Breath-Figure Self-Assembly, a Versatile Method of Manufacturing Membranes and Porous Structures: Physical, Chemical and Technological Aspects

The review is devoted to the physical, chemical, and technological aspects of the breath-figure self-assembly process. The main stages of the process and impact of the polymer architecture and physical parameters of breath-figure self-assembly on the eventual pattern are covered. The review is focused on the hierarchy of spatial and temporal scales inherent to breath-figure self-assembly. Multi-scale patterns arising from the process are addressed. The characteristic spatial lateral scales of patterns vary from nanometers to dozens of micrometers. The temporal scale of the process spans from microseconds to seconds. The qualitative analysis performed in the paper demonstrates that the process is mainly governed by interfacial phenomena, whereas the impact of inertia and gravity are negligible. Characterization and applications of polymer films manufactured with breath-figure self-assembly are discussed.


Introduction
Well-defined micro-and nanoscaled porous polymeric architectures are in great demand for their use in advanced devices, including membranes [1][2][3][4], sensors [5,6], bio-engineering [7,8] and water-oil and size-selective separation processes [9][10][11][12][13]. One of the most versatile, simple, single-stage, and inexpensive methods, enabling manufacturing of porous polymer films with finely controlled topography, is so-called "breath-figure self-assembly" [14,15]. Historically the breath-figures method has been used since the 1850s by photographers as a simple and effective way to detect oil contamination on glass substrates [16]. An extended review of the numerous applications of the ordered, microporous films obtained with breath-figure self-assembly may be found in [16].
The formation of water droplets on solid surfaces was first investigated by Aitken in 1893 [17] and later in 1911 by Rayleigh [18,19]. The notion of "breath-figures" refers to the set of fog droplets that forms when water vapor comes into contact with a cold surface (solid or liquid). Breath-figures are a phenomenon commonly observed in daily life. One example is the surface fog that appears on a window when we breathe on it [20]. Another example is the formation of dew [21]. Knobler and Beysens investigated the formation of breath-figures under water droplet condensation [22][23][24][25] and found that they formed not only on solid surfaces but on liquids as well, specifically on paraffin oil [24]. The main features of the process of breath-figure self-assembly have been revealed through detailed study of water condensation [22][23][24][25], including the intriguing effect of non-coalescence of sessile droplets, to be discussed later. The interest in breath-figures self-assembly was revived when Widawski Francois and Pitois et al. demonstrated in a series of papers [26][27][28][29][30] that breath-figure self-assembly allows for the formation of microporous reliefs with well-controlled topography when polymer solutions are rapidly evaporated in a humid atmosphere. It should be emphasized that However, results from different researchers often seem to conflict [16]. Consider first the impact of the polymer structure on the resulting pattern. Bolognesi et al. reported that polar groups of polymers play a fundamental role in the process [37]. These results were supported by findings However, results from different researchers often seem to conflict [16]. Consider first the impact of the polymer structure on the resulting pattern. Bolognesi et al. reported that polar groups of reported in [38], where the authors reported that the hydrophilic end-groups can dramatically improve the film-forming property of polystyrene, and that the regularity of the film is mainly influenced by the interaction of film-forming polymers with condensed water droplets. Amirkhani et al. reported that the end-functional polymer produced a large area of regular spherical bubbles, whereas adding particles to the polymer solution leads to smaller arrays of flattened bottom bubbles. The separation length between pores was larger for the polymer/particle sample than for that of the end-functional polymer films [39].
On the contrary, linear polystyrene without any polar end-group also led to ordered honeycomb structures when dissolved in toluene and CHCl3 solvents [40,41]. At first, it was suggested that star-like and hyperbranched polymers promote the formation of well-ordered structures [26,42]; however, later it was demonstrated that linear polymers also give rise to patterns typical of breath-figure self-assembly, such as depicted in Figure 2 [40,41]. It was suggested that coiled polymers (polystyrene) promote the formation of breath-figure patterns [42,43]; on the other hand, rigid-rod conjugated polymers also gave rise to well-ordered honeycomb reliefs [36]. The impact of the molecular weight of the polymer on the resulting pattern remains obscure, and the data reported by various groups are controversial [44][45][46]. The concentration of the solution definitely impacts the self-assembly process; however, its impact remains unclear [47]. The typical patterns for breath-figure self-assembly were obtained with amorphous [46] and crystalline polymers [48]. An additional difficulty in identifying the impact of the polymer architecture on the resulting pattern arises from the affinity of the processes of phase separation and breath-figure self-assembly, giving rise to similar eventual honeycomb patterns [49]. Breath-figure self-assembly has been carried out with thermoplastic [14][15][16]41,[44][45][46] and cross-linked polymers [50][51][52]. Several groups reported breath-figure patterns obtained with high performance, engineering polymers, such as polyimide [53], polyetherimide [54], polysulfone [45,55], and siliconcontaining graft copolymer poly(dimethylsiloxane)-graft-polyacrylates (PDMS-g-PAs) [56].
