单位时间内该平面的两侧会不断地受到气体分子从各个不同方向的撞击和动量积累，在这一过程中气体分子失去并重新获得动能，每个分子的动能为 mc ² / 2 。由此，施加在单位表面上的压力 p 平均为 nmc ² 。考虑到气体分子的真实速度 [4] 和实际相撞速度，经严格的推算可得到压力的表达式为 p = nmc ² /3，其中系数1/3修正了由于我们简化模型估算而带来的误差。从而我们得到了气体分子参数（ n ， m ， c ）和宏观属性（压力 p ）之间的 第三个关系 。

由上面的几个关系式我们知道压力和密度都与分子的参量相关，可由理想气体状态方程表示出温度与压力的关系： p / ρ = RT / M ，其中 R = 8.314 m ³ ·Pa·mol ^-1 ·K ^-1 ， R 为理想气体常数， M 为气体混合物的平均摩尔质量。这样我们就得到了 第四个关系式 [5] ： T = Mc ² /3 R ，这个表达式强调了温度 T 和分子的速度平方 c ² 之间的正比例关系（详见文章 《压强、温度和热量》 ）。

除此之外，还有一个表示气体动力学粘度的关系式： μ = nmcλ /3，这个关系式将会在本文第5节中说明。由此， 海平面处的空气 ，通常具有以下值：

p = 1.01310 ⁵ Pa ， T = 288 K ， ρ = 1.22 kg·m ^-3 ， ν = 10 ^-5 m ² ·s ^-1 ，

由上述的五个关系式可以得到：

n = 2.510 ²⁵ m ^-3 ， m = 4.810 ^-26 kg ， c = 497 m·s ^-1 ，

d = 3.410 ^-9 m ， σ = 0.4610 ^-9 m ， λ = 610 ^-8 m 。

为了对分子世界的尺度有一个更准确的认识，我们可以从上述计算结果中得知，边长为1微米的小立方体包含约2500万个气体分子，且两个相邻分子之间的距离（ d ）约为其直径（ σ ）的7倍，平均自由程（ λ ）约为60 nm，是分子直径的130倍，并且分子的平均速度（ c = 497 m·s ^-1 ）比声速（340 m·s ^-1 ）略高。

3. 气体扩散机制

假设在静态的气体环境某处有一个平面（如图3），在单位时间内有许多分子从两个相反的方向同时穿过该平面， 宏观尺度上静止的平面 也具有这样的特征。假设一个参量 G （例如给定化学物质的温度或浓度）不均匀地分布在该平面附近，在 x >0侧较大，在 x <0侧较小（x表示到该平面的垂直距离）。在这样的条件下， 气体不再处于热力学平衡状态 ，由于在 x > 0侧的分子比 x <0侧的分子具有更大的 G值 ，因此分子搅动会使 G 穿过平面 x = 0，产生从 G 大（ x > 0）一侧到 G 小（ x <0）一侧的净流动，使 G 的分布倾向于均匀化。我们把这种不均匀分布量的传输能力称之为 扩散 。正是这种能力使气体具有建立 热力学平衡 的趋势。

然而，建立这种平衡所需的时间比分子两次碰撞的间隔时间（ λ / c ≈ 1.4×10 ^-10 s）长得多（几小时或更久）。因此在非平衡状态下气体分子的碰撞会非常频繁，如果碰撞后的分子都具有相同的 G 值，即 G / n （ G 为单位体积的值），我们就说气体处于 局部平衡 的状态。

让我们想象图4示意的实验。在初始状态下，两种气体被隔板隔开。迅速去掉隔板后容器内的分子分布并不均匀，如图4（2）所示，扩散作用使得两种气体逐渐混合。最初聚集在左侧的灰色分子到达右侧，而最初聚集在右侧的黑色分子到达左侧。这种扩散一直持续到分子均匀分布为止。

穿过平面 x = 0（如图3）的分子的 G 值平均为 G / n ， 其运动距离约等于平均自由程 λ 。因为与宏观尺寸相比 λ 极小，所以 G(x) 在 x = 0附近可用其斜率为 Γ ₀ 的切线替代，这样就可以得到一个简单的分布： G = G ₀ + Γ ₀ x 。我们进一步假设：对于来自 x > 0侧的分子 G ₊ = G ₀ + Γ ₀ λ ，对于来自 x <0侧的分子 G _– = G ₀ – Γ ₀ λ 。

要估算 G 通过该单位面积截面的净流量，须将 G 的前后之差 γ ₀ λ 乘以单位时间内穿过该截面的分子数。在该单位面积上方的体积中（如图5），有 nλ 个分子，每个分子在单位时间内发生 c / λ 次碰撞，所以在该体积内单位时间发生的碰撞次数约为 nc 。但是只有碰撞后朝向该截面的分子才会穿过该面，由此进一步计算得出单位时间内穿过该截面的分子数为 nc /3（系数也是1/3）。

最后，可以通过关系式 φ = (cλ/3)dG/dx 来估计平面 x = 0上单位面积的 G 的流量 φ 。由此我们可以得到如下结论：如果 G 的分布不均匀，则 通过分子搅动传输的 G 的净流量与该区域 G 的局部梯度 d G /d x 成正比 ，可以用 φ = -DdG/dx 表示。通常，我们在式中加上负号“-”，这样当 dG/dx 为负时（比如热量从热侧流向冷侧），流量则为正。其中系数 D = cλ / 3，称为气体的 扩散系数 ，它是纯粹的运动学参量，单位是m ² /s。扩散系数反映了气体扩散能力的大小，与气体自身的性质无关。

