This study examines the translation and rotation of a spherical colloid straddling the (upper) air/liquid interface of a thin, planar, liquid film bounded from below by either a solid or a gas/liquid interface. The goal is to obtain numerical solutions for the hydrodynamic flow in order to understand the influence of the film thickness and the lower interface boundary condition. When the colloid translates on a film above a solid, the viscous resistance increases significantly as the film thickness decreases due to the fluid-solid interaction, while on a free lamella, the drag decreases due to the proximity to the free (gas/liquid) surface. When the colloid rotates, the contact line of the interface moves relative to the colloid surface. If no-slip is assumed, the stress becomes infinite and prevents the rotation. Here finite slip is used to resolve the singularity, and for small values of the slip coefficient, the rotational viscous resistance is dominated by the contact line stress and is surprisingly less dependent on the film thickness and the lower interface boundary condition. For a colloid rotating on a semi-infinite liquid layer, the rotational resistance is largest when the colloid just breaches the interface from the liquid side. When the colloid surface is only partially wetting (cos θ ≠ 1), the energy can be substantially negative and much larger than the thermal energy kBT when a is order hundreds of nanometers or larger. Hence the colloids become energetically trapped at the interface, while having the freedom to move along the surface in their own version of Abbot’s Flatland1. Their permanence on the surface armors the interface against coalescence with other surfaces, and generates a resistive surface rheology. Both of these effects play central roles in traditional colloid-related technologies in the chemical, oil and mineral flotation industries as well as in new emerging technologies. For example, nano- and micron sized colloids (10 nm–10 μm) are added to the liquid phases in the preparations of foams and emulsions; the colloids adsorb to the interfaces of the dispersed phase where they stabilize the dispersion from coalescence (i.e “Pickering” emulsions2) and thereby increase the lifetime of the dispersion. Water droplets naturally occur in crude oil recovered from reservoirs, and the droplets are stabilized from coalescence by the adsorption onto the droplet interfaces of clay colloids (100 nm–10 μm) found in the crude. A fundamental challenge in hydrocarbon processing is to remove these stabilized droplets which typically do not coalesce to larger droplets and sediment out. In the process of flotation, minerals (order 100 μm) suspended in a liquid phase are removed by adsorption to rising gas bubbles which are collected in a surface foam3.Colloids which adsorb from a liquid phase onto an air/liquid interface experience a reduction in interfacial energy as the surface area of the colloid, formerly in contact with the liquid, now straddles the surface see Fig. . This ch5. In these technologies, thin liquid films containing suspensions of colloids are spread onto a solid substrate; as the colloids adsorb onto the surface, the subsequent evaporation of the film draws the colloids together by capillary forces to create assembled structures. Magnetic and electrostatic forces applied to magnetized or charged colloids, respectively, are also used to tune the assembly which can further be tuned by using active colloids which propel themselves6.In most of the above applications, the fluid interface on which the colloids are located separate a droplet phase from a continuous phase, and the thickness of both adjoining phases are large relative to the size of the colloid. The goal of this study is to examine the case in which the thickness of the fluid phases which bound the interface can become of the order of the colloid size. We particularly focus on colloids straddling the air/liquid interface of a planar thin liquid film and examine the cases in which the opposing film interface, opposite to the interface that attaches the colloid, is either a solid substrate or a free liquid film see Fig. . Colloida is the colloid radius, U the velocity, μ and ρ the liquid viscosity and density respectively. The interface on which the colloids are confined is assumed to be flat  than the liquid. The fluid interface can also deform due to the viscous fluid stresses which develop in the underlying liquid as the colloid moves along the surface7. In either case, the interfacial deformation is resisted by the restoring force of the surface tension, γ. If surface tension forces are large enough relative to the normal force ). Under the above two assumptions, the equations for the colloid motion are linear (Stokes flow) and the flow can be divided into two realizations , their hydrodynamics become complicated as they translate and rotate along the surface because the colloid hydrodynamically interacts with the opposite surface Fig. . The preons Fig.  - one inU of order of 10−3 ms−1 , colloid densities of order 103–104 kg m−3, liquid density and viscosity of order 103 kg m−3 and 10−3 kg m−1 s−1, respectively, and surface tension of the order of 10 mNm−1, colloid diameters should be less than approximately 10 μm for Bo < 10−2 and Re < 10−2 which infers that the colloid size range which satisfies the restrictions of our study approach the colloid domain. For nanoparticles, the diffusion and forced motion of nanoparticles on fluid interfaces have been studied theoretically using molecular dynamics simulations14 without gravity and low to order one Reynolds number, and noticeable meniscus curvature at the contact line was not observed, either as the colloids diffused along the surface or were forced to move by an external force.While the restriction that the capillary number is small does not involve the diameter, the requirements that the Bond and Reynolds numbers be small restricts the colloid size. For typical values for the translational velocity 16, finite element18 or integral transform methods19 have focused on the translation, and have assumed that the colloid does not rotate due to the presence of surface roughness or chemical heterogeneities that pin the three-phase contact line (contact angle hysteresis). However recently, evaluating the drag exerted on non-smooth colloids14 with heterogeneous surfaces as they diffuse along an interface while specifically examining the role of contact line pinning and de-pinning have been important given the extensive interest in particle-laden interfaces. For the translational motion with velocity U, the fluid exerts a drag in the opposite direction to the motion calculated in terms of a drag coefficient d). The drag coefficient increases as the colloid becomes more immersed in the liquid phase, but the colloid is required to be of order ten diameters away from the interface before the drag coefficient of a colloid in an infinite medium is obtained. The translational motion of the colloid along the interface in the positive z direction also generates a torque along an axis parallel to the interface and perpendicular to the flow  and the stress asymmetry is the largest and then decreases to zero as the colloid becomes further immersed into the liquid phase. The drag coefficient due to translation has been used to model the pairwise interaction of two colloids atop an air-semi-infinite liquid phase at a particular immersion depth, and subject to capillary attraction forces due to the overlap of the interfacial depressions of the interface, and the theoretical predictions are in agreement with the measurements12. These results have been extended for the case of flow over a monolayer of interfacial colloids at the liquid-gas interface20.Past studies using either boundary integrallow Fig.  which acd/a = 1 or θ = π/2) by Brenner et al.22 who develop an eigenfunction solution for this immersion depth. Brenner et al. pointed out that the rotation requires the meniscus at the three phase contact line to advance  and recede  along the surface of the colloid. As the liquid advances or recedes, a non-integrable liquid shear stress develops on the colloid surface due to the imposition of the no-slip condition on the solid surface because the surface velocity at the contact line is discontinuous as it is approached from the solid/liquid and gas/liquid sides. This results in the non-physical conclusion that the colloid cannot rotate due to this non-integrable stress. This contact line singularity due to no-slip can be removed by the inclusion of several physicochemical hydrodynamic effects23: (i) Inclusion of slip at the contact line by a Navier slip condition, (ii) the introduction of intermolecular (disjoining pressure) forces in the fluid in the vicinity of the contact line which can lead to precursor films ahead of the macroscopic contact line, (iii) modelling the contact line motion as a thermally activated process using molecular kinetic theory in which progression of the contact angle is in steps that overcome energetic barriers and (iv) the inclusion of evaporation and condensation of liquid at the contact line, allowing the liquid to move over an existing thin layer. Here, the Navier slip model is used as experiments have revealed the presence of small slip coefficients (order 1–10 nm) in the motion of liquids over solid surfaces which are smooth on the scale of order tens of nanometers or smaller27. Brenner et al. use the Navier slip model to account for slip at the contact line. For a colloid rotating about the x-axis with angular speed Ω , the Navier slip condition is of the form λ is the slip coefficient. As λ → 0, the no-slip condition is recovered. Closed form solutions for the translational coefficients were obtained, z direction, as well as create a resistive torque (directed clockwise), cf. .If the force acting on the colloid is large enough, the torque generated by the motion can overcome the pinning forces and the colloid is able to rotate. This rotation has only been studied for the case in which half of the colloid is immersed in the liquid phase  (see Methods and Supplementary Information). Similar finite element models (FEM) using COMSOL Multiphysics has been implemented to solve Stokes equations for a sphere in bulk of a rectangular channel under Poiseuille flow conditions28. The cases of a film on a solid support, and on a free-interface are being separately studied, and in both the effect of the film thickness h/a is considered, with the contact angle being that of a neutral wetting colloid (cos(θ) = 0, d/a = 1). Finally, a more detailed study is given on the rotation of a colloid on the surface of a semi-infinite liquid, and the rotational coefficients are computed as a function of the immersion depth. The discussion uses the drag and torque coefficients to detail sliding and rotating regimes for a colloid moving under a constant force applied along the interface of a free film, and a film on a solid support.Numerical solutions are developed for the translation and rotation of a colloid moving on the surface of a planar thin film which is bounded from below by a solid wall or a free surface, accounting for both the interfacial motion (as in the semi-infinite studies) and the colloid-lower surface hydrodynamic interaction (as in the studies in which the colloid is fully immersed and in proximity to an adjacent no-slip or zero-stress interface). As in the earlier studies, inertialess flow and a flat interface with a circular contact line will be assumed. Both the translational and rotational parts of the problem are being addressed. To alleviate the contact line singularity and allow the colloid to rotate, the Navier slip condition is used on the colloid surface with a small slip coefficient U in the z direction in the reference frame in which the substrate is stationary. The thickness of the