The impact of the solvent also remains unclear. It is generally agreed that the rate of a solvent evaporation and the associated concentration-and temperature-dependent properties of a polymer solution define to a large extent the topography of the honeycomb pattern arising from the breathfigures process [57,58]. Ferrari et al. noted that the thermodynamic affinity between polymer and solvent is a key parameter for breath-figure formation, along with other solvent characteristics such as water miscibility, boiling point, and enthalpy [41]. A model predicting the evaporation profile of the casting solution in this process for low concentrations of polymer was proposed in [59]. By adding Breath-figure self-assembly has been carried out with thermoplastic [14][15][16]41,[44][45][46] and cross-linked polymers [50][51][52]. Several groups reported breath-figure patterns obtained with high performance, engineering polymers, such as polyimide [53], polyetherimide [54], polysulfone [45,55], and silicon-containing graft copolymer poly(dimethylsiloxane)-graft-polyacrylates (PDMS-g-PAs) [56].
The impact of the solvent also remains unclear. It is generally agreed that the rate of a solvent evaporation and the associated concentration-and temperature-dependent properties of a polymer solution define to a large extent the topography of the honeycomb pattern arising from the breath-figures process [57,58]. Ferrari et al. noted that the thermodynamic affinity between polymer and solvent is a key parameter for breath-figure formation, along with other solvent characteristics such as water miscibility, boiling point, and enthalpy [41]. A model predicting the evaporation profile of the casting solution in this process for low concentrations of polymer was proposed in [59]. By adding a small amount of a surface active compound, it is possible to create ordered arrays from other solvents and, thus, markedly broaden the applicability of this patterning process [60]. It was recently demonstrated experimentally that a broad diversity of solvents, including acetone, dichloromethane, chloroform, carbon tetrachloride, tetrahydrofuran, toluene, xylenes, carbon disulfide, and N-N dimethylformamide, give rise to porous patterns when dissolved polystyrene and polycarbonate were evaporated in a humid atmosphere [61]. It was demonstrated that all solutions, when sufficiently pre-cooled, gave rise to typical "breath-figure" patterns [61]. Thus, the decisive factor affecting the formation of the breath-figures pattern turns out to be the temperature of the evaporated solution [16,61].
Hence, the impact of the substrate used for the breath-figure self-assembly may be decisive due to the fact that the substrate may serve as a thermal bath stabilizing the temperature of the evaporated polymer solution; the slide thickness is also shown to be a crucial parameter in this process [59,[61][62][63]. However, the experimental data related to the impact of the substrate on the breath-figure self-assembly remain scarce [42,61,62,64,65]. Valiayaveettil and co-workers used clean glass, epoxy, amine-terminated, and dendrimer-functionalized glass as well as silicon wafers to cast a poly(p-phenylene) with pyridine chloroform solution [65]. In this work, honeycomb membranes were obtained with glass and silicon wafers. In contrast, ring patterning, low pore density, or net-type structures were obtained from the epoxy-treated, amine-terminated, and dendrimer-functionalized glasses, respectively [65].
When breath-figures-inspired patterns are formed under drop-casting, a drop of polymer solution is deposited by a precise micro-syringe (or micro-injector) on a solid substrate and exposed to air flow with a controlled humidity [66][67][68][69][70][71]. The spin-coating process involves putting a polymer solution on the center of the substrate, which is either spinning at low speed or not spinning at all. The substrate is then rotated at high speed in order to spread the evaporated polymer solution by centrifugal force. The whole process is performed under controlled humidity [72][73][74][75][76][77][78]. In the doctor-blade technique, an immobilized blade applies a unidirectional shear force to the polymer solution that passes through a small gap between the blade and the substrate [79][80][81]. When honeycomb patterns are obtained with dip-coating, the solid substrate is pulled with a constant speed from the evaporated polymer solutions [46,54,58,[82][83][84]. In all aforementioned processes, the impact of air humidity and the physico-chemical properties of a solid substrate may be decisive in constituting the topography of the resulting honeycomb relief [14][15][16]59,64,67].