用第2节中给出的数值可以估算出空气的扩散系数。当 c = 497 m·s ^-1 和 λ = 6×10 ^-8 m时，可得出 D ≈ 10 ^-5 m ² ·s ^-1 。为了使该数值更具有实际意义，让我们回到图4所示的实验中：设容器的长度为 L ，因为扩散系数是长度的平方除以时间（ L ² T ^-1 ），所以达到最终平衡状态所需时间为 L ² / D 。如果 L ≈ 10cm，则扩散持续的时间约为一个小时。倘若要使这个过程的持续时间约为1秒，则图4中的容器长约为1毫米，与小气泡相仿。

4. 湍流扩散

我们在生活中遇到的大多数流体中，如空气和水，湍流几乎是无处不在的。这是一种与分子运动完全无关的紊乱现象。在诸如风或洋流之类的流动中，湍流大范围存在，有显著的、可测量到的速度脉动。即使没有强加的流动，仅由于流体的动力学不稳定性 [6] ，湍流也会以一系列大小不一的涡的形式存在。这与前面几节所谈到的分子搅动有相似之处，然而，宏观涡的大小要远大于分子的平均自由程，于是普朗特（Prandtl） [7] 定义了一个新的量： 混合长度 [8] ，类似于气体分子的平均自由程。

目前用来测量混合长度的技术有很多，混合长度代表涡的特征尺寸。如图6中烟雾的涡，还有从烟囱或火山上方升起的烟雾的涡，从中可以看出涡的尺寸始终可延展到整个流动范围。值得注意的是，烟雾的下部（层流部分）由于分子扩散的作用会略微变宽，而烟雾上部（湍流部分）变宽的速度更快，这更体现了因烟羽的不稳定而产生的随机的涡对烟雾各种成分的输运作用。

混合长度的概念非常模糊，但它的数量级却可以有很高的精确性，特别是存在边界层的情况下（如图7所示）。普朗特（Prandtl）提出在边界层内混合长度的表达式为 l = κy ，其中 y 是到边界的距离， κ 是根据不同条件确定的常数。有了混合长度 l 的表达式，就可以用 u’ ≈ l d U /d y 估算距边界为 y 处速度脉动的量级。类比气体扩散，引入了类似分子扩散系数（ D = cλ /3）的概念，用 u’ 代替 c 、 用 l = κy 代替 λ ， 则边界层内的湍流扩散系数 D _t 为： D _t ≈ u’l ≈ κ ² y ² d U /d y 。

举例来说，在我们所处的厚度约为几百米的大气边界层内，距地面 y ≈ 10 m处，假设空气的平均速度约为10 m·s ^-1 ，d U /d y ≈1 s ^-1 ，由实验得到的常数κ的值为0.1，则可计算出湍流的扩散系数 D _t ≈ 10 m ² ·s ^-1 ，它是分子扩散系数的一百万倍。所以 获得湍流混合物所需的时间约为十分之一秒 ： t ≈ l ² / D _t ≈ 0.1s。但是这种混合是不完全的，因为各成分仅能扩散到最小的涡的尺度，这个尺度取决于流体的粘度。在大气中，边界层的厚度约为1毫米。我们知道 分子搅动达到分子尺度的混合 所需要的时间为 L ² / D 。在现实中，分子扩散达到毫米尺度时就开始于与湍流的涡的扩散并存了。由此我们可以进行简化，假设分子扩散仅在毫米量级以下起作用。当 L ≈ 10 ^-3 m， D ≈ 10 ^-5 m ² · s ^-1 时，我们得到 扩散的时间尺度仍 为十分之一秒 左右。

从这些估算中我们可以得到如下结论：

湍流扩散只能到达最小的涡的尺度。

分子扩散将继续完成混合过程，两种机制的时间尺度均在十分之一秒。

如果不存在湍流，完全混合所需的时间将达到数小时。

5. 质量扩散和热扩散

通常用质量扩散率来表示化学物质在其所处流体中的扩散。假设流体（气体或液体）中的某些分子与其它分子不同，例如气态的水分子（H ₂ O），处于较大的液体区域（例如宁静的湖泊）上的静止空气中。记 C 为在任一高度 z 下空气中水的体积浓度。参照上面推导出的公式，水蒸气向高海拔处扩散的净流量的表达式为 φ = – D d C /d z ，这就是 菲克定律 。要计算出水从低海拔处质量扩散的量，还要补充质量平衡以及初始条件和边界条件。

热扩散率 的表达式是由通过分子搅动进行热传递的过程推导得来。假设 G 是气体单位体积的内能 E ，则有d E = ρC _v d T ，其中d T 表示无穷小的温度变化量， C _v 为气体的定容热容。所以内能，即热量的扩散传输可表示为 φ = – ρC _v D d T /d z 。通常将该等式的以下两种形式都称为 傅立叶定律 ： φ = – k d T /d z （其中 k = ρC _v D = C _v nmcλ /3为气体的 热导率 ）；或 φ = – αρC _p d T /d z （其中 α = k / ρC _p 为气体的 热扩散率 ， C _p 是气体的定压热容）。

上述气体，特别是空气的物理性质（ D ， k ， α ， μ ， ν ）与分子参数 n ， m ， c 和 λ 有关。其中，对于给定密度的气体，只有分子的速度 c 可能随温度 T 变化，可以表示为： T = Mc ² /3 R = mc ² /3 kB 。由此可以推断出 气体的扩散系数 （ D ， k ， α ， μ ， ν ） 随着其温度的平方根一同变化 。

