⇦Chinese Physics Letters, 2019, Vol. 36, No. 9, Article code 094501 Phase Transition and Critical Phenomenon Occurring in Granular Matter * Yao-Dong Feng (冯耀东)1,2, Tao Su (苏涛)2, Qing-Fan Shi (史庆藩)1, Gang Sun (孙刚)2,3** Affiliations 1Department of Physics, Beijing Institute of Technology, Beijing 100081 2Beijing Key Laboratory for Nanomaterials and Nanodevices, Institute of Physics, Chinese Academy of Sciences, Beijing 100190 3School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190 Received 9 May 2019, online 23 August 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 10875166 and 11274355.
**Corresponding author. Email: gsun@iphy.ac.cn Citation Text: Feng Y D, Su T, Shi Q F and Sun G 2019 Chin. Phys. Lett. 36 094501    Abstract We investigate the granular flow states in a channel with bottleneck by molecular dynamics simulations. Our study is restricted only on a selected key area rather than on the whole system to focus on the flow properties of a single granular state. A random force field is introduced to control the granular temperature. It is also pointed out that the flow rate in the granular flow can be correlated with the pressure, which leads us to carry out a comprehensive study similar to the classical study for general liquid-gas phase transition. Our results show that the dilute flow state and the dense flow state of the granules are similar to the gas state and the liquid state of general substances, respectively, and the properties of phase transition and critical phenomenon are also similar to those occurring in general substances. DOI:10.1088/0256-307X/36/9/094501 PACS:45.70.Mg, 87.15.Ya, 75.40.Mg © 2019 Chinese Physics Society Article Text We know that substances made up of a large number of molecules appear in forms of solids, liquids, and gases, and the detailed forms are related to the environment in which they are located. The most basic parameters describing the environment are temperature and pressure. As the temperature and pressure change, the form of the substance can also change (such as from a gaseous state to a liquid state), and some physical quantities vary discontinuously during the change of the form of the substance, which is generally called a phase transition.[1,2] In solid state, each molecule has an equilibrium position. Meanwhile, in liquid and gaseous states, molecules do not have equilibrium positions. The main difference between liquid state and gaseous state is that the liquid is relatively dense and substantially incompressible, while the gas has a relatively lower density and is compressible. However, this is only a primarily inaccurate statement. Strictly speaking, there should be a phase transition between liquid state and gaseous state, and the density of the substances is discontinuous at the phase transition point. This discontinuity will exist up to a critical point, exceeding the critical point liquid and the gas will no longer be distinguishable. Granular matter[3–5] is composed of macroscopic particles (i.e., granules). The interaction between them is similar to that of microscopic particles (i.e., molecules); for example, both of them can meet momentum conservation. From the aspect of energy, granules will have a certain energy loss during the collision process, and usually the loss is weak, thus the difference of the collision between the molecules is limited. Therefore, when the granules are sparse and the collision frequency of the granules is not very high, the state of the granular matter is very close to the state of the gas composed of molecules, which is generally called a granular gas. There have been many studies of granular gases and their properties are relatively well understood.[6,7] However, when the density of the granules (or packing fraction) is increased, the state of the granules changes significantly. In a certain environment, the state of the granular matter can become a substantially incompressible dense state, which is similar to the state of liquid composed of molecules. In the granular flow, the two granular states exhibit as two different flow states; i.e., dilute flow state and dense flow state.[8–10] In a previous study,[8] we have investigated the granular flow in a channel with a specific bottleneck by both experiments and simulations. We found that there is a crucial relationship between the local flow rate and the local packing fraction in a selected choke area under the fixed particle number condition, which can define the granular flow state clearly. In the relationship, the flow rate has a maximum value at a moderate packing fraction. When the packing fraction is lower than this moderate value, the flow rate increases with the packing fraction. When the packing fraction is higher than this value, the flow rate decreases with increasing the packing fraction, and eventually stops at a maximum packing fraction. All states are stable under the fixed particle number condition, which allow us to carry out the statistical calculations for a long time. However, under the fixed inflow rate condition, which is more general condition in experiments, some states become metastable or even unstable. By considering the feedback mechanism of the relationship, we have divided the granular flow state under the fixed inflow rate conditions into stable dilute flow state, metastable dilute flow state, unstable dense flow state, and stable dense