The depletion effect was first predicted theoretically by Sho Asakura and Fumio Oosawa in the 1950s Since then, computer simulations of depletion in various systems have been conducted using molecular dynamics and Monte Carlo methods. Although computer simulations can provide a detailed description of depletion attractions in different systems, the accuracy of the simulation results can depend on the defining model used (Jiménez-Ángeles et al., 2011; Wolters et al., 2017; Wan and Glotzer, 2018; Calero and Pagonabarraga, 2020).

Based on density generalized function theory, Dietrich et al. (Roth et al., 1999) calculated the depletion potential in a small hard sphere system with a volume fraction of η _s and a radius of R _s , and a large hard sphere with a radius of R _b on a curved surface with a radius of R _c . The depletion potential W c A O a was expressed mathematically as a function of the radius of curvature R _c as follows: β W c A O a R c = − η s 1 + 3 R c R b / R s R c + γ R b (1) where β ⁻¹ = k _B T, and γ indicated whether the large hard sphere is inside (−1) or outside (+1) of the surface.

As shown in Figure 1A, the solid line indicated the results calculated from the density functional theory and the dashed line corresponds to the A-O approximation. It can be shown from the figure that the calculation results are closer when η _s is smaller. That is, the model description is more accurate when η _s → 0.

Krüger et al. (Krüger et al., 2011) proposed a simple mathematical model that combines A-O theory with Derjaguin’s approximation using differential geometry to evaluate the non-isotropic depletion potential of ellipsoids in terms of the mean and Gaussian curvatures of ellipsoidal particle surfaces. This method yielded an excellent agreement between the theoretical results and the simulated data at low concentrations of depletion agents. In a separate study, Yang et al. (Yang et al., 2015) investigated the depletion attraction of spherical colloidal particles using Monte Carlo simulations in the presence of temperature-sensitive depletion agents on a substrate with a pattern. The simulations produced overlapped excluded volume equations between spherical colloidal particles and between spherical particles and a patterned substrate, allowing for an understanding of how the surface microstructure of the substrate affected the local depletion interactions of spherical particles. Castañeda-Priego et al. (Castañeda-Priego et al., 2006) conducted Monte Carlo simulations to explore the impact of particle shape, system dimensionality, and container surface pattern on depletion attraction. Their findings indicated that the A-O model was not applicable to depletion attractions between two large spheres induced by rod particles as depletion agents, due to the A-O model’s neglect of the contribution of rotation to entropy, which was significant for non-isotropic rod particles. Compared to spherical particles, rod-shaped particles as depletion agents produced stronger and longer-range depletion attraction under similar conditions (Castañeda-Priego et al., 2006). Additionally, some results suggested that interactions between convex regions on the surface of the vessel wall and large particles were weaker, while the attraction between concave regions and large particles was stronger. Figure 1B shows the overlapped excluded volumes between the large particles and the surface in both convex and concave regions, and it could be seen that under the same conditions, the concave vessel wall generated more V _ov with large particles (Li and Ma, 2002). These results have significant implications for the utilization of depletion attractions in colloids.

Through the introduction of several methods for calculating depletion forces, we present images of depletion interactions and general concepts of equivalent interactions, followed by a detailed presentation of several practical methods for calculating hard sphere depletion interactions and various approximate analytical formulas. Due to the scope of this article, the details of calculations and formula derivations are not discussed. For interested readers who wish to delve deeper, a fairly comprehensive list of references is provided for convenience. It is important to note, however, that the references presented here are limited to supporting and supplementing this article, and not a comprehensive list of literature related to colloid depletion interactions.

In addition to theoretical calculations, direct or indirect experimental measurements of the depletion attraction/potential have been carried out (Evans and Needham, 1988; Lee et al., 1991; Richetti and Kékicheff, 1992; Andrew and Simon, 1995; Ohshima et al., 1997; Rudhardt et al., 1998; Crocker et al., 1999; Piech and Walz, 2002; Knoben et al., 2007; Gong et al., 2011; Kumar et al., 2013). However, possible interactions between large spheres and small spheres, such as van der Waals forces, electrostatic forces, and hydrodynamic effects, hinder the accurate measurement of depletion attractions when two large spheres (plates) are in close proximity. In 1988, Evans and Needham (Evans and Needham, 1988) directly measured the depletion potential by dissolving dextran (molecular weight 10,000–150,000 Da) in water as depletion agents, reducing the distance between two lipid vesicle membranes, and measuring the concentration of dextran in the solution to obtain the adhesion force between two lipid vesicle membranes immersed in it, calculated as: ∆ F = A ∫ 0 Z g Π B − Π d z (2) where A is the surface area of the vesicle, Z _g is the distance of between two lipid vesicle membranes. Π is the osmotic pressure of the solution, and Π _B is the osmotic pressure in the gap between the vesicles. The magnitude of the adhesion force is the magnitude of the depletion attraction.

