Colloidal adsorption in planar polymeric brushes

The design of nano-functionalised membranes or channels, able to effectively adsorb pollutants in aqueous solutions, is a topic that is gaining a great deal of attention in the materials science community. With this work we explore, through a combination of scaling theories and molecular dynamics simulations, the adsorption of spherical non-deformable colloidal nanoparticles within planar polymeric brushes. Our strategy is twofold: first, we generalise the Alexander-de Gennes theory for planar homopolymeric brushes to the case of diblock copolymer brushes, then we map the adsorbing homopolymeric brushes onto a diblock copolymer system, where the adsorbed colloids and all interacting monomers are considered monomers in bad solvent and we apply the generalised scaling theory to this effective diblock copolymer. This allows the prediction of the average conformation of the grafted substrate, i.e. its average height, as a function of the amount of loaded particles, as well as the introduction of a continuous mapping between a homopolymeric brush, the fraction of loaded particles and the average height of the adsorbing substrate.


Introduction
Polymer brushes are assemblies of polymer chains, tethered by one end to a solid surface via either chemical bonding or physical adsorption to the surface; 1,2 the resulting polymer lms are typically tens to hundreds of nanometres thick.These polymer-based monolayers can be integrated into devices for nano-technological applications: advances in the development of controlled polymerisation processes, combined with the renement of supramolecular strategies, allow a high degree of control over the composition, architecture and length of the anchored macromolecular chains, as well as the graing density. 1,3Polymer brushes have hence attracted the attention of the materials science community for the wide range of their possible applications which include sensing, drug delivery, anti-fouling coatings and lubrication. 4Moreover, coatings of polymer brushes can also be exploited to impart stimuliresponsive properties to surfaces, depending on the system used for functionalisation. 5,6he most common application of polymer brushes today is the functionalisation of membranes (and/or channels) for separation and purication processes. 4,7Examples include protein separation/purication (i.e.affinity membrane chromatography), 8 dehydration of organic solvents (i.e.pervaporation), 9 gas separation 10 and water remediation.
Indeed, membrane adsorption has become one of the preferred methods for removing toxic contaminants from wastewater.Alternative methods include nanoltration and ultraltration, reverse osmosis, chemical precipitation, electrochemical treatments and extraction.These techniques are oen more expensiveas they typically require large quantities of reagentsand less effective than the use of adsorptive membranes.In fact, membranes are typically highly permeable to water uxes, do not require high pressures to operate, can be regenerated and take up little space.In addition, the use of membranes does not suffer from the problem of recovery of solvent or surfactants or polymers in solution. 11,12rom a microscopic point of view, membranes are characterised by a large adsorption area (implying a large number of adsorption sites per unit area) and they can be specically designed and functionalised to optimise their efficiency, cost and versatility.However, a critical issue for the use of these nano-functionalised membranes is the optimisation of the amount of adsorbent species that can be incorporated into the membrane, either as the number of adsorption sites per polymer chain or as the number of anchored chains.In fact, the density of the graed macromolecules should be optimised to maximise the number of adsorption sites while avoiding agglomeration and clogging of the membranes.
From a theoretical as well as experimental perspective, considerable interest has been devoted to the limiting case in which there is no chemical affinity between monomers and nano-particles.In such a scenario, adsorption is possible via Nanoscale Advances PAPER entropic mechanisms. 13However, for most applications it is desirable to also optimise the enthalpic interactions between the polymer and the adsorbate; coupling it with other properties, e.g.thermo-sensitivity, endows the brush with stimuliresponsive mechanisms that can be exploited in many ways.Theoretically, a mean-eld free energy description of such a system has been proposed; [14][15][16] however, due to its mean-eld nature, it compares quantitatively against simulation results only in the limit of weak attraction.
We present here a theoretical and computational study of the adsorption of waste particles or small ligands in adsorptive membranes in solution.Using a combination of scaling theories and molecular dynamics simulations (MD), we study the adsorption of spherical colloidal nanoparticles (mimicking waste particles/ligands) within model polymer brushes by varying the strength of the interaction between the polymer and colloids and the graing density.The central core of this work is the denition of a novel theoretical framework that allows the derivation of a simplied description of the process of adsorption of colloidal nanoparticles within planar homopolymeric brushes.In particular, the scope of this work is to nd simple measurable macroscopic changes that can be used as an insight to quantify adsorption.Starting from the known results for homopolymeric brushes in good solvent, we introduce the adsorption process as a sort of perturbation to the system; when colloids are adsorbed within a polymeric brush, the nanoparticles interact with a selection of monomers belonging to the graed macromolecules.The colloids act as an effective glue between the monomers, whose collapse around the adsorbed particle can be foreseen as a local change (worsening) in solvent quality.The paper is structured as follows.In Section 2, we report the description of the numerical model and the simulation details; then, in Section 3, we present the generalisation of the Alexander-de Gennes scaling theory to diblock copolymer brushes.Aer validating our reference simulations in Section 4.1, we discuss the results of the simulations and what we can learn from them in Section 4.2.Using this insight, we successfully compare theoretical prediction with numerical simulations in Section 4.3, detailing the mapping between the monomer-resolved system and the homopolymeric effective representation that allows the correct scaling.Finally, we show that the Alexander-de Gennes scaling relation remains valid, if the proper scaling factor (i.e. the gyration radius of an equivalent diblock copolymer) is considered.