film relative to the colloid radius is h/a = 2.14 (so that film thickness is slightly large than the colloid diameter 2a) and contact angle θ = π/2. The slip length at the colloid surface λ divided by the colloid radius a is 0.01 y–z plane) which is perpendicular to the substrate and cuts the sphere in half. Hence in this plane the flow field is symmetric with respect to x, and the velocity in the x direction is equal to zero. The displayed magnitude of the vectors are proportional to the magnitude of the velocity. The color scheme in the figure provides the magnitude of the velocity in the y–z plane scaled by the colloid velocity U, i.e. z = 0 because the flow Reynolds number is very small (creeping flow). The surface flow at the gas/liquid interface is evident. The upward and downward shapes of the flow streamlines aft and forward of the moving sphere is due to the forward motion of the sphere, and the fact that the free interface is required to remain flat. So, fluid is drawn up aft of the sphere, and is pushed down in the forward direction. The flow field in the reference frame of the moving colloid is shown in Fig. z direction yields the flow field depicted in Fig. In this section, the hydrodynamic drag and torque coefficients exerted on a translating and rotating colloid straddling an air/liquid interface above a thin liquid film on a solid substrate are being calculated. Starting with the kinematics, Fig. x axis with angular velocity Ω in the symmetry plane x = 0 is given in Fig. θ = π/2, h/a = 2.14 and λ/a = 0.01. . The figure makes clear that the disturbance flow caused by the rotation is more localized around the sphere than in the case of the translating colloid  for the translating case, and 1/r2 for the rotating case. Of more interest is the flow detail in the immediate vicinity of the contact line  has to sharply change direction from normal to tangential to the surface. This change gives rise to large velocity gradients in the immediate vicinity of the contact line. Note that the magnitude of the fluid velocity at the sphere surface in the contact line vicinity is much smaller than the velocity of the rotating sphere itself. However, further away from the contact line (i.e. towards the bottom of the colloid), the velocity of the fluid in the immediate vicinity of the rotating sphere is much larger and equals the rotating velocity of the sphere. The reason is due to the large shear stresses on the rotating colloid at the contact line. These large stresses give rise at the corner to a difference between the fluid velocity adjacent to the rotating colloid and the velocity of the colloid surface itself (slip) in accordance with the Navier slip condition x axis) at the surface and ϕ) direction. The velocity difference is proportional to the slip coefficient multiplied by the stress, and even though the slip coefficient in the simulations is small (λ/a = 0.01) the large stresses account for the observable slip. Away from the contact line, but still in the vicinity of the surface, the shear stresses are smaller and the fluid hardly slips. This accounts for the higher velocities observed for the fluid in the immediate vicinity of the rotating colloid and away from the contact line.The flow streamlines for the case in which the colloid is rotating (but not translating) counterclockwise about the ine Fig.  where thx = 0 exerted on the rotating colloid  is plotted in Fig. ϕ along the surface measured from the bottom pole of the colloid. Because of the symmetry about the x = 0 plane, this is the only component of the shear in this plane and it is clear that as the three phase contact line is approached (ϕ = π/2) the stress increases by two orders of magnitude relative to the value at the bottom of the sphere ). This increase reflects the large velocity gradients at the contact line region that creates a large viscous dissipation. In contrast, also plotted in the figure is the shear stress for the case in which the sphere is totally immersed in the liquid phase and μΩ) compares to the theoretical value λ/a = 0.01. . Also plotted in Fig. h/a = 15.0; for larger values the stress profile changes by less than one percent). From the figure, it is clear that the stress distribution is slightly less relative to the case of a finitely thick film (h/a = 2.14) in the vicinity of the wall, but is approximately equal as the contact line is approached. The larger value near the wall reflects the small (1/r2) interaction of the rotating surface with the wall. However, as the contact line is approached, and the stress becomes elevated due to the large velocity gradients, the effect of the wall becomes negligible and the stress is determined by these large velocity gradients. When the slip coefficient is increased, the shear stress at the contact line is reduced due to the increased slip at the surface of the rotating colloid. In Fig. ϕ for λ/a = 0.05, and note that the increase in slip by a factor of five from λ/a = 0.01  the shear stresses are identical.The shear stress , the hydrodynamic drag and torque exerted by the viscous liquid film on the colloid due to the liquid stresses generated by the motion are being computed. In the case of translation in the z direction  the drag on a colloid with d/a = 1 (θ = π/2) is exactly half that of the translational drag in an infinite medium, or h/a is being plotted in Fig. h/a decreases, the drag increases due to the stronger hydrodynamic interaction with the solid substrate as the dominant resistance is the shear stress exerted on the colloid by the liquid. As the gap between the colloid and the substrate becomes very small, the drag increases dramatically due to the strong viscous lubrication forces in the intervening layer29. For Fig. d/a = 1, this dramatic increase begins at h/a ≈ 3 which represents a gap of one colloid diameter. For larger values of the thickness, the interaction with the wall disappears, and the normalized drag approaches the value for the drag for a semi-infinite medium, i.e. one due to the normalization.From the solutions for the velocity field for pure translation  or pure rotation , i.e. the colloid translates over an infinitely thick film. The latter torque was computed by Brenner et al.21, who calculated it analytically using series solutions of the Stokes equations in spherical coordinates (only appropriate for d/a = 1) as a function of the slip length λ. Figure h/a , and see that the torque due to the translation increases as the film thickness decreases due to the increase in the shear stress as the translating colloid becomes closer to the wall due to the larger velocity gradients which develop as the translating colloid comes in proximity to the wall.As noted in the Introduction, the colloid translates along the surface, the shear stress exerted on the colloid by the shear flow in the underlying liquid Fig.  exerts ax axis parallel to the interface, see Fig. d/a = 1 . The latter value was computed by Brenner et al.21 for d/a = 1 and arbitrary λ/a, and is used for the normalization. Plotted in Fig. h/a and note that although an increase in the normalized coefficient is observed, the increase is much smaller than the increase in the normalized drag coefficient h/a = 2.14, is that most of the resistance to the rotation arises from the corner of the three phase contact line where the shear stress is the largest due to the large velocity gradients. Note that in the vicinity of the contact line this large shear stress is essentially the same for both the case of a finite liquid layer (h/a = 2.14) and a semi-infinite layer. Away from the contact line and in the vicinity of the wall, the surface shear stress is slightly larger than the value for the semi-infinite case due to the hydrodynamic interaction with the wall. This small increase accounts for the increase observed in the torque coefficient. When the slip on the surface is increased from λ/a = 0.01 to λ/a = 0.05 and the stress at the surface of the rotating sphere is consequently reduced in the vicinity of the contact line . Although the shear stress has decreased in proportion to the increase in the slip, the torque only decreases logarithmically as the increase in the shear is restricted to the vicinity of the contact line. This logarithmic dependence on the slip coefficient was also recognized by Brenner et al.21 for the case of a colloid rotating at an interface for d/a = 1 where they show that In the case of pure rotation of the colloid at the surface . This normalized coefficient, h/a in Fig. h/a ≈ 3) due to the large velocity gradients which develop in the gap between the rotating sphere and the wall. More importantly, Fig. The resistive stress exerted on the surface of the floating colloid as it rotates counterclockwise around the ion Fig. . This viSimilar to the above, the hydrodynamic drag and torque exerted on a translating and rotating spherical colloid straddling an air/liquid interface on a thin liquid film bounded by a free interface on the other side e.g. a foam lamella has been calculated. Because the opposite interface is stress free, the hydrodynamic interaction of the colloid with the opposite film face is weaker relative to the case of a film on a solid substrate. As will become apparent, the presence of a bounding stress-free interface allows the colloid to translate and rotate with a hydrodynamic resistance which is reduced even with respect to the translation or rotation of a colloid on an interface above a semi-infinite liquid, and this resistance decreases considerably as the film becomes thinner.U in the z direction in the reference frame in which the liquid is stationary, for the same values of the film thickness (h/a = 2.14), contact angle (θ = π/2) and slip coefficient (λ/a = 0.01) as in the case of the solid support. Compared to the flow over a solid substrate  about the ine Fig.  is similine Fig.  and 5c.h/a for λ/a = 0.01 and θ = π/2. Both coefficients show a reduction in hydrodynamic resistance relative to the semi-infinite case because of the presence of the stress free interface and absence of any strong viscous lubrication forces in the gap between the colloid and the bottom interface. In the case of the drag coefficient the reduction is significant and becomes more pronounced as the film thins. The percent decrease is approximately the same as the percent increase in drag for the translation on a film over a solid substrate. However, in the case of the resistive torque, the decrease in the hydrodynamic friction  with a decrease in h/a is small as this resistance is dominated by the stresses at the contact line. This is similar to the case for the rotation of a colloid on the surface of a thin film above a solid substrate, where the increase in the resistive torque with h/a is again marginal because of the dominance of the contact line stresses. Figure θ = π/2) as a function of h/a for two values of λ/a (0.01 and 0.05), normalized by the bulk value (8πμa3). As in the case of the solid substrate, the resistive torque is smaller for the case of larger slip for any h/a. Interestingly, for rotation on the surface of a lamella, the resistive torque coefficient decreases for small h/a to a very low value for the colloids with a relatively large slip (λ/a = 0.05).The normalized drag coefficient for translation  are smaller than those for colloids on the interface of a semi-infinite liquid. The figure also shows that the two numerically computed coefficients are approximately equal; their equality is required by the reciprocity theorem.The normalized cross coefficients - the torque due to translation and the drag due to rotation - are plotted in Fig. θ = π/2 or d/a = 1. From the asymptotic behavior of the drag or torque coefficients as a function of h/a , neutral  and hydrophilic θ = π/8, d/a = 1.8 colloid, respectively, translating along the interface with velocity U in the positive z direction