Subsequently, a variety of measurement methods have been used, including atomic force microscope (AFM) (Andrew and Simon, 1995; Piech and Walz, 2002; Knoben et al., 2007), the laser radiation method (Ohshima et al., 1997; Crocker et al., 1999), the total internal reflection microscopy (TIRM) method (Higurashi et al., 1998; Gong et al., 2011), neutron diffraction method (Lee et al., 1991; Kumar et al., 2013). Experimental measurements of sphere-to-sphere and sphere-to-wall depletion attractions have been conducted. AFM can be used to directly measure the depletion attraction between a large sphere and a surface. The method entails immersing a probe with a colloidal sphere in the solution of depletion agents, scanning the substrate surface continuously, and monitoring the deflection of the cantilever with a split photodiode detector and a total reflection laser beam. The curve of the depletion attraction with distance can be obtained directly since the spring constant of the cantilever is known. Figure 2A shows a schematic illustration of the principle of AFM measurement.

Utilizing focused lasers, it was possible to construct optical traps in which minuscule particles were confined through the pressure exerted by the laser (Higurashi et al., 1998; Wilking and Mason, 2008). Ohshima et al. (Ohshima et al., 1997) introduced polystyrene (PS) microspheres possessing a diameter of 1.0 µm into a polyethylene oxide (PEO) solution containing spheres with a radius of 0.101 µm. The PS spheres experienced depletion attractions while approaching the surface of the cuvette and ultimately adhered to it. Subsequently, a focused laser beam was employed to manipulate the PS spheres, with the laser’s intensity gradually increased until the point at which it could displace the PS sphere, at which point the laser’s power was equal to the depletion attraction potential of the PS spheres and the cuvette. Crocker et al. (Crocker et al., 1999) employed poly (methyl methacrylate) (PMMA) particles with a diameter of 1,100 ± 15 nm as their larger spheres and PS particles with a diameter of 83 nm as their depletion agents. A moderate amount of sodium dodecyl sulfate (SDS) and salt were added to the solution to prevent the larger PMMA spheres from aggregating due to van der Waals attraction. Two distinct optical tweezers beams were focused on the two larger spheres, and the depletion attraction between them was directly measured (Osterman, 2010), the main schematic of the experiment is shown in Figure 2B. Consequently, a series of similar works were successively initiated (Perry et al., 2012; Julie et al., 2014; Zosel et al., 2020). Utilizing the energy harnessed from optical tweezers, the researchers immobilized the spherical colloids within the solution matrix and measured the potential of mean force between two large colloidal particles. These results paved the way for a systematic examination of the intricate relation between depletion forces and the spatial separation distance among the large spheres. By calculating the dynamics of the large spheres, assistances were provided in elucidating the nucleation mechanism influenced by depletion attraction, the stability of the solution and the conformational issues of biomacromolecules in crowded environments. Similarly, there is the work of Amano et al. (Amano et al., 2016), who directly measured the depletion attraction potential (Φ_LL) between two large colloidal particles using linear optical tweezers, and then proposed and tested a theory that translates Φ_LL into the number density distribution (g _LS) of small particles around large particles. Their theory would contribute to the structural analysis of colloidal suspensions, which was important for both understanding colloidal properties and developing colloidal products. A recent work by Liu et al. (Liu et al., 2019) captured two silica particles in a point-like optical trap and a linear optical trap using dual-optical tweezers technology, and by controlling the linear trap brought one particle close enough to the other. By modulating the concentration of the ionic surfactant sodium dodecyl benzene sulfonate (SDBS), and concurrently regulating the spatial separation distance, the depletion attraction between the two particles could be measured as the separation distance varies.

