Article pubs.acs.org/JPCB
Reentrant Stabilization of Grafted Nanoparticles in Polymer Solutions Huanhuan Feng,†,‡ Marcel Böhmer,§ Remco Fokkink,† Joris Sprakel,† and Frans Leermakers*,† †
Physical Chemistry and Soft Matter, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands Dutch Polymer Institute (DPI), P.O. Box 902, 5600 AX Eindhoven, The Netherlands § Philips Research Laboratories Europe, High Tech Campus 4, 5656 AE Eindhoven, The Netherlands ‡
S Supporting Information *
ABSTRACT: Polymer chains grafted onto nanoparticles may facilitate the dispersion of such particles in a polymer solution. We explore the optimal strategy for stabilizing polymer-grafted nanoparticles using self-consistent field theory and experiments. The best results are obtained for relatively low grafting densities and for chain lengths of the brush polymer NB larger than that of the freely floating polymers Nf. When Nf > NB, one finds a compatibilization gap and re-entrant stabilization: At both very low and very high polymer concentrations particles disperse in the polymer solution, while at intermediate concentrations the particles lose their colloidal stability. At low grafting densities the underlying surface is in contact with the solvent. Particles covered by a bidisperse brush can combine a low grafting outer region with full coverage of the surface by a densely grafted inner layer. Using classical colloidchemical stabilization criteria the region in the phase diagram for which the particles are expected to mix with a concentrated polymer solution opens up. Now, also upon an increase in the length of the freely dispersed polymers, a re-entrant colloid-chemical stabilization is found for particles on the nanometer length scale: At both short and long polymer chains in solution the particles will not aggregate, whereas at intermediate lengths the colloidal stability is marginal. This multi re-entrant behavior is found from numerical self-consistent field calculations, and these predictions are consistent with corresponding experiments.
■
INTRODUCTION It is a long-term topic of research in both academia and industry to create composite materials through mixing some colloidal particles, some filler, with a polymer matrix.1−4 The key target is to have a homogeneous system wherein the filler is dispersed as primary particles rather than in the form of aggregates. Typically the particles are not compatible with the polymer melt or polymer solution, and in the absence of additional repulsive forces the particles will aggregate in a process known as flocculation. Mechanistically, the flocculation is in part due to van der Waals interactions (always attractive) assisted by some attractive contribution that originates from the polymers. When the polymers adsorb onto the particles attraction is caused by bridging; in the case in which the polymers do not adsorb onto the particles, the process is known as depletion. In polar solvents one can anticipate repulsive electrostatic contributions to counteract the attraction. As a result, one may find a DLVO-type potential5−7 with a primary and a secondary minimum near the particle surface and further away from the surface, respectively, with a local maximum at an intermediate distance that endows the colloidal stability of the particular solution. The key idea is that when the local maximum exceeds the value of thermal energy significantly the particles cannot come close enough to reach the primary minimum. Such a system is said to be colloidally stable © 2015 American Chemical Society
(provided that the secondary minimum is not deep with respect to the thermal energy). Typically the height of the barrier can be tuned by the size of and the charge density on the particles as well as the ionic strength in solution. In apolar media the route to use electrostatics is cut off because there is no significant surface charge. Hence another strategy must be used to fight against the attractive interactions. A frequently used strategy is to tether polymers with one of their ends to the particle surface.5−10 Depending on the grafting density we consider two regimes. As long as the distance between the grafting points exceeds the radius of gyration of the grafted chains, we have isolated mushrooms on the surface. In this case, there is only a marginal stabilization effect because the surface is largely in contact with the solution. Alternatively, when the distance between the grafted chains is small compared with the coil size, there is a polymer brush. In a brush the polymers are stretched due to lateral interactions between the chains. When two such polymer brushes are pushed into each other, we find a significant repulsive, usually called steric, contribution to the pair potential. The range and strength of the repulsion may be tuned by the polymer chain length, NB, Received: June 9, 2015 Revised: September 7, 2015 Published: September 14, 2015 12938
DOI: 10.1021/acs.jpcb.5b05504 J. Phys. Chem. B 2015, 119, 12938−12946
Article