Materials and methods
We consider a polymeric brush, composed of linear chains graed onto a perfectly at impenetrable surface; we also consider colloids, acting as "adsorbates", diffusing in the system.We set the thermal energy k B T = 1/b = 1 as the unit of energy.All monomers and colloids interact via the Lennard-Jones (LJ) interaction potential Indicating with the subscript c the colloids, m the monomers and cm the mixed case, the LJ-sizes are a m = a = 1.0, and is thus set as the unit of length, a c = 1.5a and We choose a colloid size that is only slightly larger than the monomer size.Such a size ratio has been shown to reproduce, at least qualitatively within a model system, adsorption of heavy metal ions, whose removal is of capital importance in water remediation, by polymer functionalised nanoparticles. 17The parameter 3ã is chosen such that the value of the potential energy is zero at r cut a , ensuring a smooth transition of the piecewise dened potential without affecting the force.We consider a polymer brush in good solvent, whose monomers act as adsorption sites for the suspended colloidal particles.Thus, monomer-monomer as well as colloid-colloid interactions are chosen as purely repulsive r cut c = 2 1/6 a c and r cut m = 2 1/6 a m ; instead, we introduce attraction in the mixed case by setting r cut mc = 2a mc .The depth of the LJ potential is 3 c = 3 m = 1.0kB T while, for 3 mc , we explore a range of values 3 mc ˛{1.0, 1.5, 2.0, 4.0}k B T in order to assess the role of the monomer-colloid interaction energy.Neighbouring monomers along the graed polymer chains are held together by means of the nite extensible nonlinear elastic (FENE) potential: 18 where K = 303 m /a 2 and R 0 = 1.5a.We simulate a system in a box of the size V = L x L y L z , with L x = L y = 20a and L z = 200a, with periodic boundary conditions in the x-and y-directions; the polymer chains are graed at z = 0 and effectively occupy a fraction of the available space.The perfectly at, impenetrable plane at z = 0 is implemented using again a LJ potential, with 3 w = 3 m and a w = a, acting on all monomers (except, for simplicity, the graed ones) and all colloids.We construct the system by rst placing the graing points of the polymers in a random manner using Poisson disk sampling 19 in order to obtain a spatially homogeneous graing distribution in the x-y plane.
We then grow the chain from the graing points using a selfavoiding and boundary-avoiding random walk, in order to grow the chain away from the bottom wall.We further place N c colloids in the upper half of the box again using Poisson disk sampling.
We perform full monomer molecular dynamics (MD) simulations using LAMMPS. 20Alongside the units of length and energy, we set m = 1 for both monomers and colloids, for simplicity; this also sets the MD time unit s = 1.We integrate the equations of motion using the velocity Verlet algorithm, with integration time step Dt = 0.001s; simulations are performed in the NVT-ensemble employing the Langevin thermostat, with friction coefficient g = 1s −1 .We neglect hydrodynamic interactions, as they are not affecting the static and thermodynamic properties in equilibrium.We simulate polymer brushes of M = 200 monomers at different graing densities s g a 2 = 0.032, 0.048, 0.064, 0.08.Furthermore, we span a wide range of values for the colloid packing fraction, dened

