A comprehensive review on flow field properties in polymer mixing processes: a focus on applications in energetic materials

The mixing process is a critical step in the production of energetic materials and has a profound impact on product performance. As modern formulations for energetic materials continue to advance, the needs placed on the mixing process have become increasingly complex. Understanding and mastering the properties of the mixing flow field are essential for achieving optimal mixing function, ensuring process safety, and optimizing the parameters of both the mixing process and equipment specifically for energetic materials. In this comprehensive review, we analyze the influence of flow field properties on the mixing process of energetic materials by examining the mixing mechanism of two types of flow within the flow field. Additionally, we provide evidence to support the advantages of elongational flow in achieving effective mixing. We also discuss the application of mixing flow field properties in the processing of energetic materials, including advancements in mixing equipment and methods designed to optimize flow fields. Finally, we address the current shortcomings in energetic material mixing and offer an outlook for future developments in this field.


Introduction
Energetic materials are a class of materials with high chemical reactivity and energy release properties, including explosives, propellants, and pyrotechnics.These materials play important roles in applications such as explosions, propulsion, and flame generation.Modern energetic materials are mainly being developed with a focus on low sensitivity, high energy release efficiency and enhanced reliability [1].The advancements in performance are predominantly observed through formulation optimization, particularly in terms of particle size refinement.This optimization facilitates the concentration of high-energy particles, allowing them to approach the maximum permissible stacking fraction and ensuring more effective inter-component contact [2].However, the pursuit of formulation optimization raises the standards for manufacturing and processing energetic materials.Among all the phases, mixing plays a key role, exerting a significant influence on both processing safety and product performance.
The efficient mixing of suspensions containing energetic materials currently presents a critical challenge.Energetic materials typically comprise highly viscous liquid additives combined with various solid additives, exhibiting non-Newtonian fluid behavior and limited flow and heat transmission characteristics until cured.Their flow state during the mixing process is comparable to that of plastics, rubber, and other polymers in civil applications.Furthermore, due to their high solid content and suspension nature, energetic materials display high viscosity and complex rheological responses, requiring substantial effort to achieve the desired mixing effect.Additionally, energetic material formulations often include high-energy solid explosives like HMX, RDX, and aluminum powder, alongside other oxidizers and combustibles, resulting in sub-stable properties characterized by heightened thermal susceptibility, mechanical sensitivity, and vulnerability to external environmental and process conditions [3].This characteristic also entails risks for efficient mixing.
In addition to the components, there is also a challenge from the perspective of equipment and methods.Currently, mixing equipment for energetic materials predominantly operates within shear flow-dominated flow fields.This encompasses batch mixing methods like kneaders and continuous mixing techniques such as screw mixers.These processes are characterized by directing the fluid through high-stress zones in narrow gaps, promoting dispersion at a high shear rate [4].Nevertheless, with the ongoing evolution of energetic materials, attaining desired mixing quality within the traditional shear flow-dominated mixing flow field is progressively posing greater difficulties.
Multiple studies have illustrated that elongational flow, when combined with shear flow in the mixing flow field, can significantly enhance mixing efficiency [5,6].In civil applications, elevating the presence of elongational flow has been proven effective in optimizing the mixing flow field.Therefore, it is crucial to analyze the properties of the flow field and incorporate elongational flow into the energetic materials mixing processes, while appropriately allocating the ratio of the two flows, in order to optimize the flow field.This paper offers a brief overview of the mixing process, delves into the impact of flow field properties on various mixing processes, and evaluates the current utilization of flow field properties.These endeavors are geared towards laying the groundwork for the future optimization of energetic material mixing processes.

Description of the mixing process 2.1. Mixing principle for high viscosity fluids
Mixing is a critical process that aims to evenly distribute and refine the components within a system's entire volume while preserving their fundamental properties.In polymer mixing, the dominant mechanism is volumetric diffusion, as described by Brodkey's theory [7].This mechanism involves the movement of fluid masses, droplets, or solid particles from one spatial location to another within the system.It also encompasses the intermingling of multiple components within the same space, ultimately leading to a uniform distribution of all constituents [8].
Energetic materials are typically high-viscosity suspensions that require external forces to induce either shear or elongational flow for effective mixing.When mixing energetic materials, it is crucial to consider the rheological properties of polymer materials.The effect of the rheological properties of polymer materials on the mixing process can mainly be outlined in four key points: (1) Viscosity affecting mixing performance.The viscosity of polymer materials plays a key role in influencing the flow behavior of the mixture during the mixing process.High-viscosity polymers may hinder the flow of the mixture, reducing the diffusion and interaction between different components, thus affecting the uniformity and stability of the mixing; (2) Shear force transmission efficiency and shear thinning/thickening effects.The rheological properties of polymer materials also affect the transmission efficiency of shear forces during the mixing process.Some polymers exhibit shear thinning or shear thickening effects, which can influence the dispersion and aggregation of particles, thereby affecting the uniformity and stability of the mixing; (3) Deformation behavior impact on the mixing process.The deformation behavior of polymer materials is another important factor to consider.Different polymers exhibit different deformation characteristics, such as plastic deformation or elastic recovery.These properties can affect the flow behavior of the mixture during the mixing process, potentially leading to issues of uneven mixing or instability; (4) Temperature sensitivity.The rheological properties of polymer materials are often temperature-dependent.Changes in temperature directly affect the viscosity, flow behavior, and deformation characteristics of polymer materials, thereby influencing mass transfer and reaction rates during the mixing process.Therefore, by properly controlling and optimizing these properties, a more uniform, efficient, and stable mixing process can be achieved, ultimately enhancing the performance and quality of the final product.
The mixing process for energetic materials consists of dispersion mixing and distribution mixing.Dispersion mixing involves breaking up agglomerates or liquid clusters in the mixture, which modifies the physical characteristics of the ingredients, such as changes in particle size distribution.Distribution mixing, on the other hand, involves convective transport of particles to distribute them evenly within the binder of the suspension.Figure 1 illustrates the mixing process in the energetic materials mixing system [8].Both dispersion and distribution mixing rely on plastic flow of the mixture to transform it into a highly viscous material with appropriate rheological properties.These processes not only ensure the expected plasticization of the initial mixture but also facilitate the merging of solid and fluid elements contained within each particular energetic material's chemical composition.