The Journal of Physical Chemistry B and the grafting density, σ, and one can ensure that the van der Waals attraction is overcompensated. In particular, when the grafted polymers are chemically identical to the polymers in solution one then would intuitively expect that such particles will be accepted by the polymer solution; however, the situation is still delicate because the particle solution remains vulnerable to depletion flocculation. Depletion flocculation has been extensively studied over several decades for hard spheres or semi-hard particles (particles grafted with short polymers).8−11 In the absence of an affinity of the freely dispersed polymers for the surface of the particles, one will find a so-called depletion layer around the particles where the polymer density is lower than in the bulk. Overlap of depletion zones gives an attractive contribution to the pair potential. The range of the interactions is proportional to the coil size as long as the polymer solution is dilute and is given by the mesh size in semidilute solutions. In the latter case the range of the depletion interaction decreases with increasing polymer concentration and becomes on the order of the segment size in melts. The strength of the depletion interaction is proportional to the osmotic pressure of the polymer solution, which increases with polymer concentration. In the dilute regime it is proportional to the number density of freely dispersed polymers and is for given weight concentration a decreasing function with molecular weight. In semidilute solution the osmotic pressure increases roughly quadratically with the weight concentration and becomes independent of the molecular weight. When the product of the overlap volume and the osmotic pressure is sufficiently large, the depletion interaction may be sufficient to flocculate the particles. At both very low and very high polymer concentrations the particle solution is stable. In the lower limit this is because the osmotic pressure is not high enough to push the particles together. At high concentrations the depletion zone is too small to have a significant depletion contribution. In between these extremes the particle solution is subject to depletion flocculation. In other words, depletion restabilization or stability re-entrance is intrinsically coupled to the polymer concentration control parameter.12 Depletion flocculation studies of very soft spheres or semisoft particles (particles grafted with much longer polymers) have received less attention.13 With respect to depletion of polymers due to the presence of an impenetrable hard surface, a polymer brush is to some extent penetrable for polymers in solution. In particular, when the freely dispersed polymers are short with respect to the brush chains, the depletion zone above the brush is negligible. In this argument the grafting density is expected to be low so that the brush is well-solvated by the solvent and there is plenty of space to penetrate the brush (polymers exchange with solvent molecules). For this reason we expect only a significant depletion zone when the chain length for the freely dispersed polymers exceeds that of the brush chains. It is known that long polymers have nothing to gain to go deep into the brush due to the entropy unfavorable, and hence they will form a depletion zone above the brush. Hence the depletion effect most prominently occurs for particles covered with a short brush. The situation with respect to the colloidal stability is less studied and less obvious when a bidisperse brush is grafted onto the particles. Our interest in this paper is in a composite brush wherein a proximal relatively dense brush of short chains is admixed with a few longer chains that form a distal layer of mushrooms. We may hypothesize that when two of such
surfaces are pressed toward each other the compression of mushrooms gives a weak repulsive contribution to the pair potential and the compression of the inner dense brush gives a robust steric protection (screening of the underlying surface). Free polymers in solution hardly see the mushrooms and generate a depletion zone starting from the inner brush. The attraction due to the overlap of depletion zones is largely compensated by the repulsion due to mushroom overlap. Hence the overall interaction potential remains modest and the particles are expected to be accepted by the polymer solution, especially at high polymer concentrations where the range of the depletion attraction does not exceed the height of the mushroom layer. We present experimental results on the colloidal stability of particles decorated by a brush in dilute and semidilute polymer solutions. These results strengthen the equilibrium modeling results for these systems. Experimentally, we opted to study quantum dots (QDs) onto which we grafted PDMS chains. These particles were dissolved in toluene that contains PDMS polymers as well. Light-scattering studies give information on the colloidal stability of these resulting solutions. In the modeling we consider both unimodal as well as bimodal polymer brushes on these QDs and focus on the re-entrance phenomena, as previously discussed. Our modeling work is based on the Scheutjens−Fleer selfconsistent field (SF-SCF) method. Within this method we can predict the structure of the brush layer that decorates these small particles and investigate the stability phase diagram with coordinates (i) the free polymer concentration and (ii) the chain length ratio Nf/NB. In the case of a bidisperse brushes, NB is the length of the longest chain on the particles. A few basic features of the theory and the models that we implemented are discussed later in more detail. In the SF-SCF modeling the depletion restabilization as a function of the concentration of free polymer is recovered. On top of this the theory predicts a