in the laboratory frame with slip coefficient λ/a = 0.01. As would be expected, the more hydrophilic the colloid is, the greater its penetration in the liquid phase and larger is the hydrodynamic disturbance and the net viscous stresses exerted on the colloid.In this section. the hydrodynamic drag and torque exerted on a translating and rotating spherical colloid straddling an air/liquid interface above a semi-infinite liquid have been calculated, and the effect of the immersion depth (contact angle) on the hydrodynamics has been studied which in the calculations in the prior sections was fixed at d/a. For immersion into a semi-infinite liquid, the drag coefficient is normalized by the Stokes drag on a colloid which is completely immersed  by Brenner et al.21 and Pozrikidis16 and for a sphere fully immersed in the liquid (d/a > 2)30. As evident in the figure, their results are in agreement with the prior calculations over the complete range of d/a. . However, as a result of the stress singularity at the vicinity of contact lines which recede or advance over a solid surface under no slip-conditions, a small slip on the solid (sphere) surface has a strong effect on the resistive torques due to rotation.In Fig. d/a is plotted in Fig. d/a = 1) colloid translating in the liquid phase. In this case, when the colloid is just immersed in the liquid, the torque exerted by the translation is very small since the wetted contact area is close to vanishing. The coefficient then increases rapidly with immersion depth until the colloid is half immersed (d/a = 1) and then starts decreasing as the colloid becomes completely immersed (1<  d/a< 2). As the torque is driven by the asymmetry in the shear stress (in the y direction) exerted by the liquid on the colloid due to the translation  on the resistive torque exerted by the liquid on the colloid. Figure see Figs .λ/a = 0.01). As the immersion depth increases beyond 2, the contact line disappears and its contribution to the resistive torque vanishes as well. Because of this, the resultant resistive torque for d/a > 2 is due only to the viscous drag around the rotating colloid. As the now completely immersed colloid descends further into the liquid phase (d/a = 2). Plotted in Fig. et al.22, who obtained solutions for zero slip. For this case, the resistive torque becomes infinite as the colloid just touches the interface from the completely immersed side due to the stress singularity at the contact line as liquid advances on one side and recedes on the opposite side of the rotating sphere. The resistive torque remains infinite as the colloid straddles the interface (0 < d/a < 2). Figure d/a tends to 2 from the liquid side, the computed coefficients are smaller than those of Davis et al.22 because of the assumption of a small but finite slip (λ/a = 0.01). Far enough from the interface in the liquid (d/a = 2.1) or a gap separation distance of one-tenth of a radius, the effect of the finite and small slip is not important and the calculations agree with the no-slip calculations of Davis et al.22.The normalized resistive torque due to pure rotation of the colloid, a) translating and rotating along a planar interface atop either (i) a thin liquid film on a solid support, (ii) a thin free liquid lamellae or (iii) a semi-infinite liquid phase  is of primary interest. The drag exerted on a translating colloid was found to be a strong function of the thickness of the film relative to the colloid radius (h/a), and whether the opposite face of the film was solid boundary or an air/liquid interface. For a colloid translating on an interface of a liquid film above a solid substrate, the drag coefficient (increases significantly as the film thickness decreases to the order of the colloid radius (h/a → 1) due to the large velocity gradients (and hence large viscous lubrication forces) as a result of the strong hydrodynamic interaction of the moving colloid with the stationary bottom wall  because the mobility of the bottom free interface does not allow strong velocity gradients to develop . For a colloid with a no-slip surface which straddles the surface and rotates, viscous stresses become infinite along the contact line since this boundary moves relative to the surface. Hence the net resistive torque exerted on the colloid by the fluid becomes infinite and the colloid with a no-slip surface cannot rotate. Even for this small slip coefficient, the resistive torque exerted by the liquid on the rotating colloid was dominated by the stresses developed at the contact line as the coefficient for the resistive torque  gradually experiencing the Stokes drag for a colloid in an infinite medium for d/a of order 10 . This increase results from the development of the large viscous stress at the contact line as the contact angle decreases and the increasingly smaller wedged shape region creates increasingly large velocity gradients  is being used to obtain solutions for the flow field for a spherical colloid , p is the pressure normalized by μΩ), where μ is the viscosity of the liquid.As shown in Fig. y direction) is zero and the tangential shear stress (x and z directions) is zero, nf is the normal vector to the upper air/liquid interface, y direction) and the tangential shear stresses are zero. The bounded computational domain is detailed in the Supplementary information , the velocity normal to the surface  is equal to z direction  normal to the colloid. Denoting by t a unit tangent vector to the colloid surface and At the surface of the solid colloid, the kinematic boundary conditions are formulated to include slip at the interface, as is required to allow the colloid to rotate. In the case of translation, the mentclass2pt{minimNote that this tangential slip condition is formulated for two orthogonal tangential directions on the surface.x direction . The normal component of the fluid velocity at the surface is zero (n · vs = 0), and the tangential slip condition is given byFor the purely rotational case, the (dimensionless) rotational velocity vector of the surface is mentclass2pt{minimP denotes the portion of the colloid surface in contact with the liquid, and The hydrodynamic flow generates a traction and torque on the surface of the translating or rotating colloid, and for each flow realization these are given  by:The hydrodynamic equations and boundary conditions for each of the flow realizations were solved using the COMSOL Multiphysics numerical software which uses the finite element method to solve the field equations and boundary conditions. Briefly, in this technique, the variables are discretized as sums over a set of basis (approximating) functions defined over local spatial domains  for which the computational domain is partitioned, and the tessellation of the domain is designed to merge smoothly with the boundaries of the domain.Supplementary Information


Measurements of the surface tension, density and viscosity of sodium dodecyl sulfate (SDS) and rhamnolipid (RL) mixtures were carried out in aqueous solution. From the obtained results, composition of mixed surface layer at the water–air interface, mixed micelles, parameter of intermolecular interactions, activity of SDS and RL in the surface layer and micelles, Gibbs standard free energy of adsorption and micellization as well as Gibbs free energy of SDS and RL mixing in the surface layer and micelles were established. These parameters were discussed in the light of independent adsorption of SDS and RL and the size of their molecules as well as the area in contact with water molecules. A correlation between the number of water molecules in contact with those of SDS and RL and standard free energy of adsorption as well as micellization of these surfactants was observed. A correlation between the apparent and partial molar volumes of RL and SDS in their mixture and size of surfactant molecules as well as the average distance between molecules was also found. The parameter of intermolecular interactions indicates that there is a synergetic effect in the reduction of water surface tension and micelle formation.The online version of this article (doi:10.1007/s11743-016-1916-6) contains supplementary material, which is available to authorized users. Surfactants are amphiphilic compounds which have very wide application due to their ability to reduce surface tension, increase solubility of many substances connected with micelle formation or detergency power , 2. Two In many practical applications, a mixture of different kinds of surfactants is used , 2. CareThus, to explain accurately the synergetic effect of the surfactant mixture in the reduction of water surface tension and micelle formation, the size of the surfactant, its contactable area with water molecules, the kinds of functional groups in the structure of surfactant molecule, the relative mole fraction of the surfactants in the mixed monolayer and the fraction of the surface occupied by the surfactant mixture and individual components should be known. To achieve such information, the measurements of surface tension, density and viscosity of aqueous surfactant mixtures at constant concentration of one surfactant while changing the other can be helpful.Recently special attention has been paid to biosurfactants which are produced by microorganisms and have very interesting properties . They haCRL) was in the range from 0.0002 to 40 mg/dm3, SDS from 1 × 10−8 to 1 × 10−2 M. All solutions were prepared using doubly distilled and deionized water (Destamat Bi18E) with an internal specific resistance equal to 18.2 MΩ.R-95 rhamnolipid (RL) and sodium dodecyl sulfate (SDS) were purchased from Sigma-Aldrich and were used without further purification. The surfactant structures are shown in Fig. γLV) of the aqueous solutions of RL and SDS mixtures was measured by the Krüss K100C surface tensiometer using the du Noüy ring detachment method as well as a Wilhelmy plate. Before the measurements, the platinum ring and plate were cleaned with distilled water and heated to a red color. The average value of the surface tension for each mixture was obtained on the basis of more than 10 measurements. The standard deviation depending on the methods and surfactant concentration was in the range from ±0.1 to ±0.25 mN/m.The equilibrium surface tension (3.The density of the studied solutions was measured using a densitometer DMA 5000 Anton Paar. The precision of the density measurements given by the manufacturer is ±0.000001 g/mThe dynamic viscosity measurements of the aqueous solutions of RL and SDS mixture were performed with the Anton Paar viscosimeter (AMVn) with a precision of 0.0001 mPa·s.All the experiments were carried out at 293 K within ±0.1 K.H and ΔS are the changes of the enthalpy and entropy at a given temperature (T), respectively.During the adsorption and aggregation processes of surfactants at constant temperature and pressure, the changes of Gibbs free energy of the system are observed, which satisfies Eq. Gadso) and micellization (ΔGmico).The changes of Gibbs free energy of aqueous solutions of surfactants expressed by Eq.  are connγT is the surface free energy of the tail, γTW is the interfacial free energy of the water-tail, AT is the contactable area of the surfactant tail or its part, the γWH and γWH1 are the surface free energies of water-head at different degrees of head hydration, AH is the contactable area of the surfactant head or that with a part of the tail and N is the Avogadro number.The standard Gibbs free energy of adsorption of the surface active ion or molecule at the water–air interface results from the transfer of hydrophobic part of surfactant from water to air phase and changes of hydration degree of hydrophilic part of surfactant. Thus, it was suggested that this energy can be expressed by Eq. Gint) of the surface active ion or molecule through the water phase. This energy is expressed by