Type of mixed flow field composition
A comprehensive understanding of the composition of the mixed flow field serves as a fundamental prerequisite for studying the mixing process.The mixed flow field can be classified into shear flow and elongational flow based on their distinct modes of action [8].Shear flow creates a velocity gradient field perpendicular to the flow direction, while elongational flow generates a velocity gradient field parallel to the flow direction.To illustrate the disparities between a shear-flow-dominated flow field and an elongational-flow-dominated flow field, simplified two-plate and three-plate models are depicted in the schematic diagrams presented in figure 2 [9].
In figure 2(a), the materials primarily undergo shear deformation due to the drag effect of the moving plate, while in figure 2(b), the materials experience periodic changes as a result of the sliding plate and moving plate moving simultaneously, resulting in a dominant elongational flow.For shear flow, the flow field is usually described by the shear stress τ and the shear rate ,  g which is expressed by the power law equation as in equation (1): where s h is the apparent viscosity in shear, K is the fluid consistency coefficient, which reflects the magnitude of the viscosity, and n is the non-Newtonian exponent.The viscosity decreases with increasing shear rate, due to the rheology of most energetic material fluids are pseudoplastic.Many mixing equipment rely on this phenomenon called 'shear thinning' to achieve improved polymer flow properties through shearing action, which in turn facilitates mixing.
For elongational flow, which is formed by an elongational deformation, the direction of the elongational velocity gradient is parallel to the direction of the deformation, and is usually described by the tensile stress σ and the elongational rate  e by equations (2)-(3).Conventional polymer mixing is primarily based on shear flow, leading to a well-developed theory of the rheology of polymer materials on their response during the shear flow.In contrast, elongational deformation is more complex and less studied.Understanding tensile viscosity is crucial for investigating elongational deformation and polymer rheological behavior under a mixed flow field.Trouton defined tensile viscosity as the resistance of a fluid to tensile deformation under tensile flow.At low strain rates, shear viscosity and tensile viscosity of polymer melt remain stable, but tensile viscosity is three times greater than shear viscosity.As the strain rate increases, unlike the 'shear thinning' action of shear viscosity, the tensile viscosity of polymer may undergo more complex transformations including an increase, no change, or a decrease [10].Equation (3) shows that achieving uniform and stable elongational behavior requires exponential growth of material deformation with time, making it challenging to impose a stable elongational flow during the mixing process.
Many researchers try to measure elongational viscosity and obtain an elongational constitutive equation [11,12].Cogswell's convergent flow model [13] estimates tensile viscosity and is widely used due to its simplicity and ease in obtaining experimental parameters.However, the model assumes that tensile viscosity conforms to Newtonian fluid law, which does not accurately reflect the tensile rheology of most polymer fluids.Binding modified Cogswell's model by arguing that tensile viscosity is similar to shear viscosity in conforming to the power law, leading to Binding's model [14][15][16].However, the tensile power law exponent is not always constant during flow, rendering Binding's model inaccurate at times.In addition to the two planar constriction flow channel models mentioned above, Gibson developed a conical converging flow channel model [17], which accurately predicts fluid flow behavior in polymer processing equipment when the entrance angle is not too large.
With the fast advancements in science and technology, there has been a surge in the development of tensile rheometers and test devices to meet the demands of tensile testing.This, in turn, has provided favorable conditions for studying tensile flow.The Goettfert Rheotens Tensile Rheometer, designed by Meissner et al has gained widespread popularity for testing the tensile flow behavior of polymers [18].This specific rheometer elongates the sample vertically through capillary extrusion and rotor pull.Additionally, Meissner and Hostettler proposed the design of an RME tensile rheometer that enables horizontal tensile testing of polymer melt samples [19].Sentmanat et al further contributed to this field by designing and constructing the Sentmanat Extensional Rheometer (SER) tensile rheology test device, which allows for tensile testing by incorporating a tensile fixture attachment to a rotational rheometer [20].The SER is widely used for testing polymer melts due to its simple drive hub design.Furthermore, the later-developed Extensional Viscosity Fixture (EVF) and Horizontal Extensional Rheometer (HER) are additional tensile test fixture systems based on similar principles as the SER and RME [21,22].
These various tensile rheological test devices offer valuable support in predicting the tensile flow behavior of highly viscous polymers within a flow field.However, the influence of elongational flow behavior on polymer mixing calls for further discussion.