re-entrant behavior as a function of the ratio between the lengths of the free polymer and that of the (longest chain of the) brush in the case of a bidisperse brush. This is of interest because it opens up an experimental window to have colloidal particles solubilized in concentrated polymer solutions. It is of no surprise that in the literature similar systems have received some attention already. Phenomena such as the depletion effect of PDMS grafted silica particles have been shown to depend on the details of the brush interface.14 Zukoski et al. reported that particles with a bidisperse bush have an enhanced compatibility at high polymer concentrations.15 A direct proof of the prediction that a restabilization of particles at very high polymer concentrations can take place is not yet in the literature. The remainder of the paper is as follows. We will first outline the basics of SF-SCF theory and give full details of the model that is implemented. Then, we will present the experimental methods that are used. In the Results section the theoretical predictions are discussed systematically and the corresponding experimental data are presented as supports of the predictions. In the final parts of the paper we will summarize our conclusion. Self-Consistent Field Modeling. We like to understand how a decoration of nanosized particles by a polymer brush can influence the colloidal stability of such particles in a polymer solution of chemically similar chains. Ideally in the modeling one would like to take all molecular details of such system into 12939
DOI: 10.1021/acs.jpcb.5b05504 J. Phys. Chem. B 2015, 119, 12938−12946
Article
The Journal of Physical Chemistry B
Conformational degrees of freedom are accounted for by a lattice implementation of the freely jointed chain (FJC) model.22 In the FJC model two neighboring segments occupy neighboring lattice sites, but there are no angular constraints between two consecutive bonds along the chain. Long range along the chain those correlations are ignored. As a result these chains are not self-avoiding. Adverse effects of this approximation are counteracted at least in part by the incompressibility constraint.19−21 The FJC model allows for an efficient propagator formalism to evaluate the single-chain partition functions.19−21 For the brush chains the propagators are easily modified to include a positional constraint on the first segment of the chain to sit next to the solid surface.17 The remainder parts of the chains do not have such constraint, and these can distribute throughout the system as far as they can go, of course. The formalism is straightforwardly implemented for a spherical coordinate system.18,24 The free energy of the system F can be computed by multiplying all molecular partition functions of the molecules in the system (the brush polymers, the freely dispersed polymers, and the monomeric solvent). Here and below we will normalize the free energy19−21 by the thermal energy kBT and per particle. Our interest is in the optimized free energy of our system. An iterative procedure23 is used to find this free energy (above referred to as solving the SCF equations) and corresponding best distributions of the molecules in the calculation volume. Here we do not get into the optimization formalism and mention that typically the density profiles and the free energies are optimized up to seven significant digits. The pair interactions are found by recording the free energy as a function of the system size D/2 and by subtracting the value at large separations results in the free energy of interaction ΔFint(D)
account. Because exact analytical solutions are not available, one may consider the use of computer simulations to generate relevant information for these complex systems; however, such simulations are extremely challenging due to the large length and long time scales. A more realistic strategy for simulations is to use a set of particles in a volume where the solvent and the polymers in the solvent are not explicitly accounted for but represented by effective pair potentials.16 One way to generate relevant pair potentials for such coarse-grained simulation is by using the self-consistent field method. Interestingly, by inspection of the SCF pair potentials we can already predict in some detail the fate of the particles in a polymer solution. We therefore focus on the pair interactions predicted by the SCF method and discuss the expected consequences of these potentials and do not present corresponding simulation results. The free energy of interaction between two spherical particles requires us, in principle, to solve SCF equations in a two-gradient cylindrical coordinate system.17 A computationally less expansive result is obtained by using the cell model (refer to Figure 1b).18 In this case we can use a spherical coordinate
Figure 1. (a) 2D scheme of the lattice model used in Flory−Huggins as well as in SF-SCF theory. In this case there is one polymer chain of 26 segments in a monomeric solvent. (b) Schematics of the cell model. A central particle (orange) decorated by a brush (hairs on the sphere) in the presence of a solvent (small dots) and freely dispersed polymers (green coils). The reflecting boundary condition (blue dashed line) mimics the symmetry plane between 12 neighboring particles. The radial position of this boundary is at r = D/2 (schematically indicated by the blue double-headed arrow).