AFM and laser radiation represent interventional measurements that unavoidably influence the thermal motion of colloidal spheres. The magnitude of the depletion attraction potential is equivalent to the thermal motion energy ∼ k _B T (Gong et al., 2011), thereby making it effortless for the thermal motion to interfere with the depletion potential measurement. In contrast, total internal reflection microscopy (TIRM) serves as a non-interfering method in essence gauge the inter-surface interactions with the thermal energy k _B T and thus very sensitive to weak forces. The potential measurements are not reliant on laser intensity, which eliminate the impact of light forces. When light is reflected at a solid/fluid interface above the angle of total internal reflection θc (Figure 2C), an evanescent wave is formed which leaks into the fluid. Briefly, in a TIRM measurement, an evanescent wave is engendered at the interface between the solid material and the liquid medium through the mechanism of total internal reflection within the glass substrate. This evanescent wave exhibits an exponential decay pattern relative to the distance from the interface with a characteristic decay length β ⁻¹ which depends on the incident angle θ, the wavelength of the light beam, and the refractive indices at the interface. When an object which scatters light, e.g., a colloidal sphere, approaches the surface close enough to enter the evanescent field frustrated total reflection will occur. The scattering intensity of a colloidal sphere I is related to the distance z between the colloidal sphere and the surface; it can be written as I ∝ e ^−βz . Measuring the scattered intensity over a long period provides a histogram of the separation distance, the probability p(z) of where the colloidal sphere stays are obtained. Using the Boltzmann distribution function p(z) = e^{−[Φ(z)/ k} _B ^T], the potential of depletion attraction Φ(z) between the ball wall can be obtained (Gong et al., 2011). A schematic representation of the principle is illustrated in Figure 2C.

In addition to the above methods, small-angle neutron scattering (SANS) could also be used to measure the depletion attraction in solution (Kumar et al., 2013; Ray and Aswal, 2014; Yadav et al., 2014). Kumar et al. (Kumar et al., 2013) used nano-sized silica dispersed in a solution of 0.1 M NaCl, adding poly(ethylene oxide-poly(propylene oxide-poly(ethylene oxide) triblock copolymers as depletion agents. The depletion attraction caused the aggregation of nanoparticles. The differential coherent scattering intensity per unit volume cross section of silica particles was measured by SANS. It can be determined the magnitude of the potential of depletion attraction.

In the aforementioned experimental measurement systems, the colloidal particles suspended in a polymer matrix provide a model system for studying the behavior of depletion attractions. Measuring depletion forces not only helps in understanding the phase behavior of colloidal particle-polymer systems, but also finds applications and attention in specialized functional material design such as targeted drug delivery and diagnosis, where some colloidal particles have demonstrated the practicality of protein sensor arrays. Many of these applications require an understanding of the interaction parameters in these complex systems. Despite interdisciplinary intersections, a comprehension of the depletion attraction forces and their complex interactions among the microstructure are still necessary.

When the size of colloidal particles in a dispersion is significantly different, the smaller particles can act as depletion agents, leading to depletion attractions between the larger particles. If these depletion attractions are strong enough, phase separation can occur as demonstrated by several studies (De Hek and Vrij, 1981; Pusey and van Megen, 1986; Yasrebi et al., 1991; Bartlett et al., 1992; Sanyal et al., 1992; Calderon et al., 1993; Duijneveldt et al., 1993; Kaplan et al., 1994; Buitenhuis et al., 1995; Ilett et al., 1995; Imhof and Dhont, 1995; Steiner et al., 1995; Herzfeld, 1996). The effect of overlapped excluded volume has been studied in colloidal solutions for a long time (Hobbie, 1998; Hobbie, 1999; Abbas et al., 2009; Hu et al., 2013; Ray et al., 2015). It has been successfully applied to the separation and purification of substances. For example, the separation of carbon nanotubes and graphene oxide (Padovani et al., 2015), and the separation of “isotopic” particles from colloidal particles (Perry and Manoharan, 2016).

Bibette developed a method for the purification of polydisperse emulsions into monodisperse emulsions using depletion attraction (Bibette, 1991). The addition of SDS above the critical micelle concentration (CMC) to a silicone oil micro-drop emulsion stabilized with surfactant SDS caused droplet flocculation when the SDS micelles act as depletion agents. Since the overlapped depletion volume (V _ov ) between large droplets was larger, flocculation was more likely. The volume fraction of the cream formed in the solution was high, and the larger droplets were the primary component while the smaller droplets remained in the solution for further separation. The cream was removed, redispersed in SDS, and a slightly lower SDS concentration was added. The newly flocculated cream was removed again, and the process was repeated to obtain large monodispersed droplets. By adding more SDS to the emulsion after the first removal of the cream, the remaining monodisperse droplets could be obtained by continuously removing the cream (Bibette, 1991).