Influence law of flow field properties on the mixing process for energetic materials
Based on the aforementioned analysis, it can be deduced that both elongational flow and shear flow play pivotal roles in the mixing process of high-viscosity fluids.In this section, we aim to compare the influence of flow field properties by examining four crucial processes from the perspective of energetic materials.These processes encompass the mixing between liquid components, droplet fragmentation, agglomerate fragmentation, and energy consumption.

Mixing between the liquid components
The high viscosity of concentrated suspensions and the relatively low velocity during the mixing process of energetic materials lead to distinct interfaces between the binder and the formulation, which are often implicated in the mixing process [23].The creation of interfaces is considered one of the key mechanisms for mixing, as it significantly influences the control, rate of mass and energy transfer, and chemical reactions [7].
Consequently, measuring the increase in interfacial area per unit volume can serve as an indicator of the degree of mixing.It has been widely utilized in numerous studies to assess the mixing effect.For instance, Bigg and Middleman [24] investigated a more realistic flow system in a rectangular channel (as shown in figure 3), where the motion of the upper surface induced partial mixing of fluids.The interfacial area, calculated over time, was used as a quantitative measure of laminar mixing.Tsung et al [25] explored a novel parallel laminar micromixer with two-dimensional staggered curved channels and tapered structures to separate and merge fluids in microchannels, thereby increasing the interfacial area and significantly enhancing the mixing effect.Perugini et al [26] simulated the mingling process of magmas using a three-dimensional chaotic dynamical system consisting of stretching and folding processes.The results suggested a linear correlation between mixing intensity and the logarithm of the interfacial area, implying that magma mixing can be regarded as a chaotic process.Birendra [27] simulated a series of processes where a more viscous fluid (dark) was replaced by a less viscous fluid (light) (as depicted in figure 4).This included the formation of interfaces, presenting as fingers originating from the unstable interface, growing within the spots of the more viscous fluid, splitting into multiple fingers, merging together, and connecting the spots to achieve a mixing effect between the two fluids.It was concluded that an optimal viscosity ratio is required to maximize the generation of interfacial regions between the fluids, thus maximizing mixing velocity.
Current research focuses on developing techniques for calculating and quantifying interfacial area in laminar fluid-liquid mixing.One such technique involves deriving a simple interfacial area equation to predict the initial growth of interfacial area [28].Additionally, Seck et al [29] created a growth model that characterizes the mixing behavior in a continuous biaxial kneader by measuring the amount of product generated after interface contact, which acts as an indicator of the contact area.
The increase in interfacial area varies significantly across different flow types.Table 1 displays the average and maximum values of the increase in interfacial area (A A 0 / ) for three common flow types: planar elongational flow, uniaxial elongational flow, and simple shear flow [30].The value of 0 l is the elongation ratio in the x-direction, , y z l l are the elongation ratios in they and z axis directions.In planar elongational flow, the material is stretched along the x-direction with restrictions in the z-direction, while uniaxial elongational flow involves stretching along the x-direction without restrictions in other directions.The symbol e represents the total strain ( exp ).For simple shear flow, the initial shear direction occurs along the x-axis.The interfacial growth functions for these flows reach their maximum values at this point.A comparison of the interfacial increase functions reveals that the growth of interfacial area in elongational flow is exponentially related to the elongational strain, resulting in a much faster increase rate.This indicates that planar elongational flow and uniaxial elongational flow are more effective in promoting mixing compared to simple shear flow.