ΔFint(D) = F(D) − F(∞)
(1)
where it is understood that the free energy accounts only for the contributions within the cell volume; that is, it is normalized per particle. Moreover, here and below we will focus on the polymer contribution to the free energy of interaction only. Again, a possible contribution from the van der Waals interaction must be added on top of this. In the spherical coordinate system the lattice sites are arranged in spherically symmetric layers numbered r = 1, 2, ..., D/2. The value of D/2 is varied to find the pair potentials (eq 1). The size of a lattice site is advised to be a = ∼0.5 nm. In our default system the particle size is R = 5 nm, and thus the particle occupies in the lattice coordinates the first 10 lattice layers. The brush chains are pinned with the first segment to r = 11. In the default system the brush chains are NB = 100 segments long. The amount of grafted polymer segments per unit area (that is, per lattice site) is given by θ = σNB = 5 unless otherwise specified. The length of the free chains is varied over a large range, both shorter and much longer than the brush chains. The concentration of the free chains is a control parameter to study the re-entrance behavior in the depletion problem. The solvent is modeled as single monomers. Experimental Materials. The quantum dots are purchased from Crystalplex (NC 605A-1). They are CdSe/ZnS core/shell nanocrystal quantum dots bearing oleate functional ligands suspended in n-heptane (50 mg/mL). PDMS (polydimethylsiloxane trimethylsiloxane terminated) are purchased from Gelest. The average Mw (molecular weight) of the PDMS samples are 2 × 103, 4 × 103, 6 × 103, 1 × 104, 1.4 × 104, 1.4 ×
system and solve the SCF equations in a one-gradient approximation, where r refers to the distance from the center. At r = 0 we have a central particle with radius R, decorated by a polymer brush with given grafting density σ (number of chains with length NB per unit area, or in the case of a bidisperse brush there are two grafting densities and two chain lengths) surrounded by freely dispersed polymers with length Nf, dissolved with a volume fraction φfb in a monomeric solvent. We consider the polymers to be in an athermal solvent and disregard specific affinities with the particle surface. At the upper boundary of this system at r = D/2 we have reflecting boundary conditions. As a result we mimic the surrounding of ∼12 image particles, which are a distance D away from the central particle. van der Waals interactions between the particles are omitted, and in a superposition Ansatz should be superimposed on top of the SCF predictions. The SF-SCF method19−21 considers models with polymer chains composed of segments similarly as in the Flory− Huggins theory (refer to Figure 1a); each segment occupies a lattice site, and sites that are not occupied by segments are taken up by the solvent (incompressible system). The segments of a chain of type i have ranking numbers s = 1, 2, ..., Ni. 12940
DOI: 10.1021/acs.jpcb.5b05504 J. Phys. Chem. B 2015, 119, 12938−12946
Article
The Journal of Physical Chemistry B 104, and 2.4 × 104 g/mol. The ligand (side-chain carboxylfunctionalized linear polysiloxane) is synthesized within the Philips laboratories, and the Mw is 1 × 104 g/mol. Both the PDMS and ligand PDMS structure are given in Scheme 1. The Scheme 1. Structures of PDMS (Left) and Ligand (Right)a
a
w, x, and y are the value of repeat units. The ratio of x/y is 80/1.
ligand exchange on quantum dots is performed in Philips, and the products are purified by Philips as well; more details of these brushed particles will be published elsewhere. The ligands exchange quantum dots are suspended in toluene after ligands exchanging. The rest of the agents are purchase from Sigma and used without further purification. Experimental Characterization. The particles size is measured by using an ALV goniometer and correlator equipped with a 632.8 nm 22 mW Uniphase 1145P HeNe laser using avalanche photodetector (Exelitas Technologies). Measurements are performed at a fixed scattering angle of 150°. The measured decorrelation functions are fitted using the standard cumulant methods. Samples are prepared by mixing 100 μL of quantum dots dispersed in toluene into 900 μL of mixture of PDMS of varying molecular weight and toluene. Presented measurements represent the average of 100 independent DLS measurements for each sample. The viscosity of polymer solution is roughly estimated by ln η = X1 ln η1 + X2 ln η2 as in Victor’s paper.24 The X1 is the volume fraction of composition 1, and η1 is the viscosity of composition 1. The X2 is the volume fraction of composition 2, and η2 is the viscosity of composition 2. The ligands exchanging is also characterized by DLS through the analysis of the difference of the particle size before and after exchange, which is named Rshell. It was found that Rshell exceeds the radius of gyration of the grafted chains Rg25 by a factor of 2. This value is comparable to unity, and this implies that the ligands form a rather weak brush and that the chains are only slightly stretched due to lateral interactions. See the details in Figure S1.
