Nanopores as Switchable Gates for Nanoparticles: A Molecular

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Nanopores as Switchable Gates for Nanoparticles: A Molecular Dynamics Study Cheng-Wu Li,† Holger Merlitz,*,‡ Chen-Xu Wu,*,† and Jens-Uwe Sommer*,‡ †

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Research Institute for Biomimetics and Soft Matter, Fujian Provincial Key Laboratory for Soft Functional Materials Research, Department of Physics, Xiamen University, Xiamen 361005, China ‡ Leibniz-Institut für Polymerforschung Dresden, 01069 Dresden, Germany ABSTRACT: Using molecular dynamics simulations and scaling theory, we present a systematic study of the function of cylindrical nanopores which are decorated with polymer brushes. Our focus is on the regimes in which these systems are able to function as switchable gates for bulky nanoparticles. The process of switching is triggered through the addition of a smaller component to the solvent, which acts as a co-nonsolvent, causing a discontinuous collapse of the brush and opening a central passway through the nanopore. We quantify the conditions under which agents of different diameters are allowed to pass through the nanopore, through direct simulation of their fluxes as well as evaluations of the free energy profiles during a forceful inclusion of these particles into the pore. Our results quantitatively confirm how these nanopores serve as semipermeable gates which grant passage of small compounds, while blocking larger ones under certain conditions.

1. INTRODUCTION

Molecular dynamics (MD) simulation studies of polymer brushes in cylindrical pores have first been presented by Dimitrov et al.28 They have shown that these systems can form structures which exhibit a central passway, depending on chain length and pore diameter, or a roughly homogeneously filled cylinder. Subsequent studies21,29 have analyzed the phase diagram of these brushes in good solvent with the help of scaling theory. Opferman, Coalson, and co-workers have recently simulated the collapse of planar and cylinder-grafted brushes in the presence of attractive nanoparticles (NPs) of different sizes,30−32 pointing out that this might present a possible way of controlling passability through such a nanochannel. Monte Carlo (MC) simulations and SCF calculations of polymers inside cylindrical pores and in poor solvent have recently been presented by Osmanović et al.33,34 According to these studies, a brush layer of low density tends to break up to form “clumps” which break the rotational symmetry of the brush. Further important simulation studies have focused on a rather detailed modeling of NPCs, in which a variety of different brush morphologies are displayed.35 Several hypotheses regarding the transport mechanism including cargo particles have been discussed as well.36−38 A selection of modeling techniques applied to NCPs has been presented in recent reviews.39,40 In our present work, we are picking up several threads that are left loose in these previous studies, trying to arrive at a

Polymer brushes tethered on substrates have attracted increasing attention in both theoretical and laboratory studies. The first classical studies about brushes on planar substrates were based on self-consistent-field (SCF) approaches1−4 or scaling theory.5,6 The following rapid developments in this field are summarized by Azzaroni et al.7 and Binder et al.8 Meanwhile, brush research is increasingly branching out into specialized topics, including brushes made of dendrimers,9−11 brush-decorated macromolecules for drug delivery,12,13 or switchable (environment-responsive) surface layers.14−18 If brushes are grafted onto nonplanar surfaces, such as spheres19,20 or concave surfaces21 of considerable curvature, additional features emerge as a result of geometrical restrictions. Brush-decorated nanopores are among those systems in which the curvature of the substrate plays a key parameter for the morphology as well as the function of the brush. In biological systems, nanopores such as the nuclear pore complex (NPC) are anchored by functional proteins inside the core membrane and serve as selective gates.22 To mimic those nanosieves, holes of diameter of 30 nm have been drilled into silicon23 or gold24 plates and subsequently grafted with functional protein chains. The interaction forces imposed by these grafted chains have been measured by using an atomic force microscope,25 and the passive diffusion of cargo particles through these decorated nanopores has been tuned by modifying the degree of swelling of the brush layer, which is controlled by the properties of the external environment, such as the pH value, or by external fields.26,27 © XXXX American Chemical Society

Received: June 4, 2018 Revised: July 31, 2018

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DOI: 10.1021/acs.macromol.8b01149 Macromolecules XXXX, XXX, XXX−XXX