Fragmentation of droplets
In the molten state, many polymers are thermodynamically immiscible due to unfavorable interactions between their components and the minimal gain in entropy resulting from the mixing of high-molecular-weight species [31].As a result, polymer blends typically exhibit a heterogeneous morphology defined by the size, form, and dispersion of the domains that comprise the dispersed phase.While fiber-like or lamellar structures can be generated, the majority of blends exhibit a droplet-like morphology, with droplets of the dispersed phase embedded in the matrix.In certain energetic material formulations, there may be incompatible liquid additives that form droplets with interfacial tension and can only be mixed after breaking under stress [1].When droplets fragment, they decompose into smaller droplets or particles, thereby increasing their surface area.This increased surface area allows more liquid or solid phases to come into contact with other surrounding components, promoting material transfer and diffusion processes.Furthermore, due to the increased surface area, the chemical reaction rate also increases, indicating that reactions between components occur more rapidly, thus contributing to achieving a more uniform mixture.
The force exerted on a droplet by the flow field deforms and disperses it, while the interfacial tension represents the cohesive force between the two dispersed phases.The competition between these two forces has the potential to break up the droplet.To represent this relationship, Taylor introduced the concept of capillary number from a microrheology perspective in equation (4) [32].
where t is the shear stress, s is the interfacial tension, R is the local radius.The critical capillary number C acrit is the minimum critical value for a droplet to maintain its shape.If the capillary number exceeds its critical value (C C a a c r i t > ), the shape of the droplet will be out of equilibrium and be elongated, then it will finally break up into small droplets.The value of the critical capillary number C acrit depends on the flow form and the viscosity ratio P of the dispersed phase to the continuous phase, and P determines the deformation rate and breakup time of the droplet.
The research pointed out that the effect on droplet fragmentation is different in different flow forms and gives the relationship between C acrit and P as shown in figure 5 [33].Compared with shear flow, elongational flow has small fluctuation of values and changes in a wide range of P, no matter how the viscosity of the dispersed phase of the mixed system changes.It is proved that the droplets are easy to break up under shear flow only when 0.3 < P < 1.5.Utracki & Shi concluded that elongational flow was more effective than shear flow at P>3, and that only elongational flow was able to rupture the droplets at P > 3.8 [34].
In practical processing, the action of elongational flow induces a mechanism of droplet dispersion mechanisms, which explains the droplet-to-fiber transition observed in miscible blends under the action of elongational flow, as shown in figure 6 [6].This mechanism for the fragmentation of fluid clusters proposes that not only the shear flow can lead to the dispersion of droplets, but that the elongational flow can stretch the droplets into a state of elongated fibers, which is conducive to the dispersion and mixing of fluids [35].Researchers has already noted that the critical capillary number for threads is generally smaller than that for drops [31].This suggests that achieving a finer dispersion by the temporary mechanism of thread breakage during extension is preferable than repeatedly drop breakup at C .acrit The Lee and Park controlling interface equations may qualitatively anticipate the shift from an initial spherical morphology to a fibrillar structure in elongational flows, as observed by Lacroix using a nozzle shaped die, where the flow is primarily extensional [36].
Zhang et al prepared a PLA/PVDF hybrid media with high melt flow index ratio based on a novel twin-eccentric rotor extruder (TERE) [37], the mechanism of equipment action and the morphological changes of the blends are shown in figure 7. The results indicate that the elongational flow facilitates the dispersion and orientation of the dispersed phase in blends compared to shear flow.And a relationship between temperature and the amount of droplet deformation under elongational is established based on morphology analysis and droplet deformation theory [38].Besides, another advantage of elongational flow is that it does not generate a rotational effect which no contribution to droplet breakup, this feature allows for more effective dispersion [33].
Elongational flow efficiently reduces the cross-sectional diameter of elongated strip droplets and is more effective than shear flow at dispersing the bubbles and droplets formed during the practical processing of energetic materials.

Fragmentation of agglomerates
During the processing of energetic materials, the high surface energy of high-energy solid particles and metal powders makes them prone to cohesion, leading to the formation of agglomerates.These agglomerates can negatively impact the properties of the final products and hinder mass transfer and thermal diffusion in reactions if they are not dispersed by the mixed flow field.Additionally, the binder fluid trapped within the confines of the agglomerates does not contribute to the flowability of the suspension [1].From a processing safety perspective, the agglomeration behavior reduces the coating effect of the binder on solid particles, resulting in collisions and friction between particles within the agglomerate and between particles and equipment, posing processing hazards [39].It is worth mentioning that, in the mixing process, shear flow typically increases the likelihood of electrostatic accumulation.This is because shear flow results in friction and collisions between particles, leading to the generation and accumulation of electrostatic charges, which can pose safety hazards.On the other hand, elongational flow helps reduce electrostatic accumulation.When materials flow under elongational flow, particles are elongated, reducing the frequency of their contact and thus lowering  the generation of electrostatic charges.Therefore, by selecting the appropriate flow field mode, especially utilizing elongational flow, it is possible to mitigate the safety risks associated with electrostatic effects to some extent, which is crucial for the production and handling of energetic materials.
Agglomerates are typically formed by the aggregation of multiple particles or particle clusters.When they fragment, the internal particles or clusters disperse into smaller individual particles.This process increases the contact surface area between particles, facilitating interactions between components.The increased contact surface area helps enhance the diffusion and mass transfer rates between various components in the mixture, thereby promoting a more uniform mixing.Most research aimed at enhancing safety by reducing the agglomeration of energetic solid particles has focused on novel mixing techniques.Zhou et al [40] successfully prepared Al-HMX energetic microspheres with uniformly distributed particle sizes using a microfluidic technique, effectively reducing the agglomeration of aluminum powder.This method demonstrates significant potential for the continuous, safe, and efficient preparation of aluminized explosives.Zhigach et al [41] achieved the uniform distribution of components in an aluminum/HMX nanocomposite through the spray-drying method.Van et al [42] observed that reducing the particle size of energetic solids enhances their sensitivity to friction and impact, thereby reducing the risks associated with agglomerates.However, fewer studies have focused on the dispersion of agglomerates by the mixed flow field during processing in the field of energetic materials.
Currently, the dispersion mechanism of the solid phase in the flow field primarily consists of three phases: fragmentation of agglomerates, exfoliation of debris, and further distribution and refinement of debris in the continuous phase [43,44].The fragmentation stage plays a critical role in dispersive mixing, and it has been demonstrated that an elongational flow field disperses particles more effectively than a shear flow field under the same applied stress.This results in smaller fragments in a shorter time, as illustrated in figure 8 [45].Tadmor [8] proposed a model assuming the dispersion of the solid phase in the flow field as two small spheres and theoretically analyzed the forces acting on the solid phase particles in the flow field.Calculations based on this model indicate that, for the same deformation rate, the maximum force on solid-phase particles in an elongational flow is twice that in a shear flow.
Tadmor also joins the modeling on the bond strength of agglomerates with the analysis of fluid dynamics to give the dimensionless number as the following equation (5): where the dimensionless number Z represents the magnitude of the force that breaks the agglomerate, c is a geometric scalar reflecting the agglomerate rupture process, m is the viscosity of the suspended liquid, and T is the tensile strength of the agglomerate.Z needs to exceed 2 in simple shear flow, 0.5 in uniaxial stretching and 1 in biaxial stretching.Theoretical calculations pointed out that elongational flow fields enhance the process of agglomerate dispersion in comparison to simple shear flows [46].A power law relationship between the number of agglomerate fragments and the stress in the flow field is derived from discrete element numerical calculation, and results concluded that the agglomerates in the elongational flow can be dispersed into small and numerous fragments [47].Fanelli et al developed a working model to simulate the agglomeration behavior during hydrodynamic shear, and the findings reveal that lowering cluster size (with a constant agglomerate-cluster size ratio) results in a considerable drop in shear flow ease of dispersion [48].Kao et al found that elongational flow is more likely to break up agglomerates, and similar to droplet breakage, the agglomerates under shear flow may do pure rolling and be poorly dispersed, while elongational flow is able to better apply the force on the agglomerates [49].Afshar et al applied the population equilibrium method to study the dispersion of agglomerates in shear flow field (SFF) and elongated flow field (EFF), and it can be seen from figure 9 that due to the ability of the EFF to disrupt agglomerate fragmentation, it can disrupt the agglomerates even at lower deformation rates, whereas the SSF relies more on the deformation rate because of agglomerates' rotation [50].Hartmann et al found that the magnitude of stresses on the aggregate surface is increasing with time monotonously and shear stress is maximum on the outer parts of the aggregate from the numerical solution of the Navier-Stokes equation and the continuity equation based on finite volume methods.Normal stress takes on maximum values on the upstream and downstream oriented faces [51].