The shape of larger particles affects V _ov and the magnitude of depletion attraction. Mason revealed the relationship between the shape of large particles and the depletion attraction and used these differences of depletion attractions to separate particles of different shapes (Mason, 2002). For example, spheres could be successfully separated from disks by exploiting the significant differences in the depletion attraction between disk-disk, disk-sphere and sphere-sphere configurations (Figure 3A). Compared with the configuration of the sphere-sphere, there were larger V _ov in rod-sphere system. Phase separation in the rod-sphere system was experimentally investigated by Koenderink et al. (Koenderink et al., 1999). By mixing smaller silica rods with larger silica microspheres, phase separation of the spheres occurred under the effect of depletion attraction from the rods (Figure 3B). In agreement with theoretical predictions, this phase separation could occur at very low concentrations of rods and spheres, which confirmed the theoretical model of A-O that depletion agents of the rods resulted in phase separation faster and better compared to the depletion agents of spheres. In nonactive colloidal systems with Brownian motion, phase separation will occur when the thermal disturbance (∼k _B T) is unable to overcome the binding of the aggregates by the depletion attraction. Therefore, in the system of active matter, efficient depletion attraction is required to facilitate phase separation.

Colloidal crystals are structures that are arranged in periodically ordered by colloidal particles, which can be used for the modulation of light and have promising applications in the development of new optical devices, enhanced anti-counterfeiting technology, information storage, sensors and other fields. Depletion attraction has been extensively used in the preparation of colloidal crystals.

Colloidal crystals that are patterned at specific locations can further broaden their applications. Dinsmore et al. (Dinsmore et al., 1996) etched microscopic steps on the container wall added smaller particles as depletion agents and successfully achieved preferential aggregation of colloidal particles at the step edges by exploiting the larger V _ov between the step edges and the colloidal particles. Similarly, using the differences in shape between different parts of the cell wall, it was possible to preferentially form aggregates at specified locations, e.g., V _ov (for large particles in a corner) > V _ov (for its surface on the vessel) > V _ov (between two spherical particles) in the presence of depletion agents, as shown in Figure 4A. Dinsmore et al. (Dinsmore and Yodh, 1999) succeeded in causing large spheres to preferentially form crystal nuclei in the corners of the vessel and grew along them to become crystals using this principle. Subsequently, nucleation and crystallization also occurred on the vessel walls. Ordered microstructures could trap or induce aggregation of particles at specific locations, thus forming specific microstructures that could aid in the preparation of highly ordered microarray structures. Du et al. (Du et al., 2020) successfully achieved the controlled deposition of monolayer colloidal particles on substrates using depletion attractions, combined with patterned substrates.

The fabrication of large-area multilayer crystals presents a challenging task due to the tendency of wrinkles and defects in one layer to affect particle alignment in subsequent layers, especially when compared to single-layer crystallization. To overcome this challenge, Kim et al. (Kim K. et al., 2019) prepared colloidal assemblies with multilayer crystal structures at the air-water interface using depletion attractions. As shown in Figure 4B, polyvinyl alcohol (PEG) was added to the aqueous dispersion of silica colloidal microspheres as depletion agents. The depletion attraction generated by PEG in the solution caused the silica microspheres to aggregate into a single-crystal film at the gas-liquid interface, which continuously reduced the distance between the mica flakes at both ends, and the single film was transformed into a multilayer structure. The thickness of the multilayer structure was related to the concentration of PEG and could be simply regulated by the concentration of PEG. This experimental phenomenon could be explained by the classical theory of elastic films at fluid-fluid interfaces, providing a theoretical basis for the fabrication of stable three-dimensional crystals and experimental support for understanding the role of colloidal particles at the interface.

Rod-shaped particles, unlike spherical ones, have the potential to form diverse crystal structures. Baranov et al. (Baranov et al., 2010) assembled nanorods into a two-dimensional hexagonal rotator crystal structure in organic solvents using depletion attractions. This ordered structure could be transferred from the solution to the substrate, yielding a monolayer of vertically standing rod-like particles stacked in a single crystal. Using the similar method, they then prepared three-dimensional crystals in the shape of octopod/tetrapod (Zanella et al., 2011). Dogic and co-workers have explored the structure of the rod-shaped fd virus assembled using glycolic anhydride (Dextran) as depletion agents (Barry and Dogic, 2010). It was found that when the concentration of depletion agents was high, due to the direction dependence of depletion interaction, the single-layer disks assembled by fd virus were superimposed to form smectic filaments. As the concentration of depletion agents decreased, the disks were no longer stacked on top of each other but instead combine laterally to form a single layer of film with a diameter of hundreds of microns. Li et al. have investigated the structure of rod-shaped active matter assemblied by depletion agents in the solution consisting of rods of tobacco mosaic virus (TMV) and methylcellulose (MC) (Li et al., 2013a; Li et al., 2013b). The results showed that TMV was able to form a super-lattice by MC induced depletion attraction, and that the lattice spacing was responsive to temperature and TMV concentration (Figure 4C). This lattice spacing was controllable over a wide range, up to five times the TMV diameter due to the use of depletion attractions.