6. 动量扩散：粘度

假设上述概念对沿着壁面流动的宏观流体仍然适用。只要这个宏观流动的时间（通常为数秒或数分钟）远大于分子两次碰撞间隔的时间（约10 ^-11 s），就可以证明这个假设是正确的。

记 u 为沿壁面流动的流体质点的局部速度。单位体积流体的 动量 为 ρu ，其中 ρ 为流体的密度。让我们将该量 ρu 记为 G ，它表示由扩散传输的流体在垂直于 z 方向平面上的通量。参照上文所述，由分子搅动传输的动量通过平面 z = 0的速率可以表示为 φ = – ρD d u /d z 。这个通量的方向与切应力 [9] ，即单位面积上流体对壁面施加的力 τ 相反（详见 《物质是如何形变的——流体和固体》 ），其表达式 [10] 通常写为 τ = μ d u /d z ，称为 牛顿粘性定律 ，满足该定律的流体称为 牛顿流体 。 μ = ρD = nmcλ /3是流体的 动力粘度 ，另一种常用的形式是 运动粘度 ν = μ / ρ = cλ /3，其中涉及到了扩散系数。只要不发生边界层的分离现象（如图7），沿着运动壁面总 存在 着非常薄的 边界层 ，这显示了通过流体粘性进行的动量传输。在边界层中流体的速度分布为从与移动壁面接触处的 u 变化到边界层外的0。

当 z = 0不是壁面与流体的界面，而是两个流层之间的界面时，上面的结论也同样适用。当两个流层之间的界面以速度 u 流动时，牛顿粘性定律也可以表示流层之间相互作用的切向力，通常称之为 摩擦力 。这个沿速度方向的切向力会与和界面垂直方向的压力叠加（详见 《压强、温度和热量》 ）。

7. 液体中的扩散机制

实际上所有液体都有自己的体积，这说明组成它们的分子彼此靠得很近。相邻分子之间的平均距离取决于施加在该液体上的压力，分子间距在通常情况下（ p ≈ 10 ³ hPa）比分子尺寸稍大，为10 ^-10 ~10 ^-9 m。这些分子的热运动，或称分子振动，使液体介质具有了一定的扩散能力，但是其扩散能力远弱于气体。这主要是因为液体中分子的作用半径小于其自身大小，而在气体中分子的作用半径即平均自由程，约为分子大小的25倍。这就解释了为什么在咖啡里加糖后需要用勺子搅拌。

存在于液体介质中的任何与介质分子不同的粒子都会不断地受到整体振动的影响，与气体介质中类似，这些粒子会从多的一侧流向少的一侧。该通量仍可以用菲克定律来表示：φ = -DdC/dx，其中D表示液体中粒子的扩散系数，C为其体积浓度。

为了了解液体的 质量扩散率 ，从而反映其传输另一种物质的能力，爱因斯坦在1905年提出了如下定律 [11] ： D = k _B T / 6πμr ，其中 k _B = 1.38×10 ^-23 J·K ^-1 是玻尔兹曼常数， T 是绝对温度， μ 是液体的动力粘度， r 是被输运粒子的半径。在一个 r ≈10 ^-9 m、 μ ≈10 ^-3 Pa·s的液体环境中，例如水，当环境温度为300 K时，我们得到 D ≈ 2×10 ^-10 m ² ·s ^-1 ，比气体的扩散率小了约10000倍。爱因斯坦的公式中含有温度 T ，这表明分子热运动是扩散背后的真正驱动力。而分母中包含 μ 和 r 则意味着粘度与扩散率呈反比例关系，粒子的半径越大，扩散就越困难。

对于水和所有电绝缘液体而言，它们的热扩散率和质量扩散率差不多。但是 对于液态金属 来说，电子的热传导也是一种主要的传热机制，比质量扩散要重要得多。最著名的例子是液态钠用于冷却超再生器型核反应堆，如“凤凰号”，这一反应堆位于马尔库尔（Marcoule），1973年至2010年期间一直在运行。在300 K左右的温度下，液态钠的热扩散系数约为7×10 ^-5 m ² ·s ^-1 ，比它的质量扩散率要高出10000倍。

8. 总结

混合物中特定参量或物质的扩散，如热量和动量，是由其基本粒子即分子或原子的搅动引起的。这是一种非常缓慢的输运机制，但却是微观尺度上的主导机制。

在空气和水等粘度不太高的流体中，湍流会促进质量、热量或动量的分散。但是这种机制下的最小尺度不能小于由粘度控制的最小的涡的尺度（通常大于一微米），小于这个尺度的扩散过程则由分子扩散继续作用来完成混合。

扩散定律起初是凭经验发现的，然后才发展成为理论模型。在稀薄气体（例如空气）中相对来说比较简单，本文已进行了概述。而在液体环境中则复杂得多，目前仍在研究中。

参考资料及说明

封面图片： 在日出之前乡村宁静止的空气中，植被呼吸作用释放的物质通过扩散作用传输到空气中。

[1] 纳米是米的十亿分之一（10 ^-9 m），微米是米的百万分之一（1μm= 10 ^-6 m）

[2] 路德维希·玻尔兹曼（Ludwig Boltzmann，1844-1906），奥地利物理学家、哲学家，提出了一种名为“硬球”的分子模型，由该模型推导出了一个以他名字命名的方程。