Main Stages of Breath-Figure Self-Assembly
The main stages of breath-figure self-assembly are: nucleation of water droplets, condensation on the polymer solution/vapor interface, interaction between droplets, and final removal of water through its evaporation. Condensation is the formation of a liquid phase from the gaseous (vapor) one, which starts with "nucleation". Nucleation is the formation of an embryo or nucleus of a new phase in another phase [32,85]. Homogeneous and heterogeneous nucleation scenarios should be distinguished. Heterogeneous nucleation takes place in the presence of foreign particles or surfaces, whereas homogeneous nucleation occurs while growing small clusters of molecules [86]. If it is thermodynamically favorable for these clusters to grow until they become recognizable droplets of the liquid phase [32,85,86]. It should be emphasized that the precise mechanism of nucleation during the breath-figure self-assembly remains unclear, due to the fact that the nuclei of water droplets are formed in the presence of a solvent vapor [87]. The processes of nucleation and condensation have a decisive influence on the formation of the eventual breath-figures-inspired pattern [87]; hence, additional experimental and theoretical efforts are necessary for elucidating the kinetics of nucleation and condensation taking place under the conditions of the breath-figure self-assembly.
It is also possible that nucleation occurs at the polymer solution/vapor interface. In this case the nucleation rate I (for its accurate definition, see [32,[88][89][90][91]) is modified through the function depending strongly on the equilibrium (Young) contact angle θ Y , namely: max is the value of the potential barrier to be surmounted for heterogeneous nucleation, T is the temperature, and k B is the Boltzmann constant. The value of ∆G het max for a contact angle hysteresis-free substrate is given by: where ∆G hom max is the potential barrier of the homogeneous nucleation supplied by Equation (2): where r c is the size of the critical nucleus and γ is the surface tension [86,88]. Beysens et al. studied the formation of breath-figure patterns formed on cold borosilicate substrates, either pristine or hydrophobized by a solution of octadecyltrichlorosilane [22]. The treatment by octadecyltrichlorosilane enabled control of the apparent contact angle of the cooled solid substrate [22]. The pattern of water on glass was studied by direct observation and light scattering as a function of the contact angle θ Y , the velocity of vapor volume transfer "flux", denoted as Φ vol , the degree of supersaturation ∆T, and time t. It was established that when θ Y = 0 0 , a uniform water layer forms whose thickness grows as t increases at a constant Φ vol and ∆T. For θ Y = 0 0 , droplets are formed at a constant Φ vol and ∆T; the radius of an isolated droplet grows as t 0.23 , but as a result of coa1escences the average droplet radius grows as~t 0.75 [22]. The most important conclusion following from these considerations is expected, namely that the eventual "breath-figure" pattern depends on the apparent contact angle θ Y , as predicted by Equations (1) and (2). The growth process is accompanied by coalescence of droplets and turns out to be similar; coalescences simply rescaled the distances and left the basic droplet pattern unaltered [22]. The details of the coalescence were addressed in [23]; the authors showed that the number of coalescences undergone by a given droplet grows logarithmically with time; the total distance traveled by this droplet is proportional to its size [23]. The experiments, reported by Beysens et al., supported the important information about the kinetics of formation of the "breath-figure" patterns [22,23]. However, these experiments were carried out under model conditions in the absence of evaporated polymer solutions, which essentially complicates the physics of the process; hence, novel experimental data shedding light on the kinetics of formation of breath-figure patterns occurring at the polymer solution/vapor interface are necessary.

Multi-Scale Patterning Observed under "Breath-Figure Self-Assembly"
Multi-scale, hierarchical patterning is featured in breath-figures [14,76,77,[92][93][94]. The characteristic spatial scales of patterns vary from nanometers to dozens of micrometers. Consider the large-scale patterning observed in rapidly evaporated polymer solutions [94][95][96][97][98][99], depicted in Figure 3, observed during the dip-coating of substrates with rapidly evaporated polymer solutions (similar large-scale patterns were also observed under other experimental techniques [96][97][98][99]). In particular, it was demonstrated that the characteristic dimensions of cells constituting the pattern grow with the concentration of the polymer solution [95]. The physical mechanism of the patterning remains debatable. For film thickness less than about 100 nm (thin layers), effective molecular interactions between the liquid layer surface and the substrate dominate all other forces (like thermo-or soluto-capillarity or gravity) and thus determine the film stability and patterning under dewetting [100][101][102].
