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Surface tension of nanoparticle dispersions unravelled by size-dependent non-occupied sites free energy versus adsorption kinetics

Surface tension of nanoparticle dispersions unravelled by size-dependent non-occupied sites free... www.nature.com/npjmgrav ARTICLE OPEN Surface tension of nanoparticle dispersions unravelled by size-dependent non-occupied sites free energy versus adsorption kinetics 1,2,3 Hatim Machrafi The surface tension of dispersions presents many types of behaviours. Although some models, based on classical surface thermodynamics, allow partial interpretation, fundamental understanding is still lacking. This work develops a single analytical physics-based formulation experimentally validated for the surface tension of various pure nanoparticle dispersions, explaining the underlying mechanisms. Against common belief, surface tension increase of dispersions appears not to occur at low but rather at intermediate surface coverage, owed by the relatively large size of nanoparticles with respect to the fluid molecules. Surprisingly, the closed-form model shows that the main responsible mechanism for the various surface tension behaviours is not the surface chemical potential of adsorbed nanoparticles, but rather that of non-occupied sites, triggered and delicately controlled by the nanoparticles ‘at a distance’, introducing the concept of the ‘non-occupancy’ effect. The model finally invites reconsidering surface thermodynamics of dispersions and provides for criteria that allow in a succinct manner to quantitatively classify the various surface tension behaviours. npj Microgravity (2022) 8:47 ; https://doi.org/10.1038/s41526-022-00234-3 INTRODUCTION developed. The condition expressing the tangential stress balance, including surface-tension-induced convection, i.e. Marangoni Fluid dynamics of complex fluids represent a field of study that convection, at the interface is given by concerns a series of energetic, medical and industrial engineering applications. Since these applications concern, in many cases, ðσ  nÞþðσ  nÞþ γð∇  nÞ n  γ ∇ T  γ ∇ φ ¼ 0 (1) g l Σ Σ Σ T φ fluids wherein dispersions are used for material printing or separation processes at the surface level, it is important to control where σ and σ are the stress tensors at the interface on the gas g l the behaviour of surface-related mechanisms . Stability require- and liquid sides, respectively, n the normal vector on the interface, ments during the dispersion processing and the printing process γ the surface tension, (∇∙n) stands for the curvature of the depend on the physical properties, such as the viscosity, and its interface, ∇ = ∇ − nn ∙ ∇ for the surface gradient, T and φ stand deposition quality depends on controlling the fluid dynamics of for the temperature and nanoparticle volume fraction at the 2,3 the deposited fluids and the underlying mechanisms . Moreover, interface, respectively, whereas γ defines the surface tension the wettability is a relevant physical property for processes where derivative with respect to the temperature and γ the surface droplet impingement, thin film flows, microfluidics, surface tension derivative with respect to the volume fraction. Generally, 4,5 speciation and heat transfer are implied . γ is readily available and reasonably approximated to be In order to focus on surface-related mechanisms of complex constant. However, the behaviour of γ is not so clear. Since at fluid dynamics, microgravity experiments are useful, cancelling microgravity, convection responsible for fluid patterns is of the thereby the interference of buoyancy. A sounding rocket surface-tension type, it is crucial to not only have a physics-based experiment took place under the framework of the Advanced analytical expression for the surface tension of nanoparticle-laden Research on Liquid Evaporation in Space (ARLES) experiment fluids but also to understand the underlying mechanisms that supported by the European Space Agency (ESA). ARLES was part govern the surface tension of such complex fluids. of the payload in a SubOrbital Express rocket (MASER 14) and aims It appears from several experimental studies that apparently at studying the evaporation of pure and complex sessile droplets. contradictory tendencies of the surface tension as a function of It also serves as a preparation of an experiment to be performed in nanoparticle content are observed: the surface tension is observed the near future at the European Drawer Rack 2 on board the to increase, decrease or pass through a minimum as a function of 6–15 International Space Station. The evaporation of the complex the nanoparticle content in the dispersions . Even for the same droplets resulted into pattern depositions of nanoparticles, nanoparticle, e.g. SiO , a constant and increasing trend is 9,13 interesting for future printing applications. These experiments observed , while for Al O both decreasing and increasing 2 3 allowed studying the depositions, but not how fluid dynamics, trends, in two separate cases, are observed . Due to the diversity surface effects and particle–fluid interactions controlled those of nanodispersions, there is no universal relation yet that could depositions. Another sounding rocket experiment is planned to be comprehend and clarify quantitatively such observed trends. performed in the near future, where one of the focuses will be to Fitted correlations or empiric relations may give good comparison monitor the fluid dynamics of the complex droplets. In order to to experimental values, but are specific to experiment conditions, prepare the flight scenario, a numerical model has been do not explain physically why certain surface tension phenomena 1 2 3 Université de Liège, Institut de Physique, Liège 4000, Belgium. Université libre de Bruxelles, Physical Chemistry Group, Bruxelles 1050, Belgium. Sorbonne Université, UFR Physique, Paris 75005, France. email: H.Machrafi@uliege.be Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; H. Machrafi 7,10 occur and lack often generality . One may find some explana- begin with the latter. The surface concentration of the tions, but mostly based on a qualitative assessment of the Gibbs nanoparticles Γ is defined by the surface coverage θ times p p free energy or on mere intuition. Some works have explained the maximum surface concentration Γ : p;max surface tension behaviour by an energy variation upon nanopar- Σ Σ Γ ¼ θ Γ ; (2) ticle transfer to the interface , attractive van der Waals or p p p;max 6,17 attractive capillary forces or even by analogy with electrolyte the surface coverage θ stems from principles concerning solutions . Interestingly, it is just the presence of nanoparticles at p thermodynamic equilibrium and adsorption kinetics and it is the surface that is given as reason for surface tension increase in more appropriate to discuss it later in a proper context. For now, ref. , while the nanoparticle adsorption is suggested to cause a we will focus on the framework of the dividing surface and how decrease in the surface tension elsewhere . Others explain the surface concentration is represented within its context. surface tension decrease by a high ionic strength of the base In Eq. (2), Γ is the maximum surface concentration of the fluid countering the otherwise repulsive force between the p;max nanoparticles, assumed to be determined by the principle of nanoparticles and the liquid–gas interface . This, however, does 20–22 maximum stacking via a maximum coverage fraction f . not explain the decrease of surface tension of nanoparticle-laden Other geometrical considerations of particle adsorption have been fluids with low ionic strength, such as nanoparticle dispersions in 21,22 distilled water nor for surface tension minima. The initial treated in refs. , but we only need here their results for decrease of the surface tension is suggested to be due to the maximum coverage. In the presence of nanoparticles, the large spacing between the nanoparticles, favouring electrostatic liquid–gas layer can be defined as the layer where the forces between the nanoparticles or to initial adsorption of nanoparticles go gradually from a bulk concentration c to a 17 Σ nanoparticles at the liquid–gas interface . purely surface concentration Γ . The surface concentration is then The apparently contradictory explanations for surface tension usually obtained by integrating the concentration profile over the behaviour are often intuitively provided and many existing thickness of that layer. It has therefore, generally, a thickness that models, useful as they might be, only predict part of the is larger than the size of the nanoparticles, a thickness that is tendencies, which is a consequence of universal underlying defined by the difference between the dividing surface and an mechanisms still remaining unelucidated. This work develops an imaginary parallel surface, beyond which the bulk concentration is analytical model proposing a new insight in surface thermo- attained. The degree of strength of the interaction energy dynamics, surface energy and, more particularly, in the interac- between the fluid molecules and the nanoparticles in that layer tion between the dispersed phase and the liquid–gas interface. will determine the amount of nanoparticles that are ‘trapped’, i.e. The model mainly aims at elucidating the underlying mechanisms adsorbed, or allowed to disperse in the bulk. Each nanoparticle of the surface tension of nanoparticle dispersions, both correctly that adsorbs will push away fluid molecules. In analogy to the predicting and explaining thereby the different experimentally bulk, according to the lattice model, (where each lattice fits a observed tendencies. We start by formulating the framework liquid molecule), we can define that the surface area that is within which the liquid-gas interface is defined. This will also occupied by a fluid molecule harbours a possible adsorption site. allow introducing definitions of nanoparticle (excess) surface As such, the adsorption sites are geometrically equivalent to the concentrations based on geometrical and size considerations. The fluid molecules in the interfacial layer. In order to represent this nanoparticles are modelled as being incompressible, non- framework in a manner that fits Gibbs’ isotherm, we have defined stretchable and the only material that can be adsorbed on the a dividing surface. Speaking of the maximum surface concentra- liquid–gas interface. Then, an analytical expression for the surface tion, this also necessitates to project the real maximum tension of nanoparticles will be calculated and compared to concentration in the interfacial layer onto the dividing surface several experimental data of different nanoparticle dispersions. (that has zero thickness). This projection method is depicted in The model will be used to explain the different observed Fig. 1a, while Fig. 1b focusses on the projection of an adsorbing phenomena. Finally, it will be shown that two parameters can nanoparticle with the corresponding parameters that will be used predict the type of surface tension behaviour for all the in the model. Figure 1c shows the surface that is deemed to considered material systems. participate to the adsorption. In fact, upon adsorption, it can be imagined that not the whole surface of the nanoparticles participates in the process. It is quite difficult to assess the portion METHODS of surface that participates to this process and not much is known Representation of an interfacial layer on a dividing surface about this. In this work, we will heuristically assume that only half of a nanoparticle’s surface, i.e. the half that faces the dividing The surface energy of a nanoparticle dispersion can commonly Σ Σ be described by Gibbs adsorption isotherm dγ ¼Γ dη  Γ dη , surface, participates in its adsorption. The reason for this will be p f p f Σ Σ an exact differential. Here, η and η are the surface chemical discussed later. We will call this the participating surface. Later in potentials induced by nanoparticle and fluid surface coverage, ‘Three counter-intuitive effects of K on surface tension’,a respectively, whereas Γ and Γ stand, respectively, for the excess verification will be discussed to show that such an assumption p f surface concentrations of the nanoparticles and the base fluid. appears to be quite reasonable. Let us, at maximum coverage, order the nanoparticles into Choosing the Gibbs dividing surface to be there where the several layers that are parallel to the dividing surface. As we are excess surface concentration of the liquid equals zero, we set Γ ≡ 0. This leaves us with dγ ¼Γ dη . We will start discussing the in maximum coverage, each layer contains the same amount of excess surface concentration first. This inherently entails the nanoparticles. Let us then project, in each such layer,the definition of a framework that explains how to deal with the participating surface of the nanoparticles (as if a nanoparticle representation of a three-dimensional interfacial layer, whilst was a balloon that is cut in half, of which one half is spread over Gibbs adsorption isotherm imposes to work with a two- the surface) on the dividing surface (for the first layer at the dividing surface) and on subsequent imaginary layers parallel to dimensional one, i.e. the Gibbs dividing surface. As excess surface concentrations of nanoparticles Γ are not widely the dividing surface. This gives in each layer the same maximum documented, such a framework will allow us to propose surface concentration per unit surface, so that considering only analytical expressions for Γ . The excess surface concentration the layer adjacent to the dividing surface is sufficient to is composed out of a surface equivalent of the bulk concentra- determine the maximum surface concentration perunitsurface, tion, discussed later, and an actual surface concentration. Let us as Fig. 1ashows. npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; H. Machrafi Fig. 1 The projection of nanoparticles on the dividing surface. a Principle of projecting nanoparticles on dividing surface at maximum coverage. Note that at a coverage below the maximum one, the principle is the same. b Projection of nanoparticles (with volume V and surface A ) on a two-dimensional surface. The projected circles are oval because they are drawn in perspective, but they should be seen as circular for the spherical and disk nanoparticles, and as a square for the cubic nanoparticles. The fluid molecules (with volume V and surface A ) at the surface Σ, on which a nanoparticle adsorbs, also participate to the adsorption and are illustrated as fluid molecules that become projected (on the dividing surface) as two-dimensional adsorption sites, depicted for simplicity as a flat plane Σ at exactly the dividing surface. c Illustration of the surfaces that come into play in the volume-to-surface ratio for the spherical, cubical and disk nanoparticles. The images are not in scale. This will result in a molar concentration of nanoparticles per The projected surfaces depend on the size of the nanoparticles unit surface on the dividing surface that would be equivalent to and the fluid molecules (geometrically equivalent to that of the the corresponding real molar concentration in a realistic adsorption sites). This means that the difference in sizes between adsorption layer, through scaling by a certain defined character- the nanoparticles and the adsorption sites can be expected to istic length, so-defined as L .AsFig. 1b shows, the projected have a great impact on the adsorption process. Each adsorbed maximum surface concentration per unit surface times the nanoparticle will induce a change in the possible entropic characteristic length gives the same volume as that of the configurations of a great number of adsorption sites. As these nanoparticle. It follows then that that characteristic length must sites are not occupied, yet have a large influence on the entropy, be a volume-to-surface ratio. they are named as non-occupied sites, because their non- We define this volume-to-surface ratio by the total volume occupancy matters entropically. We can say that the total area of divided by half of the total surface (the participating surface, as is these non-occupied sites (denoted by subscript NO) per total shown in Fig. 1c). The fluid particles that surround this surface area is given by participating surface of a nanoparticle in a real interfacial layer Σ Σ Σ ς m ¼ ς m  ς m (3) NO f p NO f p are fully projected on the dividing surface as they represent geometrically the adsorption sites on that surface (see the where m and ς are, respectively, the number of particles of a schematic representation in Fig. 1b). The characteristic length of constituent per unit interface surface and the specificsurface these fluid particles, or adsorption sites, is calculated by a standard area perparticleofthatconstituent,whilstthe superscript Σ volume-to-surface ratio. indicates that it concerns a surface property. The specificsurface Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi Fig. 2 Projection of bulk concentration. Analogy of an imaginary projection of the nanoparticles at an imaginary surface at z = z with respect to the real projection on the dividing surface at z = 0. The two resulting molar concentrations per unit surface should be equal in the definition of the surface excess concentration. for disk nanoparticles. This leads to area of a constituent is given by ς  ,with i = NO, f, p, ρ L N i i A standing for non-occupied sites, the fluid and nanoparticles, ρ Γ ¼ f L 1 p (4) respectively. Equivalently, the specific surface area of a p;max constituent may conveniently also be given per mole,i.e. i Σ ς  .In thatcase, m would simply be the number of mole ρ L It should be noted that when nanoparticles are coated or i i of a constituent per unit surface via scaling by N and Eq. (3) surface treated, the maximum coverage might be less due to remains valid. Expressions for the characteristic length L will be i possible repulsive forces or more if the nanoparticles have a soft developed later (with L ¼L as geometrically the adsorption NO f compressible coating with interparticle attractive forces. Note that 20,23 sites are equivalent to the fluid molecules). It is important to similar expressions have been proposed in refs. . Nanoparticles notice here that ς introduces a size effect, i.e. the number may come in various shapes, the main ones kibeing often of densities depend on the size of the nanoparticles and the spherical, cubical or disk shape. We will, for the demonstration, surface fluid particles. limit ourselves to such undeformable shapes. The volume and We have now defined the framework for calculating the surface participating surface, i.e. V ; ,asdefined in Fig. 1c, would concentrations and may proceed with proposing a formula 4π 3 2 3 2 2 2 that allows calculating these concentrations. The maximum then be a ; 2πa ,8a ; 12a and πa h ; πa þ πa h p p p p p p p p p surface concentration is given by the maximum number of m for a spherical, cubical and disk nanoparticle, respectively. The p;max nanoparticle moles, , divided by a unit surface, i.e. volume-to-surface ratio L can then be calculated. For a spherical Σ 1 p;max Γ ¼ , where A is a total (arbitrary portion of unit) 2a t p p;max N A A t nanoparticle, L ¼ (with radius a ), for a square-like nanopar- p p surface. This can conveniently be rewritten as 2a ticle, L ¼ (with 2a the side of the cube) and for a disk-like V m A =2 p p Σ p p;max p 1 3 Γ ¼ ,where A and V are the surface and p p p;max a h A =2 A N V p p p t A p nanoparticle, L ¼ (with radius a and thickness h ). Note p p V a þ h p p p volume per nanoparticle, respectively. It can be noted that is A =2 that it is not straightforward to define a molar mass of a nothing else than two times the nanoparticle’s volume to surface nanoparticle, since it is not a molecule nor an atom. We will ratio, defined as L (see Fig. 1). Moreover, is the mole of approximate the molar mass of a nanoparticle, in analogy with N V A p that of a polymer constituted by many monomers, as an ensemble nanoparticles per unit volume, also given by ,where ρ and M p p of atoms or molecules chemically connected to one another. The are the nanoparticle’s density and molar mass, respectively. Also m A =2 molar mass of a nanoparticle equals then M ¼ M 0f , where f p;max p p p p p is simply the maximum geometric coverage fraction f , p represents the maximum stacking factor of spheres in a three- being f ≈ 0.547 for spherical non-overlapping hard particles on 21,22 pffiffi dimensional setting, assumed here to be f ¼ (that of an fcc or a two-dimensional surface .Thisvalue isobtainedby 3 2 considering random packing of spheres after their projection on hcp structure), M the molar mass of one atom or molecule, V p′ p 4π 3 the interface. It is therefore not the same as the random packing 0 the volume of one nanoparticle and V ¼ ℓ the volume of one of circles as the latter would neglect the purpose of the projection atom or molecule, assumed to be of spherical form, with ℓ the qffiffiffiffiffiffiffiffiffiffiffiffi method, where it is sought to obtain expressions on a 2D surface M 0 3 p radius of that atom or molecule ℓ ¼ . The effective radius whilst preserving the information from realistic 3D interfaces as ρ N 4π p A Fig. 1 shows. For cubic nanoparticles, it is reasonable to expect of a fluid molecule, assuming sphericity, can