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substrate to form a brush of grafting density σ = M/A, where M denotes the total number of chains and A the area of the channel wall. Pair interactions between any beads are LennardJones (LJ) potentials ÅÄÅ ÑÉ ÅÅi d y12 i d y6 ij d yz12 ij d yz6ÑÑÑ Å j z j z j z j z ULJ(r ) = 4ϵÅÅÅjj zz − jj zz − jj zz + jj zz ÑÑÑÑ j rc z j rc z ÑÑ ÅÅk r { kr { k { k { ÑÖ ÅÇ (1)

rather general and unified understanding of these nanopores as switchable gates. The line of thought is going to be as follows: The decorating brush can be in good solvent and swollen. The addition of a rather small amount of co-nonsolvent(CNS) causes a collapse of the brusha result of a first-order collapse transition41−43making the brush behave like being immersed in an (effective) poor solvent. Phase diagrams are set up, using universal scaling theory, to identify the parameter regimes in which the nanopore is closed in good solvent but open in the presence of CNS. We then quantify, through simulations, the permeability of these pores for NPs of different sizes. When the NPs are sufficiently large, their fluxes through the pore are diminishing. Here, we simulate a forceful inclusion of these agents into the pore and determine the free energy profiles, which allow estimates for the permeability in the case of very low flux rates. The rest of the work is organized as follows: In section 2, we present our nanopore model and briefly introduce our MD simulation method. Section 3 then derives scaling regimes of brushes in athermal solvent and nonsolvent. We determine the parameter range in which the brush may either close the nanopore or collapse and open a central passage way, depending on the solvent quality. The simulation results, such as density profile and brush thickness, are shown in section 4. After an addition of NPs, we quantify in section 5 the size dependence of passive diffusion and determine the free energy barriers imposed by the brush by pushing single particles into the decorated nanopore. Finally, we summarize and discuss our findings in section 6.

where ϵ is the depth of the potential (in units of kBT) at its minimum rmin = 21/6d. For pairs containing beads of different diameters, arithmetic mixing rules are applied. The parameter rc defines the cutoff distance. When the potential is truncated at rmin, a short-range repulsive potential arises which approximates a system inside an implicit athermal solvent.45 In contrary, to retain attractive contributions between selected components of the system, the LJ potential was truncated at a distance of 2.5d. Connectivity between bonded monomers is enforced by a finite extensible nonlinear elastic (fene) potential,44 defined as UFENE(r ) = −0.5KR 0 2 ln[1 − (r /R 0)2 ]

(2)

with a spring constant K = 30kBT and a maximum allowed bond length of R0 = 1.5d. With this setup, average bond lengths vary between 0.97d and 1.0d. The entire simulation box is cylindrical, and its curved boundaries are contourless LJ walls similar to eq 1 to prevent beads from leaving the system (Figure 1). In the direction of the symmetry axis (which lies on the z-axis), the system is either periodic or bounded with a planar LJ wall. If periodic, and its boundaries coinciding with the boundaries of the grafted area, then an infinite brush is simulated; otherwise, an infinite sequence of alternating grafted and empty sections are specified. Apart from the grafted polymers which form the brush, additional nonbonded beads are presented in some of the simulations, representing either a CNS or NPs. While interactions among monomers are repulsive (athermal implicit solvent), the monomer−CNS interactions are attractive to represent an explicit component which is a better solvent to the polymer than the implicit component. Co-nonsolvent beads are as small as dc = dm/2 to minimize volumetric effects of this component. In some cases, larger NPs are added to the system to model bulky components and to analyze their abilities to penetrate the channel. Those NPs have short-range repulsive interactions with the remaining components of the system. All simulations are performed using the open source LAMMPS molecular dynamics package.46 Details regarding the (rather standard) settings of the simulation parameters do not differ from our previous simulation studies47,48 and are not repeated here. Figure 1b displays snapshots of MD simulations from side view (left) and top view (right). Monomers are red, and CNS beads are green. The center of the nanopore is void of monomers and thus accessible to bulkier components of the system: The “gate” is open. In the following section, we are going to analyze under which conditions such a gate exists and what is required to make that gate switchable between open and closed states.