In addition to rod-like particles, ellipsoidal particles with different chemical properties of the surface can also be assembled in richer structures of colloidal crystals by depletion attractions. Wang et al. (Wang L. et al., 2019) synthesized columnar spherical silica particles with a PS girdle in the middle, as shown in Figure 4D, which have theta-shaped surface chemistry. When situated on a quasi-two-dimensional plane, these particles could form two types of ordered structures through depletion attraction. The first was a steady-state “standing” crystal phase that was predominantly governed by the depletion attraction between the particle and the substrate, while the second was a sub-steady “prone” crystal phase dominated by depletion attractions between particles. When the relative strength of the depletion attraction between the particle and the substrate was adjusted, the two crystal phases could be converted into each other. Liu and co-workers have synthesized ellipsoidal colloidal particles with polystyrene-poly(3 (isobutenyloxy)propyltrimethoxysilyl)-polystyrene (PS-TPM-PS) (Liu et al., 2020). By modifying the properties of their surface chemistry, the colloidal particles bonded to specific parts of each other under the combined effect of depletion attractions and electrostatic repulsion, forming colloidal crystals with special structures such as square lattices and brick-wall type lattices. In addition, the study has explored the assembly of binary mixed colloidal systems of different shapes into crystalline alloys by depletion attractions (Young et al., 2012; Ye et al., 2013; Karas et al., 2016; Brown et al., 2018; Xie et al., 2018; Sung et al., 2019; Li et al., 2020; van Ravensteijn et al., 2020; Liu H. et al., 2021).

The magnitude of the depletion attraction depends not only on the shape of the large particles in the system, but also on the surface roughness of the large particles (and the vessel walls) (Zhao and Mason, 2008a; Zhao and Mason, 2008b). Figure 5A illustrates the basic principle. For instance, non-spherical and plate-like particles produced through photolithography technology have rough surfaces. If the size of the depletion agent was smaller than the surface roughness of these particles, then there would be a relatively small concentration difference between the depletion agent inside and outside two particles. Consequently, this results in weak depletion attraction that was insufficient to cause particle aggregation. Conversely, if the size of the depletion agent exceeded that of surface roughness on these particles, then there would be a significant concentration difference between its interior and exterior parts leading to strong depletion attraction causing particle aggregation. Therefore, by adjusting the ratio between surface roughness and size of depletion agents we could regulate their strength of attraction. By exploiting the surface roughness of the larger particles, it was feasible to create colloidal assemblies with distinct structures.

Zhao and Mason experimentally demonstrated the effect of surface roughness on the depletion attraction of large particles (Zhao and Mason, 2008a; Zhao and Mason, 2008b). They used roughness controlled depletion attractions (RCDAs) to assemble particles prepared by photolithography. The operation steps were as follows: Firstly, the non-spherical colloidal particle system was mixed with the depletion agent, and the non-spherical colloidal particles were confined to the bottom surface of the container by the attractive force of the depletion agent. Then, by tilting the sample, the non-spherical colloidal particles were concentrated by the force of gravity. As the roughness of the upper and lower surfaces of the particles was much smaller than that of the side surfaces, and the size of the depletion agent selected was between these two roughness values, the particles could be adsorbed on the bottom surface of the container by the attractive force of the depletion agent to form a monolayer two-dimensional colloidal system. If the thickness of the particles was not considered, the non-spherical colloidal system constructed by this method was a true two-dimensional colloidal system, and the interaction between the particles in the system was similar to that of hard spheres. Therefore, this system provided a very good research platform for studying some basic thermodynamic problems, such as how shape affects the colloidal thermodynamic properties.

They prepared pentagonal platelets and successfully attracted the two smooth surfaces of the two pentagonal platelets to each other, resulting in a dimer by RCADs, as shown in Figure 5A. To better understand how surface roughness and sub-particle characteristics affected depletion attractions, Zhao and Mason (Zhao and Mason, 2008a) conducted a simulation using platelets with controlled prolate hemi-spheroidal asperities on their faces. They found that taller asperities in the distribution of asperity heights had a significant impact on RCDAs, and that the specific type of lateral distribution and concentration of asperities also affected RCDAs. The simulation also showed that lock-and-key arrangements of mating surface asperities could enhance depletion attractions rather than suppress it. This suggested that precise control over the geometrical features on colloidal building blocks’ surfaces could either suppress or enhance depletion attractions depending on these features’ exact nature. This simulation work (Zhao and Mason, 2008b) confirmed the theoretical concept introduced earlier by Zhao and Mason that shape complementary lock-and-key colloids could increase attractive interactions between portions of their surfaces with complementary geometrical features (Zhao and Mason, 2008a), including positive and negative relief features (protrusions and indentations).

