1 Introduction

Granular matter has strongly nonlinear dynamical characteristics, and has been widely used in agriculture, pharmaceutical, and chemical industry^[1-2]. Granular complex solid-liquid-like dual states and the transition of solid-like and liquid-like states under vertical vibration make the studies on granular materials more and more popular in recent years. When vertically vibrated, granule matter can exhibit a variety of behaviors such as granular convection^[3-4], oscillons^[5], arching^[6], size separation^[7-9], and surface waves^[10-11]. A new phenomenon about granular matter has been found in recent experiments^[12-16] that the granular particles in a static container can climb up to a certain height along a vertically vibrating tube inserted into the granular bed or if the tube is static and the container vibrates vertically. Such a phenomenon is the reminiscent of the capillary phenomenon of liquid, and thus we can call it the capillary phenomenon of granular particles. Liquid capillary phenomenon is caused by surface tension, which is a consequence of the intermolecular force^[17]. Obviously, this explanation does not apply to the capillary phenomenon of granular particles, because the intermolecular force between the granules is so smaller than the gravity of particles and the granular contact force during impact that it can be neglected.

In 1990s, Akiyama and Shimomura^[18-19] found that there was a difference in the surface levels between the granules in the container and those in the tube if the tube was stationary and the container vibrated. About twenty years later, Liu et al.^[12] concentrated on the height difference, and conducted the experiments further. They found that the vertically vibrating tube also caused granules to climb up to an equilibrium height h_eq. Additionally, when the vibration strength was not strong enough, there would be a critical height h_c for the granular capillary phenomenon. This means that if the initial granular height h₀ is less than h_c, granules will fall into the bed, and if h₀ is over h_c, granules will climb. Moreover, with the increase in the vibration strength, the critical height h_c would decrease down to zero. They explained that there would be a cavity when the tube moved upwards. However, they did not present the effects of the experimental parameters on the phenomenon. In the experiment of Liu et al.^[13], granules fell to the equilibrium height if the initial granular height was higher than h_eq, indicating that the equilibrium height h_eq was decided only by the experimental parameters but not the initial granular height. By fitting the experimental curve, they got a formula to calculate h_eq. From their fitting formula, the effects of some parameters could be understood mathematically. However, they did not give the physical explanation for these effects. To have a better understanding in the physical origin of this phenomenon, some researchers numerically simulated the mesoscopic process for the granular climbing. With the help of the discrete element method, Fan et al.^[20] concluded that the effect of granular capillarity was caused by the convection of the granules in the container, while Xu et al.^[21] concluded that the main cause was the particle compaction in the middle section of the tube.

There is significant interest with regard to the investigation on the effects of the experiment parameters on the capillary phenomenon of granular particles in recent years. Zhang et al.^[14] studied tubes and granules with different dimensions, and concluded that if the inner diameter of the tube increased, the granules would climb higher but more difficultly. This means that we need stronger vibration strength to motive the granules to climb if we use a tube with a large diameter. They also showed that granular capillary was harder to happen when the granular diameter increased. Later, Zhang et al.^[16] found that h_eq non-monotonically depended on the vibrating frequency of the tube, and had a peak value at a certain frequency. Different from the frequency, h_eq would increase when the vibration amplitude increased. The tube insertion depth h_in also had a positive effect on the experimental result. However, the effect fell when h_in reached a saturation value. Liu et al.^[15] also studied the effect of the shape of the tube. They found that climbing in the tapered-tip tube was easier due to the force chain.

If a vertically vibrating tube is inserted into water, besides the capillary phenomenon of liquid, the surface of water will only have some waves and no climbing motion will happen^[12]. Different from water, granules can bear larger shearing force, which plays an important role when granular particles climb. The shearing force between granules or granules and tube walls also called friction force, which can cause the granular particles to move synchronous with the tube. That is to say, when the tube moves upwards, the granules contacting with the tube will move upwards, too. Then, the region in the bottom of the tube will become sparse because of the granular upward motion, and the granules outside the tube will flow in under the effect of the density gradient. Similarly, when the tube moves downwards, some granules inside the tube will flow out. In the beginning vibration stage, the volume of the inflow granules V_in in a vibration period will be more than the volume of the outflow granules V_out. Therefore, granules will climb in general. When reaching a certain height, at which V_in is equal to V_out, however, granules will arrive at an equilibrium state and the height will stop increasing.