2. SIMULATION METHOD AND NANOPORE MODEL The nanopore is a cylindrical substrate which is on its inner wall decorated with polymer chains to form a channel (Figure 1). Polymers are flexible coarse-grained bead−spring chains44 with monomer diameters of dm = 1 (which thus defines the unit length). Each chain is grafted with one end onto the

Figure 1. (a) Schematic illustration of cylindrical pore which is decorated by linear chain brush. Linear chains (cyan) which are immersed in an invisible (= implicit) solvent (red dashed symbols) plus an explicit CNS component (black solid symbols). In axial direction (z-axis), the system boundary may be either periodic or fixed through a LJ wall. D, σ, N, and Lg stand for the cylinder diameter, the grafting density, degree of chain polymerization, and axial length of the grafting area, respectively. (b) Snapshots of MD simulations in side view (left) and top view (right) for the case of N = 20. Monomers are red, and CNS beads are green.

3. POLYMER BRUSHES IN A CYLINDRICAL PORE To introduce the equilibrium morphology of the surface layer, we identify the scaling regimes, based on the geometric B


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dimensions of the cylindrical pore, grafting density, and degree of polymerization of the chains. 3.1. From Mushroom to Surface Brush and Squeezed Mushroom. To arrive at the scaling regimes in the cylindrical pore, we start from the mushroom state in a dilute solvent. As pointed out by Dimitrov et al.,29 at sufficient large values of the diameter, each isolated flexible chain occupies an area of diameter ξb ∼ N ν

(7)

which implies the condition Ds − f ∼ 4Nσ 1/2ν − 1/2

(8)

for the boundary between a surface brush (with an empty area along the central axis) and a fully closed pore. Note that we do not consider concentration blobs of varying sizes along the radial direction (as e.g. done by Daoud and Cotton19) because here we are just interested in the condition in which the brush is fully filled and assuming a uniform monomer density for that state. It can easily be shown that blobs of varying sizes lead to identical scaling predictions. Actually, the Daoud−Cotton model for the cylindrical brush gives the same scaling relation as in eq 8. A second way to fill up the cylindrical pore is to start with the squeezed mushroom state. The axial length l of an individual chain (see Figure 2b), which is compressed in the narrow tube, can be estimated as49

(3)

as shown in Figure 2a and fully analogous to the planar mushroom state. Here, ν is Flory’s scaling exponent. With

ij N yz l = ξbjjj zzz ∼ ND1 − 1/ ν j Nb z k {

where Nb represents the number of monomers inside one compression blob, as defined by the Flory scaling law ξb ∼ Nbν. Once those independent cigars nestle together, l = σ−1/2, the condition

Figure 2. Schematic illustrations of different equilibrium morphologies of the decorating surface: (a) mushroom state, (b) squeezed mushroom state, and (c) surface brush state. Circles stand for (a) mushroom blobs (with the size of ξb and containing N monomers), (b) compression blobs, and (c) concentration blobs (with the size of ξb and containing Nb monomers). D, σ, and N denote diameter of the cylinder, grafting density, and degree of polymerization of the chains, respectively.

Dsm − f ∼ N ν /(1 − ν)σ ν /(2 − 2ν)

(4)

σm − s ∼ N −2ν , mushroom to surface brush

If instead the diameter of the pore is reduced to D = ξb, a squeezed mushroom state, consisting of compression blobs of diameter D, is formed (Figure 2b). The boundary between mushroom state and squeezed mushroom state lies at Dm − sm ∼ N

ν

Dm − sm ∼ N ν , mushroom to squeezed mushroom Ds − f ∼ 4Nσ 1/2ν − 1/2 , surface brush to filled

(5)

Dsm − f ∼ N ν /(1 − ν)σ ν /(2 − 2ν) , squeezed mushroom to filled

Note that lengths are expressed in terms of the unit length, so that the parameters discussed in this section are dimensionless quantities. 3.2. Conditions for Filling Up the Cylindrical Pore. We assume that a filled pore exhibits a uniform density of monomers and show later with the help of MD simulations that this assumption is approximately satisfied. The average number concentration of a uniformly filled cylinder amounts to cf =