Eq. AT* and AH* are the contactable areas between the tails and heads of two surfactant molecules or ions, respectively and ΔGHEL is the Gibbs free energy of  the electrostatic interactions between two ions.In turn, the standard Gibbs free energy of micellization is connected with Gibbs free energy of the interaction (ΔAs follows from Eqs.  and 3)  3) the c12H25– and the other with –OSO3. In the case of the rhamnolipid, the volume was expressed by four cubes dealing with the four functional groups. In the rhamnolipid there are two CH3(CH2)6– groups, C6O4H6– and –CHCH2COOCHCH2COOH. Using this approach, the calculated contactable area of SDS and RL is equal to 429.24 and 721.12 Å3, respectively. If we assume that one water molecule occupies 10 Å2, then at the first approximation, 72 molecules of water can come in contact with one surface of the RL and 43 with SDS. The ratio of the number of water molecules contacted with RL and SDS molecules is equal to 1.67. The Gibbs standard free energy of mono-rhamnolipid adsorption calculated from the Langmuir equation is equal to −85.04 kJ/mol . O. OΓSDS rms Fig. b 20]. M. MΓSDS ΓSDS + ΓRL for independent adsorption is also presented (curve 2). It is interesting that in the range of RL concentration corresponding to its unsaturated monolayer in the absence of SDS, the “real” sum is nearly the same as the “hypothetical” one (curves 1 and 2). The “real” sum was obtained from the data presented in Fig. ΓSDS + ΓRL increases from vice versa [To explain this maximum, change after the addition of RL to the SDS solution was considered Fig. , curve 1e of SDS . This materface) . It meance versa .Fig. 4A ΓSDS + ΓRL decreases to a value somewhat lower than that of the limiting area of the RL molecule (ΓRLmax) (curve 6). It should be also pointed out that the changes of “real” ΓSDS + ΓRL are between the values corresponding to those obtained for the limiting area of studied surfactants. The same conclusion can be drawn if the changes of ΓSDS + ΓRL are considered as a function of SDS concentration. The independent adsorption of one component from the surfactants mixture takes place if its concentration is lower than that corresponding to the saturated monolayer of the individual surfactant.Above this concentration, the RL molecules remove those from the mixed surface layer and the sum curve 4)  but highThis is documented by comparison of the measured and calculated from Eqs. S2 and S3  values of surface tension. It appeared that both isotherms of surface tension calculated on the basis of independent adsorption of SDS and RL (Eq. S2) as well as by using the Fainerman and Miller equation (Eq. S3) , 24 are ΓSDS + ΓRL under the above-mentioned conditions are higher than the maximal Gibbs surface excess concentration of both individual surfactants [Thus, the RL tendency to adsorb at the water–air interface is higher than SDS. It means that if the monolayer is saturated by SDS molecules, the RL ones should replace them. However, the addition of RL causes the increase of total adsorption of the surfactants mixture. This increase is observed in the concentration of RL from 0 to the concentration in the bulk phase at which its saturated monolayer in the absence of SDS is formed. This is probably possible because of the decrease in repulsive interactions between the head of RL and SDS in comparison to the RL–RL and SDS–SDS interactions as a result of the hydrogen bond formation between the –OH group in the RL molecule and the oxygen atom of the SDS one. For this reason the values of factants , 20.ΓSDS and ΓRL do not clearly explain the mutual influence of RL and SDS on their adsorption at the water–air interface, particularly at the concentration corresponding to the mixed saturated monolayer. For this reason the mole fraction of the area occupied by RL and SDS molecules at the water–air interface  as well as those of these surfactants in the mixed monolayer were determined on the basis of ΓSDS and ΓRL . If we consider the changes of the area occupied by RL molecules at the water–air interface in the presence of SDS, then we can state that this area is the same as for the solution of individual RL in the range of its concentration lower than that corresponding to its saturated monolayer . However, at the RL concentration corresponding to its saturated monolayer in the absence of SDS and the concentration of SDS corresponding to its unsaturated monolayer in the absence of RL, the area occupied by RL molecules (curves 2′ and 3′) is larger than that in the absence of SDS (curve 1′). In the case the concentration of SDS corresponds to its saturated monolayer in the absence of RL, the area occupied by RL molecules (curve 4′) is nearly the same as for the individual RL (curve 1′). Thus, the replacement of SDS molecules by RL ones takes place at the RL concentration corresponding to that when its saturated monolayer in the absence of SDS is formed. The same conclusion can be drawn when the area occupied by SDS is considered . On the basis of the changes of the area occupied by RL and SDS molecules, it can be stated that replacement of SDS molecules by RL ones takes place when the concentration of both surfactants is close to the saturated monolayer of individual surfactants.The considerations based on the measured and calculated values of the surface tension of the aqueous solutions of RL and SDS mixture as well This is also confirmed by comparison of the summary area occupied by RL and SDS molecules at the water–air interface to those of “hypothetical” ones Figs. S5, S6). It means that as mentioned above the independent adsorption of SDS and RL takes place if the concentration of one surfactant is lower than that at which its individual statured monolayer starts to form. It can be clearly seen from Fig. S7 in which the mole fraction of RL in the mixed monolayer is presented . At the concentration of SDS and RL corresponding to the saturated monolayer of individual surfactants, the “real” mole fraction of RL in the mixed monolayer is higher than the “hypothetical” one. It should be emphasized that at high concentration of SDS and RL in the mixture, the mole fraction of RL is close to 1. It is interesting to know the relation between the mole fraction of RL in the mixed monolayer calculated on the basis of the Gibbs surface excess concentration of particular components and that obtained on the basis of Hua, Rosen and Rubingh theory , 22. For, S6. It Γ12) (Eq. S1c)  (Eq. S9), the parameter of intermolecular interactions (βδ) (Eq. S10) and the activity coefficients of SDS (fSDS) (Eq. S11a) and RL (fRL) (Eq. S11b) in the mixed monolayer at the water–air interface were calculated.On the basis of the data presented in Fig. 1c) Fig. , the molβδ parameter is negative and second condition (|βδ| > | ln (C10/C20)|) is also fulfilled. However, the minimum of the dependence between the surface tension and the mole fraction of RL in the mixture is observed only at low concentrations of the mixture . This energy can be determined on the basis of many approaches but the Langmuir equation modified by de Boer is very often applied [RT or 2RT should be used. In the literature this problem is still not clearly explained and some authors use RT but others 2RT. In our opinion it is more correct if RT is used, so we calculated ΔGads0 in this way. From the obtained data it results that , when the concentration of one surfactant corresponds to its unsaturated monolayer at the water-interface in the absence of another, the values of ΔGads0 are constant. This indicates that there are no interactions between the RL and SDS molecules or these interactions have an insignificant effect on ΔGads0 values. The values of ΔGads0 in this concentration range are close to those of ΔGads0 calculated for individual RL and SDS in the absence of another surfactant at the same assumption [vice versa.The adsorption of surfactants at different interfaces is one of their characteristic properties. The measure of surfactants tendency to adsorb is the standard Gibbs free energy of adsorption   are different from those calculated using the ΔGads0 values of RL in the absence of SDS and vice versa and the mole fraction of surfactants in the mixture in the bulk phase (Eq. S13a) . It means that there is no ideal mixing of RL and SDS in the surface layer and on the basis of ΔGads0 of the individual SDS and RL, it is impossible to predict ΔGads0 of the mixture. On the other hand, ΔGads0 of the mixture should be comparable to that determined on the basis of the RL and SDS mole fraction in the mixed layer (Eq. S13b). Taking into account the mole fraction of these surfactants in the surface layer determined on the basis of the Hua, Rosen and Rubingh theory [Gads0 of individual surfactants, ΔGads0 of the mixture was determined . It appeared that there are some differences between the ΔGads0 values determined from the Langmuir equation (Eq. S12) and those based on the mole fraction of surfactants in the monolayer (Eq. S13b). Therefore, ΔGads0 mixing of RL and SDS in the surface layer was calculated (Eq. S13c) and added to the Gibbs surface free energy of adsorption obtained on the basis of the mole fraction of RL and SDS in the mixed layer . In such case a good agreement between ΔGads0 values of SDS and RL mixture at the water–air interface obtained in this way and those obtained from the Langmuir equation (Eq. S12) is observed. This confirms our suggestion that there are strong interactions between the SDS and RL molecules in the surface layer.To explain more precisely the mutual influence of RL and SDS on their adsorption at the water–air interface, the systems in which the mole fraction of RL in the mixture with SDS is equal to 0.2; 0.5 and 0.8  are taken into account. The values of Δion Eq. S2 .On the basis of our consideration mentioned above, it can be concluded that the tendency to adsorb of a given surfactant depends on the number of water molecules which are removed from contact with surfactant molecules during the adsorption process. It should depend on whether dehydration of head of surfactants takes place and part of hydrophobic chain which can transfer from a liquid to air phase. It is known that in the micelles only a part of the hydrocarbon tail is not in contact with water molecules. Thus if for example it is assumed that only half of hydrophobic tail of SDS and RL molecules is present in the air phase, then Δfrom Eq.  at the aXSDSM) and RL (XRLM) (Eq. S15), the parameter of intermolecular interactions (βM) (Eq. S16) as well as the activity coefficients of SDS (fSDSM) Eq. (S17a) and RL (fRLM) (Eq. S17b) in the micelle were determined using the Hua, Rosen and Rubingh theory [The critical micelle concentration (CMC) is the concentration at which surfactants can form micelles. This concentration, for the same surfactant can have somewhat different values depending on the method of its determination. The CMC values of RL and SDS mixtures were determined on the basis of surface tension Fig. a, b, denLM) Eq. S5, the pah theory , 22.Fig.