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Nanoscale Advances 3 /6 being the volume occupied by a colloid and H 0 being the average height of the unperturbed brush at a given s g .We consider a colloid "adsorbed" if there is at least one monomer below a certain cut-off distance, that will be introduced later.To compare our simulation results with the theoretical predictions, we introduce the packing fraction of the adsorbed colloids h a c = N a c V c /(L x L y H 0 ), where N a c is the number of adsorbed colloids.We veried that, for the chosen chain length and graing densities, the brushes are always in the scaling regime (see Section 4.1).For each set of parameters we perform 5 independent realisations: for each one, from the initial conguration, we equilibrate the system until the average height of the adsorbed colloids is constant.Aerwards, we perform a production run of 800 000s (8 × 10 8 time steps).

Generalising the Alexander-de Gennes theory to adsorbing brushes
2][23][24] Within this approach, it is possible to link the average properties of the brush to a combination of the single molecule radius of gyration and the adimensional quantity s g /s*, where s* = 1/R g 2 is the graing density at which two polymers represented through their radius of gyration would begin to overlap, x g is a model dependent prefactor and n is the Flory exponent.In particular by introducing the Alexander-de Gennes notation, 21,22 the average height of the brush can be expressed as: In order to generalise the AdG scaling theory to account for adsorption, we rst remap a single polymer chain, that has adsorbed one or more colloids, onto a diblock copolymer; we will then use such a generalisation to rescale the height proles of the brushes for all graing densities aer adsorption.
As mentioned, we assume that the monomers act as adsorption sites for the colloids; this, from a physical perspective, can be recast into an attractive interaction between the two species.Adsorption thus induces a local change in solvent quality as, effectively, monomers crowd around the colloidal particle and expel the solvent.This implies that the effective size of the graed chain, upon adsorption, should account for the adsorbed colloid.
In the spirit of the AdG formalism, we aim at linking the properties of the brush to the single chain properties.Given a brush of N chains and N a c adsorbed colloids, we name n c = N c / N and n a c = N a c /N the number of colloids and adsorbed colloids per chain, respectively.Upon adsorbing a colloid, a graed homopolymeric macromolecule, initially of molecular weight M, can be mapped onto an "equivalent" diblock copolymer chain, made of M A monomers in good solvent, and M B monomers in bad solvent.The latter quantity has two contributions.The rst comes from M int = M − M A monomers, interacting with the colloids.In addition, a single nanoparticle can be mapped onto a homopolymeric chain in bad solvent whose radius of gyration satises the relation R g where a c is the radius of the colloid and m B is the number of equivalent monomers in bad solvent. 17In other words, for each chain, every adsorbed colloid can be treated as an additional contribution to the total number of monomers in bad solvent.Thus, as adsorption takes place, each original homopolymeric chain made by M monomers is mapped onto a diblock copolymer made by M A monomers in good solvent and M B monomers in bad solvent It is important to stress that in all considerations made, the total number of effective monomers constituting the diblock c m B grows upon increasing the amount of adsorbed colloids.This ensures that the brush will be in the scaling regime at any point of the adsorption process.To generalise the results obtained using AdG, each A-B diblock copolymer is mapped onto an "equivalent homopolymer" i.e. a homopolymer whose radius of gyration is equal to that of one of the diblock copolymers.
The AdG theory requires two parameters: the graing density and the radius of gyration of the graed chains.We start by analysing the radius of gyration of the diblock copolymer, by approximating it as: where R gA and R gB are given by Eqn ( 6) can therefore be rewritten as: It is now possible to determine the "corresponding homopolymer": imposing that its gyration radius is identical to that of one of the diblock copolymers R eq g = R db g , we set the number M eq of equivalent monomers.In this way, the reduced graing density s g /s* remains the same in the two representations.
For convenience, the equivalent homopolymer will be considered to be made of monomers in good solvent, i.e. monomers that follow the same scaling laws as the A part of the diblock copolymer chain.We can therefore rewrite the total radius of gyration of the equivalent homopolymer as