Zhao et al. successfully formed a series of two-dimensional assembled structures using differently shaped platelet colloidal particles selected depletion agents with smaller sizes than the roughness of the sides of the lithographed particles by RCDAs. The interactions between these colloidal particles were hard, and the depletion attraction between the colloidal particles and the substrate could form a better two-dimensional assembly structure. Guided by the RCDAs, triangle, square, rhombic and kite-shaped colloidal platelets were assembled to form rich liquid crystal, crystalline and glass on the plane (Zhao and Mason, 2009; Zhao et al., 2011; Zhao et al., 2012; Zhao and Mason, 2012; Zhao and Mason, 2015; Hou et al., 2019; Liu M. et al., 2021; Liu et al., 2022). The study of the assembly model had been of immense help in advancing the theory of condensed matter physics (Zhao et al., 2012; Zhao and Mason, 2012; Zhao and Mason, 2015). Badaire et al. (Badaire et al., 2008) have prepared cylindrical colloidal particles using photolithography. Similarly, using the different roughness of the upper and lower surfaces of the particles compared to the sides, choosing the right-sized particles as depletion agents could cause the upper and lower surfaces of the particles to attract each other to form chain-like assemblies (Figure 5B). Roughness-controlled depletion attractions could also be used for the self-assembly of colloidal particles on hard curved surfaces (Liu et al., 2022). Thereby providing experimental supported for the development of the theory and applications of curved surface self-assembly. Besides photolithography, colloidal particles with varying surface roughness could also be synthesized chemically. Kraft et al. (Kraft et al., 2012) synthesized PS-dimer particles with one smooth end and one rough end. When depletion agents of the right size were added to the dispersion solution, the smooth surfaces of particles were subjected to a strong depletion attraction, creating a directional attraction between each other. In contrast, the rough surfaces between particles and between rough and smooth surfaces of particles have no depletion attractions, resulting in a specially assembled structure. Similarly, they also synthesized colloidal particles in the shape of “Mickey Mouse,” two rough “ears” set at 90° to a smooth “head.” Particles could form more diverse assemblies structures in the presence of appropriate sized depletion agents (Wolters et al., 2015) (Figure 5C).

Apart from the applications mentioned above, depletion attraction could also be used to prepare micelles (Abbas et al., 2009; Lin et al., 2010; Repollet-Pedrosa and Mahanthappa, 2013; Yang et al., 2019), Pickering emulsions (Kim et al., 2017), and limit crystal growth (Ishibe et al., 2019), among others, but we will not list them all here. Depletion attraction has attracted widespread attention and research since its theoretical proposal in the 1950s, and it has been less than a century since then. With the emergence of numerous colloidal particle preparation techniques, the library of colloidal particle shapes that we have mastered continues to expand. We anticipate utilizing depletion attraction to assemble even more structures, which will undoubtedly drive its application in colloidal systems. From a fundamental research perspective, one of the future research priorities is still the self-assembly of colloidal particles using depletion attractions. Through a combination of experiments and computer simulations, we can study the different assembly structures of colloidal particles under depletion attraction, attempt to summarize general principles, and ultimately answer the fundamental question of the role of depletion attraction in the self-assembly of colloidal particles. Another research hotspot is in interdisciplinary areas, especially in the direction of biology, such as simulating collective behavior of bacterial aggregation. From an applied research perspective, applied research has been relatively lacking over the past few decades. Therefore, in the next 10–20 years, applied research needs to be strengthened, and better guidance of basic research directions through practical needs. From a technical perspective, utilizing depletion attraction to prepare new functional materials is a future development direction, especially in the fields of environment, energy, and communication control. Examples include new energy storage materials for electric vehicles, new photonic bandgap materials for controllable light transmission, and smart materials capable of sensing changes in the environment, all of which can help to address increasingly pressing issues related to people’s production and life, such as environmental pollution and information security.