[3] 通常用运动粘度 ν 代替，运动粘度 ν 是动力粘度 μ 与密度 ρ 之比，即 ν = μ / ρ 。

[4] 所有这些速度矢量，包括其各自的方向和值，都可用麦克斯韦分布函数表示。但计算较复杂，超出了本文涉及的范围。

[5] 通过引入与R相关的玻尔兹曼常数 k _B 和阿伏伽德罗常数 N _A = 6.0248×10 ²³ ， k _B = R / N _A = 1.380×10 ^-23 J/ K，该表达式可更好地表示为 k _B T = mc ² /3（焦耳热用开氏温度表示）。

[6] 例如在黎明时太阳升起，地表开始变暖，地面就会通过热扩散加热最低层的空气，使低层的空气比上方的空气更轻。这种较重流体位于较轻流体上方的情况是不稳定的。

[7] 路德维希·普朗特（Ludwig Prandtl，1875-1953），德国物理学家，慕尼黑大学教授，对流体力学做出了重要贡献。他以边界层理论、混合长度的概念和有限翼展机翼升力计算的简单理论而著称。

[8] Lesieur, 湍流（Turbulence）, 法国EDP Sciences出版社, 格勒诺布尔科技出版社（Grenoble sciences collection）, 2013, p. 128.

[9] 通过建立位于壁面 z > 0侧的小的气域的整体平衡，得出 φ = – τ ，该小气域对流体施加- τ 的切向力，这种整体平衡的表述通常称之为 动量定理 。

[10] 这种表述是由纳维叶（Navier）和傅里叶（Fourier）等人建立的（详见《扩散之父》（《Fathers of the diffusion concept》））。

[11] 阿尔伯特·爱因斯坦（Albert Einstein），布朗运动理论的研究（Investigations on the Theory of the Brownian Movement），多佛出版社（Dover Publications），Inc. (1985), (ISBN 0-486-60304-0). 重新发行了爱因斯坦关于布朗运动理论的原文。

环境百科全书由环境和能源百科全书协会出版 ( www.a3e.fr )，该协会与格勒诺布尔阿尔卑斯大学和格勒诺布尔INP有合同关系，并由法国科学院赞助。

引用这篇文章: MOREAU René (2024年4月12日), 扩散，充分混合过程中极为关键的一步, 环境百科全书，咨询于 2024年9月13日 [在线ISSN 2555-0950]网址： https://www.encyclopedie-environnement.org/zh/physique-zh/diffusion-ultimate-step-good-mixture/ .

环境百科全书中的文章是根据知识共享BY-NC-SA许可条款提供的，该许可授权复制的条件是：引用来源，不作商业使用，共享相同的初始条件，并且在每次重复使用或分发时复制知识共享BY-NC-SA许可声明。

MOREAU René , 莫罗·雷内，法国科学院和工程院两院院士，格雷诺布尔工程师大学INPG/SiMAP卓越教授 。

文章目录

• 1. 静止的空气：一群躁动不安的分子
• 2. 气体的分子参数与宏观属性之间的关系
• 3. 气体扩散机制
• 4. 湍流扩散
• 5. 质量扩散和热扩散
• 6. 动量扩散：粘度
• 7. 液体中的扩散机制
• 8. 总结

要了解更多关于它的信息

R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, J. Wiley and sons, 1965.

05-02-2019

Chemical species such as carbon dioxide ( _CO2 ) in the air, or sugar in coffee, end up, after a while, being intimately mixed with the surrounding fluid, even if it is at rest. The same mechanism transports heat through the stationary bodies until their temperature is uniform. Similarly, in a fluid layer subjected to friction, the force exerted on the surface is transmitted in its full thickness. In fluid environments, turbulence disperses the constituents, but does not reach molecular scales; it is the disordered agitation of molecules that completes this transport process called diffusion. We will see that it alone would be extremely slow but that turbulence helps it effectively.

The word diffusion is used with relatively varied meanings. What is common to all situations is the fact that a certain quantity is transported through a medium dispersed by the agitation of the elementary particles that constitute it. In physics, we distinguish two families of diffusion phenomena:

A transport phenomenon based on the properties of the medium under consideration, consisting of atoms or molecules in disordered agitation. It is characterized by the fact that the distribution of the transported quantity tends to become uniform; this quantity may be the concentration of a particular chemical species, heat or the amount of movement. The uniform end state is the thermodynamic equilibrium of the environment.

The phenomenon of diffraction of waves in a non-homogeneous environment, whether these waves are electromagnetic like light or acoustic like sound. These phenomena are described in other articles in this encyclopedia: The Colours of the Sky , Emission, Propagation and Perception of Sound . In the English language, this phenomenon of wave diffraction is referred to as scattering , which distinguishes it from diffusion itself.

In everyday language, the same word, diffusion , is used with even different meanings. Thus, it can refer to the transfer of knowledge to a large number of people; it can also refer to the transmission of various types of information by radio (called radio broadcasting) or television (called television broadcasting).

1. The air at rest: an agitated set of molecules

In the article The Earth’s Atmosphere and Gaseous Envelope , the main components of dry air are mentioned: nitrogen (71%), oxygen (28%), argon (less than 1%), as well as some other gases in much smaller proportions. This set is made up of molecules, which occupy all the available space, a specific property of gases, which distinguishes them from the condensed states of matter (liquid or solid). These have their own volume, the solids also having their own shape.

