Effects of Interface Velocity, Diffusion Rate, and Radial Velocity on Colloidal Deposition Patterns Left by Evaporating Droplets

An understanding of transport and deposition of material in evaporating droplets is critical for optimizing applications in medical diagnostic [1–5], printing [6], nanoparticle self-assembly [7–10], containerless materials processing [11], and advanced manufacturing of flexible electronic devices [12–14]. Many of these practical applications are influenced by the coffee-ring effect, where material is deposited in a ringlike structure after evaporation. This effect has been the subject of extensive research efforts. For example, coffee-ring suppression has been achieved by addition of surfactant [15], manipulation of the thermal gradient [16,17], surface modification [18], increasing the aspect ratio of colloidal particles [19], and application of an electric field [20–22].

When evaporation of a colloidal droplet is limited by diffusion of vapor away from the interface, the majority of material is often transported to the periphery where it forms a ring deposition [16,23–28]. Deegan et al. [23–25] identified contact line pinning and evaporation as the mechanisms for these patterns. When the contact line is pinned, conservation of mass drives a radially outward flow that carries colloids to the contact line. An extensive body of literature has followed, with works quantifying different physical effects on particle transport and how these effects can be used to manipulate pattern formation. Relevant highlights for this study are presented in what follows. A comprehensive review of this phenomenon is presented by Larson [29].

Knowledge of fluid flow in evaporating droplets is required to build an understanding of particle transport. While the radial flow described by Deegan et al. [23–25] dominates in many evaporating droplets, it can be significantly reduced when the contact line is moving. Contact line mobility of an evaporating droplet provides information about fluid flow within the droplet by characterizing different modes of evaporation. If the contact line remains pinned as the droplet evaporates, the height and contact angle of the droplet decrease. This mode of evaporation has been referred to as the constant contact radius (CCR) regime [9,30,31]. If the contact line is mobile, the contact angle of the droplet can remain constant while the radius and height of the droplet are diminished. This mode of evaporation is referred to as the constant contact angle (CCA) regime [9,30,31]. A mixed regime occurs when both contact angle and radius are diminished simultaneously. It has been shown experimentally [9,30,31] and numerically [32,33] that the radial flow is diminished in the CCA regime. Transition to the mixed mode has been attributed to Marangoni instability, which can drive an outward radial flow during this regime [31].

Variation in temperature across the droplet can affect particle transport during evaporation. Hu and Larson [16] showed that thermal Marangoni effects can play a significant role in particle transport. The direction of the Marangoni recirculation loop is dependent on the relative conductivity of the droplet and the surface on which the droplet is evaporating [6,34]. The presence of significant Marangoni recirculation in the droplet will generally suppress the formation of the ring pattern by forcing particles toward the center of the deposit [6,16,28,34]. This suppression can be observed independent of the circulation direction of the Marangoni loop [34].

The flow inside the droplet is not the only effect contributing to particle transport during evaporation. Particle deposition can be dominated by surface interactions that trap particles on the substrate [4,22,28,35]. Bhardwaj et al. [28] demonstrated that uniform depositions can be attained when attractive DLVO forces dominate over both evaporative and Marangoni effects. The DLVO effect examines the net contribution of van der Waals' and electrostatic forces between particles and the surface. When these short-range forces dominate, particles near the substrate can become trapped before being transported to the edge or the center of the deposition [28]. In addition to the DLVO effect, the free energy in the electric double layer (EDL) of an electrolytic droplet on a charged surface has been shown to lead to a hydrophilicity-inducing tendency in both the Wenzel and Cassie–Baxter states [36]. Advection of mobile ions in the EDL during evaporation has also been shown to promote rapid particle deposition at the end of the lifetime of an electrolytic droplet on a charged surface [37]. Particle capture on the substrate has also been achieved by leveraging antibody–antigen reactions [35], as well as magnetic [4] and electrophoretic forces [22].

Advective transport can also be overcome when particles are captured on the descending interface of evaporating droplets [24,38,39]. Li et al. [38] observed uniform colloidal monolayers when water droplets were evaporated at elevated temperatures. In their investigation, the interface velocity (⁠$ui$⁠) was controlled by manipulating the substrate temperature, and the diffusion rate (⁠$x˙p$⁠) was held constant by using a fixed particle size. At low temperatures, the particle diffusion rate was greater than the interface velocity and particles migrated to the contact line to form a ring deposition. At high temperatures, the interface collapsed faster than the diffusion rate and the resultant depositions were uniform. In these cases, they argued that particles were captured on the interface and deposited uniformly on the surface at the conclusion of evaporation. A transitional region where neither rate dominated was observed when $0.5<ui/x˙p<2$⁠. In these cases, platelike depositions were reported with both a ring structure and a nonuniform central deposition. Interface capture, or the formation of a skin of particles on the droplet interface, was also observed by Deegan et al. [24] and modeled by Maki and Kumar [39].