For evaporated films with a thickness above 100 nm, thermo-capillarity forces become dominant, resulting in instability caused by the Marangoni convection, either thermo-or solutocapillary [103][104][105][106][107][108][109][110][111][112][113]. It should be mentioned that the analysis of the pattern inspired by thermo-and soluto-capillarity is an essentially non-linear one and involves complicated mathematical models [103][104][105][106][107][108]. It was demonstrated experimentally that the patterns observed in the evaporated (cooled from above) layers are different from those formed in a layer heated from below without evaporation [109]. Thus, the kinetics of evaporation, studied in [110], turns out to be crucial for constituting largescale patterns. An apparatus utilizing self-organized liquid flow for the targeted modification of macromolecular systems in a solution was suggested in [112]. Marangoni-flow-induced patterns, observed under spin-coating of evaporated polymer solutions, were reported in [114]. When the polymer solution is evaporated, thermo-and soluto-capillary flows occur and both may contribute to the eventual pattern. It remains debatable what kind of Marangoni flow has a decisive impact on the pattern. The experimental data reported in [95,115] indicate that it is the solutal Marangoni flow that causes the pattern. Indeed, heating the substrate from below destroyed the pattern [115]. This contradicts the idea that self-organization is due to the jump in surface tension caused by a temperature gradient (temperature-gradient-driven Marangoni instability).
However, when evaporation is present, thermo-and soluto-capillarity represent only a few of a diversity of destabilizing mechanisms: vapor recoil, differential evaporation (the dependence of the evaporation rate on the thickness of the film), or, sometimes both contribute to the development of interfacial instability and as a result exert an influence on the pattern's makeup [116]. de Gennes proposed an alternate mechanism of patterning. He showed that in an evaporating film, a "plume" of solvent-rich fluid induces a local depression in surface tension, and the surface forces tend to strengthen the plume. His calculations led to the conclusion that this kind of instability should dominate over the classic Bénard-Marangoni instabilities [117,118]. It should be emphasized that in all the aforementioned instabilities (namely Marangoni and de Gennes) the tangential vector field of velocities drives the liquid. This makes possible the topologically-based approach to patterning, as demonstrated in [119]. The "hairy ball theorem" of algebraic topology states that any continuous tangent vector film on a surface topologically equivalent to a sphere must have at least one point where the vector is zero [120]. Remarkably, the "hairy ball theorem" predicts the existence of at least one zero tangential velocity point at the surface of the evaporated polymer solution [119]. Indeed, these zero velocity points were observed experimentally [119]. The physical mechanism of the patterning remains debatable. For film thickness less than about 100 nm (thin layers), effective molecular interactions between the liquid layer surface and the substrate dominate all other forces (like thermo-or soluto-capillarity or gravity) and thus determine the film stability and patterning under dewetting [100][101][102].
For evaporated films with a thickness above 100 nm, thermo-capillarity forces become dominant, resulting in instability caused by the Marangoni convection, either thermo-or soluto-capillary [103][104][105][106][107][108][109][110][111][112][113]. It should be mentioned that the analysis of the pattern inspired by thermo-and soluto-capillarity is an essentially non-linear one and involves complicated mathematical models [103][104][105][106][107][108]. It was demonstrated experimentally that the patterns observed in the evaporated (cooled from above) layers are different from those formed in a layer heated from below without evaporation [109]. Thus, the kinetics of evaporation, studied in [110], turns out to be crucial for constituting large-scale patterns. An apparatus utilizing self-organized liquid flow for the targeted modification of macromolecular systems in a solution was suggested in [112]. Marangoni-flow-induced patterns, observed under spin-coating of evaporated polymer solutions, were reported in [114]. When the polymer solution is evaporated, thermo-and soluto-capillary flows occur and both may contribute to the eventual pattern. It remains debatable what kind of Marangoni flow has a decisive impact on the pattern. The experimental data reported in [95,115] indicate that it is the solutal Marangoni flow that causes the pattern. Indeed, heating the substrate from below destroyed the pattern [115]. This contradicts the idea that self-organization is due to the jump in surface tension caused by a temperature gradient (temperature-gradient-driven Marangoni instability).
However, when evaporation is present, thermo-and soluto-capillarity represent only a few of a diversity of destabilizing mechanisms: vapor recoil, differential evaporation (the dependence of the evaporation rate on the thickness of the film), or, sometimes both contribute to the development of interfacial instability and as a result exert an influence on the pattern's makeup [116]. de Gennes proposed an alternate mechanism of patterning. He showed that in an evaporating film, a "plume" of solvent-rich fluid induces a local depression in surface tension, and the surface forces tend to strengthen the plume. His calculations led to the conclusion that this kind of instability should dominate over the classic Bénard-Marangoni instabilities [117,118]. It should be emphasized that in all the aforementioned instabilities (namely Marangoni and de Gennes) the tangential vector field of velocities drives the liquid. This makes possible the topologically-based approach to patterning, as demonstrated in [119]. The "hairy ball theorem" of algebraic topology states that any continuous tangent vector film on a surface topologically equivalent to a sphere must have at least one point where the vector is zero [120]. Remarkably, the "hairy ball theorem" predicts the existence of at least one zero tangential velocity point at the surface of the evaporated polymer solution [119]. Indeed, these zero velocity points were observed experimentally [119].