Let’s start by sketching the main parameters of this molecular world which constitutes the relatively diluted gaseous media like air. The characteristic scales of molecular agitation (nanometric [1] ) are completely beyond our senses. A few hypotheses, due to Boltzmann [2] , make it possible to identify quite simply the orders of magnitude typical of this environment. Following Boltzmann, let us assume that the molecules are all spheres of the same diameter , which we will note σ, and of the same mass as we will note m. This already indicates the level of simplification proposed, since in reality the nitrogen (N2) and oxygen (O2) molecules are diatomic and look more like dumbbells than spheres. Let’s simplify even further by assuming that all these spheres have the same speed c . In fact, between collisions suffered by a molecule, its speed cannot be constant. Nevertheless, we will see that this scheme can lead to a suitable representation of the molecular world and its effects at macroscopic scales, with fairly accurate orders of magnitude.

Let us also note n the number of molecules contained in a unit of volume and isolate one of them by thought. For one time unit (1 s), it travels the distance c along its trajectory in a broken line (Figure 1). Since we limit ourselves to orders of magnitude, let’s assume that all the other molecules with which it will have a collision are stationary. The molecules it will encounter are those whose centre is located in a cylinder whose diameter is twice that of the particle, 2σ . The volume of this cylinder is equal to the area of its base (a disc of diameter σ ) multiplied by the distance c , i.e. ^cπσ2 , and the number of molecules present in this cylinder is therefore ^ncπσ2 (Figure 2). The average distance between two collisions is usually referred to as the average free run and denoted λ ; it is necessarily λ = 1/nπσ2 . The average adjective implies that the real free paths between collisions of a molecule, visible in Figure 1, vary around λ.

2. Molecular parameters with macroscopic properties of a gas, or the reverse

It is quite simple to link these parameters of the molecular world to some macroscopic quantities, measurable and accessible to our senses. These quantities – analyzed in the article Pressure, Temperature and Heat – are pressure p , temperature T , mass of the volume unit (or density) ρ and viscosity [3] μ For this purpose, we have an almost obvious first relationship between molecular and macroscopic scales: mass m is the quotient m = M/NA between the molecular weight M and the Avogadro number (number of molecules per mole) _{NA =} 6.022×1023 mol-1. In addition, the density ρ must be equal to the product of the number n of molecules present in the volume unit multiplied by the mass of each molecule. We therefore have a second relationship between the parameters of the two worlds: ρ = nm .

To connect the pressure p at a point in the gaseous medium to its molecular parameters, let us imagine that a flat solid surface of unit area (S = 1 , whatever the chosen system of units) is placed somewhere in this gaseous medium. Each of the two sides of the surface will receive and accumulate the impulses of all the molecules that will hit it during the unit of time, coming from all directions, losing and regaining their kinetic energy, in the order of mc2/2 for each of them. On average, the force exerted on the unit surface and equal to p must therefore be of the order of nmc2 . A rigorous calculation, taking into account the real velocities of the molecules encountered [4] , would lead to the expression p = nmc2/3 . In passing, we obtain an evaluation of the error due to the simplifying assumptions of the model adopted: the coefficient 1/3 . The important thing here is that we have obtained a third relationship between the molecular parameters (n, m, c ) and a macroscopic property, the pressure p .

Since pressure and density are related to molecular parameters, it is sufficient – to express temperature – to identify the pressure p expressed above with the one that verifies the state equation of perfect gases: p/ρ = RT/M , where R = 8.314 ^m3 .Pa.mol-1.K-1 denotes the universal constant of perfect gases and M the average molar mass of the gas mixture. We obtain the fourth relationship [5] T = Mc2/3R which highlights the proportionality between the temperature T and the square of the velocity of the molecules c2 (Read Pressure, Temperature and Heat ).

To these four relationships should be added the one that expresses the dynamic viscosity of the gas, which is established later in Section 5: μ = nmcλ/3 . So, with the following values, common for sea-level air

p = 1.013 ¹⁰⁵ Pa, T = 288 K, ρ = 1.22 kg.m-3, ν = 10-5 ^m2 .s-1,

the five relationships mentioned above lead to:

n = 2.5 ×1025 m-3, m = 4.8 × 10-26 kg, c = 497 m.s-1,

d = 3.4 × ^10-9 m, σ = 0.46 × ^10-9 m, λ = 6 × 10-8 m.

To get a fairly accurate idea of the scales of the molecular world, we can see that the number of molecules contained in a small cube of one micron on each side is about 25 million. We will deduce that the typical distance between two neighbouring molecules (d ) is about 7 times their diameter (σ ), that their average free path (λ) is about 60 nanometers, or about 130 times their diameter, and that the average speed of the molecules is slightly higher than the sound velocity (340 m.s-1).

3. The mechanism of diffusion in gases

If we imagine a flat section somewhere in the middle of the gas domain at rest (Figure 3), we can say that so many molecules cross this section in both directions during the same unit of time; this is precisely what characterizes rest at macroscopic scales . Let us suppose that a certain quantity G, such as the temperature or concentration of a given chemical species, has a non-uniform distribution in the vicinity of this flat section, larger on the x>0 side , smaller on the x<0 side (x denotes the distance to this flat section counted on its normal). Under these conditions, the gas is no longer in thermodynamic equilibrium . Since molecules from x>0 carry more G than those from x<0 , this molecular agitation generates a net flow of G through the plane x = 0, from the richest side (x>0) to the poorest side (x<0). It tends to standardize the distribution of G . It is this ability to transport any non-uniformly distributed quantity that is called diffusion ; it is this ability that tends to establish the thermodynamic equilibrium of the gas.

However, the time required to establish this balance is much longer (hours or more) than the time between two molecular collisions (in the order of λ/c ≈ 1. 4×10-10 s). Under these conditions, collisions are so frequent around a given point that it can be assumed that molecules that leave the vicinity of that point after a collision all carry the same value of G , i.e. G/n since G refers to the value per unit volume. It is then said that this gas is in a state of local equilibrium.