Energy consumption
Energy consumption is a critical indicator of mixing efficiency.On the one hand, it determines the product processing cost, prompting researchers to optimize process and equipment parameters to minimize energy consumption [52].On the other hand, shear, friction, and extrusion are the primary sources of heat in the mixing unit, converting some of the energy consumed during the mixing of energetic materials into heat energy,  leading to a temperature rise.As energetic materials contain decomposable oxidizing and flammable groups that can cause combustion and explosion when the temperature exceeds the formula's decomposition temperature, it is crucial to regulate the temperature rise appropriately [53].Additionally, shear viscosity is highly dependent on temperature, significantly affecting material processing characteristics [54].Thus, it is essential to regulate energy consumption at various extruder sections.The energy required for mixing materials varies depending on the flow type [55].As shown in table 2, Erwin [56] compared the energy required per unit volume for different flows to achieve the same interfacial increase.After normalizing the energy consumption of the various flows, the relationship shown in figure 10 emerged.For a unit volume of fluid, simple shear flow consumes five orders of magnitude more energy than elongational flow for the same interfacial area increase.This proves that shear flow is a relatively inefficient mixing method that generates more heat.This also indicates that elongational flow is more beneficial in reducing safety risks associated with temperature.
The comparison of the four aspects mentioned above clearly demonstrates the superiority of elongational flow in mixing energetic materials: (1) Elongational flow exhibits excellent performance in mixing high-viscosity liquids, as it can significantly increase the interfacial area, resulting in effective mixing.
(2) When it comes to dispersing droplets and solid particle agglomerates, elongational flow surpasses other flow types by offering superior dispersion and mixing capabilities.
(3) Compared to shear flow, elongational flow requires less energy, generates less heat, and ensures a safer mixing process while achieving the same level of mixing efficiency.Table 2. Relationship between the energy required for different flows per unit volume and the increase of the interface (h is the fluid viscosity, t 0 is the time that the tensile deformation endures).

Simple shear flow E
Elongational flow has been observed to provide better dispersion for finer nanoparticles and facilitates the preferential orientation of polymer macromolecules along the flow direction [57].This characteristic is particularly advantageous for the preparation of anisotropic materials [58].