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Paper Hence Within the AdG theory, a homopolymeric brush of graing density s g can be seen as a chain of non-overlapping blobs of radius R blob g ¼ ffiffiffiffiffiffiffiffiffiffi s g =2 p . 21,22Each blob contains a number of . According to our representation, the total number of blobs that will constitute the equivalent brush is given by the ratio between the equivalent length M eq and the number of monomers per blob: The height of the brush can be approximated as the product of the number of blobs and the diameter of the blobs: or, equivalently: where h 0 = 1.4 is obtained by tting the data, reported in Fig. 2.
The key point to apply eqn (15) is to know M A and M B as a function of the density of adsorbed colloids (per chain).We will propose, in Section 4.3, a way to estimate them, based on the phenomenology observed from the numerical simulations.

Calibrating the reference system via the Alexander-de Gennes scaling
In order to compare the prediction of the scaling theory, generalised to the case of adsorption, with results from numerical simulations, one needs to study, for all values of s g considered, polymeric brushes well in the scaling regime.
We report, in Fig. 2, the average height of the brush, normalised by the gyration radius of a single chain, as a function of the scaled graing density (s g R g 2 ) (1−n)/(2n) .The average height hzi is obtained as the average z-component of the chain monomer positions. 24,26,27Comparing Fig. 2 with results reported in the literature 25 we nd the correct scaling.We thus choose to take as a reference system a brush with M = 200 monomers per chain, which ts the scaling regime for a reasonable range of s g values.In particular, the range considered in this paper is 0.032 # s g a 2 # 0.08 corresponding to 1.60 We further verify that the shape of the brush density prole is parabolic, as predicted by self-consistent eld theory (see the ESI †).These results allow us to move further towards the generalisation of the AdG theory to the case of an adsorbing brush, with the certainty that all the corresponding homopolymeric systems reproduce the expected scaling predictions.

Phenomenology of colloidal adsorption from numerical simulations
In this section, we discuss the emerging phenomenology of brush adsorption, examining the results of the numerical simulations.As already mentioned, we performed an extensive investigation, by varying the monomer-colloid interaction strength 3 mc ˛[1.0, 4.0] for the xed graing density s g a 2 = 0.064 as well as by varying s g for the xed value 3 mc = 1.5kB T. We  highlight that all the observables shown in this section are measured at equilibrium.As mentioned in Section 2, a colloid is considered to be adsorbed if it is close enough to at least one monomer; vice versa, a monomer is dened as an adsorption site if it is interacting with at least one colloid.Equilibrium is reached when the average height of the adsorbed colloids displays a stationary value in time.
For the sake of clarity, we chose to report one case (s g a 2 = 0.064, 3 mc = 4k B T) in the main article and refer to the ESI † for all the other analysed cases.In fact, although details can remain system dependent, the general features are common.Specically, 3 mc = 4k B T is the interaction strength that presents more prominent adsorption and, as mentioned, s g a 2 = 0.064 reproduces the parabolic regime for the M = 200 case prior to adsorption, as shown in Fig. 3a. 21,22,26,27or both species, we measure the probability of nding either a monomer or a colloid at a certain height z with respect to the graing plane.In the reported case (see Fig. 3b), when a single colloid is inserted in the simulation box (h c = 0.0003), the adsorption probability is one, but the average shape of the brush remains almost unperturbed.Upon adsorption of a sufficient amount of colloids, the prole of the brush changes and develops a maximum in the region close to the bottom plane.At the same height we notice the concurrent accumulation of the colloids (Fig. 3c).
This co-localisation highlights the formation of a condensed phase, a droplet, made of a mixture of collapsed monomers and colloids; it is visible from the snapshots reported in Fig. 1.The droplet grows upon increasing the number of adsorbed colloids and, eventually, percolates through the periodic boundaries.At rst, the brush maintains a top layer totally exposed to the solvent (that is, in good solvent), as shown in Fig. 1c and 3d; however, upon increasing adsorption, the brush collapses completely and the colloidal particles become uniformly distributed inside the (collapsed) brush (Fig. 1d and 3e).Intriguingly, in the minimal coarse-grained model here considered, even when all monomers become adsorbing sites, the brush still shows the capability of rearranging the monomers around the colloidal nano-particles and allows for further adsorption, within a range of interaction strengths.Such an increase in the number of adsorbed colloids causes the formation of a layer of colloids that deposit on the exposed surface of the collapsed brush, as shown in Fig. 3f; the average height of the loaded brush is thus increased in the process.Interestingly, the same phenomenology has been observed with another theoretical model 14 and has also been hypothesized to explain the experimental results, obtained via quartz crystal microbalance measurements. 28Furthermore, the results obtained at small values of 3 mc are in agreement with previous simulations. 29ore quantitatively, we characterise adsorption at different colloid packing fractions h c by means of four quantities.We consider (i) the average fraction of adsorption sites (i.e.interacting monomers) over the total number of monomers hM int i/M; (ii) the average fraction of adsorbed colloids over their total number hn a c i/n c ; (iii) the average number of adsorption sites over the number of adsorbed colloids hM int /n a c i (iv) the average number of contacts or bonds between monomers and colloids hM bonds /n a c i.The sketch in Fig. 4 illustrates the difference between these four quantities.In the sketch, the dashed and dotted circle represents the distance cut-off below which we have adsorption; the distance between any two particles (colloid or monomer) is, in the coarse-grained model used, always taken as the distance between their centres of mass.Both colloids (the blue circles) are adsorbed; instead, only the striped monomers