The role of water droplets in the formation of large-scale patterns is shown in Figure 3. It seems that water droplets work as "tracers", enabling visualization of the boundaries and separating the large-scale cells depicted in Figure 3. Pores appearing after the droplets' evaporation are accumulated in zero-velocity points, demonstrating the "hairy ball theorem", as discussed in [119][120][121]. Now consider the mesoscopic, micro-scaled patterning observed by various groups [14-16, 26-30,35,36,53,67,68]. These mesoscopic patterns are built from well-ordered micro-scaled or sub-micro-scaled pores, demonstrating long-range 2D and sometimes 3D ordering [35,[122][123][124]. The mechanism of this patterning was discussed in [48,66]. Govor et al. related the mesoscopic ordering to capillary interaction between droplets, discussed in detail in [125][126][127][128]. Kralchevsky et al. demonstrated that between two particles placed on the liquid/vapor interface a force (which may be either attractive or repulsive) similar to the Coulomb interaction acts between two endless wires charged with constant linear charge densities [125][126][127][128]. It is reasonable to suggest that this capillary interaction between droplets, condensed at the polymer solution/humid vapor interface, is responsible for the long-range ordering inherent to breath-figure self-assembly. It was already demonstrated by Bragg, Nye, and Lomer (in [129,130]), that the capillary interaction of bubbles promotes the assemblage of bubbles, representing the crystal structure of real metals.
Capillary interaction is not the only kind of physical interaction between particles placed at the liquid/vapor interface. It was demonstrated by Pieranski that electrostatic interactions between floating particles may be no less important than capillary ones [131]. The role of the Marangoni thermo-capillary convection in the formation of ordered honeycomb patterns was mentioned in [66,132,133]. The details of the physical mechanism giving rise to the long-range self-assembly of pores, inherent to breath-figure patterning, remain unclear. It is noteworthy that the formation of ordered ensembles of droplets under the drop casting method occurs in the vicinity of the triple (three-phase) line, as shown in Figure 4 [134]. Self-assembly of colloidal particles (not droplets!) taking place in the vicinity of the triple line was studied extensively in [135,136]. However, the approach developed in [135,136] could hardly be extended to the explanation of the self-assembly of condensed water droplets, due to their coalescence (to be discussed below).  It was also demonstrated that some additives such as PEG and dendrons promote the ordering occurring under breath-figure self-assembly [62,137,138]. The well-ordered honeycomb patterns resulting from breath-figure self-assembly are evidence of non-coalescence or delayed coalescence of sessile water droplets condensed on the polymer solution/vapor interface. This means that a so-called "capillary cluster" built from micro-scaled water droplets exists on the polymer solution/vapor interface. The physical behavior of non-coalescent capillary clusters, in which capillary interactions prevail or play an essential role, has drawn the attention of investigators recently [139][140][141].
When droplets of the same liquid touch one another, one expects coalescence [23,142,143]. The mechanism of the non-coalescence observed in capillary clusters remains disputable. Karpitschka et al. showed that sessile droplets from different but completely miscible liquids do not always coalesce instantaneously upon contact: the drop bodies remain separated in a temporary state of non-coalescence, connected through a thin liquid bridge [144][145][146]. Karpitschka et al. suggested that the delay originates from Marangoni convection [144][145][146]. Systematic study of Marangoni-convection-inspired non-coalescence was undertaken by Dell'Aversana et al. ( [147,148]), who performed both laboratory experiments and molecular dynamics simulations. In the case of a pair of sessile droplets, a locally hotter region is formed in the center at the top of droplet, as takes place under the coffee-stain effect [149][150][151][152]. These surface-temperature variations not only give rise to thermo-capillary Marangoni convection within the droplets, as depicted in Figure 5, but also may drag air surrounding the drops into the space between them. This gas film serves to lubricate the space between the liquid surfaces, preventing them from coming into