Tine
Curk
*a,
Francisco J.
Martinez-Veracoechea
a,
Daan
Frenkel
a and
Jure
Dobnikar
*ab
aDepartment of Chemistry, University of Cambridge, Lensfield Road, CB2 1EW, Cambridge, UK. E-mail: tc387@cam.ac.uk
bDepartment for theoretical physics, Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia. E-mail: jd489@cam.ac.uk
First published on 25th April 2013
We present Monte Carlo simulations of colloidal particles pulled into grafted polymer layers by an external force. The insertion free energy for penetration of a single colloid into a polymer layer is qualitatively different for surfaces with an ordered and a disordered distribution of grafting points and the tendency of colloidal particles to traverse the grafting layer is strongly size dependent. In dense colloidal suspensions, under the influence of sufficiently strong external forces, colloids penetrate and form internally ordered, columnar structures spanning the polymer layer. The competition between the tendency for macro-phase separation of colloids and polymers and the elastic-like penalty for deforming the grafted layer results in the micro-phase separation, i.e. finite colloidal clusters characterized by a well-defined length scale. Depending on the conditions, these clusters are isolated or laterally percolating. The morphology of the observed patterns can be controlled by the external fields, which opens up new routes for the design of thin structured films.
Here we report Monte Carlo computer simulations of polymer-insoluble particles in grafted polymer layers. We demonstrate that sufficiently strong external forces give rise to collective ordering with a rich variety of morphologies. We studied polymer layers under good solvent conditions between the “mushroom” and the “brush” regime,10 with the mean spacing between the grafting points (i.e. the de Gennes blob size4) being roughly similar to the radius of gyration of the polymers and to the colloid diameter. The polymers generally repel the particles from the surface. We find that the repulsion at a given grafting density is largest when the distribution of the grafting points at the surface is ordered, while in the case of disordered grafting with large-enough monomer density fluctuations at the surface, the free energy profile resembles a barrier and is attractive close to the surface. This enables reversible adsorption of particles and their slow release. The nature of the constant external force suitable for pulling the colloids into the polymer layer depends on the particle size: large micro-particles can sediment under the influence of gravity, which, however, plays no role in the case of polymers or nanoparticles in aqueous solutions. An effect equivalent to pulling by gravity would be obtained by placing the system in a centrifuge.
(1) |
The linear potential energy term decouples from the polymeric degrees of freedom, and Fp(z) holds the key to understanding the penetration of a single colloid into a polymer layer.
In a good solvent the polymers can be modelled as self-avoiding random walks. We follow a coarse-grained model19 where the polymer chains are represented by soft repulsive blobs with a radius of gyration rb. Each chain is composed of lp blobs, which are connected via harmonic springs:
βUch = 0.534(r/rb − 0.730)2. | (2) |
Individual blobs repel each other via a Gaussian repulsion
(3) |
βUbc = 3.20e−4.17(r/rb−0.50) | (4) |
The grafting density = Np/Rg2 determines whether the polymer layer is in the dilute “mushroom” ( < 1) or in the dense “brush” scaling regime ( ≳ 3).20,21 The relevant physical length scale in the mushroom regime is the radius of gyration Rg of the chains, while in the brush regime it is the so-called de Gennes blob: ξ ∝ ρ−1/2.4 All our simulations are in the intermediate regime where the two length-scales are similar ξ ≈ Rg. In the following, rather than per Rg2, we will express the grafting density as the number of anchoring points per colloidal diameter squared, therefore ρ = Rg2/σ2. We will explore the regime 0.5 < ρ < 5, where the probability of inserting a colloid with σ ≈ Rg is non-vanishing. We first describe the model and simulations on single-colloid insertion, followed by a study of collective ordering in dense colloidal systems.
As demonstrated in Fig. 1(a), the free energy profiles in the three grafting arrangements are qualitatively different: in the ordered case the free energy monotonically decreases with height,6,10 while in the disordered case it features a barrier (also reported in ref. 9 and 12). In the inset, Fig. 1(b), we plot the 2D distribution of the hole sizes23 for the three arrangements, which clearly illustrate the above discussion. The difference between the quenched and annealed disorder is rather small, which is due to the fact that the anchoring points (monomers) are vanishingly small compared to the polymer Rg and therefore do not repel each other as the blobs do in our coarse-grained model. Close to the grafting surface the monomer fluctuations are “frozen” because the anchoring points are immobile, while away from it they relax. Any effect of the surface-imposed density fluctuations should therefore vanish at a height roughly equal to Rg – well confirmed in Fig. 1(a). In the experimentally relevant situations (disordered grafting), we therefore expect to observe a maximum in the free energy profile around z ≈ Rg and a metastable minimum at the surface. A strong-enough external force can transform the metastable local minimum at the surface into a stable free-energy minimum.