Application of flow field properties in the mixing of energetic materials
Previous study on traditional mixing equipment revealed that the fraction of elongational flow in the mixing flow field is relatively minor and that quantitative methods are lacking.As a result, improving the geometry to increase the proportion of elongational flow in the mixing flow field is a hot research topic.This section provide a brief review covering the latest advancements in methods for characterizing the role of elongational flow and optimization of mixed flow fields for increasing the percentage of elongational flow based on the mixing process of energetic materials.Following that, advancements in mixing equipment for energetic materials are highlighted.A common and effective method for quantifying elongational flow in the flow field involves calculating the stretching deformation or strain experienced by blends during mixing.In laminar flow systems such as extruders, the total strain (shear or elongational strain) can be utilized to evaluate the degree of mixing.It is equal to the product of the strain rate and the residence time.Huang [59] deduced analytical solutions for the velocity field and total strain based on a comprehensive analysis of a physical model of vane plasticizing and conveying units.The analytical solution enables the investigation of the influence of structural parameters on mixing effects.Ottino proposed another kinematic approach to analyze distributive mixing by tracking the amount of deformation or stretching experienced by fluid elements [60].The instantaneous mixing efficiency, which quantitatively characterizes the stretching rate during the mixing process, is calculated using the following equation ( 6): where D is the rate of strain tensor, l is defined as the length of stretch of an infinitely small material line.This efficiency can be thought of as the fraction of the energy dissipated locally that is used to stretch a fluid element at a given instant in a purely viscous fluid and falls in the range [−1, 1].A value of −1 indicates that all of the energy expended was used to decrease the length of the material line, effectively totally unmixing it, whereas a value of 1 shows that all of the energy dissipated was used to lengthen the material line, resulting in the highest mixing effect [61].
Another parameter to characterize the percentage and distribution of tensile deformation was named the tensile deformation effect index, which was used to investigate the proportion of tensile deformation in mixing equipment and the effect of structural parameters [59].Based on the scalar form of the second invariant of the deformation rate tensor, the tensile deformation as a percentage of the total deformation is defined as the tensile deformation effect index , h as in equation (7):  g is the rate of shear deformation in the x y-plane.In terms of application, the tensile deformation effect index is now commonly used to predict the magnitude of the elongation effect in elongational mixing equipment for the improvement of structural and process parameters.

Mixing index
The mixing index is a more generally used parameter to quantify elongational flow, which could be used to estimate which flow is more dominant in a polymer mixing flow field, has already been integrated into the mixing process of energetic materials.The value of the mixing index is often expressed as , old l with the following expression (8) [62,63]: where D is the deformation rate tensor, w is the vorticity tensor.The mixing index reflects the interrelationship between shear and tensile effects.When the mixing index is 0, the material mixing flow field does not exist in the tension and shear effect, the material only occurs through pure rolling; A mixing index of 0.5 indicates that the mixing process is shear-only, while a mixing index of 1 indicates that the mixing process is tensile-only.Usually, in the mixing process, the mixing index between 0 and 0.5 means that the material is only affected by shear, only after the mixing index exceeds 0.5 does it mean that stretching is involved [64].
The larger the mixing index, the more significant the elongational effect in the mixing process, the higher the percentage of elongational flow, which indicates that the mixing effect is better in experimental and actual processing.With the development of simulation technology, mixing index has been widely used in civil polymer research, usually used to measure the strength and distribution of mixing in each device, which is more frequently used in twin-screw mixers and kneaders.
The mixing index is related to many factors, of which the structural parameters are often the most important influences.Researchers compared the mixing ability of different threaded elements using the mixing index, proved that the screw mixing element has the best mixing ability [65], and found the optimal paddle configuration for a screw [62].Process conditions also have a certain effect on the mixing index, but the effect is less compared to the geometry, normally, an increase of screw speed or inlet flow rate will cause the mixing index to increase [66].Prakash found that the magnitude of the mixing index is related to the rheological properties of the fluid, especially for non-Newtonian fluids [67].The region with the lowest mixing index in the mixing device is distributed in the region with the lowest shear stress, and Connelly found that the flow velocity here is a local maximum in the screw flow field with a constant value of broadband [68], deducing that the material in this region is moving as a plug around the flow field center of rotation.Jiang analyzed the mixing process of composite solid propellant slurry in kneading mixers and discovered the area between the impellers appears a state with high mixing index, while the impeller tip zone exists low mixing index, indicating pure rotational flow [69].
However, the mixing index can only be used for prediction and estimation at present, and the precise results need to be verified and illustrated by practical experiments combined with theoretical modeling.In order to explain its relationship with dispersion mixing, the index needs to be combined with the changes in shear stress and rheological properties of the material.