Ntot πDLσN 4σN = = V D πD2L /4

(10)

holds, which denotes the boundary between the squeezed mushroom and the fully filled state. The filled-up state can be further divided into different scaling regimes21,29 which are of no interest in the context of the present work, in which merely the switching between open and closed gates is of relevance. Such a switch has to take place upon a change of the solvent condition and will be discussed in the following sections. 3.3. Scaling Regimes in Athermal Solvent. The regime boundaries of section 3.2 are summarized below:

increasing grafting density σ, an overlap occurs when ξb ∼ σ−1/2. Depending on the diameter D of the pore, the now overlapping chains form a closed layer on top of the substrate (a “surface brush”, Figure 2c), at the boundary

σm − s ∼ N −2ν

(9)

(11)

The scaling exponent ν is approximated as ν = 3/5 in good solvent and ν = 1/2 in theta solvent.50 To avoid unnecessary complications of additional, transient scaling regimes due to thermal blobs, we just consider the extreme case of athermal solvent, in which the thermal blobs equal the dimension of the monomer size.49 Otherwise, the coarse-graining level and the resulting unit length may be chosen as the thermal blob size without fundamental changes of the resulting phase diagrams. In athermal solvent we thus obtain

(6)

σm − s ∼ N −6/5

where Ntot and L stand for the total number of monomers in the system and the axial length of cylinder. Recall that in the surface brush state, the overlap blobs in the first layer above the substrate have the size of σ−1/2 in semidilute solvent. When the cylinder is just filled, then it is closely packed with concentration blobs of the same property as those in the surface brush, i.e., σ−1/2 ∼ Nbν. Hence

Dm − sm ∼ N3/5 Ds − f ∼ 4Nσ 1/3 Dsm − f ∼ N3/2σ 3/4 C

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mushroom is preferred if the grafting distance is shorter than the critical value, of the dimensionless order of N2/3. Then the transition from a single-chain mushroom to micelles takes place under the condition

Figure 3 summarizes the scaling regimes in athermal solvent (eqs 12). The mushroom regime is bounded by σm−s and

σm − mi ∼ N −4/3

(14)

Chains that are grafted in an area of size Rc collapse to form a micelle with the radius of Rm (as schematically sketched in Figure 4), where R c ∼ N 2/5σ −1/5 R m ∼ N3/5σ 1/5

(15)

Figure 3. Scaling regimes in good (athermal) solvent with boundaries as defined in eqs 12. The red line indicates the close packing limit (cf = 4σN/D = 1, i.e., Df = 4Nσ). Different background colors denote the existence of a central passage through the nanopore (open pore, green) or its absence (closed pore, red). The white circular area denotes “swampland” in which scaling theory remains indecisive due to the lack of prefactors in eqs 12.

Dm−sm. Starting here and increasing the grafting density, the resulting surface brush retains a central hole until the pore is filled. A further increase of the grafting density is possible until the packing limit is reached at a monomer density of c = 1 (red line), at which Df = 4Nσ

Figure 4. Scaling regimes in poor (non)solvent with boundaries as defined in eqs 18. Green and red background colors denote regimes with or without open central passage, respectively. The inserted sketch shows a single micelle of size Rm, which is feed by chains from the surrounding area of radius Rc.

(13)

On the other hand, at low graft densities, the reduction in diameter squeezes the mushroom and eventually closes the central pore through each single polymer. However, the pore is not entirely filled until the diameter is reduced to D < Dsm−f. Because of the missing prefactors in eqs 12, the precise order of cross-links between some of the regimes has to remain undetermined on the level of scaling theory. This “swampland” is denoted as a white circular area in the phase diagram. We want to point out that a similar scaling analysis of the phase diagram has been performed before by Manghi, Dimitrov, and co-workers,21,29 with results that are consistent with our presentation. However, for the purpose of our analysis, we had to focus on a couple of details which could be dismissed in previous studies. For example, the boundary between the brush and the compressed brush state, defined as D = Nσ1/3 in the previous work, serves as the boundary between the brush and the filled state in our analysis (eq 8). This modification has been necessary to allow for a precise distinction between open and closed pores. 3.4. Scaling Regime in Nonsolvent. For brushes in poor solvent, or the extreme case of nonsolvent, the approximation50 ν = 1/3, which is still valid for Dm−sm ∼ N1/3, is no longer applicable to the other boundary conditions in eqs 11. The boundary between the mushroom state and the surface brush does not actually exist: Because of the surface tension in poor solvent, the layer tends to break up to form octopus micelles which break the rotational symmetry of the system.34 Williams51 has indicated that the stretching of a single-chain