Fig. 1 Isolated colloids in polymer layers with grafting density ρ = 1.0 and number of blobs per chain lp = 20. (a) The insertion free energy profile Fp(z) for three realizations of the surface grafting: quenched disorder (black solid line), square crystalline order (red solid line), and annealed disorder (green dashed line). (b) The distribution of (2D) hole sizes for the three scenarios. The solid lines represent the analytically derived expressions (see ESI†), symbols are numerically determined. For both cases of disordered grafting there is an appreciable probability of finding large holes, while the ordered surfaces, characterized by small density fluctuations, have a cut-off in the hole size. |
In Fig. 2(a) we present the insertion free energy profiles for various parameters in the case of disordered grafting. We also show (Fig. 2(b)) the matching monomer density profiles, which are monotonic (apart from the oscillating behavior very close to the surface due to the finite size of the blobs). In Fig. 2(c)–(e) the scaling of the position and height of the free energy maximum are analyzed. The barrier height Fmax scales linearly with ρ and with σ2 (see also ESI†); in our system it is typically of the order of 10 kBT.
Fig. 2 (a) The insertion free energy profiles for disordered grafting. Each of the three family of curves show insertion free energy for various ρ = 0.5, 1, 2, 3, 4. (b) The monomer (blob) vertical density profile for lp = 60 and ρ = 4 (bold green), ρ = 1 (dashed green). The position of the free energy barrier (in the case of disordered grafting) as a function of (c) polymer size and (d) colloid size is presented. (e) shows the barrier height as a function of the grafting density for lp = 10 (red) and lp = 60 (green). |
Fig. 3 (a) Normalized probability distributions p(z) = exp(−βF(z))/∫exp(−βF(z))dz for lp = 40, g′ = 3 and various grafting densities ρ = {0, 1, 2, 3, 4}. (b) The mean distance of the centre of the colloid from the surface z as a function of the grafting density ρ at various parameters lp and g′. Due to the hard-core of the colloids, the lowest accessible distance is z = 0.5. The dotted lines are guides for the eye. |
Fig. 4 (a) Total free energy of sedimenting colloidal particles obtained by simulations at lp = 40 and ρ = 1.0 for silica colloids with three sizes: σ1 = 0.67 μm (red line), σ2 = 1.00 μm (black line), and σ3 = 1.33 μm (green line). Respectively, g′2 = 2.0 (corresponds to micrometer sized silica particles in water), g′1 = 0.40 and g′3 = 6.3 (the grafting density is equal in all three cases and defined as ρ = Np/σ22). The dashed lines represent the insertion free energy Fp(z). (b) Normalised height distribution P(z) ∝ e−βF(z) for the same system. The histograms illustrate the probability of finding the colloids adsorbed to the surface or above the brush (left set of histograms: z < Rg and right set: z > Rg). (c) Snapshots from Monte Carlo simulations of dilute colloidal suspensions under conditions corresponding to (a) and (b). |
The free energy barrier height scales with σ2, while in the case of gravity, the external force scales with the particle volume (σ3). Therefore, larger particles will preferentially sediment on the surface. In case other types of external forces are applied that do not depend on the particle size (this is the case for, e.g., osmotic gradients), the quadratic scaling Fmax ∝ σ2 results in an inverted size selectivity: at a given value of the pulling force, only sufficiently small nano-particles will penetrate, a scenario particularly important in biology. The size-selectivity for electric field driven insertion might be much less pronounced than in the case of gravity, since the electric charge typically scales with the particle surface (σ2), which cancels out with the barrier height scaling. It must be noted that electromagnetic forces would also induce particle interactions, thus deviating from the hard-sphere model used in this work.