be calculated in the qffiffiffiffiffiffiffiffiffiffiffiffi that the maximum coverage will be close to unity, to that we take M 3 f 3 same manner as ℓ ¼ , where ρ and M are the fluid’s 21,22 f f f ρ N 4π f ≈ 1 for cubic nanoparticles . For disk-like nanoparticles, of 4π density and molar mass, respectively. With V ¼ ℓ and which the circular part faces the dividing surface, we take a f maximum coverage that corresponds to maximum standard A ¼ 4πℓ , the volume-to-surface ratio for a fluid particle is given pffiffi π 3 ℓ hexagonal stacking of circles on a surface, i.e. f ¼  0:907 by L ¼ . With these definitions, the geometric definition of the 1 f 6 3 npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi maximum surface concentration Γ can be calculated from densities of adsorbed nanoparticles, the total number of micro- Σ Σ m þm ! material properties and size values. ðÞ NO p states would equal W ¼ . Boltzmann’s equation of Σ Σ m !m ! The difference between the excess surface concentration Γ and p p NO Σ Σ m þm ! the surface concentration Γ is usually defined as ðÞ p NO p entropy would give S ¼ k lnðÞ W ¼ k ln . For a pure d B B Σ Σ Z m !m ! z p NO Γ  Γ  c dz ¼c λ (5) p p p b fluid, only one undistinguishable combination exists, i.e. S = p f k ln(1) = 0. The corresponding surface fraction is given by where c is the nanoparticle concentration in the bulk layer, and p y ¼ . The surface coverage is defined to be equal to the p Σ Σ m þm λ = z − z , with z the position at which Γ = 0 and z at which NO b b 0 0 f b surface fraction, θ ≡ y , so that θ + θ = 1. Defining the i i p NO we can consider conceptually to have a bulk concentration. In configurational entropy of dispersing, due to nanoparticle cover- order to determine λ , let us make a preliminary remark. The age and non-occupied sites, as Δs = S N − S N , and using d d A f A maximum surface concentration has been obtained by projecting Stirling’s approximation for the logarithm of factorials, gives a layer of adsorbed nanoparticles on a two-dimensional surface at the dividing surface, which we defined as Γ = 0. When doing this, Δs ¼k N θ ln θ þ θ lnðÞ θ (9) B A p p NO NO it has been explained that the characteristic length of nanoparticle where k and N are, respectively, Boltzmann’s constant and projection equals L . For consistency, we should do the same B A Avogadro’s number. In dilute conditions, the enthalpy of here. In the definition of the excess surface concentration, dispersing can be neglected. It should be noted that this enthalpy the term  c dz denotes a deduction from the surface z0 results from heat liberated or absorbed due to new interactions concentration of an imaginary extrapolation of the bulk concen- that stem from the mixing process, while it is not the same as the tration, integrated over the interfacial thickness λ . So, it should b enthalpy of adsorption, which plays a role in the equilibrium rather be seen as an imaginary surface-equivalent of the bulk adsorption constant. In such conditions, we deal with an ideal concentration, Γ  c dz ¼ c λ ,defined at an imaginary dispersion, being consistent with the Langmuir’s adsorption p;b p p b isotherm, of which a detailed deduction is presented in the next surface at z = z . This also means that it is analogous to the section. The Gibbs surface free energy of dispersing is then given projection of the bulk nanoparticle concentration on the dividing Σ Σ by Δg ¼TΔs resulting into surface, named here Γ , so that Γ ≡ Γ . It remains to find Γ . p* p* p,b p* d d Imagine at the dividing surface Γ = 0 a slab V of thickness L ,of Σ f p p Δg ¼ k TN θ ln θ þ θ lnðÞ θ (10) B A p p NO NO which the contents are projected on that surface. If the projected p ς nanoparticle surface concentration is given by Γ , then the f p* We define ω  and ω  . The chemical potentials of a p f ς ς Γ NO NO corresponding nanoparticle concentration in V would be as p component i are defined by defined by the projection procedure in Fig. 1. If the projected Σ Σ m þ m NO p Σ Σ specific surface area per mole of nanoparticles is given by ς , then p η ¼ N Δg A (11) i Σ d ∂m N the corresponding volume per mole of nanoparticles in V would A i Σ T ;p;m 8j≠i be ς L . The same could be done for the fluid particles, so that the p p φ ς Γ with i = p, f and j = p, f. The number ω can also be understood as p p volume fraction φ in that slab would be described by ¼ . 1φ ς Γ f f the number of adsorption sites per nanoparticle. We then use the Within the slab V , the projected surface concentration for the aforementioned definition θ ¼ , fill this in in Eq. (10), apply i Σ Σ fluid particles Γ would simply be equal to the bulk concentration m þm f* NO c times the thickness of V , i.e. Γ ¼ c L . We then have Eq. (11) and rewrite the result back in terms of θ . This finally gives f p f f p φ ρ f f Σ Γ ¼ c L . Note that c ¼ðÞ 1  φ . Filling in the definitions η ¼ k TN ω ln 1  θ and p f p f B A f p ς 1φ M f p f ρ L Σ ρ M M p p p f f η ¼ k TN ln θ  ω ln 1  θ (12) of ς and ς leads to Γ ¼ L φ ¼ φ. Figure 2 B A p p p p f p p M ρ L ρ L M L f f p p f f p ς ς p p ς illustrates the analogy that we have discussed here. where ω   (as ω  and ω ≡ 1 because the 2 p f f L ς ς ς ρ NO f NO p p As c ¼ φ , it follows that λ ¼ . Filling this in (5) gives, with p b adsorption sites are within the present framework geometrically M L p f Eq. (2), for the excess surface concentration equivalent to the projected liquid molecules, see Fig. 1b and Σ Σ Σ corresponding discussion) is given by Γ  θ Γ  φΓ ¼ θ K  φ Γ (6) p p p Σ p;max b b M ρ L p f with ω ¼ (13) ρ M L f p ρ L p p p (7) Γ ¼ λ ¼ φ M L p f Surface adsorption isotherm p;max K ¼ (8) Equilibrium of the adsorption process is described by a net zero change of the total Gibbs free energy of the system: Σ Σ Σ b Σ b where Γ is given by Eq. (4) and φΓ is the surface equivalent dΔG ¼ dΔG þ dΔG þ dΔG þ dΔG  0, where the subscripts p;max a ad ad d d of the bulk concentration and K a constant that measures the Σ ‘ad’ and ‘d’ denote the adsorption (due to translational or potential of the nanoparticles to rather adsorb at the interface confinement effects, or effects related to particle surface energies, than stay dispersed in the bulk. As will be seen later, K is a Σ dipole-dipole and coulomb interactions , for instance) and the function of nanoparticle size, maximum packing and fluid dispersion (mixing) free energies, respectively, and ‘Σ’ and ‘b’ molecule size. The surface coverage θ in Eq. (6) will be treated p the surface and bulk phases, respectively. We focus first on the Σ Σ m þm in the context of surface kinetics, but we will first deduce the Σ NO Σ dispersing. We can then define ΔG ¼ A Δg and d d surface chemical potentials and surface adsorption. b b m þm b p b b b ΔG ¼ V Δg , where m and m are the number of bulk d d l p fluid particles and nanoparticles per unit volume of the dispersion, Surface chemical potential Σ whereas A and V are an arbitrary unit surface and volume, If we have m number densities of adsorption sites, containing Σ Σ m þm Σ Σ NO p m number densities of non-occupied sites and m number respectively. Note here that has unit moles per unit surface NO p N Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi b b m þm θ p d p ln c K þ ln  ln x ¼ 0, which (keeping in mind that and unit moles per unit volume. Note also that we can t p p 1θ N p p p d x c = c )simplifies to ¼ K c , which is the well-known define a mole fraction of nanoparticles in the bulk as x ¼ . p b b p t p p 1θ p m þm Langmuir’s adsorption isotherm, subject to further discussion in Neglecting the enthalpy of dispersing as mentioned before, an the next section. This can be rearranged into addition of nanoparticles to the surface would result into a change Σ Σ Σ Σ d dΔG m ; m , which can mathematically be written as dΔG ¼ K c d p f d p θ ¼ (14) Σ Σ Σ Σ m þm m þm ∂ NO p Σ Σ ∂ NO p Σ Σ 1 þ K c N A Δg dm þ N A Δg dm ¼ p A Σ A Σ N d p N d f ∂m A Σ ∂m A Σ T ;p;m f T ;p;m Σ Σ Σ Σ The equilibrium adsorption constant K could be calculated A η dm þ η dm . From (3) and the total number density, it can p p f f thermodynamically via Van’t Hoff’s relation. However, experimental Σ Σ Σ be derived that dm ¼ ω  1 dm . With the definitions for η p p values for the molar adsorption enthalpies and entropies are not and η (see Eq. (12) and text above), this leads with ω = 1to f readily available for the studied nanoparticle dispersions and Σ Σ especially not for various concentrations. Other expressions and dΔG ¼ k TN ln A dm . An equivalent procedure can be B A d 1θ p methods make use of more available surface energies and surface performed for the bulk phase leading, under the approximation tensions. However, even if one may perform such measurements, b b of diluted dispersion, to the relation dΔG ¼ k TN ln x V dm . B A p d p such a procedure would not allow an analytical physics-based Mass conservation stipulates that the net mass change is analysis of the behaviour of the surface tension and would not Σ b Σ b zero, i.e. A dm þ V dm ¼ 0. This leads to dΔG þ dΔG ¼ offer the understanding of the underlying mechanisms for the p p d d Σ various surface tension behaviours. Therefore, it would not align k TN ðlnð Þ lnðx ÞÞA dm . B A p 1θ with the purposes of this work. In order to obtain theoretical A change in the nanoparticles number in both phases upon parameters, independent of the experimental surface tension data, adsorption also induces a change in the free energy of the experimental regression procedures or any fitting methods, one of adsorption, which can be symbolically (as an already existing the often-used ways is to use a kinetic model. Adsorption and thermodynamic relation for the free energy of adsorption will be desorption are often described kinetically. Material properties for used, there is no need to enter into details as we did for the kinetic models are readily available for solid–liquid interfaces and Σ Σ free energy of dispersion earlier) written as dΔG Am þ ad p the methods are widely used and understood. As less data are b b Σ Σ b b available for liquid–fluid interfaces, it is the question whether dΔG Vm ¼ Δg A dm þ Δg V dm . We can use the mass ad p ad p ad p similar kinetic models would be applicable. One can argue that the Σ b conservation principle A dm þ V dm ¼ 0 and write p p adsorption of surface-charged nanoparticles (an important method Σ b Σ b Σ Σ dΔG þ dΔG ¼ Δg  Δg A dm  Δg A dm ,where Δg to obtain stable dispersions) on liquid-fluid interfaces (often ad ad p p ad ad ad ad charged with the same sign) can be approximated by adsorption stands for the net difference of the free energy of adsorption per on solid–liquid interfaces. Although subject to more investigation, mole of adsorbed nanoparticles. At equilibrium, Δg is related to ad it has already been applied successfully for liquid–fluid interfaces . the thermodynamic equilibrium constant K via Van‘t Hoff’s This motivates that within such a reasonable assumption the equation for adsorption Δg = −RTln(K )with K the thermo- ad e e equilibrium adsorption constant for the nanoparticle dispersions dynamic equilibrium constant of adsorption. The total Gibbs free can be obtained without fitting. The interpretation of underlying energy of the system dΔG becomes finally dΔG ¼ a a mechanisms would benefit from such a physics-based approach. p Σ Σ k TN ðlnð Þ lnðx ÞÞA dm  RTlnðK ÞA dm  0 at equilibrium. B A p e p p 1θ Surface kinetics This leads finally to lnK þ ln  ln x ¼ 0, which is known e p 1θ It remains to find the equilibrium adsorption constant K or for as an adsorption isotherm for ideal dispersions or solutions. The p later convenience, a dimensionless version K thereof. ‘Surface equilibrium constant K is for ideal cases related to the dimensional adsorption isotherm’ presented the thermodynamic theory behind Langmuir equilibrium constant K , which can be traditionally the Langmuir adsorption isotherm. It was mentioned that described by the equilibrium adsorption reaction: c + [*] ⇆ [P − *], unavailable experimental data for the nanoparticle dispersions where c is the nanoparticle molar bulk concentration, [*] the studied here and the aim to provide for a physics-based model surface molar concentration of empty adsorption sites and encourage the use of another way. Commonly, the equilibrium [P − *] the surface molar concentration of adsorbed nanoparticles, adsorption constant is determined kinetically, where material respectively. If we define Γ as the maximum surface p;max properties necessary for the model are readily available. The concentration, we can write ½  þ½ P ¼ Γ , which is kinetic model is based on an equilibrium between standard p;max ½ P adsorption and desorption kinetics and is treated in details in the equivalent to defining the surface coverage as θ ¼ and p Σ Γ 27–30 p;max literature . We mention the main points here. Note that therefore  1  θ . Note that later, we will use the notation Γ desorption becomes relevant when the energy of particle Σ p p p;max trapping is of the order of the thermal energy. Adsorption (with ½ P θ d p for [P − *]. Thermodynamically, K ¼ ¼ .As c has unity c ½ p c 1θ standard rate k ) depends on the bulk concentration c and the pðÞ p a p available adsorption sites (1 − θ ). Desorption (with standard moles per unit volume, K has unity volume per mole. Furthermore, p rate k ) depends on the adsorbed nanoparticles θ per specific d p as K is dimensionless, this means that we can define K ¼ c K , e t surface area of adsorbed nanoparticles ς . This writes as where c must have unity moles per volume. Similar discussions on j ¼ k c 1  θ (15) the various definitions of K and K have been performed in the a a p p literature, indicating that K in Van’t Hoff’s equation is dimensionless d 1 and that K in the Langmuir’s adsorption equation has a dimension p j ¼ k θ d d p (16) 25,26 depending on the concentrations, confirming this analysis .We p can also deduce (in dilute systems, c ≈ c ,with c and c the molar f t f t From kinetic considerations, it can be stated that nanoparticle concentrations of the base fluid and the bulk phase, respectively) accumulation, through a flux balance equation, at the interface is that c can be represented by the molar concentration of the bulk ∂θ 25,26 1 p phase . Filling this in the adsorption isotherm gives finally given by ¼ j  j , where we remind that here ς is the a d p ς ∂t npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi specific surface area per mole of nanoparticles. At quasi- permittivity of the phase at the opposite side of the interface ∂θ opposed to the relative permittivity of the phase where the stationarity, i.e. ¼ 0, we have from Eqs. (15) and (16) that ∂t nanoparticles are dispersed, ε . The van der Waals potential energy ς c k c p p a p between the nanoparticle and the interface is given by kd θ ¼ ¼ (17) 1 k k þ k c 1 þ ς c d a p ς p p p k A a a z d pΣ p p vdW Φ ¼ þ þ ln (23) pΣ 6 z z þ 2a z þ 2a p p d ka Comparison with (14) learns that K ¼ ς . As the molar p k concentration can also be expressed into the volume fraction φ by where A is the non-retarded Hamaker constant for the p−Σ c  φ, we can rewrite (17) as particle–interface interaction, where the particle (p) interaction ρ with air (a) through the base fluid (f) is assessed. This constant is k p ς φ k M d p derived by the theory of London-dispersion forces and is often θ ¼ (18) k p approximated by the combining relation A ¼ A ¼ 1 þ ς φ pΣ pfa k M d p pffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffi 39,40 A  A A  A . Non-DLVO interaction energies pp ff aa ff This allows defining a dimensionless Langmuir’s equilibrium may be considered as one potential energy, being either repulsive k p k a a 1 36 constant K  ς ¼ , and relating the surface coverage p p or attractive depending on the solid–water contact angle .In k M k L d p d p with the bulk volume fraction as another work, a Hydra parameter, depending on the hydropho- bicity of the surface, was introduced in one expression, being K φ θ ¼ (19) either negative or positive, defining, respectively, a hydrophilic 1 þ K φ repulsive or hydrophobic attractive potential energy. The Hydra 36,37,41,42 potential energy is given by refs. If we assume that particle transport occurs under a quasi-linear z z and stationary diffusional regime (this is a valid approximation Hy 0 (24) Φ ¼2πa λ γðÞ 1  CosðÞ ϑ e p 0 pΣ 0 because of the very small relaxation time), it has been shown that general analytical solutions for the adsorption and desorption where λ is a decay length, ϑ the radial liquid–solid static contact qffiffiffiffiffiffiffi 0 Φm 41,42 k πk T 27–30 a B k T angle and z a constant of the value of 0.16 nm . The total constants can be obtained ,i.e. ¼ δ e , which finally 0 k jj Φ d m EDL EDL vdW Hy potential energy is given by Φ ¼ Φ þ Φ þ Φ þ Φ . pΣ pp0 pΣ pΣ leads to sffiffiffiffiffiffiffiffiffiffi δ πk T m Material properties m B k T K  e (20) Table 1 shows the material properties of the used nanodispersions L jj Φ p m at ambient temperature. Effect of temperature on solid properties where Φ is the total potential energy, Φ , at a distance z = δ , i.e. m t m is neglected. For the base fluids, only the densities are adapted for Φ ¼ Φ j . The total potential energy stands here for the m t z¼δ m temperatures different than ambient. Since these values are well potential energy between a particle and the liquid-air interface, being composed of many mechanisms. The DLVO theory Table 1. Material properties and physical constants. mentions that the most important interactions are the electro- EDL vdW a 3 1 static Φ and van der Waals Φ interaction ener- Component Density [kg m ] Molar mass [kg mol ] pΣ pΣ 24,31–33 gies . Image charge effects in the form of a repulsive Al O 3950 0:102 2 3 EDL particle-image Φ potential energy are esteemed to be of pp0 Al 2700 0:027 importance, the reason being that in cases of particles being B 2370 0:011 oppositely charged to the interface, repulsion was still MgO 3580 0:040 Hy 24,34 observed . Non-DLVO interaction energies Φ , suggested pΣ SiO 2650 0:060 to be of the Lewis acid-base type, also appear to be of great Ag 10490 0:108 importance, such as hydrophilic repulsive interactions and Laponite 2530 2:287 35–38 hydrophobic attraction energies . The electrostatic double ZnO 5610 0:0814 layer interaction potential between a nanoparticle and a flat Dodecanethiol-ligated Au 4720 0:198 fluid–air interface is given by Water (W) 997 0:018 zþ2a k T ζ e ζ e z B p e Σ e EDL λ λ D D Φ ¼ 64πε ε Tanh Tanh a e þ e r 0 p n-decane (D) 730 0:142 pΣ e 4k T 4k T e B B Ethanol 789 0:046 zþ2ap λ λ D D Ethylene glycol (EG) 1110 0:062 þλ e þ e Þ (21) Tri-ethylene glycol (TEG) 1120 0:150 where ε , ε , e , ζ , ζ , and λ , are, respectively, the relative r 0 e p Σ D n-dodecane (DD) 750 0:170 permittivity, the absolute permittivity, the elementary charge, n-hexadecane (HD) 773 0:226 the zeta potential of the nanoparticles, the zeta potential of the liquid–air interface and Debye length. Debye’s length is given by a First nine rows concern the nanoparticle densities ρ and molar masses qffiffiffiffiffiffiffiffiffiffiffi p ε ε k T r 0 B M . The tenth to sixteenth-row concern the base fluid densities ρ and p' f λ ¼ , where I stands for the ionic strength of the base D 2 2N e I A e molar mases M . fluid. The potential energy between a particle, p, and its image, p′, b Volume-averaged and mole-averaged values are given for the density and in the phase at the other side of the fluid-air interface is given by molecular weight, respectively, based on the dimensions of the core gold 24,34 refs. nanoparticle and the dodecanethiol ligand shell. Note that the molar mass M ' given here is the one of an atom or molecule. To obtain the molar mass of 2 p z V k T ζ e ζ e p 4π 3 B p e p0 e 2 EDL pffiffi λ a nanoparticle, one must make the conversion M ¼ M 0 f ¼ a ρ N . p p p A D (22) p p Φ ¼ 32πε ε Tanh Tanh a e V 0 9 2 r 0 p p pp0 23 −1 e 4k T 4k T The values of the used physical constants are N = 6.02 * 10 [mol ], R = e B B A g −1 −1 −12 −1 −1 −19 8.3145 [J mol K ], ε = 8.854*10 [C V m ], e = 1.602 * 10 [C] and 0 e where ζ stands for the zeta potential of the image particle, given −23 −1 p′ k = 1.38 * 10 [J K ]. ε ε ζ e 2k T r 0 p e 33 B r by ζ ¼ ArcSinhð Sinhð ÞÞ . Here, ε is the relative p r′ e ε þε 0 2k T e r r B Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi Table 2. Data needed for calculation of equilibrium constant K . −20 −20 NP-L A [10 J] A [10 J] ε [−] ϑ [°] λ [nm] −ζ [mV] T [K] 2a [nm] Ref. pp ff r 0 p p a b c d w 6 Al O -W 15 3.7 80 35 0.72 64 300 50 2 3 a b e f w 6 Al O -D 15 5.45 2 26 1.16 55 300 50 2 3 g b e h 6 Al-D 15 5.45 2 33 0.87 57 300 18 i b e h x 6 B-D 6.23 5.45 2 33 0.35 55 300 46 a b i j w 6 Al O -E 15 4.2 25.3 23 1.83 38 300 50 2 3 g b i k 6 Al-E 15 4.2 25.3 36 0.98 61 300 18 l b i k x 6 B-E 6.23 4.2 25.3 36 0.71 63 300 46 a b m n 10 Al O -TEG 15 5.8 23.3 30 0.8 59 298 20 2 3 o b m n 10 MgO-TEG 12.1 5.8 23.3 30 0.67 46 298 20 a b c p 9 SiO -W 6.5 3.7 80 20.7 0.8 50 298 30 g b c q 9 Ag-W 50 3.7 80 40 0.78 45 298 100 r b c s y 19 Lap-W 1.06 3.7 80 24 0.57 49 300 25 (1.5) a b t u 11 ZnO-EG 9.2 5.6 40 36.4 0.57 60 300 67 v b e v z 38,47 (dl)Au-D 28 5.45 2 30.5 1.95 30 303 5 (1.7) v b e v z 38,47 (dl)Au-DD 28 5.8 2 33 1.71 35 303 5 (1.7) v b e v z 38,47 (dl)Au-HD 28 5.2 2 36 1.5 70 303 5 (1.7) a b c d 48 Al O -Ws 15 3.7 80 35 0.72 75 300 40 2 3 The base fluids W, D, DD, HD E, TEG, and EG stand for water, n-decane, n-dodecane, n-hexadecane, ethanol, tri-ethylene glycol, and ethylene glycol, respectively. Ws