α from 0 to 0.2 but from 0.2 to 0.8 the mole fraction of RL in the micelle is almost constant and close to 0.8. Knowing the mole fraction of RL and SDS in the mixed micelle, it was possible to calculate the CMC of mixed micelles (CMC12) from the following equation [As expected the mole fraction of RL in the micelle was higher in comparison to that in the bulk phase Fig. . A consiequation :Fig. 10A12 and the mole fraction of one component in the mixed micelle exists  is negative. In our case the βM parameter is negative in the whole range of RL mole fraction in the bulk phase |). It appeared that the second condition is fulfilled only in the range of α below 10−4 and above 0.8 . The changes of the values of ΔGmico calculated from Eq. S18  and partial . There is small drop in the apparent molar volume of SDS and RL isotherms in the range of their concentration at which the aggregation process of the mixture takes place . There is no mutual influence of RL and SDS on their partial molar volume . The isotherms of partial molar volume of RL and SDS are different from those of apparent ones. The partial molar volumes of RL and SDS in the range of their concentrations corresponding to those in which individual surfactants are present in the monomeric form in the solution are constants. At the concentration of surfactant close or higher than their CMC, a significant increase of their partial molar volumes is observed.On the basis of the density measurements Fig. , it is polecules , 27. ForExplanation of the changes of the apparent and partial molar volumes of RL and SDS was achieved on the basis of the size of their molecules and the average distance between the surfactant molecules and water in solution and between the surfactant molecules in the micelles. For this purpose it was assumed that the average distance between the surfactant hydrophilic group being in the monomeric and aggregated forms and water is nearly the same while the average distance between the hydrophobic group of surfactant and water or between the hydrophobic group in micelles can vary. Taking into account the length of the bonds between the atoms in the RL and SDS molecules and the angle between their bonds, the volume of particular groups in the molecules of studied surfactants was assumed to be equal to the volume of cubic at proper sizes. For calculations of molar volumes of surfactant molecules it was assumed that the minimal average distance cannot be lower than 1.56 Å  and the On the basis of the data obtained from the measurements of surface tension, density and viscosity of aqueous solution of SDS and RL mixtures and their discussion the following can be stated.In the concentration range of one surfactant in the mixture corresponding to the unsaturated monolayer at the water–air interface in the absence of another, there is independent adsorption of mixture components.If the concentration of both surfactants corresponds to their individual saturated surface layer at the water–air interface, then RL replaces SDS molecules in the mixed layer.The mole fraction of the surfactants in the saturated mixed monolayer at the water–air interface determined on the basis of SDS and RL Gibbs surface excess concentration is nearly the same as that determined from the Hua, Rosen and Rubingh theory , 21, 22.The standard Gibbs free energy of the mixture can be predicted on the basis of the Gibbs standard free energy of adsorption of individual surfactants, the mole fraction of surfactants in the mixed monolayer and the Gibbs free energy of surfactants mixing in the monolayer.There is synergism in the reduction of the water surface tension and the micelle formation by the SDS and RL mixture.The synergism in the mixed micelle formation was proved by the Hua, Rosen and Rubingh , 21, 22 The ratio of the standard Gibbs free energy of adsorption and micellization of RL to SDS, is close to that of the number of water molecules in the contact with the surfactant ones.The changes of apparent and partial mole volume of the surfactants in their mixture can be predicted on the basis of the molecules size and average distance between the surfactant molecules in the monomeric and aggregated forms.Supplementary material 1 (DOC 2328 kb)Below is the link to the electronic supplementary material.


We demonstrate that quality factors up to 5000 can be obtained in Tamm-like hybrid metal/semiconductor structures. To do this, a Bragg mirror is covered by a thin transparent layer and a metallic film. The reduced losses of these modes are related to an intermediate behavior between conventional Tamm plasmon and Bragg modes lying deeper in the semiconductor medium. One of the most striking features of this approach is that these super Tamm modes can still be spatially confined with the metal. Confinement on micrometric scale is experimentally demonstrated. The simplicity and versatility of high-Q mode control by metal structuration open perspectives for lasing and polaritonic applications. Plasmonic structures present a very small mode volume and allow high emission rate enhancement up to 10003. Nevertheless their relatively low quality factor can be a drawback for lasing or for strong light matter coupling applications with semiconductors4. On the other hand, dielectric micro/nano structures have larger mode volumes but very high quality factors7. They are particularly well suited for lasing, and polariton Bose-Einstein condensation has been evidenced in these systems. Nevertheless, the mode shape remains difficult to control or manipulate. Methods based on direct milling of the structures9 or mesa etching11 have been developed and have proven their efficiency at the cost of complex technological processes. However new developments in lasers, such as orbital angular momentum microlasers12, or in polariton physics13 15 require an increased complexity in term of control of the mode confinement. The hybrid metal dielectric optical Tamm structures have been proposed to overcome this issue17. These modes are formed at the interface between a Distributed Bragg Reflector (DBR) and a metal film19. The shape of the mode is easily controlled by a structuration of the metallic part allowing an easy realization of complex geometries, while the losses are reduced due to the dielectric part of the system. Strong coupling with quantum well exciton21 as well as single photon emission control22 or lasing action25 has been evidenced in these structures. Nevertheless, the quality factor of the Tamm mode remains of the order of one thousand, which is too low for Bose condensation in strong coupling26 and do not allow a strong reduction of lasing threshold27.The spatial confinement of optical modes is a key point for emission control in nanostructures as it directly affects the Purcell factorIn this paper, we propose a structure based on a geometry similar to the Tamm plasmon and allowing at the same time an increased quality factor and a confinement purely achieved by metal patterning. For this purpose, a thin dielectric film is inserted between the DBR and the metal in a conventional Tamm plasmon structure. The electromagnetic structure of the mode is then strongly modified, with a delocalization in the dielectric DBR. Playing on the spacer thickness it is possible to obtain a mode with better quality factor than both the DBR mode and the Tamm mode. To unambiguously separate this mode from Tamm plasmon (TP) mode, we will refer to it as super Tamm (ST) mode. We will first describe theoretically and experimentally the main properties of 2-dimensional ST modes. We will then demonstrate that this mode can be confined by a proper patterning of the metallic layer.Transfer matrix simulations were performed to calculate the squared electric field and the reflectivity associated to Tamm structures including a dielectric spacer of variable thickness. All the structures comprise the same 65 pairs GaAs/AlGaAs DBR centered at 880 nm. A 45 nm silver film is added on the top of the DBR to form a TP structure, and a 20 nm transparent spacer (n = 1.485) is inserted between the metal and the DBR to form a ST structure. The normalized squared electric fields associated to the TP and SP modes are plotted in Fig. The reflectivity of the TP, ST and bare DBR structures are shown in Fig. 28. At the same time the quality factor strongly increases up to 5100 for e = 20 nm, then slowly decreases from this value to a value of Q = 4000 for thicker spacers. This value remains always larger than the one associated to both conventional Tamm plasmon and Bragg modes. For comparison, the same dielectric thickness was also added on the DBR without metal on top , and in a bare DBR to the coupling to radiative modes at the upper and lower interfaces. Compared to the Tamm plasmon, the super Tamm is less bounded to the interface. The reduction of the losses can thus be associated to the reduction of the electric field extension in the metal compared to the dielectric part, in a similar way as other low losses surface plasmon-like modes (e.g. long range surface plasmon)In order to experimentally evidence the formation of ST modes, a 65 pairs GaAs/AlGaAs DBR containing InAs/GaAs quantum dots (QDs) was fabricated (see Methods). These QDs have a broadband emission around 920 nm. Part of this DBR is covered by a 35 nm thick PMMA layer (n = 1.485). A 45 nm silver layer was deposited in both areas of the DBR, i.e. directly on the last GaAs layer to form a TP mode, and on the PMMA layer to form a ST mode. Photoluminescence experiments were performed by exciting the sample with a continuous-wave Ti:Sapphire laser focused on the surface with a microscope objective . The emitted light is collected by the same objective and sent to a spectrometer associated to a CCD camera. The Fourier plane of the objective can be imaged on the spectrometer entrance slit in order to record the emission dispersion relation. The dispersion relation of the TP mode is presented in Fig. in and arm width L have been defined in the PMMA resist by e-beam lithography  is presented in Fig. The emission dispersion relation recorded on a ring structure  layers on a GaAs substrate. The spacer consists in a transparent layer of refractive index 1.485. A 45 nm silver layer is added either directly on the last GaAs layer to simulate a Tamm mode, or on the spacer to simulate the super Tamm mode. The theoretical quality factors where estimated by a Lorentzian fit of the calculated reflectivity spectra. The experimental sample was grown by Molecular Beam Epitaxy (MBE) on a GaAs substrate. The difference between the experimental and modeled sample is that during the growth three pairs of highly dense self-assembled InAs/GaAs quantum dots where embedded in the DBR: one pair is located in the penultimate GaAs layer, and two pairs in the topmost GaAs layer. It should be noted that these QDs layers represents typically 4% of the GaAs layer thickness, and exhibit a very close refractive index (nInAs = 3.7 @ 880 nm): The GaAs layer refractive index variation due to the presence of InAs is thus about 0.1%. It is therefore reasonable not to take these QDs layers into account in the transfer matrix simulations. A 35 nm thick layer of poly(methylmethacrylate) was then spin-coated on the top of the sample, and annealed at 170° for 10 min. In order to compare Tamm and super Tamm modes, the PMMA resist layer was then removed by electron beam lithography (e-beam) on large circular areas (diameter 50 µm). In order to confine the super Tamm modes, on another region of the same sample the PMMA layer was removed by e-beam lithography on annular patterns of various inner and outer diameters. A 45 nm thick silver layer was then deposited by sputtering on the whole sample surface: in the areas where the PMMA was removed (Tamm areas) and on the areas where the PMMA layer remained (super-Tamm or confined super-Tamm areas). Photoluminescence experiments were performed at 77 K. The devices are excited from the top trough the silver layer with a continuous-wave Ti:Sapphire laser at 760 nm, focused on a 1.8 µm diameter spot on the sample using a microscope objective . The emitted light is collected by the same objective and sent to a spectrometer associated to a CCD camera. The Fourier plane of the objective can be imaged on the spectrometer entrance slit in order to record the emission dispersion relation. The experimental quality factors were estimated by a Lorentzian fit of the dispersion relation at k = 0. When removing the Fourier lens from the optical path, the sample surface is imaged on the vertical entrance slit of the spectrometer which results in a spatially  and spectrally  resolved image on the CCD.Transfer matrix methods were used to calculate the reflectivity and electric field of the structures. For the simulations, the DBR centered at 880 nm was formed by 65 pairs of GaAs  and Al