Let’s imagine the experiment schematized in Figure 4. In its initial state, two gases are separated by a partition. Immediately after the removal of the partition, the distribution of molecules in the cavity is not uniform. Gradually, as shown in Figure 4(2), the diffusion leads to the mixing of the two gases. The grey molecules, initially concentrated in the left part of the enclosure, reach the right part, and black molecules, initially concentrated in the right part, reach the left. This distribution continues until the distribution is uniform.

Molecules that pass through a flat surface unit perpendicular to the x direction (back to Figure 3) have, on average, the size G/n . The distance from where they come is in the order of an average free journey λ. The smallness of this length λ compared to the macroscopic dimensions justifies assuming linear, with a slope _Γ0 , the distribution G(x) in the vicinity of the cut located at x = 0 . This leads to the simple distribution: G = G0 + _Γ0 x . Let’s even assume that the molecules that come from the x>0 side all carry G+ = G0 + _Γ0λ, and that those that come from the x<0 side carry G- = G0 – _Γ0λ.

To estimate the net flow of G through the unit area of the section, multiply the difference _γ0λ by the number of molecules that cross this area per unit time from one side or the other. Note that, in the volume above the unit of area (see Figure 5), there are nλ molecules, and that each undergoes c/λ collisions per unit of time. As a result, the number of collisions, i.e. departures, is necessarily in the order of nc . But only departures oriented towards the unit of surface must be taken into account. The calculation leads to the number nc/3 (again a coefficient 1/3 ).

Finally, the flow rate φ of G through each unit of surface of the plane x = 0 can be estimated by the relationship φ = (cλ/3) dG/dx . The conclusion of this estimate is as follows: if a quantity G is distributed non-uniformly, the net flow of G transported by molecular agitation is proportional to the local gradient dG/dx and is expressed by a relationship of the form φ = -D dG/dx , where, by convention, the sign is placed – so that the flow is positive when dG/dx is negative (heat goes from the warm side to the cold side). The coefficient D appearing in this relationship is D = cλ/3 . It is called the diffusivity of this gas; it is a purely kinematic quantity measured in m2/s. It is remarkable that it was possible to assess the diffusivity of the gas without specifying the nature of the size transported.

The numerical values given in section 2 are used to estimate air diffusivity. With c = 497 m s-1 and λ = 6 × 10-8 m, we obtain : D ≈ 10-5 ^m2 s-1. To give this value a practical meaning, let us return to the experience in Figure 4: If we note L the length of the enclosure, since diffusivity has the dimension of a square length divided by time (L2T-1), the typical duration to reach the final state in equilibrium is necessarily in the order of L2/D . If L ≈ 10 cm , this duration is about one hour. For this duration to be in the order of a second, the enclosure in Figure 4 would have to be about a millimeter long, as would be the case with a small bubble.

4. The turbulent diffusion

In most of the fluid environments in our environment, including air and water, turbulence is almost omnipresent. It is a phenomenon of agitation totally unrelated to the movement of molecules. Within a flow such as the wind, or as a marine current, it manifests itself in the presence of large scale, fairly large and easily measurable speed fluctuations. Even without imposed flow, in the presence of hydrodynamic instability [ 6] , turbulence is present as a set of vortices of all sizes entangled in each other. We can see a certain analogy with the molecular agitation described in the previous sections, but at the vortex scales, much larger than the average free path of the molecules. This analogy led Prandtl [7] to suppose that it was possible to define a mixing length [8] , similar to the average free path of the molecules of a gas.

Experimentally, various techniques have been used to evaluate this length of mixture, which represents the typical size of eddies. The smoke volutes in Figure 6, such as those rising above chimneys or volcanoes, give an idea of this and show that this size always adapts to the overall size of the flow. Note the difference between the lower (laminar) part of cigarette smoke, which widens slightly under the effect of molecular diffusion, and the upper (turbulent) part, whose much faster widening highlights the transport of various constituents of this smoke by the random eddies generated by the instability of the smoke plume.

The concept of mixing length, size noted below , is rather vague but it has the merit of allowing satisfactory orders of magnitude to be established. The situation where it is best justified is that of a boundary layer (see Figure 7). The expression suggested by Prandtl in such a boundary layer and generally adopted is l = κy , where there is the distance to the wall and κ a numerical constant determined by experience. With this expression of the length l , the order of magnitude of the velocity fluctuations at the distance y from the wall can be estimated by u’ ≈ l dU/dy . The analogy with gas agitation suggests reusing the expression of molecular diffusivity (D = cλ/3 ) by replacing c by u’ and λ by l = κy . The turbulent diffusivity Dt within a boundary layer then becomes: Dt ≈ u’l ≈ ^{κ2y2 κ2y2} dU/dy .

For example, let us place ourselves within the atmospheric boundary layer, whose thickness is about a few hundred meters, at a distance from the ground y ≈ 10 m where the average speed is about 10 m.s-1 and where dU/dy ≈ 1 s-1. By adopting the value 0.1 for the constant κ , derived from experiments, these estimates lead to a turbulent diffusivity Dt ≈ 10 ^m2 .s-1, i. e. one million times higher than the molecular diffusivity. This value reduces the time required to achieve the turbulent mixture to about one-tenth of a second : t ≈ l2/Dt ≈ 0.1 s. But this mixing is not complete because the components are only dispersed to the scale of the smallest vortices, which depends on the viscosity of the fluid. In the case of the atmospheric boundary layer, it is about one millimeter. It is the molecular agitation that completes the mixture to the molecular dimensions , and we know that this requires an L2/D duration. In reality, molecular scattering begins within millimetre eddies where it coexists with turbulent dispersion. To simplify, let us assume that molecular scattering acts alone below the millimeter. With L ≈ 10-3 m and D ≈ 10-5 ^{m2 .} s-1, we still obtain a duration of about one tenth of a second .