The objective of this paper is to try to give a quantitative physical model to explain the granular flow near the bottom of the tube and the capillary phenomenon of granular particles, and to investigate the effects of the experimental parameters. In Section 2, V_in and V_out are calculated, and the flow is described based on some assumptions. In Section 3, the results with different parameters are contrasted to prove the accuracy of the model, and the effects of the main factors on this phenomenon are discussed. In Section 4, some experiments are presented in order to prove the assumption proposed in the model, that granules move in the same form as the tube. In Section 5, how to utilize this technology is discussed for a possible application during asteroid exploration. Finally, the conclusions are summarized in Section 6.

2 Modeling of the phenomenon

Because the connection between the granular bed and the tube is only in the bottom of the tube, it is vitally important to have a clear mind of granular flow in this region. Therefore, we can look for a way to solve this problem by calculating V_in and V_out. At first, we will discuss how to calculate V_in. Since the tube vibrates vertically in the form of sine, we assume that the granules at the bottom of the tube move in the same form as the tube moves upwards. Due to the loose structure in this region and gravity of granules, the motion amplitude of the granules is smaller than the vibration amplitude of the tube. If the motion of the tube bottom is expressed as

according to the above stated assumptions, the granular movement formula can be obtained as follows:

(1)

where s₀ represents the initial granular position relative to the tube. If s₀= 0, s|_t=0=s_tube|_t=0. This means that this formula stands for the granules having the same initial height as the bottom of the tube. Similarly, if s₀> 0, s|_t=0> s_tube|_t=0. This shows that granules are inside the tube at the beginning. Because the tube moves faster than the granules, the granules in the tube will leave the tube at the time t₀. According to

we have

(2)

which means that the granules with the initial height s₀ will leave the tube at the time t₀. Since there is a one-to-one match between s₀ and t₀, t₀ can also stand for a layer of granules. After leaving the tube, granules are only affected by their gravity. Therefore, the motion formula of the granules after leaving the tube can be established as follows:

(3)

In Fig. 1, the separation point stands for the time when granules leave the tube, and then they do a free falling motion until reaching the return point.

Based on the motion form of the granules, let us work out the variation of the granular bulk density ρ. ρ is a function of time and space, and can be expressed as ρ(t, s₀) or ρ(t, t₀). At the beginning, when the tube moves upward, or at the end when the tube moves downward, granules are dense and well-distributed. Therefore,

where ρ₀ is constant. If we do not consider the inflow from the bed to the tube, we have

(4)

For the inflow to the tube, we give an assumption that, in a unit period, the density of granules flowing into the tube is constant, kρ₀. This assumption is proposed only for the concision of calculation, and will be discussed later. Based on this assumption, we get the final expression of the density as follows:

(5)

Then, V_in can also be calculated as follows:

(6)

where S is the cross-sectional area of the tube. The integration interval of Eq. (6) is hard to be established in mathematical form because it is necessary to consider whether granules will fall back on the bed or crash into the granules above when the tube is slowing down. In Fig. 2, the domain, in which the value of V_in/S is positive, is the right integration interval. The upper limit is formed because the granules, which just leave the tube, will crash into the granules above and then return to the tube again when the downward acceleration of the tube is larger than g. From this picture, we can find that the upper limit has an effect in the range from 0.3T to 0.5T, where T is the vibration period. The low limit is formed because the throw-up granules will fall back into the bed under the effect of gravity. Obviously, when the leaving velocity is larger, the time at which there is an inflow will be longer and the volume of the inflow will be larger.

Different from the loose structure of the tube bottom during its upward motion, the particles would be dense when the tube moves downwards. Hence, we will calculate V_out with a similar method to that used in Ref. [22]. In Ref. [22], the weight supported by the bottom of the silo was only a part of the total weight of the granules, i.e., most weight was absorbed by the silo wall. This is different from our problem where there is a downward acceleration. Because the vibration frequency is higher and the time for the downward motion of the tube is short, it is expected that the acceleration associated with the downward motion of the granules is constant, expressed as a. When moving downwards, the granules are dense and the density of the granules does not vary. Considering a layer of granules, we have

(7)

where t_w is the shear force by the tube wall, and σ_zz is the normal stress in the direction of the height. Moreover, there are two restrictions along the tube wall, i.e.,

(8)

(9)

where σ_rr is the normal stress in the radial direction, μ_w is the friction coefficient between the tube wall and the granules, and γ is the ratio of σ_zz to σ_rr. On the top of the granules in the tube, there is no area force in the vertical direction, i.e.,

(10)

At the bottom of the tube, the pressure is proportional to the insertion height h_in. Considering the granular pressure in the container, we have

(11)

From Eq. (11), we have

(12)

As shown in Fig. 3, a increases with the increase in h, which is consistent with our intuition that granules will decline more quickly when the granular gravity increases. When h> 0.4 m, a reaches an upper limit g, and then nearly keeps constant. It is similar to such a phenomenon that a small wood block on a fast moving board will also move forward but its acceleration is limited by the friction force between block and board. Therefore, the gravity acceleration g is a similar upper limit in the problem we discuss.