Similar to the case of mushroom states, micelles are squeezed by the narrow pore as soon as the diameter of the pore decreases to D = Rm, and the boundary between micelles and squeezed micelles is calculated as Dmi − sm ∼ N3/5σ 1/5

(16)

Conversely, if the grafting density raises, the inner surface of the pore becomes fully covered by the increasing size of micelles under the condition of Rm = Rc, at which a micelle occupies the same area as the chains which contribute to that micelle, so that σmi − s ∼ N −1/2

(17)

The condition above corresponds to the crossover from squeezed to extended chains in a dry brush. The resulting regime boundaries in nonsolvent are summarized below: σm − mi ∼ N −4/3 , mushroom to micelles σmi − s ∼ N −1/2 , micelles to surface brush Dm − sm ∼ N1/3 , mushroom to squeezed mushroom Dmi − sm ∼ N3/5σ 1/5 , micelles to squeezed micelles Df = 4Nσ , close packing limit D

(18)


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the CNS causes a collapse transition of the brush43 and an opening of the gate. The CNS, being a good solvent for the polymers in its pure form, causes an attractive interaction between the monomers in the mixed solvent state. This can be explained by the formation of transient bridges between two monomers and one CNS induced by the preferential wetting of the polymer by CNS in the mixture. In particular, for small volume fraction of the CNS the competition of the monomers for the favorable interaction partners is strong. In the theoretical model presented in ref 43 it was also shown that this can be mapped to a concentration-dependent χ-parameter for the polymer in the solvent mixture. The latter is responsible for the first-order nature of the brush collapse which can occur for CNS-induced collapse but which cannot occur for simple degradation of a one-component solvent because of the missing translation entropy of the immobilized chains in the brush. This feature makes the CNS-induced collapse highly interesting to obtain a switchable gates upon minimal variation of the solvent composition. We note that the CNS-induced phase transition cannot be mapped to a simple change in solvent quality according to an effective Flory parameter, and excluded volume effects and nonhomogeneous distribution of CNS in the brush can lead to unique phase morphologies and coexistence states. Figure 6 displays the density profiles of the monomers in axial (a) and radial direction (b) for various values of the chemical potential μ of the CNS, which is approximated as43

In nonsolvent, the brush phase is composed of packed monomers (although, more realistically, it would consist of closely packed thermal blobs), and at D = Df (eq 13), the pore is filled. Figure 4 displays the scaling regimes in nonsolvent (eqs 18). The micelle state, as an intermediate state, separates the mushroom and surface brush states by the conditions σm−mi and σmi−s, contrary to the situation in athermal solvent. Once again, the precise locations of the intersection points between these regimes remain undefined on the level of scaling theory, giving rise to another “swampland” area as indicated by the white circle. 3.5. Regime of Solvent-Switchable Gates. In the context of a switchable passage through the nanopore, the regime is of interest in which the pore is both filled (in good solvent) and alternatively open (in poor solvent). When combining the relevant scaling regimes of Figures 3 and 4, we arrive at Figure 5 with the blue area, in which the

i ρ yz zz μ ≡ lnjjjj z − ρ 1 k {

(19)

Here, ρ denotes the volume fraction of the CNS in the empty (i.e., nongrafted) area of the cylindrical pore. The ideal gas approximation is valid at low concentrations of the CNS at which two-body interactions between its molecules remain negligible. The strength of monomer−CNS interaction is fixed to ϵ = 1.8 and the number concentration of monomers is denoted by c. Figure 6 shows how the brush collapses with increasing chemical potential to reach a plateau at values above μ ≈ −5.5. During this process, a central hole with a radius of more than 5 units is produced (Figure 6b) which allows for a free passage of bulky molecules such as NPs. Note that in the absence of CNS the filled nanopore exhibits an approximately uniform monomer density (black curve) which serves as an a posteriori justification of our assumption made in eq 6. Also visible in Figure 6a is a spillover of the swollen brush phase into the nongrafted sections, which disappears in the presence of CNS when the brush is effectively in a poor solvent. To inspect the process of phase transition, we extract the center-of-mass thickness

Figure 5. Combination of the relevant scaling regimes of Figures 3 and 4. The blue area is the regime in which the pore is closed under good solvent conditions but open in poor solvent.

aforementioned condition holds. This “solvent-switchable area” is determined by the filled state in good solvent, which is further truncated by the squeezed mushroom and squeezed micelles states in poor solvent (marked in red background color in Figure 4), where a change of the solvent quality from good to poor does no longer open a central gap but merely triggers a segregation into clusters which align along the central axis. Such a configuration would not be able to serve as a switchable gate.