stands for fully stabilised water dispersion . The temperatures for which the experimental data are obtained from the literature are indicated in the table. If in the literature it is mentioned that the experimental data are obtained at ambient temperature, the value of 300 K is used. a 52 ref. . b 53 ref. , the value of TEG is approximated as that of di-ethylene glycol. c 54 ref. . d 55 ref. . e 56 ref. , same value assumed for n-dodecane and n-hexadecane. f 57 ref. . g 58 ref. . h 59 ref. , the value of B is approximated as that of Al. i 60 ref. . j 60–63 refs. , interpolative estimation. k 64 ref. , the value of B is approximated as that of Al. l 65 ref. . m 66 ref. . n 67 ref. , assumed from values of EG on mixed ceramic substrates. o 68 ref. . p 69 ref. . q 70,71 refs. , averaged value. r 72 ref. . s 73,74 refs. , averaged values. t 75,76 refs. , averaged values. u 77 ref. , approximated. v 38 ref. . w 6 TEM images in ref. show agglomeration so that the size of the nanoparticles is ~2 times that of the initial one (25 nm). x 6 SEM images in ref. show cubic-like particles with an averaged size of 80 nm so that, taking this size between opposite corners of a cube, one cube side would be 80/√3≈46 nm. The value between the brackets is the thickness of the nanodisks. z 38,47 The core diameter of Au is 5 nm andthe ligand shell thickness is 1.7 nm withan overall reported diameter of 2a = 8.4 nm . The B nanoparticles were approximated as cubic particles, evidenced from SEM images in ref. and the Laponite nanoparticles are nanodisks of a flat (the thickness is much smaller than 19 6,7,9–11,38,47,48 the radius) cylindrical shape , while the rest are spherical nanoparticles . tabulated, not more attention is given. Some data are reported in It should be noted that it is difficult to obtain precise values for the literature as a function of the mass fraction. If ξ is the the parameters ϑ, δ , ζ , I, λ ,and ζ , which need some m int 0 p nanoparticle mass fraction, ρ the nanoparticle density and ρ discussion. Reasonable values can be obtained from experimental p f the fluid density, then the nanoparticle volume fraction φ can be data for ϑ, δ , ζ , I. The minimum thickness between the m int nanoparticle and the interface at adsorption, δ , is often assumed ξ ξ 1ξ m calculated as φ ¼ þ . Table 1 also shows general 27,43 ρ ρ ρ p p f to be of the order of δ = 0.5 nm . For the interface zeta physical constants used in the model. potential, ζ , the approximated mean value of ζ = −40 mV is int int 44,45 Equations (20)–(24) allow calculating the equilibrium constant taken for water . For ethanol, tri-ethylene glycol, ethylene K . Several data are needed for this calculation. These data are p glycol and glycerol the same value is assumed, while n-decane, collected from the literature and tabulated in Table 2. A summary n-dodecane and n-hexadecane are considered to be an oily liquid of the variables and their meaning is given in Table 3. as hexane and a value of ζ =−10 mV is taken . The ionic int npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi Table 3 continued Table 3. Variables used in the model and their meaning. Symbol Description Unit Symbol Description Unit λ Debye’s length [m] a Nanoparticle radius [m] −3 ρ Density of fluid [kg m ] A Non-retarded Hamaker constant [J] p−Σ −3 ρ Density of nanoparticles [kg m ] −3 p c Bulk concentration [mol m ] −2 σ Stress tensor gas-side of interface [N m ] D Ratio excess surface to surface-equivalent [−] −2 σ Stress tensor liquid-side of interface [N m ] of bulk concentrations ς Specific surface area of fluid molecule [m per particle] e Elementary charge [C] ς Specific surface area of non-occupied site [m per particle] NO f Maximum geometric coverage fraction [−] ς Specific surface area of nanoparticle [m per particle] h Disk nanoparticle thickness [m] p −2 φ Volume fraction [−] H Non-occupancy effect [J m ] NO −3 Φ Total potential energy at first minimum [J] I Ionic strength [mol m ] m EDL −1 Φ Repulsive particle-image potential energy [J] k Adsorption rate [m s ] pp0 EDL −1 Φ Electrostatic particle-interface [J] k Boltzmann constant [J K ] pΣ potential energy −1 k Desorption rate [s ] Hy Φ Hydra potential energy [J] pΣ K Equilibrium adsorption constant [−] vdW Φ Van der Waals potential energy [J] pΣ K Ratio surface to bulk preference [−] Φ Total potential energy [J] ℓ Equivalent size of fluid molecule [m] ω Number of adsorption sites for one [−] ℓ Equivalent size of nanoparticle [m] nanoparticle L Characteristic length of fluid molecule [m] Subscript L Characteristic length of nanoparticle [m] p Nanoparticle Σ −2 m Number of fluid molecules per unit surface [particles m ] NO Non-occupied site Σ −2 m Number of non-occupied sites per unit [particles m ] NO f Fluid surface Σ −2 m Number of nanoparticles per unit surface [particles m ] −1 strength of a fluid is somewhat an unknown. However, works M Molar mass fluid molecule [kg mol ] −1 have indicated that for deionized water, typical ionic strength M Molar mass nanoparticle molecule unit [kg mol ] values are measured of the order of I = 1mol/m .Thisvalue is −1 M Molar mass nanoparticle [kg mol ] p' assumed for all the fluids used. The values for λ and ζ depend 0 p n Normal vector [−] strongly on the experimental conditions and only ranges can be −1 N Avogadro’s number [particles mol ] A indicated. Decay lengths, λ , of values up to 2.2 nm are reported 20,37,41 −1 −1 for several systems . The zeta-potentials ζ of nanodisper- R Universal gas constant [J mol K ] p sions were typically found to be approximately between −75 and T Temperature [K] 20,34,43 −25 mV . Educated guesses, not affecting the analysis in Greek symbol this work, for these two parameters within these indicated ranges −1 γ Surface tension [N m ] are implemented in Table 4 for the calculation of the equilibrium −1 γ Surface tension of fluid [N m ] f constant. The obtained equilibrium constants for the nanoparticle −1 −1 γ Temperature derivative of the surface [N m K ] dispersions are shown later in Table 5. tension −1 γ Volume-fraction derivative of the surface [N m ] φ Reporting summary tension Further information on research design is available in the Nature −2 Γ Excess surface concentration of fluid [mol m ] Research Reporting Summary linked to this article. −2 Γ Excess surface concentration of [mol m ] nanoparticles Σ −2 RESULTS AND DISCUSSION Γ Surface concentration of nanoparticles [mol m ] Σ −2 Comparison of model with experimental data Γ Surface-equivalent of bulk concentration [mol m ] Σ −2 Gibbs adsorption isotherm dγ ¼Γ dη can now be integrated. Γ Maximum surface concentration [mol m ] p p;max We use Eq. (6) for Γ (with (19) for θ ) and Eq. (12) for η . The δ First minimum of potential well [m] p p m p −1 −1 surface tension of nanoparticle dispersions is finally given by ε Absolute electric permittivity [C V m ] ω þ ω KðÞ φ þ K  K K ω þ ω K K  1 p p p Σ p Σ p p p Σ ε Relative electric permittivity [−] Σ γ ¼ γ þ R TΓ φ  ln 1 þ K φ g p f b 1 þ K φ K p p ζ Zeta-potential interface [V] int (25) ζ Zeta-potential nanoparticles [V] Σ −1 where γ is the surface tension of the base fluid, R the universal η Surface chemical potential fluid [J mol ] f g gas constant, T the temperature, K given by (8), K given by (20), Σ −1 Σ p η Surface chemical potential nanoparticles [J mol ] ω given by (13) and φ the nanoparticle volume fraction. The θ Non-occupied site coverage [−] NO equilibrium constant is a kinetic parameter obtained by models θ Nanoparticle coverage [−] from the literature, summarised in ‘Surface kinetics’. The other ϑ Contact angle [°] parameters are developed in this work using geometric principles and characteristic length scales, which would, for clarity, benefit λ Decay length in hydra potential [m] from a summary in Table 4. Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi γ − γ = Δγ + Δγ , where f p NO Table 4. Definitions, necessary for the calculation of the surface Z Z φ φ ∂η ∂η tension. p p p Δγ ¼ Γ dφ ¼ Γ dφ (26) b Σ ∂φ ∂φ 0 0 a,b b Symbol Definition Z Z φ φ ∂η ∂η Γ 2ap 2ap aphp NO NO p L − p Δγ ¼ Γ dφ ¼ Γ dφ 3 3 a þh (27) p p NO p ∂φ ∂φ 0 0 f 0.547 1 0.907 − L −− − Equations (26) and (27) show the multiplication of two terms in 4π 3 ∂η pffiffi Σ M a ρ N NO p p A 9 2 the integral. The term ðΓ Þ in Eq. (27) stands physically for ∂φ Σ ρ f L p;max 1 p the change in the surface chemical potential of non-occupied Mp Σ 2 adsorption sites per unit surface upon a change in the nanoparticle ρ L b p p Mp L f bulk concentration. It is worthy to note that this emphasises M ρ L p f f ω the influence that non-adsorbed bulk nanoparticles have on the ρ M L p f p surface chemical potential of non-occupied sites, called here the K p;max non-occupancy effect. For later use, we assign for this term the following symbol qffiffiffiffiffiffiffi Φm K δ πk T p m B k T L jj Φ p m ∂η Σ NO H ¼Γ (28) NO a ∂φ For the first two rows, the first column stands for spherical nanoparticles, the second column for cubical nanoparticles and the third column for disk- A larger absolute value of H means a greater non-occupancy NO like nanoparticles, while for the third row only the fourth column is used, effect, i.e. one bulk nanoparticle will have more impact on the standing for the fluid molecules. For the fourth to ninth rows, the definitions are general for all surface energy (and thus the surface chemical potential) of nanoparticle shapes and fluid molecules. The symbol a stands either for p the non-occupied sites. Note that an equivalent analysis can be the radius of a spherical nanoparticle, half of the side of a cubical made for the adsorbed nanoparticles contribution (see Eq. (26)) nanoparticle or the radius of a disk nanoparticle, the latter of which has ∂η Σ p through the term ðΓ Þ¼H , called the occupancy effect. The thickness h . Further, ℓ is the equivalent radius of a sphere corresponding ∂φ p f to the volume of a fluid molecule, while ρ , ρ , M , and M stand for the p f p f term ð Þ stands for the excess surface concentration normalised nanoparticle and fluid densities and the nanoparticle and fluid molar by the surface-equivalent of the bulk concentration. We will assign masses, respectively. It is worthy to note that it is not necessary to know the following symbol to it the molar mass and density of the nanoparticles to calculate the definitions in this Table and that it is mainly a question of size. D ¼ Nevertheless, the values of ρ and M (from which M is obtained) are (29) p p' p p Σ Σ Γ still given in Table 1 should one need to know the values of Γ and Γ b p;max in terms of unit mass per unit surface. The values of f have been adapted A positive value of D means a high degree of adsorption of for the (dl)Au nanoparticles due to the presence of ligands at the gold nanoparticles (decreasing the surface energy), while a negative surface inducing possible repulsion or blocking mechanisms. In ref. ,it has been established that the (dl)Au nanoparticles occupy 0.2, 0.34, and value indicates a preference of nanoparticles to remain dispersed 0.72 of the theoretical maximum coverage when dispersed in D, DD, and in the bulk. In summary, the sign of D indicates whether the HD, respectively. Therefore, for the (dl)Au-D, (dl)Au-DD and (dl)Au-HD surface tension will increase or decrease and the value of H NO systems, f has been multiplied by 0.2, 0.34, and 0.72, respectively. It is with what amplitude. As both depend on φ,itiseasyto recalled that δ and Φ represent the primary minimum of the total m m understand that the magnitude and variation of the surface potential energy and Φ its value, whereas k and T are Boltzmann’s m B tension might be different as a function of φ, generating the constant and the temperature. different observed trends. More interestingly, Table 5 shows that the several nanoparticle dispersions considered here have quite different values for the nanoparticle equilibrium adsorption constant K .Thisimplies thatthisproperty playsanimportant Table 5 shows the nanoparticle dispersions that we consider in role in determining the behaviour of the surface tension. Note this work. For completeness, the calculated numerical parameters that the parameters D and H also depend on the surface p NO that are necessary for determining the surface tension as a coverage θ , which is linked to φ through K . This encourages to function of the volume fraction, i.e. γ , K , ω and Γ , are given in p p f Σ p consider φ and the property K as suitable parameters for the Table 5 for these nanoparticle dispersions. p present analysis. Different kinds of behaviours for the surface tension of We should define a certain reference system that represents a nanoparticle dispersions are represented by several experimen- nanoparticle dispersion of which we can change freely the tal case studies, representing different materials (for both the 6,9–11,19,38,47,48 parameters φ and K and monitor their influence on the behaviour nanoparticles and liquids) with different sizes . of H and D and therefore on that of the surface tension. To NO p The surface tension is calculated from Eq. (25)for these perform numeric demonstrations, allowing the quantification of our nanoparticle dispersions and compared to the experimental analysis, we may choose data from any dispersion. Only because data in Fig. 3a–f. The experimental data in Fig. 3 show different the B-D system is an example of an interesting decrease-increase types of behaviours and the present model has an overall good behaviour, its data are used for the present demonstration. As agreement with those data. This motivates to use the model to the discussion should be followed in a general sense, and we only explain these observations. use the physical properties of this dispersion but changing freely K , it is appropriate to name it differently: the reference system R1. Non-occupancy contribution Figure 4a shows the surface tension of the R1 system as a We candividethe surfacechemicalpotential, η  η þ η ,in function of φ for various imposed K values and two different p p NO p a partthatstandsfor thecontributionbynon-occupied sites nanoparticle sizes. η ≡ −ω R Tln(1 − θ ) and a part that represents the Note that for small K values, the surface tension remains NO p g p p contribution of the adsorbed nanoparticles η ≡ R Tln(θ ). We significantly constant. As this is counter intuitive (usually p g p can also split the surface tension change into two parts as non-adsorption should lead to an increase in the surface tension npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi Table 5. Calculated parameters used in Eq. (25) for the nanoparticle dispersions. −1 −3 2 3 Σ −2 NP-L γ [mN m ] K [10 ] K [10 ] ω [10 ] Γ [nmol m ] T [K] f Σ p p b Al O -W 72.3 2.11 0.0562 6.23 148 300 2 3 Al O -D 23.8 4.66 11.5 1.28 67.1 300 2 3 Al-D 23.8 12.9 3.19 0.165 186 300 B-D 23.8 9.22 4.16 2.08 38.0 300 Al O -E 22.4 3.12 0.294 2.85 100 300 2 3 Al-E 22.4 8.66 0.103 0.369 279 300 B-E 22.4 6.17 0.0988 4.65 56.8 300 Al O -TEG 44.45 10.3 10.3 0.262 190 298 2 3 MgO-TEG 44.25 10.3 4.34 0.262 190 298 SiO -W 72.5 3.51 O (0) 2.24 247 298 Ag-W 68.0 1.05 0.209 24.9 74.1 298 Lap-W 73.6 43.5 193 0.872 85.1 300 ZnO-EG 47.3 2.29 0.0964 5.27 75.9 300 (dl)Au-D 22.96 27.7 8726 0.036 399 303 (dl)Au-DD 24.75 29.2 8.55 * 10 0.033 380 303 (dl)Au-HD 26.96 31.8 2.23*10 0.027 349 303 Al O -Ws 72.3 2.64 O (0) 3.99 185 300 2 3 The symbols W, Ws, D, DD, HD, E, TEG, EG and G stand for the base fluids water, extra stabilised dispersion, n-decane, n-dodecane, n-hexadecane, ethanol, tri- ethylene glycol, ethylene glycol, and glycerol, respectively. Lap stands for laponite and (dl)Au for dodecanethiol-ligated gold. Note that K values that are orders of magnitude smaller than unity have been considered here as being virtually zero, O(0), i.e. negligibly small. as this entails that Γ < 0), special attention will be given to the that K has the same type of effect on the surface tension for p p small K -case later. Figure 4a shows that, as K increases, γ(φ), for a whatever nanoparticle dispersion’s physical properties. Figure 4e p p given φ, first increases and then starts to decrease for small φ, shows H , D and D H (combined contribution of the latter NO p p NO two) versus φ for three K values for the R1 system. followed by an overall decrease in the depicted φ -range. This first For small K ð¼ Þ, Fig. 4e shows that D is significantly increase is also counter intuitive (usually more adsorption should p p lead to a decrease of the surface tension), a second point given negative over the whole volume fraction range. Figure 4e shows special attention later. As K continues to increase, even a that at small K there is a negligible contribution of the absolute value (being, by the way, always negative) of H (dotted blue minimum in γ(φ) as a function of φ is observed, a third point NO line), it is significantly constant over the φ range. Although D is discussed later as well. For even higher K , the surface tension p clearly negative (dotted red line in Fig. 4e), which stands for a shows a decreasing trend, which is what one would expect. Figure negative surface excess concentration and would conventionally 4a also shows that a smaller nanoparticle size tends to favour a imply an increase in the surface tension, the resulting surface decrease in the surface tension. The latter effect can be understood tension remains significantly constant as shown by Fig. 4a by noticing that smaller nanoparticles will increase, for the same φ, (straight solid lines). To understand why this is, we take the limit the number of nanoparticles and therefore also the number of of Eq. (25) for K → 0, which gives adsorbed nanoparticles, which leads eventually (for sufficiently p small nanoparticles) to a decrease in the surface tension. γ ¼ lim γ ¼ γ þ R TΓ φ K !0 f g p b (30) Let us, before entering into such an analysis, first determine K !0 what contribution to the surface tension is more important, Δγ or Δγ . Figure 4b shows, through Δγ (i = p,NO) scaled by R T, that NO i g Filling in Eq. (30) the data for the R1 system reveals that 5 4 the contribution of the non-occupied sites (i = NO, the solid lines) γ  γ ¼ Oð10  10 Þφ. This explains the seemingly (in K !0 f is the main one, especially at larger K -values. The main reason for reality, very weakly increasing) constant value of the surface this is size-related. The nanoparticles are much larger than the tension. The reason behind the seemingly constant value of the fluid molecules, which constitute the adsorption sites. This means Σ surface tension at K → 0 lies in the value of Γ . From Table 4 p b that the number of fluid molecules involved in the adsorption of a one may easily deduce that Γ / . Nanoparticles have generally a ℓ p f nanoparticle is quite large, expressed in large ω -values, i.e. ω ≫ p p a larger size than fluid molecules and apparently large enough for 1, as Table 5 shows. So, it is now evident that the non-occupied Γ to be sufficiently small and hence a seemingly constant site contribution of the surface chemical potential will be a key behaviour of the surface tension can be predicted. This explains part in the following discussions. the first counter-intuitive observation. At K = K , Fig. 4e shows that for a small range of φ we have p p* Three counter-intuitive effects of K on surface tension D > 0 (equivalent to Γ > 0, let us recall), whilejj H becomes p p NO Figure 4c shows Δγ and Γ for two volume fractions for the NO p bigger than for the previous case (see solid line in Fig. 4eas R1 system as a function of K . To facilitate the discussion three opposed to the dotted line). As H <0 (always), this leads to NO markers have been introduced for Δγ : one corresponding to NO D H < 0 (solid line in Fig. 4f). As we increase φ, Fig. 4e shows p NO the K from Table 5 for the B-D system (≡K ), a smaller one that D changes sign, i.e. D < 0, withjj H still being significantly p p* p p NO larger than zero, resulting into D H > 0. As the surface tension and a larger 100K one. Figure 4d represents Δγ and Γ for two p* NO p p NO volume fractions as a function of K for a so-called R2 system, depends on the integration of D H from 0 to φ, the surface p NO where we use the data from the Ag-W system, merely to illustrate tension will decrease as long as D H < 0 and increases as long p NO Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi Fig. 3 Modelled surface tension as a function of the volume fraction of nanoparticle dispersions, compared to experimental 6,9–11,19,38,47,48 data .a Al O -W ( , ), Al O -Ws ( , ), Laponite-W ( , ) and ZnO-EG ( , ), b Al O -D ( , ), Al-D ( , 2 3 2 3 2 3 ) and B-D ( , ), c Al O -E ( , ), Al-E ( , ) and B-E ( , ), d Al O -TEG ( ,*) and MgO-TEG ( , ), e SiO -W ( , ), Ag- 2 3 2 3 2 W( , ), (dl)Au-D ( , ), (dl)Au-DD ( , ) and (dl)Au-HD ( , ), the inset being a zoom of the (dl)Au-D system concentrating at the region of smaller φ-values, f a magnification of the systems (dl)Au-D ( , ), (dl)Au-DD ( , ) and (dl)Au-HD ( , ). The studied systems are indicated in the form “nanoparticle-fluid”. The model values are indicated by lines, while the experimental data are given by markers in the form (line,marker). K K 1 p Σ as D H > 0, passing thus through a minimum. This analysis p NO when φ ¼ or φ = 0, but the latter is a trivial solution not implies that the sign of the integrated surface of D H as a p NO considered further. The sign of D depends on the values of K p p function of φ will determine the existence and positioning of a and K . With respect to this, two cases can be considered: K Σ p surface tension minimum. It is then logical to elaborate further on and K > . These cases will depend on the parameter K . From p Σ the dependence of H and D on φ. From Eqs. (6) and (19), we NO p Σ have that Table 4, we can deduce that K / f . The values of parameter Σ 1 f (see Table 4) are constant for a certain shape and K will K φ ∞ Σ D ¼ K  φ ℓ (31) f p Σ depend on the ratio much more than on f . Therefore, for the 1 þ K φ a p p discussion of Eq. (31) we will only take into consideration K / and from Eqs. (19) and (28) that K K 1 1 1 p Σ We treat the case K  . When K  , we have  0 and p p K K K Σ Σ p K ω p p Σ it can be verified that this means that for all φ > 0 we have D < 0. H /Γ (32) NO 1 þ K φ As H is always negative, the result is a strictly increasing surface p NO tension. Depending on the amplitude ofjj H , this increase will NO The analysis of Eqs. (31) and (32) needs some mathematical be significantly measurable or not. Focussing mainly on the value considerations. From Eq. (31), we can easily deduce that D ¼ 0 of K (the value of which may vary orders of magnitude more than p p npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi Fig. 4 Analysis of surface tension behaviour. a γ of the R1 system vs φ for theoretically imposed K values ( =~0, = 10, = Δγ 2 3 3 10 , = 10 , = 5*10 for 2a = 46 and 5 nm. b with i = p,NO vs φ of the R1 system, where thin and thick lines stand for K = 10 and p p R T 5*10 , respectively, and dashed and solid lines stand for i = p and i = NO, respectively, for 2a = 46 and 5 nm. Note that all the dashed lines Δγ p;NO are significantly horizontal. c of the R1 system vs K for φ = 0.001 (dashed line) and φ = 0.025 (solid line) and 2a = 46 with three specific p p R T values of K based on the value from Table 5 (K = 416): ( for φ = 0.001 and for φ = 0.025), K ( for φ = 0.001 and for φ = 0.025) p p* p* and 100K ( for φ = 0.001 and for φ = 0.025). On the second axis, Γ of the reference system vs K for φ = 0.001 (dashed line) and φ = p* p p Δγ p;NO 0.025 (solid line) and 2a = 46, d of the R2 system vs K for φ = 0.001 (dashed line) and φ = 0.025 (solid line) and 2a = 100. On the p p p R T second axis, Γ of the R2 system vs K for φ = 0.001 (dashed line) and φ = 0.025 (solid line) and 2a = 100, e H vs φ, and D vs φ, for three p p p NO p specific values of K : (dotted line), K (solid line) and 100K (dashed line) for the R1 system, f D H vs φ for three specific values p p* p* p NO of K : (dotted line), K (solid line) and 100K (dashed line) for the reference system and D H for K = K (dotdashed line) for the p p* p* p NO p p* Ag-W system. ω and Γ , see Table 5), two cases are thus possible: vanishing be verified thatjj H has a significant value, the increase of the p b NO K values (K → 0) and non-vanishing K values (Oð0Þ K  , surface tension will be measurable. A real example for this is the p p p p where O(0) stands for a value that is so low that considering it zero Ag-W system, where Table 5 shows that K ≫ O(0) and K < . p p would reflect a measured reality). Moreover, Fig. 4f illustrates this as well by a continuously increasing D H (brown dot-dashed line). Departing from a p NO o K ! 