This work further examines the transitional region proposed by Li et al. [38] in an effort to better understand the role of interface capture on colloidal deposition patterns left by evaporating droplets. Interface velocities are varied by evaporating droplets on multiple surfaces. Diffusion rates are varied by changing the size of the particles in solution. The magnitude of the radial velocity is shown to play an important role in determining the deposition pattern when the interface velocity and the particle diffusion rate are of similar magnitude.

Experiments performed in this investigation consisted of a colloidal droplet, a substrate, and a Ramé-Hart Model 250 Goniometer (Fig. 1(a)). The solvent in the droplet was deionized (DI) water with a conductivity between $17.9$$MΩ cm$ and $18.2$$MΩ cm$⁠. Photostable fluorescent carboxylate-modified polystyrene microspheres from Life Technologies (FluoSpheres) with diameters (D_p) of $1.1 μm, 100 nm,$ and $25 nm$ were used as colloidal material. All particles were spherical to avoid the effects of aspect ratio on deposition patterns [19]. Excitation and emission wavelengths for these particles were $505 nm$ and $515 nm$⁠, respectively. In all cases, a 2% aqueous stock particle solution was diluted to a volume fraction of 0.05%. The stock solution was placed in a room temperature ultrasonic bath (Bransonic 1800) for 15 min prior to dilution. Diluted solutions were also sonicated for 15 min before droplets were drawn and deposited by micropipette (Eppendorf Research Plus 0.5–10 $μL$⁠) onto the substrate. Solutions were kept refrigerated between trials and allowed to warm up to room temperature during preparation.

Colloidal droplets were evaporated at room temperature on (i) glass, (ii) SU8 3005, (iii) polyimide, and (iv) polytetrafluoroethylene (PTFE). Glass substrates and polyimide films (⁠$25.4 μm$ Kapton HN) were purchased commercially. Thin films of SU8 and PTFE (Teflon AF) were spun on glass substrates to thicknesses of $4 μm$ and $50 nm$⁠, respectively. The droplet volume was one microliter in the majority of cases; however, droplets of this size did not repeatedly deposit on the PTFE surface and $2 μL$ droplets were used instead. The Bond number of droplets evaporated on PTFE (0.13) was similar to those evaporated on glass (0.06), SU8 (0.08), and polyimide (0.09). This suggests that all droplets in this investigation are physically comparable. Bond numbers are calculated as described in Ref. [29]. Uncertainty in the deposited volumes is reported at $±3%$ by the manufacturer. Ambient conditions in the lab were measured using an Inkbird THC-4 data logger. The average temperature during experiments was $24±1.2 °C$⁠. Relative humidity was measured to be $47%±12%$⁠.

The evolution of the droplet interface during evaporation was tracked using a Ramé-Hart Model 250 Goniometer with a backlit five-axis stage (x–y–z–pitch–roll) and a charge-coupled device camera (⁠$659×494$ pixels). Transient images of the droplet profile were processed as shown in Fig. 1(a). Images from at least five trials of each case were monitored at a minimum of one frame per second. Ramé-Hart DROPimage Advanced software calculated the contact diameter and the average of the contact angles on the left and right side of each droplet. Evaporation end time was determined by identifying the final frame where the droplet was visible. Data from multiple trials of the same case were aligned based on this final frame in order to determine the mean value and two standard deviations for each measured quantity. In order to compare data in different cases, contact diameter and time were scaled by initial contact diameter and total evaporation time of the longest trial for each case.

Receding contact angles (Table 1) were measured for noncolloidal droplets on each substrate as described by Li and Mugele [40]. Surface roughness measurements of each substrate were measured using a Taylor Hobson Surtronic 3+ Profilometer (Table 1) as roughness can influence deposition by contributing to contact line pinning during evaporation [41,42]. All surfaces had a roughness of approximately $235 nm$⁠.