From these estimates, it can be concluded that:

the turbulent diffusion only disperses the constituents to the scale of small eddies,

molecular diffusion completes this mixture,

the durations of the two mechanisms can have orders of magnitude in the order of one tenth of a second,

without turbulence, the time required to make a good mixture could reach several hours.

5. Mass and thermal diffusion

It is by the expression mass diffusivity that we usually refer to the diffusivity of a chemical species contained in the fluid medium. Suppose, therefore, that some of the molecules of the fluid considered, gas or liquid, are different from the others. Let’s take the example of water molecules (H20) in the form of steam, in absolutely calm air over a large liquid area, such as a peaceful lake. And let us now note C the volumetric concentration of water in the air at any altitude z, a quantity initially noted as G before its nature was specified. To evaluate the net flow of water vapour to high altitudes, the law seen above leads to the expression φ = -D dC/dz . It’s Fick’s law . It is essential to be able to calculate the quantity of water extracted by diffusion from low altitudes, but it must be supplemented by a mass balance and by initial and boundary conditions.

Thermal diffusivity expression is reserved for heat transport by molecular agitation. Let us now assume that quantity G is the internal energy E per unit volume of the gas, any infinitesimal variation of which is manifested by a temperature variation dT , according to the law dE = _ρCv dT , where Cv denotes the constant volume calorific capacity of the gas. Then the diffusion transport of this internal energy, i. e. heat, becomes φ = -ρCvD dT/dz . It is common to write this expression called Fourier’s Law in one of the following two forms: φ = -k dT/dz , where k = _ρCvD = Cv nmcλ/3 is called the thermal conductivity of gas, or φ = -α _ρCp dT/dz , where α = k/ρCp is the thermal diffusivity of gas. The quantity Cp then refers to the heating capacity at constant gas pressure.

The physical properties of the gases, especially air, introduced above (D, k, α, μ, ν ) are related to the molecular parameters n, m, c and λ . Among them, for a given density gas, only the velocity of molecules c is likely to vary with temperature T . And we saw that this dependence could be written: T = Mc2/3R = mc2/3kB . It can be deduced that the diffusivities of a gas (D, k, α, μ, ν ) vary like the square root of its temperature .

6. Diffusion of movement: viscosity

Let us now assume that the above notions remain valid in a fluid set in motion at macroscopic scales by moving a wall in its own plane. This hypothesis is very well justified as long as the typical durations of macroscopic motion (seconds or minutes, in general) are considerably longer than the time between two collisions of a molecule (about 10-11 s).

Note u the local velocity of a fluid particle set in motion by the movement of the wall. The amount of movement of any unit of volume of the fluid is then ρu , where ρ refers to the density of this fluid. Let us now choose this quantity ρu as the quantity G whose flux transported by diffusion through the planes perpendicular to the direction z can be expressed. As before, the rate of movement quantity transported by the molecular agitation through the plane z = 0 can be written φ = -ρD of the/dz . This flow rate of movement quantity is none other than the opposite of the tangential stress [9] τ , or force per unit area, exerted by the fluid on the wall (Read How the material is deformed: fluids and solids ). Its expression [10] , generally written τ = μ of the/dz, is called Newton’s law and the fluids that satisfy this law are called Newtonian fluids . The quantity μ = ρD = nmcλ/3 is the dynamic viscosity of the gas; its variant, the kinematic viscosity ν = μ/ρ = cλ/3 , which has the dimension of a diffusivity, is also very often used. The transport of movement quantity by viscosity is manifested in particular by the presence of a very thin boundary layer along the moving wall, except in the event of disbonding (Figure 7). The velocity distribution in this boundary layer changes from the value u in contact with the moving wall to zero beyond the boundary layer.

The above reasoning also applies when, instead of being materialized by a wall, the plane z = 0 is an interface between two fluid layers, one of which is moving in its own plane, at the velocity u , between the fluid above it and the fluid below. Newton’s law thus makes it possible to express the tangential force, often called friction , that the fluid layers exert on each other. This tangential force, oriented in the velocity direction, is added to the pressure force, which is oriented according to normal at the interface (Read Pressure, Temperature and Heat ).

7. The mechanism of diffusion in liquids

The fact that all liquids have their own volume implies that the molecules that compose them are close to each other. The average distance between neighbouring molecules depends on the pressure exerted on this liquid; under normal conditions (p ≈ ¹⁰³ hPa) it is slightly larger than the size of the molecules, typically 10-10 to ^10-9 m. The thermal agitation of all these molecules, which can be compared to a vibration, gives this liquid medium a certain diffusivity, but this is much lower than that of gases. The difference is essentially due to the fact that, in a liquid, the radius of action of a molecule is less than its size, whereas in a gas, it is the average free path, about 25 times the size of the molecules. This weakness explains why, in order to dissolve the sugar in a cup of coffee, it is necessary to add a macroscopic stirring by stirring the whole with a spoon.

Any particle with a difference from the molecules and present in this medium is constantly subjected to the vibrations of the whole. A balance similar to that for gases in section 3 necessarily shows a net flow of these particles from the richest to the poorest side. This flow rate can still be written with Fick’s law: φ = -D dC/dx , where D denotes the diffusivity of the particles in the liquid and C their volume concentration.