0 fully desorbed case (K → 0), we can say that upon enhancement (K ≫ O(0)) of the surface adsorption kinetics (up to the limit Equation (32) shows that for small K (e.g. K → 0), we have p p p K  ), the surface tension behaviour becomes a strictly jj H ! 0, so that the surface tension increase is not noticeable. p NO K This has been discussed previously around Eq. (30) for small K increasing one due to the combination of D <0 and a significant p p values and a real example for this is the SiO -W system (see Fig. 3e value ofjj H . So, initially, a higher adsorption appears not to lead NO 1 p Table 5 where indeed K ). p to a lower but rather to a higher surface tension. As / , KΣ KΣ 1 nanoparticles (having much higher size than the fluid molecules) o Oð0Þ K allow for a much larger limit for K for which D remains negative. K p p So, within this limit, upon increasing K , the strength of the non- When K is significantly non-zero but not too large, i.e. Oð0Þ occupancy effectjj H becomes significant, whilst the excess NO K  (defined as the lower-intermediary region), so that it can surface concentration remains D <0, resulting into a surface p p Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi tension increase and not decrease. This explains the second Table 6 shows that as one goes from left to right, the value of counter-intuitive observation. K K increases by several orders of magnitude. This is by either p Σ 1 1 We treat the case K > . When K > , we have a particular increasing K , K or both. Table 5 shows that K for all the p p p Σ Σ K K Σ Σ −2 −3 situation where D >0 (and therefore a decreasing surface tension) nanoparticle dispersions is of the order of 10 −10 . This means K K 1 p Σ that when K K increases several orders of magnitude, this is for 0 < φ < and D < 0 (increasing surface tension) for p Σ Kp mainly due to K . Nevertheless, for quantitative assessments, it is K K 1 p Σ φ > , where we limit φ to a certain maximum value φ , max p more convenient to mention K K . p Σ considered reasonable for typical nanoparticle dispersions (as In order to put the results in perspective, some additional 1 1 discussed later). It can be verified that the surface tension is strictly comments are in place here. In the case K > and K < 1 þ ,we p Σ K K Σ p K K 1 p Σ decreasing if 1, which means for K 1 þ . This gives have made a distinction for the surface tension behaviour K K p p between three ranges of orders of magnitude for K K . Mathema- two regions, the one given by the latter condition and K <1 þ . p Σ tically, these three cases represent all a minimum in the surface tension somewhere in the range 0 < φ < 1. The reason for making o K < 1 þ p the three distinctions is on a conceptual level, involving measured data and a defined framework. As Fig. 3 shows, most of The values given in Table 5 show that for nanoparticle nanoparticle dispersions that are used for engineering purposes dispersions typically K ≪ 1. This means that K ≫ 1. The smaller Σ p (one might also think of medical ones as well, for that matter) K K 1 p Σ ℓ (due to larger nanoparticles as K / or due to lower K ) is, present operating conditions that involve φ values that are often Σ p K a p p −2 limited by a value φ that is of the order of φ ≈ O(10 )or the closer the value of φ for which D changes sign will be to zero. max max −1 slightly higher, but still φ < O(10 ). In Table 6, φ is max max As a consequence, this fits within the typical operating φ-ranges 1 1 schematically indicated for the case K > and K <1 þ by blue p Σ (φ < φ ), resulting into an observable minimum for the surface max K K Σ p tension. This is the case for e.g. the B-D, Al-D, Al O -D systems (see 2 3 vertical dotted lines, set to a same hypothetical value for the three K K 1 p Σ images in question. It shows that as K K increases the minimum Fig. 3b and Table 5 for the values). As, however, becomes p Σ of the surface tension becomes less pronounced and shifts somewhat larger (smaller nanoparticles or higher K ) the φ for towards higher φ values (not on scale), falling out of the range which D changes sign will increase and may fall out of the limited by φ .At φ values beyond φ it is the question max max aforementioned typical operating φ-ranges (φ > φ ). This results max whether we can still speak of dispersions and we then might have into a minimum that is no longer observed (mathematically still to deal with another type of “fluid” with additional phenomena at present, but experimentally not observed within typical φ-ranges) the surface. When working with nanoparticle dispersions, we and the surface tension is virtually decreasing. This can also be have limited the analysis within the range 0 < φ < φ (named max numerically verified in Table 5 and visually in Fig. 3a for e.g. the the “operating range”). As such, depending on the value of K K , p Σ Lap-W system. For systems with even higher K , Eq. (32) shows the mathematical minimum of the surface tension may well be out thatjj H as well as the negative part of D H become more NO p NO of that range and therefore not observed nor experimentally important, confirmed by the 100K case for the R1 system in p* measured. Then, it is justified to indicate conditions (that is, within Fig. 4(e) and (f) (dashed lines). The (dl)Au dispersions (see again the range 0 < φ < φ ), where we can observe a minimum in the max Table 5 and also Fig. 3(e) and (f)) illustrate this situation by 0 1 surface tension (for K K ≈ O(10 −10 )) and where we observe a p Σ presenting surface tensions that decrease quickly for very low virtual decrease. Even for the virtual decrease of the surface volume fractions. In summary, this means that an observable tension, we have made a distinction between a “soft” decrease surface tension minimum is the result of a delicate balance 2 3 4 6 (K K ≈ O(10 −10 )) and a “steep” decrease (K K ≈ O(10 −10 ), p Σ p Σ between a sufficiently large, but not too small, nanoparticle the upper limit 10 being indicative with respect to the observed (through K / ) and a sufficient amount of adsorption (through experiments, but may conceptually be even higher). The soft K > ). This effect is therefore not an external one but stems from decrease is defined as the surface tension having a steady the same parameters that cause strictly increasing or decreasing decrease over the whole operating range, such as the Lap-W case. behaviours, merely because the conditions are right. This explains The steep decrease is characterised by a strong decrease of the the third counter-intuitive observation. surface tension for φ ≪ O(φ ) with a seemingly constant value max afterwards, such as the dl(Au) dispersions. For the parameter K , we have mentioned that for nanopar- o K 1 þ Σ Σ p ℓ ticles we have K / ,not considering f in the discussions. Σ ∞ There are, however, cases where this parameter may play a role. Mathematically speaking, a strict decrease (over the whole When strong repulsive forces are present or when the range 0 < φ < 1) in the surface tension would occur if, next to 1 1 nanoparticle surfaces (because of their nature or their functio- K > , we have K 1 þ . We have mentioned earlier that p Σ K K Σ p nalization) are such that we cannot consider them as hard typically K ≫ 1. This means that, as approximation, we are spheres, the maximum coverage may, respectively, decrease or practically dealing here with the condition K ≥ 1, which entails increase, whereas the shape may also be altered by the that a  Oðℓ Þ. This would besides possibly quantum dots or p f stretching or compressing of the adsorbed nanoparticles. In surfactants, be rather untypical for nanoparticle dispersions. such cases, additional considerations should be made in order Therefore, this case can be disregarded as well for nanoparticle to include these forces between nanoparticles .One maysay dispersions in general. that these forces will be effective there where the nanoparticles are present at the interface, so that the surface tension of Trends and comments particle-laden interfaces is argued to be an effective magni- In fine, it seems that the right combination between adsorption tude . In some cases, the nanoparticles are grafted with strength (K ) and nanoparticle size (a of which the main effect is polymers, which may cause additional effects on the surface p p represented by the parameter K ) is responsible for the different tension due to the dangling chains of the polymers .When ions behaviours. Table 6 shows a summary of the different surface are present (one may think of electrolytes or charged organic tension behaviours as a function of the parameters K and K ,in molecules) strong coulomb interactions may also influence the p Σ the form of the product K K . maximum coverage. p Σ npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi Table 6. Surface tension behaviours as function of K (adsorption) and K (size). p Σ Theeffect of thenanoparticlesize has been mainly expressed be studied in more details. In addition, it would be encouraged to through the parameter K . It should be recalled that the provide benchmark studies with experimental data on the nanoparticle size also figures in the parameter ω . The parameter adsorption coefficients of various nanoparticle adsorption on ω (standing for the number of adsorption sites) can also liquid–air (fluid) interfaces. quantitatively interfere with the magnitude of the surface Finally, the present model considers the adsorption of tension change through its linear relation with the non- nanoparticles alone in order to focus on this phenomenon. It occupancy effect Eq. (32). Moreover, the parameter ω ,when would be interesting to generalise or adapt Eq. (3) for the much larger than unity, is responsible for the non-occupancy inclusion of the adsorption of molecular species, which could effect to outnumber the occupancy effect through the surface generalise the model for the application of studying the surface chemical potential (see Eq. (12)). Should it be around unity, the tension of solutions containing surfactants or other (in)organic free energy contribution of the adsorbed nanoparticles would molecules. Should multiple adsorption occur, the thermodynamic also be important for the chemical potential and our discussion model presented in this work lends itself to be extended starting, would be different. Nevertheless, once it is established that for most importantly, from an adaptation of Eq. (3). nanoparticles generally ω ≫ 1, and that the variation of K is, as p p mentioned earlier, of far more importance for the non- Mapping of the surface tension behaviour occupancy effect, the variation of the parameter ω is not given The previous analysis has shown a general dependence of the more attention in our analysis. surface tension behaviour on K K , where K and K stand for the p Σ Σ p The size of the nanoparticles also matters from another point effect of nanoparticle size and adsorption strength, respectively. In of view. The projection method necessitates that the radius of order to quantify this dependence and map these behaviours, we curvature (reciprocal of the curvature) should be much larger choose four representative systems, having, respectively, see- than the nanoparticle radius. In other words, the interface should mingly constant, strictly increasing, minimum containing and be “flat” with respect to the size of the nanoparticles. If the virtually decreasing behaviours for the surface tension. Figure 5a pressure difference over the interface is negligible, Young’s shows the values of D , jj H and D H for these four p NO p NO equation (where the pressure difference is related to the surface nanoparticle dispersions, at two volume fractions, that have tension and the interface curvature) predicts that such an distinct behaviours with low to high K K in the following order: p Σ assumption would be realistic. SiO -W < Al O -W < B-D < Lap-W. Figure 5a shows that, although 2 2 3 In ‘Representation of an interfacial layer on a dividing surface’, SiO -W and Al O -W have comparable negative D values, D H 2 2 3 p p NO we mentioned that we used half the surface to calculate the is only significant for Al O -W due to a much higherjj H , NO 2 3 volume-to-surface ratio. A heuristic reason was employed for confirming the analysis in the previous section, which means a this, assuming that only half the surface facing the dividing non-measurable increasing surface tension for SiO -W and a surface would matter in the adsorption process for particles that measurable one for Al O -W. As K K increases, i.e. for B-D, we can 2 3 p Σ are much larger than the fluid molecules that constitute the see a positive D for φ = 0.005 and a negative one for φ = 0.01. As adsorption sites. As a verification, we performed surface tension the valuejj H is significant enough, this results into a visibly NO calculations using cases with a fourth, a sixth and the whole negative D H for φ = 0.005 and a positive one for φ = 0.01, p NO particle’s surface to calculate the volume-to-surface ratio. It meaning first a decrease and then an increase in the surface appeared that the heuristic choice we have made for the tension. For even larger K K , i.e. for Lap-W, we can see a positive calculation of the volume-to-surface ratio, i.e. using half the p Σ D for both φ’s. With a largejj H , D H is considerably nanoparticle’s surface, was the most appropriate one with p NO p NO negative for both φ’s, corresponding to a virtually decreasing respect to the experimental data. It would be interesting to surface tension behaviour that was observed for Lap-W. investigate the degree of this participating surface experimen- In the present study, we aimed at proposing a framework, tally. However, for this work, the heuristic choice we have made appeared to be sufficient. model and explanation dealing with the different behaviours of It should be noted that the way K has been calculated assumes the surface tension of nanoparticle dispersions. We have seen that that it is enough to take into account the wettability of the the adsorption strength (K K ) and the nanoparticle size (through p Σ nanoparticles in the potential energy. The DLVO theory is known K ) collaborate or compete in determining these different 24,51 to be used for adsorption on solid-liquid interfaces. In refs. as tendencies. It is then interesting to map the surface tension well as in the present work, it is assumed that the DLVO theory, behaviours of all the nanoparticle dispersions that were presented albeit extended, is applicable for liquid-fluid interfaces as well. in Fig. 3 as a function of K and K . Such a mapping is presented in p Σ 24,51 Although already used by others , such a kinetic model should Fig. 5b and gives the opportunity to tailor nanoparticle Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi = . = + = . Fig. 5 Mapping of surface tension behaviour. a Competition of key parametersjj H , D and D H for the nanoparticle dispersions SiO - NO p p NO 2 W(1), Al O -W (2), B-D (3) and Lap-W (4) for two volume fractions: φ = 0.005, φ = 0.01. Note that for a better visualisation both the values of 2 3 D and D H for only the Lap-W system have been divided by 5 and 2 for the cases φ = 0.005, φ = 0.01, respectively. b All the considered p p NO nanoparticle dispersion systems are resumed in a K − K map, representing five observed γ-vs-φ behaviours: “significantly constant” where p Σ the change (proven to be mathematically an increase) in γ is not observable (less than 1%) on the 1 mN/m range, “strictly increasing” where ∂ γ > 0 over any φ-range, “distinct minimum” where a clear minimum is visible at operating φ ranges, “virtually decreasing” where ∂ γ <0at φ φ operating φ ranges and “stronger decrease” where ∂ γ decreases distinctly steeper than the previous case. It is to be reminded that the latter three cases are distinguished within the φ-range under which typical nanoparticle dispersions are used. Moreover, the latter two cases are conceptually the same but are distinguished for application or engineering purposes: much higher adsorption kinetics and/or smaller nanoparticles induce (although theoretically having the same tendency) for the observer a decrease that is much steeper and occurs at much lower nanoparticle concentrations, which justifies to make a distinction between them. The colours indicate qualitatively the transition from one region to another, whereas the model gives two mathematical limits: the limit K K = 1 (red dashed line) designates formally the p Σ crossover from ∂ γ| >0to ∂ γ| < 0, while the limit K ¼ 1 þ (blue dashed line) stands for the crossover from ∂ γ| <0to ∂ γ| <0. φ ∀φ φ φ→0 Σ φ φ→0 φ ∀φ Interestingly, b shows that the dispersions seem to correspond to sets of simultaneously increasing K and K , indicated by the left-to-right p Σ diagonally up-going set of points. 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Contact angles of aluminosilicate Correspondence and requests for materials should be addressed to Hatim Machrafi. clays as affected by relative humidity and exchangeable cations. Colloids Surf. A 353,1–9 (2010). Reprints and permission information is available at http://www.nature.com/ 74. Li, W. et al. Oil-in-water emulsions stabilized by Laponite particles modified with reprints short-chain aliphatic amines. Colloids Surf. A 400,44–51 (2012). 75. Sengwa, R. J., Kaur, K. & Chaudhary, R. Dielectric properties of low molecular Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims weight poly(ethylene glycol)s. Polym. Int. 49, 599–608 (2000). in published maps and institutional affiliations. 76. Pradhan, S., Mishra, S. & Acharya, L. Ethylene glycol as entrainer in 1-propanol dehydration: scrutiny of physicochemical properties of ethylene glycol+1-pro- panol binary mixture at different temperature. Int. J. Inn. Technol. Expl. Eng. 8, 12S (2019). Open Access This article is licensed under a Creative Commons 77. Patel, K. H. & Rawal, S. K. Contact angle hysteresis, wettability and optical studies Attribution 4.0 International License, which permits use, sharing, of sputtered zinc oxide nanostructured thin films. Ind. J. Eng. 24, 469–476 (2017). adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party ACKNOWLEDGEMENTS material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the Financial support from BelSPo and the MAP Evaporation programme from ESA is article’s Creative Commons license and your intended use is not permitted by statutory acknowledged. regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http:// creativecommons.org/licenses/by/4.0/. AUTHOR CONTRIBUTIONS H.M. developed the model, performed the calculations and comparison with experimental data, analysed the results, wrote and approved the manuscript. © The Author(s) 2022 npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png npj Microgravity Springer Journals