The mean behaviour was expressed by the quasi-linear viscoelastic (QLV) material formulation, combining the Yeoh material model  optimisation algorithm was developed and used, in conjunction with quasi-static and dynamic tests performed to five cadaveric heel specimens, to derive specimen-specific and mean hyper-viscoelastic material models able to predict accurately the response of the tissue at compressive loading of strain rates up to 150 s These fThe material and structural behaviour of the heel fat pad regulates the amount of load that is transmitted to the bones and joints of the lower limb across a range of loading scenarios; from low rate activities such as standing and running, to high rate incidents that can cause injury such as sport and vehicular accidents. Therefore, the characterisation of the tissue across a range of strain rates can be used for a variety of applications such as understanding the pathophysiology of related diseases , shoe anIn vivo studies have used imaging , with no known pathology that could affect the properties of the fat pad, were obtained from a licensed human tissue facility. Ethical approval was obtained from the Tissue Management Committee of the Imperial College Tissue Bank ethics committee .2 (voxel size of 0.16 mm3) in order to check for any pre-existing orthopaedic pathology and to obtain geometric data for the FE models.The specimens were stored fresh frozen at −20 °C and thawed prior to dissection and testing. Each specimen was Computed Tomography (CT) scanned  with a slice thickness of 1 mm and an in plane pixel size of 0.4×0.4 mmFeet were dissected from the lower extremities by sectioning along a transverse plane proximal to the distal end of the tibial diaphysis, such that both distal tibia and fibula were preserved. To isolate the calcaneus with the fat pad attached and to prepare the sample for potting, a custom-built rig was used to ensure that the sample was positioned in a typical standing, or seated posture. This involved resting the sole of the foot flat against the bottom of the rig and positioning the exposed distal tibia perpendicular to the bottom of the rig. At this stage, soft tissues covering the medial and lateral sides of the calcaneus were removed to permit bolts to be squeezed against the bone to hold the specimen in place. The distal tibia, the fibula, the forefoot anterior to the calcaneus, the talus, and the cartilage of the posterior and anteromedial facets were removed to reveal the proximal surface of the calcaneus.Each sample, still secured on the dissection rig, was turned upside down and fixed with clamps such that approximately half of the calcaneal body was below the edge of a 45 mm deep cylindrical pot. The sample was fixed into position within the pot using polymethyl methacrylate (PMMA) bone cement a. Four u2.2Each sample was subjected to both quasi-static and dynamic testing. Although similar methods were used for both types of testing, the testing rig and loading protocols differed and are therefore described separately in the following sections.2.2.1Quasi-static compression tests were carried out using a screw-driven uniaxial materials testing machine . Each sample was centred beneath a cylindrical, flat tup, 50 mm in diameter that was connected to a 10 kN load cell (resolution ±1 N) that was incorporated in the machine b. Load, R2>0.99). The displacement required to achieve the target strain was calculated using the undeformed thickness of the tissue measured from the CT scans.For the quasi-static compression tests, three preconditioning compressive cycles were performed up to 5 N before the fat pad tissue was compressed to 50% strain at a speed of 0.01 mm/s. This protocol was repeated twice for each sample; between tests, samples were allowed to rest for 15 min. Preliminary investigations demonstrated that 15 min resting time and three preconditioning cycles were sufficient to ensure that the behaviour of the fat pad was consistent; the force–displacement data from the repeated tests after the second preconditioning cycle were similar for all samples  on the same day after the quasi-static tests. The drop rig incorporates a load cell  above a cylindrical, curved 50 mm diameter tup c. A curvTests were performed at increasing drop heights of 2, 4, 8, 16, 32 and 64 cm, corresponding to target velocities at impact of 0.6, 0.85, 1.2, 1.7, 2.4 and 3.4 m/s. To confirm that the sample was not damaged during the tests, after each increase in drop height a repeat of the initial 2 cm drop test was performed and the peak force, time to peak force and slope of the force–time curve were compared. If the difference in any of these parameters between initial and repeated 2 cm drop was greater than 10%, the sample was deemed to have failed and testing was stopped.All data were recorded using the same data acquisition system that was used for the quasi-static tests, however, at a frequency of 25 kHz. A low-pass Butterworth filter was used to filter the force measurements. The cut-off frequency (1 kHz) was selected based on the frequency analysis of the signal.2.32.3.1Specimen-specific finite element models were developed to simulate the quasi-static and dynamic tests a. The geThe extracted geometries were imported as stereolithography (.stl) files in Geomagic Studio  to form solid objects that were then processed in Autodesk Inventor  to perform Boolean operations and achieve a good match between contacting surfaces. The heel fat pad was meshed with Herrmann tetrahedral finite elements  using Hy2.3.2The cortical calcaneal bone was fixed while the tup, modelled as a rigid body, was restricted to move in the direction of the impact only. For the quasi-static compression simulations the tup was displacement driven to compress the samples up to the target displacement. The input for the dynamic simulations was the initial velocity of the impactor prior to coming into contact with the sample b; for eaContact between fat pad and calcaneus was set to ‘glued’ thereby slipping was not allowed. ‘Touching’ contact was implemented between tup and fat pad; the fat pad was allowed to slide on the rigid surface with a low coefficient of friction (0.01). A sensitivity analysis showed that coefficient of friction values between 0.005 and 1.5 – a physiological range of the coefficient of friction between palmar skin and several types of metal, reported by 2.3.3An inverse FE algorithm was used to determine the strain rate dependent material properties. The algorithm is based on the derivative free Nelder-Mead or downhill simplex method for function minimisation  and was ents Eq. . A facto. δ) Eq.  was alsoThe formulation that was selected to represent the hyperelasticity of the tissue was a subcategory of the generalised Mooney–Rivlin material model, the Yeoh material formulation described in Eq. The material constants the same , or othethe same . The optIn order to investigate the strain rate dependency of the above formulation, only the initial part of the force–time history curves from each dynamic experiment were used in the objective function (up to until the velocity of the impactor had dropped by 10%); during this time the strain rate of the sample can be assumed to be constant. By using this method, each test corresponded to a different strain rate and each run of the optimisation algorithm gave a set of specimen-specific material properties for each sample.By the end of the optimisation procedure, a set of material constants The properties were averaged across all samples in order to provide a material formulation described by strain rate dependent relationships of the material constants (was used . Five tetion Eq. .(4)W=∫0tThe specimen-specific and the average QLV material formulations were finally implemented in MSC. Marc to simulate each dynamic test for its whole duration and compare the result against the experimental data. During the dynamic tests the tissue experienced a range of strain rates. A fit between numerical and experimental results on each sample for the highest, non-catastrophic dynamic test was used to test the derived formulation for validity since the largest range of strain rates was expected in this test.33.1The compressive force–displacement curves for all samples are shown in 3.2The force–time history curves for all samples at all drop heights are presented in 3.3The derived values for the material parameters R2=0.84) to the average strain rate dependent The 4−1. An inverse FE method was used such that the material continuity of the tissue was not disturbed. This method combines the benefits of in situ and in vitro testing as stress–strain curves can be obtained through testing the whole area of interest and not isolated components. Although previous studies also utilised inverse FE methods for the same purposes   b&c strai studies . In the  studies . The avein vivo protocol, testing of cadaveric tissue is the most appropriate methodology. Compared to results from an in vivo quasi-static experiment utilising imaging techniques . The suggested formulations describe the combined response of the soft tissue underneath the calcaneus . This is due to the fact that the plantar soft tissues of the heel, attached to the calcaneus, were not disrupted prior to testing and were segmented as a single, homogenous structure from the CT scans. This is not a limitation for the suggested applications where the structural response of the soft tissues of the heel region are of interest.−1 the FE models overestimate the experimental response. The physiological explanation of this mismatch is associated with the balance between the response of the fibrous tissue and the fat globules. When compressed slowly, the fat globules expand circumferentially and stretch the chambers that mainly restrict this motion. The response is expected to be different when the tissue is compressed slowly at the end of a dynamic test after the tissue had been deformed rapidly and mainly the fat globules have been restricting the deformation up to this point. Numerically, this difference can be explained by the lack of energy dissipation terms for both material formulations. Although the strain rate dependence is taken into account, when the specimen-specific formulations are implemented, the tissue unloads like a spring and returns all the stored energy. Although the implemented QLV model includes energy dissipation terms (Prony series), it was best fitted to the specimen-specific, strain-rate dependent formulations and therefore the accuracy did not improve. This limitation can be tackled by adopting a different optimisation strategy where the QLV material parameters are optimised directly for all dynamic tests of each sample simultaneously.The average and specimen-specific material models predict well the slope of the force–time history curve for the drops on each sample but not the peak force and the unloading curve of the graph. When the strain rate of the simulated test becomes less than 10 sThe results from this study are important for understanding heel biomechanics at various conditions, from daily activities such as walking and running to high rate loading scenarios involving injury. The derived properties can be implemented readily in FE models of the foot and ankle used for a wide range of applications; from shoe and insole design to injury prediction and design of protection. The accurate description of the material behaviour of the fat pad tissue also permits better selection of materials that can be used for reconstruction. Finally, the novel inverse FE method developed can be used to characterise the material behaviour of other complex biological tissues, such as brain tissue or the intervertebral disc, across strain rates and for various types of loading.