At the beginning, the tube moves faster than the granules, while the granules move downwards at the acceleration a. Later, the tube slows down, while the granules move in consist with the tube, i.e., the granular velocity is equal to that of the tube. Therefore,

(13)

where t^* (0 < ωt^* < π) is decided by

which means that the granular velocity is equal to that of the tube. Similar to a, V_out also increases with the increase in the granular height h. When h> 0.4 m, V_out almost arrives at its limit (see Fig. 3).

3 Results

In the proposed model, the relationship of B and h has not been given. However, the relationship could decide the values of V_in with different granular heights. The parameter B can stand for the granular sparsity at the bottom of the tube. The larger B is, the sparser the granules are, and the larger V_in is. According to Janssen's model^[22], the pressure at the bottom of the tube is proportionate to

Therefore, B should satisfy

(14)

where C is an empirical constant, and is set to be 0.505^[13]. From Eq. (14), we can see that B decreases with the increase in h, which means that the granules at the tube bottom will be denser when the granule amount of the inner tube increases. When the granular gravity in the tube increases, the pressure at the bottom of the tube will increase, and the granules in this region will be denser. Moreover, V_in decreases with the increase in h, and the granules will arrive at an equilibrium state when V_in is equal to V_out (see Fig. 3).

To prove the accuracy of the new proposed model, we give some examples. Firstly, we verify the relationship of the granular climbing height h and the time t by comparing the calculation result with the experiment in Ref. [13]. From Fig. 4, we can see that the equilibrium height h_eq is about 0.6 m under the same condition with that in Ref. [13] (A = 8.6 mm), no matter what the initial height we choose. When the initial height h₀ is zero, the time for the granules to arrive at the equilibrium height is about 50 s. If the initial height is above the height of the equilibrium position, granules will decline to h_eq from the calculation result. When we use a smaller amplitude, e.g., A = 5.7 mm, the equilibrium height will be smaller.

Then, we investigate the effects of the main factors on the equilibrium height with this method. From Fig. 5, we can see that there is an approximately linear relationship between A and h_eq, which is in agreement with the experimental result in Ref. [16]. According to our physical model, when the vibration amplitude increases, the amount of the inflow granules increases, while that of the outflow granules keeps constant, i.e., V_in increases while V_out hardly changes. Figure 5 shows that the equilibrium height will be larger if we choose a higher frequency. Note that the granular declining acceleration a is limited by the friction force and hardly increases with the increase in the frequency. The time at which the tube and granules decline has been shortened, resulting in the great decrease in the outflow volume of granules. The frequency effect on V_in is less than that on V_out. Therefore, we can get a higher equilibrium height when the tube vibrates at a higher frequency. From Ref. [16], we can see that the frequency really has a positive effect on this climbing within a certain range. However, there is a negative effect in case of exceeding a certain value, which cannot be explained in our model. Perhaps a further study is needed to consider this condition. From Fig. 6, we can see that the inner diameter of the tube also has a positive effect on the equilibrium height. This relationship can also be seen in Ref. [14]. From this formula, we can see that a larger tube diameter corresponds to a smaller pressure at the bottom of the tube under the same conditions. Moreover, a smaller pressure decides a looser granular region for granules. Therefore, granules can climb higher when the tube diameter increases within a certain range.

4 Experimental validation

To prove the assumption in Section 2 that granules will move in consist with the tube and there is no phase difference, we use a 250 fps high-speed camera to observe this phenomenon and take almost twenty photos to record the granular height in a vibration period. The silo used in the experiment is a cylindrical container with a diameter of 110 mm and a granular layer with a height of 80 mm, where the granular particles have an average diameter of 0.5 mm and a bulk density ρ₀ of 1 210 kg·m³. The tube we used is made of transparent plexiglass, and has an 11 mm inner diameter and a 2 mm thickness. A motor drives the tube to move in the form of sine. The frequency in the experiment is 13.5 Hz and the amplitude is about 5 mm.