4. SIMULATION RESULTS: DENSITY PROFILES AND LAYER THICKNESS In the following simulations, a cylindrical simulation box of diameter D = 40 and length L = 187.2 is used. The grafted section has a length of Lg = 62.4, and polymers of length N = 60 are grafted at a density of σ = 0.05. Each system is initially equilibrated using up to 108 timesteps (which corresponds to 2 × 105 LJ times) before data acquisition begins. The polymers are embedded into an implicit athermal solvent, fully swollen, and in this state completely filling the grafted section of the cylinder. Then, explicit CNS beads with varying interaction parameters ϵ are added to the simulation box at various concentrations. Within certain parameter ranges,

D /2

Hcm

∫0 r 2c(r ) dr D ≡ − D /2 2 ∫0 rc(r ) dr

(20)

as a function of the chemical potential, μ, varying the depth ϵ of the monomer−CNS pair potential. c(r) is the density profile of the brush layer in radial direction. Figure 7 indicates that in the case of weak interactions, ϵ ≤ 1.2, the brush height remains unaffected by the CNS, apart from a slight increase at high CNS concentrations as a result of volumetric effects. Once ϵ ≥ 1.4, the CNS causes an increasingly steep collapse of the brush E


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strong inhomogeneities of the monomer densities in axial direction, which break up into sections of high densities, separated by areas of low occupation. In some of these examples (indicated as red triangular symbols in Figure 7), the system might appear to be already in a squeezed micelles state. However, any interpretation of states at these strong interactions should be taken with a grain of salt, since the system may not be in thermal equilibrium but frozen in a local, partially random state. Hence, to avoid the possibility of nonequilibrium states, for the rest of the work we apply ϵ = 1.8 which allows for a smooth vertical collapse of the brush. Yet, we note that the (possibly) squeezed micelle state observed at ϵ = 2.0 emerges only at sufficiently high CNS concentrations, while a switching of the gate from closed to open occurs at lower number concentrations and remains unaffected by this anomaly.

5. FACILITATED PASSAGE OF NANOPARTICLES 5.1. Size-Dependent Stationary Flux of Nanoparticles in Co-nonsolvent. As long as the brush stays in athermal solvent and in its swollen state, the nanopore is closed and the passage of NPs is hindered. In the simulations, this is verified as follows: NPs of different diameter are initially placed on one side of the nanopore, which is located in the middle of the simulation box with fixed boundary conditions. The diffusion through the nanopore is then monitored (Figure 8b), while only the complete passage through the pore is counted as a translocation event. The figure indicates that small NPs are still able to pass. The stationary fluxes are fitted to the asymptotic slopes (dashed lines) and plotted as a function of NP diameter in Figure 9 (red circles). Once dnp > 2.5, the flux is close to zero and ceases entirely with dnp > 3.5. The nanopore is thus serving as a passive barrier to bulky objects. The situation changes when the CNS (μ = −5.62, ϵ = 1.8) is added and the brush is collapsed (Figure 8a): The passage of large NPs is now possible; the corresponding stationary fluxes are far from zero, as shown in Figure 9a (black squares). In particular, the ratio of the fluxes, open gate versus closed gate (see snapshots in Figure 9b), as shown in the inset of Figure 9a, indicates the strong switching effect caused by the CNS: Being close to unity in the case of small NP diameters, it reaches values above 20 at dnp = 3.5 and increases beyond limits when the NP turns larger. Note that for dnp = 1 the diffusive flux is somewhat higher with the swollen brush. This is so because the closed gate is highly porous to objects of monomer size. Once the brush collapses, the NP paths are restricted to the central opening, since the collapsed phase has turned too compact. In this way the collapse of the brush actually slows down the exchange of small components. 5.2. Energy Barrier of Nanoparticle Inclusion. A complementary approach to the efficiency of the nanopore as a gate is to determine the free energy profile of a NP that is gradually pushed into the pore. In these simulations, the NP is placed on the central axis in front of the nanopore and fixed with the help of a harmonic potential. Its tether point is then gradually shifted toward the pore, sufficiently slow so that the brush polymers in contact with the NP relax into equilibrated conformations. The average force acting on the NP is determined by the bias it causes to its position and then integrated over the entire path into the pore, to yield the minimum work and thus the free energy of inclusion.52 More specifically, the NP is initially tethered at the position z/Lg = −0.5, well clear of the brush. With each simulation run,