Development of mixing flow fields in mixing equipment
The above quantification of the mixing flow field is solely based on theoretical predictions, and further investigation is required to understand the mixing flow fields of specific energetic materials in mixing equipment.Most mixing methods for energetic materials rely on batch mixers, which means the entire process is conducted in batches [70,71].Although batch mixers have evolved from early horizontal kneaders to vertical kneaders with improved mixing results and higher reliability, the handling of large quantities of explosive materials in kneaders and the complexity of the flow field make it challenging to establish an accurate flow field model for predicting and controlling the mixing process.This can lead to safety and quality issues [72].
Continuous processing for energetic materials was initially introduced in Europe and the United States, where binders, plasticizers, solvents, oxidizers, and curing agents are fed into a screw extruder for continuous mixing and blending using the screw element [73,74].Single screw extruders were primarily used in the production of propellants, but their mixing capacity is not outstanding due to the properties of the flow occurring within the Archimedean screw, which limits the repositioning of the interfacial region for mixing.Special threaded elements are often introduced to improve the mixing performance of single screw extruders.In the early 80 s, the United States began developing twin-screw continuous mixing technology for energetic materials, which has been applied to various applications such as composite propellants, homogenous propellants, high-energy propellants, energetic thermoplastic elastomers, PBXs, decoy flares, and thermobaric explosives [75][76][77][78].
Twin-screw extruders offer a high degree of flexibility for a wider range of energetic materials.As shown in figure 11, the screws in a twin-screw extruder can rotate in concert or in reverse, and can be fully engaged or separated from each other [79].Each screw configuration represents a different mixer with different functions, allowing for intensive mixing and applicability to a wider range of material viscosities compared to single screw mixers.Twin-screw extruders also provide increased safety due to smaller quantities of energetic materials in the hazardous mixing zone at any given time and a relatively large proportion of elongational flow in the mixed flow field.Studies have shown that twin-screw extrusion processes result in higher mixing index values compared to other methods, indicating their powerful mixing ability (as shown in figure 12).They are also suitable for breaking the agglomeration potential of ultrafine materials due to their higher surface-to-volume ratio, making them ideal for future mixing of more refined and advanced energetic materials [80,81].
Recently, several advanced mixing processes have emerged.For instance, the resonant acoustic mixing (RAM) technology, developed by ResoDyn Corporation (USA), has been proven to exhibit a favorable thermal profile and flow field [82].The microfluidic mixing method leverages diminutive scale flow channels in microfluidic systems to increase the surface-to-volume ratio, thereby offering advantages for mixing [83].Electrohydrodynamic atomization (EHDA) enables the construction of unique microstructures for energetic materials, including core-shell, laminate, microcapsule, and porous microstructures, enhancing both high energy output and processing safety [84].Moreover, supercritical fluid mixing technology in the field of energetic materials utilizes supercritical fluids to enhance mixing reactions, control particle size and structure, with features such as high efficiency, precise control, and environmental friendliness [85].This technology not  only improves reaction rates and enables particle size and structure control, but also exhibits good solubility and selectivity, allowing for extraction and separation of specific components.Despite their high potential, the aforementioned technologies are still in the experimental or small-scale synthesis stage.
It is clear that for large-scale production of energetic materials, both batch mixing and continuous mixing processes are primarily governed by the shear flow field.The shear flow field used in the mixing of energetic materials is associated with certain drawbacks, like high energy consumption, restricted mixing and dispersing effects, and limited adaptability to various material systems.Furthermore, the intense shear effect during the mixing process can result in localized heat generation and the formation of hot spots, consequently elevating the risk of deflagration in energetic materials.
In summary, there is a need to increase the proportion of elongational flow in current mixing equipment and develop mixing processes that strike a balance between safety performance and mixing effectiveness.It is crucial to optimize the flow field in existing energetic material mixing equipment by appropriately increasing the percentage of elongational flow.Additionally, there is a demand to explore new mixing technologies that consider both safety performance and mixing effectiveness.

Optimization of the mixing flow field in mixing equipment
Numerous research efforts have been focused on improving the plasticization and mixing of materials by optimizing the mechanical structure to increase the tensile deformation effect.These efforts include introducing extensional mixing elements in twin-screw extruders [86] and enhancing the current screw structure [87].However, generating effective and stable elongational flow fields for high-viscosity, sub-stable energetic materials remains a challenge.
Designing a mechanical structure capable of producing large-scale elongational flow is demanding due to the complexity of the process.To address this issue, Qu [88][89][90] developed a vane extrusion (VE) device that generates volume-stretch deformation, as shown in figure 13.The VE device consists of a cylinder-shaped hollow barrel as the stator, a columned eccentric rotor, and multiple vanes installed on the rotor.When the rotor rotates, the space between the vanes and the stator is restricted, causing periodic volume changes that produce volume elongational mixing, plasticizing, and conveying.Qu [91][92][93] also invented the uniaxial eccentric rotor extruder (ERE), based on the principle of volumetric tensile deformation, as shown in figure 14.The rotation of the spiral-shaped eccentric rotor forces the materials in the space between the rotor and stator to undergo tensile deformation through multiple convergence-divergence zones.Biaxial and triaxial eccentric rotor processing methods and equipment have also been developed [94].
Several smaller devices have emerged that need to be used in conjunction with other equipment.Utraki [95] investigated a new type of elongational flow mixer (EFM) that has a circular stretching block protruding from the bottom surface of the device.As the filler system passes through the device, it is simultaneously stretched in both radial and circumferential directions, achieving better mixing results [96,97].Pandey [98] developed a new elongation mixing element (EME) for elongational flow, using a smooth hyperbolic runner structure in conjunction with a screw extruder to achieve standard dispersion mixing efficiency.Suzaka [99] designed a device that realizes elongational flow by opening multiple converging holes with tensile action, promoting dispersion of the dispersed phase when the tensile ratio (the ratio of the large diameter to the small diameter of the converging flow channel) is 10:1.
Several innovative devices have been employed to mix energetic materials, such as the peristaltic continuous conveyor mixer developed by Japanese researchers based on the principle of intestinal motility, which achieves solid propellant processing at low shear flow occupancy ratios (figure 15) [100].Additionally, Brad [101] developed a centrifugal mixing process for solid propellants, where the maximum shear of the mixing flow field in the centrifugal mixer is much lower than that in traditional kneading processes, resulting in less hazardous conditions.
By optimizing the mixing flow fields and inventing new mixing devices, the mixing process can benefit from lower processing energy consumption, higher mass and heat transfer efficiency, improved mixing and dispersion, and reduced thermal degradation of molecular chains.In addition, the introduction of the elongational flow reduces the shear force in the flow field, making the flow field softer and realizing a reduction in the pressure of the fluid on the equipment as well as the internal pressure of the fluid.These advancements have created opportunities for introducing elongational flow in the mixing of energetic materials.
However, there are still some challenges with the methods and equipment mentioned above that require further optimization and improvement.For instance, with EME modified screw extruders, repeated insertion of EME can lead to significant pressure losses, making it difficult for the elongational section to replace the screw section entirely.In vane extruders, the stiffness of the vanes is a limiting factor for extreme conditions, and flow dead zones in the flow channel indicate that the structure needs optimization.Similarly, eccentric rotor extruders encounter difficulties in applying elongational flow, leading to increased power consumption as the eccentricity and speed increase, ultimately resulting in higher costs [94].