We use the high-speed camera to record the granular height in the tube when the granules are climbing or arrive at an equilibrium state. Figure 7 shows the situation that the granules are climbing. We can see that the granules move up and down along with the tube, but have an increase in general. We use a sine function to fit the experimental data when the granules are in an equilibrium state (see Fig. 8). The fitting function we get is

and the motion function of the tube is

Little difference of frequency, perhaps caused by experimental and fitting errors, can be neglected. Therefore, we can conclude that the granules and tube move with the same period. However, there are obvious differences in the phase and amplitude. Compared with the tube motion, the motion of the granules has a smaller amplitude and a phase lag of 90°. Our assumption above is that granules move in consistence with the tube and there is no phase difference. Obviously, a smaller amplitude and a phase lag are not what we expected. According to our observation, the differences in the phase and amplitude are caused by the effect of the granular surface. The particles on the top of the granular column can be thrown up with the only force of gravity when the whole granules move up and the downward acceleration is more than g under the effect of the friction force of the tube. An upward velocity of the granules on the surface of the whole granular column still exists when the whole granules stop moving down in this situation. Therefore, we can see a lag effect when the height of the granular surface is recorded. This explanation also suits for the difference in the amplitude.

Due to the aforementioned effects, it is difficult to make a judgment on our assumption if we continue to observe the height of the whole granular column. Therefore, we attempt to observe the motion of granules in the middle of the granular column by using colored granules in this region.

In Fig. 9, the shadow region stands for the colored granules, and the bright region stands for the granules without color. From the figure, we can see the shadow region moves up from the picture. Because of the effects of the tube wall, some colored granules will stay behind when climbing, which causes the length of the shadow region to become longer as time goes on.

We record the height of the top colored granules in Fig. 10 in order to exclude the effect of the granules staying behind. Then, we fit the experimental data with a linear function and a sine function. The sine function is

and the motion of the tube is

From these functions, we can see that both the amplitude and phase between the granule motion and the tube motion are almost the same. This means that our assumption is reasonable to some extent.

5 A possible application to the asteroid sampling

The capillary of granular particles could provide a possible way to transport granules. In aerospace engineering, the problem how we take samples on an asteroid regolith is sometimes discussed before the structure of the spacecraft with the task of exploring the asteroid is designed. For example, the JAXA Hayabusa-2 spacecraft, which launched in 2014 and has successfully arrived at the near-Earth asteroid 162173 Ryugu, aims at taking samples from the surface of Ryugu. Ryugu is near-spherical, has an effective diameter in the range from 850 m to 880 m. Its grain size ranges from 1 mm to 10 mm^[23]. Due to the weak gravitation (about 7×10^-4 m·s²) on the surface of Ryugu, the traditional method to take samples is difficult to realize, in which we can make an attempt to design a machine to complete this job based on the granular capillary. What we need to do is to construct the condition of granular capillary by fixing this machine on the asteroid surface and inserting the tube with a proper length into the soil of the asteroid. Through vibrating the tube, the particles of the asteroid regolith will climb until they flow into the sampler along the tube. In this process, the sampling rate and efficiency are what we care about. The sampling rate V_t can be considered as the volume of the granules flowing into the sampler in a vibration period, and the sampling efficiency is the ratio of the sampling rate to the vibration power or vibration strength, which is expressed as

From Fig. 11, we can find that the sampling rate V_t, which is also called the volume of transporting granules, linearly increases with the increase in the vibration amplitude. This is because that V_in increases with the increase in A while V_out varies little. Additionally, the sampling efficiency η arrives at a saturated state when the amplitude is over 10 mm. Figure 12 shows that the frequency also has a positive effect on sampling but the effect falls with the increase in the frequency because both V_in and V_out are affected by the frequency. There is a competing relationship in the final results of the frequency effect. A most effective frequency exists, at which we can transport more particles with the same vibration energy (see Fig. 12). We can see that the most effective frequency decreases with the increase in the vibration amplitude. The effects of the vibration amplitude and frequency on the sampling rate and efficiency facilitate the choices of the suitable amplitude and frequency when particles are transporting with the method of granular capillary.

6 Conclusions

A theoretical model is given to explain the capillary phenomenon of granular particles by analyzing the flow at the bottom of the tube. The volumes of the inflow and outflow are calculated. The equilibrium heights under different conditions are obtained, and the effects of the vibration amplitude, the frequency, and the diameter of the tube are discussed. The experimental results show that the middle of the granular column almost coincides with the tube, which, to some extent, gives support to the assumption we propose. However, the conditions when granular capillary can happen are still not solved in this model, and need to be further studied.