To predict the mass diffusivity of a liquid, which reflects its ability to transport another material species, we have the following law proposed by Einstein in 1905 [11] : D = kBT/6πμr , where kB = 1.38×10-23 J.K-1 , is the Boltzmann constant, T the absolute temperature, μ the dynamic viscosity of the liquid and r the radius of the transported particle. With r ≈ ^10-9 m, in a liquid such as water or μ ≈ 10-3 Pa.s, at an ambient temperature close to 300 K, we obtain D ≈ 2×10-10 ^m2 .s-1, a value about 10 000 times lower than the diffusivity of a gas. In Einstein’s formula, we will notice that T intervenes at the numerator, which reflects the fact that thermal agitation is the real driving force behind diffusion. And the fact that μ and r are involved in the denominator means that viscosity is opposed to diffusion and that the larger the particles, the more difficult it is to diffuse.

For water and for any electrically insulating liquid, heat and mass have similar diffusivities. On the other hand, for liquid metals, electronic conduction is another heat transport mechanism that can be much more important than mass diffusivity. The most well-known example is liquid sodium, used to cool nuclear reactors of the superregenerator type such as Phoenix, which operated at Marcoule from 1973 to 2010. Its thermal diffusivity at a temperature of about 300 K is about 7×10-5 ^m2 .s-1, more than 10,000 times higher than its mass diffusivity.

8. Messages to remember

The diffusion of a particular species within a mixture, like the diffusion of heat and the diffusion of the amount of movement, results from the agitation of elementary particles, molecules or atoms. It is a very slow transport mechanism, but without a competitor on microscopic scales.

In fluids such as air and water, which are not very viscous, turbulence disperses pollutants, heat and the amount of movement. But this mechanism cannot reach scales lower than those of the smallest vortices, controlled by viscosity, generally greater than one micron. Molecular diffusion relays it to complete the mixture.

The laws of diffusion were first discovered empirically before being the subject of theoretical models. Relatively simple in the case of diluted gases such as air, it could be sketched in this article. Its equivalent in the case of liquids is much more complex and still under investigation.

References and notes

Cover image. Before sunrise, in the countryside, in the still motionless air, it is by diffusion that contaminants released by vegetation are transported into the air. [Source: pixabay, royalty-free].

[1] A nanometre is one billionth of a metre ( ^10-9 m), a micrometre, or micron, is one millionth of a metre (1 μm = 10-6 m).

[2] Ludwig Boltzmann (1844-1906), an Austrian physicist and philosopher, introduced this molecular model called “hard spheres” that leads to a famous equation bearing his name.

[3] It is often substituted by the kinematic viscosity ν, which is the quotient between the viscosity μ and the density ρ, i.e. ν = μ/ρ.

[4] All these velocities, with their respective orientations and values, can be represented using Maxwell’s distribution function. Taking them into account would lead to lengthy calculations beyond the scope of this article.

[5] This expression is better known as kBT = mc2/3 , by introducing the Boltzmann constant kB , related to R and the Avogadro number _{NA =} 6.0248 × ¹⁰²³ , per kB ₌ R/NA= 1.380 × 10-23 J/K (joule by Kelvin).

[6] For example, at dawn, as soon as the sun begins to warm the ground, by diffusion it heats the lowest layers of air, which become lighter than those above them. This situation where the heavy fluid is located above the light fluid is unstable.

[7] Ludwig Prandtl (1875-1953) was a German physicist, professor at the University of Munich, who made important contributions to fluid mechanics. He is credited with the founding ideas of boundary layer theory, the concept of mixture length and a simple theory for calculating the lift of a finite wing span.

[8] M. Lesieur, Turbulence, EDP Sciences, Grenoble sciences collection, 2013, p. 128.

[9] The property that φ = -τ can be established by writing the global equilibrium of a small gas domain located on the z>0 side of the wall that exerts the tangential force on the fluid -τ. The expression of this global equilibrium is often referred to as the quantity of movement theorem .

[10] The establishment of this expression is actually due to Navier, Fourier’s congener (see the focus on the Fathers of the diffusion concept ).

[11] Albert Einstein , Investigations on the Theory of the Brownian Movement , Dover Publications, Inc. (1985), (ISBN 0-486-60304-0). Reissue of Einstein’s original articles on Brownian motion theory

环境百科全书由环境和能源百科全书协会出版 ( www.a3e.fr )，该协会与格勒诺布尔阿尔卑斯大学和格勒诺布尔INP有合同关系，并由法国科学院赞助。

引用这篇文章: MOREAU René (2019年2月5日), Diffusion, the ultimate step in a good mixture, 环境百科全书，咨询于 2024年9月13日 [在线ISSN 2555-0950]网址： https://www.encyclopedie-environnement.org/en/physics/diffusion-ultimate-step-good-mixture/ .

环境百科全书中的文章是根据知识共享BY-NC-SA许可条款提供的，该许可授权复制的条件是：引用来源，不作商业使用，共享相同的初始条件，并且在每次重复使用或分发时复制知识共享BY-NC-SA许可声明。

MOREAU René , 莫罗·雷内，法国科学院和工程院两院院士，格雷诺布尔工程师大学INPG/SiMAP卓越教授 。

文章目录

• 1. The air at rest: an agitated set of molecules
• 2. Molecular parameters with macroscopic properties of a gas, or the reverse
• 3. The mechanism of diffusion in gases
• 4. The turbulent diffusion
• 5. Mass and thermal diffusion
• 6. Diffusion of movement: viscosity
• 7. The mechanism of diffusion in liquids
• 8. Messages to remember