Fluorescent images of deposition patterns were attained using a Leica SP5 spectral confocal laser scanning microscope. Image acquisition parameters were based on the size and fluorescent properties of the particles. An argon laser was adjusted using an acousto-optic tunable filter to image each sample. A Leica HyD detector was used to image the excited depositions. The gain was adjusted between cases to present the greatest amount of fluorescent data without oversaturation of the images.

Fluorescent images were analyzed in matlab to produce deposition fluorescence intensity profiles as a function of radial position. Since the deposition radius can vary in size with azimuthal position, the pattern was divided into $M$ sectors $(A1,A2,…,AM)$ of equal angle. The mean radius for each sector (⁠$Rdepm$⁠) was computed from each sector's area, perimeter, and angle. This value was used to normalize the local radial position. Each sector was then divided into $N$ equally spaced radial positions $(A1m, A2m,…, ANm)$⁠, and fluorescence intensity for each was quantified. Intensities in the same radial position of each sector $(An1, An2,…, AnM$⁠) were averaged to determine the mean intensity at that relative radial position (⁠$In$⁠). This intensity was scaled using the average fluorescence intensity of the entire deposition (⁠$I¯$⁠). A statistical analysis of the radial fluorescence intensity was performed across all sectors in at least four depositions for each case. The process is represented in Fig. 1(b).

Previous investigations have shown that DLVO effects can have a profound effect on particle transport and deposition in evaporating droplets [28]. To examine their role in this investigation, the magnitude of the DLVO force was calculated based on the droplet solution zeta potentials and the substrate surface charges as detailed in Ref. [28] (Table 2). For all cases, electrostatic forces dominate the van der Waals' forces when both terms are calculated a Debye length away from the surface. Beyond this distance, both forces diminish rapidly. The values for zeta potentials of all solutions used in this investigation were measured after evaporation testing using a Malvern Zetasizer Nano ZS and found to be negative. Published values for the surface potentials of all materials [43–46] were also negative for pH values between 6.5 and 7.5. As such, the net DLVO force was negative for all cases examined here.

where the diffusion constant (⁠$D$⁠) is estimated from the Stokes–Einstein relation (⁠$D=kBT/6πηRp$⁠), $kB$ is Boltzmann's constant, $T$ is the temperature, $η$ is the viscosity of the solvent, $Rp$ is the radius of the particle, and $t$ is a characteristic time [38]. The time in Eq. (2) was then set to 1 s to provide an estimate for the diffusion distance per second, or the diffusion rate (⁠$x˙p$⁠) [38].

In an effort to understand the effect of particle diffusion rate on colloidal deposition in evaporating droplets, deposition patterns were observed for droplets containing $25 nm$,$100 nm,$ and $1.1 μm$ particles evaporated on SU8 (Fig. 2). The range of particle sizes here provides diffusion rates between 0.7 $μm/s$ and 4.68 $μm/s$ (Table 2). While the diameters of all three depositions on SU8 were similar, the deposition patterns were quite different. Depositions left by $1.1 μm$ particles were uniform, while the $25 nm$ particles left cracked ring patterns like those seen by Deegan et al. [24]. The stark contrast in fluorescence between the deposited ring and the cracks contributed to the variability seen in Fig. 2(a). Here, the inner and outer diameters of the ring were clearly defined and little fluorescence intensity was detected toward the center of the deposition. The $100 nm$ particles also left a ring pattern, but the inner diameter and cracks were less distinct than in the $25 nm$ case. The peak intensity in the ring was diminished as a uniformly small fluorescence intensity was observed toward the center of the deposition.

The patterns observed in Fig. 2 support the observation by Li et al. [38] that depositions can be explained by the ratio of the interface velocity to the particle diffusion rate. While the interface velocities on SU8 were similar for all particle sizes (⁠$0.96−1.08 μm/s$⁠), the particle diffusion rates increased from $0.70 μm/s$ to $4.68 μm/s$ as particle diameter was reduced from $1.1 μm$ to $25 nm$⁠. As such, the ratio of the interface velocity to the diffusion rate (⁠$uint/x˙p$⁠) decreased from $1.36$ to $0.23$ (Table 2). This supports the hypothesis that the large diameter particles are captured by the interface as it descends, while the high diffusion rates of small particles allow them to escape to form a ring pattern.

The uniform deposition observed for $1.1 μm$ particles on SU8 is not due to substrate trapping by DLVO effects. The repulsive DLVO force (Table 2) in these cases opposes particle deposition on the surface and provides additional time for particles to be carried by the flow in the droplet. This is not dissimilar from the work presented in Ref. [47], which showed that charged particles in a flow tend to deposit on oppositely charged regions of heterogeneously charged surfaces.

