Coffee rings and ridges: predicting late-time deposit profiles in evaporating droplets

当洒在桌面上的一滴咖啡干了，大家都知道，它会留下一个边缘颜色更深的污点。这被称为“咖啡环”效应，并不是咖啡所特有的，而是在许多情况下发生的，包括含有惰性（不蒸发）溶质的蒸发液滴。液滴的边缘——被称为接触线——被固定在固体表面上，这样，当液体变干时，为了补充在接触线上失去的液体，液滴中形成了一股流动，将液体从液滴内部带到接触线。这种流动携带着溶质。溶质的平流随后被接触线附近的扩散抵消，从而推动咖啡环的发展。咖啡环效应可用于许多工业问题，例如印刷微型电路或胶体图案，或利用向外流动排列DNA。工程师们也可能在需要均匀沉积的应用中，如喷涂或喷墨印刷中，寻求对抗这种影响。因此，自从咖啡环效应被发现以来，它就受到了极大的关注。然而，一个不太为人所知的现象是，作为同一过程的一部分，内部沉积物可能会增加。在干燥的后期阶段，液滴的形状会发生显著变化，液体表面开始向中心倾斜，变得非常接近固体表面。这可能导致溶质被困在液体表面和固体之间。此外，液滴形状的改变改变了流动模式，进一步增加了溶质向液滴内部的运动。这些内部沉积物被称为“咖啡脊”或“咖啡眼”，以前在含有聚合物的液滴的实验中发现过。然而，尽管咖啡脊在最终的残余模式中起着关键作用，但与它的环状结构相比，人们对它的了解相对较少。事实上，在一些应用中，咖啡脊可能比咖啡环更有问题，例如在QLED屏幕的印刷中。因此，更好地了解咖啡脊背后的物理特性，以及准确预测和理解它们的形成方式，是一项重要的数学和工程挑战。该项目旨在通过推导咖啡脊形成的数学模型来解决这一挑战。我将首先考虑一个问题，一个液滴在浅井中蒸发。这种配置有两个优点。首先，它会抑制咖啡环的形成，这样我就可以专注于液滴内部的流动动力学，从而关注咖啡脊。其次，这种配置用于OLED/QLED屏幕的打印，因此该模型具有直接的工业相关性。我将通过利用井的浅度和液滴中表面张力的主要影响，系统地推导出一个简化模型。然后，我将使用一种结合匹配渐近分析和数值模拟的混合方法来探索液滴内溶质分布的演变。特别令人感兴趣的是不断演变的沉积物的关键特征，如咖啡脊的大小和位置。然后将这些结果与文献中现有的实验数据进行比较。然后，我将在这些结果的基础上考虑液滴在平面上蒸发的更常见的配置。通过仔细分析咖啡环和咖啡脊的同时形成，我将发现流动模式是如何随时间演变的，并研究两种特征之间的相互作用。该模型将用于液滴干燥在工业和工程中的应用。

When a spilled droplet of coffee dries on a tabletop, it is well known that it leaves behind a stain that is darker towards its edges. This is called the 'coffee-ring' effect and is not unique to coffee, but occurs in a multitude of situations involving an evaporating liquid droplet that contains an inert (non-evaporating) solute. The edge of the droplet - called the contact line - becomes pinned on the solid surface so that, as the liquid dries, to replace fluid lost at the contact line, a flow develops in the droplet that takes liquid from the droplet interior to the contact line. This flow carries solute along with it. The advection of the solute is then counteracted by diffusion near the contact line, which drives the development of the coffee ring. The coffee-ring effect can be exploited in many industrial problems, for example in printing microscale circuits or colloidal patterning, or aligning DNA using the outward flow. Engineers may also seek to counter the effect in applications where a uniform deposit may be desired, such as in spray coating or inkjet printing.The coffee-ring effect has therefore seen a significant amount of attention since its discovery. However, a less well-known phenomenon is the possibility of enhanced internal deposits developing as part of the same process. At later stages of drying, the droplet shape can alter significantly so that the liquid surface begins to dip in the centre, getting very close to the solid surface. This may lead to solute becoming trapped between the liquid surface and the solid. Moreover, the change in the droplet shape alters the flow pattern, further increasing the movement of solute to the droplet interior. These internal deposits are called 'coffee ridges' or 'coffee eyes' and have been seen previously in experiments involving droplets containing a polymer. However, coffee-ridge formation is comparatively poorly understood compared to its ring counterpart, despite its key role in the final residual patterns. In fact, in several applications, coffee ridges may be more problematic than coffee rings, for example in the printing of QLED screens. A better understanding of the physics behind coffee ridges alongside a means to accurately predict and understand their formation is therefore an important mathematical and engineering challenge.This project seeks to address this challenge by deriving a mathematical model for coffee-ridge formation. I will begin by considering a problem where a droplet evaporates in a shallow well. This configuration has two advantages. First, it will inhibit coffee-ring formation, so as to allow me to focus on the flow dynamics in the droplet interior and, hence, the coffee ridge. Second, such a configuration is used in the printing of OLED/QLED screens, so the model has direct industrial relevance. I will systematically derive a reduced model by exploiting the shallowness of the well and the dominant effect of surface tension in the droplet. I will then explore the evolution of the solute distribution within the droplet using a hybrid approach that combines matched asymptotic analysis with numerical simulations. Of particular interest are key characteristics of the evolving deposit such as the size and location of the coffee ridge. These results will then be compared to existing experimental data in the literature. I will then build upon these results to consider the more common configuration in which a droplet evaporates on a flat surface. By carefully analysing the concurrent formation of the coffee ring and the coffee ridge, I will discover how the flow patterns evolve in time and investigate the interplay between the two features as they grow. The model will be used to inform future applications of droplet drying in industry and engineering.