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are adsorbing sites in bad solvent, while the ones in solid color are in good solvent.Naively counting the contacts between monomers and colloids, highlighted by dashed lines in the sketch, overcounts the interacting monomers, due to the "sharing" of some adsorbing sites between colloids.This is indeed the difference between hM int /n a c i and hM bonds /n a c i. Thus, the observed clustering requires paying special attention to the way the neighbours of the colloidal particles are dened and counted.In order to properly compute M int and n a c , a suitable cut-off distance for adsorption should be identied; the most obvious choice, the cut-off distance of the LJ potential, leads to consistent overestimation of M int , as also the monomers in the second coordination shell will be counted.We employ the SANN algorithm, 30 which allows for a parameter-free identication of neighbouring particles, to determine a cut-off distance.
We report the four quantities, introduced above, in Fig. 5 for the case of 3 mc = 4k B T and s g a 2 = 0.064.As will emerge in Section 4.3, the key quantity of interest is the average fraction of adsorbing sites per adsorbed colloid, that we name g int = M int / n a c .We can observe, in Fig. 5, that the data, reported with starshaped symbols, present two distinct power law regimes, joined at c is the value of the packing fraction at which the loaded brush collapses and there are no more noninteracting monomers.We nd that, at h a c ¼ h a* c , the number of interacting monomers per adsorbed colloid is, for all analysed systems, compatible with the coordination number of a random loose packing of spheres, which is roughly 31,32 The two regimes can be rationalised as follows.When very few colloids are present, each nanoparticle interacts with the maximum possible number of monomers per colloids, g int (h a c z 0) = g 0 (see the ESI †); upon increasing h a c , but remaining below h a* c , the aggregation process imposes the colloidal particles to share some monomers, as sketched in Fig. 4.This sharing effectively lowers the ratio of interacting monomers per adsorbed colloid.Finally, above h a* c , the maximum amount of available monomers (per chain) has been reached.However, as mentioned, the brush can still adsorb additional colloids; this implies that the number of interacting sites per adsorbed colloid has to change (see Section 4.3).
The fraction of adsorbing monomers q = M int /M is, therefore, a monotonically increasing function, that saturates to 1 at the same value of h a c ¼ h a* c where the collapse of the brush is observed (see Fig. 5).This implies that, above h a* c , the brush is under bad solvent conditions and, from that point on, the average height of the loaded brush will grow only due to the additional layer of colloids that deposit on the surface of the brush.
Qualitatively, the effects of adsorption on the brush, as shown in Fig. 3 and 5, can be found for 3 mc > 1k B T and for all the values of s g considered (see the ESI †).Quantitatively, the average properties of the brushes are a function of 3 mc .This is due to the fact that the ability of the monomers to rearrange around the adsorbed colloids is strongly dependent on the interaction strength between monomers and colloids.In particular, for the 3 mc < 4k B T cases, we reach a limit in h max c above which a fraction of colloids remains unadsorbed and freely diffuses in the simulation box.For 3 mc = 4k B T we did not reach h max c ; thus the system has not yet reached its maximum loading capability within the range of h c considered.
The behaviour of the system remains qualitatively similar also upon changing s g at xed 3 mc .However, quantitatively, varying s g has two main effects.First, the absolute height of the collapsed brush depends on the graing density and is higher at higher values of s g .Interestingly, at lower values of s g the brush has a lower adsorption capability.Indeed, aer the collapse, brushes with a lower value of s g are less able to further adsorb colloids (see the ESI †).Thus, increasing s g pushes the adsorption limit further, similarly to an increase in binding affinity.