The validity of characterizing colloidal deposition patterns by comparing the ratio of the interface velocity to the particle diffusion rate was further tested by varying the interface velocity (⁠$ui$⁠). This was done by evaporating water droplets containing $1.1 μm$ polystyrene particles on glass, polyimide, SU8, and PTFE. All of these cases were within the transitional region identified by Li et al. [38] where $0.5<ui/x˙p<2$ (Table 2).

Surface selection clearly had an effect on the observed deposition patterns (Fig. 3). Ring depositions were observed for droplets on glass and polyimide. The maximum fluorescence intensities for glass and polyimide both occurred at the periphery and were 264% and $176%$ of the average fluorescence intensity, respectively (Figs. 3(a) and 3(b)). Despite the presence of a clear ring in these cases, some particles were deposited toward the center of the region. The average fluorescence intensity away from the periphery was higher on polyimide than on glass. Interestingly, in some locations, the intensity within the perimeter on polyimide was as high as it was in the ring. This variation led to large uncertainties in this case. This deposition pattern is similar to the transitional cases identified by Li et al. [38], which exhibited the presence of a ring structure and nonuniform central deposition.

Deposition patterns on SU8 and PTFE were more uniform than those on glass and polyimide (Fig. 3). Here, each deposition had a maximum intensity toward the center of the deposition and a minimum intensity at the periphery. Maximum intensities were 130% and 109% of the average intensities for SU8 and PTFE, respectively (Figs. 3(c) and 3(d)). Deposition profiles in these cases were also more consistent across multiple trials.

Interestingly, the diameters of the depositions left on SU8 and PTFE were significantly smaller than their initial contact diameters (Figs. 3(c) and 3(d)). Representative evolution of droplet profiles for each case is presented in Fig. 4, and evolutions of contact angle and diameter are presented in Fig. 5. Note the similar shapes and initial diameters of the droplets on polyimide and SU8 where the initial contact angles agreed to within $3 deg$ (Figs. 4, 5(c) and 5(e)). In all cases, the contact line was pinned for the first 30% of the evaporation time (⁠$τ=0.3$⁠). On glass and polyimide, it remained pinned for the majority of the process. Here, mass loss due to evaporation resulted in a consistent decrease in the contact angle. Since the contact line remained pinned or receded slowly for the majority of the evaporation time, radial flow within the droplet consistently carried particles toward the contact line to form the ring patterns observed in Figs. 3(a) and 3(b). This is consistent with work by Kuncicky and Velev [30], which showed that ring patterns from similar particle concentrations are more likely on surfaces with low receding contact angles. The larger variation in the droplet diameter on glass was likely due to the tendency of droplets to pin in place as they contacted the surface and remain in noncircular shapes.

For droplets on SU8 and PTFE, the contact line depinned between $τ=0.3$ and $τ=0.4$ after reaching the receding contact angle (Figs. 5(e) and 5(g)). While a clear CCA regime was not observed in either case, the reduction in contact angle with time was less pronounced when the contact line began to recede (from $τ≈0.35$ to $τ≈0.65)$⁠. Previous investigations have found that the radial velocity is often diminished in cases where the contact line recedes [9,30–33]. This suggests that radial flow in the droplet during evaporation may be an important parameter when examining the role of interface capture in evaporating colloidal droplets.

Evaporation and Marangoni recirculation can both contribute to the net radial velocity during evaporation. Interestingly, many experimental works have observed that Marangoni flows are negligible in evaporating water droplets [16,23,38]. Specifically, Hu and Larson found that the measured flow was more than two orders of magnitude lower than the theoretically calculated value due to surfactant contamination, even at concentrations as low as $300 molecules/μm2$ [16]. While these observations were made in cases where the contact line was pinned for the majority of the evaporation time, the current work assumes that Marangoni effects can also be neglected in cases where the contact line recedes. A consequence of this assumption is that fluid flow in the droplet is entirely due to evaporation.

where $Rdep$ is the radius of the observed colloidal deposition and $tevap$ is the evaporation time. Values for all cases examined in this investigation are presented in Table 2.