Figure 2. Free energy of interaction per particle and in units of kBT as found in the cell model. The distance between the particles D is in lattice units. Different curves are for specified ratio Nf/NB that is the chain length of freely dispersed polymers and that of the grafted chains. The bulk volume fraction of freely dispersed polymer with Nf = 100 is φbf = 0.1. The grafted amount per unit area is θ = 5. The particle size is R = 5 nm, equivalent to 10 lattice units. In the inset zoomed-in interaction potentials are shown.
repulsion correlates with the height of the polymer brush on the particles. Hence the repulsion is due to the overlap of the brushes (steric interaction). As soon as the mobile chains are equal to or longer than the grafted ones, it is noticed that the interaction curves are nonmonotonic. At very close proximity the free energy of interaction is a decreasing function with distance, indicating repulsion. At larger distances the interaction free energy increases with distance between the particles. This effect is attributed to the depletion effect. As a result a minimum in the interaction potential is found. The location of the minimum again correlates with the height of the brush, indicating that a small brush overlap already overcompensated the depletion effect. For the high polymer concentration used in these calculations, the free chains are in the semidilute regime. In the case of a rigid wall, the depletion effect should be independent of the molecular weight. For depletion against a brushed surface, the depletion zone remains a function of the length of the free chains until the ratio Nf/NB ≈ 10. In accordance with this we find that the depth initially increases when the free chains are only marginally longer than the brush ones and the depth saturates when the free chains exceed the brush chain lengths by a factor of >10. A minimum in the interaction curve can lead to the aggregation of the particles and thus to a rejection of the particles from the polymer solution. The relevant energy scale for this is the thermal energy. Here we adopt the ad hoc Ansatz that a depth of −3kBT is sufficient of arrest the particles permanently in the aggregated state. Hence when the depth is less than this value we expect that particles still will disperse in the polymer solution, and we will say that the system is colloidally stable, but when the depth is below this value the particles are rejected from the polymer solution and the system is colloidal unstable. In Figure 3 we present the value at the minimum of the interaction curve ΔFmin int as a function of the ratio Nf/NB. In this Figure the horizontal dashed line represents our −3kBT
■
RESULTS AND DISCUSSION The primary target for the SCF calculations is to predict the free energy of interaction between particles covered by a brush. So let us start to discuss how these interaction curves depend on the ratio between the length of the free chains and that of the brush. In Figure 2 we present both the overall interaction curves and a zoomed-in variant for a fixed polymer volume fraction of φbf = 0.1, while the variable for the different curves is the ratio of free polymer chain length and grafted polymer chain length. Various other parameters have the default value (see also the legend). It is seen that as long as the free chains are short compared with the grafted chains that the free energy of interaction is a continuously decreasing function with distance exemplified for the case of Nf/NB = 0.1. This means that the particles repel each other for all of the interparticle distances. The range of the 12941
DOI: 10.1021/acs.jpcb.5b05504 J. Phys. Chem. B 2015, 119, 12938−12946
Article
The Journal of Physical Chemistry B
Figure 4. Set of stability phase diagrams of monografted particles in coordinates where on the y axis the volume fraction of the free polymers is given and on the x axis the ratio between the free and grafted chain lengths as found by SCF calculations is given. The stable and unstable regions are indicated. The grafted amount per unit area on the particles is varied; θ = 0.2, 0.5, 2.0, and 5.0 are indicated. The points on the horizontal and vertical lines represent solutions that were tested experimentally for colloidal stability (see Figure 5). The green spheres indicate that a colloidally stable state and the red triangles indicate that colloidally unstable states were found by dynamic light scattering.
Figure 3. Maximum depth of the free energy of interaction ΔFmin int as a function of the ratio between the free and grafted chain lengths (Nf/ NB). The volume fraction of the free polymer in the bulk is indicated. The green dashed line at ΔFmin int = −3kBT demarcates the stable from the colloidally unstable systems. Parameters are the same as in Figure 2.