Figure 6. Density profiles of monomers from MD simulations in axial (a) and radial direction (b). The interaction between monomers and CNS beads is specified by the energy depth of the LJ potential ϵ = 1.8. In axial direction, Lg indicates the axial length of the grafting area, and linear chains are grafted inside 0 ≤ z/Lg ≤ 1. The parameters dmon, dcs, and μ represent the diameters of the monomers, CNS beads, and the chemical potential of eq 19. In radial direction, r = 0 and r = 20 denote the locations of the central axis and the substrate, respectively.

Figure 7. Rescaled center of mass thickness, Hcm/H0 (eq 20), of the brush as a function of the cononsolvent’s chemical potential, μ (eq 19). H0 stands for the center-of-mass thickness in the absence of CNS, and ϵ is the depth of the monomer−CNS pair potential. Solid symbols indicate conformations with irregular morphologies at high interaction strengths that may not be fully equilibrated.

as a result of the effective attraction between the monomers caused by the formation of CNS bridges.43 At very strong pair interaction (ϵ = 2), the brush height appears to display irregular jumps (marked with solid symbols) in systems of different CNS concentrations. Here we observe F


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Figure 9. (a) Stationary fluxes of NPs, rescaled by the number Nnp of NPs, as a function of NP’s diameter, with open gate (black squares) and closed gate (red circles). Inset: the ratio of fluxes, open versus closed gate (at dnp = 4.0, that ratio is undetermined, since no passage was observed with closed gate). (b) Morphologies of an open gate (up) and a closed gate (down) in MD simulations. Red and green beads represent monomers and NPs, respectively. CNS beads are not shown in the open gate pattern.

Figure 8. Number of translocation events of NPs as a function of time in cases of brush with CNS (a), with μ = −5.62 and ϵ = 1.8, and athermal solvent (b), as a function of NP diameter. Dashed lines are fits to the asymptotic slopes (s) to determine the stationary fluxes.

the nanopore is closed. This free energy barrier reduces the probability of finding such a NP inside the nanopore by the Boltzmann factor exp(−Eb/kBT) ≈ 10−3 and thus explains the practically unobservable diffusive flux of such NPs through the closed pore (Figure 9).

it is then shifted forward by 5 length units. It follows a new relaxation of 5 × 106 simulation steps and then a data acquisition over 107 steps, during which the average position ⟨z ⟩ and its displacement ⟨z ⟩ − z0 from the tether point z0 are measured. The tether force fs(z) = λ(z − z0) with a spring constant λ then yields the average force acting at the particle’s position. Figure 10 displays the resulting force profiles for a small particle of monomer diameter (a) and a bulky NP of diameter dnp = 4.0 (b). In the former case, hardly any systematic forces are visible, regardless of whether the gate is open or closed. The bulky NP, however, passes into the channel without effort only when the gate is open (μ = −5.62 or μ= −6.0). Otherwise, a repulsive force (of negative sign) is observable upon entering the swollen brush. Note that as a result of the spillover (Figure 6a), the repulsion does not start at the location z = 0 at which the grafted section begins, but quite a bit in front of that. Once the NP is entirely embedded into the brush, the repulsive force ceases because there no longer exists any density gradient. The forces may now be integrated to obtain the free energy of inclusion: Eb =