Surface tension of nanoparticle dispersions unravelled by size-dependent non-occupied sites free energy versus adsorption kinetics

npj Microgravity , Volume 8 (1) – Nov 3, 2022

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www.nature.com/npjmgrav ARTICLE OPEN Surface tension of nanoparticle dispersions unravelled by size-dependent non-occupied sites free energy versus adsorption kinetics 1,2,3 Hatim Machrafi The surface tension of dispersions presents many types of behaviours. Although some models, based on classical surface thermodynamics, allow partial interpretation, fundamental understanding is still lacking. This work develops a single analytical physics-based formulation experimentally validated for the surface tension of various pure nanoparticle dispersions, explaining the underlying mechanisms. Against common belief, surface tension increase of dispersions appears not to occur at low but rather at intermediate surface coverage, owed by the relatively large size of nanoparticles with respect to the fluid molecules. Surprisingly, the closed-form model shows that the main responsible mechanism for the various surface tension behaviours is not the surface chemical potential of adsorbed nanoparticles, but rather that of non-occupied sites, triggered and delicately controlled by the nanoparticles ‘at a distance’, introducing the concept of the ‘non-occupancy’ effect. The model finally invites reconsidering surface thermodynamics of dispersions and provides for criteria that allow in a succinct manner to quantitatively classify the various surface tension behaviours. npj Microgravity (2022) 8:47 ; https://doi.org/10.1038/s41526-022-00234-3 INTRODUCTION developed. The condition expressing the tangential stress balance, including surface-tension-induced convection, i.e. Marangoni Fluid dynamics of complex fluids represent a field of study that convection, at the interface is given by concerns a series of energetic, medical and industrial engineering applications. Since these applications concern, in many cases, ðσ  nÞþðσ  nÞþ γð∇  nÞ n  γ ∇ T  γ ∇ φ ¼ 0 (1) g l Σ Σ Σ T φ fluids wherein dispersions are used for material printing or separation processes at the surface level, it is important to control where σ and σ are the stress tensors at the interface on the gas g l the behaviour of surface-related mechanisms . Stability require- and liquid sides, respectively, n the normal vector on the interface, ments during the dispersion processing and the printing process γ the surface tension, (∇∙n) stands for the curvature of the depend on the physical properties, such as the viscosity, and its interface, ∇ = ∇ − nn ∙ ∇ for the surface gradient, T and φ stand deposition quality depends on controlling the fluid dynamics of for the temperature and nanoparticle volume fraction at the 2,3 the deposited fluids and the underlying mechanisms . Moreover, interface, respectively, whereas γ defines the surface tension the wettability is a relevant physical property for processes where derivative with respect to the temperature and γ the surface droplet impingement, thin film flows, microfluidics, surface tension derivative with respect to the volume fraction. Generally, 4,5 speciation and heat transfer are implied . γ is readily available and reasonably approximated to be In order to focus on surface-related mechanisms of complex constant. However, the behaviour of γ is not so clear. Since at fluid dynamics, microgravity experiments are useful, cancelling microgravity, convection responsible for fluid patterns is of the thereby the interference of buoyancy. A sounding rocket surface-tension type, it is crucial to not only have a physics-based experiment took place under the framework of the Advanced analytical expression for the surface tension of nanoparticle-laden Research on Liquid Evaporation in Space (ARLES) experiment fluids but also to understand the underlying mechanisms that supported by the European Space Agency (ESA). ARLES was part govern the surface tension of such complex fluids. of the payload in a SubOrbital Express rocket (MASER 14) and aims It appears from several experimental studies that apparently at studying the evaporation of pure and complex sessile droplets. contradictory tendencies of the surface tension as a function of It also serves as a preparation of an experiment to be performed in nanoparticle content are observed: the surface tension is observed the near future at the European Drawer Rack 2 on board the to increase, decrease or pass through a minimum as a function of 6–15 International Space Station. The evaporation of the complex the nanoparticle content in the dispersions . Even for the same droplets resulted into pattern depositions of nanoparticles, nanoparticle, e.g. SiO , a constant and increasing trend is 9,13 interesting for future printing applications. These experiments observed , while for Al O both decreasing and increasing 2 3 allowed studying the depositions, but not how fluid dynamics, trends, in two separate cases, are observed . Due to the diversity surface effects and particle–fluid interactions controlled those of nanodispersions, there is no universal relation yet that could depositions. Another sounding rocket experiment is planned to be comprehend and clarify quantitatively such observed trends. performed in the near future, where one of the focuses will be to Fitted correlations or empiric relations may give good comparison monitor the fluid dynamics of the complex droplets. In order to to experimental values, but are specific to experiment conditions, prepare the flight scenario, a numerical model has been do not explain physically why certain surface tension phenomena 1 2 3 Université de Liège, Institut de Physique, Liège 4000, Belgium. Université libre de Bruxelles, Physical Chemistry Group, Bruxelles 1050, Belgium. Sorbonne Université, UFR Physique, Paris 75005, France. email: H.Machrafi@uliege.be Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; H. Machrafi 7,10 occur and lack often generality . One may find some explana- begin with the latter. The surface concentration of the tions, but mostly based on a qualitative assessment of the Gibbs nanoparticles Γ is defined by the surface coverage θ times p p free energy or on mere intuition. Some works have explained the maximum surface concentration Γ : p;max surface tension behaviour by an energy variation upon nanopar- Σ Σ Γ ¼ θ Γ ; (2) ticle transfer to the interface , attractive van der Waals or p p p;max 6,17 attractive capillary forces or even by analogy with electrolyte the surface coverage θ stems from principles concerning solutions . Interestingly, it is just the presence of nanoparticles at p thermodynamic equilibrium and adsorption kinetics and it is the surface that is given as reason for surface tension increase in more appropriate to discuss it later in a proper context. For now, ref. , while the nanoparticle adsorption is suggested to cause a we will focus on the framework of the dividing surface and how decrease in the surface tension elsewhere . Others explain the surface concentration is represented within its context. surface tension decrease by a high ionic strength of the base In Eq. (2), Γ is the maximum surface concentration of the fluid countering the otherwise repulsive force between the p;max nanoparticles, assumed to be determined by the principle of nanoparticles and the liquid–gas interface . This, however, does 20–22 maximum stacking via a maximum coverage fraction f . not explain the decrease of surface tension of nanoparticle-laden Other geometrical considerations of particle adsorption have been fluids with low ionic strength, such as nanoparticle dispersions in 21,22 distilled water nor for surface tension minima. The initial treated in refs. , but we only need here their results for decrease of the surface tension is suggested to be due to the maximum coverage. In the presence of nanoparticles, the large spacing between the nanoparticles, favouring electrostatic liquid–gas layer can be defined as the layer where the forces between the nanoparticles or to initial adsorption of nanoparticles go gradually from a bulk concentration c to a 17 Σ nanoparticles at the liquid–gas interface . purely surface concentration Γ . The surface concentration is then The apparently contradictory explanations for surface tension usually obtained by integrating the concentration profile over the behaviour are often intuitively provided and many existing thickness of that layer. It has therefore, generally, a thickness that models, useful as they might be, only predict part of the is larger than the size of the nanoparticles, a thickness that is tendencies, which is a consequence of universal underlying defined by the difference between the dividing surface and an mechanisms still remaining unelucidated. This work develops an imaginary parallel surface, beyond which the bulk concentration is analytical model proposing a new insight in surface thermo- attained. The degree of strength of the interaction energy dynamics, surface energy and, more particularly, in the interac- between the fluid molecules and the nanoparticles in that layer tion between the dispersed phase and the liquid–gas interface. will determine the amount of nanoparticles that are ‘trapped’, i.e. The model mainly aims at elucidating the underlying mechanisms adsorbed, or allowed to disperse in the bulk. Each nanoparticle of the surface tension of nanoparticle dispersions, both correctly that adsorbs will push away fluid molecules. In analogy to the predicting and explaining thereby the different experimentally bulk, according to the lattice model, (where each lattice fits a observed tendencies. We start by formulating the framework liquid molecule), we can define that the surface area that is within which the liquid-gas interface is defined. This will also occupied by a fluid molecule harbours a possible adsorption site. allow introducing definitions of nanoparticle (excess) surface As such, the adsorption sites are geometrically equivalent to the concentrations based on geometrical and size considerations. The fluid molecules in the interfacial layer. In order to represent this nanoparticles are modelled as being incompressible, non- framework in a manner that fits Gibbs’ isotherm, we have defined stretchable and the only material that can be adsorbed on the a dividing surface. Speaking of the maximum surface concentra- liquid–gas interface. Then, an analytical expression for the surface tion, this also necessitates to project the real maximum tension of nanoparticles will be calculated and compared to concentration in the interfacial layer onto the dividing surface several experimental data of different nanoparticle dispersions. (that has zero thickness). This projection method is depicted in The model will be used to explain the different observed Fig. 1a, while Fig. 1b focusses on the projection of an adsorbing phenomena. Finally, it will be shown that two parameters can nanoparticle with the corresponding parameters that will be used predict the type of surface tension behaviour for all the in the model. Figure 1c shows the surface that is deemed to considered material systems. participate to the adsorption. In fact, upon adsorption, it can be imagined that not the whole surface of the nanoparticles participates in the process. It is quite difficult to assess the portion METHODS of surface that participates to this process and not much is known Representation of an interfacial layer on a dividing surface about this. In this work, we will heuristically assume that only half of a nanoparticle’s surface, i.e. the half that faces the dividing The surface energy of a nanoparticle dispersion can commonly Σ Σ be described by Gibbs adsorption isotherm dγ ¼Γ dη  Γ dη , surface, participates in its adsorption. The reason for this will be p f p f Σ Σ an exact differential. Here, η and η are the surface chemical discussed later. We will call this the participating surface. Later in potentials induced by nanoparticle and fluid surface coverage, ‘Three counter-intuitive effects of K on surface tension’,a respectively, whereas Γ and Γ stand, respectively, for the excess verification will be discussed to show that such an assumption p f surface concentrations of the nanoparticles and the base fluid. appears to be quite reasonable. Let us, at maximum coverage, order the nanoparticles into Choosing the Gibbs dividing surface to be there where the several layers that are parallel to the dividing surface. As we are excess surface concentration of the liquid equals zero, we set Γ ≡ 0. This leaves us with dγ ¼Γ dη . We will start discussing the in maximum coverage, each layer contains the same amount of excess surface concentration first. This inherently entails the nanoparticles. Let us then project, in each such layer,the definition of a framework that explains how to deal with the participating surface of the nanoparticles (as if a nanoparticle representation of a three-dimensional interfacial layer, whilst was a balloon that is cut in half, of which one half is spread over Gibbs adsorption isotherm imposes to work with a two- the surface) on the dividing surface (for the first layer at the dividing surface) and on subsequent imaginary layers parallel to dimensional one, i.e. the Gibbs dividing surface. As excess surface concentrations of nanoparticles Γ are not widely the dividing surface. This gives in each layer the same maximum documented, such a framework will allow us to propose surface concentration per unit surface, so that considering only analytical expressions for Γ . The excess surface concentration the layer adjacent to the dividing surface is sufficient to is composed out of a surface equivalent of the bulk concentra- determine the maximum surface concentration perunitsurface, tion, discussed later, and an actual surface concentration. Let us as Fig. 1ashows. npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; H. Machrafi Fig. 1 The projection of nanoparticles on the dividing surface. a Principle of projecting nanoparticles on dividing surface at maximum coverage. Note that at a coverage below the maximum one, the principle is the same. b Projection of nanoparticles (with volume V and surface A ) on a two-dimensional surface. The projected circles are oval because they are drawn in perspective, but they should be seen as circular for the spherical and disk nanoparticles, and as a square for the cubic nanoparticles. The fluid molecules (with volume V and surface A ) at the surface Σ, on which a nanoparticle adsorbs, also participate to the adsorption and are illustrated as fluid molecules that become projected (on the dividing surface) as two-dimensional adsorption sites, depicted for simplicity as a flat plane Σ at exactly the dividing surface. c Illustration of the surfaces that come into play in the volume-to-surface ratio for the spherical, cubical and disk nanoparticles. The images are not in scale. This will result in a molar concentration of nanoparticles per The projected surfaces depend on the size of the nanoparticles unit surface on the dividing surface that would be equivalent to and the fluid molecules (geometrically equivalent to that of the the corresponding real molar concentration in a realistic adsorption sites). This means that the difference in sizes between adsorption layer, through scaling by a certain defined character- the nanoparticles and the adsorption sites can be expected to istic length, so-defined as L .AsFig. 1b shows, the projected have a great impact on the adsorption process. Each adsorbed maximum surface concentration per unit surface times the nanoparticle will induce a change in the possible entropic characteristic length gives the same volume as that of the configurations of a great number of adsorption sites. As these nanoparticle. It follows then that that characteristic length must sites are not occupied, yet have a large influence on the entropy, be a volume-to-surface ratio. they are named as non-occupied sites, because their non- We define this volume-to-surface ratio by the total volume occupancy matters entropically. We can say that the total area of divided by half of the total surface (the participating surface, as is these non-occupied sites (denoted by subscript NO) per total shown in Fig. 1c). The fluid particles that surround this surface area is given by participating surface of a nanoparticle in a real interfacial layer Σ Σ Σ ς m ¼ ς m  ς m (3) NO f p NO f p are fully projected on the dividing surface as they represent geometrically the adsorption sites on that surface (see the where m and ς are, respectively, the number of particles of a schematic representation in Fig. 1b). The characteristic length of constituent per unit interface surface and the specificsurface these fluid particles, or adsorption sites, is calculated by a standard area perparticleofthatconstituent,whilstthe superscript Σ volume-to-surface ratio. indicates that it concerns a surface property. The specificsurface Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi Fig. 2 Projection of bulk concentration. Analogy of an imaginary projection of the nanoparticles at an imaginary surface at z = z with respect to the real projection on the dividing surface at z = 0. The two resulting molar concentrations per unit surface should be equal in the definition of the surface excess concentration. for disk nanoparticles. This leads to area of a constituent is given by ς  ,with i = NO, f, p, ρ L N i i A standing for non-occupied sites, the fluid and nanoparticles, ρ Γ ¼ f L 1 p (4) respectively. Equivalently, the specific surface area of a p;max constituent may conveniently also be given per mole,i.e. i Σ ς  .In thatcase, m would simply be the number of mole ρ L It should be noted that when nanoparticles are coated or i i of a constituent per unit surface via scaling by N and Eq. (3) surface treated, the maximum coverage might be less due to remains valid. Expressions for the characteristic length L will be i possible repulsive forces or more if the nanoparticles have a soft developed later (with L ¼L as geometrically the adsorption NO f compressible coating with interparticle attractive forces. Note that 20,23 sites are equivalent to the fluid molecules). It is important to similar expressions have been proposed in refs. . Nanoparticles notice here that ς introduces a size effect, i.e. the number may come in various shapes, the main ones kibeing often of densities depend on the size of the nanoparticles and the spherical, cubical or disk shape. We will, for the demonstration, surface fluid particles. limit ourselves to such undeformable shapes. The volume and We have now defined the framework for calculating the surface participating surface, i.e. V ; ,asdefined in Fig. 1c, would concentrations and may proceed with proposing a formula 4π 3 2 3 2 2 2 that allows calculating these concentrations. The maximum then be a ; 2πa ,8a ; 12a and πa h ; πa þ πa h p p p p p p p p p surface concentration is given by the maximum number of m for a spherical, cubical and disk nanoparticle, respectively. The p;max nanoparticle moles, , divided by a unit surface, i.e. volume-to-surface ratio L can then be calculated. For a spherical Σ 1 p;max Γ ¼ , where A is a total (arbitrary portion of unit) 2a t p p;max N A A t nanoparticle, L ¼ (with radius a ), for a square-like nanopar- p p surface. This can conveniently be rewritten as 2a ticle, L ¼ (with 2a the side of the cube) and for a disk-like V m A =2 p p Σ p p;max p 1 3 Γ ¼ ,where A and V are the surface and p p p;max a h A =2 A N V p p p t A p nanoparticle, L ¼ (with radius a and thickness h ). Note p p V a þ h p p p volume per nanoparticle, respectively. It can be noted that is A =2 that it is not straightforward to define a molar mass of a nothing else than two times the nanoparticle’s volume to surface nanoparticle, since it is not a molecule nor an atom. We will ratio, defined as L (see Fig. 1). Moreover, is the mole of approximate the molar mass of a nanoparticle, in analogy with N V A p that of a polymer constituted by many monomers, as an ensemble nanoparticles per unit volume, also given by ,where ρ and M p p of atoms or molecules chemically connected to one another. The are the nanoparticle’s density and molar mass, respectively. Also m A =2 molar mass of a nanoparticle equals then M ¼ M 0f , where f p;max p p p p p is simply the maximum geometric coverage fraction f , p represents the maximum stacking factor of spheres in a three- being f ≈ 0.547 for spherical non-overlapping hard particles on 21,22 pffiffi dimensional setting, assumed here to be f ¼ (that of an fcc or a two-dimensional surface .Thisvalue isobtainedby 3 2 considering random packing of spheres after their projection on hcp structure), M the molar mass of one atom or molecule, V p′ p 4π 3 the interface. It is therefore not the same as the random packing 0 the volume of one nanoparticle and V ¼ ℓ the volume of one of circles as the latter would neglect the purpose of the projection