Challenges and future scope
Elongational flow has proven to be advantageous in theory and practice for achieving effective dispersion mixing processes with low energy consumption, efficient heat transmission, and a more uniform temperature field.These characteristics make elongational flow ideal for processing energetic materials.However, due to limitations in device design, shear flow-based mixing methods and devices remain the primary technology.Adapting to future energetic products with characteristics such as multi-species, high energy, high solids content, and high performance will pose a challenge.Currently, there is a lack of research on the characteristics of the mixing flow field in the mixing process for energetic materials.Although several mixing devices are available to increase the proportion of elongational flow, most remain in the experimental stage and are primarily used in civilian polymer applications.
Given the prevailing circumstances and challenges, it is urgent to improve the mixing flow field to meet the needs of future energetic products.In order to appropriately increase the amount of elongational flow, the flow field should be matched based on the rheological properties, loading characteristics, and agglomerative segregation characteristics of energetic materials, with mixing effect being the primary goal and safety serving as the boundary.The next step is to establish the productive process according to the mixed flow field model, since the mixing step is crucial to the choice of processing technology.Currently, the batch mixing process is unable to adapt to the new type of energetic material processing, given its shortcomings in discontinuity and low mixing efficiency.Therefore, there is a need to develop a safer, flexible, and efficient continuous process with an increasing proportion of elongational flow.Finally, increasing investment in the research of new advanced mixing process methods or equipment (such as RAM, microfluidic method, EHDA and so on) by combining material properties, equipment parameters, process parameters, and the optimized mixed flow field model is necessary.

Conclusion
In conclusion, this review provides a comprehensive overview of cutting-edge research on energetic material mixing processes.We delineate the mixing principles for high-viscosity fluids and present a thorough understanding of the mixed flow field composition.By delving into the impact of flow field properties on mixing processes for energetic materials through the analysis of key processes-mixing between liquid components, droplet fragmentation, agglomerate fragmentation, and energy consumption-we uncover essential insights.Furthermore, the review explores recent advancements in characterizing elongational flow's role and optimizing mixed flow fields to enhance the elongational flow percentage in energetic material mixing processes.The discussion extends to advancements in mixing equipment tailored for energetic materials.Lastly, the paper outlines existing challenges and future prospects.Overall, this review furnishes valuable perspectives on energetic material mixing technology and underscores the significance of augmenting the proportion of elongational flow within the mixing flow field.The authors envisage that ongoing progress in processing techniques and materials science will propel innovation in this domain, ushering in novel devices and technologies with enhanced performance and safety.

Figure 1 .
Figure 1.Main processes in mixed systems of energetic materials.Reprinted from [8], Copyright (2013), with permission from John Wiley & Sons.

Figure 4 .
Figure 4. Snapshots of the concentration field at increasing time steps from a numerical simulation of miscible viscous fingering.Reprinted from[27], Copyright (2011), with permission from American Physical Society.

Figure 5 .
Figure 5.Effect of viscosity ratio on critical capillary number in shear and tensile flow.Reproduced from [33].CC BY 4.0.

Figure 6 .
Figure 6.Schematic representation of the possible mechanism of the droplet-to-fibril transition in immiscible blends subjected to elongational flow: (A) Deformation of the original particles of the dispersed phase and (B) formation of microfibrillar structures.Reproduced from [6].CC BY 4.0.

Figure 9 .
Figure 9.The effect of different agglomerate structure on dispersion process in (a) shear flow field (b) elongational flow field.Reproduced from [50].CC BY 4.0.

Figure 10 .
Figure 10.Energy required for mixing in different flows.

4. 1 .
Quantification methods of flow forms for flow fields in mixing equipment 4.1.1.Tensile deformation Given the challenges in directly measuring the elongational and shear behavior of fluids during the mixing process, simulation techniques have emerged as effective means for analysis.By setting appropriate parameters, simulations can predict the proportion and distribution of different flow forms in the mixed flow field, offering a foundation for optimizing the mixing process.

2  g and yy 2 
g are the rates of tensile deformation in the x and y directions, xy 2

Figure 12 .
Figure 12.Scatter of the thermo-gravimetric analysis data for mixtures prepared with conventional and twin-screw extrusion processes and their mixing indices.Reprinted from [80], 06 Jun 2007, reprinted by permission of the publisher (Taylor & Francis Ltd, http://www.tandfonline.com).

Figure 13 .
Figure 13.(a) Structure of a vane extruder (VE) with different working sections.(b) Photograph of the solid.Reprinted from [90], John Wiley & Sons.© 2014 Society of Plastics Engineers.

Table 1 .
Comparison of the increase in interfacial area for different flow forms.