contact [147,148]. It is noteworthy that under breath-figure self-assembly the experimental situation is essentially complicated by the fact that sessile water droplets are located at the rapidly evaporated polymer solution/vapor interface. This may strengthen thermo-capillary Marangoni flows [151,152]. However, as will be demonstrated below in Section 6, the condensed droplets relatively rapidly come to thermal equilibrium; thus, the true role of the thermo-capillary Marangoni flows in preventing coalescence remains unclear. Other mechanisms of non-coalescence were discussed in [148]. We conclude that the details of the non-coalescence of droplets in capillary clusters remain unclear and call for further experimental and theoretical insights. When droplets of the same liquid touch one another, one expects coalescence [23,142,143]. The mechanism of the non-coalescence observed in capillary clusters remains disputable. Karpitschka et al. showed that sessile droplets from different but completely miscible liquids do not always coalesce instantaneously upon contact: the drop bodies remain separated in a temporary state of noncoalescence, connected through a thin liquid bridge [144][145][146]. Karpitschka et al. suggested that the delay originates from Marangoni convection [144][145][146]. Systematic study of Marangoni-convectioninspired non-coalescence was undertaken by Dell'Aversana et al. ( [147,148]), who performed both laboratory experiments and molecular dynamics simulations. In the case of a pair of sessile droplets, a locally hotter region is formed in the center at the top of droplet, as takes place under the coffeestain effect [149][150][151][152]. These surface-temperature variations not only give rise to thermo-capillary Marangoni convection within the droplets, as depicted in Figure 5, but also may drag air surrounding the drops into the space between them. This gas film serves to lubricate the space between the liquid surfaces, preventing them from coming into contact [147,148]. It is noteworthy that under breathfigure self-assembly the experimental situation is essentially complicated by the fact that sessile water droplets are located at the rapidly evaporated polymer solution/vapor interface. This may strengthen thermo-capillary Marangoni flows [151,152]. However, as will be demonstrated below in Section 6, the condensed droplets relatively rapidly come to thermal equilibrium; thus, the true role of the thermo-capillary Marangoni flows in preventing coalescence remains unclear. Other mechanisms of non-coalescence were discussed in [148]. We conclude that the details of the non-coalescence of droplets in capillary clusters remain unclear and call for further experimental and theoretical insights. Figure 5. Scheme of non-coalescence of sessile droplets is depicted (see [147,148] for details).
Nanoparticles also prevent the coalescence of droplets and enable the formation of the additional nanoscale in the hierarchical topographies obtained under breath-figure self-assembly (see the extended review of the state of the art of nanoparticles in breath-figure self-assembly in [16,[153][154][155][156]. Saunders et al. demonstrated that the superlattice of mono-dispersed gold nanocrystals formed under the breath-figures process had an ordered structure at the nanometer scale [156]. The interaction between self-organization processes at the nano-and micrometer length scale, especially through the formation of a water droplets/evaporating polymer solution interface and droplets' collective motions, was addressed in [157].

Main Physical Processes Involved in Breath-Figure Self-Assembly and the Hierarchy of their Temporal Scales
Now consider the dimensionless numbers describing breath-figure self-assembly, namely the Bond (Bo), capillary (Ca) and Reynolds (Re) numbers:  Nanoparticles also prevent the coalescence of droplets and enable the formation of the additional nanoscale in the hierarchical topographies obtained under breath-figure self-assembly (see the extended review of the state of the art of nanoparticles in breath-figure self-assembly in [16,[153][154][155][156]. Saunders et al. demonstrated that the superlattice of mono-dispersed gold nanocrystals formed under the breath-figures process had an ordered structure at the nanometer scale [156]. The interaction between self-organization processes at the nano-and micrometer length scale, especially through the formation of a water droplets/evaporating polymer solution interface and droplets' collective motions, was addressed in [157].