demarcation line. The different curves are for different values of the free polymer volume fraction in the bulk. For all polymer concentrations the value of ΔFmin int is a decreasing function of the ratio Nf/NB. This means that the longer the chains in the bulk the lower is the interaction minimum. In all cases the curves level off when the free chains are more than ten times the length of the brush chains. The limiting value ΔFmin int for very long chains in the bulk is, however, a nonmonotonic function of the volume fraction of the free polymer. There exists a minimum around φbf ≈ 0.5. The critical ratio Nf/NB, below which the system is colloidally stable and above which it is not (dashed line), is also a nonmonotonic function of φbf . For both very low and very high bulk volume fractions, the critical ratio is close to unity; however, at intermediate concentrations of the free polymers in the bulk, the system becomes unstable, especially for the case that the free chains are significantly longer than the brush chains, that is, for Nf > NB. The instability of the system for large free polymer chain lengths in semidilute and concentrated polymer was experimentally reported by Dutta and Green.14 Collecting the crossing points with the dashed line as discussed in Figure 3, we can construct a phase diagram wherein on the y axis the volume fraction of the polymers in the bulk is plotted and on the x axis the critical ratio Nf/NB is plotted. A line in such phase diagram splits the parameter space into two: In one region, the free chain lengths are short enough so that the particles can disperse in the solution. This region is marked as “Stable”. For high polymer chain lengths the particles are not accepted by the polymer solution and the system is expected to flocculate; therefore, we marked the region as “Unstable”. In Figure 4 we present such phase diagrams for three values of the grafting density of polymers onto the particles. For the experimentally accessible range of grafting densities σ = θ/NB used in Figure 4, the U-shaped phase diagram shifts slightly to lower values of Nf/NB for larger values of the grafting density. For a planar brush the limits of σ = 0 and 1, both give a solid boundary; therefore, one might expect that for high grafting densities the U-shaped phase
diagrams should shift to the right again; however, for small particles the limiting value of σ = 1 does not correspond to a sharp brush−solvent interface. Even in this limit the brush− solvent interface is fluffy and the effect of the grafting density on the shift of the U-shaped phase diagram is monotonic. Polymer brushes are often used to change surface properties of particle. One is inclined to think that more chains (high grafting density) are better. Inspection of Figure 4 illustrates that this is not the case for the compatibilization of particles in a polymer solution. Here our results at very low grafting densities may be slightly influenced by the fact that we ignored the surface properties of the bare particles and we ignored the van der Waals interactions, but the trend with increasing grafting density is clear: The U-shaped phase diagram shifts to the left, implying that the stable region decreases at the expense of the unstable region. With increasing grafting density it becomes harder and harder for the free chains to interpenetrate the brush, and the system becomes progressively more vulnerable for depletion flocculation. There are two ways we can cross through the phase diagram. At fixed volume fraction of polymer we might increase the disparity between free and grafted chain lengths (horizontal line) and go from a stable at short lengths of the free chains to the unstable state at large lengths. Alternatively, we can for a fixed ratio of the two chain lengths, vary the volume fraction of polymer. Starting with a very low polymer concentration, the system is stable and becomes colloidally unstable when the concentration is increased; however, in the melt regime when the polymer volume fraction is close to unity we recover the stable state. We decided to take this reentrant behavior to a test experimentally. The estimated points in the phase diagram for our test samples are indicated by the points on the horizontal and vertical lines in Figure 4. Dynamic light scattering was performed as mentioned in experimental part. From such experiments we obtain a 12942
DOI: 10.1021/acs.jpcb.5b05504 J. Phys. Chem. B 2015, 119, 12938−12946
Article
The Journal of Physical Chemistry B
polymer volume fraction was fixed to 0.1, as mentioned in experimental part. The length of the free polymer chains was varied (molecular weights were given above). The ratio Nf/NB varied in the range 0.2−2.8. The radius of QDs is 5 nm without grafted polymer and 9 nm with grafted polymer in toluene. The QD radius decreases a little bit from 9 to 7 nm when suspended in a polymer solution when Nf/NB = 0.4 or 0.6. This shrinkage due to the increased osmotic pressure of the polymer solution compared with the monomeric solvent is in line with SCF predictions (not shown). In this regime, where the free chains are short compared with the brush chains, the colloidal stability is good. For Nf/NB = 1, the light scattering reports a particle size of 100 nm, and this implies that aggregation has started, even though visual inspection still gives a clear solution (see inset). Increasing Nf/NB > 1, that is, when the free chains exceed the length of the brush chains, results in reported particle sizes of thousands nanometers, and now the samples are turbid (see right inset). In this limit the size of the aggregates grows in time, proving that the aggregation continues. The vertical cut through the phase diagram is given by the blue line in Figure 4, which is positioned for Nf/NB = 0.6, that is, for the case that the length of the free polymer chain is significantly smaller than the length of the grafts. Again the samples that were tested are represented by green spheres when the solution is colloidally stable and red triangles signal unstable samples. In Figure 5b the corresponding particle size distributions are presented. In the regime from pure solvent to a polymer volume fraction of 0.2 the particle solution is perfectly stable (blue curves in Figure 5b). Again we find that the particle size slightly decreases with increasing polymer concentration. With increasing polymer concentration we enter the unstable regime until the free polymer concentration is larger than ∼0.8. In this regime light scattering reports particle sizes larger than 100 nm. For polymer volume fractions of 0.85 and 0.9 the light scattering once again gives a single peak consistent with single dispersed particles. The results clearly support the