Introducing adsorbed colloids in the generalised Alexander-de Gennes scaling theory
In this section, we use the insight gained by the analysis of the numerical data to model the adsorption process in equilibrium, within the AdG scaling theory described in Section 3. As already mentioned, the key point to apply eqn (15) to a brush, that has adsorbed n a c colloids per chain, is to know M A and M B , i.e. the number of monomers in good and bad solvent per chain, respectively.
We recall that, within the diblock copolymer mapping introduced in Section 3, the number of monomers per chain in bad solvent M B is given by eqn (5).We can formally rewrite eqn (5) as where, as described in the previous section, n a c g int = M int accounts for the interacting monomers and m B ∼ (a c /a m ) 1/nB encodes for the size difference between each adsorbed colloid

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Nanoscale Advances and the "replacement" monomers, as introduced in eqn (5).Furthermore, g int introduced in Section 4.2, has a non-trivial dependence on the number of adsorbed colloids and displays two distinct regimes (see Section 4.2).It will be convenient, from now on, to discuss the two adsorption regimes separately.The rst regime deals with the properties of the brush prior to the saturation of the interacting monomers, i.e. when n a c # n a* c .The second regime instead refers to the properties of the brush once all interacting sites are saturated i.e. when n a c .n a* c .As noticeable in Fig. 5, data suggest that the function g int (n a c ) shows two different power law behaviours in the two regimes (for a more in depth discussion of the functional dependence see the ESI †).
For n a c ¼ n a* c the brush reaches its collapse.The number of adsorption sites equals the total number of monomers in the brush, i.e.M int = M and g int = g 1 .
For n a c # n a* c we can dene where g 0 is the number of interacting monomers in the limit of a single adsorbed colloid (see Section 4.2 and the ESI †) and z = ln(g 0 /g 1 )/ln(M/g 1 ).As n a c .n a* c , the number of interacting monomers remains constant and is equal to M. As the brush still adsorbs colloids, the average number of interactions between monomers and colloids has to diminish so that: Throughout adsorption, the number of monomers in good solvent per graed chain is given by M A = M − M int , while the number of monomers in bad solvent becomes M int + m B n a c .This leads to the following expression for the radius of gyration as a function of adsorption: We can make the dependence of the fraction of interacting monomers q on g int explicit, obtaining: where, obviously, q ˛[0, 1].In particular, for n a c \n a* c , q reads while q = 1 for n a c $ n a* c .We can now combine eqn (11) and ( 21) to obtain for the case while for n a c .n a* c we obtain: We can now plug the results obtained for the effective diblock radius of gyration in eqn (15) to get a prediction for the height of the brush in the different regimes, obtaining, for and for n a c .n a* c : We remark that the scaling has to reproduce the free brush case for q = n a c = 0.This leads to a further assumption on the numerical pre-factor h 0 ; for continuityirrespective of the regime analysedit has to be the same.We will then use h 0 = 1.4 as done in eqn (15).
Finally, we compare the results from the numerical simulations with those of the developed scaling theory eqn (15).We rst address the case of 3 mc = 4k B T: we report the comparison between numerical data and theoretical prediction for the height of the brush (Fig. 6a) and for g int = M int /n a c (Fig. 6b), as a function of h a c .We start from the latter panel: eqn ( 17) and (18) (full line) are compared with the numerical data (symbols) without any tting parameter.The agreement is remarkable and shows that the proposed functions capture the adsorption process very well.Considering now the average height of the brush (Fig. 6a), we compare the numerical data with the predictions of eqn (24) and (25).In this case, we perform, for convenience, a t of the data to eqn (25) (dashed line) in the second regime h a c .h a* c , in order to get an estimation of the prefactor x B (see the ESI † for further details), assuming that it