flow state. Here the terms of stable, metastable and unstable are directed against the properties under the fixed inflow rate condition. Using the nature of these four states, we successfully explained the sudden change of the flow rate observed in our experiments. The phenomenon of a sudden change in physical quantity usually leads us to think about a phase transition. From the typical characteristics of the dilute and the dense granular flow state, the stable dilute flow state is close to the gaseous state in the substances made up of molecules, and the stable dense flow state is close to the liquid state. Thus, there is reason to compare the transition from the dilute flow state to the dense flow state with the phase transition between the gas and liquid state. However, in granular matter, the fluctuation of kinetic energy, which is sometimes defined as the granular temperature,[11,12] is difficult to control due to the loss of energy during the collision of the granules.[3] In addition, the properties of the granular matter are spatially non-uniform, thus the physical quantities, including granular temperature, may vary with space. Generally, the granular temperature is lower at positions where the collisions happen frequently due to the loss of energy. Recently, to control the granular temperature, we introduced a normal distributed random force field in the molecular dynamics simulation study of the granular matter.[13] In the study of a quasi-one-dimensional granular flow system, it was found that the random force field can increase the fluctuation of the velocity while remain other physical quantities almost unchanged. This result gave us a way to control the granular temperature in the simulation study of the granular matter, so that we may observe the transition of the granular matter from a dilute flow state to a dense flow state at different granular temperatures. In addition to temperature, another external environmental variable in the solid-liquid-gas phase transition of a general substance is the pressure. In the case of the granular flow, we can correlate the flow rate of the granular flow with the pressure. Because the granular flow rate can be expressed as the flow rate density multiplied by the acreage of the cross section, and the flow rate density is defined as the product of the average packing fraction and the average velocity, which can also be considered as momentum density. Therefore, we suggest to study the granular flow in the flow rate-granular temperature space, and the results can be compared with the properties of the general substance in the pressure-temperature space. In this Letter, we study the granular flow through a channel with a specific bottleneck by molecular dynamics simulation. Our simulation study was carried out in a two-dimensional space. The shape of the granules was circular with a diameter of $d_0=2.0$ mm and a mass of $m_0=0.0327$ g. In the simulation, we consider the translation of the circular granules in both directions in the two-dimensional space and the rotation perpendicular to the two-dimensional plane. The system we studied is in a two-dimensional channel with small exit (Fig. 1). In the channel, there is gravity ($g \sin 20^{\circ}$) along the channel direction, where $g$ is the gravity acceleration. The particles are produced on the entrance at the top and flow out from the exit at the bottom. The height and width of the channel are $L_{\rm s}=400.0$ mm and $W_{\rm s}=60.0$ mm, respectively, and the width of the small exit is $D=14$ mm. To control the granular temperature of the granular flow, a random force field is applied in a rectangular region of height $H_{\rm r}=25.0$ mm above the exit. An additional calculation is maintained below the exit with a height $H_{\rm s}=25.0$ mm. To take into account all kinds of granular flow state (including the dense flow state), a soft sphere approach is used to simulate the collision between granules. The detailed interactions form between granules or side wall are the same as those we used in the previous work,[8,13] but in addition to the gravity and the collision between granules or side wall, a normal distributed random force (${\boldsymbol f}$) is considered to apply to each granules at every time step $\Delta t$. The average of this normal distributed random force is zero in both $x$ and $y$ directions ($\bar{f_x}=\bar{f_y} =0$), and the standard deviation is $F_{\rm r}$. Here $F_{\rm r}$ represents the strength of the random force field, and is an important new parameter in this study. Through molecular dynamics simulation, the evolution of each granule can be calculated from its generating at the entrance to the exiting at the outlet. However, as a study of granular flow, we need to calculate the corresponding macroscopic physical quantities concerned with fluid mechanics, including average packing fraction, average granular velocity, fluctuations of the granular velocity, and granular flow rate, etc. The calculation of these quantities can be performed by statistical analysis for all the granules in the selected area.[8,13]
cpl-36-9-094501-fig1.png
Fig. 1. Illustration of the geometry of the channel with a bottleneck. The selected fan-shaped areas around the exit are also sketched, where the granular state determines the whole outflow rate of the system and the main calculations are carried out in this area.