At any given value of the external field strength g′ and polymer size lp, there is a “critical” value of the grafting density ρ = ρ0, at which the external pressure is balanced by the repulsion of the polymers. The colloids can only penetrate the polymer layers with small-enough grafting densities ρ < ρ0. The insertion of multiple colloids into a polymer layer is a many-body problem: when one is inserted, it compresses the polymers and increases their effective density that is felt by the next colloid. In our simulations, μ controls the equilibrium number of colloids within the layer. Note that, due to a constant external force, the chemical potential of inserted colloids depends linearly on their height; we chose to define the chemical potential at the bottom surface (z = 0). For small μ, the penetrating particles form clusters at the bottom surface that ”grow” towards the top of the brush as μ increases. Fig. 5 compares typical snapshots on disordered and ordered surfaces. At small μ there is a difference: the shape of the clusters above the ordered surfaces is narrower at the bottom (resembling a table-top) due to the strong repulsion of colloids from the surface, while in the case of disordered grafting the local free energy minimum enables pyramid-like structures. In both cases, however, similar structures emerge as μ is increased: clusters with a uniform vertical profile are formed – spanning the brush from bottom to top resembling straight cylindrical towers. Once such towers reach the critical conditions for fully loaded layers, increasing the chemical potential further leads to accumulation of colloids on top of the layer.
Fig. 5 Cluster growth as a function of the number of colloids (chemical potential) in the system. The sequence on ordered (left-hand side) and disordered (right-hand side) surfaces is depicted. When the number of inserted colloids is small, “pyramid” and “table-top” structures are formed on disordered and ordered surfaces, respectively. Adding more particles, in both cases “critical” clusters with uniform vertical profiles spanning the polymer layer are formed. The number of blobs per chain is lp = 40, the anchoring density ρ = 4.0, g′ = 6.0. The chemical potentials (μ) and number densities (per unit area) of colloidal particles (n) corresponding to the three rows of snapshots are: (top) βμ1 = 41, ncolsq = 0.54, ncolrnd = 0.82; (middle) βμ2 = 46, ncolsq = 1.91, ncolrnd = 1.98; (bottom) βμ3 = 51, ncolsq = 3.20, ncolrnd = 3.23. |
The maximum amount of colloids that can be inserted into a layer depends on the grafting density: it is zero at the ρ ≥ ρ0 and grows linearly with decreasing density. Once the maximum load is achieved, the average density of polymers in the space around the colloids equals the critical value, regardless of the grafting density ρ. Consequently, the thickness of a fully loaded layer should not depend on the grafting density. Insertion free energy scales approximately linearly with the brush grafting density (see Fig. 2 and a discussion in ESI†), so the polymer pressure should also scale approximately linearly with the grafting density. External (hydrostatic-like) pressure scales linearly with the strength of the force field g′ and brush height hb, the balance of pressures gives
ρ0 = κhbg′. | (5) |
The proportionality constant κ in general depends on the solvent properties and the polymer and colloidal interactions. In our model, κ is defined by the choice of the interaction potentials, which represents hard-sphere colloids and self-avoiding walk polymers. From the simulation data (see ESI†), we obtain κSAW ≈ 0.11. In experiments, as long as they are in the good solvent regime and there are no specific colloid–colloid or colloid–polymer interactions, the value of κ should be similar. For large-enough external forces, the value of the critical effective density is well in the brush scaling regime. Therefore, the height of the fully loaded layer is expected to scale as hb ∝ lpρ0ν with the exponent ν = 0.35 for self-avoiding chains.4 Using eqn (5) this leads to an interesting scaling relationship for the fully loaded brush height:
(6) |
From the simulations such scaling is roughly confirmed. Insertion of colloids can thus potentially be used to control the thickness of polymer brushes: experimentally, adding an excessive amount of colloids and washing off the ones above the brush could be attempted.