We present a novel approach called ChemMaps for visualizing chemical space based on the similarity matrix of compound datasets generated with molecular fingerprints’ similarity. The method uses a ‘satellites’ approach, where satellites are, in principle, molecules whose similarity to the rest of the molecules in the database provides sufficient information for generating a visualization of the chemical space. Such an approach could help make chemical space visualizations more efficient. We hereby describe a proof-of-principle application of the method to various databases that have different diversity measures. Unsurprisingly, we found the method works better with databases that have low 2D diversity. 3D diversity played a secondary role, although it seems to be more relevant as 2D diversity increases. For less diverse datasets, taking as few as 25% satellites seems to be sufficient for a fair depiction of the chemical space. We propose to iteratively increase the satellites number by a factor of 5% relative to the whole database, and stop when the new and the prior chemical space correlate highly. This Research Note represents a first exploratory step, prior to the full application of this method for several datasets. Amongst the multiple methods to explore chemical space, principal component analysis (PCA) of pairwise similarity matrices computed with structural fingerprints has been used to analyze compound datasets3. A drawback of this approach is that it becomes impractical for large libraries due to the large dimension of the similarity matrix4. Other approaches use molecular representations different from structural fingerprints, such as physicochemical properties or complexity descriptors, or methods different from PCA, such as multidimensional-scaling and neural networks6.Visual representation of chemical space has multiple implications in drug discovery for virtual screening, library design and comparison of compound collections, among others10. Also, a related and more recent approach, Similarity Mapplet, makes possible the visualization of very large chemical libraries, by considering PCA of different molecular features, including structural11.In representation of the chemical space based on PCA there have been “chemical satellite” approaches, such as ChemGPS, which select satellites molecules that might not be included in the database to visualize, but have extreme features that place them as outliers, with the intention to reach as much of the chemical space as possibleAlthough we concur with the fact that not all compounds in a compound data set should be necessary to generate a meaningful chemical space, there are still obvious limitations of using a fixed set of satellites to which the user is blinded. Also, until now there was no proposal of such a method based on structural similarity.We therefore suggest the hybrid approach, ChemMaps, in which a portion of the database to be represented is used as satellite, thereby decreasing the computational effort required to compute the similarity matrix without losing adaptability of the method to any particular database. Since it is expected that more diverse sets would require more satellites, a second goal of this study was to qualitatively explore the relationship between the internal diversity of compound datasets and the fraction of compounds required as satellites, in order to generate a good approximation of the chemical space.12. Briefly, we selected focused libraries of inhibitors of DNMT1 , L3MBTL3 , SMARCA2 , and CREBBP . Datasets were selected based on their different internal diversity Forward approach: start adding compounds to the similarity matrix until finding the reduced number of required compounds  to reach a visualization of the chemical space that is very similar to computing the full similarity matrix. The second approach would be the usual and realistic approach from a user standpoint. Each method is further detailed in the next two subsections.To assess the hypothesis of this work we performed two main approaches A):17:The following steps were implemented in an automated workflow in KNIME, version 3.3.2N compounds, generate theN XN similarity matrix using Tanimoto/extended connectivity fingerprints radius 4 (ECFP4) generated with CDK KNIME nodes.1. For each compound in the dataset with2. Perform PCA of the similarity matrix generated in step 1 and selected the first 2 or 3 principal components (PCs).gold standard’. It should be noted that the “real” distances or true gold standard would consider the whole distance matrix. However, for visualization purposes it is unfeasible to render more than 3 dimensions. Therefore, we selected as reference the best 2D or 3D visualization possible by means of PCA.3. Compute all pair-wise Euclidean distances based on the scores of the 2 or 3 PCs generated in step 2. The set of distances are later used as reference or ‘N X 1 similarity matrix. The first compound was selected randomly. In this case, for example, it is only possible to calculate one PC, but as the number of satellites increases, we can again compute 2 or 3 PCs.4. Repeat steps 1 to 3 with one compound as satellite, generating angold standard) and those obtained in step 4.5. Calculate the correlation among the pairwise distances generated in step 2 obtained using the whole matrix .6. Iterate over steps 4 and 5 increasing the number of satellites one by one until7. Estimate the proportion of satellite compounds required to preserve a ‘high’ (of at least 0.9) correlation.8. The prior steps were repeated five times for each dataset in order to capture the stability of the method.The former approach is useful only for validation purposes of the methodology as a proof-of-principle. However, the obvious objective of a satellite-approach is to avoid the calculation of the complete similarity matrix e.g., step 1 in backwards approach. To this end, we developed a satellite-adding or forward approach, in contrast with the formerly introduced backwards approach. We started with 25% of the database as satellites and for each iteration we added 5% until the correlation of the pairwise Euclidean distances remains high (at least 0.9). A further description of the methods for standardizing the chemical data and integrating the dataset can be found in the Supplementary material, as well as a further description of the PCA analysis used.This file contains the six compound datasets used in this work in SDF formatNo special software is required to open the SDF files. Any commercial or free software capable of reading SDF files will open the data sets supplied.Click here for additional data file.Copyright: © 2017 Naveja JJ and Medina-Franco JL2017Data associated with the article are available under the terms of the Creative Commons Zero "No rights reserved" data waiver (CC0 1.0 Public domain dedication).In this pilot study, we assessed a few variables to tune up the method, such as the number of PCs used (2 or 3) and the selection of satellites at random or by diversity. We found that selection at random is more stable, above all in less diverse datasets . Likewiswhereas for 2D-diverse datasets i.e., DNMT1 and SMARCA2, up to 75% of the compounds could be needed to ensure a high correlation. Nonetheless, even for these datasets, using 25% of the compounds as satellites the correlation with the gold standard is already between 0.6 and 0.8; using 50% of the compounds as satellites the correlation is between 0.7 and 0.9. Hence, the higher the diversity of a dataset , the higher the number of satellites required.Therefore, from this point onwards we will focus on the results of the at random satellites selection and using 2 PCs . From thEvidently, a useful method for reducing computing time and disk space usage should not use the PCA on the whole similarity matrix to determine an adequate number of satellites for each dataset. With that in mind, we decided to design a method that starts with a given percentage of the database as satellites, and then keeps adding a proportion of them until the correlation between the former and the updated data is of at least 0.9. InIn this pilot study we applied the ChemMaps method to visualize the chemical space of two larger datasets .Dataset 1. This file contains the six compound datasets used in this work in SDF format. No special software is required to open the SDF files. Any commercial or free software capable of reading SDF files will open the data sets supplied.http://dx.doi.org/10.5256/f1000research.12095.d17163218 Although the main issue I raised in my earlier review regarding the ‘gold standard’ was not addressed, the authors did make mention of the issue, and they did address some of it in Figure S5. While I’m not totally satisfied with this, I feel that their work is ready for publication, but with the reservation regarding the issue just mentioned. This issue should be considered in more detail in their future work. The authors have made some helpful improvements in their manuscript, but there are still some issues they may want to consider to further improve it, although it is not necessary for them to do so.Step 1 in the ‘Backwards approach’ is unclear. Perhaps an example would help clarify exactly what is being done. I could not reproduce Step 1 as it is currently written.In step 3 of the ‘Backwards approach’ the authors state that:gold standard’. It should be noted that the “real” distances or true gold standard would consider the whole distance matrix.However, for visualization purposes it is unfeasible to render more than 3 dimensions. Therefore, we selected as reference the best 2D or 3D visualization possible by means of PCA. Compute all pair-wise Euclidean distances based on the scores of the 2 or 3 PCs generated in step 2. The set of distances are later used as reference or ‘ With regard to the underlined text in the authors’ statement above, I believe they are missing the point. The reason for considering the full or a significant subspace for a given compound collection and not the 2- or 3-dimensional subspaces derived from a PCA has nothing to do with the impossibility of rendering the data in more than three dimensions because this ‘gold standard’ is not going to be graphically depicted. The important point is that the distances calculated in this space are exact within the limitations of the methodology and means of data collection used. Hence, they form the best ‘gold standard’ that can be achieved within these limitations. The difficulty with this approach is the need to compute Euclidean distances in the full higher-dimensional space, which may require some additional work. However, this distance matrix need only be computed once. I feel the authors should consider this in the future work. Alternatively, the authors need not necessarily involve the complete space, but they should chose a subspace of sufficient dimension to ensure that a significant percentage of the variance, say greater than 90%, is accounted for as a basis for the gold standard.     In Step 8 the terminology ‘prior steps’ is used. Does this include all prior steps? I think it would be clear to name the actual steps as is done in earlier steps in this section, e.g. step 4 states “Repeat steps 4 and 5 increasing…”Although unnecessary, might the authors comment on less well behave character associated with the iterations of the DNMT1 inhibitors dataset compared to the other datasets depicted in Figure 1.the backwards approach.”Line 2 in the ‘Forward approach’ should read: “… e.g., step 1 inIf I understand the %variance plots in Figure S5 correctly, it seems that the %variance for two PCs are for three of the databases greater than or equal to 50%, but three are less than 50%, with the SMARCA2 database only reaching about 20%. Is the %variance in the latter three cases, but especially for the SMARCA2 database, sufficient to provide a basis for a reasonably faithful representation for all of the respective datasets?  