6. CONCLUSION In this work we have shown that a CNS can be used to trigger brush-decorated nanopores between a “closed” and an “open” state. Here, the CNS causes a collapse of the brush due to the preferential and nonspecific adsorption onto the polymer chains which induces an effective attraction between the monomers. The advantage of the co-nonsolvency effect to switch the nanogate is that neither drastic changes in the solvent quality nor temperature variation is required. Moreover, for an appropriate adjustment of parameters such as grafting density and chain length only a minimal amount of CNS is necessary to achieve the collapse of the polymer layer and thus the opening of the gate. In our simulations a volume fraction of only about 0.2% is sufficient for switching. Because the CNS induced transition is sharp, a very small variation of the volume fraction is necessary. These are ideal conditions for a potential realization of gates in nanodevices. The functionality of the switchable pore we have studied in this work is the gating of NPs. We have shown that NPs which are only 4 times larger in radius as the coarse-grained monomers are strongly blocked from diffusive translocation

Lg /2

∫−L /2 f (z) dz g

(21)

Figure 11 shows that in the case of a bulky NP of diameter dnp = 4.0 (green triangles) the barrier height exceeds 6kBT when G


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Macromolecules

using a forced insertion method for the particle. Once that energy barrier is known, its concentration inside the nanopore is estimated by the Boltzmann distribution. This in turn allows the a posteriori the calculation of the flux trough the pore. Using scaling methods, we have identified the parameter regime in which such a channel is switchable by regarding the effect of added CNS as a change of solvent quality from good to poor. Depending on the grafting density, the CNS-induced monomer−monomer attraction can lead to micellar structures, known as octopus micelles for flat brushes. If the size of the micelles reaches the pore diameter, the micelles become squeezed and form stoppers which block the passage for larger particles as effectively as the homogeneously swollen brush. The open channel corresponds either to a homogeneously collapsed brush layer on the cylinder wall or to a surfacemicelle state. The transition between the squeezed micelle state and the open channel can also cause switching of the gate and might be realized upon variation of the CNS volume fraction in the collapsed state of the brush. Although we have obtained first indications for such a second mechanism of gating, a closer analysis is necessary in future work. Our work may be extended in several ways: First, time scales for a reversion of the CNS induced collapse are interesting to know for a practical application of this mechanism to switchable gates. Second, a detailed study of the morphologies of the collapsed phases under a wider range of parameter settings is of interest to understand the conditions, under which the formation of micelles replaces the regular vertical collapse of the brush. This is of particular relevance to the function of the nanogate because a microsegregation of the brush phase along the central axis would prevent the formation of a central passage way. Finally, particles that move through the NPC may not be perfect spheres. The permeability of particles of various different shapes with corresponding different inclusion free energies53 would be interesting to study in this context as well as particles with patterned surfaces such as hydrophilic and hydrophobic patches.

Figure 10. Average interaction force acting on a single NP at different locations along the central axis of cylinder, at a NP diameter of dnp = 1.0 (a) and dnp = 4.0 (b). The dashed vertical line indicates the boundary between free and the grafting area (see Figure 1a), and Lg stands for the axial length of grafting area. The chemical potential μ of the CNS corresponds to a fully open gate (μ = −5.62 or μ = −6.0), partially (μ = −6.44, blue symbols), and fully closed gate (red and black symbols).

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (H.M.). *E-mail [email protected] (C.-X.W.). *E-mail [email protected] (J.-U.S.). ORCID

Cheng-Wu Li: 0000-0002-6278-2883 Jens-Uwe Sommer: 0000-0001-8239-3570 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partly supported by the National Science Foundation of China under Grants 11574256, 11747619, and 11747313, the 111 project (B16029), and the DFG Grant SO277/17-1. C.-W.L. thanks the IPF Dresden for the hospitality during his research visit.

Figure 11. Energy barrier (in units of kBT) for NP inclusion into the nanopore as a function of the chemical potential of the CNS, and for different NP diameters. While the small NP enters without effort, the largest NP requires almost 7kBT to enter the nanopore in its closed state. The absence of CNS is indicated as a chemical potential of −∞ (corresponding to a brush in athermal solvent).

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in the swollen state of the brush. The collapse of the brush can increase the permeability by a factor of more than 20 in this this case. To calculate the permeability even for cases where a direct observation of translocation events is impossible during the simulation time, we have calculated the free energy barrier

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