Li et al. [38] demonstrated that deposition uniformity decreased as the ratio of radial velocity to diffusion rate (⁠$urad/x˙p$⁠) increased and identified a transitional regime where $urad/x˙p≈1$⁠. Interestingly, the current investigation has found both uniform and ring depositions within that transition regime. This suggests that at least one additional parameter is required to understand the transition between these patterns.

The results presented previously suggest that colloidal transport and deposition are affected by changes in the radial velocity (⁠$urad$⁠), the interface velocity (⁠$ui$⁠), and the particle diffusion rate (⁠$x˙p$⁠). These parameters can be combined into the ratio of the radial velocity to the particle diffusion rate (⁠$urad/x˙p$⁠) and the ratio of the interface velocity to the particle diffusion rate (⁠$ui/x˙p$⁠). A phase diagram showing the depositions of all cases examined here and in Ref. [38] is presented in Fig. 6. Cases (i) and (ii) are discussed in more detail later.

The ratios $ui/x˙p$ and $urad/x˙p$ appear to describe the observed deposition patterns. For the deposition of $1.1 μm$ particles on SU8 and PTFE, $ui/x˙p$ was greater than $urad/x˙p$ and the maximum fluorescence intensity was observed toward the center of the deposition (see iii and iv in Fig. 6). For each deposition on glass (see vi in Fig. 6), $urad/x˙p$ was greater than $ui/x˙p$ and a clear ring was present at the periphery. For each deposition on polyimide (see v in Fig. 6), $ui/x˙p$ was approximately equal to $urad/x˙p (ui/x˙p≈0.93⋅urad/x˙p)$ and both a ring structure and a nonuniform central deposition were observed. This is similar to the description of the transitional cases identified by Li et al. [38] where $ui/x˙p$ was also approximately equal to $urad/x˙p$⁠.

The observation that the ratios of interface velocity and radial velocity to diffusion can be used to explain colloidal deposition patterns was further tested by revisiting the cases where droplets were evaporated on SU8 (Fig. 2). The characteristic radial velocities, as defined by Eq. (3), were similar in these cases since the initial and deposition radii are similar. However, the characteristic diffusion rate increased as particle size decreased. As previously discussed, the ring deposits observed in cases with nanoscale particles support the interface capture hypothesis. However, in both of these cases, labeled (i) and (ii) in Fig. 6, $ui/x˙p$ was approximately twice the value of $urad/x˙p$⁠. The other cases on the phase diagram depicted previously suggest that this should result in a uniform deposition pattern. As such, the characteristic radial velocity (3) does not appear to explain the presence of the ring deposition in the nanoparticle cases.

The evolutions of contact angle and diameter for all droplets evaporated on SU8 are presented in Fig. 7. Constant contact diameter, constant contact angle, and mixed regimes were observed in all cases. The presence and size of particles did not affect the early evolution of the interface shape as the initial and receding contact angles of the evaporating droplets were consistent with values measured for DI water. This agreement suggests that the particle concentration at the contact line was not high enough to impede depinning of the contact line when the receding contact angle was achieved. This is consistent with previous work, which suggests that it is more difficult for particles to impede contact line dynamics on hydrophobic surfaces [48,49].

The formation of the ring pattern in the nanoparticle cases appears to be the result of contact line self-pinning that occurs in the final 10% of drying time. While the evolution of the contact diameter in all three cases was similar for the first 90% of the evaporation time, the diameters of the nanoparticle-laden droplets remained constant over the final 10% of evaporation time (Figs. 7(b) and 7(d)). Conversely, diameter decreased monotonically for $1.1 μm$ particles (Fig. 7(f)). This is consistent with the mechanism for self-pinning described by Weon and Je [50], which predicts that nanoparticles are more likely to cause self-pinning than microparticles because they are (i) more numerous at the same volume fraction and (ii) able to get closer to the contact line. These were the only cases where evidence of repinning was present.

Here, $Rpn$ and $tpn$ are the radius and time at which repinning was observed, respectively. The radial velocity of the $25 nm$ particles increased from $0.51 μm/s$ to $2.49 μm/s$⁠, while the velocity of $100 nm$ particles increased from $0.47 μm/s$ to $2.71 μm/s$⁠. This suggests that particles had not been captured by the interface when the contact line repinned and were carried to the contact line by the increased radial velocity. The characteristic velocity after repinning is presented as a dashed circle in Fig. 6. If the pinned radial velocity is used in these cases, their deposition patterns can be described using the ratios of interface velocity and radial velocity to the diffusion rate.