Figure 2 shows the distribution of the packing fraction and the speed (absolute value of the velocity) of the granules over the entire channel for a fixed granule number of 700, which belongs to a moderate dense flow state, without random force field. We can clearly see that this system is extremely non-uniform and the physical quantities in different places can vary seriously. For example, above the outlet, the particles accumulate more, the average packing fraction is larger, and the granules move slower, by contrast, at most of the upper part of channel, there is no particle accumulation, the packing fraction is small, and the granules move faster. We can also see that in some areas the granules rarely arrive (such as below the baffle of the exit), thus the calculation of the density weighted average becomes inaccurate, which can be found by the asymmetric speed distribution below the exit (Fig. 2).
cpl-36-9-094501-fig2.png
Fig. 2. Distribution of the packing fraction (a) and the speed of granule (b) over the entire channel.
Although Fig. 2 shows the distribution of the packing fraction and the velocity of granule over the entire channel, it is not very helpful to explain to the sudden change in the flow rate observed in the experiments. In the previous work,[8] we have pointed out that the key area for determining the outflow rate is the fan-shaped area above the exit (the thickness of the fan-shaped area is set to $d=4d_0=8.0$ mm, see Fig. 1), and the discontinuous change in outflow rate observed by the experiment can result in the change of the flow state in this area. Figure 2 also shows that the packing fraction and the speed of the granules do not change much in this area. Further detailed study can show that the velocity of the granule in this area is almost along the normal (i.e., centripetal) direction, thus we can approximate that the normal (i.e., centripetal) and tangential (i.e., cambered) components of the velocity of the granule (it is notable that the components are not indicated by $x$ and $y$) are basically the same in this area. Thus, we can use these quantities as a basis to describe the flow properties of the whole channel. The calculation of these physical quantities is performed for the entire fan-shaped area. Because the area of the fan-shaped area is relatively large, the accuracy of statistical calculation is also high. In addition, the main advantage in this calculation is that it gives a single physical quantity rather than a distribution, we can discuss easily the relationship between the quantity and the physical properties of the system if it exists. We should also point out that although strictly speaking the physical quantities (packing fraction, flow rate, etc.) at different elementary locations in the fan-shaped area will be slightly different, we can ignore this difference to make the calculations operational. Figure 3 shows the flow rate-packing fraction relationship in the fan-shaped area at different strengths of the random force fields. It can be seen from Fig. 3 that there are two different types of flow rate-packing fraction relationships in the area. The first type (Fig. 3(a)) occurs in weaker random force field, which is similar to the flow rate-packing fraction relationship without the random force field we observed in the previous work. In this type of the flow rate-packing fraction relationship, the flow rate reaches a maximum at a moderate packing fraction. Below this moderate value, the flow rate increases with the increase of the packing fraction and above this value the flow rate decreases with the increase of the packing fraction. The flow rate-packing fraction relationship will eventually stop at a maximum packing fraction. The second type occurs in stronger random force field, where the flow rate increases monotonically with the packing fraction until a maximum packing fraction is reached (Fig. 3(b)).
cpl-36-9-094501-fig3.png
Fig. 3. Simulation results of the relationship between the flow rate and packing fraction in the fan-shaped area. The two different types of flow rate-packing fraction relationships are plotted separately in (a) for the first type and (b) for the second type. The inset is the pressure-volume relationship in the analysis of the gas-liquid phase transition in statistical physics (quoted from Ref.  [8]).