theoretical predictions quantitatively with respect to both the chain length dependence as well as the re-entrance effect with respect to the polymer volume fraction. In a recent paper, Schadler showed that quantum dots coated by a bidispersed polymer brush have an extraordinary good stability in a dense polymer solution.26 Inspired by this result we have extended our SCF analysis in this direction, aiming to better understand this effect. More specifically we focused on a bidisperse brush for which the majority of chains are short (NBS = 10), whereas a minority of the grafted chains are ten times longer, NBL = 100. The grafting densities were chosen such that the amount of the two species was the same, and hence the grafting density of the short ones is 10 times larger than that of the longer ones. Our interest is once again in predicting the phase diagram for the same size particle as presented in Figure 4. Before we can present these phase diagrams, it is necessary to discuss the features that are found for the free energy of interaction curves for particles covered by a typical bidisperse brush. As can be seen from Figure 6, the interaction curves for bidisperse brushed particles as found by SCF calculations in the cell-model have several features that are relatively easily traced to the brush features and the depletion phenomenon of the freely dispersed chains. Going from some large separation between the particles to shorter separations, we first see a weak
distribution of sizes in the system. When there is a single peak around the size of the primary particles, we conclude that the system is colloidally stable (see Figure 5). Upon visual
Figure 5. (a) Size distributions (radius) as found by dynamic light scattering for the brush-coated quantum dot solutions in a polymer solution with a volume fraction of 0.1. The value of the ratio of the free chain length and the length of the brush chains is indicated. (b) Size distributions for a fixed ratio of chain lengths Nf/NB = 0.6 but for varying volume fraction of the free polymer, as indicated. Red curves correspond to colloidally unstable states. Blue and green curves are colloidally stable cases.
inspection the solution remains clear (see inset picture on the left of Figure 5a); however, when a new peak shows up at much larger size than the primary particles we envision that some sort of aggregation has taken place, and we conclude that the system is intrinsically unstable. Then, a turbid solution is found, which is easily seen by visual inspection (see inset picture on the right of Figure 5a). In intermediate situations, the solution may visually still be clear, but the cumulant analysis already indicated a larger size. In somewhat more detail we refer to Figure 5a in which we present size distributions for a given polymer concentration. The selected data points are plotted in Figure 4 by green circles and red triangles along the line. The green circles mean that the particle solution was (colloidally) stable. The red triangles are used for the unstable samples. The corresponding particle size distributions as found by DLS measurement are collected in Figure 5a. For points on the horizontal line the 12943
DOI: 10.1021/acs.jpcb.5b05504 J. Phys. Chem. B 2015, 119, 12938−12946
Article
The Journal of Physical Chemistry B
from stable to unstable and back to a colloidally stable situation. The long mushroom polymer is compressed when the depletion layers start to overlap. The compression leads to a compensation that counterweighs the depletion attraction, and in the most effective case stabilizes the particles kinetically and leads the particles into a kinetically stable regime, as Figure 7
Figure 6. Free energy of interaction in units of kBT for particles with a radius or R = 5 nm covered by a bidispersed polymer brush with total amounts per unit area θ = 5, in different concentrations of the freely dispersed polymer as indicated. The chain length of the free chains is Nf = 20, 50, and 100.
attraction between the particles. Then, a local maximum in the curves is found, followed by a primary minimum and a strong steric repulsion at the shortest separations. The repulsive regions are attributed to the steric repulsions. At short distances this originates from the compression of the shortest chains. The weaker repulsion at larger distances is due to the compression of the longer chains in the brush, which are much less densely grafted. The depletion effect onto the dense inner-brush is larger than the depletion at the periphery of the brush. Indeed the outer layer of this composite brush is very dilute, and the depletion is largely suppressed. The deepest minimum at short separations is called the primary minimum; the other one is the secondary minimum. In between these minima is referred to as the local maximum. The overall interaction potential has the shape of the DLVO potential. We follow a classical colloid science strategy to identify various regimes of colloidal stability. As above, we will consider a minimum deep enough to arrest the particles from redispersing when the secondary minimum is deeper than 3kBT. When the secondary minimum is less deep, the remainder of the interaction curve becomes of interest. When the primary minimum is deeper than 3kBT, we might expect that the aggregation can take place. However, when the local maximum is high enough the system is thought to be kinetically stabilized. Here we have chosen a 5kBT threshold. Hence, as soon as the local maximum is higher than this value, it is irrelevant how deep the primary minimum is because the particles will not be able to cross the local maximum. Inspection of the three examples of the interaction curves of Figure 6 proves that the red curve is just a case, whereas the secondary minimum is deeper than 3kBT and thus this system is unstable. For the green curve the local maximum exceeds the 5kBT threshold and as the secondary minimum is not very deep, the system is kinetically stable: The system cannot reach the deep primary minimum. Finally, for the black curve, for the shortest length of the free chains, the two minima as well as the local maximum are small, and the system is stabilized by the steric repulsion of the shortest brush. These three examples show that with increasing length of the free chains we can go
Figure 7. Pair of double re-entrant stability phase diagrams in free polymer volume fraction and ratio of the lengths of the free chains and the longest chain in the bidisperse brush coordinates for particles with radius R = 5 nm. The stable, unstable, and kinetically stable regions are indicated. The values of the overall grafted amounts are indicated. The three data points on black line indicate the graphs from Figure 6.