Results in Fig. 6 suggest that a single parameter may be used to understand the deposition profile of the cases here and in Ref. [38]. Since both axes in Fig. 6 are scaled by the diffusion rate, the ratio of interface velocity to radial velocity (⁠$ui/urad$⁠) appears to characterize the deposition patterns for the cases examined here (Fig. 8). When the interface velocity is greater than the radial velocity, particles are captured on the interface and deposited uniformly on the substrate. When the radial velocity is greater than the interfacial velocity, particles migrate away from the interface and collect at the contact line. A transitional regime between these two patterns appears to exist when these velocities are similar. Here, a ring is present but it is accompanied by localized areas where a large number of particles have deposited on the surface.

It is important to note that the results here examine cases where DLVO forces are repulsive and Marangoni effects are thought to be minimal. In cases where these effects are more significant, it may be possible to generalize the results of this work by incorporating the contributions of these effects into the radial velocity of the particles or by adding an additional axis to the phase diagram proposed here.

This investigation experimentally examines the role of interface capture on colloidal transport and deposition in evaporating droplets by investigating evaporating droplets over a range of interfacial velocities, radial velocities, and particle diffusion rates. It is particularly concerned with the previously identified transitional regime where the magnitudes of the interface velocity and particle diffusion rate are similar. Results presented here suggest that while both uniform and ring patterns can be observed in this regime, the resultant pattern can be explained by studying the ratio of the radial velocity to diffusion rate. Uniform depositions were observed when the ratio of the interface velocity to the diffusion rate was greater than the ratio of the radial velocity to the diffusion rate. Conversely, ring depositions were observed when the ratio of radial velocity to diffusion was greater than the ratio of interface velocity to diffusion. A transitional pattern with high deposition at the periphery and nonuniform central deposition was observed when the ratios were similar. The transitional cases identified by previous work also fell in this regime. Since both characteristic ratios examined here are scaled by particle diffusion rate, the single ratio of interface velocity to radial velocity can be used to characterize the deposition patterns.

The cases examined in this investigation all have a repulsive DLVO force and Marangoni effects are assumed to be negligible. It may be possible to generalize the characterization method proposed here by modifying the characteristic radial velocity to include attractive DLVO and Marangoni effects. Further investigation is required for these cases.

The authors gratefully acknowledge the assistance of several professors and students at RIT: Wilkie Olin-Ammentorp, Hee Tae An, and Rakesh Chokanathan for substrate fabrication; Peter Dunning for substrate preparation and evaporation trials; Hyla Sweet, Evan Darling, Lauren Heese, and Teresa Zgoda for fluorescence microscopy; Anju Gupta for zeta potential measurements; and Michael Haselkorn, Patricia Iglesias Victoria, and Hong Guo for roughness measurements. The authors would also like to thank Daniel Attinger (Iowa State University) for valuable insights during conversations at the NTNU International Workshop for New Understanding in Nanoscale/Microscale Phase Change Phenomena. This material was based upon the work supported by the ADVANCE RIT grant, which is funded through the National Science Foundation under Award No. HRD-1209115. The authors also gratefully acknowledge the support of the Kate Gleason College of Engineering and the School of Mathematical Sciences at the Rochester Institute of Technology.

$A$ =

area

$D$ =

diffusion constant

$dep$ =

subscript denoting the deposition pattern

$DLVO$ =

subscript denoting DLVO effects

$evap$ =

subscript denoting evaporation time

$H$ =

height of a droplet

$i$ =

subscript denoting the interface

$I$ =

fluorescence intensity

$kB$ =

Boltzmann's constant

$m$ =

index for sectors of an image of a deposition pattern

$M$ =

number of sectors in a deposition image

$n$ =

index for radial section of a sector in an image of a deposition pattern

$N$ =

number of radial sections of a sector in an image of a deposition pattern

$p$ =

subscript denoting the particle

pn =

subscript denoting the repined state

$r$ =

radial coordinate

rc =

subscribe denoting the receding contact angle

$R$ =

radius

$t$ =

time

$T$ =

temperature

$u$ =

velocity

$vdW$ =

subscript denoting van der Walls effects

$x$ =

characteristic length of travel of a particle due to diffusion

$x˙$ =

characteristic diffusion rate (value of $x$⁠, when $t$ is $1 s$⁠)

$0$ =

subscript denoting the initial value of a property

$η$ =

viscosity

$θ$ =

contact angle

$τ$ =

dimensionless evaporation time (⁠$t/tevap$⁠)