I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Figure 1 andFigure 2 againstFigure S2 andFigure S3)" than with 3 PCs. When I compare Figure 1 with Figure S2, it seems to me that 3 PCs are at least not worse than 2 PCs, and maybe even slightly better on the basis of smaller difference between runs. Thus, without a quantitative measure it cannot be said that performance of 2 PCs is better, it may be only said that 3 PCs do not provide improvement over 2 PCs (though it is not obvious from the plots), and if 3 PCs scheme has higher computational demands, it may also be mentioned.The authors say that "selecting 2 PCs the performance is slightly better and more stable (compareThe compound standardisation procedure (Supplementary Methods) looks as it is taken from a different manuscript. For example, compound activity is mentioned, although not used in this study. It is also not stated that standardisation procedure was not applied to DrugBank (at least to the SDF file available in the supplementary dataset).I thank the authors for addressing most of the issues. However, two problems still appear.I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. et al. 2015). Thanks for citing our paper on similarity maps. However the key point is missing: in our 2015 paper  we computed similarity values for all ChEMBL molecules (> 1 million) to only 100 molecules used as reference (the same as "satellites" here), and not the full similarity matrix. We then performed a PCA of the resulting similarity fingerprint to obtain a 2D-map . We have also published another implementation of similarity maps to visuzalize the Protein DataBank, using the same principle of choosing a limited set of sattelites only , we have acknowledged your approach. Indeed, ChemMaps is highly related to your work (as well as to Chem-GPS). A key difference, however, ChemMaps is meant to analyze individual data sets of compounds that might be included or not in ChEMBL or Protein Data Bank. This would be useful for corporate databases. We are not proposing a defined set of standard compounds or satellites. The set of satellites would be "dynamic" and dependent on the data sets. et al present a methodology for representation of chemical space of small sets of compounds. In general, the approach involves selection of satellite compounds from the database, computing the similarities of all compounds in the database to these satellites, and finally projection of the resulting similarity matrix using principal component analysis. J. Jesús Navejaet al further report various methods for selecting satellite compounds  and show how the number of selected satellite compounds influence the quality of projection.J. Jesús NavejaThe authors are completely hiding the fact that similarity mapping is quite well-known and absolutely not new, the authors should read and cite Awale et al., J. Chem. Inf. Model., 2015, 55 (8), pp 1509–1516 and the detailed discussion of literature precedents on similarity mapping presented therein.The authors compare their satellites to the satellite compounds used by T. Oprea in his 2001 approach to mapping chemical space. Obviously either they did not read Oprea’s paper or they misunderstood it: Oprea’s satellites are artificial molecules with extreme properties such as to orient the PCA projection and stretch its dimensions in reproducible directions. However the projection is simply PCA, and does not involve similarity mapping. In similarity mapping the satellites are molecules from within the database to which similarities are calculated.In the abstract, author mentioned that “3D diversity played a secondary role, although it becomes increasingly relevant as 2D diversity increases”. However, I didn't found the relevant explanation in main text supporting this statement.Figure 1 and Figure 2: The five random sets in the legend. Its not clear exactly what the author meant by five random sets. As per my understanding the author used the complete set of compounds for each target and what is changing is the random selection of satellites, which is repeated for five times.In case of forward selection approach: “..With that in mind, we decided to design a method that starts with a given percentage of the database as satellites, and then keeps adding a proportion of them until the correlation between the former and the updated data is of at least 0.9. ” The correlation between projections obtained from the current set of satellites and projections obtained from former set of satellites might well be high, but still the correlation to the projection obtained from the complete similarity matrix is low. How one can assure the quality of projection in this case?For all plots axis labels are too small to read. Comments:I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Dear Dr. Reymond, thank you for your comments to this Research Note.In the Introduction we briefly discuss other similarity approaches to visualize the chemical space. We have expanded that discussion there with a reference to the Similarity Mapplet approach.We did not intend to imply that Oprea’s and Gottfires’ ChemGPS approach is based on structural similarity. To clarify this point we rephrased that in the introduction.In the corresponding Figures legends we changed “random sets” with “iterations”.We added to the Supplementary Information a discussion on the correlation of the complete similarity matrix Euclidean distances and using only 2 and 3PCs. However, we would like to highlight that our approach is intended to approximate the best possible chemical space visualization using PCA. This last is given by the first 3PCs at most.We augmented the font size in all figures.Regarding the modifications we have done considering your comments: The paper under consideration presents an elegant approach to efficient mapping of chemical space using principal component analysis. Being technically sound in general, well-written and easily understandable, the paper lacks several technical details without which it is not complete. In particular:The concept of 'chemical satellites' is discussed in a rather concise manner, a bit more details may be added and the seminal paper by Oprea & Gottfries [1] needs to be cited. The approach suggested here is rather different from the Oprea's one, because satellites are defined there as intentional outliers, whereas in the current work they are just extracted from the mapped dataset. This difference should be stated in a clearer way.Dataset processing routine is not presented. Although the suggested technique would work on totally random datasets , standardization of structures should be performed for consistency and for more informative application of similarity measures. Targeted datasets in the supplement look standardized, but DrugBank contains metal ions, unconnected molecules, and macromolecules, all of which may significantly distort the comparison. For HDAC1 inhibitors the procedure to obtain this dataset from ChEMBL should be provided, because simple target keyword search for 'hdac1' gives 9 different datasets.Diversity of datasets may be additionally illustrated by any of currently available visualization methods. A method that clearly shows compound clustering or diversity of the dataset would be preferred.Visual comparison of figures is not sufficient to make conclusions about preference of random selection over diversity-based . Differences are visible, but their importance and significance are not obvious (maybe just for me), so use of a quantitative measure would be highly appreciated. Random selection shows sometimes lower stability of the backwards analysis (larger difference between the iterations), and this observation could be discussed.Some analysis of the technique applicability domain would significantly improve the conclusions of the paper. One parameter that deserves attention is dataset diversity threshold above which the technique becomes unstable or less useful. Will it work good for totally random or intentionally diverse compounds or for datasets with two or three large congeneric series? A slightly more thorough characterization of example datasets would be useful to deal with this question.  I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Dear Dr Osolodkin, thank you, we highly appreciate your comments to this Research Note.We added a citation to the first publication related to ChemGPS by Oprea and Gottfries. In the Introduction, we further, although briefly (given the extension limit of a Research Note), explained the differences among these two approaches.We added a Supplementary Information file describing the data curation methodology used. -We also added the HDAC1 dataset to the supplementary files.Supplementary Figure 1 should address the visualization of the diversity of the datasets.We find quite interesting your observation about quantifying the stability of the iterations, as well as that about determining the applicability domain of the approach (including defining a diversity threshold). Based on this Research Note we are planning an extensive study fully addressing these concerns.Regarding the modifications we have done considering your comments: Graphically representing coordinate-based chemical spaces requires some type of dimensionality reduction. One method involves the use of similarity matrices treated as data matrices that are subsequently subjected to principal component analysis (PCA). The first two or three PCs are then used as a basis to graphically depict the chemical space. Although this approach works reasonably well, the size of chemical spaces that can be treated is somewhat limited, since the PCA transformation requires diagonalizing a matrix whose dimension is equal to the number of molecules in the chemical space of interest. The work of Naveja and Medina-Franco seeks to overcome this limitation by building a lower dimensional representation of chemical space in a stepwise manner using “backwards” or “forward” procedures. While the method has the potential for accomplishing their goals, it does not in my estimation provide a sufficiently rigorous test of the approximations that are the foundation of their approach. For this reason additional work needs to be done before their method can be applied with confidence. My objection is based on the authors’ use of the first 2 or 3 PCs as the ‘gold standard’ for representing of the entire chemical space, and as a basis for all subsequent comparisons of the approximate chemical spaces. I would at least like to see what percent of the total sample variance is accounted for by these PCs. If it is an insignificant amount, then approximating these PCs by whatever method will not produce a sufficiently accurate model of the chemical space and their model will have to be improved. The true ‘gold standard’ is the original set of column vectors in their data matrix from which the PCs are obtained. This will produce the ‘true’ distance between ‘molecular points’ in the full dimensional chemical space, but because of its very high dimension computing distances in the original chemical space can be a problem. An alternative is to carry out the PCA and choose a larger subset of PCs (say 6 or 8) that do account for most of the sample variance and then use these in the correlation or error analysis.I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Dear Dr. Maggiora, We thank you for your feedback on this Application Note. We entirely agree with your comment that if the variance captured by the first 2 or 3 PCs is not high enough, the visual representation of the chemical space will not be meaningful. For the data sets included in this work, we have seen that the variance is high. We also agree that formally speaking the ´true gold standard´ would involve computing the distances for the full matrix. Based on your feedback we are preparing a revised version of this manuscript.