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remains the same throughout the whole adsorption process.Again, the comparison between numerical data (symbols) and theoretical prediction (full line) is very favourable.Remarkably, the theory is unaware of the phase separation that is occurring at small values of h a c ; albeit the system being heterogeneous, the picture, based on a single-chain scaling, captures the height variation very well.
We report the same comparison for the other systems considered in the ESI; † the theory always captures how the number of interacting monomers changes upon increasing the number of adsorbed colloids and yields quantitative predictions of the height of the brush, except for the case 3 mc = 1k B T. Indeed, in this case, the interaction affinity between monomers and colloids is not sufficient to drive the complete collapse of the brush.As the latter is one of the hypotheses of the theory, the match is not expected.
Last, we perform a nal check on our theoretical framework.Pushing the AdG theory even further, we draw a master curve, similar to that reported in Fig. 2. We report in Fig. 7 the results obtained by re-mapping all adsorbing brushes for all values of h c and s g and interaction strength 3 mc between monomers and colloids, remapped onto their corresponding homopolymeric brushes and re-scaled by means of the correct AdG rescaling factor.The linear dependence of all measured hzi/R g values, hzi being the height of the brush, on the expected scaled variable 2n is striking, thus corroborating and strengthening the validity of the methodology proposed here.

Conclusions
In this work we introduced a generalisation of the Alexander-de Gennes theory that allows the effects of adsorption on the average height of a brush to be quantied.The followed strategy is twofold: at a single chain level, the adsorption process is mapped onto an iterative local change in solvent quality.While interacting with colloidal particles, monomers, initially in good solvent, undergo a "phase transition", giving rise to a diblock copolymer.The bad solvent fraction grows with the number of adsorbed colloids, that also participate in the said bad solvent fraction with a certain number of "replacement" monomers, chosen to match the colloids' occupied volume.This iterative local change in solvent quality ends when all monomers are in bad solvent.The secondand laststep in the re-mapping process consists in dening a homopolymeric brush that is "equivalent" to the newly dened diblock copolymer brush, in terms of re-scaled graing density s g /s*.This continuous mapping between two descriptions of the system, the adsorbing brush and its homopolymeric "twin", allows dening, through the AdG scaling for the "twinned" homopolymeric brush, the instantaneous value of the average height of the brush, as a function of the adsorbed colloids.
We explored systems characterised by a series of monomercolloid interactions and diverse graing densities.The remapping is then tested upon a wide set of simulations performed for a range of values of the monomer-colloid interaction affinity and of the graing density s g /s*; it is shown to reproduce quite well the numerical results for the average height of the brush as a function of adsorbed monomers.
As mentioned, the results obtained are in agreement with those of previous theoretical approaches [14][15][16] as well as with experimental observations. 28However, to the best of our knowledge, there are no further experimental studies that have explored this effect in detail.Rather, efforts have been focused on the applications of the brushes in other contexts. 33he strategy presented in this paper can thus be seen as a promising tool to monitor, via mesoscopic measurements, the evolution of adsorption within a homopolymeric brush through its change in average height.Such a measurement can be performed, for example, using AFM on the solvated brush; [34][35][36] other techniques, such as dynamic light scattering or spectroscopic ellipsometry may be employed. 33Once calibrated on a specic system, universal scaling can be exploited to predict the height of the brush, given a certain concentration of colloids in the solvent.Such information is of capital importance in the design of functionalized membranes, as it allows the adsorbing capability to be maximized while, at the same time, avoiding clogging.In particular, our study suggests that, depending on the interaction affinity, an "overloaded" brush may even grow taller than in the unloaded (or unperturbed) stage.This should be avoided as, in a pore, excessive growth would lead to clogging.
We focused, in this study, on a specic size ratio between the adsorbate molecule and the adsorbing monomers; as mentioned, we chose a mc = 1.5a as our case study because, typically, solvated heavy metal ions are only slightly larger than the monomers.However, we expect the results to be valid for larger size ratios within the protein limit. 37inally, it could be interesting to extend a similar remapping, i.e. from the original system to a diblock copolymer to an Fig. 7 AdG scaling obtained by remapping every adsorbing brush to its own homopolymeric equivalent brush.Strikingly all systems reproduce the expected linear scaling that would be proper for the homopolymeric brush.The reported results refer to, at fixed 3 mc = 1.5kB T, s g a 2 = 0.032 (blue crosses), s g a 2 = 0.048 (orange stars), s g a 2 = 0.064 (green diamonds), s g a 2 = 0.08 (red left triangles) and, at fixed s g a 2 = 0.064, 3 mc = 2.0k B T (purple right triangles), 3 mc = 4.0k B T (brown up triangles).