shows later. In Figure 6 we can see three cases of different free polymers. The black circle shows the primary and secondary energy drops are both not deep enough for the particle trapping when Nf/NBL is 0.2. So it is in stable regime, as Figure 7 shows later. The secondary energy drop is deeper than 3kBT when the Nf/NBL is 0.5, as the red triangle shows in Figure 6. It means the particles will be already trapped due to the depth of enough secondary energy drop. So it is in the unstable regime, as Figure 7 shows later. Interestingly, the secondary energy become smaller than 3kBT, while the energy barrier also increases to 5kBT when the Nf/NBL is 1. It indicates that the particles will not be trapped by the secondary energy drop due to its shallowness nor able to conquer the energy barrier due to its height. The particles are just in the regime in which it is kinetically stable, as Figure 7 shows later. Although it is imaginable that the particles may be able to conquer the energy barrier by giving more thermal energy such as heating in experiments, it indeed opens a window for particles suspension research and application. In Figure 7 a few typical phase diagrams are presented for particles decorated by a bidisperse brush with coordinates similar to the phase diagrams of Figure 4, namely, the volume fraction of the free polymer and the length ratio between the free chains and the longest brush chain. Now three regions are indicated. First there is the region wherefore the particles are expected to disperse in the solution. This region is labeled “stable”. Again there is also a region in which the particles are expected to flocculate, which is labeled “unstable”. More specifically, in this region the secondary minimum is deeper than 3kBT. In a third region, labeled “kinetically stable”, the local maximum exceeds the 5kBT threshold, while the 12944
DOI: 10.1021/acs.jpcb.5b05504 J. Phys. Chem. B 2015, 119, 12938−12946
Article
The Journal of Physical Chemistry B
brush. The issue is that a layer of mushrooms on a bare particle surface will, in general, give compatibility problems because of the surface-specific adsorption effects. In the bidisperse case the role of the dense inner brush is simply to hide the surface from the solution. This layer should be thick enough to mask the van der Waals or solvophobic effects. We note that our results strongly depend on the particle size. We evaluated the height of the free-energy barrier and the depth of the attractive wells for small particles. As the depletion and the steric repulsion contributions to the free energies of interaction per particle scale with the surface area, we should expect for larger particles not only that the flocculation in the secondary minimum is more pronounced but also that the local maximum is higher as well. Both effects obviously influence our phase diagram. The generic effects that represent themselves in our SCF calculations remain relevant, of course.
secondary minimum is not deep enough to render the system unstable. The boundary between stable and kinetically stable is plotted with a dashed line. To the left of the dashed line the primary minimum is deeper than 3kBT, and the system should have phase-separated if the local maximum would not have prevented this. An increase in the grafted amount on the particles for the bidisperse brush has relatively small effects, as in the case of a single brush. With increasing grafting density the demarcation line shifts to the left. This effect is caused by the fact that the freely dispersed chains have more difficulty in penetrating a dense brush than a dilute one. Interestingly the dashed lines cross each other, indicating that the depth of the primary minimum is influenced by the compression of the longer chains and that this effect depends on a nontrivial way on the volume fraction of the free polymer. When we compare for the same amount of grafted polymer the phase diagrams of Figure 4 (monodisperse brush) with Figure 7 (bidisperse brush) we see that the unstable region is much smaller in the latter case. Indeed, for the high polymer concentrations, the bidisperse brush is now kinetically stable. We can understand this effect by inspection of the phase diagram in Figure 4 for the lowest grafting densities. More specifically, for θ = 0.2, we see that the unstable region of the phase diagram is limited to free polymer volume fractions of (in this case)
