Experimental study of liquid droplets impact on powder surface: The application of effective dimensionless parameters in analysis

Spreading dynamics of liquid droplets impacting onto powder bed are experimentally studied using high-speed photography. Dimensionless numbers—We, Re, the modified We∗ and Re∗ corrected by substrate deformation—are used to analyze the impact behaviors of droplets. The spreading time and the maximum spreading factor are further analyzed. The spreading time is accurately described by a universal scaling law that is obtained from the modified dimensionless time vs. the effective Weber number (We∗), and the maximum spreading factor is found to follow the modified classic scaling law β f We∗, Re∗ .


Introduction
Droplet impact on powder beds occurs extensively in industrial and technological applications, such as granulation in the pharmaceutical industry [1,2] ; granule formation by spraying for powder coating and spray drying [3,4] ; and in three-dimensional printing processes [5][6][7] . It is important to understand the dynamics of the droplet impact process on powder beds, which is critical for controlling such processes. Various impact behaviorse.g., spreading, retraction, rebounding, and splashing-and the resulting crater morphology as well as, liquid marbles have been investigated in detail [8][9][10] . The behaviors of droplets upon impact have been studied comprehensively by experimental [11,12] , theoretical [13,14] , and numerical methods [15] .
In the experimental and theoretical study, the impact dynamics of liquid droplets and powder surfaces have been systematically investigated. The influences of initial impact velocity and liquid properties on spreading time and maximum spreading have been examined. By using modified Weber number and Reynolds number, we apply empirical expressions for non-deformable solid substrate to powder bed, and reveal their similarities for maximum spreading and spreading time. Scaling analyses were adopted to reveal three characteristic parameters.

Experimental setup and mathematical models
Alumina beads were supplied by the manufacturer Hebei Xinda Alloy material, China. The initial packing fraction, Φ, was reproducible for successive repeat experiments. And in the experiments, Φ is controlled in the range of 0.58 -0.60 by using tight compression.
Liquids and their properties used in the experiments are displayed in Table 1. Dimensionless parameters used in the analysis are Weber number , and the Reynolds number : / ; / ; where , , , and are, respectively, liquid density, liquid dynamic viscosity, droplet impact diameter, and liquid surface tension factor; g is the gravitational acceleration and 2g ℎ is the velocity at the time of impact. The droplet is released onto powder bed from different heights ranging from 3 to 600 mm, resulting in impact velocity range from 0.2 to 3.2 m/s. Droplets can be considered as sphere upon impact. The equilibrium contact angle, , is also measured.
The experimental setup (Fig. 1) includes a high-speed video camera (Qianyanlang 5KF20), which can capture the impact, spreading, and penetration in a single video sequence.  is typically a few milliseconds, which is of the same order of magnitude as the characteristic advective time [16] / 2 or the capillary-inertial time [17] ρ /8σ . . Fig.  2(b) shows that t s decreases sharply as the impact velocity increases for * ≤ 40 but slowly decreases for * 40. Since and ignore influences such as viscosity and surface wettability, a modified capillary-inertia time /8 . is adopted to study the spreading [18] , where d max is related to liquid viscosity and wettability. Fig. 2(c) shows a loglog plot of normalized spreading time / vs. We * . This shows that the function / * gives a good fit to the experimental data. The fitting coefficient and the exponent for different viscosities are shown in Fig. 2(d)

Maximum spreading factor βmax
An effective Weber number * / 2 , where Z m is the maximum vertical deformation measured [11] , is used to replace We. It has been shown that We * collapses the maximum spreading data for various packing densities [19] , indicating that We * can also characterize the effect between liquid droplets with various viscosity and the powder bed is not applicable to the analysis of viscosity effect in this paper, Fig. 4 also proves this point) respectively for various viscosity liquids. As shown in Fig. 3 (a) and (c), a scaling law behavior, ~ * , is observed at We * > 1. It comes as no surprise to see that β max increases with We * . The data set exhibits three other distinguished features: (1) The maximum spreading factor is suppressed with increasing viscosities (different color symbols in Fig. 3 (a)). (2) In the doubly logarithmic scale, there are two asymptotes in the extension of data: the upper is We *1/4 and the lower is We *1/10 . (3) The value of α 1 decreases with increasing viscosities (Fig. 3(c)). On the other hand, it seems that the spreading is dominated by viscous force within the liquid for all experimental data. In Fig. 3(b), β max grows with * following the power law, ~ * .The exponent α 2 is ~0.5 for lowly viscous liquids with 10 mPaꞏs, and it decreases to ~0.33 for liquids with 1430 mPaꞏs. Because of the quantitative relation We 1/10 ~ D -1/4 Re 1/5 , a law of the type β max ~ We *1/10 leads to β max ~ Re 1/5 . According to the analysis of the transition to a viscous regime, we thus define the effective number * ≡ * / * / for deformable substrate ( ≡ / / for non-deformable substrate [20] ). The transition between the capillary regime and viscous regime is plotted in Fig. 4(a), where the dimensionless viscous extension / * / is plotted as a function of * . While for ϯ ϯ / ϯ / in Fig. 4(b), the result is surprisingly that the transition is opposite to Fig. 4(a)-the viscous droplet spreading changes from viscous regime to the capillary regime as ϯ increases. Consequently, ϯincorporates the bulk wettability of a granular substrate-doesn't fit to interpret the influence of liquid viscosity on spreading. The use of * and * reveals the hidden similarities between a droplet impact on deformable substrate and that on a non-deformable substrate for maximum droplet spreading. Same as the non-deformable substrate, the influence of internal inertial flow at maximum spreading phase also exists, and causes the maximum spreading smaller than the theoretical value.

Conclusions
We have performed an experimental study of various liquid droplets impacting alumina powder beds. Weber numbers ( ) and Reynolds numbers ( ) were obtained by adjusted the droplet fall height. Ten liquids were used that provided various viscosities and various surface tensions. Then we adopted spreading time t s and maximum spreading factor β max to quantitatively analyze the spreading behavior of the droplets after impacting the powder bed. The modified . Besides, the effective weber number We * , replacing traditional weber number, is adopted to attain the scaling law ~ * , and α 1 decreases with the increasing viscosity. The same thing happens for the scaling law ~ * . The capillary regime considered implies a small P * (P * = We * /Re *4/5 < 1), the viscous regime refers to large P * (P * = We * /Re *4/5 > 10). Thus, the use of * and * reveals the hidden similarities between a droplet impact on deformable substrate and that on a non-deformable substrate for maximum droplet spreading. This study is supported by Grants from National Natural Science Foundation of China (No. 51876071).