# Dynamical density functional theory for the dewetting of evaporating thin films of nanoparticle suspensions exhibiting pattern formation 

Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK 

A.J. Archer, M.J. Robbins, and U. Thiele 

Recent experiments have shown that the striking structure formation in dewetting films of evaporating col- loidal nanoparticle suspensions occurs in an ultrathin ‘postcursor’ layer that is left behind by a mesoscopic dewetting front. Various phase change and transport processes occur in the postcursor layer, that may lead to nanoparticle deposits in the form of labyrinthine, network or strongly branched ‘finger’ structures. We develop a versatile dynamical density functional theory to model this system which captures all these structures and may be employed to investigate the influence of evaporation/condensation, nanoparticle transport and solute trans- port in a differentiated way. We highlight, in particular, the influence of the subtle interplay of decomposition in the layer and contact line motion on the observed particle-induced transverse instability of the dewetting front. 

How surface patterns and structures evolve over time is of great interest for a wide range of scientific fields. Striking examples include river network patterns [1], the growth of rocks around geothermal springs [2] evaporation-caused cof- fee stain patterns [3], and the patterns in the distribution of living organisms [4]. Many such structures are generated by the interaction of fluid motion over the surface and deposition and/or abrasion of material. A particular process of high re- cent interest that concerns us in this Letter is the formation of structures during the (evaporative) dewetting of nanoparti- cle suspensions on solid substrates [5–7]. The patterning is generic to a wide class of dewetting evaporating suspensions and solutions [8–12] and depends crucially on the interplay of several competing phase-change and transport processes. 

The rapidly expanding study of such systems currently re- ceives strong impetus from research in two distinct areas. On the one hand, studies from the last couple of decades of the dynamics of dewetting of surfaces by non-volatile liquids [13, 14] have been extended to investigate the interplay be- tween (de)wetting and evaporation of volatile liquids [15, 16]. On the other hand, there is interest in the non-equilibrium thermodynamics and rheology of the respective phase and flow behaviour of bulk suspensions and solutions – see e.g. Ref. [17] and references therein. A thin film of pure non- volatile liquid that is deposited upon a smooth substrate (e.g., a polystyrene film having a thickness of a few tens of nanome- ters, deposited on silicon oxide [13]) may rupture due to ef- fective molecular interactions between the film surface and the solid substrate. The rupture mechanism can be (i) via a sur- face instability (often called spinodal dewetting) that occurs spontaneously and results in patterns of a certain characteristic wavelength, or (ii) via nucleation at randomly distributed de- fects [18, 19]. The resulting holes then grow to form a polyg- onal network of liquid rims that may subsequently decay into drops. All stages of this process are intensively studied: the rupture mechanisms [13, 14, 18], the hole growth [20], the morphologies and evolution of the resulting patterns [21], and the stability of receding liquid fronts [22, 23]. For reviews of this body of work, see Ref. [24]. 

The dewetting processes of solutions and suspensions are more involved than those of a pure liquid because they in- volve several interdependent dynamical processes: transport of solute or colloids, transport of the solvent and evapora- 

tion/condensation of the solvent. As a consequence, one must distinguish between ‘normal’ convective dewetting and evaporative dewetting. Experimental studies performed with volatile solutions/suspensions of polymers [8, 9], macro- molecules [10–12] and colloids a few nanometers in size (re- ferred to as ‘nanoparticles’) [5–7, 25] describe a variety of richly structured deposits of the solutes. One may observe labyrinthine and polygonal network structures similar to the structures observed following ‘classical’ dewetting. As the solvent evaporates, the solute remains dried onto the sub- strate and therefore ‘conserves’ the transient dewetting pattern [6, 10]. However, the solute is not just a passive tracer: it may influence the thresholds and the rates of the initial film rup- ture processes. Most importantly, it may also destabilise the straight dewetting fronts and trigger the creation of strongly ramified structures – a process observed in many different sys- tems [7–9, 26]. 

For instance, the dewetting of films of a suspension of thiol- coated gold nanoparticles in toluene may result in the depo- sition of the nanoparticles in branched finger patterns. The precise properties of these depend on the strength of the attrac- tion between the colloidal particles [7]. Employing contrast- enhanced video microscopy to study the dynamics of the sys- tem, initially the receding of a mesoscopic dewetting front (equivalent to the receding three phase contact line) is ob- served, leaving behind an unstable ultrathin ‘postcursor’ film, having a thickness similar to the diameter of the nanoparticles. Subsequently, the postcursor film breaks, forming a pattern of holes that themselves grow in an unstable manner, resulting in an array of branched structures. Note that the mesoscopic front may also be unstable – an effect that does not concern us here. 

Theories for modelling such processes are rather limited at the present time. Hydrodynamical models of dewetting by thin films based on a long-wave approximation [24, 27] provide a mesoscale description of the liquid film, and can account for evaporation of volatile liquids [28] and for the presence of a solute [29]. However, they are not able to describe the pro- cesses occurring in the postcursor film as they do not account for the interactions between the solute particles and between the solute particles and the solvent. Alternatively, to describe these processes one may employ two-dimensional (2d) kinetic Monte-Carlo (KMC) lattice models that focus solely on the 