Main Physical Processes Involved in Breath-Figure Self-Assembly and the Hierarchy of Their Temporal Scales
Now consider the dimensionless numbers describing breath-figure self-assembly, namely the Bond (Bo), capillary (Ca) and Reynolds (Re) numbers: where ρ is the density (the densities of water and polymer solutions are very close), η and γ are the viscosity and surface tension of the polymer solution respectively, v is the characteristic velocity of droplets and pores, and L is the characteristic spatial scale. Assuming for numerical values of physical parameters appearing in Equation (3): (the viscosity of the solution is taken for the initial stage of the evaporation, and the velocity equals the maximal velocity of pores, established experimentally in [121]), we conclude that inequalities take place for all lateral characteristic scale lengths inherent to breath-figure self-assembly, namely: 10 −9 m < L < 10 −5 m. This means that the effects due to gravity, inertia, and viscosity are negligible, and breath-figure self-assembly is mainly driven by interfacial phenomena. However, the viscosity, growing with the evaporation of the polymer solution layer, helps to stabilize the eventual honeycomb pattern, as will be shown below. Indeed, the breath-figure patterns were observed on horizontal [14][15][16] and vertical [46,[95][96][97] substrates. Moreover, multi-layered porous structures were observed for vertically driven substrates under the dip-coating process, evidencing the crucial role of interfacial processes, and the negligibility of gravity for the breath-figures process [62]. It is seen from estimations supplied by Equation (4) that breath-figure self-assembly is a "slow" process in which effects due to inertia and viscosity are negligible. In order to clarify the precise meaning of the wording "slow process", it is instructive to elucidate the hierarchy of time scales inherent to the process, namely: τ sol therm and τ drop therm , which are the characteristic times necessary for attaining thermal equilibrium in the evaporated polymer solution and condensed water droplets, respectively; τ inter f /visc , which is the characteristic time necessary for viscous stresses (developed in the evaporated polymer solution layer) for balancing interfacial ones (at the length scale of a single droplet R), and eventually the temporal scales τ sol ev ∼ = 10s and τ drop ev ∼ = 1s, which are the characteristic times of evaporation of the polymer solution and water droplets, respectively. The estimations for the time scales yield: where α sol and α w are the thermal diffusivities of a polymer solution and water, respectively; L sol and R are the characteristic spatial scales of the evaporated layer of the polymer solution (in other words its typical thickness) and the condensed droplet (i.e., its radius), respectively; and η and γ are the viscosity and the surface tension of the polymer solution at the initial stage of the evaporation, as noted above. Assuming: α sol ∼ = 7.7 × 10 −8 m 2 s (as it is taken for the chloroform, as a typical solvent); α w ∼ = 1.47 × 10 −7 m 2 s , L sol ∼ = 10 −4 m, R ∼ = 10 −6 m, yields following rough estimations: Finally, we estimate the hierarchy of time scales inherent for the breath-figure self-assembly: This hierarchy may be interpreted as follows: the condensed water droplet almost immediately comes to thermal equilibrium, whereas the polymer solution layer, fixing the pattern, is far from thermal equilibrium on the time scale of its evaporation. Equation (6) also implies that at the lateral scale of a single droplet, interfacial stresses are immediately balanced by viscous ones, which are developed in the polymer solution layer. So, the behavior of a single droplet is quasi-static, and the formation of pores is stabilized by the viscosity of the polymer solution layer, growing with the evaporation, whereas the dynamics and thermodynamics of large (~10 µm) cells, depicted in Figure 3, are essentially at non-equilibrium.

Characterization of the Ordering of Patterns
Ordered honeycomb structures arise from breath-figure self-assembly. How may the 2D ordering of patterns be quantified? Two approaches to the quantifying of surface ordering have been reported. The first exploited the statistical properties of the autocorrelation functions [158], calculated over the pixels of SEM images taken of honeycomb patterns [62]. The correlational analysis of the SEM images indicated short-range and mesoscopic ordering of the honeycomb structures on the characteristic scale of 5 µm for the typical, weakly ordered patterns, represented in Figure 2. The second approach to the problem involved the use of Voronoi diagrams (or Voronoi tessellations), described in detail in [159,160]. Perfect ordering of pores on the scale of 1-10 mm was registered [14][15][16]. Voronoi tessellation and Voronoi entropy were successfully applied for the quantitative characterization of the ordering of capillary clusters (see [141]) and the pores constituting honeycomb reliefs in [73,161,162]. The study of defects (including line defects), observed under breath-figure pattering and possibilities of their elimination was reported in [163].

Surface Characterization of Patterns Obtained with Breath-Figure Self-Assembly
The simplest (and an extremely useful) method of characterization of micro-rough surfaces is based on the measurement of the apparent contact angle [32][33][34]164]. A detailed study of the apparent contact angles inherent for honeycomb, porous surfaces arising from breath-figure self-assembly was reported in [165][166][167]. It was demonstrated that honeycomb surfaces produced by breath-figure self-assembly demonstrate a pronounced heterogeneous (air-trapping) Cassie-Baxter wetting regime [79,[168][169][170]. The interfaces characterized by stable Cassie-Baxter wetting have high apparent contact angles and low contact angle hysteresis [32][33][34][168][169][170]. Thus, they are suitable for a broad range of applications, where self-cleaning properties of the interface are demanded [171,172]. Somewhat surprisingly, polymer surfaces manufactured by breath-figure self-assembly showed the Cassie-like wetting, even when hydrophilized by a metal coating [167]. The mechanism of stability of the Cassie wetting states observed on honeycomb polymer and metallized reliefs was addressed in [173].
At the same time, the mechanical properties of microporous films obtained with the breath-figure self-assembly remain obscure. The non-trivial mechanical properties of these films are expected, owing to their reduced size and dimensionality compared with those of their macroscopic counterparts [174].