atom or molecule, assumed to be of spherical form, with ℓ the qffiffiffiffiffiffiffiffiffiffiffiffi method, where it is sought to obtain expressions on a 2D surface M 0 3 p radius of that atom or molecule ℓ ¼ . The effective radius whilst preserving the information from realistic 3D interfaces as ρ N 4π p A Fig. 1 shows. For cubic nanoparticles, it is reasonable to expect of a fluid molecule, assuming sphericity, can be calculated in the qffiffiffiffiffiffiffiffiffiffiffiffi that the maximum coverage will be close to unity, to that we take M 3 f 3 same manner as ℓ ¼ , where ρ and M are the fluid’s 21,22 f f f ρ N 4π f ≈ 1 for cubic nanoparticles . For disk-like nanoparticles, of 4π density and molar mass, respectively. With V ¼ ℓ and which the circular part faces the dividing surface, we take a f maximum coverage that corresponds to maximum standard A ¼ 4πℓ , the volume-to-surface ratio for a fluid particle is given pffiffi π 3 ℓ hexagonal stacking of circles on a surface, i.e. f ¼  0:907 by L ¼ . With these definitions, the geometric definition of the 1 f 6 3 npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi maximum surface concentration Γ can be calculated from densities of adsorbed nanoparticles, the total number of micro- Σ Σ m þm ! material properties and size values. ðÞ NO p states would equal W ¼ . Boltzmann’s equation of Σ Σ m !m ! The difference between the excess surface concentration Γ and p p NO Σ Σ m þm ! the surface concentration Γ is usually defined as ðÞ p NO p entropy would give S ¼ k lnðÞ W ¼ k ln . For a pure d B B Σ Σ Z m !m ! z p NO Γ  Γ  c dz ¼c λ (5) p p p b fluid, only one undistinguishable combination exists, i.e. S = p f k ln(1) = 0. The corresponding surface fraction is given by where c is the nanoparticle concentration in the bulk layer, and p y ¼ . The surface coverage is defined to be equal to the p Σ Σ m þm λ = z − z , with z the position at which Γ = 0 and z at which NO b b 0 0 f b surface fraction, θ ≡ y , so that θ + θ = 1. Defining the i i p NO we can consider conceptually to have a bulk concentration. In configurational entropy of dispersing, due to nanoparticle cover- order to determine λ , let us make a preliminary remark. The age and non-occupied sites, as Δs = S N − S N , and using d d A f A maximum surface concentration has been obtained by projecting Stirling’s approximation for the logarithm of factorials, gives a layer of adsorbed nanoparticles on a two-dimensional surface at the dividing surface, which we defined as Γ = 0. When doing this, Δs ¼k N θ ln θ þ θ lnðÞ θ (9) B A p p NO NO it has been explained that the characteristic length of nanoparticle where k and N are, respectively, Boltzmann’s constant and projection equals L . For consistency, we should do the same B A Avogadro’s number. In dilute conditions, the enthalpy of here. In the definition of the excess surface concentration, dispersing can be neglected. It should be noted that this enthalpy the term  c dz denotes a deduction from the surface z0 results from heat liberated or absorbed due to new interactions concentration of an imaginary extrapolation of the bulk concen- that stem from the mixing process, while it is not the same as the tration, integrated over the interfacial thickness λ . So, it should b enthalpy of adsorption, which plays a role in the equilibrium rather be seen as an imaginary surface-equivalent of the bulk adsorption constant. In such conditions, we deal with an ideal concentration, Γ  c dz ¼ c λ ,defined at an imaginary dispersion, being consistent with the Langmuir’s adsorption p;b p p b isotherm, of which a detailed deduction is presented in the next surface at z = z . This also means that it is analogous to the section. The Gibbs surface free energy of dispersing is then given projection of the bulk nanoparticle concentration on the dividing Σ Σ by Δg ¼TΔs resulting into surface, named here Γ , so that Γ ≡ Γ . It remains to find Γ . p* p* p,b p* d d Imagine at the dividing surface Γ = 0 a slab V of thickness L ,of Σ f p p Δg ¼ k TN θ ln θ þ θ lnðÞ θ (10) B A p p NO NO which the contents are projected on that surface. If the projected p ς nanoparticle surface concentration is given by Γ , then the f p* We define ω  and ω  . The chemical potentials of a p f ς ς Γ NO NO corresponding nanoparticle concentration in V would be as p component i are defined by defined by the projection procedure in Fig. 1. If the projected Σ Σ m þ m NO p Σ Σ specific surface area per mole of nanoparticles is given by ς , then p η ¼ N Δg A (11) i Σ d ∂m N the corresponding volume per mole of nanoparticles in V would A i Σ T ;p;m 8j≠i be ς L . The same could be done for the fluid particles, so that the p p φ ς Γ with i = p, f and j = p, f. The number ω can also be understood as p p volume fraction φ in that slab would be described by ¼ . 1φ ς Γ f f the number of adsorption sites per nanoparticle. We then use the Within the slab V , the projected surface concentration for the aforementioned definition θ ¼ , fill this in in Eq. (10), apply i Σ Σ fluid particles Γ would simply be equal to the bulk concentration m þm f* NO c times the thickness of V , i.e. Γ ¼ c L . We then have Eq. (11) and rewrite the result back in terms of θ . This finally gives f p f f p φ ρ f f Σ Γ ¼ c L . Note that c ¼ðÞ 1  φ . Filling in the definitions η ¼ k TN ω ln 1  θ and p f p f B A f p ς 1φ M f p f ρ L Σ ρ M M p p p f f η ¼ k TN ln θ  ω ln 1  θ (12) of ς and ς leads to Γ ¼ L φ ¼ φ. Figure 2 B A p p p p f p p M ρ L ρ L M L f f p p f f p ς ς p p ς illustrates the analogy that we have discussed here. where ω   (as ω  and ω ≡ 1 because the 2 p f f L ς ς ς ρ NO f NO p p As c ¼ φ , it follows that λ ¼ . Filling this in (5) gives, with p b adsorption sites are within the present framework geometrically M L p f Eq. (2), for the excess surface concentration equivalent to the projected liquid molecules, see Fig. 1b and Σ Σ Σ corresponding discussion) is given by Γ  θ Γ  φΓ ¼ θ K  φ Γ (6) p p p Σ p;max b b M ρ L p f with ω ¼ (13) ρ M L f p ρ L p p p (7) Γ ¼ λ ¼ φ M L p f Surface adsorption isotherm p;max K ¼ (8) Equilibrium of the adsorption process is described by a net zero change of the total Gibbs free energy of the system: Σ Σ Σ b Σ b where Γ is given by Eq. (4) and φΓ is the surface equivalent dΔG ¼ dΔG þ dΔG þ dΔG þ dΔG  0, where the subscripts p;max a ad ad d d of the bulk concentration and K a constant that measures the Σ ‘ad’ and ‘d’ denote the adsorption (due to translational or potential of the nanoparticles to rather adsorb at the interface confinement effects, or effects related to particle surface energies, than stay dispersed in the bulk. As will be seen later, K is a Σ dipole-dipole and coulomb interactions , for instance) and the function of nanoparticle size, maximum packing and fluid dispersion (mixing) free energies, respectively, and ‘Σ’ and ‘b’ molecule size. The surface coverage θ in Eq. (6) will be treated p the surface and bulk phases, respectively. We focus first on the Σ Σ m þm in the context of surface kinetics, but we will first deduce the Σ NO Σ dispersing. We can then define ΔG ¼ A Δg and d d surface chemical potentials and surface adsorption. b b m þm b p b b b ΔG ¼ V Δg , where m and m are the number of bulk d d l p fluid particles and nanoparticles per unit volume of the dispersion, Surface chemical potential Σ whereas A and V are an arbitrary unit surface and volume, If we have m number densities of adsorption sites, containing Σ Σ m þm Σ Σ NO p m number densities of non-occupied sites and m number respectively. Note here that has unit moles per unit surface NO p N Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi b b m þm θ p d p ln c K þ ln  ln x ¼ 0, which (keeping in mind that and unit moles per unit volume. Note also that we can t p p 1θ N p p p d x c = c )simplifies to ¼ K c , which is the well-known define a mole fraction of nanoparticles in the bulk as x ¼ . p b b p t p p 1θ p m þm Langmuir’s adsorption isotherm, subject to further discussion in Neglecting the enthalpy of dispersing as mentioned before, an the next section. This can be rearranged into addition of nanoparticles to the surface would result into a change Σ Σ Σ Σ d dΔG m ; m , which can mathematically be written as dΔG ¼ K c d p f d p θ ¼ (14) Σ Σ Σ Σ m þm m þm ∂ NO p Σ Σ ∂ NO p Σ Σ 1 þ K c N A Δg dm þ N A Δg dm ¼ p A Σ A Σ N d p N d f ∂m A Σ ∂m A Σ T ;p;m f T ;p;m Σ Σ Σ Σ The equilibrium adsorption constant K could be calculated A η dm þ η dm . From (3) and the total number density, it can p p f f thermodynamically via Van’t Hoff’s relation. However, experimental Σ Σ Σ be derived that dm ¼ ω  1 dm . With the definitions for η p p values for the molar adsorption enthalpies and entropies are not and η (see Eq. (12) and text above), this leads with ω = 1to f readily available for the studied nanoparticle dispersions and Σ Σ especially not for various concentrations. Other expressions and dΔG ¼ k TN ln A dm . An equivalent procedure can be B A d 1θ p methods make use of more available surface energies and surface performed for the bulk phase leading, under the approximation tensions. However, even if one may perform such measurements, b b of diluted dispersion, to the relation dΔG ¼ k TN ln x V dm . B A p d p such a procedure would not allow an analytical physics-based Mass conservation stipulates that the net mass change is analysis of the behaviour of the surface tension and would not Σ b Σ b zero, i.e. A dm þ V dm ¼ 0. This leads to dΔG þ dΔG ¼ offer the understanding of the underlying mechanisms for the p p d d Σ various surface tension behaviours. Therefore, it would not align k TN ðlnð Þ lnðx ÞÞA dm . B A p 1θ with the purposes of this work. In order to obtain theoretical A change in the nanoparticles number in both phases upon parameters, independent of the experimental surface tension data, adsorption also induces a change in the free energy of the experimental regression procedures or any fitting methods, one of adsorption, which can be symbolically (as an already existing the often-used ways is to use a kinetic model. Adsorption and thermodynamic relation for the free energy of adsorption will be desorption are often described kinetically. Material properties for used, there is no need to enter into details as we did for the kinetic models are readily available for solid–liquid interfaces and Σ Σ free energy of dispersion earlier) written as dΔG Am þ ad p the methods are widely used and understood. As less data are b b Σ Σ b b available for liquid–fluid interfaces, it is the question whether dΔG Vm ¼ Δg A dm þ Δg V dm . We can use the mass ad p ad p ad p similar kinetic models would be applicable. One can argue that the Σ b conservation principle A dm þ V dm ¼ 0 and write p p adsorption of surface-charged nanoparticles (an important method Σ b Σ b Σ Σ dΔG þ dΔG ¼ Δg  Δg A dm  Δg A dm ,where Δg to obtain stable dispersions) on liquid-fluid interfaces (often ad ad p p ad ad ad ad charged with the same sign) can be approximated by adsorption stands for the net difference of the free energy of adsorption per on solid–liquid interfaces. Although subject to more investigation, mole of adsorbed nanoparticles. At equilibrium, Δg is related to ad it has already been applied successfully for liquid–fluid interfaces . the thermodynamic equilibrium constant K via Van‘t Hoff’s This motivates that within such a reasonable assumption the equation for adsorption Δg = −RTln(K )with K the thermo- ad e e equilibrium adsorption constant for the nanoparticle dispersions dynamic equilibrium constant of adsorption. The total Gibbs free can be obtained without fitting. The interpretation of underlying energy of the system dΔG becomes finally dΔG ¼ a a mechanisms would benefit from such a physics-based approach. p Σ Σ k TN ðlnð Þ lnðx ÞÞA dm  RTlnðK ÞA dm  0 at equilibrium. B A p e p p 1θ Surface kinetics This leads finally to lnK þ ln  ln x ¼ 0, which is known e p 1θ It remains to find the equilibrium adsorption constant K or for as an adsorption isotherm for ideal dispersions or solutions. The p later convenience, a dimensionless version K thereof. ‘Surface equilibrium constant K is for ideal cases related to the dimensional adsorption isotherm’ presented the thermodynamic theory behind Langmuir equilibrium constant K , which can be traditionally the Langmuir adsorption isotherm. It was mentioned that described by the equilibrium adsorption reaction: c + [*] ⇆ [P − *], unavailable experimental data for the nanoparticle dispersions where c is the nanoparticle molar bulk concentration, [*] the studied here and the aim to provide for a physics-based model surface molar concentration of empty adsorption sites and encourage the use of another way. Commonly, the equilibrium [P − *] the surface molar concentration of adsorbed nanoparticles, adsorption constant is determined kinetically, where material respectively. If we define Γ as the maximum surface p;max properties necessary for the model are readily available. The concentration, we can write ½  þ½ P ¼ Γ , which is kinetic model is based on an equilibrium between standard p;max ½ P adsorption and desorption kinetics and is treated in details in the equivalent to defining the surface coverage as θ ¼ and p Σ Γ 27–30 p;max literature . We mention the main points here. Note that therefore  1  θ . Note that later, we will use the notation Γ desorption becomes relevant when the energy of particle Σ p p p;max trapping is of the order of the thermal energy. Adsorption (with ½ P θ d p for [P − *]. Thermodynamically, K ¼ ¼ .As c has unity c ½ p c 1θ standard rate k ) depends on the bulk concentration c and the pðÞ p a p available adsorption sites (1 − θ ). Desorption (with standard moles per unit volume, K has unity volume per mole. Furthermore, p rate k ) depends on the adsorbed nanoparticles θ per specific d p as K is dimensionless, this means that we can define K ¼ c K , e t surface area of adsorbed nanoparticles ς . This writes as where c must have unity moles per volume. Similar discussions on j ¼ k c 1  θ (15) the various definitions of K and K have been performed in the a a p p literature, indicating that K in Van’t Hoff’s equation is dimensionless d 1 and that K in the Langmuir’s adsorption equation has a dimension p j ¼ k θ d d p (16) 25,26 depending on the concentrations, confirming this analysis .We p can also deduce (in dilute systems, c ≈ c ,with c and c the molar f t f t From kinetic considerations, it can be stated that nanoparticle concentrations of the base fluid and the bulk phase, respectively) accumulation, through a flux balance equation, at the interface is that c can be represented by the molar concentration of the bulk ∂θ 25,26 1 p phase . Filling this in the adsorption isotherm gives finally given by ¼ j  j , where we remind that here ς is the a d p ς ∂t npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi specific surface area per mole of nanoparticles. At quasi- permittivity of the phase at the opposite side of the interface ∂θ opposed to the relative permittivity of the phase where the stationarity, i.e. ¼ 0, we have from Eqs. (15) and (16) that ∂t nanoparticles are dispersed, ε . The van der Waals potential energy ς c k c p p a p between the nanoparticle and the interface is given by kd θ ¼ ¼ (17) 1 k k þ k c 1 þ ς c d a p ς p p p k A a a z d pΣ p p vdW Φ ¼ þ þ ln (23) pΣ 6 z z þ 2a z þ 2a p p d ka Comparison with (14) learns that K ¼ ς . As the molar p k concentration can also be expressed into the volume fraction φ by where A is the non-retarded Hamaker constant for the p−Σ c  φ, we can rewrite (17) as particle–interface interaction, where the particle (p) interaction ρ with air (a) through the base fluid (f) is assessed. This constant is k p ς φ k M d p derived by the theory of London-dispersion forces and is often θ ¼ (18) k p approximated by the combining relation A ¼ A ¼ 1 þ ς φ pΣ pfa k M d p pffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffi 39,40 A  A A  A . Non-DLVO interaction energies pp ff aa ff This allows defining a dimensionless Langmuir’s equilibrium may be considered as one potential energy, being either repulsive k p k a a 1 36 constant K  ς ¼ , and relating the surface coverage p p or attractive depending on the solid–water contact angle .In k M k L d p d p with the bulk volume fraction as another work, a Hydra parameter, depending on the hydropho- bicity of the surface, was introduced in one expression, being K φ θ ¼ (19) either negative or positive, defining, respectively, a hydrophilic 1 þ K φ repulsive or hydrophobic attractive potential energy. The Hydra 36,37,41,42 potential energy is given by refs. If we assume that particle transport occurs under a quasi-linear z z and stationary diffusional regime (this is a valid approximation Hy 0 (24) Φ ¼2πa λ γðÞ 1  CosðÞ ϑ e p 0 pΣ 0 because of the very small relaxation time), it has been shown that general analytical solutions for the adsorption and desorption where λ is a decay length, ϑ the radial liquid–solid static contact qffiffiffiffiffiffiffi 0 Φm 41,42 k πk T 27–30 a B k T angle and z a constant of the value of 0.16 nm . The total constants can be obtained ,i.e. ¼ δ e , which finally 0 k jj Φ d m EDL EDL vdW Hy potential energy is given by Φ ¼ Φ þ Φ þ Φ þ Φ . pΣ pp0 pΣ pΣ leads to sffiffiffiffiffiffiffiffiffiffi δ πk T m Material properties m B k T K  e (20) Table 1 shows the material properties of the used nanodispersions L jj Φ p m at ambient temperature. Effect of temperature on solid properties where Φ is the total potential energy, Φ , at a distance z = δ , i.e. m t m is neglected. For the base fluids, only the densities are adapted for Φ ¼ Φ j . The total potential energy stands here for the m t z¼δ m temperatures different than ambient. Since these values are well potential energy between a particle and the liquid-air interface, being composed of many mechanisms. The DLVO theory Table 1. Material properties and physical constants. mentions that the most important interactions are the electro- EDL vdW a 3 1 static Φ and van der Waals Φ interaction ener- Component Density [kg m ] Molar mass [kg mol ] pΣ pΣ 24,31–33 gies . Image charge effects in the form of a repulsive Al O 3950 0:102 2 3 EDL particle-image Φ potential energy are esteemed to be of pp0 Al 2700 0:027 importance, the reason being that in cases of particles being B 2370 0:011 oppositely charged to the interface, repulsion was still MgO 3580 0:040 Hy 24,34 observed . Non-DLVO interaction energies Φ , suggested pΣ SiO 2650 0:060 to be of the Lewis acid-base type, also appear to be of great Ag 10490 0:108 importance, such as hydrophilic repulsive interactions and Laponite 2530 2:287 35–38 hydrophobic attraction energies . The electrostatic double ZnO 5610 0:0814 layer interaction potential between a nanoparticle and a flat Dodecanethiol-ligated Au 4720 0:198 fluid–air interface is given by Water (W) 997 0:018 zþ2a k T ζ e ζ e z B p e Σ e EDL λ λ D D Φ ¼ 64πε ε Tanh Tanh a e þ e r 0 p n-decane (D) 730 0:142 pΣ e 4k T 4k T e B B Ethanol 789 0:046 zþ2ap λ λ D D Ethylene glycol (EG) 1110 0:062 þλ e þ e Þ (21) Tri-ethylene glycol (TEG) 1120 0:150 where ε , ε , e , ζ , ζ , and λ , are, respectively, the relative r 0 e p Σ D n-dodecane (DD) 750 0:170 permittivity, the absolute permittivity, the elementary charge, n-hexadecane (HD) 773 0:226 the zeta potential of the nanoparticles, the zeta potential of the liquid–air interface and Debye length. Debye’s length is given by a First nine rows concern the nanoparticle densities ρ and molar masses qffiffiffiffiffiffiffiffiffiffiffi p ε ε k T r 0 B M . The tenth to sixteenth-row concern the base fluid densities ρ and p' f λ ¼ , where I stands for the ionic strength of the base D 2 2N e I A e molar mases M . fluid. The potential energy between a particle, p, and its image, p′, b Volume-averaged and mole-averaged values are given for the density and in the phase at the other side of the fluid-air interface is given by molecular weight, respectively, based on the dimensions of the core gold 24,34 refs. nanoparticle and the dodecanethiol ligand shell. Note that the molar mass M ' given here is the one of an atom or molecule. To obtain the molar mass of 2 p z V k T ζ e ζ e p 4π 3 B p e p0 e 2 EDL pffiffi λ a nanoparticle, one must make the conversion M ¼ M 0 f ¼ a ρ N . p p p A D (22) p p Φ ¼ 32πε ε Tanh Tanh a e V 0 9 2 r 0 p p pp0 23 −1 e 4k T 4k T The values of the used physical constants are N = 6.02 * 10 [mol ], R = e B B A g −1 −1 −12 −1 −1 −19 8.3145 [J mol K ], ε = 8.854*10 [C V m ], e = 1.602 * 10 [C] and 0 e where ζ stands for the zeta potential of the image particle, given −23 −1 p′ k = 1.38 * 10 [J K ]. ε ε ζ e 2k T r 0 p e 33 B r by ζ ¼ ArcSinhð Sinhð ÞÞ . Here, ε is the relative p r′ e ε þε 0 2k T e r r B Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi Table 2. Data needed for calculation of equilibrium constant K . −20 −20 NP-L A [10 J] A [10 J] ε [−] ϑ [°] λ [nm] −ζ [mV] T [K] 2a [nm] Ref. pp ff r 0 p p a b c d w 6 Al O -W 15 3.7 80 35 0.72 64 300 50 2 3 a b e f w 6 Al O -D 15 5.45 2 26 1.16 55 300 50 2 3 g b e h 6 Al-D 15 5.45 2 33 0.87 57 300 18 i b e h x 6 B-D 6.23 5.45 2 33 0.35 55 300 46 a b i j w 6 Al O -E 15 4.2 25.3 23 1.83 38 300 50 2 3 g b i k 6 Al-E 15 4.2 25.3 36 0.98 61 300 18 l b i k x 6 B-E 6.23 4.2 25.3 36 0.71 63 300 46 a b m n 10 Al O -TEG 15 5.8 23.3 30 0.8 59 298 20 2 3 o b m n 10 MgO-TEG 12.1 5.8 23.3 30 0.67 46 298 20 a b c p 9 SiO -W 6.5 3.7 80 20.7 0.8 50 298 30 g b c q 9 Ag-W 50 3.7 80 40 0.78 45 298 100 r b c s y 19 Lap-W 1.06 3.7 80 24 0.57 49 300 25 (1.5) a b t u 11 ZnO-EG 