The utilization of depletion agents for bacterial aggregate formation is widespread. Currently, they mainly include biological substances (e.g., succinose, alginate, non-adsorbent EPS, Mucin, DNA, Phage, F-actin, Hyaluronan) (Liu et al., 2008; Ghosh et al., 2015; Secor et al., 2018; Secor et al., 2021), chemical polymers (e.g., PEG, sodium polystyrene sulfonate, cellulose nanocrystals) (Eboigbodin et al., 2006; Schwarz-Linek et al., 2010a; Sun et al., 2012). Researchers have studied the particular mechanisms and physiological significance of depletion aggregates utilizing different agents. The type, shape, molecular weight, concentration, hydrophilicity, and salt responsiveness of the polymers impacted the aggregation state and morphology of bacterial aggregates. Sun et al. discovered that rod-shaped nanoparticles were highly effective in depleting flocculation of colloidal-sized bacteria and that phase separation of bacteria could occur in very low concentrations of rod-shaped CNC particles (Figures 6C, D) (Sun et al., 2012). Secor used different mutants of Pseudomonas aeruginosa by employing various depletion agents and producing phase diagrams. It was discovered that high molecular weight DNA had a stronger depletion attraction than low molecular weight DNA, and the mixture of DNA and mucin as depletion agents usually required lower concentrations to allow bacterial aggregation than when DNA alone was the depletion agent. Furthermore, the addition of DNA enzymes to degrade DNA, when DNA was the depletion agent, could cause the depletion aggregation of bacteria to disappear (Figures 6E–H) (Liu et al., 2008; Secor et al., 2018). Lobo-Cabrera found that larger polymer sizes or concentrations led to the development of interesting striated and dendritic colonies, as shown in Figure 6I. The analysis of the results of Brownian kinetic simulations showed the existence of a threshold polymer concentration that distinguished between the two growth mechanisms. Below this threshold, the main force driving polymer-induced compaction was hindering the spread of bacterial cells, while collective effects played a smaller role. Above this threshold, especially for large polymers, polymer-induced compaction was a collective phenomenon driven by depletion forces. Well beyond this concentration threshold, severely limited diffusion drived the formation of filamentous and dendritic colonies (Lobo-Cabrera et al., 2021). Additionally, the introduction of dispersants could also lead to changes in the viscoelasticity of bacterial cultures. Wang et al. added polyethylene glycol (PEG) with different molecular weights and concentrations to two types of Bacillus subtilis cell cultures and observed the cell clusters in cell suspensions. It was found that the viscoelastic changes in cell cultures were the result of depletion interactions and shear force competition. When depletion interactions dominated, cells tended to aggregate more, which led to an increase in viscosity; when shear forces were greater than stripping forces, this leads to dispersion of cell clusters and reduction in viscosity (Wang X. et al., 2019).

In addition to bacterial depletion aggregates being influenced by depletion agents, the different properties of bacteria can also affect the formation of aggregates, such as bacterial shape, surface properties, bacterial products and motility, and bacterial growth stage (Hayashi et al., 2003; Dorken et al., 2012; Secor et al., 2016; Ebrahimi et al., 2019; Azimi et al., 2021). Secor found that cells overexpressing the extracellular polysaccharides Pel and Psl maintained aggregation after a short period of exhaustive aggregation, whereas wild-type and mucinous P. aeruginosa did not aggregate. It was demonstrated that after bacteria form aggregates by depletion mechanism, intercellular interactions could be promoted to further form irreversible aggregates (Secor et al., 2021). Dorken et al. found that succinose secreted by Sinorhizobium meliloti promoted the formation of evacuated aggregates, as shown in Figure 7A. A mutant that overproduced the extracellular polysaccharide succinoglycan exhibited enhanced aggregation, resulting in phase separation of the culture. Applying purified succinoglycans from cultures to different concentrations of cells, the same “crowding mechanism” occurred, causing cells to form laterally arranged clusters (Dorken et al., 2012). Azimi et al. found that changes in the O-antigen glycans covering the LPS resulted in different types of aggregate assembly by altering the physicochemical properties of the bacterial cell surface (Figure 7D). In contrast to the lateral aggregates formed by wild-type PAO1, they found that loss of the O-specific antigen (OSA) and alteration of the lipopolysaccharide (LPS) core glycoforms resulted in increased cell surface hydrophobicity that overcame the entropic forces exerted by the host polymer, leading to disordered, irreversible aggregation (Azimi et al., 2021). Secor discussed the implications of the accumulation of filamentous Pf phages produced by P. aeruginosa in the biofilm matrix, as shown in Figure 7B. They interacted with matrix polymers to organize the biofilm into a highly ordered liquid crystal. This structural structure promoted bacterial adhesion, dry survival and antibiotic resistance-all features associated with biofilms (Secor et al., 2016). Eboigbodin used 0.5 to 2.5 wt% of Escherichia coli AB1157 harvested at different growth stages, adding different concentrations of non-adsorbent polymers sodium polystyrene sulfonate (SPS) for depletion aggregation studies. E. coli AB1157 was negatively charged and the cell surface properties changed at different growth stages (Eboigbodin et al., 2006). The cell surface charge was closely related to the non-adsorbent polymer concentration induced depletion aggregation. The differences in bacterial aggregation properties were due to changes in the amount of charge during bacterial growth (Hayashi et al., 2003). Meanwhile, Secor found different effects of depletion aggregation on the interaction of bacillus P. aeruginosa and spherical Staphylococcus aureus in co-culture of two different shapes of bacteria (Figure 7C). As each shape formed different communities, S. aureus enhanced its resistance to P. aeruginosa virulence factors through the formation of aggregates (Secor et al., 2021).