Novel Applications of Breath-Figure Self-Assembly
We already considered the numerous applications of interfacial honeycomb structures arising from breath-figure self-assembly in Section 1 (see also the comprehensive review in [16]). Some very recent applications of these structures are noteworthy. Microporous functional polymer surfaces arising from breath-figure self-assembly have been proven to be selective surfaces for attracting eukaryotic cells while maintaining antifouling properties against bacteria [175]. Polymer scaffolds prepared by the breath-figure technique enabled differentiation of human mesenchymal stem cells, as demonstrated in [176].
A process of manufacturing microspherical particles with the breathfigures process was reported in [177]. Breath-figure self-assembly was successfully employed in manufacturing solid-state supercapacitors [178]. It was shown that honeycomb polymer films supported by steel meshes enable the manufacturing of electrically controlled membranes [179]. Low friction and bubble-repellent surfaces with micro-dimple arrays, obtained with the breath-figures process, were reported in [180,181].

Breath-Figure Self-Assembly and Manufacturing of Membranes
Honeycomb polymer films manufactured by breath-figure self-assembly are not directly suitable for membrane applications for two reasons, namely: (a) pores obtained with this process are too large (see Figure 2); (b) pores are not through-pores across the films [182]. However, recent studies have demonstrated that casting a very thin film of polymer solution on ice allows for the preparation of a thin polymer membrane with through-pores, formed by the breath-figures method; the membrane was then transferred onto a porous substrate to form a thin selective layer for microfiltration [183]. Highly ordered porous membranes of cellulose triacetate prepared successfully on ice substrates using breath figure method were reported in [184]. Wan et al. reported manufacturing membranes with breath-figures-inspired honeycomb patterns transferred onto a dense electrospun nanofiber mesh. The alternate approach to manufacturing membranes with the breath-figures technique was discussed in [1], where multilayer interpenetrating porous structures were reported. A similar strategy involving the use of SiO 2 , TiO 2 , Co and Cd nanoparticles for manufacturing membranes demonstrating hierarchical porous structures with nanometrical pores (with dimensions of 2-50 nm) was reported in [185]. Layer-by-layer deposition of carbon nanotubes onto the honeycomb membrane surface, arising from the breath-figures process, allowed for manufacturing voltage-activated membranes demonstrating potential as humidity sensors and microclimate regulators [186]. Breath-figure templating was successfully involved in the preparation of low-resistance microfiltration membranes having a uniform size of pore opening using polysulfone (a regular membrane polymer) [187]. A highly permeable brominated poly(phenylene oxide) microfiltration membrane with binary porous structures, fabricated by a combination of the breath-figure and colloidal crystal template methods, was recently reported in [188]. We conclude that a combination of breath-figures assembly, giving rise to microscaled porous polymer films, with other techniques enabling the formation of nanopores connected to micro-scaled ones has potential in the manufacturing of membranes [183][184][185][186][187][188]. Another useful application of the method is related to the manufacturing of so-called breathing cathodes for fuel cells [189,190].
Breath-figure self-assembly is a robust, single-stage process; however, it involves almost all events inherent to interface science, namely evaporation of a solvent, condensation of water droplets, instabilities developed in the evaporated polymer solution, delayed coalescence of closely packed droplets forming the capillary cluster, and their interaction [16,[125][126][127][128][129][130][131]. Thus, the process of breath-figure self-assembly may be understood only within the broader context of surface science [31][32][33][34].
Several basic questions related to the physical and chemical mechanisms of breath-figure self-assembly remain obscure; even the impact of the polymer architecture on the resulting pattern is unclear and calls for future investigations. In this situation the qualitative macroscopic approach to the analysis of the process becomes preferable. Such an analysis is proposed in this review. The estimation of dimensionless numbers describing breath-figure self-assembly, namely the Bond (Bo), capillary (Ca), and Reynolds (Re) numbers, provides evidence that the effects due to gravity, inertia, and viscosity are negligible, at least at the first stage of evaporation of a polymer solution, and that the process is mainly driven by interfacial phenomena. The viscosity of a polymer layer contributes to the stabilization of the pores' radii.
The hierarchy of spatial and temporal scales inherent to breath-figure self-assembly is elucidated. The characteristic spatial scales of patterns range from dozens of micrometers to nanometers. The temporal scales of the process vary from microseconds to seconds. Analysis of the hierarchy of temporal scales demonstrates that the condensed water droplet almost immediately comes to thermal equilibrium, whereas the polymer solution layer is far from thermal equilibrium. It is also shown that the behavior of a single droplet is quasi-static, whereas the dynamics and thermodynamics of large-scale cells are essentially at non-equilibrium. The topological aspect of self-assembly is considered [119].