Paper
Nanoscale Advances equivalent homopolymer, in other settings where polymer and (in a loose sense) colloids interact to form complex structures at the nano-scale.For example, it is known that some proteins participate in the organisation of chromatin, regulating the activation and de-activation of active genes via a coil-globule transition that makes them more or less accessible. 38Furthermore, these transitions are believed to be accompanied by the condensation of the proteins, 39,40 which was also observed in this case.As mentioned, our scaling theory, unaware of this "phase transition", manages to reproduce the data very well.The unbalance of the protein condensation can lead to diseases, such as the Kabuki syndrome. 41,42Linking the protein concentration to the collapsed state of the chromatin ber could potentially open an avenue for improving the treatment of such diseases.

Fig. 2 2 )
Fig. 2 Height of the brush over the bulk gyration radius of a single chain as a function of the scaling parameter (s g R g 2 ) (1−n)/(2n) for brushes of different lengths M and different grafting densities, as in ref. 25.The dashed blue line indicates the beginning of the stretched regime at s g s * ¼ 1.The dotted gray lines are guides to the eye and indicate the two regimes, mushroom and stretched, predicted using the theory.Fitting the data returns the numerical prefactor h 0 = 1.4.

Fig. 1
Fig. 1 Snapshots of the simulated systems at fixed 3 mc = 4.0k B T, s g a 2 = 0.064 and (a) h c = 0.002, (b)h c = 0.006, (c) h c = 0.013, and (d) h c = 0.027.The polymer beads that act as grafting points are indicated in blue, monomers in red and colloids in magenta.The grey volume indicates the space occupied by the colloids; it is obtained with the open-source software OVITO as the minimal surface encompassing all the colloids.

Fig. 3
Fig. 3 The probability density P(z) to find either a colloid (orange dashed line) or a monomer (blue full line) at height z as a function of the reduced height z/H 0 , H 0 being height at h c = 0, for the reference system 3 mc = 4k B T, and s g a 2 = 0.064 at different values of the colloid packing fraction (a) h c = 0 (b) h c = 0.0003 (corresponding to, for this specific system, a single colloid) (c) h c = 0.02, (d) h c = 0.04, (e) h c = 0.19 and (f) h c = 0.32.

Fig. 4
Fig.4Sketch, exemplifying the difference between the number of interacting monomers M int and the number of bonds M bonds between colloids and monomers, drawn as dashed lines.The blue larger particles identify the colloids, while the orange smaller and tethered particles belong to the grafted chains.The dashed and dotted circle refers to the cut-off radius, employed in the analysis.

Fig. 5
Fig. 5 Average fraction of adsorption sites M int /M (orange crosses), average fraction of adsorbed colloids n a c /n c (green triangles), average number of adsorption sites per adsorbed monomer M int /n a c (red stars), and average number of contacts per adsorbed colloid M bonds /n a c (blue diamonds) as a function of h a c for the reference system 3 mc = 4k B T, s g a 2 = 0.064.The alternative axis, with the corresponding values of h c , is reported here for completeness, as h c = h a c for 3 mc = 4k B T in the range of values considered in this work.

Fig. 6
Fig.6Comparison between the numerical results and theoretical predictions for both H (panel a) and g int (panel b) as a function of h a c for the reference system 3 mc = 4k B T, s g a 2 = 0.064.In panel (a) the full line refers to the theoretical predictions of eqn (24) and (25), crosses refer to numerical data, and the dashed line refers to the fit of the data with eqn (25) (dashed line) in the second regime h a c .h a* c .The error bars in (a) refer to the standard deviation, determined on the five independent realisations, simulated for each value of h c .