In the inset of Fig. 3 we present the most typical pressure-volume relationship in the analysis of the gas-liquid phase transition in statistical physics.[1] We mentioned previously that the granular flow rate corresponds to the pressure, and now we point out that the packing fraction corresponds to density, which is the reciprocal of the volume. Thus, the illustration of the pressure-volume relationship should be right-left reversed to compare with our granular flow rate-packing fraction relationship. From this point of view, our granular flow rate-packing fraction relationship is similar to the pressure-density relationship in the analysis of the gas-liquid phase transitions, which also have two types characterized by having maximum and monotonously rising. In statistical physics, the type of the pressure-volume relationship with a maximum appears in the low temperature, which indicates that there is a gas-liquid phase transition, and the density will change discontinuously during the phase transition. Analogously, in the case of granular flow the type of the flow rate-packing fraction relationship with a maximum means that there is a transition between the dilute flow state and the dense flow state, and the packing fraction is discontinuous during the transition. The type of the monotonically rising pressure-density relationship appears in higher temperature, which indicates that there is no gas-liquid phase transition, and the density will change continuously, thus the gas phase and the liquid phase become indistinguishable at this temperature. Analogously, in the case of granular flow, the type of the monotonically rising flow rate-packing fraction relationship means that there will be no discontinuous change in the packing fraction. In this case, all changes become continuous, thus the dilute flow state and the dense flow state become indistinguishable. By this analysis, there is a critical granular temperature (expressed by the strength of the random force field in this work). Lower than this temperature, there is a phase transition between the dilute flow state and the dense flow state, and higher than this temperature the dilute flow state and dense flow state become indistinguishable.
cpl-36-9-094501-fig4.png
Fig. 4. Phase diagram in granular temperature-flow rate space for the granular system. The inset is a typical phase diagram in temperature-pressure space. The solid green line applies to most substances, and the dotted green line gives the anomalous behavior of water. The green lines mark the freezing point and the blue line the boiling point.
A phase diagram is generally used in describing comprehensively the form of the substance in various environmental conditions. The phase diagram depicting the solid-liquid-gas state is usually drawn in the temperature-pressure space. According to our aforementioned correspondence, the diagram for the granular matter should be drawn at granular temperature (equivalent to the strength of the random force field)-flow rate space. In Fig. 4, we plot the flow rate of the stable dense flow or similarly the flow rate of the maximum stable dilute flow state as a function of the strength of the random force fields. The flow rate is plotted by circles for the first type of the flow rate-packing fraction relationships and by the lower triangles for the second type of that in Fig. 4. The curve linked by the circles represents the dilute-dense flow transition line, above which the granules are in the dense flow state and below which they are in the dilute flow state. This curve ends at a point with strong random force field and high flow rate. This indicates that there is a critical point, exceeding the critical point the dilute flow state and the dense flow state become undistinguishable. In the inset of Fig. 4, we show a typical phase diagram for general substances.[14] We can see that the portion of the gas-liquid phase transition (up-right part) of the phase diagram is very similar to the phase diagram of our granular flow. This again supports the view that the dilute flow state is similar to the gas state, while the dense flow state is similar to the liquid state. We also plot the flow rate of the maximum dilute flow state as a function of the strength of the random force fields in Fig. 4 (up-triangle). However, because this state is not a stable state, we cannot find the correspondence in the phase diagram of the water. In summary, we have shown that at low granular temperature there are two stable states; i.e., the stable dilute flow state and stable dense flow state. In the transition between them, the packing fraction is discontinuous. This amount of the discontinuity decreases gradually as the granular temperature increases, and finally disappears completely when the granular temperature exceeds a critical point. These features are the typical properties of phase transition and critical phenomenon for the form of general substances. By further detailed analysis, we have also found that the dilute flow state in the granular flow is similar to the gas state of general substances, while the dense flow state is similar to the liquid state. Finally, we have given the phase diagram of the granular flow in the random force field-flow rate space, and it is shown that the phase diagram is similar to that of the gas-liquid phase transition for general substances. Our results are partially supported by the experiments of granular flow in a channel with a bottleneck similar to that used in the simulations.[9] The experiments show discontinuous change in the outflow rate under the fixed inflow rate condition, which can be successfully explained by the stability of each states defined by the relationship between the local flow rate and the local packing fraction. References Granular solids, liquids, and gasesBuilt upon sand: Theoretical ideas inspired by granular flowsRapid Granular FlowsScales and kinetics of granular flowsRelationship between the flow rate and the packing fraction in the choke area of the two-dimensional granular flowGlobal Nature of Dilute-to-Dense Transition of Granular Flows in a 2D ChannelIntroduction to granular temperatureGranular material flows – An overviewControl of the fluctuation in the uniform granular flow by a random force field
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