9.2 5.6 40 36.4 0.57 60 300 67 v b e v z 38,47 (dl)Au-D 28 5.45 2 30.5 1.95 30 303 5 (1.7) v b e v z 38,47 (dl)Au-DD 28 5.8 2 33 1.71 35 303 5 (1.7) v b e v z 38,47 (dl)Au-HD 28 5.2 2 36 1.5 70 303 5 (1.7) a b c d 48 Al O -Ws 15 3.7 80 35 0.72 75 300 40 2 3 The base fluids W, D, DD, HD E, TEG, and EG stand for water, n-decane, n-dodecane, n-hexadecane, ethanol, tri-ethylene glycol, and ethylene glycol, respectively. Ws stands for fully stabilised water dispersion . The temperatures for which the experimental data are obtained from the literature are indicated in the table. If in the literature it is mentioned that the experimental data are obtained at ambient temperature, the value of 300 K is used. a 52 ref. . b 53 ref. , the value of TEG is approximated as that of di-ethylene glycol. c 54 ref. . d 55 ref. . e 56 ref. , same value assumed for n-dodecane and n-hexadecane. f 57 ref. . g 58 ref. . h 59 ref. , the value of B is approximated as that of Al. i 60 ref. . j 60–63 refs. , interpolative estimation. k 64 ref. , the value of B is approximated as that of Al. l 65 ref. . m 66 ref. . n 67 ref. , assumed from values of EG on mixed ceramic substrates. o 68 ref. . p 69 ref. . q 70,71 refs. , averaged value. r 72 ref. . s 73,74 refs. , averaged values. t 75,76 refs. , averaged values. u 77 ref. , approximated. v 38 ref. . w 6 TEM images in ref. show agglomeration so that the size of the nanoparticles is ~2 times that of the initial one (25 nm). x 6 SEM images in ref. show cubic-like particles with an averaged size of 80 nm so that, taking this size between opposite corners of a cube, one cube side would be 80/√3≈46 nm. The value between the brackets is the thickness of the nanodisks. z 38,47 The core diameter of Au is 5 nm andthe ligand shell thickness is 1.7 nm withan overall reported diameter of 2a = 8.4 nm . The B nanoparticles were approximated as cubic particles, evidenced from SEM images in ref. and the Laponite nanoparticles are nanodisks of a flat (the thickness is much smaller than 19 6,7,9–11,38,47,48 the radius) cylindrical shape , while the rest are spherical nanoparticles . tabulated, not more attention is given. Some data are reported in It should be noted that it is difficult to obtain precise values for the literature as a function of the mass fraction. If ξ is the the parameters ϑ, δ , ζ , I, λ ,and ζ , which need some m int 0 p nanoparticle mass fraction, ρ the nanoparticle density and ρ discussion. Reasonable values can be obtained from experimental p f the fluid density, then the nanoparticle volume fraction φ can be data for ϑ, δ , ζ , I. The minimum thickness between the m int nanoparticle and the interface at adsorption, δ , is often assumed ξ ξ 1ξ m calculated as φ ¼ þ . Table 1 also shows general 27,43 ρ ρ ρ p p f to be of the order of δ = 0.5 nm . For the interface zeta physical constants used in the model. potential, ζ , the approximated mean value of ζ = −40 mV is int int 44,45 Equations (20)–(24) allow calculating the equilibrium constant taken for water . For ethanol, tri-ethylene glycol, ethylene K . Several data are needed for this calculation. These data are p glycol and glycerol the same value is assumed, while n-decane, collected from the literature and tabulated in Table 2. A summary n-dodecane and n-hexadecane are considered to be an oily liquid of the variables and their meaning is given in Table 3. as hexane and a value of ζ =−10 mV is taken . The ionic int npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi Table 3 continued Table 3. Variables used in the model and their meaning. Symbol Description Unit Symbol Description Unit λ Debye’s length [m] a Nanoparticle radius [m] −3 ρ Density of fluid [kg m ] A Non-retarded Hamaker constant [J] p−Σ −3 ρ Density of nanoparticles [kg m ] −3 p c Bulk concentration [mol m ] −2 σ Stress tensor gas-side of interface [N m ] D Ratio excess surface to surface-equivalent [−] −2 σ Stress tensor liquid-side of interface [N m ] of bulk concentrations ς Specific surface area of fluid molecule [m per particle] e Elementary charge [C] ς Specific surface area of non-occupied site [m per particle] NO f Maximum geometric coverage fraction [−] ς Specific surface area of nanoparticle [m per particle] h Disk nanoparticle thickness [m] p −2 φ Volume fraction [−] H Non-occupancy effect [J m ] NO −3 Φ Total potential energy at first minimum [J] I Ionic strength [mol m ] m EDL −1 Φ Repulsive particle-image potential energy [J] k Adsorption rate [m s ] pp0 EDL −1 Φ Electrostatic particle-interface [J] k Boltzmann constant [J K ] pΣ potential energy −1 k Desorption rate [s ] Hy Φ Hydra potential energy [J] pΣ K Equilibrium adsorption constant [−] vdW Φ Van der Waals potential energy [J] pΣ K Ratio surface to bulk preference [−] Φ Total potential energy [J] ℓ Equivalent size of fluid molecule [m] ω Number of adsorption sites for one [−] ℓ Equivalent size of nanoparticle [m] nanoparticle L Characteristic length of fluid molecule [m] Subscript L Characteristic length of nanoparticle [m] p Nanoparticle Σ −2 m Number of fluid molecules per unit surface [particles m ] NO Non-occupied site Σ −2 m Number of non-occupied sites per unit [particles m ] NO f Fluid surface Σ −2 m Number of nanoparticles per unit surface [particles m ] −1 strength of a fluid is somewhat an unknown. However, works M Molar mass fluid molecule [kg mol ] −1 have indicated that for deionized water, typical ionic strength M Molar mass nanoparticle molecule unit [kg mol ] values are measured of the order of I = 1mol/m .Thisvalue is −1 M Molar mass nanoparticle [kg mol ] p' assumed for all the fluids used. The values for λ and ζ depend 0 p n Normal vector [−] strongly on the experimental conditions and only ranges can be −1 N Avogadro’s number [particles mol ] A indicated. Decay lengths, λ , of values up to 2.2 nm are reported 20,37,41 −1 −1 for several systems . The zeta-potentials ζ of nanodisper- R Universal gas constant [J mol K ] p sions were typically found to be approximately between −75 and T Temperature [K] 20,34,43 −25 mV . Educated guesses, not affecting the analysis in Greek symbol this work, for these two parameters within these indicated ranges −1 γ Surface tension [N m ] are implemented in Table 4 for the calculation of the equilibrium −1 γ Surface tension of fluid [N m ] f constant. The obtained equilibrium constants for the nanoparticle −1 −1 γ Temperature derivative of the surface [N m K ] dispersions are shown later in Table 5. tension −1 γ Volume-fraction derivative of the surface [N m ] φ Reporting summary tension Further information on research design is available in the Nature −2 Γ Excess surface concentration of fluid [mol m ] Research Reporting Summary linked to this article. −2 Γ Excess surface concentration of [mol m ] nanoparticles Σ −2 RESULTS AND DISCUSSION Γ Surface concentration of nanoparticles [mol m ] Σ −2 Comparison of model with experimental data Γ Surface-equivalent of bulk concentration [mol m ] Σ −2 Gibbs adsorption isotherm dγ ¼Γ dη can now be integrated. Γ Maximum surface concentration [mol m ] p p;max We use Eq. (6) for Γ (with (19) for θ ) and Eq. (12) for η . The δ First minimum of potential well [m] p p m p −1 −1 surface tension of nanoparticle dispersions is finally given by ε Absolute electric permittivity [C V m ] ω þ ω KðÞ φ þ K  K K ω þ ω K K  1 p p p Σ p Σ p p p Σ ε Relative electric permittivity [−] Σ γ ¼ γ þ R TΓ φ  ln 1 þ K φ g p f b 1 þ K φ K p p ζ Zeta-potential interface [V] int (25) ζ Zeta-potential nanoparticles [V] Σ −1 where γ is the surface tension of the base fluid, R the universal η Surface chemical potential fluid [J mol ] f g gas constant, T the temperature, K given by (8), K given by (20), Σ −1 Σ p η Surface chemical potential nanoparticles [J mol ] ω given by (13) and φ the nanoparticle volume fraction. The θ Non-occupied site coverage [−] NO equilibrium constant is a kinetic parameter obtained by models θ Nanoparticle coverage [−] from the literature, summarised in ‘Surface kinetics’. The other ϑ Contact angle [°] parameters are developed in this work using geometric principles and characteristic length scales, which would, for clarity, benefit λ Decay length in hydra potential [m] from a summary in Table 4. Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi γ − γ = Δγ + Δγ , where f p NO Table 4. Definitions, necessary for the calculation of the surface Z Z φ φ ∂η ∂η tension. p p p Δγ ¼ Γ dφ ¼ Γ dφ (26) b Σ ∂φ ∂φ 0 0 a,b b Symbol Definition Z Z φ φ ∂η ∂η Γ 2ap 2ap aphp NO NO p L − p Δγ ¼ Γ dφ ¼ Γ dφ 3 3 a þh (27) p p NO p ∂φ ∂φ 0 0 f 0.547 1 0.907 − L −− − Equations (26) and (27) show the multiplication of two terms in 4π 3 ∂η pffiffi Σ M a ρ N NO p p A 9 2 the integral. The term ðΓ Þ in Eq. (27) stands physically for ∂φ Σ ρ f L p;max 1 p the change in the surface chemical potential of non-occupied Mp Σ 2 adsorption sites per unit surface upon a change in the nanoparticle ρ L b p p Mp L f bulk concentration. It is worthy to note that this emphasises M ρ L p f f ω the influence that non-adsorbed bulk nanoparticles have on the ρ M L p f p surface chemical potential of non-occupied sites, called here the K p;max non-occupancy effect. For later use, we assign for this term the following symbol qffiffiffiffiffiffiffi Φm K δ πk T p m B k T L jj Φ p m ∂η Σ NO H ¼Γ (28) NO a ∂φ For the first two rows, the first column stands for spherical nanoparticles, the second column for cubical nanoparticles and the third column for disk- A larger absolute value of H means a greater non-occupancy NO like nanoparticles, while for the third row only the fourth column is used, effect, i.e. one bulk nanoparticle will have more impact on the standing for the fluid molecules. For the fourth to ninth rows, the definitions are general for all surface energy (and thus the surface chemical potential) of nanoparticle shapes and fluid molecules. The symbol a stands either for p the non-occupied sites. Note that an equivalent analysis can be the radius of a spherical nanoparticle, half of the side of a cubical made for the adsorbed nanoparticles contribution (see Eq. (26)) nanoparticle or the radius of a disk nanoparticle, the latter of which has ∂η Σ p through the term ðΓ Þ¼H , called the occupancy effect. The thickness h . Further, ℓ is the equivalent radius of a sphere corresponding ∂φ p f to the volume of a fluid molecule, while ρ , ρ , M , and M stand for the p f p f term ð Þ stands for the excess surface concentration normalised nanoparticle and fluid densities and the nanoparticle and fluid molar by the surface-equivalent of the bulk concentration. We will assign masses, respectively. It is worthy to note that it is not necessary to know the following symbol to it the molar mass and density of the nanoparticles to calculate the definitions in this Table and that it is mainly a question of size. D ¼ Nevertheless, the values of ρ and M (from which M is obtained) are (29) p p' p p Σ Σ Γ still given in Table 1 should one need to know the values of Γ and Γ b p;max in terms of unit mass per unit surface. The values of f have been adapted A positive value of D means a high degree of adsorption of for the (dl)Au nanoparticles due to the presence of ligands at the gold nanoparticles (decreasing the surface energy), while a negative surface inducing possible repulsion or blocking mechanisms. In ref. ,it has been established that the (dl)Au nanoparticles occupy 0.2, 0.34, and value indicates a preference of nanoparticles to remain dispersed 0.72 of the theoretical maximum coverage when dispersed in D, DD, and in the bulk. In summary, the sign of D indicates whether the HD, respectively. Therefore, for the (dl)Au-D, (dl)Au-DD and (dl)Au-HD surface tension will increase or decrease and the value of H NO systems, f has been multiplied by 0.2, 0.34, and 0.72, respectively. It is with what amplitude. As both depend on φ,itiseasyto recalled that δ and Φ represent the primary minimum of the total m m understand that the magnitude and variation of the surface potential energy and Φ its value, whereas k and T are Boltzmann’s m B tension might be different as a function of φ, generating the constant and the temperature. different observed trends. More interestingly, Table 5 shows that the several nanoparticle dispersions considered here have quite different values for the nanoparticle equilibrium adsorption constant K .Thisimplies thatthisproperty playsanimportant Table 5 shows the nanoparticle dispersions that we consider in role in determining the behaviour of the surface tension. Note this work. For completeness, the calculated numerical parameters that the parameters D and H also depend on the surface p NO that are necessary for determining the surface tension as a coverage θ , which is linked to φ through K . This encourages to function of the volume fraction, i.e. γ , K , ω and Γ , are given in p p f Σ p consider φ and the property K as suitable parameters for the Table 5 for these nanoparticle dispersions. p present analysis. Different kinds of behaviours for the surface tension of We should define a certain reference system that represents a nanoparticle dispersions are represented by several experimen- nanoparticle dispersion of which we can change freely the tal case studies, representing different materials (for both the 6,9–11,19,38,47,48 parameters φ and K and monitor their influence on the behaviour nanoparticles and liquids) with different sizes . of H and D and therefore on that of the surface tension. To NO p The surface tension is calculated from Eq. (25)for these perform numeric demonstrations, allowing the quantification of our nanoparticle dispersions and compared to the experimental analysis, we may choose data from any dispersion. Only because data in Fig. 3a–f. The experimental data in Fig. 3 show different the B-D system is an example of an interesting decrease-increase types of behaviours and the present model has an overall good behaviour, its data are used for the present demonstration. As agreement with those data. This motivates to use the model to the discussion should be followed in a general sense, and we only explain these observations. use the physical properties of this dispersion but changing freely K , it is appropriate to name it differently: the reference system R1. Non-occupancy contribution Figure 4a shows the surface tension of the R1 system as a We candividethe surfacechemicalpotential, η  η þ η ,in function of φ for various imposed K values and two different p p NO p a partthatstandsfor thecontributionbynon-occupied sites nanoparticle sizes. η ≡ −ω R Tln(1 − θ ) and a part that represents the Note that for small K values, the surface tension remains NO p g p p contribution of the adsorbed nanoparticles η ≡ R Tln(θ ). We significantly constant. As this is counter intuitive (usually p g p can also split the surface tension change into two parts as non-adsorption should lead to an increase in the surface tension npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi Table 5. Calculated parameters used in Eq. (25) for the nanoparticle dispersions. −1 −3 2 3 Σ −2 NP-L γ [mN m ] K [10 ] K [10 ] ω [10 ] Γ [nmol m ] T [K] f Σ p p b Al O -W 72.3 2.11 0.0562 6.23 148 300 2 3 Al O -D 23.8 4.66 11.5 1.28 67.1 300 2 3 Al-D 23.8 12.9 3.19 0.165 186 300 B-D 23.8 9.22 4.16 2.08 38.0 300 Al O -E 22.4 3.12 0.294 2.85 100 300 2 3 Al-E 22.4 8.66 0.103 0.369 279 300 B-E 22.4 6.17 0.0988 4.65 56.8 300 Al O -TEG 44.45 10.3 10.3 0.262 190 298 2 3 MgO-TEG 44.25 10.3 4.34 0.262 190 298 SiO -W 72.5 3.51 O (0) 2.24 247 298 Ag-W 68.0 1.05 0.209 24.9 74.1 298 Lap-W 73.6 43.5 193 0.872 85.1 300 ZnO-EG 47.3 2.29 0.0964 5.27 75.9 300 (dl)Au-D 22.96 27.7 8726 0.036 399 303 (dl)Au-DD 24.75 29.2 8.55 * 10 0.033 380 303 (dl)Au-HD 26.96 31.8 2.23*10 0.027 349 303 Al O -Ws 72.3 2.64 O (0) 3.99 185 300 2 3 The symbols W, Ws, D, DD, HD, E, TEG, EG and G stand for the base fluids water, extra stabilised dispersion, n-decane, n-dodecane, n-hexadecane, ethanol, tri- ethylene glycol, ethylene glycol, and glycerol, respectively. Lap stands for laponite and (dl)Au for dodecanethiol-ligated gold. Note that K values that are orders of magnitude smaller than unity have been considered here as being virtually zero, O(0), i.e. negligibly small. as this entails that Γ < 0), special attention will be given to the that K has the same type of effect on the surface tension for p p small K -case later. Figure 4a shows that, as K increases, γ(φ), for a whatever nanoparticle dispersion’s physical properties. Figure 4e p p given φ, first increases and then starts to decrease for small φ, shows H , D and D H (combined contribution of the latter NO p p NO two) versus φ for three K values for the R1 system. followed by an overall decrease in the depicted φ -range. This first For small K ð¼ Þ, Fig. 4e shows that D is significantly increase is also counter intuitive (usually more adsorption should p p lead to a decrease of the surface tension), a second point given negative over the whole volume fraction range. Figure 4e shows special attention later. As K continues to increase, even a that at small K there is a negligible contribution of the absolute value (being, by the way, always negative) of H (dotted blue minimum in γ(φ) as a function of φ is observed, a third point NO line), it is significantly constant over the φ range. Although D is discussed later as well. For even higher K , the surface tension p clearly negative (dotted red line in Fig. 4e), which stands for a shows a decreasing trend, which is what one would expect. Figure negative surface excess concentration and would conventionally 4a also shows that a smaller nanoparticle size tends to favour a imply an increase in the surface tension, the resulting surface decrease in the surface tension. The latter effect can be understood tension remains significantly constant as shown by Fig. 4a by noticing that smaller nanoparticles will increase, for the same φ, (straight solid lines). To understand why this is, we take the limit the number of nanoparticles and therefore also the number of of Eq. (25) for K → 0, which gives adsorbed nanoparticles, which leads eventually (for sufficiently p small nanoparticles) to a decrease in the surface tension. γ ¼ lim γ ¼ γ þ R TΓ φ K !0 f g p b (30) Let us, before entering into such an analysis, first determine K !0 what contribution to the surface tension is more important, Δγ or Δγ . Figure 4b shows, through Δγ (i = p,NO) scaled by R T, that NO i g Filling in Eq. (30) the data for the R1 system reveals that 5 4 the contribution of the non-occupied sites (i = NO, the solid lines) γ  γ ¼ Oð10  10 Þφ. This explains the seemingly (in K !0 f is the main one, especially at larger K -values. The main reason for reality, very weakly increasing) constant value of the surface this is size-related. The nanoparticles are much larger than the tension. The reason behind the seemingly constant value of the fluid molecules, which constitute the adsorption sites. This means Σ surface tension at K → 0 lies in the value of Γ . From Table 4 p b that the number of fluid molecules involved in the adsorption of a one may easily deduce that Γ / . Nanoparticles have generally a ℓ p f nanoparticle is quite large, expressed in large ω -values, i.e. ω ≫ p p a larger size than fluid molecules and apparently large enough for 1, as Table 5 shows. So, it is now evident that the non-occupied Γ to be sufficiently small and hence a seemingly constant site contribution of the surface chemical potential will be a key behaviour of the surface tension can be predicted. This explains part in the following discussions. the first counter-intuitive observation. At K = K , Fig. 4e shows that for a small range of φ we have p p* Three counter-intuitive effects of K on surface tension D > 0 (equivalent to Γ > 0, let us recall), whilejj H becomes p p NO Figure 4c shows Δγ and Γ for two volume fractions for the NO p bigger than for the previous case (see solid line in Fig. 4eas R1 system as a function of K . To facilitate the discussion three opposed to the dotted line). As H <0 (always), this leads to NO markers have been introduced for Δγ : one corresponding to NO D H < 0 (solid line in Fig. 4f). As we increase φ, Fig. 4e shows p NO the K from Table 5 for the B-D system (≡K ), a smaller one that D changes sign, i.e. D < 0, withjj H still being significantly p p* p p NO larger than zero, resulting into D H > 0. As the surface tension and a larger 100K one. Figure 4d represents Δγ and Γ for two p* NO p p NO volume fractions as a function of K for a so-called R2 system, depends on the integration of D H from 0 to φ, the surface p NO where we use the data from the Ag-W system, merely to illustrate tension will decrease as long as D H < 0 and increases as long p NO Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi Fig. 3 Modelled surface tension as a function of the volume fraction of nanoparticle dispersions, compared to experimental 6,9–11,19,38,47,48 data .a Al O -W ( , ), Al O -Ws ( , ), Laponite-W ( , ) and ZnO-EG ( , ), b Al O -D ( , ), Al-D ( , 2 3 2 3 2 3 ) and B-D ( , ), c Al O -E ( , ), Al-E ( , ) and B-E ( , ), d Al O -TEG ( ,*) and MgO-TEG ( , ), e SiO -W ( , ), Ag- 2 3 2 3 2 W( , ), (dl)Au-D ( , ), (dl)Au-DD ( , ) and (dl)Au-HD ( , ), the inset being a zoom of the (dl)Au-D system concentrating at the region of smaller φ-values, f a magnification of the systems (dl)Au-D ( , ), (dl)Au-DD ( , ) and (dl)Au-HD ( , ). The studied systems are indicated in the form “nanoparticle-fluid”. The model values are indicated by lines, while the experimental data are given by