The formation of bacterial depletion aggregates is the result of an entropic increase and thus, unlike passive colloidal particles, the presence of flagella and the specific motility of bacteria strongly influence their formation (Schwarz-Linek et al., 2012; Porter et al., 2019). Schwarz-Linek et al. combined experiments, theory and simulations to demonstrate that the use of active colloids (e.g., motile bacteria) can dramatically alter this mixture of evacuated aggregates physical properties. First, significantly stronger interparticle gravitational forces were required to induce phase separation. The force pushing each cell was ≈ 3πησv, where σ ≈ 1 μm and v ≈ 10 μm/s were the typical cell size and velocity, respectively, and η ≈ 1 cP was the viscosity of the aqueous medium; this gived a propulsive force of ≈0.1 pN. Typical forces due to losses can be estimated by the ratio contact strength starting phase separation (≈k _B T, thermal energy) and its range (approximately the size of the polymer, e.g., by its rotational radius, r _g ≈ 10–100 nm), so that k _B T/r _g ≈ 0.05–0.5 pN. Thus, the competition between activity and interparticle attraction should yield interesting and novel physics. Second, the finite-sized aggregates formed under lower interparticle attraction conditions rotate in one direction. Presumably, bacterial suspensions contained a fraction of non-motile organisms (Wilson et al., 2011; Schwarz-Linek et al., 2012). In the presence of polymers, these cells should be the first to aggregate, producing “nuclei” for subsequent aggregation of motile cells. Preliminary BD simulations of motile and non-motile attractive dumbbell mixtures supported the idea that the rotation and translation of clusters were driven by external motile cells. Porter studied the depletion of aggregation of motile and non-motile bacteria in polyethylene glycol solutions in a short time (B10 min) using experimental and numerical calculations. Surface in semi-dilute polymer concentration state and short time scales, the motility of bacteria in dilute solution (Figures 8A–D) made it difficult to form depletion aggregates while greatly promoting the collision rate to enhance the formation of aggregates in a viscous environment, as Figure 8D showed (Porter et al., 2019).

The physiological implications of bacterial depletion aggregation due to the presence of non-interacting polymers have been studied extensively. Secor found that bacterial depletion aggregation induced significant antibiotic resistance, which relied on the SOS response, a stress response activated by genotoxic stress, as Figures 8E, F showed. These findings raised the possibility that targeting the conditions that promoted depletion aggregation or depletion-mediated mechanisms of tolerance may lead to new therapeutic approaches to combat chronic bacterial infections (Secor et al., 2018; Secor et al., 2021).

Bacterial aggregates are ubiquitous and their widespread presence in seawater, freshwater, and wastewater treatment systems as well as in chronic infection sites. The formation of bacterial aggregates in marine gel particles (MGP) has been reported to be dependent on the viscosity of plankton-produced polymers (Rochelle-Newall et al., 2010); the large number of polymers secreted by the host found at chronic infection sites may drive bacterial aggregation through entropic forces independent of biofilm formation, which did not require biofilm assembly function or even bacterial viability. Importantly, depletion aggregation significantly and rapidly increased bacterial antibiotic resistance through a mechanism mediated by LexA SOS regulation (Secor et al., 2021). Although extensive research has been conducted on how solid surfaces alter biofilm initiation and formation, studies on bacterial depletion aggregation are still limited due to the complex growth and movement of microorganisms themselves. Current research has focused on the factors affecting bacterial depletion aggregation, and more research is needed to explore further physiological mechanisms of aggregation and effective control of aggregation formation. The next research directions can be focused on the following aspects: 1) Use molecular biology method to dig deeper into the mechanisms of how different polymers affect bacterial aggregation. 2) Focus on the process of bacterial aggregation while swimming. Meanwhile, the physiological processes accompanying aggregation when bacteria growing deserve further exploration. 3) Development strategies of blocking the physiological responses leading to depletion of aggregation-mediated antibiotic resistance or to modify the host environment in a way that reduces the aggregation force generated by host polymers reduce the aggregation and colonization of pathogenic microorganisms.