Given the uniform vertical profiles observed in the colloidal clusters (Fig. 5), the particle load is directly correlated with the colloidal surface coverage η of the two-dimensional horizontal cross-sections:
(7) |
Fig. 6 Phase behavior as a function of the polymer grafting density and the external pressure. Various lateral patterns are depicted by symbols: isolated towers (blue circles), “walls” (blue triangles), percolating structures (red diamonds) and inverted towers (red squares). The black lines separating the phases are lines of constant surface coverage η. The white lower region is “super-critical”: no colloids penetrate the brush. The grey-shaded region depicts the regime where the colloids do penetrate but the effective polymer density is too small to induce particle ordering. The extent of the fluid region depends on the physical parameters, especially on the colloidal excess density and size. Presented here is the case of σ = 1 μm silica colloids in water; for smaller colloids the boundary of the fluid region would shift upwards. Top-view snapshots from the simulations displaying the lateral morphology are shown at the side and marked by the same symbols. The critical density is ρ0 = 6.0 with g′ = 6.0, lp = 40. |
In order to theoretically understand the micro-phase separation leading to these various patterns we consider two mechanisms governing the stability of colloidal clusters. Based on the simulations, the clusters are assumed to be vertically uniform and spanning the entire polymer layer. The polymers are excluded from the volume occupied by the colloids, which increases their effective density in the available space to the critical density ρ0. We will specifically focus on the transition from symmetric towers to the elongated wall-like clusters. The towers have a circular cross-section with radius r. For simplicity we assume that a “wall” is composed of the middle part of length l with a rectangular cross-section and two semicircular ends (radius r). A generalized cluster is characterized by the width 2r and the aspect ratio α ≡ 1 + l/2r (α = 1 for the towers and α > 1 for the walls).
The first contribution to the free energy penalty is the “excluded volume” term Fex. We assume that the polymers are excluded from the volume occupied by the cluster and from an additional “depletion” layer with the thickness γ around it – modelling the entropic penalty of confining the polymer configurations near a solid object. The free energy penalty due to the excluded volume is proportional to the total volume inaccessible to the polymers:
Fex/kBT = Cexρ0hb[r2fα + γr(fα + π)]. | (8) |
The second contribution to the free energy penalty is the “squashing” term Fsq, an elastic-like term that measures the entropy loss of the parts of the polymer chains squashed beneath the cluster. The elastic penalty is assumed to be proportional to the length of the squashed chains, e.g. kBT per squashed de Gennes blob. We also assume that the polymer chain extends radially (with respect to the center of tower) from its grafting point in order to minimize the squashed length. For the towers, , where Csq is a constant that specifies the squashing entropic penalty per unit length of a chain, which is (see ESI†) . The total free energy penalty F = Fex + Fsq for a small number N of generalized clusters is
(9) |
Minimizing eqn (9) with respect to N, r and α subject to a constraint of fixed total colloid area, Scol = Nr2fα, we obtain the optimal number, size and shape of the clusters (Fig. 7): at small cluster volumes hbScol, a single tower with circular cross-section is formed. The theory predicts that the width of the circular tower grows from zero to L0 ≈ 4.6σ with an increasing cluster volume. In reality, clusters thinner than L0 are not layer-spanning (see Fig. 5) and we only expect to observe circular layer-spanning towers of width L0. Larger clusters then elongate into wall-like objects – keeping their width constant. The theory correctly predicts the characteristic length scale L0 and the shape of small clusters.
Fig. 7 The typical width (black line) and the optimal aspect ratio α (red line) as a function of the surface cross-section of the clusters predicted by minimizing the free energy eqn (9) assuming Cex = 4.0kBT/σ, Csq = 2.4kBT/σ and γ = 0.2σ. The smallest layer-spanning clusters are isotropic towers with the width of L0 ≈ 4.6σ, while clusters larger than that form elongated walls. Clusters smaller than hbL0 (shaded region) are not layer-spanning in the simulations. |
Deriving the final expression, we made two approximations. First, we assumed ρ ≈ ρ0. While this assumption is valid for isolated clusters (η → 0), it does not hold at larger colloid filling fractions. It overestimates the squashing term and consequently favors circular towers over elongated walls. The second approximation is the assumption that the effective polymer density is equal to ρ0 everywhere in the interstitial space between clusters. This is not strictly true since polymers are anchored and the blob density should be higher near the clusters than further away. The approximation neglecting this fact favors elongated over circular clusters. Our hand waving argument is that the two effects partially cancel out – extending the validity of eqn (9) to finite values of η. The good agreement of the theoretical predictions with simulations at small values of η supports this argument. To quantitatively predict the stability of percolated structures at large η, however, a more refined approach, explicitly taking into account the two corrections mentioned above, would be needed. The micro-phase separation happens when the squashing constant Csq is positive. If, on the other hand, anchoring points were mobile on the surface, the polymers would not get squashed because they could simply escape from beneath the clusters. This is equivalent to setting Csq = 0, where the theory predicts r → ∞, e.g. a macroscopic phase separation. This was indeed observed in the simulations.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c3sm50486g |
This journal is © The Royal Society of Chemistry 2013 |