markers in the form (line,marker). K K 1 p Σ as D H > 0, passing thus through a minimum. This analysis p NO when φ ¼ or φ = 0, but the latter is a trivial solution not implies that the sign of the integrated surface of D H as a p NO considered further. The sign of D depends on the values of K p p function of φ will determine the existence and positioning of a and K . With respect to this, two cases can be considered: K Σ p surface tension minimum. It is then logical to elaborate further on and K > . These cases will depend on the parameter K . From p Σ the dependence of H and D on φ. From Eqs. (6) and (19), we NO p Σ have that Table 4, we can deduce that K / f . The values of parameter Σ 1 f (see Table 4) are constant for a certain shape and K will K φ ∞ Σ D ¼ K  φ ℓ (31) f p Σ depend on the ratio much more than on f . Therefore, for the 1 þ K φ a p p discussion of Eq. (31) we will only take into consideration K / and from Eqs. (19) and (28) that K K 1 1 1 p Σ We treat the case K  . When K  , we have  0 and p p K K K Σ Σ p K ω p p Σ it can be verified that this means that for all φ > 0 we have D < 0. H /Γ (32) NO 1 þ K φ As H is always negative, the result is a strictly increasing surface p NO tension. Depending on the amplitude ofjj H , this increase will NO The analysis of Eqs. (31) and (32) needs some mathematical be significantly measurable or not. Focussing mainly on the value considerations. From Eq. (31), we can easily deduce that D ¼ 0 of K (the value of which may vary orders of magnitude more than p p npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi Fig. 4 Analysis of surface tension behaviour. a γ of the R1 system vs φ for theoretically imposed K values ( =~0, = 10, = Δγ 2 3 3 10 , = 10 , = 5*10 for 2a = 46 and 5 nm. b with i = p,NO vs φ of the R1 system, where thin and thick lines stand for K = 10 and p p R T 5*10 , respectively, and dashed and solid lines stand for i = p and i = NO, respectively, for 2a = 46 and 5 nm. Note that all the dashed lines Δγ p;NO are significantly horizontal. c of the R1 system vs K for φ = 0.001 (dashed line) and φ = 0.025 (solid line) and 2a = 46 with three specific p p R T values of K based on the value from Table 5 (K = 416): ( for φ = 0.001 and for φ = 0.025), K ( for φ = 0.001 and for φ = 0.025) p p* p* and 100K ( for φ = 0.001 and for φ = 0.025). On the second axis, Γ of the reference system vs K for φ = 0.001 (dashed line) and φ = p* p p Δγ p;NO 0.025 (solid line) and 2a = 46, d of the R2 system vs K for φ = 0.001 (dashed line) and φ = 0.025 (solid line) and 2a = 100. On the p p p R T second axis, Γ of the R2 system vs K for φ = 0.001 (dashed line) and φ = 0.025 (solid line) and 2a = 100, e H vs φ, and D vs φ, for three p p p NO p specific values of K : (dotted line), K (solid line) and 100K (dashed line) for the R1 system, f D H vs φ for three specific values p p* p* p NO of K : (dotted line), K (solid line) and 100K (dashed line) for the reference system and D H for K = K (dotdashed line) for the p p* p* p NO p p* Ag-W system. ω and Γ , see Table 5), two cases are thus possible: vanishing be verified thatjj H has a significant value, the increase of the p b NO K values (K → 0) and non-vanishing K values (Oð0Þ K  , surface tension will be measurable. A real example for this is the p p p p where O(0) stands for a value that is so low that considering it zero Ag-W system, where Table 5 shows that K ≫ O(0) and K < . p p would reflect a measured reality). Moreover, Fig. 4f illustrates this as well by a continuously increasing D H (brown dot-dashed line). Departing from a p NO o K ! 0 fully desorbed case (K → 0), we can say that upon enhancement (K ≫ O(0)) of the surface adsorption kinetics (up to the limit Equation (32) shows that for small K (e.g. K → 0), we have p p p K  ), the surface tension behaviour becomes a strictly jj H ! 0, so that the surface tension increase is not noticeable. p NO K This has been discussed previously around Eq. (30) for small K increasing one due to the combination of D <0 and a significant p p values and a real example for this is the SiO -W system (see Fig. 3e value ofjj H . So, initially, a higher adsorption appears not to lead NO 1 p Table 5 where indeed K ). p to a lower but rather to a higher surface tension. As / , KΣ KΣ 1 nanoparticles (having much higher size than the fluid molecules) o Oð0Þ K allow for a much larger limit for K for which D remains negative. K p p So, within this limit, upon increasing K , the strength of the non- When K is significantly non-zero but not too large, i.e. Oð0Þ occupancy effectjj H becomes significant, whilst the excess NO K  (defined as the lower-intermediary region), so that it can surface concentration remains D <0, resulting into a surface p p Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi tension increase and not decrease. This explains the second Table 6 shows that as one goes from left to right, the value of counter-intuitive observation. K K increases by several orders of magnitude. This is by either p Σ 1 1 We treat the case K > . When K > , we have a particular increasing K , K or both. Table 5 shows that K for all the p p p Σ Σ K K Σ Σ −2 −3 situation where D >0 (and therefore a decreasing surface tension) nanoparticle dispersions is of the order of 10 −10 . This means K K 1 p Σ that when K K increases several orders of magnitude, this is for 0 < φ < and D < 0 (increasing surface tension) for p Σ Kp mainly due to K . Nevertheless, for quantitative assessments, it is K K 1 p Σ φ > , where we limit φ to a certain maximum value φ , max p more convenient to mention K K . p Σ considered reasonable for typical nanoparticle dispersions (as In order to put the results in perspective, some additional 1 1 discussed later). It can be verified that the surface tension is strictly comments are in place here. In the case K > and K < 1 þ ,we p Σ K K Σ p K K 1 p Σ decreasing if 1, which means for K 1 þ . This gives have made a distinction for the surface tension behaviour K K p p between three ranges of orders of magnitude for K K . Mathema- two regions, the one given by the latter condition and K <1 þ . p Σ tically, these three cases represent all a minimum in the surface tension somewhere in the range 0 < φ < 1. The reason for making o K < 1 þ p the three distinctions is on a conceptual level, involving measured data and a defined framework. As Fig. 3 shows, most of The values given in Table 5 show that for nanoparticle nanoparticle dispersions that are used for engineering purposes dispersions typically K ≪ 1. This means that K ≫ 1. The smaller Σ p (one might also think of medical ones as well, for that matter) K K 1 p Σ ℓ (due to larger nanoparticles as K / or due to lower K ) is, present operating conditions that involve φ values that are often Σ p K a p p −2 limited by a value φ that is of the order of φ ≈ O(10 )or the closer the value of φ for which D changes sign will be to zero. max max −1 slightly higher, but still φ < O(10 ). In Table 6, φ is max max As a consequence, this fits within the typical operating φ-ranges 1 1 schematically indicated for the case K > and K <1 þ by blue p Σ (φ < φ ), resulting into an observable minimum for the surface max K K Σ p tension. This is the case for e.g. the B-D, Al-D, Al O -D systems (see 2 3 vertical dotted lines, set to a same hypothetical value for the three K K 1 p Σ images in question. It shows that as K K increases the minimum Fig. 3b and Table 5 for the values). As, however, becomes p Σ of the surface tension becomes less pronounced and shifts somewhat larger (smaller nanoparticles or higher K ) the φ for towards higher φ values (not on scale), falling out of the range which D changes sign will increase and may fall out of the limited by φ .At φ values beyond φ it is the question max max aforementioned typical operating φ-ranges (φ > φ ). This results max whether we can still speak of dispersions and we then might have into a minimum that is no longer observed (mathematically still to deal with another type of “fluid” with additional phenomena at present, but experimentally not observed within typical φ-ranges) the surface. When working with nanoparticle dispersions, we and the surface tension is virtually decreasing. This can also be have limited the analysis within the range 0 < φ < φ (named max numerically verified in Table 5 and visually in Fig. 3a for e.g. the the “operating range”). As such, depending on the value of K K , p Σ Lap-W system. For systems with even higher K , Eq. (32) shows the mathematical minimum of the surface tension may well be out thatjj H as well as the negative part of D H become more NO p NO of that range and therefore not observed nor experimentally important, confirmed by the 100K case for the R1 system in p* measured. Then, it is justified to indicate conditions (that is, within Fig. 4(e) and (f) (dashed lines). The (dl)Au dispersions (see again the range 0 < φ < φ ), where we can observe a minimum in the max Table 5 and also Fig. 3(e) and (f)) illustrate this situation by 0 1 surface tension (for K K ≈ O(10 −10 )) and where we observe a p Σ presenting surface tensions that decrease quickly for very low virtual decrease. Even for the virtual decrease of the surface volume fractions. In summary, this means that an observable tension, we have made a distinction between a “soft” decrease surface tension minimum is the result of a delicate balance 2 3 4 6 (K K ≈ O(10 −10 )) and a “steep” decrease (K K ≈ O(10 −10 ), p Σ p Σ between a sufficiently large, but not too small, nanoparticle the upper limit 10 being indicative with respect to the observed (through K / ) and a sufficient amount of adsorption (through experiments, but may conceptually be even higher). The soft K > ). This effect is therefore not an external one but stems from decrease is defined as the surface tension having a steady the same parameters that cause strictly increasing or decreasing decrease over the whole operating range, such as the Lap-W case. behaviours, merely because the conditions are right. This explains The steep decrease is characterised by a strong decrease of the the third counter-intuitive observation. surface tension for φ ≪ O(φ ) with a seemingly constant value max afterwards, such as the dl(Au) dispersions. For the parameter K , we have mentioned that for nanopar- o K 1 þ Σ Σ p ℓ ticles we have K / ,not considering f in the discussions. Σ ∞ There are, however, cases where this parameter may play a role. Mathematically speaking, a strict decrease (over the whole When strong repulsive forces are present or when the range 0 < φ < 1) in the surface tension would occur if, next to 1 1 nanoparticle surfaces (because of their nature or their functio- K > , we have K 1 þ . We have mentioned earlier that p Σ K K Σ p nalization) are such that we cannot consider them as hard typically K ≫ 1. This means that, as approximation, we are spheres, the maximum coverage may, respectively, decrease or practically dealing here with the condition K ≥ 1, which entails increase, whereas the shape may also be altered by the that a  Oðℓ Þ. This would besides possibly quantum dots or p f stretching or compressing of the adsorbed nanoparticles. In surfactants, be rather untypical for nanoparticle dispersions. such cases, additional considerations should be made in order Therefore, this case can be disregarded as well for nanoparticle to include these forces between nanoparticles .One maysay dispersions in general. that these forces will be effective there where the nanoparticles are present at the interface, so that the surface tension of Trends and comments particle-laden interfaces is argued to be an effective magni- In fine, it seems that the right combination between adsorption tude . In some cases, the nanoparticles are grafted with strength (K ) and nanoparticle size (a of which the main effect is polymers, which may cause additional effects on the surface p p represented by the parameter K ) is responsible for the different tension due to the dangling chains of the polymers .When ions behaviours. Table 6 shows a summary of the different surface are present (one may think of electrolytes or charged organic tension behaviours as a function of the parameters K and K ,in molecules) strong coulomb interactions may also influence the p Σ the form of the product K K . maximum coverage. p Σ npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA H. Machrafi Table 6. Surface tension behaviours as function of K (adsorption) and K (size). p Σ Theeffect of thenanoparticlesize has been mainly expressed be studied in more details. In addition, it would be encouraged to through the parameter K . It should be recalled that the provide benchmark studies with experimental data on the nanoparticle size also figures in the parameter ω . The parameter adsorption coefficients of various nanoparticle adsorption on ω (standing for the number of adsorption sites) can also liquid–air (fluid) interfaces. quantitatively interfere with the magnitude of the surface Finally, the present model considers the adsorption of tension change through its linear relation with the non- nanoparticles alone in order to focus on this phenomenon. It occupancy effect Eq. (32). Moreover, the parameter ω ,when would be interesting to generalise or adapt Eq. (3) for the much larger than unity, is responsible for the non-occupancy inclusion of the adsorption of molecular species, which could effect to outnumber the occupancy effect through the surface generalise the model for the application of studying the surface chemical potential (see Eq. (12)). Should it be around unity, the tension of solutions containing surfactants or other (in)organic free energy contribution of the adsorbed nanoparticles would molecules. Should multiple adsorption occur, the thermodynamic also be important for the chemical potential and our discussion model presented in this work lends itself to be extended starting, would be different. Nevertheless, once it is established that for most importantly, from an adaptation of Eq. (3). nanoparticles generally ω ≫ 1, and that the variation of K is, as p p mentioned earlier, of far more importance for the non- Mapping of the surface tension behaviour occupancy effect, the variation of the parameter ω is not given The previous analysis has shown a general dependence of the more attention in our analysis. surface tension behaviour on K K , where K and K stand for the p Σ Σ p The size of the nanoparticles also matters from another point effect of nanoparticle size and adsorption strength, respectively. In of view. The projection method necessitates that the radius of order to quantify this dependence and map these behaviours, we curvature (reciprocal of the curvature) should be much larger choose four representative systems, having, respectively, see- than the nanoparticle radius. In other words, the interface should mingly constant, strictly increasing, minimum containing and be “flat” with respect to the size of the nanoparticles. If the virtually decreasing behaviours for the surface tension. Figure 5a pressure difference over the interface is negligible, Young’s shows the values of D , jj H and D H for these four p NO p NO equation (where the pressure difference is related to the surface nanoparticle dispersions, at two volume fractions, that have tension and the interface curvature) predicts that such an distinct behaviours with low to high K K in the following order: p Σ assumption would be realistic. SiO -W < Al O -W < B-D < Lap-W. Figure 5a shows that, although 2 2 3 In ‘Representation of an interfacial layer on a dividing surface’, SiO -W and Al O -W have comparable negative D values, D H 2 2 3 p p NO we mentioned that we used half the surface to calculate the is only significant for Al O -W due to a much higherjj H , NO 2 3 volume-to-surface ratio. A heuristic reason was employed for confirming the analysis in the previous section, which means a this, assuming that only half the surface facing the dividing non-measurable increasing surface tension for SiO -W and a surface would matter in the adsorption process for particles that measurable one for Al O -W. As K K increases, i.e. for B-D, we can 2 3 p Σ are much larger than the fluid molecules that constitute the see a positive D for φ = 0.005 and a negative one for φ = 0.01. As adsorption sites. As a verification, we performed surface tension the valuejj H is significant enough, this results into a visibly NO calculations using cases with a fourth, a sixth and the whole negative D H for φ = 0.005 and a positive one for φ = 0.01, p NO particle’s surface to calculate the volume-to-surface ratio. It meaning first a decrease and then an increase in the surface appeared that the heuristic choice we have made for the tension. For even larger K K , i.e. for Lap-W, we can see a positive calculation of the volume-to-surface ratio, i.e. using half the p Σ D for both φ’s. With a largejj H , D H is considerably nanoparticle’s surface, was the most appropriate one with p NO p NO negative for both φ’s, corresponding to a virtually decreasing respect to the experimental data. It would be interesting to surface tension behaviour that was observed for Lap-W. investigate the degree of this participating surface experimen- In the present study, we aimed at proposing a framework, tally. However, for this work, the heuristic choice we have made appeared to be sufficient. model and explanation dealing with the different behaviours of It should be noted that the way K has been calculated assumes the surface tension of nanoparticle dispersions. We have seen that that it is enough to take into account the wettability of the the adsorption strength (K K ) and the nanoparticle size (through p Σ nanoparticles in the potential energy. The DLVO theory is known K ) collaborate or compete in determining these different 24,51 to be used for adsorption on solid-liquid interfaces. In refs. as tendencies. It is then interesting to map the surface tension well as in the present work, it is assumed that the DLVO theory, behaviours of all the nanoparticle dispersions that were presented albeit extended, is applicable for liquid-fluid interfaces as well. in Fig. 3 as a function of K and K . Such a mapping is presented in p Σ 24,51 Although already used by others , such a kinetic model should Fig. 5b and gives the opportunity to tailor nanoparticle Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 47 H. Machrafi = . = + = . Fig. 5 Mapping of surface tension behaviour. a Competition of key parametersjj H , D and D H for the nanoparticle dispersions SiO - NO p p NO 2 W(1), Al O -W (2), B-D (3) and Lap-W (4) for two volume fractions: φ = 0.005, φ = 0.01. Note that for a better visualisation both the values of 2 3 D and D H for only the Lap-W system have been divided by 5 and 2 for the cases φ = 0.005, φ = 0.01, respectively. b All the considered p p NO nanoparticle dispersion systems are resumed in a K − K map, representing five observed γ-vs-φ behaviours: “significantly constant” where p Σ the change (proven to be mathematically an increase) in γ is not observable (less than 1%) on the 1 mN/m range, “strictly increasing” where ∂ γ > 0 over any φ-range, “distinct minimum” where a clear minimum is visible at operating φ ranges, “virtually decreasing” where ∂ γ <0at φ φ operating φ ranges and “stronger decrease” where ∂ γ decreases distinctly steeper than the previous case. It is to be reminded that the latter three cases are distinguished within the φ-range under which typical nanoparticle dispersions are used. Moreover, the latter two cases are conceptually the same but are distinguished for application or engineering purposes: much higher adsorption kinetics and/or smaller nanoparticles induce (although theoretically having the same tendency) for the observer a decrease that is much steeper and occurs at much lower nanoparticle concentrations, which justifies to make a distinction between them. The colours indicate qualitatively the transition from one region to another, whereas the model gives two mathematical limits: the limit K K = 1 (red dashed line) designates formally the p Σ crossover from ∂ γ| >0to ∂ γ| < 0, while the limit K ¼ 1 þ (blue dashed line) stands for the crossover from ∂ γ| <0to ∂ γ| <0. φ ∀φ φ φ→0 Σ φ φ→0 φ ∀φ Interestingly, b shows that the dispersions seem to correspond to sets of simultaneously increasing K and K , indicated by the left-to-right p Σ diagonally up-going set of points. 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Open Access This article is licensed under a Creative Commons 77. Patel, K. H. & Rawal, S. K. Contact angle hysteresis, wettability and optical studies Attribution 4.0 International License, which permits use, sharing, of sputtered zinc oxide nanostructured thin films. Ind. J. Eng. 24, 469–476 (2017). adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party ACKNOWLEDGEMENTS material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the Financial support from BelSPo and the MAP Evaporation programme from ESA is article’s Creative Commons license and your intended use is not permitted by statutory acknowledged. regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http:// creativecommons.org/licenses/by/4.0/. AUTHOR CONTRIBUTIONS H.M. developed the model, performed the calculations and comparison with experimental data, analysed the results, wrote and approved the manuscript. © The Author(s) 2022 npj Microgravity (2022) 47 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA

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